MIT Radiaton Lab Series V8 Principles of Microwave Circuits
MIT Radiaton Lab Series V8 Principles of Microwave Circuits
MIT Radiaton Lab Series V8 Principles of Microwave Circuits
\\
PRINCIPLES OF
MICROWAVE CIRCUITS
Edited by
C. G, MONTGOMERY
ASSOCIATE PROFESSOR OF PHYSICS
YALE UNIVERSITY
R. H. DICKE
ASSISTANT PROFESSOR OF PHYSICS
PRINCETON UNIVERSITY
E, M. PURCELL
ASSOCIATE PROFESSOR OF PHYSICs
HAIWARD UNIVERSITY
Iv
~s, INST. TF
#
c+
MAY la 1956 ‘
c’)
PRIN~IPLES OF MICROWAVE CIRCUITS
EDITORIAL STAFF
c. (1. MONTGOMERY
D. D, MONTGOMERY
Il. R. BERINGER
R. H. DICKE
N. MARCUVITZ
c. G, MONTGOMERY
E. M. PURCELL
o
a
L.
Foreword
L. A. DUBRIDGE
Prejace
THE AUTHORS
NEW HAVEN, CCJNN.,
February, 1947.
lx
Contents
PRF;FAC!E. ix
1.1. }ficrowalres 1
1.2. Microwave Circuits 3
1,3. iUicrOwave Lfeasurements 5
14. The Aims of hficrorvave ( ‘ircuit Analy.is 8
1.5. Linearity 9
16. Dissipation 9
17. Symmetry 9
TH~FI~L~ RELATIONS . . . . 10
83
T-JuNcTIoNs . . . . . . . . . . ..283
9.1. General Theorems about T-junctions. 283
CONTENTS xv
INDEX 481
CHAPTER 1
INTRODUCTION
BY E. M. PURCELL
(a) (e)
.
(b) (f)
I’IG,1,1.–-Resonantcircuitsat low frequenciesand at microwavefrequencies.
ELECTROMAGNETIC WAVES
BY C. G. MONTGOMERY
F = p[E + (V X B)].
dB
div B = O. (1)
curl E = – z’
The second pair of field vectors are determined by the charges and
currents present and satisfy the equations
where J is the current density and p the charge density. The set of
Eqs. (1) and (2) is known as Maxwell’s equations of the electromagnetic
field. Sometimes the force equation is included as a member of the set.
Tke connection between D and E, and B and H, depends upon the
properties of the medium in which the fields exist. For free space, the
connection is given by the simple relations
D = eoE, B = ~,H.
For material mediums of the simplest type the relations are of the same
form but with other characteristic parameters
D = cE, B = ~H.
The symbol e denotes the permittivity of the medium, and ~ the perme-
ability. For crystalline mediums which are anisotropic, the scalars c and
p must be generalized to dyadic quantities. Then D and E no longer have
the same direction in space. This more complex relationship will ?ot
be dealt with here. If the medium is a conductor, then ohm’s law is
10
SEC.21] .tiAX WILLL’S EQUATIONS 11
aH aE
curl E = —Y%, curl H=uE+c X,
(3)
div (AH) = O, div (cE) = p.
1
The units in which these equations are expressed are rationalized
units; that is, the unit electric field has been so chosen as to eliminate the
factors of 47rthat occur when Maxwell’s equations are written in electro-
static units, for example. The practical rationalized system, the mks
system of units, will be used here. Thus E and B are measured in volts
per meter and webers per square meter respectively, H in amperes per
meter, D in coulombs per square meter, p in coulombs per cubic meter,
and J in amperes per square meter. In the mks system y and c have
dimensions, and have numerical values that are not unity. For free
space p and c will be written as po and ~0, and these quantities have the
values
MO= 1.257 X 10–6 henry per meter,
co = 8.854 X 10–12 farad per meter.
The velocity of light in free space is l/~p,c, and is equal to 2.998 X 108
meters per second. A quantity that frequently appears in the theory is
-. For free space, this is equal to 376.7 ohms. The mks system of
units is particularly suitable for radiation problems. As will be shown
later in this section, a plane wave in free space whose electric field ampli-
tude is 1 volt per meter has a magnetic field amplitude of 1 amp per
meter, and the power flow is ~ watt per square meter. Of more impor-
tance, perhaps, is the fact that in practical units, the values of impedance
encountered in radiation theory are neither very large nor very small
numbers but have the same range of values as the impedances encoun-
tered in the study of low-frequency circuits. The dimensions of quanti-
ties in the mks system may be chosen in a convenient manner. To the
12 ELECTROMAGNETIC WAVES [SEC.21
basic dimensions of mass, length, and time is added the electric charge
as the basic electrical dimension. Table 2.1 shows the dimensions and
practical units for the quantities that are of most importance. It is of
interest to note that no fractional exponents occur in this table. It iS
this circumstance, in fact, which urges the adoption of the charge Q
as the fourth basic dimension for physical quantities.
TABLE2.1.—DIMENSIONS
ANDUNITSOF ELECTRICAL
QUANTITIES
Length. . . . . . . . . . . . . . . . . . . . L Meter
Mass. . . . . . . . . . . . . . . . . . . . . . M Kilogram
Time, t. . . . . . . . . . . . . . . . . . . . . T Second
Power. . . . . . . . . . . . . . . . . . . . . ML2T-a Watt
Charge. . . . . . . . . . . . . . . . . . . . . Q Coulomb
Current,I . . . . . . . . . . . . . . . . . . QT-1 Ampere
Resistance,R. . . . . . . . . . . . . . . JfL2T-lQ-2 Ohm
Electric potential, V. ML2T-2Q-1 volt
Electric field, ,? i’ . . . . . . . . . . &rLT-2Q-1 Volt/meter
Displacement,D..... L-2Q Coulomb/square meter
Conductivity, a...... M-lL-3TQ2 Mho/meter
Dielectric constant,c. J,-lL-3T2Q2 Farad/meter
Capacitance,C. M-lL-2T2Q2 Farad
Magnetic intensity, H. L-lT-lQ Ampere/meter
Magnetic induction, B MT-IQ–1 Weber/square meter
Permeability,p. . . . . . . . . . . . . MLQ-2 Henry/meter
Inductance, L. ML2Q-2 Henry
eEm— c’E: = $.
E, = O, H. = O, H, = K. (6)
S= EXH, (12)
,g = g,
27rr
If 1 is the length of the surface, then the normal flux of S’ will be simply
EZ1. This represents, then, the rate of energy flow into the wire, and it is
just what would be calculated from more elementary considerations of
the Joule heating produced by the electric current.
It is known that this energy is transferred to the wire by collisions of
the electrons with the atoms of the wire. It is useful, however, to dis-
regard considerations of this mechanism and to think of the energy as
being stored in the electromagnetic field and flowing into the wire at the
definite rate given by Eq. (1 1).
In a similar manner, for periodic fields represented by complex quan-
tities, the follo~ving expression can be formed
From Eqs. (9), after transformation to the integral form and division by
2, there results
v*E + u2PdZ = O,
If H, is eliminated,
d2E. —
—— jcop(u + j(Jc)Ez = 0. (15)
a22
An identical equation exists for H.. These equations are known as the
“ telegrapher’s” equations.
The solution of Eq. (15) is E. = Ee–yz, where E may be taken as
real, and
~= djupu – U’ql = a + j~. (16)
H. = – ~~ E..
The ratio EJHU is called the “wave impedance” and may be denoted
by Zw.
(17)
When u = O, Z,. = v’P/~; and for free space, Z. = 377 ohms. For u
not equal to zero,
z. = <a, \ ~,e, (P + ~~).
z. = (.
and a similar equation can be written for HV. These equations represent
two waves traveling in opposite directions.
A general solution of Maxwells equations may be built up out of a
set of elementary uniform plane waves, the directions and amplitudes
being chosen in such a way as to satisfy the boundary conditions. This is
an excellent artifice for some purposes, but because the description of the
field in these terms becomes quite complicated except in the simplest
cases, this method will not be discussed further here. In a similar
manner, the six components of E and H might be replaced by wave
impedances Zii = Ei/Hi, i, j referring to some coordinate of the space
that is being considered.
In general, the problem is not simplified in this way. However, in
certain cases the wave mpedance is a useful concept, and a complete
description of a situation may often be obtained in terms of wave imped-
ances and propagation constants. A very important example is the
treatment of the problem of reflection and refraction of plane waves at
the boundary between two mediums. For a complete discussion of thi~
problem the reader is referred to other texts. 1
2.6. Nonuniform Transverse-electromagnetic Plane Waves.—Let us
remove the restriction that the fields be uniform in a transverse plane,
but let H, = E. = O. The electromagnetic equations may be broken
up into two sets, the first of which governs the propagation in the
zdirection
~+$=fj,
aE,
g+$$=o,
(19)
aEz _—t3EU ~= __—dHv..
~– (3X’ ay ax 1
From the first of these sets, identical equations in any of the four
variables E., Eti, H., or Hv can be obtained by elimination. For example
a2E=.
— = jup(u + jcw)Ez.
8Z2
Hence the wave impedances of all plane TEM-waves are identical, and
equal to the intrinsic impedance of the medium.
If E. had been found from Maxwell’s equations, and if it had been
assumed that E“ = E (x,y)e-vz, then the wave impedance would have
been found to be
zvz=g.
.
–LyAyl = –Zw = –{.
.3E= d% .
aEy _ ———
—.— —
dy M ax ay
SEC.25] NONUNIFORM TEM PLANE WAVES 21
Thus E is determined as though the field were a static one, and all of
the techniques for solving static problems become available. Since
Zw = – Z,z,
&=_Hu
E. F=’
or the electric lines of force must be perpendicular to the magnetic lines
of force. However, the electric lines are perpendicular to the lines
$ = constant; hence the equipotential lines must represent magnetic
lines.
One very important consequence of these conditions may be shown
easily. Suppose that there is a cylindrical metal tube whose wall is
represented by the curve @ = a. Now if @ has no singularities within
the region bounded by 4 = a, then 4 must be a constant throughout the
region and E must be zero. Therefore no purely transverse electromag-
netic wave can propagate down a hollow pipe. If there is another con-
ductor, however, within the region 4 = a, then a finite value of E is
possible, since 1#1
may have singularities within the inner conductor which
allow the boundary conditions to be satisfied. This mode of propagation
will be studied in more detail in Sec. 2.7.
Suppose that there is a field distribution representing a TEM-wave.
Then without disturbing the field, a conductor can be inserted along any
curve @ = constant so that it is everywhere perpendicular to the electric
lines. This is a useful device and will often be employed.
A stream function may also be employed to specify the fields. If
~ =_g,
H.=%, .
ax
then the equation
dH=
~+dd+==–;’+y=o
dx dy
/
+
FIG.2.2.—Infiniteconductingplanes.
The power flow between the plates in the z-direction is found by calcula.
tion of the Poynting vector
where Re ({*) represents the real part of the intrinsic impedance. If the
medium is lossless,
P = ~KK*b Re ({).
21 = bRe ({).
It should be noted that the dimensions of this quantity are not ohms but
ohm-meters. In a similar and wholly arbitrary fashion an impedance
may be defined in terms of voltage. Let us take V for the voltage accord-
ing to
b
v= E, dy = bEU.
/ o
Since there are no longitudinal components of the fields, this integral is
independent of the choice of path of integration between the plates.
Thus V is uniquely defined. The “ voltage impedance” may now be
defined as
Zv . Re
()
!&F* = b Re ({*),
and is thus equal to Z1. It should also be noticed that a third kind of
impedance may be defined by
v bEy
~= Hz=bC”
Thus, for this simple case of a parallel-plate transmission line, there are
several definitions of impedance that lead to the same result, in a fashion
identical with that which prevails for low-frequency transmission along
a wire. For the more complicated modes of propagation it will be found
that these simple relations are no longer true.
2.7. Z’Eil.l-waves between Coaxial Cylinders. —Another simple case
of considerable interest is the propagation of waves between conducting
<,;.
24 ELECTROMAGNETIC WAVES [SEC. 2.7
c%
respectively (Fig. 2“3). Then
Ar
E,= H,= H,= E4 =0,
and the equations
? ;..’
.
0 c9E, dHq
= –jwpH4, ~ = – (u + j~)E,
b z
/ z
determine the propagation in the z-direction with
FIG. 2.3.—TEM -
waves between the same propagation constant ~ and wave imped-
coaxial
cylinders.
ante as those obtained in general for TEM-waves.
The transverse variation of the fields is determined from
or
$ (rH*) = O,
K = l?+(b).
The current I is then
2=
Kb d~ = 2~bHe(b),
/ o
and hence
The total current on the inner conductor is also 1. The radial electric
field is
The directions of E,, H+, and I for a wave traveling in the z-direction are
shown in Fig. 2.3.
The Poynting vector is
S.=;E,H;=~{~
2 (271T)’
and the total power is
sm. 24q SPHERICAL TEM-WA VES 25
Re(~)lnb= bE,
— dr_V
Z,=F = z“,
a.I—~/
which are again equal only because of the simplicity of the field con-
figuration. The impedances defined here are all measured in ohms.
Here also the integral in the definition of V is unique, since the fields are
entirely in the transverse plane.
The fields can also be expressed in terms of the fields at the surface of
the center conductor. Let these fields be Ea and Ha. Then
H, = : H..
z,
P = 2r2a2ZOH~ = 2r2a2 —
[Re(~)]2 ‘2”
If these are compared with the set of Eqs. (18), it is seen that they are
identical provided that
These are again exactly analogous to Eqs. (19) and show that either E
or H or both are derivable from either a potential function or a stream
function. It can therefore be concluded that spherical TEM-waves have
the same wave impedance and propagation constant as plane TEM-waves.
aa=o
F&=z ‘
E,= H,=O.
-— . _.
SEC.29] UNIFORM CYLIN1)RIL’AL 1+’AVES 27
and
aHz
$ (rE@) = –jwprH,, ~ = – (u + jcoc)E$. (21)
The solution of this equation is a Bessel function of order zero with the
argument jx 1 Any pair of independent solutions may be chosen, in
terms of which the fields may be expressed. Let us take
E, = AHjl’ (jr) + ~~,(j~),
where A and 1? are arbitrary constants and H$l) is a Hankel function of the
first kind
H$’ (X) = Jo(Z) + j~o(X)
7=a+j13,
I See, for eXaIII
ple, KuKeI~
e Jalmke and Fritz Erode, Tables oj Functions wi/}/
Formulas and C’i,rws, Dover Publications,New York, 1943,p. 146,
b
28 ELECTROMAGNETIC WAVES [SEC.2.10
then
E= = C
J ~e-e’e-j~’,
ci.
“= –y J Fe-a’e-’R”
H, =~gcos(jyr-~)
ZW=j~cot jyr–~.
()
Thus this solution does not represent a propagating wave unless some loss
is present. If ~ is purely imaginary, ZWis likewise purely imaginary and
no propagation of energy takes place. It should be emphasized that for
neither solution is Z~ independent of r, although for the outward-travel-
ing wave it approaches a constant value. All the cases in which the
electromagnetic vectors are purely transverse to the direction of propaga-
tion have now been described briefly. The longitudinal modes will be
described in the portion of this chapter following Sec. 2“10.
201O. Babinet’s Principle. -Maxwell’s equations have a symmetry in
the electric and magnetic fields that is extremely useful in the discussion
of electromagnetic problems. One aspect of this symmetry can be
expressed as a generalization of Babinet’s principle in optics. If E and
H in Maxwell’s equations are replaced by new fields E’ and H’, according
to the relations
SEC.2101 BABINET’S PRINCIPLE 29
(22)
fruitful examples of the application of this principle can be given for waves
propagating in pipes.
c3EV 8E.
= jupH,, –jwHv, (23)
az az =
aHz aH aH. 8H.
—U = jweE,, = jucEV, (24)
ay – az az – ax
aEu aE=
— –jwpH,, (25)
ax – ay
aHu 8H.
= o, (26]
ax – ay
8H.
+ = 0. (28)
dz
From these equations the first important reslllt is obtained. The wave
SEC,211] GENERAL PROCh’DUtW 31
impedance ~H k
(30)
(31)
This result is significant. The set of lines in the transverse plane that
give the direction of the transverse electric field Et at any point have the
slope dy/cZZ = EJE.. A similar statement is true for the magnetic
lines of force. Thus Eq. (31) is equivalent to
@ =_ 1 ,
(-)
dx electric
()
dy
z mlletic
and the lines of electric and magnetic force are therefore mutually per-
pendicular in the transverse plane.
If the assumed variation with z and the values of E= and E. given by
Eqs. (29) are inserted in Eqs. (24), it is possible to solve for dH./ax and
aHz/ay,
aH. _ # + C&p
Hz,
ax = Y
(32)
aHz _ ?2 + Ozw
H..
ay = -Y 1
Equations (25) and (28) become equivalent, because of Eqs. (29), and
lead to
aH=
~-+a$–jHz=O.
a2Hz a2H.
—+— ayz + (T2 + U2W)H. = O. (33)
ax*
d2E.
~ + :; + (-Y’ + AL)EZ = o, (36)
(38)
(39)
I
SEC. 212] THE NORMAL MODES OF RECTANGULAR PIPES 33
E=, E., H., and H, determined from the value of E, by Eqs. (37) and
(38). In a similar way, by the use of H., the Z’E-mode portion can be
separated out by means of Eqs. (29) and (32). The remainder of the
field will then be purely transverse. It has already been shown that no
Z’Eikf-mode is possible in a hollow pipe with no central conductor.
Hence, any field in a hollow waveguide, however complicated, may be
represented by a combination of TE- and TM-modes.
Once an expression for the fields has been found, the energy flow down
the hollow pipe can be computed by integrating the value of Poynting’s
vector over the cross section of the waveguide. The power flow P is
found to be
where Zw has been written for the E- or H-mode wave impedance, as the
case may be. To maintain the fields in a hollow pipe, currents must flow
in the walls, and the surface current density is equal to the tangential
component of the magnetic field. For E-modes, the tangential com-
ponent of H is equal to the total magnetic field at the guide walls. Since
His purely transverse, K is purely longitudinal. If a small slit is cut in a
waveguide such that it is parallel to K, the field in the guide is not dis-
turbed and there is no radiation from the slit.
2.12. The Normal Modes of Rectangular Pfpes.-Let us consider a
waveguide of rectangular cross section, which has dimensions b in the
y-direction and a in the x-direction, as shown in Fig. 2.6, and which has
-r
I
b
L m
+“-1—---’
/
z
FIG.2.6.—Coordinatesfor a rectangularwaveguide.
perfectly conducting walls. Let us first consider the Ti!l-modes. Equa-
tion (33) for Hz is obviously separable in rectangular coordinates and leads
to simple sinusoidal solutions. Let
H. = COSkzx COSkulJ,
k: = k: + k: = yz + Uzep.
34 ELM’ THO:VAGNETIC WAVES [SW, 212
The quantities k= and kU are called the wave numbers in the .z- and
y-directions, respectively. Following thestandard procedure, itis found
that
(40)
and
2
or
(-)
mr
b’
(41)
SEC.2.12] THE .VORhfA 1. .t10.OJIS OF RECTA NG1lLA R PIPES 35
where
(42)
k=(#+(~Y
and L is called the cutoff wavelength. A little consideration will show
that Eq. (40) is universally applicable for both types of modes and all
shapes of pipe; only h differs in the different cases. The value of x in
Eq. (41) is the wavelength of a plane !i’’E’kf-wave in the medium that fills
the hollow pipe. In terms of XO, the wavelength of a Z’EM-wave in
free space,
lt will be noted that the cutofi wavelength defined in this way is indep-
endent of the dielectric material filling the Waveguide; the critical
frequency defined in Eq. (40) is not.
For frequencies below the critical value, 7 becomes real and the
waves are attenuated. For very low frequencies,
‘2=(9 ‘w”
For m = 1 and n = O, y = ~/a, which corresponds to an attenuation of
27.3 db in a distance equal to the width of the pipe.
A pair of values of n and m suffice to designate a particular mode
that is called, according to the, accepted notation, a Z’E~n-mode. The
mode that has the lowest critical frequency for propagation is the
!l’Elo-mode, if a > b. The critical freql~ency is
(44)
This lowest mode is called the dominant mode. Equations for the fields
and diagrams showing the lines of electric and magnetic force for various
modes may be found in Sec. 2“19.
It is of interest to examine in more detail the case of the lowest H-mode
in rectangular guide, since this is by far the most important case. The
fields in the guide have the values
ELECTROMAGNETIC WAVES [Sllc. 2.12
HU=EZ=O,
H==~sink~=j~sin~z,
z a
Ev=_3; .2a
. k.z = —~~ p . xx .2a (46)
am ~ sm –a– = —~— Z~ sin ‘~1
z J A.
H. = COSk~ = COS‘;>
I
where it is assumed that there are no losses and hence ~ is purely imagi-
nary. The power flow is
(47)
for a unit amplitude of H.. In terms of the maximum value of Illul, this
becomes
p = a~ IEWI*
4ZH”
Since H= = O at the side walls of the guide, the current density on the
side walls has only a y-component which is independent of y and which is
of unit amplitude if H. has unit amplitude. The current in the top and
bottom of the guide has both longitudinal and transverse components
(48)
H-+---------- ..
-+----+ -----+--
HIi
.-+ --- -- ~--- ---F -- view
Side
lowing a current line can be com-
pleted only by including the
-—+ --- -- ---- —-—> --- displacement current on a portion
FIG.,2,7,—Linesof current flow (solid of the path. The total longitudi-
lines) for the lowest H-modein rectangular nal current is
waveguide. The dotted lines are the mag-
neticlines.
SEC.2.12] THE NORMAL MODES OF RECTANGULAR PIPES 37
h=~=29H.
Thus impedances Zm, Z,, and Z, are all different, and there is no unique
way to define a useful quantity in the nature of an impedance for a single
waveguide. Thk point will be discussed at length in later chapters.
The general solution of Maxwell’s equations in terms of plane waves
may be used with profit for the particularly simple case of the dominant
H-mode in rectangular guide. Let us consider two plane wavefronts
inclined at an angle 6 to the z-axis, with the electric field in the direction
of the y-axis, as shown in Fig. 2“8. If conducting plates are inserted at
y = O and y = b, they will cause
no distortion of the fields, since
they are everywhere perpendicular X
to the electric field. Now, if ver-
tical metal walls are placed at
z = O and z = a, the plane waves FIG.Z.S.—Propagationof the dominant
H-mode in rectangularwaveguidein terms
will be successively reflected at a of Dianewaves,
constant angle 0 and will thus be “
propagated down the guide. If the plane waves are taken to be in phase at
the point A, then the electric fields of the two waves add to produce a
max~mum intensity at thk point which we shall call E. The electric field,
at a point such as B, is equal to the sum of the amplitudes of the two waves
taken in the proper phase and is
—
where c is the velocity of the plane waves and A B is the distance between
the points A and B. Since A—B= (a/2) – Z,
which is just the value for the H-mode field. In a ~imilar manner the
38 ELECTROMAGNETIC WAVES [SEC.213
components of the magnetic field can be found, and they will agree with
the H-mode values. Thus the H-mode field has been decomposed into
that of two plane waves and a useful concept in describing some of the
properties of the H-mode is gained. The point .4 moves with the
velocity
c h.
—=C—)
sin e x
which is just the value of the phase velocity of the H-mode waves.
The results for the rectangular waveguide may be applied to the
higher modes between parallel plates. If the height of the guide, b, is
allowed to become very large, the solution for parallel plates, when the
electric field is parallel to the plates, is obtained. A series of modes
exists, for all integral values of n excluding n = O, corresponding to the
Z’E~-modes in the rectangular guide.
The TM-modes may be treated in a very similar manner. The
equation for E, is again separable in rectangular coordinates, and the
following values are found for the fields:
\vhere as before
lc~ = k: + k: = ~z + W2ep,
The modes are designated as l’.lf~. -modes, and it is evident that the
lowest mode is the Z’.lf,,-mode, since the zero values of m and n arc
,
excluded. The cutoff frequencies are given by Eq. (40), and the cutoff
wavelength of the lowest mode is
h, = &!!T2.
E, = R(r)@(@),
the equation for @ is
~2@
W+ XW=O,
and hence x must be an integer or zero. The complex form of the func-
tion @ is interpreted to mean that two solutions are possible: One is
@ = cos xO; the other is @ = sin x6. Thus the modes are degenerate
in pairs. The two modes may be thought of as two states of polarization
of the field.
The equation for R is
(52)
E. = e~X’JX(kCr), (53)
where JZ is the Bessel function of the first kind of order x. The solution
NX(k.r), the Bessel function of the second kind, is excluded because of the
singularity at r = O. The boundary conditions may be applied immedi-
ately, since E. must vanish at r = a, the radius of the tube, Thus k. is
determined by
JX(k.a) = O,
or
kca = tX., 1
where Lz. is the nth root of the Bessel function of order x.
(54)
The rlltoff
frequencies are then determined by
(55)
The smallest value of tXnis hl = 2.405. Other values are given in Table
2.2 at the end of this section.
40 ELECTROMAGNETIC WAVES [SEC.213
The TM-modes are distinguished by the subscripts TMXm, and the low-
est TM-mode is the TM O1-mode. Equation (41) is valid for these modes
if the values of A, given by
~=~a
. (57)
tXn
are used.
The TE-modes are treated in the same fashion. The solution of the
wave equation for H, leads to the same solution as that for E. in the TM
case:
H, = e~X’JX(k,r), (58)
This component of the field must vanish when r = a. If s,” are the roots
of J;, then
k,a = SX,,.
The cutoff frequency is
~2=_ 1s2 x’, (59)
c
w () a
and the r~ltoff ~~avelengt,h is
(60)
The modes are designated by !fE ,., and the dominant mode for round
pipe is TE,,, for which s,, = 1.841. Table 2“2 gives other values of the
roots of J;. It should be noted that the TE-modes also have two states
of polarization except when x = O. The other components of the fields
are
The principal formulas and pictures of the electric and magnetic lines
are collected together in Sec. 2.19.
SEC.2.14] HIGHER MODES IN COAXIAL CYLINDERS 41
Jx ANDJ:
TABLE2 2.—Room OF THEBESSELFUNCTIONS
TE,, J; 1,841
TM,, J,I 2,405
TE,, J; 3.054
TEo,, TM,, J:, J, 3.632
TE,, J: 4.200
TM,, Je 5.135
TE,, J: 5.30
TE,, J; 5.330
TMo, Jo 5.520
TM,, J, 6.379
TE5 , J: 6,41
TE,, J; 6.71
TE,,, TM,, J:, J, 7.016
Hz
= ei~OIA~X(kcr) + BNX (k.~)]. (62)
E, }
The boundary conditions for an E-mode are, then, that this quantity
vanish at r = a and r = b.
For the TM-modes, therefore,
— A _ NX(ha) _ N, (k.b)
(63)
B– JX(k.a) – J, (k,b) -
N; (k.a) _ N~(lccb)
(64)
~~(k,a) – J~(k.b)”
The lowest mode is the TElo-mode, where kc is given by the first root of
‘g ‘e’’IJ1(k’)N’b:) -J’(k:)N’(kc’)l’
Er._Jk&!;~
. ‘2hk”’N’(k4 -JJ(kcw(k4) ’66)
The next higher mode will be a TM-mode and will have a cutoff wave-
length greater than the cutoff wavelength for the TE-mode by a factor
of approximately T/2.
2.15. Normal Modes for Other Cross Sections.—There are several
other cases in which it is possible to obtain a separation of the wave
equation for Hz or E.. If the cross section of the waveguide is elliptical,
the fields are expressible in terms of Mathieu functions. The solutions
have been investigated in detail by L. J. Chu. 1 The solution for a
‘L. J. ~hu, “Mectromagnetic Waves in 1311ipticHollow Pipes of hletai,”
J. Applied Phys., 9, 583 (1038).
SEC.2.16] NORMAL MODES’ FOR OTHER CROASIS’SECTIONS 43
(a) (b)
(d)
(e)
FIG.2.11.—Modesderivedby insertionof conductingsurfacesperpendicularto linesof
electricforce.
contour lines, that is, the lines of EZ = constant. Now the boundary
condition for a TM-mode is that E, vanish on the boundary. Hence
the contour EZ = O may be chosen as the boundary of the cross section.
1R. D. Spenceand C. P. Wells,PhIJs.Reo.,62, 58 (1942).
2S. A. Schelkunoff,ll~ectromagnetic
Waves,Van Nostrand, New York, 1943, Sec.
10.8,p. 393.
44 ELECTROMAGNETIC WAVES [SEC,2.15
The contour lines now represent the magnetic lines in the transverse
plane. The electric lines are then orthogonal to these magnetic lines.
In a similar way, if H. is assumed to be a solution of the wave equation,
then the boundary of the waveguide for which this solution is valid is
normal to the lines H, = constant. These contour lines represent the
lines of electric force in the transverse plane, and the transverse lines of
magnetic force are orthogonal to them. The cutoff wavelengths are
prescribed when the functions Hz or E, are specified, since they contain
k, = %/h, as a parameter.
Moreover, if a solution for a simple case has been obtained, it is pos-
sible to derive other cases from it by inserting a conducting surface that
is everywhere perpendicular to the lines of electric force. If such a sur-
face includes a portion of the original boundary of the guide, the cutoff
wavelength will remain unchanged. Figure 2.11 shows several examples
of such derived modes.
It is always possible, of course, to solve the wave equation by employ-
ing all the well-known techniques of numerical integration, perturbation
methods, and so forth. There is a general relation between the cutoff
wavelength and the solution to the wave equation. The two-dimensional
Green’s theorem is
where the surface integrals are taken over the guide cross section and the
line integral around the boundary. Let us take U equal to V, and let it
represent either E= or H.. Then
but the first term on the right may be written k~j U2 di$, since U satisfies
the wave equation, and the second term on the right vanishes, since
either [J or a [J/an is zero. Therefore,
k,=(~). L[(%)+(%)lds.
e (69)
h. –
U, ds
/ s
This quantity is always positive; therefore, for any arbitrary shape, some
transmission mode is possible.
If some approximate form for U is known, A, can be calculated from
this equation. It may be shown that this is a variational expression for
SEC.216] TRANSMISSION LOSSES 45
I&; that is, the function U that results in the minimum value of lr~ is
the correct value in the sense that it satisfies the wave equation and the
boundary conditions. Therefore, if any function for U is used, the value
of k: calculated from it will always be larger than the actual value that
is the correct solution to the problem. It is also possible to establish a
systematic method of successive approximations that converges on the
correct values of k. and U for the particular problem. This procedure has
been discussed in detail by J. Schwinger. An example of the results of
such calculations is shown in Fig. 2.12, which shows the cross sections of
several waveguides with flat tops and bottoms and semicircular sides.
The cutoff wavelengths are identical for all these shapes.
2“16. Transmission Losses. —Throughout the preceding sections it
has been assumed that the walls of the waveguide are made of perfect
conductors. A guide with real metal walls has a finite, although large,
conductivity, and thk must be
taken into account. The alter-
ation appears in the boundary
conditions. It has been assumed
that the tangential electric field
vanishes on the surface of the con-
ductor. In the case of a real
metal, the tangential electric field
does not quite vanish but has the
small value determined by the
product of the conductivity and
the current density. The current
density is equal to the tangential L———— 0.5AC ~
Since Eti is small, it is permissible to use for Hti the value that it would
have in a guide of infinite conductivity: At the metal surface the tan-
gential component of the electric field must be
Eti = Z.H~.
It will be recalled that
,.
(70)
Id&’
—— .— 1 Re (SJ d. = %) lHk12 ds,
a=–2PdZ 2P / walls \ walk
where the element of area ds is a strip of unit length in the z-direction.
For a good conductor, an approximate value of Z- may be used, since
@c << u. By expansion, it is found that
(71)
For metals, a is usually greater than 107 mhos per meter, e is of the order
of 10– 11farad per meter, and hence even for frequencies corresponding to
u equal to 1013per second, at/u is only 10-5. Thus even the first power of
uc/u may be neglected entirely, compared with unity, and
—
Hence
z. =
d
*(1 + j). (72)
(73)
Thus the metal losses are equal to those which wouldbe produced by a
uniform current K flowing through a surface layer of conductivity u and
thickness & Therefore
8= ~, (74)
J- ~w
and hence 6 is characteristic of the metal and of the frequency. Table
2.3 shows values of u for various metals, values of ~ for a frequency of
1010cps, assuming Y = go, and the relative losses per meter in waveguides
constructed of the various metals. The propagation constant in the
metal is -y = ~jtipu — u2cp, which, for large u, becomes
.y=
J-
y(l+j)=; (l+j). (75)
Hence the fields within the metal fall off to 1/e of their value at the sur-
face at a depth equal to 6.
TABLE23.-SKIN DEPTHANDRELATIVE
Loss OFVARIOUSkIETALS
6.42 X 10-7 o 97
6.60 1.00
7.85 1.19
8.11 1 23
8.26 1,25
12.7 1.92
17.0 2.5
18 5 2.8
Equation (47) gives the value of P for this mode. The attenuation a
may therefore be written
(76)
approaches zero. Figure 2.13 shows the calculated values of the attenua-
tion in a rectangular copper waveguide for four modes.
2.17. Cylindrical Cavities.-Suppose that a piece of waveguide of
length 1is closed off by metal walls perpendicular t o the axis of the guide.
If there are electromagnetic waves in the cavity so formed, they will be
reflected from the ends and will travel back and forth until they are all
dissipated in heating the metal. For certain frequencies, a cavity of this
0.1-
0.08
0.06
0.04 1 \
TM,I
y
0.02 \
<Ezo
\
0.01
0.008
0.006
4X103 6X103 ~04.
2X104 4X104 6X104
Frequency
inMe/see
FIG. 2,13.—.4ttenuationin rertangukuroppexwaveguidefor sev?ralmodes;a = 2 ill..
h=l in.
where n is an integer and & is the }vavelengtb in the guide, The discus-
sion of most of the properties of resonators \~ill be fo~md in Chap. i.
Only the losses in the cavity \vill be discussed here. These losses are
most conveniently expressed in terms of a quantity called the Q of the
cavity. This quantity Q is defined as the energy stored in the ca~-ity
divided by the energy lost per radian. If the losses occur only in the
. .
cavity Itself and not by transfer of energy to other systems, the perti-
nent quantity is the “ unloaded” Q, l~hich is denoted by QO. This con-
cept is a natural extension of lolv-frequency terminology and is usef[]l in
very much the same way, as \villbe sho~vn in more detail later.
SEC.2.17] CYLINDRICAL CAVITIES 49
If the cavity is of resonant length, the field pattern if; in the form of a
standing wa{e having nodes at the two ends and (n — 1) nodes along the
length of the cavity. The stored energy could be calculated by integrat-
ing, over the cavity, the quantities icE2 and 4PH2. Likewise the losses
could be found by integrating the square of the tangential magnetic fields
over the walls and the ends of the cavity. This calculation has already
been performed in effect, however, and a value for Q, may be derived
from the previous results. The standing-wave pattern of the fields may
be decomposed into two waves of equal amplitude traveling in opposite
directions. It will be shown in Sec. 2.18 that the waves carry energy
with the group velocity v,. If this is assumed to be true, the energy
stored, W, is seen to be
(78)
where A and c are the wavelength and the velocity in the dielectric
medium, the expression for the power flow may be written
(79)
4aPl 2naPA.
w,=—=— (80)
Ll) u“
where the integral is taken over the two ends. This is directly related
to the quantity P by
(82)
or
1 _ Al 4h~
(83)
Qo irA” ‘TMWU”
For a longitudinal H-mode (Z’E-mode)j Z. = (x,/k)~; and for a longi-
tudinal E-mode (TM-mode), Z. = (k/x,){. Thus QO is expressed in
terms of quantities already calculated. Values of Qo for the various cases
are included in Sec. 2.19. It has been assumed here that the losses in
the dielectric material in the cavity may be neglected. In Chap. 11 the
dielectric losses will be taken into account.
2.18. Energy Density and Power Flow in Waveguides.—To complete
this survey of longitudinal electromagnetic waves, it remains to prove
some general theorems regarding the normal modes and to calculate the
power flow and stored energy in waveguides. It has been shown that the
fields for both E- and H-modes are completely determined once a single
component of the field is known. If either the Iongitudhal component
of the magnetic field for H-modes or the electric field for E-modes is
designated by V., this quantity is determined by
(v; + k:) Vz = o, (84)
-,vhere k: = Y2 + C02CY,and v’; is the Laplacian operator in the transverse
coordinates [Eqs. (33) and (36)] with the boundary condition that on the
guide walls,
Vz=o for E-modes,
(3Vz o (85)
for H-modes,
x= 1
where c3/c3nis the derivative in the direction normal to the guide walls.
The values of k: are the characteristic numbers of the problem which
determine the cutoff wavelength
~=2~
c (86)
k, “
The cutoff frequency is
k.
(87)
“c = @’
and the guide wavelength is given by
(88)
(92)
1
H,.llzb ds = k:e – k:b (H.aV2H& – H.bV’Hw) dS.
\ [
By Green’s second theorem, this integral becomes
/( H dH.b
—
‘a an
aHza
– H.b ~ dl,
) (93)
where the integral is taken over the curve bounding the guide and
vanishes by virtue of the boundary conditions [Eqs. (85)]. For the
transverse components,
The first integral vanishes because of the boundary conditions, and it has
just been proved that the second integral vanishes. Because the trans-
verse electric fields are proportional to the transverse magnetic fields,
the orthogonality of the electric fields also is proved.
The proof for two E-modes is exactly analogous. For the longi -
,..,.< ,.
52 ELECTROMAGNETIC WAVES [SEC.2.18
tudinal fields theintegralin Eq. (93) again applies, with Ezwrittenfor H..
The integral again vanishes because of the boundary condition E, = O.
For the transverse components there is an expression similar to Eq. (94)
which vanishes in the same manner. For one H-mode (a) and one
E-mode (b), the longitudinal components are orthogonal;
and hence the transverse magnetic fields are orthogonal. The proof for
the transverse electric fields is almost identical. Thus it is clear that the
longitudinal components of the electric and magnetic fields and the
transverse components are all separately orthogonal for any two different
modes.
It remains now to show that the energy flow for two modes contains no
mixed terms. If two H-modes are considered. the power flow contains
terms such as
— H,a o Hi dS = O, (97)
/
as has already been shown. The argument is identical for two E-modes,
and for one H-mode and one E-mode. Thus when several modes exist
at the same time in a waveguide, the flux of power of two modes can be
computed independently and added. It should be noted that this is not
true for the loss in a waveguide, since there can be a mixed term of the
form HtiH., the integral of which does not vanish.
Expressions for the stored electric and magnetic energv in a waveguide
SEC.2.18] ENERGY DENSITY AND POWER FLOW 53
~=~eud
]H.]’ d8. (98)
4 k: /
which is equal to the total electric energy. The rate of flow of energy is
p=iRe(\EtH+=- ~ Z~ ~
\
IH.12dS.
(102)
(105)
Here again, therefore, the total stored energy is equally divided between
electric and magnetic energy. It can be shown as above that the same
expressions for Vph,UQ,and v hold for E-modes as for H-modes.
2.19. Summary of Results.—The survey of the classical electromag-
netic theory of both transverse and longitudinal waves has been com-
pleted. It remains only to summarize the results in a manner that will
be convenient for ready reference. For each of the modes that are of
practical importance, the specific form of the fields will be given, together
with the cutoff wavelength, formulas for the power flow and the attenua-
tion, and the expressions for the unloaded Q of a cavity n half-wave-
lengths long.
Coaxial TEM-mode.—The transverse cross section of a coaxial trans-
mission line operating in this mode is shown in Fig. 2.14. The fields are
given by
E.= HZ= E.= H,=O,
2 a3 + b3
u = ~ q ab(a2 + b’) [1 -(:)’1~’
12
~ = lrbo
-1 q ab;;2++b;2) ‘+:[1 -(w]”
The H I,-mode ( TE, l-mode) in Round Waveguide.—The TEl,-mode is
the dominant mode in waveguide of circular cross section. The fields
are shown in Fig. 2“18. The field and power relations are
in
rmlndw:~vcuuide. roltnd wnvt,zllide.
SEC.219] SUMitfARY OF RESIILTS 57
E,=J,
()
t“,:,
Eo = O,
“=+4’-(021)
i ‘:+W-(a’]”
The Hz I-mode ( TEzl-mode) in Round Waveguide.—The next mode in
round wave guide, in order of decreasing cutoff wavelengths, is the HZ1-
mode. The fields are shown in Fig. 220. The equations for this mode
are
ELECTROMAGNETIC WAVES [SEC,2.19
H.=Jo
()
So,: )
It should be noted that the attenuation for this mode has the unique
property that it decreases continuously with decreasing wavelength.
BY C. G. MONTGOMERY
3s1. Some General Properties of Guided Waves.—In the previous
chapter, it was shown that waves may travel in hollow pipes in many
modes of transmission and that for each of these modes there is a cor-
responding cutoff frequency. For frequencies below the cutoff frequency
the energy in the mode is quickly attenuated; above the cutoff frequency
it is freely transmitted. The most important condition in practice is
that in which the frequency lies above the cutoff frequency for the lowest
mode but below the cut off frequent y for the next higher mode. The
lowest, or dominant, mode will then propagate
(1)
F = Aeif@t–.z) + Bei(.t+u)
= #[Ae-iKS + B#z+4)],
where
B = Blei+.
The first term is the traveling wave of amplitude D, and the second term
is the standing wave of amplitude 2B 1.
It is easy to measure a quantity proportional to the amplitude of F
by inserting a small probe in the waveguide, and in this way the standing-
wave pattern can be measured. The methods for making such measure-
ments are discussed in detail in Vol. 11 of this series. The standing-wave
ratio will be denoted by r. It is given by
(3)
and
r—l
pl . — (5)
T+l”
At the position of the minimum in the stan ding-wave pattern, the phase
of r is T, and
r=_~
A“
which converges, since Irrl] < 1. The total wave amplitude traveling
to the left \vill be
Thus the whole effect of a reflection from the generator is to change the
(M WA VEGUIDES AS TRANSMISSION L1 NES [SEC. 3.2
amplitude of the incident wave from A to A‘, and the situation is other-
wise unaltered.
Some simple relations between the transmitted power and the reflected
power, in terms of r and r, may now be written. The fraction of the
power reflected, P,, is
2
P, = rr” = Irl’ = ~~ .
()
rhe fraction of the power transmitted by the obstacle, Pi, is
p,= $2.
Icl
A+ l?, ‘r
IC12
‘=l–p’= 1–(A+B#+lr1)2
. Icl’
1– (A – l?,)’ (1 – Irl)’, (7)
or
IC12
1 – (~ !’;,)2 (r ; 1)2” (8)
‘=1–(A+B,)2~2=
picious. Standing waves, short and open circuits, and other things usu-
ally associated with ordinary low-frequency circuits where currents and
voltages and not electric and magnetic fields are taken to be the funda-
mental quantities have been mentioned. These suspicions may be lulled
by establishing more explicitly the connections with low-frequency cir-
cuits. It should be emphasized, however, that up to this point only the
fact that there are waves traveling down a waveguide and being reflected
or transmitted by obstacles has been utilized, and therefore the results
are completely general. But one restriction has been made, namely,
that only the dominant mode can be propagated in the guide.
In a coaxial transmission line, energy is propagated in the principal
or Z’Elf-mode. In Sec. 2“7 the expression for the fields and the equa-
tions that they satisfy have already been derived. It was found that if
losses are neglected,
where 1 is the total current flowing in the walls of either the inner or
outer conductor. These equations can be put into a slightly different,
form. If the voltage across the line is defined as
b
v= E, dr,
\ a
this value of V is independent of the path of integration from the inner
to the outer cylinder provided only that the path be restricted to a trans-
verse plane, since H is purely transverse, If the equations are integ-
rated with respect to r over such a path, and if 1 is substituted for 2m-H4,
then
where Z and Y are the series impedance and shunt admittance per unit
length of the line. These equations are rigorously true for the coaxial
66 WA VEGUIDES AS TRANSMISSION LINES [SEC.3.2
line if
(lo)
These values could have been found, not only from Maxwell’s equations
dkectly as has been done here, but also from a calculation of the induct-
ance and capacitance per unit length between coaxial cylinders. For
transmission lines of other shapes, such as parallel wires, Eq. (9) is valid
if the usual low-frequency approximations are made. The values of
Z and Y will, of course, be different; they will be those characteristic of
the particular line under consideration. The solution of Eq. (9) may now
be written as the sum of waves traveling to the right and to the left of
the point of observation
v = ~e–’t’a + ~#z,
1 1 (11)
1 = To ‘e-’” – z ‘e’s’ 1
where 20 is the characteristic impedance of the transmission line and
y is the propagation constant; thus
(12)
Y=:= YYO.
1 (13)
av aI
—.— 7ZoI, –7YOV. (14)
a.z Fz =
“The current reflection coefficient can be defined as the ratio of the reflected
current wave amplitude to the incident current wave amplitude. Hence
1+:
A+B
“=z OA-B=z Ol_~”
A
The elimination of the ratio B/A, by means of Eq. (16), has the result
rv=;=;[;;;:=
* -:=-r,.
.
The other ratio,
v, 2Z(1)
(18)
“=m=z(q+zo=l+ r”’
1 1
Yo = *J ‘(z) = z(l)’ “i” = z“
(23)
u 6
I
left to the right. The admittance to the right from a point just to the
left of Y is Y + Yo, and the voltage reflection coefficient is
–1’
(26)
‘= Y+2YO”
Y, + Y’z
r=–
YI + Y, + 2YO”
Z=l+a+cqlc.
Likewise
-/=a+flz(l+ a).
If x is eliminated,
,=a+p(:::-’. (27)
— 27 2a 2@
l+7=–l+a– l+p”
From Eq. (26),
Y – 2r
Y- = I+r”
Thus the law of additivity of shunt admittances has been verified from
the wave picture.
The argument just stated could have been carried through using the
concept of an equivalent series impedance that combined simply with
another series impedance. Again the reflection coefficients do not com-
bine simply. This is another aspect of the importance of the admit-
tance or impedance concept for use in waveguides, where neither currents
nor voltages may be uniquely defined.
3s6. Transmission-line Charts.-It has been shown, in the preceding
section, that a reflection in a transmission line can be described in several
alternative ways. Each of these ways is convenient for certain problems;
all are in common use. A reflection can be described by any of four pairs
of variables:
These relations may be separated into their real and imaginary parts.
Thus,
l+lrl=/(R +l)’+x’+d(R–1)’+X2
~=l–lrl <(R + 1)’+ X2 – <(R – 1)2 + X’
~(G + 1)2 + B’ + <(G – 1)2 + B’
= <(G + 1)2+ B’ – v“~’ ’28)
~d=f$-m
— 1 tan–l
. ~ 2x T
2 ~2+x2_l–ij
– ~ tan–l ;, (29)
T—1
pi=-=
T+l tm:=%%k- ’30)
2x 2B
t$=2tcd+r ‘tm-’R2+X2_l = ‘an-’ G’ + B’ _ 1’ (31)
r 1 – Irl’ G
R = ~Z C052 ~d + sin’ Kd — (32)
– 1 – 21rl cos @ + lrlz = G’ + Bj’
x = (1 – r’) sin Kdcos Kd = 21rl sin @ –B
— (33)
r’ COS2Kd+ sin’ Kd 1 – 21rl cos @ + 11’12= G’ + B’
r 1 – Irl’ R
G= (34)
r’ sinz Kd + COS2Kd = 1 + 21rl cos @ + lrl’ = R’ + X2’
p — 1) sin KdC13sKd = –2)rl sin @ –x
B=( (35)
T2sin2 ud COS2Kd 1 + 21rl cos @ + lrl’ = R’ + X2”
~=— aw+b
Cw.+ 2“
SEC. 3.6] TRANSMISSION-LINE CHARTS 73
Thus it is possible to apply many general theorems which are well known
for transformations of this type.
These equations are sufficiently numerous and complicated that some
graphical method of handling them is almost essential. Fortunately,
a method exists that is convenient and easy to use, whereby these 24
relations can be represented by a single chart. This ‘chart, designed by
.0
~
FIG.3.5.—TheSmithimpedancechart.
P. H. Smi?~h,l is illustrated in Fig. 3.5. The quantities Irl and @Iare
chosen as polar coordinates, and lines of constant R and constant X are
plotted. The region of interest is within the circle of unit radius, 1P = 1.
The family of curves R = constant, X = constant consists of orthogonal
circles. In terms of rectangular coordinates u and v in the I’-plane, these
circles are given by
(u-&) +u2=(Rll)2
2
(u–l)’+ v–+ =+.
()
The R-circles all have their centers on the u-axis and all pass through the
point u = 1, v = O. The X-circles all have their centers on the line
u = 1, and all pass through the point u = 1, v = O. .411 values of R
1~, H. Smith, Electronics,January 1939,January 1944.
74 WA VEGUIDES AS TRANSMISSION LINES [SEC.3.6
from zero to plus infinity and all values of X from minus infinity to plus
infinity are included within the unit circle. Thus there is a convenient
means of transformation from Ir I and @ to R and X and inversely. If
the reference plane is moved nearer to the generator, that is, in the nega-
tive z-direction, the vector r rotates clockwise, making one revolution in
+ 1.0
+ 0.5
u-l
m
o
50
%
-0.5
-1.0
0 0.5 1.0 1.5 2.0
Ror G
FIG.3.6.—Impedancechartwithrectangularcoordinates.
half a wavelength. An auxiliary scale outside the unit circle, running
from O to 0.5 around the circumference, facilitates this transformation.
Curves of constant standing-wave ratio are concentric circles about the
origin which pass through the points r = R. The parameter d/A~ is
read on the external circular scale. The relation between impedance
and admittance is obtained in the following manner. A shift of reference
plane of one-quarter wavelength inverts the value of the relative imped-
ance; the shunt admittance equivalent to a series impedance is given,
therefore, by the point diametrically opposite the origin from the imped-
ance point at the same radius. Moreover, it is apparent from the trans-
formation equations that if II’1 is replaced by – Irl, then R m~~’ be
replaced by G and X by B. Thus the same chart maybe used for admit-
tances provided the value of o is increased by r.
SEC.37] THE IMPEDANCE CONCEPT 75
The use of a Smith chart is very similar to the use of a slide rule; many
tricks and short cuts are possible that are hard to describe but greatly
facilitate computations. The Emeloid Company, of Arlington, N. J.,
makes a chart of this kind, of celluloid, which is called the “Radio Trans-
mission Line Calculator. ”
Impedance charts of other varieties have been made and used, but
only one other is commonly encountered. In this version, R and X
are used as rectangular coordinates, and the lines of constant ~ and
d/& are plotted. The chart has the same form when G and B are used
as coordinates. The reflection coefficient cannot be read easily frcm the
diagram. The lines of constant r are a family of circles with centers on
the real axis, and the lines of constant d/x. are circles centered on the
imaginary axis and orthogonal to the r-circles. An outstanding difficulty
with a chart of this type is that the points of infinite R and X are not
accessible. This rectangular form of impedance chart is illustrated in
Fig. 3.6.
3.7. The Impedance Concept in Waveguide Problems.—It has been
shown in preceding sections that the properties of both waveguides and
low-frequency transmission lines can be described in terms of incident
and reflected waves. The state of the line or waveguide can be expressed
by means of reflection coefficients that are, with the exception of a con-
stant factor, sufficient to specify this state completely. In addition, it
has been seen that the rule of combination of reflection coefficients is
complicated even in the simplest cases.
On the other hand, the state of a low-frequency transmission line may
be expressed equally well in terms of a relative impedance or admittance,
that is, the ratio of the impedance or admittance to the characteristic imped-
ance or admittance of the transmission line. The impedance or admit-
tance combines simply with other impedances, and it is this property
which leads to a demand for an equivalent concept for the characteristic
impedance of a waveguide. It has been seen that the reflection coef-
ficient in a waveguide can be replaced, at least formally, by a relative
impedance that is completely equivalent and that expresses the state of
the fields to within an unknown factor. In any configuration of wave-
guides of a single kind, relative impedances or admittances may be
defined in terms of r and @ and combined according to the usual low-
frequency rules. It is not necessary to specify exactly what is meant by
the characteristic impedance of the guide.
Let us now consider the junction of two waveguides as illustrated in
Fig. 3.7. If radiation is incident upon the junction from guide 1, there
will be, in general, a reflected wave in guide 1. This reflected wave may
be described in terms of the reflection coefficient or in terms of an equiva-
lent relative shunt admittance or series impedance thatrterminates guide
1 at the junction. Provided the losses in the neighborhood of the junc-
76 WA VEGUIDES AS TRANSMISSION LINES [SEC.37
tion may be neglected, the power flowing in guide 2 must be equal to the
difference between the incident and reflected powers in guide 1. The
amount of reflected power will be determined by the actual electric and
magnetic fields in the aperture, which, of course., satisfy Maxwell’s
equations and the appropriate boundary conditions. In particular,
across any transverse plane, the tangential
Guide1 Cuiie2
— electric and magnetic fields must be con-
—
— tinuou~ To complete the analogy with
1 low-frequency transmission lines, quanti-
Fm. 3.7.—Junctionof two wave- ties analogous to the current and voltage
guides.
must be defined for waveguides, since it is
in terms of the values of current and voltage that the terminal ccmditions
must be specified. A few possibilities will be discussed.
The voltage and current should be linear in the magnetic and electric
fields, since it is desired that their product be a measure of the power.
Thus let
V = aEi + bHt,
-n d
1 = cE, + dH,,
where Et and Ht are some mean values of the transverse fields. The
complex power is then
V = aEt, I = dH,.
The impedance at any point is then
~=a Et
(36)
I z Tc
and the power flow
P = ~ad*E,H~. (37)
It has already been shown that all properties of reflected waves can
be expressed in terms of a relative impedance, and no condition is imposed
on the proportionality factor of Eq. (36). The second condition ~Eq.
SEC.343] EQUIVALENT T-NETWORK 77
(37)] when applied to two guides such as indicated in Fig. 3.7 does repre-
sent, however, a new condition. Any mean value may be chosen for
E, and H,, and in fact different values may be taken for two different,
waveguides, provided only that the conservation of power at a junction
between two guides in ensured. This condition cannot be written
explicitly, since it depends upon the nature of the junction. It can be
seen, however, that if the current is identified, for example, with the
transverse magnetic field at the center of the waveguide, then the con-
stant of proportionality between the voltage and the electric field is
definite and is determined so that P = iVI* represents the true power
flow. It may be pointed out that the ratio between I and Ht need never
be specified and may be chosen at will. If particular values of E, and H,
are chosen, then only the product ad* is determined, but neither a nor d
separately.
The situation is somewhat analogous to that arising from the insertion
of ideal transformers in a network. if an ideal transformer were con-
nected to each voltmeter and ammeter in a network in such a manner
that the product of the readings remained the same, the result would be
an effective change in the definition of impedance, all the power relations
being conserved.
3.8. Equivalent T-network of a Length of Waveguide.—If the con-
cept of impedance in a waveguide is to be useful, it is important to deter-
mine whether or not it can be used in the same manner as the impedance
in low-frequency circuits. It has al- Z, Z1
ready been seen that the reflections
in a-long line are correctly described
in terms of an equivalent shunt ad- 7,2 z,
~
mitt ante or series impedance. Now ““” 1 +
~1
the question is whether or not more FIG.3.8.—Sy,nn,etrlcalT-network.
com~licated
. structures can be repre-
sented by equivalent circuits. Ii a straight piece of waveguide is ter-
minated in such a way that the reflection is described by an impedance ZI
at the end of the line, then the equivalent impedance at the input terminals
of the line is
ZI + jZO tan K1
Zih = ZO (3.17)
Zo + ~zl tan .1”
or
~, z, + 2Z, + 71
Z,+Z2 ‘
z. =
1++ ZI + z,
z, z, + 2Z2
= jzo tan K!,
z, + z,
1 tan
—. Z.
Kl
—.J
z, + z,
If these equations are solved for Z, and Z,, the result is, with some
trigonometric fransl mmations,
Z, = jZO tan $
j(v’~ – 1)
j(J+ 1)
–j(/2 – 1)
o
EV = A sin ~ V(z),
where V(z) expresses the field variation along the line and may be called
the voltage. Likewise the transverse magnetic field is
where 1(z) may be called the current. Then from Eq. (2.46) the longi-
tudinal field is
H. = A ;a COS: V(z).
C?&
= ju~H.,
az
aH= ——
aHz .
jcoeEV.
a% ax
If the values of the fields are substituted in the first equation to find the
voltage and current, the result is
dv(z) _
–j@#I(z).
a%
jup = -fZo = z.
The propagation constant -y of the transmission line has the same value
as that of the waveguide
Q
72 = –U2EP + : ,
()
and the characteristic impedance is
S. dx dy = – ; O“EYH~ dx = ~ VI*.
/ /
Therefore
and for the line current a quantity proportional to the longitudinal cur-
rent flowing at the center of the broad face of the wave~ide. This
longitudinal current is equal to the maximum value of the transverse
magnetic field
I(z) = aH=(z)
Let us consider the line integral of the electric field around the rectangular
path ABCD in Fig. 3.9. As the dist antes AB = CD become infini-
tesimal, the line integral approaches dV/dz. By Faraday’s law (the
curl E equation)
dV _ jupbI
dz–a”
If likewise the limit of the line integral of H= is taken around the path
EFGH the result is
ldI
—— a . 7rX . V “~in~dZ_2K,
adz / o ‘ln Tdx=Ju’% ,
/ o a
where ICtis the transverse current density across EF or GH. Its value is
hence
The propagation
‘= -k-a”
constant of the line is given by
z;=$=_ 1 (LLyLb) 2
a’ ~2cp — .’=–;’7
—
a’
1
()
WA 2
“
82 WA VEGUIDES AS TRANSMISSION LINES [SEC.3.9
The choice of a may now be made such that ~ VI * is equal to the complex
power.
J. Schwinger has shown that it is possible to proceed in an entirely
general way and transform Maxwell’s equations directly into the trans-
mission-line equations whenever the boundary conditions are independ-
ent of the z-coordinate. Such a general case need not be considered here.
The procedure that will be uniformly adopted here is as’f ollows:
BY C. G. MONTGOMERY
4.1. Elementary Considerations.—In this chapter will be presented
some of the elementary results of network theory that are useful in the
study of microwave circuits. Theapproach will beintermsof whatmay
be called the low-frequency approximation to electromagnetic theory.
This approximation is the one usually employed in conventional circuit
theory, and the results are well kno~n and available in many standard
textbooks. For the convenience of the reader and also to aid in a more
orderly presentation of the properties of high-frequency circuits, some of
the more useful material has been collected. This material is offered, in
general, without detailed proof of its correctness. Many of the results
are proved in Chap. 5 as special cases of more general theorems. In
other cases only the method of proof is outlined. The reader will find
himself already acquainted with a large part of this discussion.
In this chapter, the concept of an impedance element, or impedor, will
be considered as fundamental. An impedance element is a device that
has two accessible terminals. It may be a simple device, such as a piece
of poorly conducting material (a resistor), or it may be a very complicated
structure. It is required, however, that it be passive, that is, that no
energy is generated within the element. Charge may be transferred to
the element only by means of the terminals; and if a current flows into
one terminal, an equal current must flow out of the other. This is the
first portion of the low-frequency approximation mentioned above.
Thus a conducting sphere is not an impeder, since it has only a single
terminal, but the equivalent impedance element can be supposed to have
one terminal at the sphere and the other terminal at the point of zero
potential or ground, perhaps at infinity. In the region between the
terminals of the impedance element there exists an electric field. The
potential difference, or voltage, between the terminals is defined as
the line integral of the electric field from one terminal to the other. The
second portion of the low-frequency approximation under which network
theory is here treated requires that this line integral be independent of
the path between the two terminals. The difference in voltage, for any
two paths, will be proportional to the magnetic field integrated over the
area enclosed between the two paths and to the frequency, and can be
made as small as desired by the choice of a sufficiently low frequency.
The ratio of the voltage across the terminals to the current entering and
83
84 ELEMENTS OF NET WORK THEORY [SEC. 41
Z=–~+R+joL, (1)
The impedance of Eq. (1) has, on the other hand, an admittance made up
of the conductance
‘= R2+(”~-iY=R2:x2
and the susceptance
~= -(”L-A) (2)
R2+(”L-SY “[:x’”
. .
and the variat~on with u is characteristic of R, L, and C in series.
SEC.41] ELEMENTARY CONSIDERA T’IONS 85
The simple relation Z = V/1 remains true if the quantities are all
replaced by their duals; that is, Y = I/V. Likewise the statement
“impedances are added in series” becomes “admittances are added in
parallel.” If the duality replacement is made in Eqs. (2), they become
(3)
the path. Kirchoff’s second law states that the algebraic sum of the
currents flowing into each branch point must be zero. This law follows
from the conservation of charge, since charge cannot accumulate at the
branch point. A sufficient number of linear relations may be set up by
means of these two laws to make possible the determination of all currents.
A simpler set of equations is obtained if the currents in the meshes
are used as the unknown variables. These currents, which are indicated
in Fig. 4“1, are sometimes called the circulating currents. When circu-
lating currents are so chosen, Kirchhoff’s second law is automatically
satisfied. Thus there is a set of equations of the from
voltages are & + V~2). Thk is known as the superpow’tion theorem for
linear networks.
The fundamental set of network equations may be established on a
node basis rather than on a loop basis as are Eqs. (4). The independent
variables are the voltages of the nodes, and the dependent variables the
currents flowing in and out of the nodes. The equations thus obtained,
. (6)
.
im = Ymlvl + “ Q . + Ym#m,
are the duals of Eqs. (4). The coefficients Yii are the self-admittances
of the network, and Yij are the mutual admittances.
4.2. The Use of Matrices in Network Theory .-Many of the results
of network theory, both in the low- and high-frequency approximations,
can be written most conveniently and concisely by the use of matrix
not ation. For the convenience of the reader, a summary of the rules of
matrix manipulation is presented in this section. Eqs. (4) are written as
v=zi (7)
where Z is the impedance matrix of the system. It is a square array of
the coefficients of Eqs. (-l),
Vm
All matrices are distinguished by the characteristic saris serif type used
above; their components are printed in the usual italic type, since they
are ordinary scalar quantities.
The operation of cornbiniug Z and i is callwl multiplication. It is
clefined by the follo[ving equati(m \\hich holds for the multiplication of
any two matrices proyided the number of columns of the first matrix is
88 ELEMENTS OF NET WORK THEORY [SEC.42
It is possible also to define a zero matrix O whose elements are all zero,
and the equation
A–B=O
means that
A,, = B,j, for all z and j.
CA = (CAi~) = Ac.
Likewise, the unit matrix I may be defined, whose elements along the
diagonal running from the upper left-hand corner to the lower right-
hand corner are unity and whose other elements are zero,
100 ...0
01 ”....
001 . . . .
l=. . or Iii = Siiy
. .
. .
[0 . . ...1 I
where &j is the Kronecker delta. For any matrix
AI= IA=A.
A matrix that has elements only along the diagonal is called a diagonal
mat rix. Two diagonal matrices always commute with each other.
If Eqs. (4) are solved for the i,, i,, . . , ifi in terms of VI, u,, ,
i = Yv,
SEC.4.2] THE USE OF MATRICES 89
zi = ZYV = v.
Vmz’”i),
‘i = detl(Z) ‘V’fi’ + ‘Zz’j “ “ “
where det (Z) is the determinant formed from the elements of Z and .2P is
the cofactor of the element Zii in det (Z). Therefore it is evident that
the reciprocal of a matrix can be defined as
(Zii)
(9)
‘-’ = det (Z)-
2 = (ZiJ, if Z = (Zij).
The product of a column vector and a row vector, taken in that order, is a
square matrix
a~ = A,
where
Aii = aibi.
90 ELEMENTS OF NET WORK THEORY [SEC.43
Sb=c= a,bi.
z
1
4.3. Fundamental Network Theorems.—The fundamental physical
principles that form the basis of network theory are embodied in Max-
well’s equations for the electromagnetic field. These equations, together
with the force equation, the appropriate boundary conditions, and Ohm’s
law, would be sufficient for all further developments. Network theory is,
however, limited to some rather special cases of all those to which the
general electromagnetic equations- may be applied. It would thus be
possible to formulate network theory from several specialized and
rather simple theorems or postulates which, in turn, are derivable from
the general equations. It will not be attempted here to erect this logical
structure, because the problems considered in waveguide networks are
more general than those treated by ordinary network theory. The dis-
cussion will be confined to a mere statement of the network theorems
without a rigorous justification for them. The more general point of
“view will be adopted in Chap. 5. The choice of theorems that are to be
regarded as the primary ones, from which all the others can be derived,
and those which are corollaries to the primary theorems is, of course, to
some extent arbitrary.
The first network theorem, the superposition theorem, was stated in
the first section of this chapter. This theorem follows directly from the
linearity of Maxwell’s equations. The second important theorem is
called the reciprocity theorem. This theorem is most concisely expressed
by the statement that the impedance matrix of a network is symmetrical;
the element Z~i is equal to the element Zii. This theorem follows from
the symmetry of Maxwell’s equations and will be proved in Chap. 5.
Since Y = Z-1, it follows that Y is also a symmetric matrix and Yij = Y;,.
The reciprocity theorem is often expressed by the rather ambiguous
statement that it is possible to. interchange the position of a generator
and an ammeter without changing the ammeter reading. An inspection
of Eqs. (4) will convince the reader that the statement is vague but
correct.
The third important network theorem is called TW%enin ‘.s theorem.
This theorem states that a network having two accessible terminals and
containing sources of electromotive force may be replaced by an electro-
motive force in series with an impedance. The magnitude of this electro-
motive force is that which would exist across the two terminals if thev
were open-circuited, and the impedance is that presented between the
two terminals when all the voltage sources within the network are
replaced by their internal impedances. This equivalence is illustrated in
SEC.4.3] FUNDA MENTAL NET WORK THEOREMS 91
Fig. 4.2. Let V be the open-circuit voltage of the network and Z the
impedance looking back into the network when the internal electromotive
forces are zero. Let Z. be the load impedance placed across the network
terminals. Then Th6venin’s theorem states that the two networks shown
in Fig. 4.2 are equivalent. To make this evident, let us imagine that a
source of voltage —V is placed in series with the load ZL. NO current
will then flow in Z.. Now, invoking the superposition theorem, this zero
current may be considered as composed of two equal and opposite cur-
pjc+@=i342
FIG. 4.2.—Equivalent networks demon-
strating Th6venin’s theorem. theorem.
Al
FIG. 4.3.—Circuit to illustrate Th6venin’s
rents, one excited by the external source and the other by the source
within the network. However, the value of the former current is
– V/ (Z + Z.); therefore, the current through Z. from the sources within
the network is V/(Z + Z.), which proves the equivalence.
As an example of the application of this theorem, let us consider the
circuit shown in Fig. 4.3. The impedance Z looking to the left of .4B
when VI is short-circuited is
Z= Z,+*,.
z,
v= –-VI
ZI + z,”
Hence
z,
v, —
12=–
Zl+z? =_ V1Z2 __
Zlz, + Z*Z3+ Z,Z3+ Z,(Z; + z,)”
z,+—
ZY22 + “
Th6venin’s theorem is one of a class of similar relations, each of which
is particularly useful in certain applications. Let us consider, for
example, a network \~ith t}vo pairs of accessible terminals with the volt-
ages and currents as indicated in Fig. 1.4. The output voltage is
v,
“ =–r
92 ELEMENTS OF NET WORK THEORY [SEC.4.3
and hence
Z1211
I,=–
z,, + z,”
An equivalent output circuit is, therefore, that shown in Fig. 4“5a. In a
similar manner, it is found from the admittance equation that
v2=– V12V1
Y,, + Y.;
therefore, an equivalent circuit can be drawn as in Fig. 45b.
Another theorem that is useful in the study of the behavior of net-
works is the compensation theorem. If a netwo;k is modified by making
a change AZ in the impedance of one of its branches, the change in the
current at any point is equal to the current that would be produced by
an electromotive force in series with the modified branch of —iAZ, where
i is the current in the branch. This is immediately evident from the
superposition theorem, since, if the network is altered by both changing
. the impedance and inserting a series
electromotive force —iAZ, no aker-
ation is caused in any of the net-
,:~—~,v,~ work currents. Consequently the
two alterations have equal and op-
posite effects. This is a statement
FIG.4.4.—Two-terminal-pair
network.
of the comKIensation theorem. It
is necessary, however, to consider the special case for which AZ is infinite,
that is, when the branch is open-circuited. Let the impedance in the Kth
branch be Zx and the current through it f~. Now let us suppose that
Z~ becomes infinite.
-422
‘=-z12’lazL’=-’2’myL=-’’=l=zLzL
(a) (b)
FIG. 4.5.—@ Jtput circuits equivalent to circuit of Fig, 4.4.
and
‘=R+
’(”’-4
,~1,
R -’(”’) -+)
Rationalized, this becomes
94 ELEMENTS OF NETWORK THEORY [SEC.4.3
?YL = +Llil%l,
WC = +c\vc\’(d,
where VCis the voltage across the capacitance. By usc of the relation
~’
-E’ = E;,
v
some algebraic manipulation v-ill show that
We).
1t is easy to see that if the load imprdance is varied, the conditions for
to be a maximllm are that,
R,. = R,
SE(:, 44] 7’HE SYNTHESIS PROBLEM 95
and
x. = –x,.
This maximum value is
1 Va
P
‘n- = % ~L”
It should be noted that here again the factor ~ would become ~ if the rms
value of V were used. The factor would, of course, be ~ in the d-c case
also,
For current generators, the dual relation is in terms of admittances.
The circuit becomes that of Fig. 4.613. For this case,
P = + Re (Y.) IV12,
1 I’
P
“x = ~ FL’
where
G. = G,, B. = – B,.
When the conditions for maximum power transfer are satisfied, the load
impedance is said to be the conjugate image impedance. The quantity
Pm is often called the “available
power” of the generator.
If the load impedance does not
satisfy the conditions for maximum
power transfer, a matching network -
is often inserted between the gen- A B
1:1~.4.7.unlatching network.
erator and the load, as shown in
Fig. 4.7. If the network is lossless, the condition for maximum power
transfer may be applied, with the same result, at either A or B or indeed
at some point at the interior of the matching netw-ork. If the network
is lossy, the two conditions are different, and the proper procedure
depends upon considerations of design.
4.4. The Synthesis Problem and Networks with One Terminal Pair.—
The problem of finding the properties of a network when its structure and
the behavior of its component inductances, condensers, and resistors are
known has been considered. The problem inverse to this, and more
often encountered in practice, is that of constructing a network having
certain specified properties. It \vill be seen that there are severe limita-
tions on the possible behavior of networks; these limitations are funda-
mental to net!vork design. A’etworks have been considered as composed
of a n~lmbcr of branches forming complete circuits and containing sources
of electromotive force and resistors in which power is dissipated. In
general, the purpose of a network is to transfer power from a generator
to one or more impedances ~vhich absorb the po\ver. It is convenient
then to rmnovc thr grncrators and the loads from the net~vork and to
96 ELEMENTS OF NET WORK THEORY [SEC.4.4
@a
For networks having more than two accessible terminals, ,. only the
transfer of power from a generator connected to one pair of terminals to
various loads connected to other terminal pairs is usually of interest.
The potential difference between one terminal that is connected to the
generator and another that is connected to the load is of no importance.
Thus a network with three accessible terminals may be regarded as a
two-terminal-pair network, one of the terminals being common to the
input and output circuits, as indicated in Fig. 4.8.
Whether or not the two lower terminals shown in the right-hand figure
are connected is usually immaterial. Thus a network with n accessible
terminals may be regarded as possessing n – 1 terminal pairs. The
general net work equations [Eqs. (4)] may thus be reinterpreted as express-
ing the linear relations between the currents that flow in and out of each
terminal pair and the voltages across the pairs of terminals. The order
of the impedance or admittance matrix is thus an index of the complexity
of the network under consideration. Networks with one, two, and more
terminal pairs will be treated in succession.
The problem is essentially one of synthesis. Given a junction that
has N pairs of terminals, the contents of the junction being specified by
the elements of the admittance or impedance matrix, it is required to find
the possible meshes inside the junctions and the values of the individual
elements in the meshes. The problem can be solved in two stages hav-
ing clifferent degrees of complexity. Fh-st, ways must be found to con-
nect individual elements and the impedances of these elements so that the
SEC. 4.4] THE SYNTHESIS PROBLEM 97
(a)
m (b)
FIQ. 4.9.—Equivalent circuit demonstrating Foster’s theorem.
AI’=–
z ,-
I; AZ,
—
v’ ‘“
Al’
ja:k,.
Au
-z =
,
Since this is greater than zero, the theorem is proved. At zero frequency,
the network must be either a pure inductance or a capacitance, and the
admittance must therefore be either
m:::! IzcD
FIG. 4.10.—Fostcr representations for a
10SS1CSS
i,r,ped:,nre clelnw,t.
Iht+. 4.1 1.– General Iossy two-terminal
network.
then change to minus infinity, and increase through zero to plus infinity
again, repeating the process pm-haps many times. From the duality
principle it is clear that, this must be true also for the impedance. This
dua,l relationship is illustrated by the two possible equivalent circuits
SEC.4,5] TWO-TERMINAL-PAIR NETWORKS 99
““JL.-!k_
‘t 1
I.lti. 4.12.—T-network.
2
and Vz may be found in a similar fashion. The current in the first mesh
flows through the impedance 212 in the same direction as the current in
the second mesh. Thus if the shunt impedance in the network is posi-
tive, the mutual impedance element in the matrix is also positive. This
is the most cogent reason for the choice that has been made of the positive
directions of the current. The choice of the positive directions will be
made, whenever possible, so that if “power is flowing into all terminals of a
complicated network, all the impedance or admittance matrix elements
y2 are positive definite if the cor-
responding network elements are
Y = z-’. (13)
(14)
where
D = Z,, Z,, – Z;2. (15)
It will be noticed from the second of Eqs. (14) that if Z12 is positive, Y,2
is negative. This arises because the positive direction of the currents
shown by the network of Fig. 4.12 would not be correct for an admitt ante
representation. In a similar way, it is possibl e to write
SEC. 45] TWO-TERMINAL-PAIR NETWORKS 101
2,1 = +
Z,* = – :,
(16)
Z*2 = p,
A = Y11Yz2–
Moreover,
DA= 1. (17)
~=~=zl,_
m (21)
I,
k= ~. (23) I
4Z,,Z,,
(24)
The parameter k thus has some value between zero and unity.
For many cases, the most useful circuit parameters are the input
impedances of the network when the output end is open- or short-cir-
cuited. The impedance looking into terminals (1) when terminals (2)
are short-circuited will be denoted by ZQJ; correspondingly, when ter-
minals (2) are open-circuited, the input impedance is Z$~J. The super-
script (2) will be used to denote the corresponding quantities for terminals
(2). From Eq. (21) it is clear that
(25)
1 D _ 2(1)
(26)
Y,, = z,, – ‘c “
2(;; z:;)
~: = ~’ (28)
and by the duality principle
(29)
(31)
Other useful parameters are the coefficients in the set of linear equa-
tions that relate the input current and voltage to the output current
and voltage. They are given by
av2 – (SU2,
1
VI =
(32)
11 = C?V2– D12.
The parameters are related to the impedance matrix elements by
(33)
There must be a relation that corresponds to Eq. (28) and that also
expresses the reciprocityy condition. This has the form
@@l
= 1. (34)
CD
‘f (::1[3
and are chosen as the column vectors, then Eq. (34) is
the determinate of the matrix. If Eq. (32) is solved for Vi and 12,
it can be written, in matrix form, as
(35)
The elements of the matrix are the same as those of Eqs. (32), but in a
different order, and the determinant of the matrix of Eq. (35) is equal to
unity. This set of parameters has been in use for a long time, particu-
larly for applications involving power transmission lines. From Eqs.
(33) it is immediately evident that if the network is lossless and the
Z’s are all pure imaginary, the elements ~ and D are pure real and @
and c are pure imaginary. If the network is symmetrical, Zll == 222
and ~ = D.
The utility of this set of parameters becomes more obvious when
networks connected in cascade are to be considered. This situation
will be t Peated in more detail in the following section. An important
example of the use of these parameters is afforded by the case of the
ideal transformer. The matrix of a transformer takes the form
104 l?I>Ei14ENTS OF NETWORK THEORY [SEC.46
(36)
where n is the turn ratio of the transformer, the direction of voltage stepup
being from terminals (1) to terminals (2). No impedance or admittance
matrix exists for the ideal transformer. The elements all become
infinitely large, and therefore the series impedances of the equivalent
T-network are indeterminate.
Three important sets of parameters and their duals have just been
presented which may be used to specify completely the properties of a
general, passive, linear network with two pairs of terminals. The set
that is most convenient to use depends on the particular application in
question.
4.6. Equivalent Circuits of Two-terminal-pair Networks.-Another
method of describing the behavior of a network is by means of an equiva-
lent circuit. It has been shown that three parameters are necessary
for the complete specification of a network with two pairs of terminals.
The equivalent circuit must therefore contain at least three circuit
elements. There is, of course, no unique equivalent circuit but an
infinite number of them. Moreover, they may contain more than three
circuit elements. Two examples have already been given—the familiar
T- and II-representations. For microwave applications other representa-
tions are also useful. Portions of a transmission line have been intro-
duced as circuit elements. In Chap. 3, lines of this type were discussed,
and the II- and T-equivalents for such lines were given there. These
lines will now be considered as convenient circuit elements, and the
electrical length and characteristic impedance to define their properties
will be specified. Although this could be done for the general case of
lossy transmission lines, the discussion will be confined to the case where
the lines are lossless, since these are by far the most important cases for
microwave applications.
A simple case of such equivalent circuits is demonstrated by the cir-
cuits shown in Fig. 4.16. These circuits consist of transmission lines,
one shunted by an arbitrary admittance and the other having an arbi-
trary impedance in series. The three parameters are thus the value of
this admittance Y, the length of the line 1, and its characteristic admit-
tance Yo. The line with the series impedance might be termed the dual
representation. These two circuits are duals of each other in the sense
that the relation between Y and the elements of the admittance matrix
corresponding to one circuit is identical with the relation between Z
and the impedance matrix elements that describe the other circuit. Not
all of the important circuits of this type will be discussed in detail, but
several are shown in Figs. 4.16 to 4.23. The relations between the cir-
SEC. 4.6] TWO-TERMINAL-PAIR NETWORKS 105
Z.
(a)
A’ (b)
FIG.4.16.
Z,l = –jZO cot @l, z = 222 – 21,,
I z]
1 Z2
IEr:
Y,
Yo=l
(a)
Y2
Zo=l
(b)
FIG. 4.17.
1
Y,, = YI – j cot 131, Yl=Y,l+j 1 + ~2~
d
1
Y22 = Y, – j cot pl, Yz=Y,, +j
J 1 + y,’
*j
Y,, = *j csc /31, @ = csc-’(+j Y12) = sin-’ ~,
“()
1=+ z,
1=+ Z2
!icr=
Y,
Y.
(a)
Y2
FIG.4.18.
(b)
A--L% .Zo=h .—
,.=l
FIG.4.20.
circuit (b), on the right, is the dual circuit. Thus in the legend of Fig.
4.17 the relation
Y,l = Y, – j cot @l
similar way,
Z,, = j csc ~1, (37)
Z22 = Z2 – j cot bl.
~ ~
Ak–_$L&. I:n.
FIG. 4.21.
Z,, = j tan [~11+ tan-l (nz tan /312)1,
ZM = j tan
[
~lz + tan-l
~
Z,,Z2, – Z;2 =
tan fill tan ~lz – n2 ‘
l+cl–az–bz+
tan fill = 1 +2ybc–_”:)– b’ 2 + 1,
2(bc – a) – J[
1
b+ca l+ aa
tan@~=_a+ a=-
c—cxb’
nz = ‘Ccl — b a—a
l+aa=–c–ab’
a = —jZ1l,
b = Z,, Z,, – Z;z,
c = —jz22,
a = tan ~ll.
1 :?L I:n
(a) (b)
FIG.4.22.
z,, = –jCot @l, @ = cot–’ jZll,
z ;2
Z22= Z – jn’ cot P1,
n = <1 + z;,’
z11z;2_.
Z,z = jn ~cot’ @ – 1, z=z22– —
1 + Z;l
108 ELEMEN1’S OF NET WORK THEORY [SEC. 46
l:n
(a) (b)
FIG.4 23.
z;,
z, = z,, – ~,,
Z2 = z,,,
z,,;
n=z<
\vhere 13and 11represent the distances from the reference planes and 11
and 12are the network parameters.
It is important to notice one fact about all these equivalent circuits,
Mthoughj at a given frequency, the elements of the circuit are perfectly
definite and can be represented by circuit elements familiar to low-
frequency practice, these elements do not have the proper variation with
frequency.
,Ic!IrIt F1~, 424,-Trallsfornlatioll from II- to T-network,
1X’. \[:iuctlYitz, “ \\”avt#ui[le II:LlltllxxIkSupplelllent ,“ 1{1, I{eport Xo. 41, Jan 23,
rw15,p. 2
110 ELEMENTS OF NET WORK THEORY [SEC.47
(39)
where
~’ = Y.y, + YAYC + YBYc. (40)
Since these circuits are duals of one another, the inverse relationships are
identical in form. For example,
z,
Y, = (41)
2122 + 2,23 + 2223”
4.7. Symmetrical Two-terminal-pair Networks.-Many waveguide
configurations are symmetrical- about some plane perpendicular to the
axis of the transmission line. If this is the case, the input and output
terminals are indistinguishable, and the number of independent param-
eters needed to specify the network is reduced from three to two. In the
matrix representation, 211 becomes identical with Zm The circuits
shown in Figs. 4“16 and 4“20 reduce simply to a transmission line with the
usual parameters: length and characteristic impedance. The relation-
ship between a symmetrical two-terminal-pair network and a line is well
known, and it will not be considered further here. We can state, how-
ever, a useful theorem known as the’{ bisection theorem. ” 1 This theorem
can be formulated in a somewhat simpler form than that in which it was
originally stated. If equal voltages are applied to the terminal pairs (1)
and (2) of a symmetrical network, equal currents will flow into the two
pairs of terminals and no current will flow across the plane of symmetry.
The input impedance is then simply (ZU + Z,,). This maybe called the
open-circuit impedance of half the network, Z~c~~J.
z, If equal voltages are applied to the two pairs of
terminals but in opposite directions, the voltage
Z2 z~ across the center line of the network must be zero
and the currents entering the terminals equal and
opposite. The input impedance under these con-
Z, ditions is (2,1 – 212). This impedance is written
x as ZSS(~). These two values of input impedance
FIG. 4.25.—Lattice net-
work. are convenient ones to use, in some cases, to specify
a symmetrical network.
A good example of the application of this theorem is the lattice form
of network shown in Fig. 4.25. If equal voltages are applied to the two
ends of the lattice, no current will flow in the impedance 21. Hence
Zm(>$l= 2, = z,, + 212. (42)
1A. C. Bartlett, The ‘l%eo~ of Ekctrical Artificial Lines and Fitters, Wiley, New
York, 1930,p. 28.
~EC.4,7] SYMMETRICAL T1l’O-TERMINAL-PA IR NETWORKS 111
2’
(44)
Z,2 _ z, – z,,
2 I
The bisection theorem as originally stated by Bartlett was phrased in
terms of cutting the symmetrical network into two equal parts. The
theorem stated the values of the input impedance of half of the net-work
when the terminals exposed by this bisection were either open- or short-
circuited. The example of the lattice network has been given because,
for this case, it is difficult to see just how the network should be divided.
The derivation that involves the application of two sources of potential
avoids this difficulty. The lattice network is particularly suitable for
theoretical investigations of the properties of low-frequency networks and
has been much used for this purpose. It can be shown that the lattice
equivalent of any four-terminal network is physically realizable in the
lattice form. “ Physically realizable” means, in this case, that it is
unnecessary to use any negative inductances or capacitances to construct
the lattice. The lattice form, on the other hand, is
Z1
quite unsuitable for the construction of practical net-
works at low frequencies, since no portion is
Z3 Zp
grounded and the inevitable interaction between the
elements of the net~vork destroys its usefulness.
This is not true of microwave applications. A con- Z,
Tbe ~lrcllit sho\vn in Fig. 4.21 also has no obvious s,vmm~try ~~hen the
112 EI,E,JTENTS OF NE TW70RK TIIEOR 1’ [Sm. 48
iapllt and olttput terminals are ifl{,ntil,al. ‘~llt matrix clementjs reclucc to
for this case. The three network parameters given in Fig. 4.21 must be
subject to one condition. This condition is that
v
transmission lines can be regarded
z,, -212 222- Z,* z;, - 2;2 2;2- 2,2
as composed of a number of such
networks connected in cascade.
The transmission line can then be
treated as a whole, or a small part,
L 1 of it can be reduced to a new T-
I(’Iu. 4.27.—Two-terminal networks in cas- network with the proper values of
cade.
the network parameters. For two
T-networks in cascade, as shown in Fig. 4.27, the matrix elements of the
combination are given by
Zy; = Z,l – ‘~’
Z12 + Z;l’
qt~= z12zl’2
(48)
Z22 + z~’
z;;
Z;J= Z.22–
Z,2 + z~,’
It is evident from this that the matrix of the combination is equal to the
product of the matrices of the components.
Let us consider an infinite chain of identical networks. At any pair
of terminals the impedance seen looking in either direction must be
independent of the particular pair of terminals chosen. It must be given
Ivy
Since the networks are identical, this impedance is commonly called the
iteratiue impedance. The two signs before the square root refer to the two
values of the impedance seen in opposite directions from the pair of ter-
minals. These impedances are alternative parameters with which to
describe the network behavior. Moreover, the ratio of input to output
currents in any network in the chain is given by
where the negative sign arises from the convention, earlier established,
that the currents always flow into the network at the upper terminals.
The third network parameter is then defined by
z.+ , = e–r
(53)
In
e., _ –(Z,,
— + 2’,,) + d(Z,, +>,)’ – 4Z;Z
(54)
2Z,2
~ =~zL+t)
m CZL+ d“
Thus, the output plane may be said to be mapped onto the input plane
by this transformation. Transformations of this form are called bilinear
transformations or linear-fractional transformations. They have the
important property that they are conformal; that is, angles in one plane
are transformed to equal angles in the other plane. Thus a grid of
perpendicular intersecting lines is transformed to two sets of circles that
are mutually orthogonal. The iterative impedance, as is evident directly
from Eq. (50), is represented by the point whose coordinates are unchanged
by the transformation. It is thus sometimes referred to as the fixed
point.
From Eqs. (25) it is evident that these impedances can also be expressed
as
z,, = V’zgzg,
z,, = 4Z;:)Z::). I (58)
(60)
‘an” =&= &
~o,h /j .-2, (61)
212
(c)
FIG.4,29.—Simple filters: (a) Low-pass filter, (b) high-pass filter, (c) bandpass filter.
The presence of attenuation does not imply that the energy of the attenu-
ated waves is dissipated in heat. It is, of course, possible to construct
a device for which this is so, in which the attenuation of unwanted
frequencies is accomplished by means of resistive elements. Such a
device is usually called an equalizer. It is difficult, however, to have zero
attenuation in the pass band when resistive elements are used. The
microwave analogues of equalizers have, as yet, no important applica-
tions, and no further discussion of their properties will be presented here.
A simple example of a filter is shown in Fig. 4.29a. Each section of
this filter may be taken to be a T-network with series inductance L/2 and
shunt capacitance C, Hence
~1, _ jwL ~-
2 UC
(64)
z,, = – $.
The characteristic impedance is real only for values of u less than the
cutoff value u.. The cutoff frequency is given by
(66)
cosh 0 = 1- ‘~,
(67)
‘= ’coS-’(’ -% 1
(70)
“ = 2~L—C’
(71)
“=’C:S-’(’ -A)
SEC.442) FILTERS 117
1.’
Z1l = ju ~ – j
Z12 = –j
–—-C–
J,!
I.’
c
T7
I
‘“-
——
WC
f
(72)
1“
uL’ — —
WC )
The pass band is given by
–1<21 <+1
– Z12 –
or
L
‘4 ‘I? – “Lc s 0“
l–~2L’C 4
w
z: = —U212 r.’ + — 1
I.
4 () (75)
1’
Z,, =~dz+—
Y dz’
(76)
z,, = *Z)
I
and
cosh 0 = ; Y(dz)z + 1
o
* Jz Y. (77)
dz =
20=
J( ) ; dz’+$
where
Y=j.C–$,
or
(78)
As before, 20 is real for W2> l/L’C but remains real for all higher values
of w. The upper cutoff frequency has therefore moved off to infinity as
the impedance of each section was decreased. The propagation constant
is
~ = $ – W2LC. (79)
SEC.410] CONNECTION OF NETWORKS 119
The shunt admittance per unit length will have two parts, the first con-
tributed by the displacement current and therefore capacitive and the
second arising from the longitudinal magnetic field. Let it be assumed
that
(81)
(a) (b)
FIG. 430.-(a) Equivalent circuit for dominant-mode transmission in rectangular wave
guide; (b) equivalent transmission line for E-modes.
Figure 4.31a shows two networks connected in parallel. Here, the admit-
tance matrices are most useful, and
y = y(l) + y(am (84)
Combinations in which the input terminals are in series while the output
terminals are in parallel, or vice versa, are also possible. They represent,
however, obvious extensions of the simple cases just discussed.
(a) (b)
F1a.4.31.—Two-terminal-pair networks connected (a) in parallel and (b) in series.
An important condition is always imposed in this type of intercon-
nection. When the impedance matrix is set up, it is assumed that the
same currents flow out of the lower terminals of the network as into the
upper terminals. For the relations given by Eqs. (83) and (84) to be
valid this must also be true of the combined network. Moreover, the
potential between, for example, the upper terminals on the input and
T ‘T=
(a) (b)
FIG.4.32.—Rightand wrongwaysof couplingT-networksin series.
output sides must be undisturbed. Thus, in the series connection, if
each network is represented by a T-network, the network must be
arranged as in Fig. 4.32a and not as shown in Fig. 4“32b. In the arrange-
ment b, the conditions are obviously violated by the short-circuiting of
the lower network. For a further discussion of this question, the reader
is referred to Guillemin 1 and the references there cited.
1E. A. Guillemin, CommunicationNetworks,Vol. II, Wiley, New York, 1935,
Chap. 4, pp. 147fl.
SEC.4,11] THREE-TERMINAL-PAIR NETWORKS 121
,l-~.iz
FIG. 4.33.—Equivalent circuit for three-terminal-pair network
II
Z,, Z,,
Z = z,, Z,, z13
z13 z,, z,,
Because many microwave junctions have a symmetry of this type, this
represents an important case. Other equivalent circuits maybe obtained
from that shown in Fig. 4.33 by the transformation of portions of the
circuit from T- to II-networks or to some of the transmission-line forms
shown in Figs. 4.16 to 4.23 inclusive (Sec. 4.6).
Since it is not required that the equivalent circuit represent the volt-
age between one terminal and another, the six input lines of Fig. 4.33 may
be reduced to four by connecting three of the lines together. If this
common point is designated as the ground point, then another circuit
may be drawn, with four terminals and six independent parameters, as
122 ELEMENTS OF NETWORK THEORY [SEC,411
.
members of terminals (I) and (2)
of Fig. 4.33 and the corresponding
points, (1) and (2), of Fig. 4.34.
v z3#!!
The impedance matrix implies
1 2
6
3
FIG. 4.34.—Circuit with FIG. 4.35.—Transformer representation of
iour terminals and six inde- series and shunt T-junctions.
pendent parameters.
nothing about this voltage difference but the circuits define it uniquely.
This fact must be kept clearly in mind when equivalent circuits for rnicr~-
wave devices are employed.
The four-terminal circuit of Fig. 4.34, rather than the more usual
T- or II-network,’ is the exact equivalent circuit for a low-frequency
transducer. In the low-frequency region this problem is usually of
importance only in connection with the exact equivalent-circuit repre-
sentation of a practical transformer.
Other circuits whose components are transmission lines are often
useful in microwave work. Figure 4.35 shows two extremely useful
forms. A transformer and a length of transmission line have been
included in each arm of each of the networks, although it is obvious that
one transformer may be assigned an arbitrary turn ratio. These two
circuits may be said to represent series and shunt T-junctions. It is to be
1See M. A. Starr,ElectricCircuitsand Wave Filters, 2d cd., Pitman, London, 1944,
Chap, VI, and references there cited.
SEC.4.11] THREE- TEILJII:V.4L-P.4 lR .VEI’J!’ORKS 123
emphasized that either circuit is equally valid for any T-junc~icm. The
choice between the two circuits may depend, for example, upon the
similarity between the physical configuration of the device and that of
the circuit, and this might be a valid reason for choosing one in preference
to the other.
Circuits of another type which may be useful in some circumstances
are shown in Fig. 436. If three two-terminal-pair networks are connected
in series or in shunt, equivalent circuits are obtained that have some
3
‘v’ 3
(c)
FIG,4.36.-—Combination of three two-terminal-pair networks to give a three-terminal-pair
network.
where the terminals (5), (6), and (7) are indicated in the figure. By
means of these three equations is, i& and iT may be eliminated from the
set of six equations that represent the three two-terminal-pair networks.
The first member of this set is, for example,
VI = Zllil + Zlsis.
The result of eliminating these parameters will be three equations in t hr
124 ELEMENTS OF NET WORK THEORY [SEC,4.12
three currents il, L, ia, and the impedance matrix is the matrix of these
three equations. It might be thought that some general matrix method
would be available to perform this elimination in a systematic fashion.
Although such a method exists, it is usually too complicated to apply. .4
straightforward manipulation with the linear equations is much quicker
and easier.
Circuits of the forms shown in Fig. 4.35 are particularly convenient
for finding the effect on the power transfer from terminals (1) to (2),
for example, of a load on terminal (3). It is immediately obvious that a
reactive load of the proper value on terminals (3) will cause no voltage
to appear across terminals (2) when voltage is applied to terminals (I).
The same value of the load will also make the voltage across terminals (I)
equal to zero when voltage is applied to terminals (2). hforeoverj if
the circuit is symmetrical about a plane through terminals (3), so that
nl = nz, then for some value of a reactive load on terminals (3), the input
impedance seen from terminals (1) and (2) is the characteristic impedance.
The matrix manipulation that corresponds to the application of a
load to one pair of terminals is again most easily seen from a considera-
tion of the corresponding set of linear equations. If a load Z~ is put on
terminals (3), then
23=–:,
where the negative sign arises from the convention that currents and
voltages are always designated as positive when they represent power flow
into the net work. If is is eliminated from the equation by means of this
relation, there results the new second-order mat rix whose elements are
us call one of these points the common terminal of all the N input lines.
Now if an impedance element is connected between each pair of points,
the number of such elements is given by the binomial coefficient
pair case. The parameters are the six characteristic impedances of the
quarter-wavelength lines and the four shunt admittances, making, in
all, the necessary 10 constants.
There is a third general approach to finding the equivalent circuit
for N pairs of terminals. Suppose that it can be arranged that the sum
of the applied potentials is zero. The lines can be arranged as shown
schematically in Fig. 4.38, where the case for N = 4 is indicated. It
should be noticed that the four-terminal network at the center of the
figure has flowing into it currents whose sum is zero. This four-terminal
device therefore satisfies the requirement that the voltages across its
terminals are linearly related to the currents, and is equivalent to a three-
terminal-pair network. The voltages v;. must be derived from the
126 ELEMENTS OF NETWORK THEORY [SEC.412
0
),
0
pb 3& t
D5-
v,
+1
3
I
% I
fore let il take a finite value while all the other i’s are zero. Adding the
equations results in
O = (Z,l + Z21 + + Zvl)il – Zlil
or, in general,
z, = z,,,.
s
A
SEC.413] RESONANT CIRCUITS 127
Now if the equations are expressed in terms of v; ~vhose sum is zero and
(ii – i,+,) are used as independent variables, the set of N equations
reduces to N — 1 equations, since one equation is redundant. 1
To repeat this process, it should be remembered that the N – 1
equations correspond to N — 1 pairs of terminals, each pair having a
member in common with all other pairs. This condition can be removed
by the use of ideal transformers. Figure 4.39 shows schematically how
this may be accomplished. The process of reduction can be continued
until only a two-terminal-pair network remains. For this network any
standard circuit form may be used. It should be noted that at each step
an impedance in each line k removed and the total number of parameters
is correct.
To conclude this section, the change in the elements of the impedance
matrix when a load is placed on one terminal pair R
may be stated. If a load Z, i< placed across the
kth pair of terminals, the new impedance elements
arc v c
Z:j . Zti – .*.
z,, + Zk Q
At resonance,
‘=R+(~L-ii’)
1
U“L =
L@
1
(JU= —–’
~LC
circuit, is
‘4’+ ’Q(H”
‘=41+ 2’Qe)
If Q is large, then the resistance and the reactance
FIG.
ED!?,
4.41 .-Shunt-resonmt circuit.
The duality principle may be invoked to obtain thr corresponding
m
F1~, 4-42. -.\ wrond sllul,t-reso,xant cil cuit.
relations for the shunt circuit shoivn in Fig. 1.41. These relations are
1’. = ;,
d“
SEC,4131 RESOKAN1’ CIRCUITS 129
A second parallel circuit, shown in Fig. 4.42, may be reduced to the pre-
ceding circuit in the low-loss case. The admittance of the inductance
and resistance in ~eries is
R – juL
Y= 1 RZ + ~2L2”
R+juL=
If R2 << WZL2,
Y=-&–;L7
G = &
““’=
R++L-4=”
in the simple series circuit. The roots of this equation are complex and
are
If
BY R. H. DICKE
where E, H, and p are complex numbers such that E’(t) = Re (Eei”~) and
similarly for H and p.
A connecting link between field theory and electrical engineering can
be found in Poynting’s energy theorem limited to periodic fields. It
was shown in Sec. 2.2 that if
div EXH*=H* .curl E–E. curl H*
— –juPH* . H – (u – &)E* .E (2)
SEC.5.2] PO YNTING’S ENERGY THEOREM FOR A PERIODIC FIELD 133
! s
EXH*. dS=
\
(E.H: – E.H:) dx dy. (4)
where j=, jti, g= and gu are real functions of the coordinates which give the
distribution of the field. It was shown in Chap. 2 that the transverse
elect ric (or magnetic) components are in phase with each other, i.e.,
E and H at a given point are constant in direction. It is assumed that
the functions are normalized in such a way that
and
ei” = jw
[/ ~H*”Hdv-J’E*”Edvl+ J”E*”Edv ‘8’
Equation (8) is rather fundamental. It may be written as
I34 GE,VI<RAI, MICROWAVE CIRCUIT THEO REJIS [SEC.53
From the linearity of Maxwell’s equations, it follows that (E, – E,) and
(Hl – H,) with the terminal parameters (el – e,) and (z1 – i,) also form
a solution. If this solution be substituted in Eq. (8), there follows
right-hand side. This requires that the average stored magnetic and
electric energies be equal and this condition may be defined as resonance.
This resonance condition, however, can be satisfied only at certain discrete
frequencies, the natural resonances of the termination. For all other
frequencies E, = E,, HI = H,, and the uniqueness theorem is satisfied.
In completely lossless terminations resonances of two different types
may be considered. In the first type there is no coupling between the
terminals and the resonance fields, as though there were an isolated cavity
somewhere inside the termination. Clearly a resonance of this type is of
no particular importance. The second type of resonance couples with
the input terminals, and in this case the terminal current is not given
uniquely by the terminal voltage. Of course, it must be emphasized that
these conditions are never encountered in practice; therefore whenever
lossless terminations are discussed, it will be assumed that the terminal
voltage is uniquely related to the terminal current and the field quantities.
It is evident from the uniqueness theorem and from the linearity of
Maxwell’s equations that a change in the argument or modulus of e,
will result in a corresponding change in the argument or modulus of the
field quantities inside the termination. Thus the field quafitities inside
the termination are proportional to the voltage or current at the terminals.
It follows from this that the terminal current i is proportional to the
terminal voltage e. Thusj as before, values Z(o) and Y(a) may be
defined such that
e
.= z(.) = +. (11)
‘1
i *iZ = ju /AH
u *“Hdv-l’E*”Edv) +JuE*”Edu ’12)
or
(13)
W~ and WE are the mean stored magnetic and electric energies; I’ is the
average power dissipated in the termination. In the same way,
(14)
and
Y*(-@) = }’(u). (16)
Several things aretobe noticed about Eqs. (13) and (14). Let
Z= R+jX
(R and X real).
1. Since P zO, R zO.
2. If P = O, then R = O, or Zis purely imaginary.
3. If X = O, WE = W~; this isthe resonance case.
4. From Eqs. (15) and (16) it can be seen that foralossless termina-
tion the reactance and the susceptance are both odd functions of
frequency.
Equations (13) and (14) indicate the steps that may be taken to
produce desired impedance effects at microwave frequencies. For
instance, any change in configuration that increases the amount of stored
magnetic energy in a termination automatically increases the reactance
at the terminals.
5.5. Field Quantities in a Lossless Termination.—One of the results
obtainable from the uniqueness theorem concerns a lossless termination.
It will be shown that in such a termination the electric field is evm--y}vhcre
in phase and the magnetic field is every ]~here 90° out of phase \\-iththr
electric field. Let us assume that E and H, lrith correspondin gtermirml
quantities e and i, are a permissible solution of the field equations for a
particular lossless termination. Substitute for E and H in Maxwell’s
equations [Eqs. (1)] the following quantities:
E= E,+ E;,
H= H,+ H,.
The subscripts r and i denote, respectively, that the qlumtities are the
Teal and imaginary portions of the field vectors. Since the termination
is assumed to be lossless,
,J =()
for
E#O.
The imaginary portion of the first equation, the real portion of the second,
the real portion of the third, and the imaginary portion of the fourth arc
curl H, — @E, = O,
curl E. + j~PHi = O,
(17)
div cE, = p,,
div pH~ = O. 1
E,, Hi there correspond terminal voltage and current e, and ii. By the
uniqueness theorem, the above particular solution is unique. Any
other solution can be obtained from this solution by the multiplication of
the field quantities by some complex number. E, is a solution with the
electric field everywhere in phase or 180° out of phase, and the phase of
the magnetic field is 90° or 270° with respect to the electric field.
5.6. Wave Formalism.-It has been shown that the field quantities
inside a termination, or single-terminal-pair network, are uniquely
determined by either the current or voltage at the terminals. Voltages
and currents are not the only useful parameters that can be used as
representations for the fields inside a termination. Another very useful
representation can be obtained from the amplitudes of the incident and
scattered waves.
The amplitude and phase of the transverse component of the electric
field in the incident wave, measured at the terminals, will be designated
by a, which will be so normalized that ~aa * represents the incident power.
In a similar way b will be used to designate the amplitude and phase of
the reflected wave. It is easily seen that the uniqueness theorem also
applies to the representation in terms of incident and reflected waves.
For any incident or reflected wave the fields inside the termination are
uniquely defined. As was shown above, the impedance Z = e/i is a
quantity that is a function only of the frequency and the shape of the
termination. In an analogous way one can define the reflection coefficient,
(18)
i=~(a–b)=~~(l–r), (20)
aa*(l
and
+ r)(l
,.
=j.u– r*)
PH*
“H~o-J’E*”Ed”)+ J”E*”EdoJ ‘2’)
(22)
l–r*r=~, (23)
~a *a
and
(24)
r*l’ ~ 1. (25)
Stated in words Eq. (27) says that for 1 watt of power incident on a
termination, the difference between the magnetic and electric stored
energy is always less than or equal to l/w joules.
If Eqs. (23) and (24), are solved for r,
~=~l–~”ej~
@Vj’- Wj) (28)
~ = sin-~
[ Vi–P’ 1‘
where
TV, = _y_E_ ,
E ~a *a
~v, = _W&
w ~u*a
P’ = *;:a.
SEC.58] EXTENSION OF THE UNIQUENESS THEOREM 139
enfl
i* = 4j0(W. – W’,) + 2P, (30)
z
n
5.9. Imoedance
. and Admittance Matrix.-Since the N terminal cur-
rents and voltages of the junction are connected by linear equations, N2
quantities Z,q can be defined such that
eP = Znqi,. (31)
2
Y
il
l’!!’!
el
iN e.W
The vectors i and e will in the future be called the current and voltage
vectors of the junction. The matrix formulation of Eq. (31) is
e = Zi. (34)
The linear relation between the terminal currents and voltages can
be expressed in another way if Eq. (31) is solved for the N currents.
Then
Yll Y12 . . .
Y21 Y22 . .
Y= . . (36)
[1 . .
. .
1
This matrix will be called the admittance matrix;
i = Ye. (37)
SEC.510] 817MMETRY OF MATRICES 141
It is worth while to notice the analogy between Eqs. (34) and (11).
They are formally identical. The impedance Z of Eq. (11) has been
generalized to an impedance matrix. The current and voltage have
been generalized to current. and voltage column vectors. For a termina-
tion, that is, a single-terminal-pair network, Eq. (34) reduces to Eq. (1 1).
6.10. Symmetry of Impedance and Admittance Matrices. -It will
now be shown that the impedance and admittance matrices, matrices
(32) and (36), are symmetrical. By a symmetrical matrix is meant one
for which
z.. —– z..,,
(39)
Y m. = Y.m. }
Let there be two solutions of Maxwell’s equations that satisfy the bound-
ary conditions imposed by the junction. The field quantities and
terminal quantities of the two solutions will be distinguished by super-
scripts 1 and 2. From Eqs. (1),
curl H(l) — (jtic + O)E[l) = O,
(40)
curl E(l) + jOPHfl) = O; I
curl H(z) — (ju + u)E@ = O,
(41)
curl E(?) + @~H(2j = O. }
Likewise
div [E(’) )( H(2) –– E(2 x H(l)]
= H(z) curl E(l) – E(l) curl H(2) – H(l) curl E(t) + E(z) curl H(i). (42)
Equation (45) holds for any two sets of applied voltages at the terminals.
In particular let
(46)
The sum given in Eq. (45) reduces, for this special case, to
(48)
zn,m
i:z.mim = 4jti(wH – WE) + 2P. (52)
It is evident that this condition is the same as that for the termination,
as indeed it should be.
Since P 20 for any values of i, and i,,
SEC.512] LOSSLESS JUNCTION 143
It is evident from inspection of the coefficient of R,, in Eq. (54) that the
minimum value of the left side of Eq. (54) occurs when ZIand iz have argu-
ments that are the same or that differ by T. Then i, ~izis real, and
(55)
The above arguments apply also to the admittance matrix and can be
extended to junctions having more than two pairs of terminals.
An extension of the theorem to junctions with more than two terminal
pairs yields the result that conditions imposed by Eq. (52) require the
determinant of the real part of the impedance or admittance matrix and
the determinant of each of its minors obtained by successively removing
diagonal elements in any order to be greater than or equal to zero.
5.12. The Polyterminal-pair Lossless Junction. —Usually in practical
microwave applications, a junction having more than one pair of terminals
is essentially lossless. This is not always true; but usually, such things
as tuners, T-junctions, and directional couplers have low loss. For this
reason the lossless case is of considerable importance. In Eq. (52), if
P = O, the equation is purely imaginary for all applied currents i,,.
Consider the special case
in = o, for }L # k.
Re (Zk~) = O. (59)
That is to say, all the diagonal terms of the impedance matrix are pure
imaginary. Consider now a special case in which all the applied currents
\’anish except two, the kth and mth, for example,
i. = 0, for n # k, m.
From Eq. (52),
Thus, for a lossless junction, all lhc terms in ll(e impedance w~ahiz are pure
imqinary. The above conditions apply also, in an analogous way, to
the admittance matrix, and all the terms of the admittance matrix for a
lossless junction are pure imaginary. It is seen that the statements in
Sec. 54 are special cases of the above.
6.13. Definition of Terminal Voltages and Currents for Waveguides
with More than One Propagating Mode.—The original definition of a
waveguide junction was limited to one excited by transmission lines sup-
porting a single propagating mode. In order to extend this definition to
transmission lines with more than one propagating mode without invali-
dating the previous results, it is natural to impose the condition that
Eq. (30) be valid in the new system and that the resulting impedance
and admittance matrices be symmetrical. Whereas previously it was
necessary to introduce a single voltage and current to describe completely
the conditions in a given transmission line, now it will be necessary to
introduce a voltage and current for each mode in the guide. There is no
unique way of introducing these voltages and currents, but there is one
way that is a little more natural than the others. It is to let each voltage
and current be a description of one particular mode in the transmission
line. To make this more definite, in the derivation of Eq. (30) a surface
integral Eq. (29) is encountered. This is the same surface integral that
occurs in Eq. (4).
To simplify the discussion, let us consider the junction to be excited
by a single transmission line along which N modes may propagate.
Equation (4) is applicable; but because of the N modes, Eq. (5) must be
generalized to
E. = e.flzn)(x,y)s
s
(62)
for one mode are orthogonal to the transverse magnetic and electric fields,
respectively, of any other mode. By the use of this fact together ~vith
Maxwell’s equations it can be shown that
If Eqs. (62) are substituted in Eq. (4) and then Eqs. (63) and (64) are
used, Eq. (30) results. In other words, the particular choice of param-
eters en and in introduced in Eq. (62) results in the same connection
IEq. (30)] between terminal energy quantities that was obtained for
single-mode guides. Thus the terminal-parameter description of a
transmission line with N propagating modes is, at least to this extent,
equivalent to the description of N single-mode guides. It can be seen
in an analogous way that the reciprocity condition is also satisfied and
that the impedance and admittance matrices are symmetrical. Thus all
the preceding results are valid for this case also.
As was pointed out earlier, the currents and voltages introduced in
Eq. (62) are not the only permissible ones. To show this, new currents
and voltages that m-e linearly related to the currents and voltages of
Eq. (62) may be defined. If i’ and e’ are column vectors representing
the new currents and voltages, then the linear relation may be expressed
as
i = Ti’,
e = Te’ 91 (65)
(66)
T*T = 1, (70)
146 GENERAL MICROWAVE CIRCE;IT THEOREMS [SEC.514
(71)
(77)
c. = G + b,,,
(78)
i,, = a,, — bn; 1
a. = *(c. + 2.), (79)
b,, = *(c,, – in). I
2 + (Zn,,,
!
a.=+ ?i,,m)im,
m
z
(80)
b,, = + (Z.n, – &nL)im,
“,
~~herc
b,,m = o, ?L # m,
(5nm =1, ” ?1 = m.
The matrix connecting a and b in Eq. (84) will be called the scattering
matrix S;
b = Sa, (85)
and
S=(z–l)(z+ I)-’. (86)
It is evident that
Z+ I=H,
S = GH-’. 1 (88)
(~$))
GH = HG.
and
‘= [3 ‘= (~:1
(107)
S22S,, S22
‘1
S12 – ~
S21 . (108)
T=
_ S1l 1
[- S21 %
If the junctions are numbered in the same order in which they occur in
the chain, then
~k = Tk’k, (109)
and
gk = ‘k+l, (110)
where the subscript refers to the number of the junction. If the equa-
tions are combined, it is found that
However, this implies that the resultant T-matrix for the chain is
7’,, 1
s =
[3
T22
1 T12
T22 T22
T22 .
(113)
bk = &’kiai. (115)
z
j
Hence, if these expressions are combined,
r’
ak = 1 – rs~~ z j
Skjaj, (116)
where the prime denotes that the kth term is eliminated in the sum.
Equation (1 16) can then be substituted in the remainder of the scattering
matrix to eliminate a,. The lcth terminal is thus completely eliminated
from the scattering matrix.
Let us now consider a solution of Eq. (117) which satisfies the boundary
conditions of the junction, and let us introduce a variation of the fre-
quency and field quantities consistent with the boundary conditions.
The variations satisfy the equations
If the quantity
is introduced, and if, in the right side of this expression, quantities from
Eqs. (117) and (118) are substituted, there results
If Eq. (120) is integrated over the volume ~f the junction and the left
side of the equation converted to a surface integral, there results
! s
(E X (5H – JE X H) . dS = jc$ti
Jv
(~11’ – d32) dv. (121)
where, as before, in and enare the current and voltage at the nth terminals.
It is to be noted that Eq. (122) relates a variation of the terminal voltages
and currents to an energy integral times a variation in frequency. It
is evident from Sec. 5.10 that if en is real for all n, then E is real, H is
imaginary, and i. is imaginary. If Eq. (122) be limited to real terminal
voltages, it becomes
zn
(e.hin – in~e.) = –j~a
— –4ja@(WE
\u
(/.IH* “ H + cE* . E) dv
+ W},), (123)
when emis real, independently of n. If the variation of Eq. (31) is taken
If this and Eq. (31) are substituted in Eq. (123), since i: = – in,
z n,m
i; 6Zn~i~ = 4j&o(wE + w.), (125)
L$Z.m dZ.. = z,
6W = du ‘m’
z
rhm
e~Y~mem= 4j(~E + ~R). (127)
Equations (126) and (127) are the starting point of the discussion of the
frequency dependence of impedance and admittance matrices.
In matrix notation, Eqs. (126) and (127) become
In matrix notation
2(b6a – ~~b) = 4jej~W 8w, (132)
where, as above, W is the total stored energy. Since E2 has the phase
angle /3, en has, except for a possible change in sign, a phase angle ~/2 for
all n. Also L has a phase angle (~/2) + (r/2).
154 GENERAL MICROWAVE CIRCUIT THEOREMS [SEC. 52.3
Therefore,
(133)
is pure real and
is pure imaginary.
From Eqs. (78) and (85),
e = (1 + S)a,
(134)
i = (1 – S)a, }
arg Ez = D.
*.i~*S*S’a = W. (139)
Equations (139) and (136) are the starting point for the investigation of
the frequency dependence of the scattering matrix of a general junction.
5.23. Transmission-line Termination. -Consider a lossless termina-
tion of a single transmission line. The matrix equations ~Eqs. (128) and
(129)] reduce to the scalar equations
(140)
and
(141)
(142)
where —@ is the phase delay in the wave after reflection. Equation (144)
states that the electrical line length into the termination and out again
always increases with frequency, and the rate of increase is equal to the
stored energy per unit incident power.
The physical significance of Eq. (144) is rather interesting. If a
pulse, represented by
1-
Equation (150) is, of course, valid only over the small range of fre-
quenciw inrll]ded in the pulse,
15$ OENERAL MICROWAVE CIRCUIT THEOREMS [SEC.5.24
q-=-.w (151)
P
Equation (151) states that the time required for a pulse of energy to
enter the termination and leave -again is just the average stored energy
per unit incident c-w power.
5s24. Foster’s Reactance Theorem.—From Eqs. (140) and (141),
~,=2wE+wH
> (152)
+i*z
~,=2WE+WH
(153)
~e*e
Thus if X(u) has a zero at a certain frequency, B(cJ) has a simple pole with
a negative residue at the same point. Conversely a zero in l?(o) leads to
a simple pole in X(u).
From Eqs. (13) and (14),
(157)
(158)
It is evident from Eq. (157) that for o = O, X = O unless the stored elec-
tric or magnetic energy becomes divergent for a given finite current.
If the stored energy becomes infinite, from Eq. (152) u = O is a singular
point. If thk singular point is a pole, it is clear from the f oregoiqg dlscus-
SEC,5241 FOSTER’S REACTANCE THEORI171f 157
sion that the pole is simple and has a negative residue. In this case
13(0) = O. Thus, at u = O, either X(u) orll(~) hasa zero.
Let it be assumed for the present that X(u) has a zero at u = ().
Remembering that X(u) = – X( – O),
x= b@l+b,u3+b5u’+ . . . . (159)
This expansion is valid for lo I < lull, where CJIis the location of the first
pole.1 Since this pole must be simple and must have a negative residue,
its principal part is
v,
(160)
w — WI’
where VI is positive.
In order for X to be odd, however, there must be a singularity at
u = – uI. If the principal parts of both these singularities are sub-
tracted from X, the remainder of the function is regular at the points
u — ULand u = —UI and can be expanded in a power series that is valid
for Iul < Iuzl where m is the next singularity.
If there are only a finite number of poles, this process can be continued
until the power-series expansion is valid for all finite frequencies. Then
X(U)=–
u +n+*
.
r.
) +al. +a3.3 +-... (161)
(162)
In order to simplify the discussion, the terminals have been located in the
transmission line at such a position that ~’1,2= 1. This represents no
important restriction.
In general none of the elements of S’ vanish, but S’ must satisfy Eq.
(139), where a satisfies Eq. (136). 1A
Sm. 5.25] FREQUENCY VARIATION 159
1
al=@ ~ ,
[)
1
az = ~’jj
[1 ~’
(164)
1
aa=d~
[1_l,
1
a4 = 42
() –~
These four column vectors satisfy Eq. (136), as can be seen by inspection,
Let W, be the average stored energy in the junction corresponding to
a~.
If a, is substituted in Eq. (139),
‘l’his reduces to
(::)s’(:)
‘(1]1) =‘1 (165)
In a similar way the remainder of the a’s may be substituted in Eq. (139)
to yield
(170)
then a; and a: represent ~~aves inrident in onc line only. They differ
tmly in the direction of transit thrwlgh the junction. Hence the stored
vnergies are wlual for a; and a~. I,et this stored energy be lV. It should
lw noted that
a~ = a; + (,j)~-laj. (171)
linear combination of the vectors of Eq. (170). If the electric and mag-
netic fields in the jLIn”ct
ion corresponding to the incident waves a; and
a; are denoted by the subscripts 1 and 2 respectively, then
(I w)
(181)
From Eqs. (166) and (168),
‘+
’(”c+-)
cJc ——
S1l = j = s,,. (188)
1
UL
At the resonant frequency,
S1l = o,
S{l = jC. 1 (189)
The value of W/P is C, and therefore the equal sign in Eq. (187) holds.
A simple shunt-tuned circuit thus has the maximum value of frequency
sensitivity.
CHAPTER 6
BY C. G. MONTGOMERY
from many higher modes, each of which differs from the others. (2)
The relative susceptance contains the frequency variation of the charac-
teristic admittance of the waveguide. The characteristic admittance is
proportional to the wave admittance of the guide and therefore contains
the factor h/Ao. Thus if an absolute inductive susceptance contained the
factor l/w or A, as the susceptance of a coil of wire at low frequencies,
the relative susceptance would be proportional to h,. A capacitance
independent of A in absolute value yields a relative susceptance propor-
tional to &/A2. However, since Foster’s theorem remains valid for wave-
guide terminations, no radical departures from the accustomed frequency
variation are to be expected.
6.3. The Inductive Slit.—If, in rectangular waveguide capable of
propagating the dominant mode only, a thin metal partition is inserted
in such a way that the edge of the partition is parallel to the electric field,
the iris formed is equivalent to a shunt inductance. The iris may be formed
symmetrically as in Fig. 6“la, or the slit may be asymmetrically placed
as in Fig. 6 lb and c. Since the electric field is in the y-direction, the
higher modes excited by the diaphragm are all H-modes and the stored
energy is therefore predominant y magnetic. According to Eq. (5.14)
the shunt susceptance is negative, and the diaphragm is a shunt induct-
ance. The value of the susceptance has been accurately calculated and
the exact formula may be found in Waveguide Handbook, Vol. 10 of this
SEC.6.3] THE INDUCTIVE SLIT 165
(2)
where f(a/k) is a small term. Equation (2) gives the susceptance relative
to the characteristic admittance of the waveguide. The principal term
is proportional to AO,and hence the susceptance has very nearly the fre-
quency dependence of the suscept-
2.5
ante of a coil of wire at low fre-
quencies. The correction term j
is, however, not proportional to Xg 2.0
but has a different frequency de-
pendence. The magnitude of j
and its variation with frequency - B 15
are illustrated by Fig. 6“2. At
large values of & f can be neg-
1.0
lected; when X, is small, the correc-
tion term contributes appreciably.
The short vertical lines indicate
the values of X,/a for which the
HZ,- and the H, O-modes may first %
propagate. Since the slit is sym- ~lG.6.2.—The variation of susceptance of
an inductive slit of width d = a/2. The
metrical, the HzO-mode is not straight line with a slope 01 unity is the
excited by it. For wavelengths cotangentterm
the ~X,CtvalueofinB,
Eq. (2); the curve give~
short enough for the ll~o-mode to
propagate, the slit no longer behaves as a simple shunt element but
excites some of the H30-mode.
The susceptances of the asymmetrical cases of Fig. 6.lb and c may also
be expressed to a good approximation by simple formulas. The suscept-
ance of the diaphragm of Fig. 6 lb is given by
B=–~cot2~
( l+sec2~cot2~
) (3)
(4)
‘= -:cot’:(’+csc’x)
The expressions given in Eqs. (3) and (4) are not so exact as the corre-
spondhg approximation for the symmetrical slit. Asymmetrical dia-
phragms excite the HzO-mode and other even modes as well as the
HSO-mode and the other odd modes. The correction terms to be added
are therefore larger and the frequency dependence correspondingly greater
166 WA VEGUIDE CIRCUIT ELEMENTS [SEC. 64
than for the symmetrical case. For practical applications the diaphragm
of Fig. 6.lc is often used, since it is the simplest possible construction.
The approximate formulas just stated and the exact curve shown in
Fig. 6“2 are all valid only for a metal partition that is infinitely thin.
It is usually necessary to use metal thick enough so that some correction
is needed. Although the theoretical correction has not been worked out,
an empirical correction has been found that is fairly exact. If the value
of d of the thick slit is reduced by the thickness t, the value of B is increased
to compensate for the thickness effect; thus
/
/4
x
(!
(a) (b)
FIG. 6.3.—Capacitive diaphragms in rectangular waveguide. The metal partitions are
shown shaded.
(6)
for the symmetrical opening of Fig. 63a. Correction terms that are
important at high frequencies are omitted from Eq. (6). The frequency
variation of B is similar to that of the susceptance of a condenser at low
frequencies except that k, is substituted for ~. It does not have the
frequency variation of the relative susceptance of a lumped capacitance,
which would be proportional to A~/A2 as mentioned in Sec. 6.2. The
importance of the high-frequency correction terms to Eq. (6) may be
judged from Fig. 64 in which the susceptance of a diaphragm is plotted
as a function of b/k~ for an opening d = b/2. The straight line represents
Eq. (6); the accurate value of the susceptance is given by the curve. It
should be remembered that the dimensions of the waveguide are usually
chosen so that l)/AOis about, ~.
SEC.6.5] THE THIN INDUCTIVE WIRE 167
~=~1nc5c&
(7)
A, 2b “
1 /
The capacitive slit is not often
used in high-power microwave ap-
plications, since the breakdown o’ 0.2 0.4 0,6 0,8 1.0
strength of the waveguide is great- !/h a
ly reduced by it. The effect of a 11~, 6.4.—Relativesusceptanceof a thin
finite thickness of the partition is symmetricalcapacitiveslit with an opening
equalto one-halftheheightof thewaveguide.
much larger than for the induc- The straight line representsEq. (6); the
tive slit and will be discussed in a curve shows the values calculated from the
accurate expression.
following section.
6.6. The Thin Inductive Wire.—A thin wire extending across a rec-
tangular waveguide between the center lines of the two broad faces of
the guide forms an obstacle that acts as a shunt inductance. If the
radius of the wire is small and the resistivit y large, the skin depth in
the wire may be ma,de comparable with the radius of the wire. The rela-
tive impedance of the wire then contains a resistive component, and power
is absorbed in heating the wire. Such a device forms a bolometer element
and is commonly used to measure microwave power. The relative
impedance of the wire is given by
where T is the radius of the wire, u is the conductivity, and e is the base
ofnatural logarithms. Thed-cresistance lt,ofth ewireisgivenby
(9)
(lo)
where
(11)
(13)
This circumstance strengthens the belief by some that the most reason-
able choice for the characteristic impedance of a waveguide is that given
by Eq. (11). It will be recognized that this impedance is the proper
value to choose in order to obtain the correct value of the power flow W
from the expression
~v=~y,
2 z,
V = bEV.
(14)
m Br
(a)
B~ 1
(b)
FIG. 6.6.– -(a) Equivalent circuit of a shunt-resonant thin diaphragm; (b) resonant dia-
phragm.
Y=l+2jQ~,
where COO is the resonant angular frequency and Ati is the deviation from
the resonant value. The Q-values for resonant apertures are low, of
the order of magnitude of 10, and i~crease as b’ decreases. This is to be
expected, since a decrease in b’ increases the capacitive susceptance Bc.
Another commonly used resonant aperture is obtained by combining
a symmetrical inductive diaphragm and a capacitive tuning screw. The
resoriant frequency may be conveniently changed by means of the screw,
and the Q altered by changing the aperture of the diaphragm. Cll,>h
o 0
FIG. 6,8.—Resonant obstacles and apertures in waveguide of circular cross section.
The obstacles are totally reflecting at resonance; the apertures totally transmitting.
metallic portions are shaded.
Tbe
sented to illustrate the variation of the susceptance with the diameter of the
aperture or chsk. No theoretical estimates of the susceptance are available.
1MicrouxweTransmission Circuits, Vol. 9, Chap. 10, Radiation Laboratory Series.
~ Microwave Mizers, Vol. 16, Radiation Laboratory Series,
172 WA VBGUIDE CIRCUIT ELEMENTS [SEC. 68
There exist also both series-resonant obstacles that are totally reflecting and
shun t-resonant apertures. A group of these is shown in Fig. 6.8. In
the use of diaphragms in waveguide of circular cross section it must be
remembered that modes of two polarizations can exist (Sec. 2.13) and
the diaphragm must be symmetrical with respect to the electric field so
that the second polarization will not be excited.
A close correspondence exists between a capacitive slit in rectangular
waveguide and a slit in a parallel-plate transmission line. In rectangular
waveguide the z-dependence of the fields, both near the obstacle and far
from it, is determined by the z-dependence of the incident field; in particu-
lar E= is zero, and E,, E. and Hz vary as sin Tx/a. Each component of
the electric field satisfies the wave equation; for example,
d2Ev d2Eu
—+—+ ~+ IC2EV= O,
axz ay’
where k = 27r/A is the wave number in free space. The z-derivative can,
however, be evaluated, and the equation becomes
a2E
dy2 (15)
‘+%+[’2-(YIE.’”
and similar equations hold for the other components. In a parallel-plate
transmission line also, E. is zero and the other components satisfy the
wave equation; thus,
a’Eu
—+ ~ + k’Eu = O. (16)
ay’
Since the boundary conditions are independent of x for a capacitive
obstacle, the solutions of Eqs. (15) and (16) differ only in that where k
occurs in a parallel-plate solution k2 — (m/a) 2 occurs in the waveguide
solution Consequently, one result may be derived from the other by
rePlacing kz — (r/a) 2 with k’ or equivalently by replacing X. with L
Thus the susceptance of a’ symmetrical capacitive slit in a parallel-plate
transmission line follows immediately from Eq. (6) and is
4b rd
(17)
‘=xlncsc%”
A coaxial transmission line with a thin disk on either the inner or the
outer conductor behaves very similarly to a parallel-plate transmission
line with a capacitive slit. Accurate values of the susceptance for
capacitive disks are to be found in Waveguide Handbook. Some values
calculated in a different manner have been given by Whinnery and
others.1 A wire extending from the inner to the outer conductor of a
1J. R. Whinneryand H. W. Jamieson,R-oc. IRE, 32, 98 (1944); J. R. Whinnery,
H. W. Jamieson,and T, E, Robbins, I%oc. IRE, 32,695 (1944).
SEC.69] I.VTERACTION BETU’EE.V DIAPHRAGMS 173
(18)
where —Bl and —B2 are the susceptances of the ttro windows and l’~. is
the admittance of the load at the second window. The magnitude and
] >’. 11, Frank, RL Report No. 197, February 1943.
174 WA VEGUIDE CIRCUIT ELEMENTS [sEc. 610
phase of Y~/ Ya are plotted in Fig. 6.9 as functions of s/& for the case
where Y~ = 1, kg/a = 1.96, and B1 = BZ = 0.5. The effect of the
1.0
1
s
y2 0.9
0.8 -
1~
/
~2 /
%
.b
%3
I
-4 /
I
5
0 0.1 0.2 0.3 0.4
s/Ag
FIG. 6.9.—The magnitude and phase angle of the relative effective admittance Y,’/ Y 2
as a functionof the separations of two symmetricalinductiveaperturesof susceptance
B = –0.5.
interaction is important at small separations and is principally a reduction
in the magnitude of the susceptance.
To describe these effects by means of an equivalent circuit, it is neces-
sary to consider each aperture as a three-terminal-pair network and the
two networks connected together
a=&-
as shown in Fig. 6.10. Terminals
2 of the two networks are con-
nected by the transmission line
A for the dominant mode, and
terminals 3 are j oined b y the trans-
Aperture 1 Aperture 2 mission line B for the next higher
FIG. 6.10.—Equivalent circuit of two mode excited. Line B is attenu-
apertures m close that interaction effects ating, since the waveguide is be-
must be taken into account.
yond cutoff for the higher mode.
Three-terminal-pair structures are discussed more completely in Chap. 9.
Unfortunately, the necessary data are not available for most situations,
and interaction effects must be determined by experiment.
6.10. Babinet’s Principle. -Useful results for waveguide structures
with a high degree of symmetry can be obtained by the application of
Babinet’s principle (Sec. 2.10). An example of this has been given by
Schwinger.’ The actual electromagnetic problem that is solved to find
1David S. Saxon, ‘( Notes on Lectures by JuliaII Schwinger, I)isccrntinuitiesiu
W@eguidcw,” February 1945.
SEC.610] BABINET’S PRINCIPLE 175
J—r Z~=2jX
I
ZL=2jX
H, - mode Babinet’s
principle
,,,~_d) Wmme@
.___ -_--;J; ~+ (c)
4
lLJ=d’— z A P
(20)
Since the magnetic field is zero along the plane through the center of the
aperture, a magnetic wall may be inserted there and the configuration of
Fig. 6.1 lb is obtained. This configuration has the same terminating
impedance 2.jX. Babinet’s principle may now be employed to obtain
Fig. 6.1 lc. The electric and magnetic walls are interchanged, and E
is replaced by H. The terminating admittance j(B/2) of this structure
must be equal to the terminating impedance of Fig. 6.1 lb, and
(21)
Babinet’s principle in the form of Eq. (21) has been applied also to
corresponding resonant structures such as those shown in Fig. 6.8.
This application is not a rigorous one, however, since the electric walls
that form the waveguide are not transformed. Nevertheless, if the size
of aperture is not too large, Eq. (21) applies approximately. If the
apertures are large or close to the walls of the waveguicle, asthose in the
capacitive and inductive obstacles of Fig. 6“7, the deviations from Eq.
(21) are large. The product of the inductive and capacitive susceptances
is not —4 as required by Eq. (21) but varies from —2 to —10 over the
range of disk diameters from 2.0 to 5.5 cm.
6.11. The Susceptance of Small Apertures.-The transmission of
radiation through an aperture may be expressed in very general termsl
if the size of the aperture is small enough. On a metallic wall, the
normal magnetic field and the tangential electric field vanish, but a
tangential magnetic field HOand a normal electric field Eo maybe present.
If there is a small hole in the wall, within the hole there will be a tangential
electric field and a component of the magnetic field perpendicular to the
wall. If a linear dimension z of the hole satisfies the relation that
z << A/27r, then the fields in the neighborhood of the hole are closely
approximated by the unperturbed fields HOand EQplus the fields from an
electric dipole and a magnetic dipole within the hole. The strength
of the electric dipole is proportional to -EO,and the dipole is directed
normally to the wall. Similarly, the magnetic dipole is in the plane of
the wall and of a strength proportional to HO. The constants of propor-
tionality are the polarizabilities of the hole. The electric polarizability
P is simply a constant, since the dipole and the field are parallel. The
directions of the magnetic dipole and the exciting magnetic field are,
however, not necessarily the same. The magnetic polarizability is
i H. A. 13ethe,Phys. Reu., 66, 163, (1944),
SEC.6.11] THE SUSCEPTANCE OF SMALL APERTURES 177
so = n. EXH~ds, (24)
/
where E is the normal-mode electric field. Equation (23) may be written
as
(25)
L78 WA VEGUIDE CIRCUIT ELEMENTS [SEC.611
where the 1- and m-components are taken in the directions of the principal
axes of the hole and the n-component is normal to the hole.
The amplitude transmission coefficient may be expressed in terms of
the susceptance of the diaphragm, since
A.l+r. ~ (26)
2 + jB’
if the waveguide is the same on both sides of the diaphragm. Since the
hole is small, B is very large and
B=–%”. (27)
Finally, since the normal-mode fields are the same on both sides of the
diaphragm, the zero subscripts may be dropped provided that the
amplitude of the fields is doubled. Hence
(28)
Circle ofradius r . . . . . . . . . . . . . 43
~r :T3 +r3
r-p = ~blcz ab’%z w abz
Ellipse*of eccentricitye = ~ 1 – ; -—
() 3(1 –,1)(F –h’) ~E–(1–c’)F SE
r
Long narrowellipse (a >> b). . ... . . ; a’ ab2 ~ ab~
,n 4a _l 3 3
() r
Slit~ of width d and length 1.. .
1
_=_ $ sinz ‘~ [Ml cosz (l)Z) + Mn cosz (m,z)], (29)
B a
where co is the z-coordinate of the center of the iris and (l,z) is the angle
between one principal axis of the aperture and the z-axis. The suscept-
ance of a narrow inductive slit that is centered may be calculated by
inserting the proper value of Ml, and it is found to be
1 47r ~
—.— (30)
B 0.955 (7rR2) ‘
which has been written to exhibit the similarity to Eq. (29). For circular
waveguide operating in the EO-mode, where the iris is at the center of the
guide and has sufficient symmetry so that no other propagating modes
are excited, the cmly field at the iris is the normal electric field. The
quantity E: in Eq. (28) is negative, since the longitudinal field in a wave-
guide is 90° out of phase with the transverse field; B is positive, and the
iris is capacitive. The susceptance is
B = 0.92R’
(31)
PA* “
It should be noted that the variation with frequency of both the inductive
and capacitive small holes is similar t o that of the larger irises.
seen through the line, and a matched condition would no longer exist.
Moreover, standing waves would exist along the transmission line even
at the correct frequency, and the resistive losses in the line would be
correspondingly great. By the insertion of a shunt susceptance at the
proper place a short distance from the load, it is possible to make the
relative admittance of the combination equal to unity, and the frequency
sensitivity of the match is small.
The load admittance usually must be determined by a standing-wave
measurement, and it is therefore convenient to express the matching
conditions in terms of the observed standing-wave ratio r and the posi-
tion of a minimum in the standing-wave pattern. If the losses in the
matching elements are neglected, the magnitude of the susceptance
necessary to match a load that produces a standing-wave ratio of T is
the susceptance that produces a voltage standing-wave ratio of r when
inserted in a matched line (See. 4.3). The relation between B and r
is given by Eq. (3.28),
(32)
or
(34)
or
‘an@l=-;G
=-$-’ (35)
1 ‘an-’ +r
(36)
~=–%
The negative sign of the square root in Eq. (35) was chosen, and therefore
1,/k, is less than ~, and a positive, capacitive susceptance must be added to
produce a match, Tf it is desired to match with a negative, inducti~e
SEC.6.13] SCREW TUNERS 181
.- tan-’ $,
1
(37)
G= Zlr
Im (Y + jB) = O. (38)
If the line is Iossless and B is finite, then the conductance of the cavity
is zero. For resonance, the line is nearly one-half wavelength long if B
is large and negative, and the length approaches one-quarter wavelength
as the magnitude of B decreases. In practice, however, the losses must
be taken into account; but since the length 1 is small, it will be assumed
that al is small compared with unity, where a is the attenuation coef-
ficient. The magnitude of the susceptance B will be assumed to be large
compared with unit y. The length of the cavity is near n half wave-
lengths, and the dimensionless quantity ~ is defined by
(W)
Y==!= (40)
al — j;’
and Eq. (38) becomes
1
B=– (41)
(al)”+ 6’ = t
For an inductive susceptance, c is
positive and the cavity is a trifle
less than n half wavelengths long. -
The conductance of the cavity is
G= “1 = B’al. (42)
(al)’ + c’ .
o
It is instructive to follow the
various transformations on an ad-
mittance chart as in Fig. 6.14. In
a lossless line, as the point of ob-
servation is mo~-ed away from the
short circuit toward the generator,
the admittance point travels along Lossless
line
FIG.6.14.—Theadmittance(fiagramof a
the outer circle in a clockwise di- lengthof short-circuitedline plus an induc-
rection. If a large negative sus- tive susceptanceof such a magnitudeas to
produceresonanre.
ceptancc is added after the point
has traveled almost half a wavelength, the total admittance can be made
Iw II”A VEGCJIDE CIRC[T17’ ELEMENTS [%c. 6.1
zero and a resonant condition exists. If the line is lossy, the admittanc
point spirals inward instead of traveling on the outside circle. It will b,
remembered that the magnitude of the reflection coefficient varies as e–z-~
After nearly a half wavelength of travel the addition of a negative suscept-
ance brings the admittance to the positive real value given by Eq. (42).
Of primary interest in resonance phenomena is the frequency sensi-
tivity of the admittance. The conductance varies as l/h~, since B varies
as 1/Xo if a is assumed t o be constant, and the derivative of G is
dG
— = Bzan, (43)
dh,
since
(44)
dBL =_@
dh,
~ CSC2(31= –
,
:Z(1 + B*), (45)
9
since B = cot @ at resonance. Therefore
dB~ _ ~ B,. (46)
dh. = ,
That it is correct to neglect the losses can be verified by differentiating
Eq. (4o) directly. The coefficient of B’ in Eq. (46) is much larger than
the coefficient of Bz in Eq. (43), and the conductance can be assumed to
be constant. The frequency sensitivity of the cavity can therefore be
described by the Q of the equivalent shunt-resonant circuit (Sec. 4.13),
(47)
(48)
which is the value of QO given in Eq. (2.83) with the losses in the end
walls omitted. The unloaded Q is independent of n. If the cavity is
matched. G = 1 and the loaded Q is
(49)
sm. 6.14] CAVITY FORMED BY SHUNT REACTANCE 185
tan /311= ~ +
tan@,=-~-
r
:+l=B,
r
:+1=–;,
I (50
where the approximate values are for B >> 1. The transformer ratio N
is given by
(51)
(52)
where the volume integral is taken over the cavity and the surface
integral over the cross se~tion of the waveguide. For a TE-wave
; E,H, = e A E:,
J i h,
186 WA VEGUIDE CIRCUIT ELEMENTS [SEC.6.14
where E. denotes the field outside the cavity. At resonance cE~ = ~H~,
where Ei is the field inside the cavity. Equation (53) can therefore be
written as
or
B tan @ = 2. (54)
The loaded Q has just one-half the value for a cavity with a single window;
G is, of course, now equal to 2. It is easy to see from the admittance
diagram of Fig. 6“15 that for a lossless line, the conductance is unity if
the two windows are equal. Ifthe
line is Iossy, the conductance is
larger. To obtain a matched
transmission cavity in a lossy line,
the input window must be slightly
larger (smaller Il?[) than the out-
put window.
It should perhaps be men-
tioned explicitly that calculations
of the kind just described neglect
the losses in the iris itself. For
accurate calculations of high-Q
cavities, some estimatd of the loss
FIG.6.15.—Theadmittancediagramof a should be included. It has been
cavityformedby two inductivewindowsone- suggested that a suitable estimate
half wavelengthapart.
for the losses produced by the
presence of a hole in a metal wall is made by assumi~g that the iosses
over the area of the hole are twice the losses in the wall before the hole
Waa cut. It is evident that this is an extremely uncertain approximation.
SEC. 6.15] DIAMETER CHANGES IN COAXIAL LINES 187
(55)
(56)
for the symmetrical change in height. The function j(b/hg) is the high-
frequency correction term. Since the susceptance is positive, the junc-
tion is sometimes termed a capacitive change in cross section. In the
larger of the two waveguides the field configuration is very similar to
that near a symmetrical capacitive slit, and in the smaller guide the field
is not greatly clifferent from that of the dominant mode. It might be
expected that the stored energy and therefore the susceptance given by
Eq. (57) should be approximately half that given by Eq. (6). Although
it is not at all evident from the form of the expressions, insertion of
numerical values shows that the difference is indeed small, of the order
of 10 percent.
By an argument identical with that given in Sec. 6.4, the junction
susceptance for the completely asymmetrical change in height can be
obtained from the symmetrical case if ~, is replaced by As/2. Similarly,
the junction effect for a change of height in a parallel-plate transmission
line is to be found by replacing & by A.
Change in Waveguide Width.—The inductive change in cross section
of a waveguide leads to a junction effect that is approximately one-half
the susceptance of the corresponding thin inductive aperture. The
proper choice of the characteristic admittance allows the equivalent
circuit to be that of Fig. 6.16. Junctions of two types are possible. If
the change in width is symmetrical and from a to a’, the waveguide of
smaller width is beyond cutoff if A > 2a’ and the characteristic admit-
tance of the smaller guide becomes imaginary. The proper value of
admittance is
(58)
SEC.60171 QUARTER-WAVELENGTH TRANSFORMERS 189
where
(59)
and
T a’
~=– 4 Cos 2 T
(60)
, 2“
‘1–:
()
Fora’/a <<l, Eq. (58) reduces to
(61)
(62)
If ~ < 2a’, the small waveguide is not beyond cutoff, and the proper
characteristic admittance is
(63)
(64)
if the junction effects are negligible (see Sec. 3.3). Such a section of line
is commonly called a transformer. If the load end of such a transformer
190 WA VEGUZDE CIRCUIT ELEMENTS [SEC. 6.17
or
(65)
(66)
rz=a–
()
b ‘->$
a
The frequency sensitivity of the match is usually not large, but this sensi-
(67)
tan ~1 = ~, (69a)
then
Bz + Y:
Y.=— (69b)
YL “
(70)
~=~ dln Z
470(T)o-+,f%),’-2J:’dz. (7,,
The subscripts O and 1 refer to the values of the quantities at the begin-
ning and end of the taper, respectively. If the derivatives are not very
different in value, a length 1 of the taper can be chosen to make r a mini-
mum. For example, if in Z varies linearly along the line, the reflection
1Slater,op. cit., pp. 71fl.
{92 WA VEGUIDE CIIWlrI T ELEMENTS [sm. 61!)
is given by
1’ = 4 in ~0 (1 – e-’j~z) (72)
43/31
if ~ = j@ does not change over the length of the taper. The quantity
r is thus zero when @ = mr or 1 = h./2.
6.19. The Cutoff Wavelength of Capacitively Loaded Guides. —Values
of impedance changes and shunt susceptances can be used to compute
the cutoff wavelengths of waveguides with complicated cross sections.
If thin metal fins are inserted from the top and bottom faces of a rec-
tangular waveguide, the cross section becomes that shown in Fig. 6.18a.
m (a)
d“
(b)
FXG. 61 S.—Capacitively loaded waveguides.
l+jl’’tank, ~=O,
Sw. 620~ SH (~.VT BRANCHES IN COAXIAL LINES 1!)3
or
tan k. ~ = ~ (75)
(7(i)
where B is the junction susceptance given by Eq. (57) with lc=/2 sub-
stituted for K.
In all calculations of this type the possible effects of interaction must
not be forgotten. Thus if the distance 1, is too small, the wall of the
waveguide will interact with the junction and the results of the calcula-
tion will be inaccurate.
produced. If the frequency of the radiation is high, the stub should not
be exactly a quarter wavelength long, but the correct length must be
found experimentally. Such a simple stub is rather frequency-sensitive.
If a standing-wave ratio of 1.05 is allowed, which is a rather large value,
the admittance Y of the stub must be less than + 0.05j. Since
the usable wavelength band di/i is only a little over 3 per cent.
A stub support that is usable over a much broader band can be made
by adding two quarter-wavelength transformers, one on each side of the
stub, as shown in Fig. 6.19. At the center frequency, where the stub
admittance is zero, the quarter-
wavelength transformers are, to-
gether, one-half wavelength long
and the whole device is reflec-
tionless. At a lower frequency,
t* the stub presents an inductive
I f susceptance, but the transformers
1.950” are less than a quarter wavelength
FIG.6.19,—Broadbandstubsupport. long and the net reflection is
again zero. This is indicated in the rec~angular admittance diagram
shown in Fig. 6.20a. The first transformer moves the admittance point
clockwise along the circle from YO to .4; the inductive susceptance of
the stub takes it from .4 to B; and the second transformer moves the
admittance from B back to YO. At a shorter wavelength, the admittance
(a) (b)
FIG,6.20.—Rectangular admittance diagram illustrating the action of the broadband
stub: (a) The path of the admittance point for a long wavelength, (b) the path for a
wavelength shorter than the center wavelength.
diagram is shown in Fig. 6.20b. The first transformer moves the admit-
tance point from Y, to A through more than a quarter wavelength; the
stub is capacitive and moves the admittance from A to B; and the second
transformer moves it from B back to Yo. Here again, to obtain com-
pensation for the junction effects, the stub length must be adjusted experi-
SEC. 6.21] SERIES BRANCHES IN COAXIAL LINEs 195
r“~””-””’ -’-~
\
(a) (b)
FIG.6.22.—Seriesbranchesin coaxiallines.
* i
(a) (b)
E!!
r!.— ‘:[;2 (c)
& (d)
FIG. 6.23.—Chokejoints in coaxialline: (a) a joint in the outer conductor showl
schematically
y, (b) a joint in the inner conductor, (c) and (d) choke joints in the outer
conductor which have reduced frequency sensitivity y.
(78)
(80)
Therefore the optimum condition is to have Z1 << ZO and Zz >> ZI. This
desirable low-impedance-Klgh-impedance condition reduces the fre-
quency sensitivity by almost a factor of 2, and it is employed in the design
“of nearly all half-wavelength chokes. A more compact choke design is
shown in Fig. 6.23d. For low frequencies, the “folding” of the choke has
little effect on the length or frequency sensitivity.
6s22. Series Branches and Choke Joints in Waveguide.-A junction
that behaves much as a series branch may be made in rectangular wave-
essure gasket
(a) (b)
FIG. 6.24.—Choke-joint connector for rectangularwaveguide.
guide with a secondary guide branching from the broad face of the main
transmission line. Since waveguide dimensions are much longer com-
pared with the wavelength than are the dimensions of coaxial line, the
junction effects are large and cannot be neglected. If the height b’ of
the branch guide and the height b of the main line are both small com-
pared with A., a pure series junction is closely approximated. A choke
joint may be made in much the same manner as in coaxial line. A gap
in the form of a vertical slit in the narrow face of the waveguide has only
a small effect, since the currents in the wall are in the direction of the
length of the slit. A gap in the broad face of the waveguide would
produce, however, a large disturbance. A short-circuited stub line one-
half wavelength long, which is broken at the quarter-wavelength point,
would form a good choke joint. A practical design of a choke connector
for rectangular waveguide is shown in Fig. 6“24. The circular choke
198 WA VEGUIDE CIRCUIT ELEMENTS [SEC. 6.23
m
‘E z, 22 i,
(a) (b)
FIQ.625.-Plungers for usein rectangularwaveguide.
principle, however, allows the matching formulas and the tuning range
that apply for the shunt circuit to be readily converted to the series case.
To vary the length of a stub line, some form of adjustable short circuit
or Plunger is necessary. Plungers are usually designed with choke
joints as indicated in Fig. 6.25a. Another design that is very similar
employs three quarter-wavelength sections as shown in Fig. 6.25b. If
the characteristic impedances of the sections, which are proportional to
the waveguide heights, are those designated in the figure, the input
impedance Z of the plunger is
z = z:.
(-)
()
z,
70
(81)
Since ZI/ZO can be made small, perhaps 0.1, and Zz/ZO can be 0.5, Z can
easily be as small as 4 X 10–4, and the power loss is therefore about 0.01
db. These values are adequately small for nearly every application.
susceptance alone, but the T-network that describes its behavior con-
tains series elements also. The electromagnetic problem that must be
solved to find the elements of the equivalent circuit may be reduced as
before totwoproblems by symmetry-arguments. Thus ~f equal voltages
are applied to the two terminal pairs of the device, a magnetic wall is
1
I
.=. —-—- -
a- *
,
I ,I
(a) (b)
..
—-
3 —
(c) (d)
Fx~. 6.26.—Decompositionof the problemof the thick post acrossa waveguide(a)
into the even (b) and the odd (c) problems. The cross sections are taken in the magnetic
plane.
For the odd case, an electric wall is in the plane of symmetry as in Fig.
6.26c and the terminating impedance is ZM(~), where
2
X,1–X12=–: ()
TD
—
a
2“ (82)
‘1+;”;
()
The shunt element of the T-network is inductive, as for the thin wire, and
the equivalent circuit is that shown in Fig. 6.26d. If the high-frequency
correction terms are omitted, XII + X12 is given b the imaginary Part
200 H’A VEG1:IDE CIRCl,TIT ELEME.VT,7 [SEC,623
of Eq. (8). For accurate results these correction terms should be incluflc~l
and the data given in Waueguidc Handbook should be used.
The thick tuning screlv must he described similarly by a ‘~-netlvork
with both shunt and series elements. For small insertions of the screw,
the circuit elements are all capacitive. With increasing insertion, the
absolute value of the shunt reactance decreases and the magnitude of
the series reactance increases. NO theoretical treatment of the behavior
is available, but experiments indicate that when the reactance of the
shunt element is zero, the series element is approximately – 0.2j. The
effective short circuit is therefore slightly in front of the reference plane.
(a) (b)
FIG. 6.27.—A thick capacitive slit and the equivalent circuit.
B,= B–~tan~t,
9
(83)
BZ = – : CSC2:,
9 I
where B is the junction-effect susceptance for the change in height of the
waveguide (Sec. 6.16). Equations (83) are based on the assumption
that the interaction of the two changes in height is completely negligible.
This assumption is certainly justified when t>> d. The experimental
data indicate, however, that these expressions are approximately true
even for irises of very small thickness.
For all thick obstacles, the choice of the reference planes is arbitrary
to some extent, as well as the form of the equivalent circuit. ‘1’wo
circumstances should be considered in the choice. (1) A certain set i)f
SW. 62S] BENDS AND CORNERS 201
T’ = 10 log,, ~.
The accurate calculations for the centered circular hole verify this
empirical relation rather closely, and Eq. (85) is extremely valuable for
design calculations. It may be relied upon to be accurate to about 1 db
when used with the formulas of Sec. 6.11 for T’.
6.25. Bends and Comers in Rectangular Waveguide.—The transition
from a straight waveguide to a smooth circular bend, either in the E- or
202 WA VEGUIDE CIRCUIT ELEi14ENTS [SEC.6.25
L
\
,.
) ,/’
–p
0
(a) (b)
(c)
7 (d)
Fm. 6.28.—Waveguide corners:(a) An E-planecornerwiththe equivalentcircuitshown
in (b). Reflectionless corners are shown in (c) and (o!).
(86)
where
‘y ‘d(+nr))
1 – (j Y)’
r=l– Y
l+Y”
junction effects can be neglected, the line section should be a half wave
length long. The resonant length when junction effects are present can
be found from Eq. (68) of Sec. 6.17 by equating Yin and Y.. The reso-
nant length is given by
(87)
l— N, z’— N z— N2 1
(89)
BY ROBERT BERINGER
EQUIVALENT CIRCUIT OF A
SINGLE-LINE, LOSSLESS CAVITY-COUPLING SYSTEM
Consider for simplicity a cavity-coupling system without loss, con-
taining only a single emergent transmission line. Such a system is
shown in Fig. 7.1. It is possible to define, at n,
reference plane A, a voltage and a current that
are uniquely determined by the electromagnetic
fields interior to A. The voltage-to-current ratio
A at this plane defines an impedance, which is the in-
a put impedance of the cavity-coupling system. It
~IG, 7.1,—Sir1gle- is pure imaginary and a function of frequency alone.
line cavity-coupling This impedance function can frequently be rep-
system.
resented by an equivalent circuit. It is desired to
find this function or representation and, in particular, its frequency
dependence.
SEC.71] LOSSLESS LUMPED CIRCUITS 209
‘*----t3- C2 c, Czn-z
FIG. 7.2.—Input-impedance representation of an n-mesh circuit with poles at w = O and
~=m.
ante function are simple, or of first order. It is seen from Eq. (1) that
two networks are equivalent, that is, they have the same impedance at
all frequencies, if they have the same poles and zeros and have imped-
ances numerically equal at a single nonresonant frequency. This latter
condition fixes the value of A.
An expression of the form
a2.–2
z(Lo)=jAu
(1+;+*;+”””+ u’ — U;._z )
(2)
L,. = A
juz-’(u)
ck=–a~=– ~,_u;
[11 ~=U, k=0,2,4, .”,2 n-2.
1
Lk=—
Ll;ck )
In Eqs. (1) and (2) the network is taken to have poles at both u = O
andw= m. In special cases, one or both of these may be removed.
I Guillemin, Communication Networks, Vol. II, Wiley, hTewYork, 1935, Chap. 5;
Schelkunoff,Ek.ti-omugnetic Waves,Van Nostrand, New York, 1943, Chap. 5.
210 RESONANT CAVITIES [SEC.71
‘=;-3 ~;i~!;{;:i::::t:
2?1-1
analogous to Eq. (2)” is obtained by
FIG.7.3.—Input-admittance repre-
sentationof n-meshciLcuit with zeros expanding Z–l(a) = Y(u) around the
atti=Oandw=m. poles of Y(u), namely, f.ol, tij, . . ,
u,._,. This yields
Y(u) = z(u)–’
,2n–
In the expansion of Eq. (3) Y(o) has been assumed to have zeros at
~=oandatu=ca. As in the impedance representation, these zeros
may be removed in special cases, corresponding to degeneracies in one or
two of the resonant elements.
Ck
The zero at a = O is removedby
letting one of the C’l = co, in which
case Y(u) -+ @ as u ~ O.
zero at o = m is removed by ‘he let- +=+Ih
ting one of the L; = O so that FIG. 7.4.—Input-impedance representation
near a pole w = wh.
Y(u)~~atu-+w.
At frequencies near one of the poles of Z(a), Eq. (2) may be written
as
(4)
an approximation that lumps all contributions from other poles into the
almost constant term X~. This approximation is good at frequencies
near the pole ti~ and far removed from any other poles. Equation (4)
can be represented by the circuit of Fig. 7-I, where
SEC.7.2] LOSSLESS DISTRIBUTED CIRCUITS 211
1
‘“k = –Tk
1
‘; ‘Lkck”
A–’bl(.c
Y(u) = –jB1 –jm (5)
●
near the pole W, and Eq. (5) can be represented by the circuit of Fig.
7.5, where
1
@f=~ Ll
-Bl
and
CL
1 D
bl=— L&l” FIG. 7,5,—In-
put-admittance
Inthe study of cavity-coupling systems, itis often rewewntation near
a pole u = u[.
necessary to know the behavior of the system at fre-
quencies near a particular resonance. There is therefore need for equiva-
lent-circuit representations of the form of Figs. 7.4 and 7.5.
7.2. Impedance Functions of Lossless Distributed Circuits.—The
impedance-function representation just discussed is of great generality
and utility in lumped circuits. It furnishes a method for finding the
equivalent circuit of any lossless, single-terminal-pair network in terms
of the frequencies at which either Z(u) or Y (co) is infinite. An extension
of the method to distributed circuits is clearly desirable.
It has been stated that a cavity has an infinite number of resonant
frequencies. This is true for all distributed circuits. The impedance
function, therefore, has an infinite number of poles and zeros, correspond-
ing to an infinite number of network meshes. This suggests an extension
of the foregoing method to the representation of a network with an infinite
number of meshes. Such an extension is formally possible, and has been
carried out. 1 The expansions of Eqs. (2) and (3) are formally the same
as those for the lumped-constant circuits, except for the fact that n ~ cc .
Schelkunoff has stated that convergence difficulties sometimes arise in
such series.
Although this formal extension is possible, it is more satisfactory to
use the methods of Chap. 5 which make use of the field equations in
defining and formulating the input impedance of a distributed circuit.
In Chap. 5, a region surrounded by a perfectly conducting surface per-
forated by a single transmission line is discussed. Nothing is specified
about the region except that the dielectric constant, permeability, and
1Schelkunoff,Proc. IRE, 32, 83 (1944).
212 RESONANT CAVITIES [SEC.7.2
[Sec. 5.24, Eq. (5.162)] that if Y(u) has zeros at w1, ~z, ”””, then
Z(u) can be expanded as
.
jw
z(w) = –2 ~. ~ + jalw, (7)
z .
n=l
which is an obvious extension of Eq. (2) to the case of an infinite number
of resonances. The question of the convergence of the series in Eq. (7),
in the most general distributed case, is not always straightforward. All
ordinary cavity-coupling systems, however, are free from convergence
difficulties. In all practical cases, Eq. (7) reduces to Eq. (4) near a
resonance.
7s3. Impedance-function Synthesis of a Short-circuited Lossless
Transmission Lme.—It is illustrative to consider a simple example of a
distributed circuit in which Eq. (7) is evidently convergent and the poles
and zeros of the impedance function are well known. A short-circuited
lossless transmission line is such an example.
Let the line operate in the fundamental TEM-mode (A = x.), and
let the characteristic admittance be YO. Then, at terminals at a dis-
tance 1 from the short-circuited end, the input admittance is
Y = –jYo cot ~ 1
(9)
(lo)
T—
ml
214 RESONANT CAVITIES [SEC.7.3
whereby
(11)
Each term of the sum in llq. (10) can be identified as the admittance
of a series I, C-circuit. Such a series circuit has an admittance
(12)
where a’. = n%? = l/L= C.. The values of L. and C. may be identified
if it is noted that
(JJ W1—
2Y,, u,
~ I —;, = ~ –W~12~2J
~~~— —2 ~2 — _,
~1 ~1
from which
and
1 2% _21%
c. = ~L—n= ;Fwl; = ~2T2c
)
The short-circuited transmission
line is evidently represented by the
circuit of Fig. 7,7, where the circuit
elements are given by Eqs. (11)
‘~mY~~ and (13). Theconvergenceof
I’T(;. 7.7. l{epre~en tation of a short-cir-
this representation is assured by
ruited transmission hne.
Eq, (10). Near a resonance, a
single term predominates and represents the admittance very well. Fig-
ure 7.8 shows the exact form of F~q. (10) and rurves for t\vo approxima-
tiol~s’ ((me and five resonant elements).
EQUIVALENT CIRCUIT OF A
SINGLE-LINE CAVITY-COUPLING SYSTEM WITH LOSS
The theory that has been given thus far is incomplete for the solution
of the problem of cavity-coupling systems, since it deals only with com-
pletely lossless systems. The treatment must be extended to include
Exactvalue,
Fiveresonant xactvalue
element5 I
I
0.2 0.8 1.0 1.2 1.4 2,0
eh -.
P“
‘/
/
4
~
/
Oneresonant
element
/
FIG, 7.8.—Curves showing exact form of admittance function and two approximations.
where Al, &, . . . , and O, AZ, Ad, . . . , are respectively the zeros and
poles of Z(A). For passive networks all decrements are negative, cor-
responding to a decrease in amplltude with time. Hence, all functions
Z(A) have negative $ at the poles and zeros; that is, all poles and zeros
of Z(A) lie in the left half of the k-plane. 1 As in the lossless case, Eq.
(14) is expanded around its poles, with the result that
.Ax+++~
z—
n
a.
A – A.’ n = 0,2,4, . . . . (15)
The poles and zeros of Z(A) occur as conjugate pairs so that one term in
the sum is of the form
(16)
The values of Z(A) that are of interest are those for real frequencies
A. = j~, for which Eq. (15) becomes
Z(A) =j~.
which reduces to
–j$+A
z n
(u: –
an(2jti
IJJ
– 2,Q
——
2 — 2jwfn + #
Z(A) =jAa–j~+A
2 —
an2jw
(u: – ;’-–
—
2j&l&)
(17)
for sfnall loss. It is seen that Eq. (17) is of the same form as in the loss-
1Guillemin,10C.cit., or S. A. Schelkunoff,E[e!ctromagnetw
Waws, Van N’ostr:uld,
New York, 1943, Chap. V,
SEC.7.5] IMPEDANCE FUNCTIONS 217
less case, except that the summation contains an imaginary term in the
denominator. At each resonance u = u“, therefore, Z(A) does not go to
infmit y but has a real contribution A a*/& from the summation.
The admittance case for small loss proceeds in an analogous fashion.
The result is obtained, for real frequencies (k = j~), that
‘(”)‘R+
’(”L-3
If the resonant frequency uo is defined as that frequency at which Z(u) is
pure real, a; = l/LC and
energy stored
0=2.
energy lost per cycle
;L12 _ WOL
‘2Tf0~ R “
In terms of this Q,
or
1
Y(fJ) = —1
R+jQR :.–:
()
CLx.oo
~R .
=j (19)
~~—~2+j&”
Q
m c LR
Fm. 7.10.—Sim-
ple shunt-resonant
circtut with loss.
where
Y(.)
“i
= ;
..
= L—c”
1.
+ T@
;+j.oc
1 + j.C
:–:,
() Wo
from which
(20)
Comparison of Eq. (20) with Eq. (17) shows that the summation in Eq.
(17) can be represented by a number of shunt-resonant circuits con-
nected in series.
7.6. The Equivalent Circuit of a Loop-coupled Cavity .-Thus far,
arbitrary cavity-coupling systems have been treated, first lossless sys-
tems and then systems having a small amount of loss introduced into
each resonant mesh. The circuit representations have been derived
either directly from field considerations or by analogy with lumped cir-
cuits. Although this approach is very fruitful, it is only qualitative in
that there are a number of arbitrary features in the equivalent circuit.
These features can be specified completely only by an actual solution of
the field problem for the cavity-coupling system. This, unfortunately,
has been done only in a few special cases. One of these is the case of a
loop-coupled cavity.
It is instructive to study this solution. The circuits are derived from
a new and more physical point of view; and in particular, the physical
form of the approximations necessary to treat the dissipative system are
made apparent. It has been shown that the treatment of such systems
by an extension of Foster’s theorem is uncertain with regard to the physi-
SEC.7.6] LOOP-COUPLED CAVITY 219
on the cavity walls. ‘They form a normalized orthogonal set, in that they
satisfy
1 E. U. Condon, Rev. Mod. Phys., 14, 341 (1942); J. Applied Phys., 12, 129 (1941.)
zJ. C, Slater, “Forced Oscillationsand Cavity Resonators,” RL Report No. 188,
Dec. 31, 1942.
? P, D. Crout, RL Report No. 626, Oct. 6, 1944.
t A. Baiios RI, Report No. 630, Nov. 3, 1944.
220 RESONANT CA VI1’IES [SEC.7.6
where v is the cavity volume. .8. and ~~ are related by the expressions
lGa&a= v x K.,
–Mea = v x s..
The electromagnetic field in the cavity can be expanded in this set
of functions by
H= k.l@a,
z
D= kiq&
2
a
where the qa are scalar functions of the time. The coefficients in the
expansion have the dimensions of electric charge. The functions q.
corresponding to a normal mode k. are the amplitude functions of the
mode and are analogous to the coordinates of a dynamical problem.
They are the so-called normal coordinates of the system.
The dynamical equation of the system in terms of these normal
coordinates is the Lagrangian equation
(21)
where T and V are the total kinetic and potential energies of the system.
The quantity T is identified as the magnetic energy, and V as the electric
energy of the system; that is,
and
La = pk;~,
(22)
f74 ‘&”
I
(23)
where io is the loop current and Mk the mutual inductance between the
loop and the normal-mode mesh, given by
222 RESONANT CAVITIES [~Ec.7.6
The loop is seen to couple to all modes except when the integral of Eq. (24)
vanishes. If & is known, Mo. can be found by integrating Eq. (24)
over the loop area.
The introduction of loss in the cavity walls adds another term to
the dynamical equation, which then becomes
where F is given by
d
Zt ()
dT
—
dqa +Z.
dV
+z.
dF
=e”’
(25)
(26)
u being the conductivity y of the walls and 6 the skin depth. It is assumed
that the dissipation does not change the normal-mode fields & and 3c..
It can be shown that Eq. (26) becomes
where
(27)
For b = a,
(28)
the loop. The mutual resistances R~ couple all meshes for which Eq.
(27) does not vanish.
It has been shown that Eq. (27) vanishes, for a cavity having the form
of a right circular cylinder, for all modes,that can be-
JfoI RI
come simultaneous y resonant. It is prcibable that
this is true also for other shapes. This fact con- L1C,
siderably simplifies the circuit representation of Eq. B
(29), since the mutual resistances vanish and the M02 R2
meshes can be separated. Figure 7.12 is the repre- LO
LZC2
sentation of Eq. (29) under such conditions. B
I !
The frequencies UCare the resonant frequencies
of the meshes of Fig. 7”12 when the terminals are I
open-circuited, that is, the frequencies at which
cd1’
the series mesh reactance vanish and the stored elec- FIG, 7.12.—Circuit
tric and magnetic energies are equal. The natural representationof a lcmp-
coupled cav~ty.
oscillation frequencies of the meshes are somewhat
different, being given by
!
U. = w. dl – +Q:,
which for high Q’s reduces to
Most cavities have such high Q’s that ti~ and u. differ at most by only a
few parts in 105.
:2F-3
FIG. 7.13.—Single normal-mode mesh of
Fig. 712.
Z-T-3
FIG. 7.14.—Alternative equivalent circuit for
a single normal-mode mesh.
The circuit of Fig. 7.12 can be transformed easily into the form’ of Eq
(17). Each normal-mode mesh is of the form of Fig. 7.1?, which is
equivalent to the circuit of Fig. 7.14. The circuit of Fig. 7.14 has an
input impedance
z. = jd, + @2M;.
(30)
jdc + R= – j AC
(31)
224 RESONANT CAVITIES [SEC.7.6
where
1
‘a = Lxt “
If Q. is defined by
If the Q is higl., the last term is negligible [at rmonance the third term is
o~(M&/LJQ~ and the last term is a~(M~./LJ]. Thus, to a very good
approximation for even moderate Q’s,
A comparison of this equation with Eqs, (20) aud (17) shows that the last
term may be represented as a shunt-resonant r:ircuit.
A comparison of Eqs. (33) ‘and (20) shows that the summation can bc
represented as a circuit of the form of Fig. 7.15, where
SEC.7.71 IMPEDANCE FUNCTIONS NEAR RESONANCE 225
and the terms of Eqs. (32) and (19) can be identified. This leads to the
following reiations between the circuit elements of Figs. 7.12 and 7.15.
11
(34a)
‘i = G. = LaCa’
“~(r ‘~[c
o
FIG.7 19.—Admittance representation of cavity-coupling system near a resonance.
near resonance the terms R’ and T’ are always very small, since they
correspond to off-resonance losses in the transmission line and coupling.
They ~vill be considered to vanish.
A very convenient tool in the study and design of cavity-coupling
systems is the Smith impedance diagram. Consider a simple series
RLC-circuit terminating a transmis-
sion line of characteristic impedance
ZO. On a Smith impedance diagram,
the variation of input impedance of
the circuit with frequency describes a
locus like the circle (a) in Fig. 7.20.
At the real axis, u = ~0 = l/tiLC,
the resonant frequency of the circuit.
If the circuit is shunted by a capaci-
tor, the locus is a circle such as (b) in
Fig, 7.20.1 The new resonant fre-
quency u{ is different from oo. The
FIG.720.-Loci of input impedances
on a Srmth impedance diagram, for a radius of the new circle is also differ-
cavity-coupling system. ent. If the simple RLC-circuit is
shunted by an inductance, the circle will be shifted in a counterclockwise
direction.
1This is obtained by transforming(a) to the admittanceplane, which is accom-
plishedby the reflectionof (a) through the point 20. The capacitive susceptanceis
then added, which rotates tbe admittance circle. This circle is then reflectedback
through2, to obtain (b).
SEC.7.7] IMPEDANCE FUNCTIONS NEAR RESONANCE 227
It has been shown that the circuits of Fig. 7.18 are the most general
impedance representations for a cavity-coupling system near resonance.
Hence, loci such as (b) are the most
general impedance contours for a
cavity-coupling system near reso-
nance.
Since a change of reference ter-
minals inthetransmission line corre-
spends, to a first approximation, toa
simple rotation of locus (b) around
20, it is evident that by a suitable
choice of reference terminals, any
cavity-coupling system can be
brought into the form of locus (a). FIG. 7.21,—Locus ot input admittance
on a Smith admittance chart, for a cavity-
Hence, any cavity-coupling system
coupling system.
near resonance behaves as a simple
series RLC-circuit at suitable terminals in the transmission line.
A discussion of the representations of Fig. 7.19 in terms of the Smith
admittance diagram proceeds in an exactly parallel fashion. The admit-
tance of a simple RLC shunt circuit describes a locus such as (a) in Fig.
721. The introduction of a series reactance rotates and changes the
scale of (a). As before, a suitable choice of terminals in the transmission
line can bring the locus of the most general representation of Fig. 719 to a
form such as (a), and hence the input admittance of any cavity-coupling
system near resonance behaves, at suitable terminals, as a simple shunt
RLC-circuit.
In the case of the loop-coupled cavity of Fig. 7.12, it is easy to derive
another near-resonance representation which is often more convenient
than the Foster representations of Figs. 12.18 and 1219. It is clear that
near the resonant frequency w = u., Fig. 712 reduces to the circuit of
Fig. 7.22, which may be represented as Fig. 7.23. It can be shown that
at a single frequency any lossless two-termmal-pair network can be
represented by a length of transmission line, an ideal transformer, and
a series reactance. Thus, Fig. 7.23 can be represented by the circuit of
228 RESONANT CAVITIES [SEC.7.8
n ‘=(’”’-l%’)
and
L’ = L. – n’L,,
-- L’ RQ
z. j
/
wr~~ LlcJe21J
C(,
frequency different from W. but near to it if La>> n2L0, which is the case
of small coupling. It is further seen that if the length of transmission
line chosen in the representation is the same kind as that physically
connected to the cavity, then new reference terminals may be chosen
such that the cavity can be rep-
resented by the lumped circuit of
Fig. 7.25. This ideal-transformer
representation is particularly con-
i
vement for cavity-couphng sys-
~~~~’” ‘~ca terns with several emerging
Ca $ Ra transmission lines.
7s8. Coupling Coefficients and
i
External Loading.-Suppose that
FIG. 7.25.—Representationsof loop-
coupled cavity near resonancew at pre- the transmission line emanating
ferredterminalplanein transmissionline. from a cavity-coupling system is
terminated in its characteristic impedance (admittance). At suitable
terminals, the equivalent circuit of the total system is as shown in Fig.
726. The Q of the cavity-coupling system is ao1/T, when w%= l/lc.
This is the unloaded Q of the system, Q., the Q when terminals A are
short-circuited. The loaded Q of the circuit of Fig. 7.26 is cod/(r + Z,).
If the loaded Q is denoted by Q., then
SEC.7.8] COUPLING AND LOADING 229
and
Q.=QL
()1+:. (36)
The coupling parameter is defined as 13= Zo/r; which is also the input
conductance of the cavity-coupling system at the terminals of Fig. 7“26,
normalized to Yo, or
1
(37)
(38)
then
Q
I
Z.
1
YORCL
@
I c
A A
FI~. 726.-Equivalent circuit of cavity- FI~. 7.27.—Equivalent circuit of cavity-
coupling system terminated in Zo at a par- couphng system terminated in Yo at a Par-
ticular set of terminals. titular set of terminals,
Qu=QL(l+RYO)=QL
()1+~
where B and X are the shunt susceptance and series reactance at the
terminals, e and i are terminal voltages and currents, and W, and W~ are
the stored electric and magnetic energies in the system interior to the
terminals. It can also be shown [Sec. 5.23, Eqs, (5.140) and (5.141)]
that
dB 4(WE + ~H)
z= ee*
(40)
ax 4(WH + w.)
au = ii* ‘1
If values for B or X at some reference terminals can be found and
dB/6’w or dX/C3ti can be evaluated, the total stored energy can be found
from Eq. (40). The Q can then be written
1 * dB
Q=2rf — “-“e %
‘~
energy dissipated/see
1 ,.*ax
–
4 ‘z au
= 2.f
energy dissipated/see”
Q.$~
—
—
w ax
(41)
2h? d.
}tquut]on (4 I ) gives the general ~ for LL system [Jhose imped:mf!e or admit-
t:lllce fllllctillll i+ kllt~i~n :Lt :LgiVetl tel’millal pilil.
SEC.7.10] IRIS-COUPLED, SHORT-CIRCUITED WA VEGUIDE 231
al + j tan @
z, = 20 (43)
1 + j(al) tan 131”
nAo
12.—
tan & =
‘an(r-”?)= -“?=%”’ ’45)
where UOis the frequency at which tan P1 = O and 6 = u — wO. Sub-
stitution of Eq. (45) into Eq. (44) yields
z. =20
(d+j$:oa)
This expression is seen to be of the same form as the input impedance of
(46)
232 RESONANT CAVITIES [SEC.7.10
‘=R+
’(wL-a
= R + j2L8 (47)
near resonance. Here 6 = co – m, and a~ = l/LC. We can, therefore,
identify the terms in Eq. (46) as
R = Z,(al), )
(48)
The iris appears in shunt with this circuit so that the complete equivalent
circuit of the cavity-coupling system at plane .4 is that shown in Fig.
7.29. The iris susceptance is – jb = –j/u~.
‘~\ D
F1~. 7.29.—Equivalent circuit of an iris-
coupled short-circuited waveguide nearreso-
nance. Terminals are at the plane of the
FIG, 7.30.—Equivalent series circuit for
an iris-coupled short-circuited waveguide
cavity.
iris.
If the waveguide of Fig. 7.28 is terminated in its characteristic imped-
ance to the left of plane .4, the impedance Zfl appears across the terminals
of the equivalent circuit in Fig. 7.29. The loaded Q under these condi-
tions may be calculated. The series impedance of the parallel S, ZO com-
bination is @ZO/(ZO + jcd). For a high-Q system d << Zo, and this
series impedance is jo~ + U2.Q2/Z0. The complete series circuit is that
of Fig. 7“30. The radiation Q is
~=mL+ulX
,
U:J32
Zo
~z ~ y;
=~5~2 (49)
ii = (50)
1
G= )
SEC.7101 lRIS-CO1,I’I’l,EI), ,SHOR7’-C’IR<:1 ‘[TEl~ 11“A I’,! W1’1OE 233
The unloaded Q is
Qu - T ‘;,
2(a/) A’
since w&/R is the unloaded Q of the circuit when the terminals of F;g.
7.29 are short-circuited.
It is seen by examination of Fig. 7.29 that resonance occurs for fre-
quencies somewhat less than o,, corresponding to n& > 21, or cavity
lengths somewhat shorter than n&/2. This is also seen by examining
the total admittance at plane A which is
Y._ b 1
Y, – ‘~ TO+ol+jtanbl
— al tan @
(51)
(al)’ + tan’@ – j !0 – j (al)’ + tan’ @“
In the high-Q case Itan D1I>> al, Thus Eq. (51) becomes
At resonance, Y,/YO is pure real, This occurs for negative cot @l, or 1
slightly less than n&/2. We see that the input conductance at resonance
is
2
()Y, :1
m ,,s – tan2 /31
.a[L.
()Y“
(53)
(54)
234 RESONANT CAVITIES [SEC.711
dB
~ + Yol(l + cot’ pi). (55)
%=–afi
Near resonance, cot’ D1 = b’/ Y: >>1 and 1 = h,/2; and from Eqs. (55)
and (54),
since b/ YO>> 1. ‘
For the radiation Q the shunt conductance is Y,, and so
(57)
This is seen to be identical with Eq. (49). For the unloaded Q, the shunt
conductance is YO(cZl) (b/ YO)2, and
Qu. —IT A: (58)
2(al) m’
CAVITY-COUPLING SYSTEMS
WITH TWO EMERGENT TRANSMISSION LINES
As in the treatment of the single-line cavity-coupling system it is
convenient to consider the general representation theory that has been
derived for lossless n-mesh net-
works and to show the equivalence
of this representation to that de-
<e rived from field theory.
@2) 7.11. General Representation
&--N
Z;2’
1 n2
of Lossless Two-terminal-pair
~p Networks.—It is not difficult to
extend the reactance theorem to
I two-terminal-pair networks of n
(1)
G 3-) represented
‘:::::r::a’:::a:; as a series combina-
FIG, 7.31.—T-section representation of two- tion of T-sections or a parallel
terminal-pair n-mesh 1ossless network. combination of II-sections. The
T-section representation is shown in Fig. 7.31, where each Z., Zb, 2. nl~~
1E. A. Guillemin, Communication A“etworks, Vol. II, Wiley ISew York, 1935?
Chap. 5, p. 216.
SEC.711] GENERAL REPRESENTATION 235
n
Consider the special case where
Xc and xb are inductances, the
ideal transformer has a ratio of
one to one, X, = O, and Za, zb and
Z. are parallel-resonant elements
at the frequency w This situa-
tion is shown in Fig. 7.33, FIG. 734,-Network equivalent of Fig, 7.33
where near the resonant frequency wo = m.
z,, =
(60)
Z22=
z,, =
S’ = L’ – n~L.,
(64)
2“ = L“ – n;L6. )
In all practical cases, L’ >> n~La and L“ >> n~L6; therefore S = L, and
the series circuit of Fig. 7.35 resonates at nearly the same freqllency as
that of Fig. 7.33 or that of Fig, 734.
SEC.7,14] TWO-LINE CAVITY-COUPLING SYSTEM 237
It is clear from Fig. 7“35 that if the lines 7, and 12are chosen to be
identical with the physical lines connected to the cavity, new terminals
may be chosen in the physical lines such that the cavity-coupling system
is simply a series LC-circuit coupled by ideal transformers at the input
and output terminals.
The transformation of various other forms of Fig. 7.32 to the form of
Fig. 7.35 will not be treated in detail. This transformation can be
performed in all cases for which X. = O.
M,
7.12. Introduction of Loss. —Just as in the case
of the single line or single terminal pair, it is possible (I) L,
to treat the small~loss approximation, in which loss n
R
L
is introduced into each mesh of a purely reactive
net work. It is clear that the general representation ,2) ~z c
of such a network is of the form of Fig. 7“31 with n D
resistive elements added to each Za, Zfi, and Z.. Mz
Near a particular resonance this reduces to the form FIG. 7,36.—Cavity
with two loop-coupled
of Fig. 7.33 and finally to that of Fig. 7.35, where a linesnearwoz= 1/LC.
resistive element now appears in the resonant mesh.
This can be verified by a straightforward analysis following that shown
for the single-terminal-pair network.
7.13. Representation of a Cavity with Two Loop-coupled Lines.—The
loop-coupled cavity, like the single-line cavity-coupling system discussed
previously, can be treated rather exactly by field m=thods, A simple
extension of the single-loop case of Fig. 7“12 shows that Fig. 7.36 is the
L1+MI L’+M1 C
T-
~,
(1)
Fm. 737.-Circuit equivalent to Fig. 7.36 near woz = 1/LC.
forming the load and generator into the resonant mesh results in the cir-
cuit of Fig. 7.39. The unloaded Q, which is obtained by putting
Ro=R. =0,
QU==QL l+n~~+n~#’.
( )
The input- and output-coupling parameters are defined, respectively, as
p, – M&,
@2 = ‘$”
Z=R
[
(l+@, +@z)+&?ti
(=)1’
and the power into the load impedance is
(65)
SEC.714] TWO-LINE CA VIT Y-COUPLING SYSTEM 239
T(wo)
T(u) = 2
l+Qfi~-:
() ~o
or, putting u = uo + (Ao/2),
T(wo)
T(u) = z“ (67)
l+Q} $
()
It is noted that T(u) = iT(ao) (i.e., half-power points of transmission
occur) for Au/@O = l/QL. The quantity Au is frequently called the
~’bandwidth ‘“ of the cavity at the resonant frequency COO.
CHAPTER 8
BY N. MARCWITZ
m
R ror R o T’ R
D“=’~=’’’:sl R
(cl) Conical antenna
(1)
a;+K2’U= O, (2)
which determines the variation of the mode amplitude along the trans-
mission system. The two independent mathematical solutions to this
equation may be expressed as
Cos /(2, Sk K.Z
and interpreted as tra~’cling ]vaves (cf. Chap. 3). Since either set of
solutions suffices to specify completely the propagation in the z-ciirertiou,
the impedance description of uniform transmission lines must be expressed
in terms of these trigonometric functions,
Similarly, in a cylindrical r~z coordinate system the wave equation
determining the complex amplitude u of one of the field components
may be written as
:1(’3+(:-%,+:+’2)”=0 (3)
which determines the variation of the mth mode amplitude along the
direction of energy propagation. As before, two independent mathe-
matical solutions to this equation exist; these are the Bessel and Neu-
mann functions of order m (Bessel functions of first and second kind,
respectively)
J_(w) Nm(KT-)
ll~(w-), ~:)(K~),
(5)
H$(x) s ~e
(
+3 .—~ “)
= (~j). 2,
*’(’-;)
(6a)
J J 7rx
or, equivalently,
2
.Tm(x) = –Cos X-y=j
J- irx ( )
~5in ~_2?7z+l
Nm(z) = —Ir ) (6b)
d- lrx ( 4 )
Physically these approximations imply that at large distances traveling
radial waves are identical with traveling plane waves save for the decrease
in amplitude of radial waves along the direction of propagation. This
Ices z
Besselfunctions Trigonometricfuyctions
FIG, 8.2.—Comparison of zero-order Bessel functions and trigonometric functions.
Nn(x)
h~(x) =
J~,
lrx
for z>> 1,
h
1.5
1.0
0.5
W
0
2 4 6 8 10 x
/’
-r
i
FIG. S.3.—The amplitudeand phaseof the Hankelfunctionof orderzeroand one.
Table 8.1. The values of the Bessel functions Jo(z) and ~l(z) are also
included in this table.’
Since in a radial transmission line the point r = O is a singular
point, it is not to be expected that the asymptotic small-argument
approximations (for z << m,m ~ 1),
m
.()
(m– 1)! 2 ~,
()
I
J.(z) = + ; J N.(z) = – for m z 1,
7r z
2 (8)
J,(z) d-; z) iVO(z) = – -in ~, m=O,
() 4r
~ = 1.781 . . .1
are closely related to those of the trigonometric functions, although there
is a qualitative resemblance. The corresponding small-argument rela-
] Cf. the tables in G. N. Watson, Theory of Be8.seiFunctions, Cambridge,London,
1944.
SEC.8.2] UNIFORM AND NONUNIFORM REGIONS 247
tions for the Hankel function may be obtained by use of the identity
of Eq. (5).
TABLE8.1.—VALUESOF THEBESSELFUNCTIONS
.Tm(z)NA(z) – .V.,(Z)JL(X) = ~!
TX 1
248 RADIAL TRANSMISSION I,INIi,? [%c. 83
where the prime denotes the derivative with respect to the argument.
Equations (9) are also quite similar to the corresponding trigonometric
relations
d
—cosx=—sinz,
dx
d ,.
z “n x = Cos “
d. d
cosz —smx-sinz-cos z= 1,
dz dz
particularly for large arguments \vhere the z“ and Z–mfactors of Eqs. (9)
may be omitted.in first approximation.
With this brief discussion of the characteristic waves in uniform and
radial transmission lines it is now desirable to turn to the impedance
description of such transmission systems. An impedance description
exists for every characteristic mode. In the following however, this
will be carried out only for the lowest or dominant mode, since this mode
is usually the most important for practical applications. The treatment
of any other mode in terms of impedances is carried through in an exactly
similar manner. As a preliminary to the impedance description of
nonuniform radial transmission systems, that of the uniform line will
be reviewed briefly (cf. Chap. 3). The corresponding treatment of radial
lines is developed in close analogy thereto.
8.3. Impedance Description of Uniform Lines. -In Chaps. 2 and 3 the
electromagnetic field within nondissipative uniform transmission sys-
tems, such as linear waveguides, was described in terms of a superposi-
tion of characteristic modes. The introduction of a voltage V and a
current 1 as measures of the transverse electric field EL and magnetic
field H, associated with the dominant mode was made quantitati~e by
the definitions
E,(z,y,z) = .~(Z)@(2,y),
H,(z,y,z) = ~(Z){ X @(Z,y),
(lo)
(11)
Since both the voltage and current satisfy equations similar to Eq. (4),
solutions to Eq. (10) can be written in terms of standing waves as
The application of the boundary conditions that the voltage and current
at z = ZOare V(zO) and l(zO) leads to
asthe complete solution for the behavior of the dominant mode. Since
many of the quantities of physical interest depend only on the ratio of
I(z) I(zo)
—
K
l’
v(z) ~
i
Zo
I
I
‘t
I
I v(z~)
—z
I I
I I
z z~
FIG. S.4.—Section of a uniform transmission line showing positive directions of I and V.
and by division to convert the solution, Eqs. (13), into th’e fundamental
admittance relation
y,(z) = ~ + Y’(z,) cot K(Z, – Z) (15)
COt K(ZII – Z) + jy’(zo) “
From Eq. (16) it is seen that the relation between the reflection coef-
ficient and relative admittance at any point z is
Equations (17) and (18) provide a method alternative to that of Eq. (15)
of relating admittances at two different points on a transmission line.
With this brief review of the description of the uniform transmission
regions of an electromagnetic system, it is now appropriate to turn to
the treatment of the discontinuity regions. A discontinuity region is
described by indication of the relations between the voltages and currents
at the transmission lines connected to its terminals. For the case of
two such terminal lines, distinguished by subscripts 1 and 2, the voltage-
current relations are linear and of the form
1, = Yllvl + Y12V2,
(19)
12 = Y12,V1 + Y22V2. 1
The positive directions of voltage and current are chosen as in the equiva-
lent circuit of Fig.. 8.5 which is a
schematic representation of Eqs.
(19). Since the principal interest
in this chapter is in transmission
systems, the discussion of the prop-
erties of the circuit parameters Y U,
$:$2 y,,Y22willbeomitted.I tisas-
FIG. S.5.—Equivslent circuit of a discon-
sumed that these parameters are
tinuity region.
known either from theoretical com-
putations or from experimental measurements on the discontinuity.
In connection with circuit descriptions of discontinuities, it is useful
to observe that equivalent-circuit descriptions exist for transmission
SEC. 8.3] IMPEDANCE DESCRIPTION OF UNIFORM LINES 251
Z(Z) = –jy, COt K(Z, – Z)~(Z) – jY, CSC K(Z, – 2)[– ~(z,)],
(20)
~(z,) = –jyo csc K(ZO – z)V(Z) – jY, cot K(ZO – 2)[– V(ZO)], I
where — V(ZO) is chosen as the positive voltage at ZOin accord with the
sign convention of Eqs. (19). Comparison of Eqs. (19) and (20) then
shows that the equivalent-circuit parameters of a length (zO – z) of
transmission line are
Ar (z) Ar(zo)
— = ~ + jz(z – ZO)AK, (22a)
r(z)
and
Ar(z) = _ 2A Y’(z)
(22b)
r(z) 1 + [jY’(z)]’”
Equation (22a) states that on a change in frequency the resulting relative
change in the input reflection coefficient of a nondissipative uniform line
differs from that of the output reflection coefficient only by a phase change
of value twice the change in the electrical length of the line. Rewriting
Eq. (22a) in terms of admittance with the aid of Eq. (22b), one finds that
and the desired relation is finally obtained by observing from Eq. (11)
that
dY’(z) dY’(zo)
dlno k’ dlnu
(23b)
1 + [jY’(z)]’ = j ()i ‘(20 – ‘) + 1 + [jY’(zO)]’”
P//f/A v /,
‘ v--f
I I b
v/J///l v/ /,-i,d-l
1
—r--’
r=o
Topview Side view
FIG.8.6.—Coordinates for the cylindrical region between two disks.
typical example of a radial transmission system is the cylindrical region
between two annular disks as shown in Fig. 8“6. The discussion starts
from a transmission-line form of the fundamental Maxwell equations and
proceeds to a characteristic-mode representation of the fields and thence
to a detailed treatment of the impedance (or admittance) properties of
several of the lowest modes. As indicated in Fig. 8.6, an r@z polar
coordinate system is most suitable for the description of the field. In
this coordinate system the electric field E and magnetic field H are given
implicitly, for the steady state of angular frequency u, by
and
1 (26a)
(26 f))
(27(1)
?Z =0 ,1,2,... 1
The mode amplitudes Vi and 11, called the mode voltage and current,
respectively, are functions only of r and have been so defined that the
average power flow in the positive r-direction is ~ Re ( VIIj). After
substitution of the mode forms (27a) or (27b) into the Eqs. (26) and
evaluation of the transverse derivatives therein, there are obtained the
transmission-line equations
dVl
—. ‘jKlzlIL,
dr
(28a)
dIz =
‘jK YIV1,
T 1
where
and
“=K’-(:Y=’2-(9-(3 (28b)
kKl 21rTtn
ZL=; ={2F for the H-type modes. (28c)
m
The superscript distinguishing the mode type has been omitted, since
the equations for both mode types are of the same form. A field represen-
tation similar to this one can also be given if the region has an angular
aperture less than 27r. Such a representation differs from that in Eqs.
(27) and (28) only in the dependence on @ and in the value of the ampli-
tude normalization required to maintain the power definition.
If neither E, nor HZ vanishes, the transverse field may be represented
as a superposition of the mode fields in Eqs. (27a) and (27 b). In this
case the two types of modes can no longer be distinguished on the basis
of vector orthogonality as in the uniform line. Nevertheless, the four
voltage and current amplitudes of the mixed ninth mode satisfy Eqs. (28)
SEC.8,4] CHARACTERISTIC MODES 255
and can be determined at any point r from the four scalar components of
the total transverse field at r.
As in the corresponding case of the uniform line, Eqs. (28) constitute
the basis for the designation of Vl and 11as the mode voltage and current.
Concomitantly they also provide the basis for the introduction of a
transmission line of propagation constant K1 and characteristic impedance
Zt to represent the variation of voltage and current along the direction of
propagation. The transmission line so defined cliffers from that for a
uniform region in that both the propagation constant and characteristic
impedance are variable. This variability of the line parameters ~st e
taken into account when eliminating 1; from Eqs. (28a) by differen&ati .)
in order to obtain the wave equation ,k
( :
Id d V;
+ K;V; = o, L (2{) i
r dr ()‘z
‘—
propagating. Conversely it is seen from Eq. (8) that for Kr < m the
waves damp out as (K~)-”’. Thus the propagating or nonpropagating
nature of a radial wave is determined from the sign of the square of the
propagation constant K~ exactly as in the case of a uniform wave. In
the case of radial wavesit shouldbe noted that the same mode maybe
both propagating and nonpropagating, the propagation being character-
istic of the wave behavior at large distances.
Uniform regions are most suitable as transmission systems if only one
mode is capable of propagating. This is likewise true for radial regions,
and therefore the following section will be concerned with the detailed
discussion of the transmission properties of a radial region in which only
one E- or H-type mode is propagating.
8.5. Impedance Description of a Radial Line.—A typical radial region
to be considered is that shown in Fig. 8.6. The applicability of a simple
transmission-line description to such a region is subject to the restrictions
that only one mode propagate and that no higher-mode interactions exist
bet ween any geometrical discontinuities in the region. These restrictions
are not essential and may be taken into account by employing a multiple
transmission-line description although this will not be done in this
chapter. The simple transmission-line description for the case of only
the lowest E-type mode propagating is based on the line equations
dV
–jkZJ,
dr =
dI
–j,tl”ol”, (30)
dr = 1
d-- )
/lb
zo =&= e 2m-’
dV p k 41rr
—. ‘jKzoI, 20=;0= -—,
dr J :Kb
2 (31)
dI
—. #=#_nl.
‘jKYo V, I
dr () b
From a comparison of Eqs. (30) and (31) it is again apparent that duality,
in the above-mentioned sense, no longer exists because of the different
characteristic impedances and propagation constants of the two modes.
.4 modified and useful form of duality, however, still obtains. If V, ZJ,
and k of the E-type mode are replaced by 1, YOV, and K of the H-type
mode, the line Eqs. (30) go over into Eqs. (31) and conversely. This is
easily seen if the line equations are rewritten in the forms
E-type H-type
dV d(Y,V) + (Yov) = _jK,,
–jkZOI, — —
z= dr r
(32)
dI
g (2,1) + * = –jkv, – = –jtiYov.
dr I
As a consequence of this modified duality, all relative impedance relations’
of the one mode become identical with the relative admittance relations
of the other mode provided the propagation constant k is associated with
the E-type and K with the H-type relations.
In both the E- and H-type modes the voltage V and current Z are
measures of the intensities of the electric and magnetic fields associated
with the propagating mode. This fact is indicated quantitatively in
Eqs. (27a) and (27b). The positive directions of V and 1 may be shown
schematically by a transmission-line diagram of the usual type as in
Fig. 87. This schematic representation of the behavior of the lowest
258 RADIAL TRANSMISSION LINES [SEC.8.5
E- (or H-) type mode differs from the corresponding representation for
the uniform line in that the characteristic impedance. being variable,
must be specified at each reference plane
l(r) I(TO
)
—
I k
I
I
1’ I Zo(r) ZO(70)
: ~r
‘1
v(r) I I V(rO)
I (K) I
The waves defined by this equation have been discussed in Sec. 8.2.
The standing-wave solution to Eq. (33) was there shown to be of the form
With the aid of Eqs. (30) and the differential properties of the Bessel
functions, the solution for the current 1 can be written as
‘(r)= v(rO)rlON::lOJOl+’zO(rO)’(rO)rOOJOs:ONOl’
(35)
where
Jo(k?-) = Jo, Jo(lc?-o) = Jo,, J,(kro) = Jlo,
-d
FIG.8.8a.
-d
Fm. 88b.
FIQ. 8.8.—(cz) Values of ct (z, ~) for y/x > 1. (b) Values of ct (x, v) for v/z < 1
SEC. 8.5] IMPEDANCE DESCRIPTION OF A RADIAL LINE 261
1,
FIG. 89a,
~lG. 89b,
FIG.89.-(a) Value. of Tn (.c, v) for ~/z > 1, (b) Values of T,, (r, y’] for vfz < 1
262 RADIAL TRANSMISSION LINES [SEC.8.5
1.0
OB
0.6
(w)
0.4
02
-02
FIG.8.10.—Values of ~(z, ~) = ((v, z).
‘“s(+sin(+ -Sin(x-$)cos(y -$
Ct (Z,g) =
(m)
SNC.8.6] REFLECTION COEFFICIENTS 265
By use of the definition [Eq. (7a)], Eq. (43a) may be put into the form
Y’(r) = –j tan
()kr – ~ , kr>> 1. (44C)
where .4 and B are the complex amplitudes of the waves traveling in the
directions of increasing and decreasing radius, ,respectively. The solutlo~
266 RADIAL TRANSMISSION LINES [SEC,86
for the current Z follows from Eq. (45a) by use of Eq. (31) and the dif-
ferential properties of the Hankel functions as
(46b)
On the elimination of the factor B/A there are obtained the fundamental
transmission-line relations
r.(r) = rV(TO)el~[~O(~r~–nO(~rO)l, (47a)
r,(r) = r,(ro)eiz[~,c~rl–n,(~ro)l, (47b)
that relate the values of the reflection coefficients at the radii r and To.
The quantities q, and ql are the phases of the Hankel functions as defined
in Eq. (7a). The transmission Eqs. (47) also provide a means of relating
the admittances at two points on a radial line. This relation may be
obtained from the wave solutions by division of Eq. (45a) by Eq. (45b)
with the result that at the radius r
](r) = H\2)(kr) 1 + r,
–jYO(r) m ~ + rw” (48)
V(r)
yj(r) . w’
l+r,
.
1 – r.
l+rv”
()
-~~
(49a)
or conversely as
Zj–1
r.(r) = -f r,=-~, (49b)
—
m(kr) = ql(lcr) = kr — ~,
r. = –r,,
and
Y~ = Y; = Yo,
Equations (47) and (49) provide a method alternative to that of Eq. (37)
of relating the admittance at two clifferent points of an E-type radial line.
The corresponding relations for the case of an H-type line are obtained by
the duality replacements discussed above.
8.7. Equivalent Circuits in Radial Lines.—The radial-transmission-
line relations so far developed permit the determination of the dominant-
mode voltage and current or, alternatively, the admittance at any point
on a radial line from a knowledge of the corresponding quantities at any
other point. These relations assume that no geometrical discontinuities
exist between the two points in question. The existence of such a non-
uniformity in geometrical structure implies that relations like Eqs. (35),
for example, must be modified. The form of the modification follows
directly from the linearity and reciprocity of the electromagnetic field
equations as”
11 = Yl, vl + Y12V2,
(52)
12 = Y12V1 + Y22V2, 1
where 11 and VI are the dominant-mode current and voltage at a reference
point on one side of the discontinuity and IS, Vz are the same quantities
at a reference point on the other side. These equations are identical
with Eqs. (19) which describe a discontinuityy in a uniform line. The
discontinuity can likewise be represented schematically by the equivalent
circuit indicated in Fig. 8“5 which shows the choice of positive directions
for the voltage and the current. The equivalent-circuit parameters
Y,,, Y, ~, and Y2, depend on the geometrical form of the discontinuityy
as well as on the choice of reference points. As in the case of the uniform
line, the explicit evaluation of the circuit parameters involves the solu-
tion of a boundary-value problem and will not be treated here. The
customary assumption that they are known either from measurement or
268 RADIAL TRANSMISSION LINES [SEC. 87
and
: AY
Cst(z)y) = Jo~oo – ~oJoo = – Cst(y,x) .
The function cst (z,y) is termed the radial cosecant function, since it
becomes asymptotically identical with the trigonometric cosecant
function for sufficiently large z and y. The value of the radial cosecant
function may be computed from the tabulated values of the radial
cotangent functions by use of the identity
1 + Ct(z,y) Ct(z,y)o
Cstz (Z,y) = (54)
{(X,Y)
This identity goes over into the corresponding trigonometric identity
CSC2(y – z) = 1 + Cotz (y – z)
for sufficiently large z and y. From Eqs. (53) it is ~o be noted that the
equivalent circuit of a length of radial line is unsymmetrical in contrast
to the case of the uniform line. The shunt and series parameters of the
~-circuit representation (cf. Fig. 8“5) for the radial line are seen to be
’11-’12=-~yo(’)[ct(zy)+{cst(z
Y22 – Y12 = –jYO(?-o)
Y12 = –j
[ ‘Ct(yx)
<YO(?-)YO(?-0)
+$CS4
Cst(z,y).
I
1
’55
SEC.87] EQUIVALENT CIRCUITS IN RADIAL LINES 269
where the various quantities are defined as in Eq. (53). These circuit
equations are of the form
L-L_!
FIG.81 1.—T-circuit for a dlscolltinuit y with positive directiolls of
The series and shunt elements of the T are given by
V and I
211 – 21? =
–Jz”(’) [.Ct ‘z’v) + ~ 2
ECst(.r,y) 1),
.
d q
(57)
22?– Z,* = –jzo(ro) –Ct(?y,.r)+ : cst(lJj.r) ,
[_____ Y
z,, = –j liao(r)z(ro) Cst[.r,y).
obtained from Eqs. (34a) and (34b) and the definitions of Eq. (36).
Forming the differential of the logarithm of this ratio at z = h and
y = Iwo and equating the results, one gets, with the aid of the Wronskian
relation, Eq. (9c),
(- AY’(r)
——-j
1 + [jY’(r)]’
+
{
Y’(r)
~x + 1 + [jY’(r)]’
Ak
k
} -)
Cl(r)
1 + [jY’(r)]’ 2
a(r) = [J,(=) – jY’(r-).J,(kr)]2 mkr”
The reason for the introduction of a coefficient a(r) lies in the fact that
a(r)/a(rO) approaches unity as kr and kr-~become sufficiently large. In
this far region Eq. (59) becomes asymptotically identical ~vith the corre-
sponding uniform-line relation [Eq. (23a)]. It should be pointed out that
I{;q. (59) can Iw eml)lt)ved tl~conlpllt.e the (“hiill~l~ill relativfi input aclmit-
SEC.8.8] APPLIC~ TIONS 271
1 Az’(ro)
Y’(ro) = m, AY’(rO) = A — —
() 2’(7.) = – [Z’(ro)]2’
and therefore Eq. (59) reduces to
(61)
where the first form pertains to the lumped-constant circuit and the
second to the extended structure. The fact that in the second case the
circuit parameters may depend on the angular frequency has been
explicitly emphasized in the expressions for the total conductance gLand
total susceptance B,.
A resonant frequency of an electromagnetic system is defined as a
frequency at which the average electric and magnetic energies within the
system are equal. Since by an energy theorem (cj. Chap. 5) the total
susceptance of an electromagnetic system is proportional to the dif-
ference between the average electric and magnetic energies stored in
the system, it follows that an angular frequency w of resonance is
identical with the frequency for which
Since the total susceptance may vanish at more than one frequency,
Eq. (62) determines, in general, a series of resonant frequencies.
The Q of a resonant electromagnetic system is defined as the product,
at resonance, of the angular frequency Q and the ratio of the total energy
stored in the system to the power dissipated or otherwise coupled out of
the system. The total energy stored within an electromagnetic system
can be expressed in terms of the frequency derivative of the total sus-
ceptance and the rms voltage V associated with the equivalent circuit
describing the system. From the energy theorem discussed in Chap. 5
this expression for the total energy is
K’+iw’=%”
SEC.8.9] A COAXIAL CAVITY 273
(63)
CIj13 o rl
Cross-sectionalview
Yz
z= &r,s 0.020” k
~=kr2=0.115”k
“’mm”=”
rl rz
Equivalent network
FIG.8.13.—A coaxial cavity.
shown in the figure the equivalent electrical network for this structure is
an E-type transmission line of length TZ— rl and propagation constant
k = 27r/A = u/c with infinite-admittance terminations at TI and Tz.
With the choice of rl as the reference radius, the total admittance at
r, is seen to be the sum of the infinite admittance of the termination
plus the input admittance of a short-circuited E-type radial line of
electrical length y – Z. By Eqs. (41) and (63) the resonant wave
nl.zmber k (or angular frequency ~0) is therefore determined from the
274 RADIAL TRA.VSMISSION LINES [SEC.8.10
resonance conditiou
Ctj(z,y) = —~. (64)
The resonant wavelength is thus about 3 per cent greater than that of a
corresponding uniform cavity of equal length. The remaining resonant
parameters Q and g are, for this nondissipative case, infinite and zero,
respectively.
8.10. Capacitively Loaded Cavity. -As a second illustration let us
consider the calculation of the resonant frequency of a loaded cylindrical
Cross-sectional view
Z= kr, = 0.020”k
y=kr2=0.115”k
b=0.W7°
b’ =0.042 “ &
o ‘L Tz
Equivalent network
FIG.8.14.=A capacitively loaded cavity.
cavity oscillating in the lowest angularly symmetric E-type mode. As
indicated in Fig. 8“14, the equivalent network that describes the fields
within such a structure consists of a junction of an open- and a short-
circuited E-type radial transmission line of unequal characteristic admit-
tances YO(r) and ~o(r-) but identical propagation constants k = 2~/h and
a junction admittance of value jlll. The network parameters are
obtained from Eq. (30) and the Waveguide Handbook as
Y,(r) = b’
(65a)
Y{(r) T’
B, _ 2kb’
— in ~, (65b)
Y~(r,) – m
where in is the logarithm to the base e. Equation (65b) is an approxi-
mation that is valid to within a few per cent for the cavity shape under
SEC.8.11] CHA.VGE IN HEIGHT 275
MC)
—_ _ B,
~ + ~ – Ct(x,v). (66)
V,(?-,)
The resonant frequency of the lowest E-type mode is obtained from that
value ko for which the total suscept ante vanishes. With the insertion of
the numerical values into Eq. (66) the. vanishing of the total susceptance
leads to the transcendental resonance condition
which can be solved graphically for lco with the aid of Fig. 8“8a. The
left-hand side of Eq. (67) is expressed as a function of
y, – Xo = ko(r, – TJ = o.095ko,
mlgIIJ l)l~z
Cross-sectional view
,,,0,
?j = 0.115”
Y; Y: b?
0
b’ :0.042 “
jBl jB2 b“= 0.050 “
m
A ‘1 “ r3
Equivalent network
FI~, 8.15.—Loaded coaxial cavity with a change in height.
joined at the radii r, and r~, with the first and last lines being open- and
short-circuited, respectively. The junction admittances jl?, and jllz at rl
and rz are capacitances whose susceptance values may be determined to
within a few per cent from the Waueguide Handbook. Higher-mode
interaction effects between the discontinuities at TI and rz are assumed
to be of negligible importance. For the cavity indicated in Fig. 8.15,
the values of the network parameters are
_ ~11
Y:(r) Y,(r) _ b’ _ 0.042,
~ – ~ = 1.19, (68a)
Y;(r) – T b
B,
–YO(7-*) = %%w$(a+’)
7r a ‘$ ‘68’)
B,
— = 0.016,
Y~(r2)
and BI is determined from Eq. (65b) which incidentally is a limiting form
of Eq. (68b) for a <<1. To determine the gap height b, it is first necessary
to compute the total susceptance at some reference point, say r = rl + O.
The total susceptance at this reference point is the sum of the suscept-
ances looking in the directions of increasing and decreasing radius. The
Susceptance in the direction d increasing radi{w is computed by a st,epwi~e
SEC.812] OSCILLATOR COUPLED TO WA VEGUIDE 277
B, Y:(r,)
— ct(z,,z,) = 0.016 – 1.334 = – 1.32
Y,(r,) – Y{(?-,)
with the aid of Eqs. (68) and Fig. 8.8a. It is noted that the change in
height at r, has a relatively minor effect in terms of the junction sus-
ceptance introduced thereby but a major effect in terms of the change in
characteristic admittance. With this knowledge of the relative sus-
ceptance at xi, the relative susceptance at z 1 + O in the direction of
increasing radius may be calculated with the aid of the radial-trans-
mission-line relation [Eq. (37)] as
2 _ 1 – 1.32 ct(z,,x,)~(zl,zz)
(69a)
Y~(rl) – ct(x,,~z) + (1.32) ((z,,z2)’
i 1 – (1.32)(4.04)(0.872) = _ ~ 45
(69b)
~= 1.36 + (1.32)(0.872) “ “
(70a)
B
— = 0.341 In ‘~ + O%. (70b)
Y~(r,)
The resonance condition [Eq. (62)] that the total susceptance ~ + ~
vanish then leads to
0.341 in% + ‘= -1.45 = O,
drical cavltles we shall consider the case of the radial cavity of Fig.
8.16 oscillating in the lowest angularly symmetric E-type mode and
coupled to a matched rectangular waveguide. Such structures, fre-
quently encountered in high-frequency oscillator tubes, are excited by an
electronic beam along the symmetry axis. The case illustrated in Fig.
I A
or, + r2
Y. Y; Yg
I A Ag “
11
:IB, - jB
I
O 7-1 ’72
FIG.8.16.—Cavityand equivalentcircuitof an osillatorcoupledto waveguide.
8.16 resembles closely the cavity of the Neher tube designed for operation
in the l-cm wavelength range. The calculation of the over-all electrical
characteristics of such an electronic-electromagnetic system requires a
knowledge of the interaction of the electronic beam and the electromag-
netic field as summed LIPin the expression for the electronic admittance
at some reference plane. This electronic problem has been considered
elsewhere in the Radiation I,aboratory Series’ and \vill be omitted in
tile following discussion, since the modification thereby introduced is
taken into account simply by inclusion of the electronic admittance in
the expression for the total admittance.
The computations as always are based on the network equivalent of
the oscillating system. As shown in Fig. 8.16, the radial-transmission-
line description of the system to the left of the radius ra is identical with
that employed in Sec. 8.10, and hence the values of the associated circuit
parameters need not be indicated again. The only additional values
necessary are those associated with the coupling network at ra and with
1Klystrons and Aficrowaue Triodes, Vol. 7.
SEC. 812] OSCILLATOR COUPLED TO WA VEGUZDE 279
(71a)
‘g =Jti”
(72)
which merely expresses the fact that the total admittance is a function
both of the frequency and the terminating impedance at r = rz. Equa-
tion (72) provides the basis for the computation of the frequency shift
caused by the perturbation AZ;. The imposition of the equilibrium
condition [Eq. (62)] that AB~ = O leads to the desired vallle for Ak/k.
280 RADIAL TRANSMISSION LINES [SW. 812
= j2.78.
.—
The corresponding frequency derivative of }“’ may be complited in a
~aFf
= j(O.098)kG = jl.39,
dk
I,ikewise from Eq, (60) the ~vwiation of the total admittance with the
o(lt,put impedance at the unperturbed freq~lency kOis
or
1
= (1.41 + j21.8)10-’,
‘; = 2.97 – j45.9
the numerical values being obtained by use of Eqs. (71a) and (71 b). The
frequency perturbation necessary to maintain the equilibrium condition
of vanishing total susceptance is obtained by setting the imaginary part
of Eq. (72) equal to zero
The actual resonant frequency of the coupled system is thus 0.705 per
cent less than the unperturbed resonant frequency, and the smallness of
this frequent y shift justifies the perturbation method of calculation.
The remaining characteristics of the system can be computed with
the aid of the results obtained above. The resonant relative conduct-
ance, for example, is the real part of the change in the total admittance
at r = rl due to the coupled perturbation AZ; and is given by the real
part of Eq. (72) as
or with the insertion of the numerical values from Eqs. (73) and (75)
x (4.17) = 1100.
Q = 2(1.9:)10-3
T-JUNCTIONS
If a waveguide junction has three arms, it will be designated as a
T-junction. Such a junction is completely characterized by a matrix
of the third order containing six independent elements.
9.1. General Theorems about T-junctions. -Three fundamental
statements that are simple and useful may be made about a T-junction.
By the arguments of Chap. 5, the behavior of the T is identical with the
behavior of an equivalent circuit. Circuits that contain the required
number of independent parameters and are therefore suitable are dis-
cussed in Sec. 4.11. From the properties of these circuits the general
theorems can be proved.
!!’lteorem I.—Tt is always possible to place a short circuit in one arm
of a l’-junction in such a position that there is no transmission of power
}wt\veen the othpr t\vo arms. The prrmf is simple. In Sec. .4.I2 the
283
284 WA VEGUIDE JUNCTIONS WITH SEVERAL ARMS [SEC.91
I 1
21, –
z= ’11– 233+ 23 23
233 + (1)
z:, z;,
’12 – 233 + z, ’22 – 233 + z,
The plunger stops all power transmission if the mutual element of this
matrix vanishes, or if
(2)
Since 23 can take any value from – m to + m, this equation can always
be satisfied, and the theorem is proved. The input impedances of arms
(1) and (2) under these conditions are given by
z12z13
2::1 = Z,* – ~,=,
(3)
z,’z,,
Zg) = z’, – 223 .
I
Since 23 may have any value either positive or negative, Z, may also
have any value, and in particular Zo may be unity. This demonstrates
t,he second theorem.
l’he theorem may also be proved from the equivalent circuits. For a
symmetrical T the circuits of Fig. 4.35 reduce to those shown in Fig.
9.1. Perfect transmission from terminals (1) to (2) results if the proper
impedance is placed on arm (3) to resonate the impedance Z. The cir-
(3)
/3 la
X. 1:)1
o S12 S’13
(1 ,2) element,
S1lST, + S12S;2 + S1JS:2 = o
or
S,3S;3 = o. (6)
Thus either Sl~ or S’Z, mllst be zero, Ho\rever, neither AS’13 nor S?, can
vanish if the diagonal elements of S % are to be unity. Th~ls the diagonal
elements of S may not all vanish.
It can also be seen, by the llse of similar arguments, that any tivo of
the diagonal elements of S can vanish only if the third arm of the junction
is completely decoupled from the junction.
286 W“AVEGUIDE JUNCTIONS WTITH SEVERAL ARMS [SEC.92
ak
i: -
—
(k-l) (~-.2)
Vk’ t
The negative signs result from the convention that positive currents flow
into the network at the positive terminal of each pair. The network N
is described by the set of equations
vl=Z1lil+...+Z1~i,, +. ...
. . .
(8)
v~=Zl~il+...+Z~Li~ +. . . .
. . .
1
(9)
(lo)
(11)
(a) (’)
. lG. 9.3.— A seriesjunction. (a) Equivalentcircuitof an E-planeT-junrtion; (b)mOSS-
sectlonal viewof an E-planeT-junction,
of all forms. .$s the frequency increases, the properties of the junction
become more complicated and depend upon the shape of the junction
and the kind of transmission line. Figure 9.3 illustrates diagrammatically
a series junction and defines the convention of positive directions of the
voltages and the currents flowing into the junction. The impedance
and admittance matrices of this system both contain infinite elements
and are conseqllently of little use. The linear equations that relate the
voltages and currents are
u~ + v~ + tj~= o,
(1:3)
il = i2 = i3. }
A rectangular waveguide, operated in the dominant mode, that has a
branch waveguide joining it on the broad face behaves, at long wave-
lengths, as a pure series j~mction of three }~a~-eguides, Such a junction
is called an E-plane T-junction, since the change in structllre at the
branch occurs in the plane of the ele(:tric field, Figllre {).31) sholvs a
cross section of a Tva].cguid(, \\”it
h :Lu ~!-plane branch. The e(l~li~alen(,
ill Fig. !),?w. The proper choicr
circuit of this E-pl:me jllntli(m is ,SIIOII-U
of the characteristic impe(lim(c of the tlIIWI linw is in~mediately ul)vio(ls.
At long lvavelengths tlw integral of the electric field ttiken around the
SEC.9.3] THE E-PLANE T AT LONG WAVELENGTHS 289
Thus if the voltages are taken proportional to the heights of the wave-
guides, Eq. (14) is equivalent to the first of Eqs. (12). Since the currents
in the three lines are equal, the characteristic impedance of each line
must be chosen proportional to b. In Fig. 9.3a the characteristic imped-
ances are denoted by Zl, Zz, Zs.
Although the impedance matrix of this system does not exist, the
scattering matrix does. By the application of elementary circuit theory, .
the scatt~ring matrix S may be shown to be
B=x’+lnal (16)
“ ‘:{l+lnal (17)
(19)
Since b/h. <<1, the difference between Eqs. (18) and (19) may be shown
to be
‘h- “=%42-$
‘joo’% (20)
Thus the two methods of calculation give the same results to a good
approximation.
This comparison is a critical test of the reliabilityy of both formulas,
since the short circuits are placed close to the junction (b << A.) and
consequently interaction effects are to be expected.
It is often convenient to transform the equivalent circuit of Fig.
SEC.9.4] E-PLANE T AT HIGH FREQUENCIES 291
9-4 to a new reference plane in the branch arm. Since the impedance
matrix is infinite, the general form of the transformation given in Sec.
9“2 cannot be used. It is easy, however, to proceed from first principles.
The junction equations are
01+ VZ+tf3 =0,
(21)
il = iz = is — jBvs. )
If a II-network is connected to arm (3), the new currents and voltages are
given by
ii = .Y1lV:— y12V3,
i: = y12V3— yZZV&
(23)
For this length, Eqs. (22) reduce to Eqs. (13). Since 1 is negative, the
reference plane is within the junction as shown in Fig. 9.6.
lent to it, the circuit becomes that shown in Fig. 9.7b. The impedances
are related to the atilttances of Fig. 9“7a by the equations
(24)
y13
z= =
y12y33– y?3’ \
Y12 – y13
Zd =
y12y33– y;3/
As b’ becomes small compared with b, zb and zd approach zero, Z-
becomes very large, and the only remaining element is the shunt capaci-
tance Zc. The circuit elements are shown as inductances or capacitances
o L
(a)
1 0 ‘U-L& (b)
FIG,9.7.—Exactequivalentcircuitsfor an E-planeT-junction at highfrequencies.
in Fig. 9.7 according to the sign of the admittances when b! = b. Typical
experimental values, for b/A, = b’/k, = 0.227, are
z. = –jlo.4,
z, = jo.50,
(25)
Zc = –j4.85j
z, = –jo.57. 1
1,=:
= 0.014, $ = 0.028,
G (26)
Z = j(.01, nz = 0.;29, )
for b’ = b = 0.2A0.
If the three arms of the T-junction have equal heights and branch at
equal angles of 120°, the junction possesses a higher degree of symmetry
SEC.9.4] E-PLANE T AT HIGH FREQUENCIES 293
than the 90° junction, and the number of circuit parameters necessary
to describe the behavior is reduced to two. The impedance matrix is
of the form
Z1l Z12 Z,2
z = z,, z,, Z12 . (27)
[1Z12 Z12 z,,
A circuit that exhibits the same symmetry as this junction is shown in
Fig. 9.8. The elements Z, and Z, are related to the elements of Z by
the equations
z,, = Z,(2Z, + 3Z,) z:
z,, =
3(Z, + z,) ‘ 3(Z, + z,)’
(28)
z, = (z,,+ Z12)(Z11– 2Z12)
z, = Z1l + .Z12,
SZI, I
Another special case of considerable interest is the 180” E-plane
junction or the bifurcation of a wave-
guide in the E-plane. Such a junction is
shown schematically in Fig. 9.9a. If the
dividing wall has negligible thickness,
Z*
certain special properties are manifest.
The equivalent circuit of Fig. 9.7b maybe
drawn as in Fig. 9.9b for the reference
planes indicated in Fig. 9.9a. Since the
(1)
reference planes of the three arms coincide,
the voltage v{ = VI + V2. The T-net-
work in Fig. 9.9b must reduce to a shunt FIG.98.-Equivalent cirruit of a
120°E-planejunction.
element, and the equivalent circuit be-
comes that of Fig. 99~. Tf the heights of the smaller guides are not equal,
an additional element must be added to the circuit. This additional ele-
~ ~;; ~;;~
V1+U2+V3 =0,
?LIil+ ?32i2—n3i3 = o,
(29)
. . 1 n1n2 U1 V2
2,–22=2= ~–;21
( )( )1
when the positive directions for the currents and voltages are those
shown on the figure.
u
(3)
? f’
‘i3
i2 12
4 V* (2)
+
o 1 0
FIG. 9. 10.—Equivalent circuit of an ,?i’. plane
T-junction with a three-winding transformer.
The series impedance Z in Fig. 9.12) may also be placed in shunt with
the transformer. If the characteristic impedance of line (3) is chosen as
n2b’/b, the circuit reduces to that shown in Fig. 9.4 at the proper reference
planes in the three lines.
9.5. H-plane T-junctions. —,Junctions with three arms in which the
branching takes place in the H-plane maybe discussed in a similar fashion
to E-plane junctions. For very long wavelengths, the junction is a pure
(1)
E a
.-:.-
(a)
I
,2) ,*
(b)
FIG.9.1I.—H-planejunctionandequivalentcmcuitat long wavelengths.
shunt junction. The coupling from the main waveguide to the branch
guide is by means of the magnetic fields. If the branch arm is at 900
(2)
to the main line, the coupling will hr from th(~ I(mgitlldind mi~gnetic [iel(l
SEC.9.6] A COAXIAL-LINE T-JUNCTION 295
in the main line to the transverse magnetic field in the branch line. Since
these fields are in quadrature with respect to the transverse electric field,
the junction behaves as a pure shunt junction only if a quarter-wave-
length line is inserted between the branch line and the main line. Figure
9.1 la shows the junction, with the Zd
reference plane indicated. The
equivalent circuit is shown in Fig. Zb
(3)
9.1 lb. The proper ratio of imped-
ances of the main line and the +x 2= Zc
Za
quarter-wavelength line is the (I)
(2)
ratio of the transverse magnetic
field to the longitudinal magnetic
field. The characteristic imped- FIG.9.12.—Exact equivalent circuit for an H-
plane T-junction.
ance of the quarter-wavelength
line relative to that of lines (1) and (2) is therefore 2a/Xg. & the fre.
quency is increased, additional circuit elements must be employed to
express the effect of higher modes at the j unct ion, and the circuit becomes
that shown in Fig. 9.12. The values of the elements on the side arm will
depart from those of a quarter-wavelength line. Experimental values Of
These parameters for A = 3.20 cm and a = 0.902 in. are
z. = jo.17, z* = jo.19,
(30)
2. = –jl.04, z, = jl .00. )
The value of 2a/h0 is 1.002 for these conditions.
9.6. A Coaxial-line T-junction. —Stubs or T-junctions in coaxial line
have equivalent circuits that are similar to those of waveguide junctions.
5
-- -- --- - Ref. plane
k
0.058” ~ 1
2.145”
m!
Ref. plane Ref. plane
1:
is
where En is the electric field in guide (3) normal to the hole, H1 and Hm
are the tangential magnetic fields in the directions of the principal axes of
the hole, P is the electric polarizability, and Ml and Mz are the two
magnetic polarizabilities. The quantity S is a normalizing factor that
is equal to ~(Xa/A)ab for unit transverse magnetic field and for the
dominant mode in rectangular waveguide. It will be recalled that
this expression neglects the reaction of the load upon the matched
generator, which is small if B is large.
The power leaking into guides (1) and (2) is given by similar expres-
sions. If the amplitudes of the waves into guides (1) and (2) are denoted
by A ~ and A,, respectively, then
1J. R. Harrison, “ Design Considerationsfor Directional Couplers,” RL Report
No. 724, Dec. 31, 1945, Fig. 56.
SEC.g.~ THE T-JUNCTION WITH A SMALL HOLE 297
where the difference in sign results from the fact -that the waves A 1
and Az are traveling in opposite directions.
If a wave is incident in guide (1) of the junction, a somewhat different
situation arises. The incident field excites a field in the hole which
radiates waves out ward from all three arms of the T-junction. The
wave from arm (3) is given by
s,
‘3 = “R,”
The wave radiated away from the hole in arm (1) is a wave reflected from
the hole. The amplitude is
The wave radiated away from the hole in arm (2) produces only a small
change in phase in the incident wave. The amplitude is
Since 1Az~ and IB,I are small comparecl with unity, -ZIZ @ large and
2,, – z,,
B1=~~2M2=A2,
= 0:
()
298 WA VEGUIDE JUNCTIONS WITIi SEVERAL ARMS [SEC.9.8
Z33- Z3
z= (31)
211 Z,Z, Y, Z, Z3(Y, + Y2) Z,Z4Y,
ZIZ2Y2 z,, Z,Z,YI Z2Z4(Y1 + Y,)
ZIZ3(YI + Y2) Z,Z3YI 233 Z3Z4Y*
Z,Z,YI z2z4(Yl + Y2) z3z4Y2 z,, I
I
where Yl and Yt are the admitt antes of the crossed circuit elements.
SEC.9.9] DIRECTIONAL COUPLERS 299
From the form of the circuit or from the impedance matrix, certain
properties are evident. (1) With respect to a given arm, there is one
opposite arm and two adjacent arms. (2) The mutual impedance
between the opposite arms (1) and (3) can be made to vanish either by
making 21 or 23 equal to zero, a trivial case, or by making Y1 + Yz = O.
For this condition, however, the mutual impedance between arms (2)
and (4) also vanishes. (3) If the mutual impedance vanishes between
two adjacent arms, for example, (2) and (3), then Y, = O, and the
mutual impedance between the other pair of adjacent arms (1) and
(4) also vanishes if the trivial cases are neglected.
(2)
Z,, -z,
)
(1)
Y,
coupling between (1) and (3) and (2) and (4). The suitability of the
name can be seen from Fig. 9.17. A wave incident on the junction in
line (1) leaves by lines (2) and (4). A wave incident in line (2) leaves
1
Destructive
inWemxe(G~&
by lines (3) and (l). Thus the powers absorbed by matched loads on
arms (3) and (4) are indicative of the powers traversing the junction in
lines (1) and (2) in the two directions. The ratio of the power emerging
from line (4) to that incident in line (1) is called the coupling coefficient
of the directional coupler.
A directional coupler has several interesting properties. One of
these is that all the terminals are matched. The waves indicated by
1 dotted lines in Fig. 9.18 may be
reversed in time and combined
with the waves indicated by solid
and (3) as shown by the solid and dotted arrows. If the phase and
amplitude of one of these waves are adjusted with respect to those of the
other wave, the two waves in line (4) can be made to cancel each other.
A reversal of time causes a wave to enter line (2) and leave at lines (1)
and (3). This is the same situation which is indicated by the dotted
arrows in Fig. 9.17, and therefore the junction is a directional coupler.
9.10. The Scattering Matrix of a Directional Coupler.—The scattering
matrix of a directional coupler is of the form
o s,, ~o s,,
!1
s,, o [ s23 O
s = -- - ;- ------ . (32)
0 s,, ~O S34
s,, o ~S43 o
The Zero elements on the diagonal indicate that the junction is com-
pletely matched, and the remainder of the zero elements indicate that
there is zero coupling between arms (1) and (3) and also between arms
(2) and (4). The remainder of the elements of S are not completely
independent but must be such as to make the matrix symmetrical and
unitary. The conditions thus imposed are that
Sjk= Skj,
(33)
1s,21’
+ 1s,41’
= 1, }
and
s21s~3 + s41sfs = O,
(34)
s12sY4 + s32s$, = 0. }
ls,211s231 = 1s1411s341,
(35)
1S,,11S,,1 = 1s2311s341,
}
and hence
1s,21= 1s341,
(36)
1s,41= 1s231.
}
Equations (36) state that the coupling from arm (1) to arm (2) is equal
to that from arm (3) to arm (4) and also the coupling from arm (1) to
arm (4) is equal to that from arm (2) to arm (3). Thus a wave incident
in arm (1) couples the same fraction of its po~ver into arm (4) that a wave
in arm (2) couples into arm (3),
There is still a great deal of arbitrariness in the phases of the ele-
ments of S. This indeterminateness can be eliminated by the correct
choice of the locations of the reference planes, ;is an example, the loca-
tion of the terminal plane (2) may be chosen in such a ]vav that S,, is
real and positive, Simihidy’ the location of the phnr (It :11,111
(1) may lM
302 WA VEGUIDE JUNCTIONS WITH SEVERAL ARMS [SEC.9.10
chosen in such a way that ~14 is Positive imaginarY, and the location of
the plane in arm (3) may be chosen so that SU is positive real. From
Eqs. (36),
S12 = S34 = ~, (37)
.-.-.--1.
0. 0 jp
jfl O
s = 1‘f.....o.- (40)
I O
jfl
(X2+62=1.
If the first of Eqs. (42) is multiplied by /S21,the second by s,,, and the
difference is taken,
(,S;, – ,~s)sls = O. (43)
lIcu(.{, t,illler ,S:),, l:lllisllcs :IIIfl t}lc jllncti(,ll is :1 (Iireclit,mil folIpll,r :1s
.
already proved, or
S21 = S43 = a, (44)
Oa
[“
\S13 jp
If the first row of the matrix (46) is multiplied by the complex conjugate
of the second column, and if the first column is multiplied by the complex
conjugate of the fourth row, then because S is unitary,
–j@313 + 313STZ= O,
(47)
as,l + d’?j = O. 1
If neither a nor/3 vanishes, Eqs. (47) imply that both SS1and S,2 vanish.
This puts Eq. (46) into the same form as Eq. (40), and the junction is a
directional coupler. If either a or ~ van-
ishes, the junction is also a directional
coupler. Thus the result has been ob-
tained that any junction of four trans- ~JL, ,
mission lines which is completely I , ,3
matched is a directional coupler.
9.11. The Arbitrary Junction of Four ___
\
Transmission Lines.—It might be 2 ~ Plww
~r
thought that any junction of four trans-
mission lines could be completely
matched by a transformer in each of the FIG. 9.20.—ArbitrarY junction of
four transmission lines.
four transmission lines and therefore
could be made into a directional coupler. This cannot be done in gen-
eral, however, as will now be shown. 1
Let us consider the arbitrary junction of four transmission lines shown
in Fig. 9.20. If plungers are inserted in lines (2) and (3) as shown in the
figure, then for any position of the plunger in arm (2) there exists a position
of the plunger in line (3) such that no coupling exists between lines (1) and
(4). The junction with the plunger in line (2) at some definite position is
1This proof is due to R. L. Kyhl.
.
~{
+~+
/’/’
b tl~
~11
.$$. 1
3
-.
‘i [ =~: /2 t
FIG. 921.-Positions of nodes in arms of FIG. 9,22.—Linear combination of solutions.
four-j unction for two positions of the
plungers.
position of the plunger in line (3) which causes line (4) to be decoupled
from line (l), and there are pure standing Tvaves in lines (1), (2), and (3).
The nodes of the standing waves in line (2) must no]v be in a new position.
However, the nodes in lines (1) and (3) may or may not be in the same
position. In Fig. 9.21, the dotted lines marked a represent one position
of the nodes, those marked b represent the other position.
Since these solutions represent pure standing waves, they are charac-
terized by the fact that the electric fields are everywhere in phase or
180° out of phase, as shown in Chap. 5. If none of the positions a and b
coincide, a linear combination of
these two solutions, taken with dif-
ferent time phases, corresponds to
running and standing \vaves in the
lines (l), (2), and (S). only if the
nodes of the t~vo solutions coincide
in one of the guides will the linear
combination be a pure standing
wave in that guide. The linear com-
I~IG. 9.23. —Four-junrtion with matching
trail.forl),ers.
bination of solutions is shown in Fig.
922. Since the waves in the guides
are not pure standing waves, the amplitude of a wave running one way
is different from that running the other way. This is indicated in Fig.
9.22 by the arrows of different length. It will be assumed that one long
arrow points into the net\\-orkand t\vo point out. If the converse condi-
tion were obtained, a time reversal would lead to the desired condition.
By the addition of mat thing transformers to the lines (1), (2), and (3)
SEC. 9.11] JUNCTION OF FOUR LINES 305
the conditions of Fig. 9.23 are obtained. The transformers set up the
standing-wave pattern of Fig. 9.22.
From a consideration of the terminals located on the transmission-line
side of the transformers in Fig. 9“23, it is seen that the new modified
junction is matched looking into line (2). Moreover, there is no coupling
between lines (2) and (4). The terminals of line (4) can now be matched
by the inclusion of a transformer in line (4). Thus both lines (2) and (4)
are matched, and there is no coupling between them. Hence, by an
earlier theorem, the junction is a directional coupler. Conversely, the
junction may be considered as a perfect directional coupler with trans-
formers that mismatch it in the four transmission lines.
Four transformers are actually not required; it is easily seen that three
are sufficient. If the linear combination of the two standing-wave solu-
1
F1~. 9.24.—Behavior of a. degenerate four- FIG. 9 25.—Junction of two T’s connected
junction. together.
tions is taken correctly, one of the lines, line (1), for example, can be made
to contain a pure running wave. Thus the transformer on line (1) is not
essential, and three transformers are sufficient to match the junction of
four transmission lines.
It should be remembered that the foregoing derivation hinges upon
the assumption that none of the nodes a and b in Fig. 9“21 coincide.
Since the plunger in line (2) is moved to a new position’ in going from
case a to case b, it is clear that the resulting nodes also move. Hence if
the nodes do coincide in one of the transmission lines, it will have to be
either in line (1) or in line (3). If the nodes a and b of Fig. 9.21 coincide
in line (3) and nowhere else, a linear combination of the two solutions
may be taken such that the waves are completely canceled in line (3).
The combination must be one with equal time phases. The resulting
linear combination is one with standing waves in lines (1) and (2) only,
as in Fig. 9.24. Figure 9.25 is an example of a junction in which standing
waves are set up in lines (1) and (2) by a plunger in arm (2). It is to be
306 HrA VEGUIDE Ju2vCT101VS WITH SEVERAL A Riws [SEC.912
noted that the junction of Fig. 9.25 is composed of two T-junctions con-
nected together by a transmission line c. The plunger in arm (2) is in
such a position as to decouple c
II
4 from arm (1).
It may turn out that the nodal
planes a and b coincide in both
~ lines (1) and (3). This is possible
1 3 only if the waves are completely
absent from line (2). This im-
plies,. however, that there are pure
2 standing waves in lines (1) and (3)
‘Y? with no waves in the other lines.
FIG. 9.26.—Behaviorof a degeneratefour-
junction. This condition is indicated in Fig
9.26.
The procedures just described could be repeated for a wave introduced
into line (4). A standing-wave solution analogous to Fig. 9.24 or 9.26
would be obtained. In each case the network is equivalent to a junction
of the type shown in Fig. 9.25 (the
lines need not be numbered in this 4
order) provided all degenerate
forms of Fig. 9.25, such as those \\
shown in Figs. 9.27 and 9.28, are
included.
~~
To recapitulate, any arbitrary 3
junction of four transmission lines l-l
can be represented either as a di- ll
rectional coupler with transform-
ers in three of the lines or as a
junction consisting of two inter- FXG. 9
II
.27.—Equivalent
2
form of a degenenite
four-junction.
connected T-junctions.
9.12. The Magic T.—A matched directional coupler with a coupling
coefficient of ~ has proved to be an extremely useful device for many
microwave applications and has become known as a magic T. The low-
frequency analogue of a magic T is the well-known hybrid coil used in
telephone repeater circuits. Such a device is indicated in Fig. 9.29.
Among the many applications of the magic T, the more important
ones are impedance bridges, 1 balanced mixers, z balanced duplexers, s
and microwave discriminators. 4
A magic T can be realized in any one of a number of forms, in wave-
guides or in coaxial lines. One of the simplest of these forms is the com-
bination of an E-plane and an H-plane T-junction which is shown in
Fig. 930. A wave incident on the junction in arm (4) has even symmetry
about the symmetry plane, and the
transmitted power is divided with
4
even symmetry between arms (1)
and (2). No power is coupled to
arm (3), since no mode that has
7bevensy3~tein
F1o. 9.2&-Another form of a degenerate FIG.9.29.—Circuitof a
four-junction. hybridcoil.
arm (3). There is also a reflected wave in arm (4). The reflected wave
can be matched out, however, by adding to the junction some post or iris
that does not destroy the-symmetry
of the junction. A wave incident
in arm (3) possesses odd symmetry
and therefore excites fields in arms
(1) and (2) which have odd sym-
Arm
metry. No power is transmitted to arm (4), since arm (4) will not support
a mode with odd symmetry. To eliminate the reflected wave in arm (3),
a second matching device must be added to the junction. Figure 931
shows one method for mat thing the junction. The dimensions are given
for a wavelength of 3.33 cm in ~- by l-in. waveguide. The post is 0.125
in. in diameter, and the iris -& in. thick.
308 WA VEGUZDE JUNCTIONS WITH SEVERAL ARMS [%c. 9.13
o 0 11
I
-1 1
z=~.o ~2 ~......._.;. J (48)
00
1 1 00
-“”-””-”-----!
for the proper choice of reference planes. It maybe shown that
Y=–z=–s,
this remaining element is also absent. The turn ratios of the trans-
formers may also be adjusted if corresponding changes are made in the
line impedances.
A ring circuit may beeasily realized inwaveguide by combining four
E-plane T-junctions as shown in Fig. 9.35 for a symmetrical case. If
(3)
/4 12
(1) 1 (2)
b
1
FIG.9,34.—Aringcircuitmadeup of four three-arm junctions.
(3) b 11 (4)
‘b
b’ b’
3P “
(1) b b“ (2)
the distances between the reference planes are so chosen that each T-junc-
tion is a pure series junction and that each is separated by &/4 from the
adjacent junctions, the device becomes a directional coupler of a common
form, sometimes called a double-stub coupler, If b’ <<b, the reference
planes may be chosen as shown in Fig. 9.6. The directional collpler is
matched it’ b“ has the value
c=—
()F
b,–2”
1+
() ~
If ’b’ is very small compared with b, b“ mav be made eq{lal o h and the
310 WA VEG~JIDE JUNCTIONS WITH SEVERAL A RMs [SEC. 9.13
(3)
u (1)
(! 1,
I:II.. 9 37u---.inlagir T in roaxid Ii]w,
FIG. 937tJ--.i magir T n!ade frozn E-plane T-junrtlons.
‘ B .>. 14ipp1na]ln.“ ‘~hv ‘~limry of Dlrrcti[)]}:!]( ‘ollpirr~,” liI. I{rp{]rt 1-o. 8(}0,
1)1(. 28, 1!)+5.
SEC, 9,14] FOUR-J UNCTIONS 14’ITH SMALL HOLES 311
_l_ —x%———+
~
M2
3 Ag
L2
(3) (4)
M
KL~
Z. Z.
‘% ~r~ d&
J-
1
KLMm
(1) E (2)
1
Zo=l
K 2L2 ,M2
I;lG. 9.35.—,4 getmlal form of directional FIG. 9.39,—A magic‘r CO,,IPOW,lor
coupler in which each line has a different E-planewavegtidejunction~.
characteristic impedance.
L2
—— ~& ~
K2L2
(3)
Lfi~ A
‘%~— KL/M %
c/M
‘Ilzx ~ (2)
—3A _
Zo=l 9? ~
1:1~,9.40,- -Iralues of impedances ncccssary to make the junction of Fig. 9.39 a matched
directional coupler.
of Fig. 934. The line lengths 1*and 14are effectively zero, and 11and 13
are one-quarter Ivavelength. The two ~va~eguides are coupled by the
longitudinal magnetic fields, !~hich excite magnetic dipoles within the
holes. If the holes are small, the dipoles are of equal strength and 90°
out of phase. The dipoles radiate into the second waveguide; but
because of spacin~ bet~veen the holes, the radiation is reinforced in the
direction of the original v’ave in the first fvaveguide and is canceled in
the opposite dirwtion. Thlls one arm of the junction (that containing
the termination in the figLlre) is decoupled from the arm in which the
po~ver is incident. The amount of radiation from the holes may be com-
plltetl by the formldas given in Sec. !),7, The interactions between the
holes arc, of course, rwglectwl in the formulas, lvhereas in practical
312 WA VEGUIDE J{lNCTIONS WITH Sh’V8RA L A Rhf8 ISW. 9 ! I
directional couplers such as the one shown in the figure, the interactions
may not be negligible.
It is also possible to couple from one waveguide to another by two
holes so that the wave in the auxiliary line travels in the opposite direction
to that in the main line. Such a device is shown in Fig. 9“42. Direc-
tional couplers of this type are called reverse couplers. The two coupling
holes are located on opposite sides of the center line, on the broad side
of one waveguide and on the narrow side of the other. The longitudinal
9u
..-
../:-” ““-
Q
FIG,9.42. —Schwingcr reverse-muphng d,rertional coupler.
The
magnetic fields that excite the holes are in the opposite directions.
holes are spaced along the Traveguide by a qluwter of a \va\’elength.
and therefore radiation from tme htde is reinformd by rtitlititi(m from t}]e
other for the uxivc traveling in the backwnrd dirwtion. The coupling
SEC. 9.15] DEGENERATE FOUR-J UNCTIO.VS 313
coefficient for this coupler also may be calculated from the small-hole
formulas if the proper relative values of the fields are inserted.
Another directional coupler, which operates on a different principle,
is shown in Fig. 9.43. This device is known as a Bethe-hole coupler.
Since the hole is in the center of the broad face of the waveguide, it is
excited both by the normal electric field and by the transverse magnetic
field in the waveguide. Both an electric dipole and a magnetic dipole
are produced in the hole, and both dipoles radiate in both directions into
the second waveguide. The electric coupling is an even coupling about
the axis of the hole, wheras the magnetic coupling produces fields with
odd symmetry. The strength of the magnetic coupling can be adjusted,
therefore, to be equal to the electric
coupling, by the rotation of one
waveguide with respect to the other.
The phases of the fields are such
that if power is incident on the cou-
To detector Mainline
pier in the lower waveguide from FIG.9.43.—Betheholecoupler,
the left side of the figure, the power
coupled into the upper waveguide proceeds in the direction of the arrow
labeled “To detector. ”
Design formulas for Bethe-hole directional couplers and for other
types as well are given in Chap. 14 of Vol. 11 of this series.
9.15. Degenerate Four-junctions. -In Sec. 9 ~11 it was shown that a
degenerate form of a four-junction that did not have the properties of a
directional coupler consisted of two three-junctions connected together
as in Fig. 9.25. To find the conditions on the elements of the impedance
mat rix that must be satisfied in order that the junct ion be degenerate,
it is most convenient to find the impedance matrix for two T-j unct ions
connected together. If the terminals of one T-junction are designated
by (l), (2), and (3) and those of the other T-junction by (4), (5), and (6),
and if terminals (3) and (4) are connected together, then
V3 = V4, is = —ii,
These relations may be used to eliminate us, vi, ia. and i, from the network
equations. The result is the impedance matrix
Z,l-Z*4, Z12.ZS4
Z,3Z45 Z,3Z46
Z33+Z4, Z33+Z44
Z;3 Z23Z4, Z23Z46
...
“’-Z 33+Z44 Z33+Z44 .Z33+Z44
z=
z;, Z45Z46
z55– Z33+Z44
—
‘56– Z33+Z44
Z6,– L
Z33+Z44
314 JVAVEGUIDE JUNCTIONS WITH SEVERAL ARMS [SEC. 9.15
Y’=;, +l,
z,
and similarly
y“ = & + ;3.
z 14-Z,2
o
J“ >
Z*l- Z14 Z12
II ~ 0
FIG. 946.-Second form of equivalent circuit of Fig. 9.45.
v’ = Zlli’ + Z12i”,
v“ = Z21i’ + Z22i”,
and
“= [::)’‘“= [::1:
‘ia
minal pairs (3) and (4), together with series generators ei, The network
equation then becomes
e’ = (Zll + Zl)i’ + Z12i”,
e “ = Zz,i’ + (Z22 + Z2)i”,
where
i’ = *( Z’)–le’, for e“ = O,
i“ = *( Z’’)–le”, for e’ = O.
From this equation, it is evident that if terminals (1) and (2) are to be
decoupled and terminals (3) and (4) are also to be decoupled, Z’ and
Z“ must he diagonal matrices,
(1
21
0
0
z, ‘
z!! =
(1
23
o
()
z,
el e2
i, = ~zi, for e“ = O,
‘2 = Zz’
e3 ed
i, = 2X for e“ = O.
“=x
Lippmann’ has extended the development of the theory to the treat-
ment of chains of four-terminal-pair networks. The method runs
parallel to the corresponding treatment of chains of four-terminal
networks. An image transfer constant and a propagation constant
may be defined, and the generalization is essentially complete. A
some\vhat similar problem arises in the consideration of a chain of
cavities connected in a ring that forms the resonator system of a strapped
magnet ron. ?
power is negligible compared with the reflected power from the antenna
itself. As an example, the particular case of an antenna consisting of
the open end of a rectangular waveguide may be considered. At the
end of the waveguide the fields are distorted to satisfy the boundary
conditions, and both electric and magnetic energy is stored near the end
of the pipe. This stored energy produces effects equivalent to those
a’E _ ~
(49)
~2 E _ ~2
at2 “
By VI is meant the vector operator
~?=vv. –Vxvx. (50)
In terms of cartesian components, ~q. (49) reduces to the three scalar
equations
1
For any sollltion of lb]. (54), ttwre is a vtwtor solution of Eq. (53) given
by
L = VIL. (!55)
It ~l~tnd(l lNI tl[)ted that E(l. (52) is not satisfiwl I)y L. Ho\~ever, L
ciin IW used to (’onstruct solenoidal vector fields. I.et R be a radills
320 11’AV,LGUIDB JliN6’TIONS l}’l TH Sh’VEKAL ARMS [SIX,. !)18
M=Rx L (56a)
and
N=vx(Rx L) (56b)
are solutions of Eqs. (52) and (53). It should be noted that M is normal
to R. Because of the occurrence of R in Eq. (56), the coordinate system
in which this equation has the most simple form is the system of spherical
coordinates shown in Fig. 9.50.
Scalar Wal~e Equation.—In the spherical coordinate system Eq. (54)
becomes
1 a 1
113W
()
du
;2 ~r rz— +r~FO smO~ +— ~“ + lc’u = o. (57)
Eh () rz sinz 0 a+z
A set of single-valued solutions to this equation is
(59)
for p # m and/or q # n.
The behavior of the various cylinder functions for large values of kr
will shed some light on the physical meaning of Eq. (58). It is easily
(62)
These are solenoidal solutions of the vector wave equation [Eq. (53)].
It is to be noticed that M~~ is normal to R. For this reasod, a solution
for which M~~ is the electric or magnetic field is called transverse-
electric or transverse-magnetic respectively. It should be noted akw
that if the electric or magnetic field is M~fi, the corresponding magnetic
or electric field is of the form N~~.
It may be demonstrated that the set of waves of Eqs. (63) and
(64) forms a complete set of solutions of Maxwell’s equations which, for
empty space, vanish at infinity.
In addition to this completeness property, the set of solutions given in
Eqs. (63) and (64) have the important property of orthogonality. This is
322 WA VEGUZDE JUNCTIONS WITH SEVERAL ARMS [SEC.9.19
the same orthogonality property which was invoked in the case of a wave-
guide with several propagating modes. Because of the orthogonahty of
the functions in Eqs. (63) and (64), the stored electric or magnetic
energy inside a simply connected charge-free region is equal to the sum
of the energies computed for the various modes. Moreover, the power
entering or leaving the bounding surface is equal to the sum of the
powers computed for each of the spherical waves.
9.19. Solutions of the Vector Wave Equations.—Since the magnetic
field may always be determined from the electric field, it is necessary
to consider only the electric field in a description of the electromagnetic
field in a charge-free space. It is convenient to use Hankel functions in
the description of the field. Let us consider the description of one of
the spherical waves of the set of Eqs. (63) and (64),
M .s = R X VU... (65)
The solutions M’ are even in +, and the solutions M“ are odd in ~. The
function Z~+}5 will be assumed to be a linear combination of the two
Hankel functions,
Z.+}j = allfl+}fifl) + bHn+!j(2). (68a)
In c~se the region includes the origin, the field at the origin must be
finite, Z~+w must be a Bessel function, and
a=b. (6Sb)
TABLE9.1.—REPLACEMENT
OF DOUBLESUBSCIUPTS
BY SXNGLE
SUBSCRIPTS
2 M:,
3 M;, Electric dipole
4 M:, )
5 N;,
6 N;, Magnetic dipole
7 N:,
J
The subscript 1 will refer to the terminals in the transmission line that
excites the antenna. As in Chap. 5, column vectors will be introduced
to represent the incident and scattered waves. Let
!2]‘ b,’
I
a= ? b= (69)
am bm
Sa = b. (70)
But ~6*b is the total power scattered by the antenna, and it must
equal the total incident power @*a. Therefore
3*S = 1, (73)
Dr
$. = s-la (74)
Thus S is a unitary matrix.
To show that S is symmetrical, it should be noticed that for any
solution of Maxwell’s equations for zero loss there is another one with
time reversed. Forexample, if
M~n =
R X V ~lG
P~”}(COS8) COSmd — [a&H(’)( k r) + E@’)(b)] ,
{ ))
S=s (80)
scattering matrix has its first row and column omitted. The relation
Eq. (68b)requires that theremainder of thescattering matrix bejust the
identity matrix
( \
....
\lo o....
~o lo.... .
so = (81)
\oo l....
: . -- .. -- ..
: -- -- -- .. .-
\ /
9.21. Scattering Matrix of a Simple Electric Dipole.—Assume that an
electric dipole radiates the mode M~l, and that it is matched to its
transmission line. It may be assumed also that the antenna does not
affect any of the other modes. The scattering matrix is
‘o eia 00 O....
@l 0000 .. ..
s= o 0 100 .. .. (82)
0 0 010 .. ..
0 0 001 .. ..
.. . . ..
It should be noted that the assumption that the antenna is matched
and radiates only mode (2) determines the first column of matrix (82).
The second column follows from the symmetry and unitary property of
S. The first two rows are fixed by the symmetry of S. The remainder
of the matrix is fixed by the assumption that the antenna does not disturb
the other radiation modes. It has the same form as Eq. (81).
Such a dipole antenna will absorb power only from the one dipole
mode that it radiates. It also absorbs all the power incident on the
antenna in this mode. If a plane wave falls on the dipole antenna, the
antenna absorbs the one dipole mode and leaves the rest of the wave
undisturbed. The absorption of the dipole mode may be described as a
dipole wave radiating out from the antenna and having the right ampli-
tude and phase to cancel the dipole component in the plane wave. This
negative wave will be called the scattered wave. Scattering of this type
may be described by the matrix
s’=s–
‘o=k----:l: (83)
326 WA VEGUIDE JUNCTIONS WITH SEVERAL ARMS [SEC.922
o j s,, .. ..
( ............................ 1
1 1“
s = s,, is,, .. .. . (84)
:‘- --
.. .
,.. .. .
There are a few interesting things to be seen about matrix (84). The
first of these is the reciprocity property. Since
H
Because of the unitary property of S, a \vave incident on the antenna such
that
o
[1
will be completely absorbed. It ;vill be recognized that a, is just b, with
a time reversal.
The general antenna is not of great interest. Its generality is so
great that any pile of tin ]~ith a transmission line exciting it may be
called an antenna. It is evident on physical grounds that such a pile of
tin does not make a good antenna, and it is worth while to search for some
distinguishing characteristics that ran be ~Medto differentiate between an
SEC.9.23] THE GENERAL SCATTERING PROBLEM 327
ordinary pile of tin and one that makes a good antenna. When the
properties of a good antenna are considered, the only one that stands
out-is thegener~l economy of metal. Agoodantenna does not have an
ensemble of metallic ears, flaps, and springs that play no useful role in
the business of radiating. In other words, there maybe two antennas
that have identical radiation patterms. One of them may have miscel-
laneous structures attached to it that are not necessary. Since the
radiation patterns are identical, the patterns cannot be used to distinguish
between the antennas. Itmight beexpected, however, that one antenna
would scatter more than the other. It is worth while therefore tosee
what can be done in the way of differentiating between a good and bad
antenna on the basis of scattering.
9.23. The General Scattering Problem.—It is convenient to break the
scattering matrix of a general antenna into two parts
s = s, + s,, (88)
where
..
1
..
> (89)
0
0
.. ..
/
Similarly, it is convenient to break the column vectors a and b into two
parts
(90)
Thus for each of these eigensolutions, the power radiated by the antenna
is
*~~lJ*ajn = ~a~j (96)
()
[1
LY21
~=. (103)
It is clear that Eq. (11 1) cannot be satisfied for all k, as this would yield
the identity matrix for the scattering matrix and this corresponds to no
antenna at all. In order for Eqs. (106), (106a), and (110) to be satisfied
for some value of k (k = 1,for example) it is necessary, however, that
(113)
330 WA VEGUIDE J UNCTIONS WITH SEVERAL ARMS [SEC. 9.24
=35
=‘a”
‘a”
It can be assumed, without loss in generality, that al = az = ~=. In
1
Then
. . .
92 —92 . . .
‘=+5 g3
.
.
–(73
.
. . .
. .
. .
SEC.924] .$lINI.lf UM-SCA TTERING .4NTE,ViVA 331
\
SI o 00
0 –s, o 0 .
0 0 10 .
s, = 0 0 01 (124)
1“
o 0 .
f(sl
(s,
(s,
–
–
–
1)
l)g,
l)g,
–(s,
(s, l)g,
+
(s, + 1)93
+1)
o 0
o 0
I 19293””:
1 –q2 –g3
1 +
s=+
\
1, (i !7)
–1 Slqz s1g3
s1g2 – 9; –g2g3
s,, = o (132)
requires that
Sl= —1, (133)
Sa = b, (134)
then
Sa* = b*.
Thus the components of b* differ only in phase from those of b, and the
gain of the antenna in one direction is identical with that in the opposite
direction.
Another property of antennas to which the scattering matrix (130)
applies is that they scatter radiation with the same pattern that they
normally radiate. To see this, it should be noted that the radiation
pattern of the antenna is given by the first column of matrix ( 13o).
On the other hand, the matrix that represents the scattering of the
antenna is
‘o i o
o’
gz
c
g3 >
)
where c is some number.
SEC.924] .JIINIMUM-SC!AT1’ERIA’GANTENNA 333
‘0
a.~
~= U3 (136)
\
the power absorbed by the antenna is
Pa = +( Ta)*(ra), (137)
where r has the same significance as in 13q. (103). ‘l’he power scattered
by the antenna is
MODE TRANSFORMATIONS
BY E. M. PURCELL AND R. H. DICKE
such as that of an impedance match looking into the cofixial inp~lt liuc
when the waveguide has a reflectionless termination.
In Fig. 10. lb a similar transformation is effected in a different way.
The antennas B and C are brought in from the same side of the guide and
symmetrically disposed but are driven exactly out of phase, the line AC
being made just one-half wavelength longer than AB. Again the
TEZ,-mode is excited, and the 7’E,o-mode is not. Whereas in the device
of Fig. 10. la the achievement of “mode purity, ” that is, the freedom from
excitation of the TE 1~-mode, depends on geometrical symmetry only,
,.
...
.,.
,i:~j
(a) (b)
FIG. 10. 1.—Excitation of tlie TE,,-mocle fro!II a roaxial lim
this is not true of the circuit of Fig. 10 lb. The extra section of line,
nominally one-half wavelength long, will vary in electricid length if the
frequency is changed. Thus, if extreme mode purity oler a band of
frequencies is a requirement, the scheme b is not a good one. This
illustrates one of the practical considerations that influence the design of
mode transducers.
There is’ another difference between the circuits of Fig. 10. la and b
which should be noted in passing. If the ~va~’eguide were large enough to
allow still higher modes to propagate, these would, in general, not respond
in the same way to excitation by the two probe arrangements a and b.
The T1121-mode, for example, the field configuration of which may be
found in Fig. 10.2, would be excited by the system of Fig. 10.1 b but not
by that of Fig. 10. la, ~~hereas the converse is t r~le for the T~, ,-mode.
SEC.10.1] MODE TRANSDUCERS 337
1
w 0
2
1
@ o
0
2
,-- ...
4.- /’ -: >...
“’;? .
W-
Q‘,‘, >>
s. .@@ r,
..L. Q
-..’..
(a) (b)
FIG.10.3.—Coupling from the !fE!,o-mode in one waveguide to the Z’Ezo-mode in another
by means of small holes.
(<
(a)
FIG. 10.4.—Matched
~:
&/ (b)
mode transducer from the TEI.- to the !fEm-rnode.
electric field strength, for a given power transfer. For some purposes,
however, this might not be objectionable.
The original transducer can be altered in another way, by enlarging
the coupling holes. If this course is pursued in an effort to obtain a
matched mode transducer, the elementary explanation of the action of
the two coupling holes soon loses its validity. Instead, there exists a
complicated field configuration not easy to analyze, and there is no longer
any assurance that only one mode will be excited in guide (2). Never-
theless, it may be possible to achieve the desired result by a cut-and-try
procedure, in which one or more geometrical parameters are systematic-
ally varied and their effect upon the impedance match ‘and upon the
excitation of the unwanted mode in guide (2) is examined. Perhaps the
simplest example of this treatment is shown in Fig. 10.4. Here the entire
wall common to the two guides has been removed. A plunger in guide (a)
provides an adjustable parameter which is found, experimentally, to
control the excitation of the TE 10-mode to a considerable degree. In one
specimen of this type, the fraction of the power incident on the junction
from guide (1) which was transferred to the TEIO-mode in guide (2) could
SEC. 101] MODE 1’RANSDUCERS 339
FIG. 10. 6.—node transducers that employ a taper from one mode to another. (a) Rec-
tangular-to-round transition, (b) transitioll from the TEI O-to the TEm-mode.
of the system, by virtue of Eqs. (1). A wave incident jrorn (3) will be
totally reflected as if, for this mode, there existed a short circuit some-
where within the transducer. The apparent position of the short circuit
is specified by 233.
In practice mode transducers are not quite perfect, either because
they are not perfectly matched or because an appreciable amount of
energy is transferred to the unwanted mode or for both reasons. A
slight mismatch is usually tolerable and in any case presents no new
problem. Excitation of one or more unwanted modes, on the other hand,
oft en proves very troublesome. It is in the examination of this problem
that the description of a transducer as an n-terminal-pair device is most
helpful.
If a transducer is considered such as the one in Fig. 10.4, it is seen to
involve only one unwanted mode and can be regarded as a three-terminal-
pair device. The general three-arm lossless junction has the property
(Chap. 9) that a short circuit suitably located in one arm, say (3), entirely
decouples the other two arms from each other. Hence, no matter how
weak the excitation of the unwanted mode, if it happens that the energy
transferred to that mode is sub-
sequently reflected back, without
loss, to the junction and in pre-
cisely the most unfavorable phase,
- there will be no transfer of energy
FIG. 10.S.—Interconnectionof two three- between guides (1) and (2) in the
terminal-pairnetworks.
steady state. A wave incident
upon the junction from guide (1) or from guide (2) will be totally
reflected. A situation like this can arise when two mode transducers
are connected together by joining the multimode pipe of one to the
corresponding pipe of the other. This is equivalent to connecting
two three-terminal-pair networks as in Fig. 10.8. The behavior of the
combined network may be expected to depend critically on the length of
the connecting pipe.
Little can be said, in general, about junctions involving more than
three modes. In special cases symmetry conditions may simplify the
solution. An example is provided by the junction sholrn in Fig. 1()-!a
which, if the plunger is removed, becomes a four-terminal-pair device.
It is a very special four-terminal-pair device, however, for inspection
discloses that in so far as the four propagating modes are involved, it has
precisely the symmetry of the magic-T four-junction. The TEjO-mode
in arm (2) plays the role of the series branch of the magic T. Thus
any reiult derived for the latter junction applies at once to Fig. 10.k.
It is interesting, although perhaps disappointing, to note that there is not
SEC.103] I’H.V PROBLEAI OF MBASUW’MIJNT 343
~rhere he’daccounts for the relative amplitude of the second wave and its
phase relative to the first ~~a\e at z = O. The distance between succes-
si~-e positions of maximum 1,, is
344 MODE TRANSFORMATIONS [SEC. 10.3
(4)
An analysis of the fields within such a guide, even when several waves
are present, can be made on the basis of a number of probe readings taken
at points distributed along the guide. In principle, six probe readings
should suffice to determine the six quantities of interest: the relative
amplitudes of the four traveling waves (three numbers) and their relative
phases at some reference point
(three numbers). This, of course,
assumes that the guide w a v e -
length for each mode is previously
known as well as the relative
degree of coupling to the probe,
that is, the field at the probe in
each mode when equal power is
flowing in the two modes. If slots
are ruled out, as they often are,
fixed probes may be used. These
may take the form of small holes
in the guide walls coupling to
external guides to which a detec-
tor can be connected. In any
case, the deduction of the desired
quantities from several p robe
FIG. 10.9.—Probes for detecting the TMoP readings is a process discourag-
mode. ingly tedious and probably inac-
curate unless the circumstances permit simplifying approximateions.
A difficulty that may arise from the use of a probe is the coupling
between the two modes caused by the probe itself. This is especially
troublesome when one of “the modes present has a low intensity. The
difficulty can be avoided, at some cost in complexity, by a scheme that
also furnishes a method for measuring the amplitude of one mode alone
in the presence of the other. Let the circular guide supporting the
T1711- and Z’MOl-modes be provided with two diametrically opposite
longitudinal slots and a traveling probe for each slot, as shown in Fig.
10.9. If the transmission lines coming from the probes are joined
together externally at a point equidistant from the two probes, the volt-
age at this junction point will be a measure of the amplitude of the
!Z’MOl-mode only. Of course, this is merely an application of the multi-
antenna mode transducer described at the beginning of Sec. 10.1. The
principle can obviously be extended to any mode configuration if the
complexity of the resulting apparatus can be tolerated.
SEC. 10.3] THE PROBLEM OF MEASUREMENT 345
When there are more than two allowed modes, the difficulty of
quantitative determination of the fields within the guide becomes indeed
formidable. In many cases, however, it is possible to ascertain the prop-
erties of the mode-transducing junction through measurements made in
the single-mode guide that usually forms one arm of the transducer.
Consider such a transducer, exemplified by the devices shown in Figs.
10.1 and 10.5, which serves to couple a guide carrying only one mode to
another guide in which more than one mode can propagate. For brevity,
the single-mode guide will be referred to as A and the multimode guide
as B. It is assumed further, for simplicity, that only two modes are
excited; the desired mode (2) is strongly excited, whereas the mode (3)
is excited only weakly. The following remarks thus apply to the TE,O-
to TMO1-mode transducer of Fig. 10.5 if the junction is free from irregu-
larities that would introduce asymmetry and result in the excitation of the
Z’E,,-mode as an elliptically polarized wave in pipe B, which is in this
case the circular guide. Let us see how the electrical properties of the
junction can be determined from standing-wave measurements made in
guide.4 only. While the procedure to be described is a practical one, it is
discussed here chiefly as an application of the ideas of the preceding
section.
It should be noted at the outset that one operation which can be
performed on the waves in pipe B in a nearly ideal manner is that of
reflecting them from a short-circuiting plunger, which fits snugly int~
pipe B and the face of which is a flat metal surface normal to the axis of B.
Such a plunger introduces no coupling between modes. Moreover,
because the phase velocities of the two modes in B are different, motion of
the plunger varies the terminations of the two modes at clifferent rates.
It was shown in the preceding chapter that an equivalent circuit of
the form shown in Fig. 10”10 can be drawn for a lossless three-terminal-
pair network, and this representation is convenient for the present pur-
pose. Three of the parameters in this general description are the line
lengths 11, 12, and 1s, which are of no immediate concern. The other
parameters are the turn ratios nla and nls of the two ideal transformers,
and the convention is adopted that n H < 1 for a voltage step~p from line
(1) to line (3), which is the situation suggested”in Fig. 1010. The ideal
transducer is then characterized by nlz = 1, n13 = O. The case of inter-
est, however, is n12 = 1, 0 < n13 <<1.
The plunger in pipe B now serves to connect to lines (2) and (3)
impedances 22 and 23 which, but for losses in pipe B, would be pure
reactance jXl and jX,. Any combination of Xl and X, can be realized
if pipe B is long enough and if the guide wavelengths of the modes in B
are incommensurate. Measurements of the impedance of the junction,
as seen from pipe A, for a sufficient number of plunger positions provides,
in principle, information from which the parameters of the equivalent
346 MOD13’TRANSFORiWA TIOXS [SEC. 103
i3b!1 L
t
Ud
:>::’+E:::E{=
T &l, +
FIG. 10.10.—Equivalent circuit of a lossless three-terminal-pair network,
mAup
—–jo
_m_i4.; -+
()~
Z1 = ,,,;3 72–. jX2. (6)
+ e’
4
Wkes —
(a) (b)
FIG, 10. 12.—Wire screens for selective transmission of (a) TMt,,-mode and (b) TE~,-mOdo.
4n~3
pA–l. — (7)
mAg,a’
*.
Now e 4 is the attenuation of the amplitude of a wave running one way
between the junction and the plunger in line (3). This might be written
more generally as e–y, including the effect of any mode absorber intro-
duced into pipe B. Then
pA–l. (rid’.
— (8)
Y
It should be noted that the quantit y (n,,) 2, which has been used to specify
the coupling of the unwanted mode, is approximately [(nls) 2<< 1] the
ratio of the power transferred into mode (3) to that transferred into mode
(2) when both are terminated without reflection. If it is required for a
given application that p. – 1 shall not exceed a specified number, regard-
less of the distance between the junction and the reflecting termination
of the parasitic mode (3), Eq. (8) prescribes the amount of attenuation
that must be provided in pipe B by a selective mode absorber.
10.6. The TE,,-mode in Round Guide.—The TEl,-mode in wave-
guide of circular cross section, which was mentioned at the beginning of
this chapter, deserves special attention. Propagation can occur in two
modes simultaneously in such a guide, but there is an essential arbitrari-
ness in the description because the two modes have identical cutoff
frequencies and because there is no unique direction to fix the
coordinate system in -which the field configuration is to be described.
350 ,VODE TRA.VSFORMA TIONS [SEC, 105
First, two mutually perpendicular planes through the axis of the guide
may be selected, and modes (1) and (2) may be defined as the two Z’1711-
modes with electric fields parallel and perpendicular respectively to
each of these planes. Any wave in this pipe, for example a wave polar-
ized in a plane making some angle yith the two chosen planes, can be
represented as the sum of waves associated with modes (1) and (2).
The wave, in other words, can be resolved into what w-ill be called two
hsic polarizations. The basic polar-
izations need not be perpendicular
to each other, but this choice is
usually convenient; and if it is made,
~ the modes (1) and (2) will be said to
be orthogonal. .
,.
Figure 10.13 shows how a plane-
4 ! polarized wave at some arbitrary
l:IG. 10.13.—Res0luti0n of a plane- polarization angle may be resolved
polarized wave into two basic waves.
into two basic polarizations. It is
clear that there is nothing fundamental about the particular directions
chosen for the basic polarizations. If these basic polarizations are rotated
to a new position, an alternative representation is obtained for the plane
wave.
In a similar way two plane waves may be superposed to make a circu-
larly polarized wave as shown in Fig. 10-14. Thus, a circularly polarized
wave may be said to be resolvable into these two basic polarizations.
Basic waves need not be taken as
plane polarized. For instance,
it is well known that two circu-
larly polarized waves can be com-
bined to form a plane-polarized
wave. Thus the two circular
polarizations form a possible set
of basic waves. FIG. 10.14.—Resolution of a circularly
polarized wave into two PIane polari zecl
For the analysis of circuits ,Vave,,
and circuit elements employing
the TE1l-modes, a description by means of scattering matrices, which
were introduced in Sec. 5.13, is helpful. As before, a waveg~ide carrying
two modes is to be regarded as equivalent to two transmission lines, and
the terminal of such a guide is represented by two terminal pairs, each pair
being associated with one of the basic polarizations. The new element
in the situation arises from the possibilityy of shifting, at will, from one
set of basic polarizations to another. Such a change in representation
involves a transformation of the scattering matrix that ‘‘ mixes” the
original terminal pairs, so to speak. Although this is a procedure that
SEC,106] PERMI&SIBLh’ l’RANSFOltMA TIONS 351
can be carried out formally for any guide carrying two modes, it usually
lacks the useful physical interpretation that can be given in this special
case.
10.6. Permissible Transformations of a Scattering Matrix.-con-
sider the system shown in Fig.
10.15. The two planes of polar-
ization are designated by the num-
bers (1) and (2). The remainder
of the terminals are located in rec- i-
tangular waveguides and are des- (3)
FIG. 10.15.—Waveguide junction with
ignated by the numbers (3) and four ~ermind~ Two terminalsare in rec-
(4). As was shown in Chap. 5, if tangular guide; the other two terminals
are two basic polarizationsin the round
the incident waves are represented ~aveguide
by the column vector
al
(lo)
then
Sa = b, (11)
al = aj cos 0 — a~ sin 0,
(12)
a? = aj sin d + a: C05 0. 1
352 MODE TRANSFORMATIONS [SEC. 106
a = Ta’, (13)
where
Cos e —sin O:
\o
sin 8 Cos e !
T= ,.. .—––.
-----1. (14)
~lo”
I
In a completely analogous way,
o
I
;0 1
b = Tb’; (15)
b’ = T-lSTa’. (16)
The matrix
S’ = T-IS.T (17)
@L () 00
& () o
P= 0 J (19)
00 ei+’ O
ej~4
[1000
where +k = 2~(dk/~k) . It should be noted that P* = P-’. The matrix
S’ obtained by the transformation
s’ = PSP (20)
(23)
/ /,’ ,, , // // / / ,
:Ie
\\\\
FIG. 1017.-Quarter-wave pipes. The steps at each cnd of the fin are quarter. wavelength
transformers for matching; the dimensions give]l are for X = 1.25 cm
II1000”
Ojoo
j
(24)
SEC.108] ROTARY PHASE SHIFTER 355
o o! 1 1
0 o~–1 1
S, = ~ e’+’ ;------l-l-~ ---o------o , (25)
II1 l! o 0
1 A. G, Fox, “ Waveguide Filters and Transformers, ” BTL Memorandum MM-41
160-25.
356 ~fODE 7’h?A.VSFOR.l[A
TIO.L-S [SEC. 108
o 0 1 –.i
0 0 –1 –j
(26)
1 –1 o“ o
-----------1
–~ –~ 0 OJ
The axis of the half-wave pipe is at an angle 6 with respect to the direction
(4). Therefore, the problem is simplified if the matrix of Eq. (26) is
transformed by a rotation through an angle d by the matrix
I
10
o
1
T = ‘- -1 (27)
I Cos o — sin 0
,[ o
)
Now passage through the half-wave pipe has the effect of changing
the sign of the fourth row and column. Therefore,
(29)
‘–M je–2,a
o
_e2,e ~“ez,e
I
(:30)
219 _ ‘Zje
“-1”
‘–
o
je-2,.9 +je2i0
I 0
o
0,
1
o
(33)
b = @+7-W
II 0.
1
0
The outgoing wave contains 0 as a phase angle. Thus, as 0 is increased,
the equivalent line length bet ween input and output terminals is increased.
(34)
0)
o
(:{.5)
10.9. A Rectangular-to-round
[1
b = @47+20) 0
o’
1
that the terminals (2) and (3) are matched. For example, there is a
position of the plunger such that a linearly polarized wave, polarized as
(2) in the figure, is reflected back, polarized as (3). .
The statements (A), (B), and (C) apply regardless of the symmetry
of the junction or orientation of the polarization. The interpretation
of the statement (A) is obvious. Theorem B states that if terminal
(1) is matched and (2) is the polarization excited in the round guide
by a wave iicident in (l), then the polarization (3) orthogonal to (2)
is completely decoupled from (1) and (2). Alternatively, if a polariza-
tion (2) is completely reflected polarized as (3), then (1) does not couple
with the round guide.
The theorem (C) has an interesting application to the junction of
Fig. 10.19. If a plunger is inserted in (1), then for any polarization (2)
there is a position of the plunger such that this polarization is reflected
back unchanged. Also, for this same position the orthogonal polariza-
tion (3) is unchanged by reflection.
10.10. Discontinuity in Round Guide.—A round guide with a dis-
continuity in the middle is an
(3)
example of a four-terminal-pair
network. As such, it satisfies the
conditions applicable to direc-
tional couplers. One of the most
interesting of these is the follow- (1)
ing: Any four-terminal-pair junc-
tion that is completely matchedl
is a directional coupler. To state
it in other words, if a junction
of four waveguides is matched at
each waveguide, then for any ~rc, 10.20.—Discontinuityin round wave-
guide.
guide there is one other guide to
\vhich it does not couple (see Chap. 9).
This theorem may be taken over completely in the case of the round-
guide junction. If, for a given set of basic polarizations, the junction is
matched, then it has the properties of a directional coupler. Referring
to Fig. 10.20 this implies that the polarization (1) (shown as a plane
polarization, whereas it may be elliptical or circular) does not couple to
one of the polarizations (2), (3), or (4).
The case of no coupling between guides (1) and (2) is rather trivial.
In this case no power is reflected in either (1) or (2) for incident power in
(2) or (l). This is also true for any linear combination of (1) and (2).
I As usual the term “matched” is intended to mean that if all terminal pairs but
one are provided with reflectionless terminations, a wave incident at the one terminal
pair is not reflected.
360 MODE TRANSFORMATIONS [SEC.1011
then
S*a = s*a. (39)
[fthecwmplex conjugate ofllq. (39) isadded to~q. (37), the result is
S(a + a’j = s(a + a*). (’lo)
But a + a* represents a pure real column ~rector, or a linearly polarized
wave, Hence, linearly polarized wales can be found that satisfy Eq.
(37). These particular planes of polarization afford convenient basic
polarizations because the resulting scattering matrix has only diagonal
elements.
If the termination of the waveguide is lossy, the scattering matrix is
no longer unitary and Eq. (40) is no longer valid. It is still possible to
obtain solutions of Eq. (37), but these solutions, in general, represent
elliptically polarized waves.
10.12. Resonance in a Closed Circular Guide.—The problem to Im
{liscussed in this section derives its importance from the use of waveguide
rotary joints employing the Z’Mol-mode. It has already been pointed
out in Sec. 10.3 that even very weak excitation of one of the Z’1111-modes
by a transducer such as that shown in Fig. 10.5 may have a pronounced
effect on the behavior of a rotary
<
joint made from two such trans
ducers connected in cascade in the
y-:j ,“
4
event of a TE 1l-resonance in the
pipe connecting the two. To find
the conditions under which such a ~
‘Y’ “)
(
‘!:’”>. .
‘=(2)\
resonance can occur, it is neces- ~ ~
b
sary to take into account the fact r x// ‘“. (2)
,,
that a TE1,-wave incident on one ,’
‘, 1
of the transducers is reflected with /v
a phase that depends, in general, (1)
on the polarization of the wave. ~
FIG. J0.22.—Round waveguide closed at both
That is to say, the transducer, as ends.
regards TE1 l-waves, acts v e r y
much like the junction of Fig. 10.21. There is, of course, some “loss”
representing the leakage of power through the transducer into the rectan-
gular guide, but this can be disregarded in the search for the resonance
condition, as can the presence of the TMO,-wave in the pipe. The prob-
lem can thus be reduced to that of the section of round guide closed at
both ends, which is shown in Fig. 1022.
The waveguide is cut in the middle to allow the two sections to be
rotated with respect to each other, The two halves of the guide are
identical. Their terminations are totally reflecting and are identical
3ti2 .J1ODE7’R.4.YSFOE’JIATIO.VS [SEC.1012
also, although of some irregular shape. The principal axes of the ends
are indicated by the directions a and b. ‘I’he median plane is taken as a
common terminal plane for the t\ro hall-es of the system, and the polariza-
tions (1) and (2) are common to the two halves of the system. The
problem is to find the conditions for which, at a fixed frequency, a lvave
can exist in this closed lossless system. This problem was first treated
by H. 1<. Farr.’
The t\vo ends are designated by the numbers 1 and 2. If the scatter-
ing matrices for the two ends are SI and S?, then
S,a, = b,,
S,a, = b,. (41)
Since the incident ~~ave for one end is the scattered wave coming from
the other end, it is clear that
a, = b?,
(42)
a? = b,. i
(45)
If now the two halves of the guide are rotated by o and – 0, respectively,
we have
s, = s(e)
and
s, = S(–e),
and SZ is the same except for a char,ge in sign of the off-diagonal elements.
It is merely a matter of straightt’orl~-urd algebra to substitute these
matrices in lXI. (4+) and thus obtain a relation connecting t?, o,, and I#Iz.
[f, instead of 01 and o?, the angles c and 6 are introduced, these angles
lwing defined by
6=41+6, 15=r#, -f#h, (47)
(48)
a wave polarized along the other principal axis, then ehu/4ir is the sum of
the distances from the midplane to each of the t~vo reflectors and c$x,,/4r
is the dist ante between the two reflectors. For the structure shown in
Fig. 10.23,
:111(1
assuming that the edge of the fin acts as a short circuit at ~hat point for a
wave polarized parallel to the fin.
If the guide were terminated at each end by a flat plunger, 6 would be
zero and solutions of Eq. (48) would exist only for c = 2nm, (n = 1,
2, .). This means that the total length of the pipe between plungers
must be n&/2, the familiar condition for resonances in a closed uniform
guide,
364 MODE TRA :Y,$FOR.lf A TIOJYS [SEC, 1012
If 6 is not zero, ~q. (48) shows that resonance can occur at some value
of @ for any c within a limited range. The situation is best illustrated
by curves of constant 6 on the (d – c)-plane, as in Fig. 1024. If 6 = m/4,
for example, resonance will occur for some value of 0 if c lies within the.
z E— 2r r
F1o. 10.24,—Theangleof rotation @ for the occurrenceof resonanceof a closed round
waveguide,
DIELECTRICS IN WAVEGUIDES
BY C. G. MONTGOhIERY
ancl
7=a+j13=. +.j~J
() 2T
(1)
366 DIELECTRICS IN WA VEGUIDES [SW’. 111
(4)
(8)
The parameters n and k are most often used when dealing with optical
frequencies but occasionally are used in the microwave region.
When a dielectric is introduced into a waveguide, the transmission loss
is increased because of the losses in the dielectric and the increase is
measured by a as calculated above. The Joule heating losses in the walls
of the waveguide are also altered. The change in the metal losses
caused by a change in the real part of the dielectric constant is a simple
one. Since the propagation constant is given by
where Hkm is the magnetic field tangential to the metal walls and H~ is
the transverse magnetic field. Two cases must now be treated. First,
~-modes may be considered, where no longitudinal magnetic field exists,
and HL = H~.. on the boundary of the waveguide. The ratio of the two
integrals in the expression for am depends only on the shape and size of
the guide, and the dielectric constant occurs only in Re (Z.). Thus
1 1 _ (U,’)’ + (u,”)’
Re (Z,) = ~e ,y – au~” + 13u~”
]m
(-)
368 I) IEI. E(:TRI(’JS Iv If’.’t 11EG1JII)178T [s1,(. 1I 1
Hence, the fractional change in the metal losses resulting from an imagi-
nary term in the dielectric constant can be written as
1 1
Aen,
— — Re (Z,~ – Re (Z’J
—
am 1
Re (Z.)
or
Aa.
= -w]
‘sin’+ (9)
It is seen from this equation that the effect is of the second order in
E“/E’. Moreover, the change can be either positive or negative, depend-
ing on whether k. is less or greater than ~2 X.
For H-modes, the situation is somewhat more complicated, because
there is a longitudinal magnetic field. On the boundary of the wave-
guide
lHtanl’ = lzztl’ + IH,I’,
pup
since
Re (Z.) = Re
(“)
‘~~ = —
a’ + @’
,712 = ~, + p?,
Hence
As,,, _a?~
— —. —-. “
Qm
@-.4+B
?2
A and B are positive quantities, and therefore As., la- is always positive
Aam — tanz v ho 4 A
(lo)
a. –( 4 x)
A+$2B
()
is of the second order in c’~/cf. Here again, because am has been calcu-
lated only under the assumption that a/P <<1, the total attenuation is
given by a + am.
11.2. Reflection from a Change in Dielectric Constant.-The bound-
ary conditions that are to be satisfied at the surface of a discontinuity
in the properties of the medium are that the tangential components of
the electric and magnetic fields must be continuous. Suppose that there
is such a discontinuity in a waveguide and that the plane of the dis-
continuity is perpendicular to the axis of the guide. The power flow
Re (Z.)
down the guide is P = ~ 111,1’ds, where the integral is taken
/
over the cross section of the guide. Now H, must be continuous across
the interface of the two dielectrics,
whereas the wave impedance Zm 0 0
to satisfy the conditions at the FIG. 11.1 .—Equivalent circuit for a change
in dielectric constant.
boundary. Moreover, the equiv-
alent impedance that will correctly describe this reflection should be
chosen proportional to the wave impedance. Hence, the equivalent
circuit representing the discontinuity is simply that shown in Fig. 11.1.
The standing-wave ratio r is the ratio of the impedances taken in
such a way that r > 1. The position of the minimum is either at the
junction or a quarter wavelength away from it, depending on the loca-
tion of the observation point with respect to the reference plane.
The value of the wave impedance, as derived in Chap. 2, differs for
E- and H-modes, and the two values are
ZH = J? Z* = L.
julp
(2.91)
(235)
(2.39)
370 DIELECTRICS IV WA VEG(’IDES [SEC. 112
(11)
and
Zg)
— (12)
z~i
If ~[z) > c(IJ, then the relative H-mode impedance decreases monotoni-
callyas E‘2) increases. The relative E-mode impedance is not a monotonic
function but may increase, pass through a maximum, and then decrease.
It is easy to show that the maximum value occurs when
~(z)
— =2;,2, (13)
tO (,)
:md the maximum value is
& Jo
.=qiJ&J (14)
Thus, for sufficiently large values of XO/A.j there are two values of
~(z)that make Z~)/Z~l) equal to unity, namely,
~(2j = Jl)
and
The second value corresponds to the case of Brewster’s angle for plane
waves, that is, the angle at which, when the electric vector lies in the
plane of incidence, there is no reflected ray from the boundary between
two mediums. Figure 11.2 shows values of the relative wave impedance
for (kO/k.)z = 0.8 when C(l) = cO.
When the dielectric material is 10SSY,the wave impedance becomes
complex. Its value can be expressed in terms of the complex dielectric
SEC. 112] REFLECTIO.V FROM DIELEC’ TRIC BOL’.\ DAR]’ 371
ZH=—
jkvl
a+J3
—
—
()
f.J/.4
~2 + P2
I+ ’j:
B
—
,+%’C’’J(’
“%)’ (15)
and
It is seen from these expressions that the change in the real part of
ZH or Z. is of the second order in e“. For 7’11-modes, the presence of
“1 2 3 4 5 6 7 8 9 10
Dielectric
constant
e ‘2~~o
FIG.11.Z.—Relativewaveimpedanceas a functionof dielectricconstant.
loss makes the wave impedance slightly inductive; for !f’M-modes, the
reactive part is either positive or negative depending on the frequency.
Near cutoff Z. is capacitive; at higher frequencies it becomes inductive.
The expressions for Z~ and Z. can be transformed to a somewhat mrwe
useful form in the following way:
372 DIELECTRICS IV WAVE
GUIDES [SEC, 11.2
(17)
1 (18)
!1
ZE = $.
J-
(19)
(:)-:[
1
V2(1 + tanz f?) + (1 – tan~ L9)21--~iv~ + v4t,an~ e = 0, (20)
l–r)~
c’
— 1–V2 ~l–v~ ‘2+2 R’
~o = “ + ‘j’--- – B R,-- ;-+ — 1 :-;2 ‘
‘-R,
and (22)
l–v~
‘–R,”-
~=zo .- —
1 – ;’”
“ + RT
(23)
:
gJ 1.4 \
~
3 1.2
u)
1.0
0.9 1.0 1.1
Wavelength ratio A/k
o
FIG. 11.4.—Variation of standing-wave ratio with wavelength for a dielectric plate one-
half wavelength thick,
I
SEC. 11,3] DIELECTRIC PLATES IN WA VEGUIDES 375
The thin dielectric plate is thus equivalent toa small shunt capacitance
across the waveguide. By means of uninductive iris placed at the face
of the plate, the circuit may be made resonant and reflectionless.
It is possible to construct a tuning device of two movable dielectric
slabs each one-quarter wavelength thick. When the two slabs are in
contact, the combination is reflectionless. When they are separated by
()
[!
l-p”
c
R= 7
e — XQ 2
—
~o () <
for H-modes, at the face of the first dielectric slab. Thus, referring to
Fig. 115, the impedance at the face of the first slab can have values along
the boundary of the shaded circle. If the combination is moved, as a
whole, along the guide, all impedances within the larger circle can be
attained at a given point.
It is also possible to insert a quarter-\vavelength transformer to match
376 DIELECTRICS IN WA VEGUIDES [SEC. 11.4
from an empty guide to one filled witha dielectric, the transformer see-
tion being composed of the guide filled with a dielectric having a dielectric
constant of an intermediate value. This intermediate value, for H-modes,
issuchas tomake the guide wavelength the geometric mean of the guide
wavelengths in the full and empty guides. It should be noted that no
end corrections are necessary, since the junction effect is absent.
A plate composed of a lossy dielectric can be treated in a manner simi-
lar to that for the lossless plate. The expressions become much more
complicated i n form, but their
derivation is straightforward.
The general nature of the behavior
may be seen from Fig. 11.6 which
shows some experimental values of
the transmitted and r e f 1e c t e d
power as a function of the thick-
ness of a piece of plywood in wave-
guide. The observations were
taken at a wavelength of 10 cm in
1.5- by-3-in. waveguide.
11.4. The Nature of Dielectric
Phenomena.—In a homogeneous
isotropic dielectric medium, the
electric displacement differs from
its value in free space by the polar-
ization P which is the electric
moment per unit volume in the
medium;
P = D – ,,E = (, – ,,)E. (26)
Thicknessin cm The electric susceptibility x. is
F1.,. 11.6.—Experimental values of related to P by
transmitted and reflected power as a func-
tion of the thickness of a plywood plug in P =5_
waveguide. x6=_ 1 = k. – 1. (27)
cOE eO
If there are N molecules per unit volume and the electric moment of one
molecule is m, then P = Nm. To explain the observed value of the
dielectric constant and its variation with frequency, it is necessary to
consider the nature of the mechanisms whereby molecules can acquire an
electric moment. If a small conducting sphere of radius a is placed in
an electric field 1’, the conduction electrons will distribute themselves in
such a way that the sphere acquires an electric moment equal to AwOCZV’.
In a similar manner the electrons in a molecllle \vill redistribute them-
selves in such a manner that the molecule \villacquire an electric moment
whose magnitude is proportional to F We write m = a,F and call a,
the polarizahility of the molrmlle. ‘Illis pt)l:~rizability \vill bc indr-
pcn(lent of temperature. on the otllfr lIan{l, tl]e mfllf,clllc may have.
by virtue of its structure, aperrnancnt elcrtric mommlt of magnitude m,
When the field isapplicd, the molecule \villtencl toturnand align itself
with the moment in the direction of the field. This alignment ~vill be
destroyed by the collisions and other random forces that the molecule
experiences. Since the energy of the electric moment when it makes an
angle 8 with the field F is —mF cos 6’,the mean value of the moment may
be calculated by means of Boltzmann’s distribution law,
(28)
%= (a,+~,)F. (29)
It is now necessary to find the value of F, the total field acting upon the
molecule. Thk field is made up of a contribution from the external
applied field E PIUSthe contributions from the fields of the other dipole
moments in the medium. The contribution of the dipoles is very difficult
to estimate. This is evident from the fact that the number of dipoles
at a distance r from the point under consideration is proportional to
47n-2dr, and the field of a dipole is proportionalto (COS0)/r3. Hence the
total effect is proportional to J (4T cos 8/r)dr. For large r this integral
vanishes, since there are equal contributions from those regions where
cos 8 has opposite signs. For small r, however, the integral diverges
and the value of the field is extremely sensitive to the particular assump-
tions made about the nearest neighbors of the dipole under consideration.
1 For a more complete discussion of the details of the quantum-theory calculation,
the reader is referred to J. H. Van Week, Electric and Magnetic Susceptibilities, Oxford,
New York, 1932.
378 DII?I,L’C’TRI(’, V l.} 1[’.4[’lWl’[D~.T [SEC. 11.4
1n the case of a gas, no very large error is made by neglecting the field of
the other dipoles entirely :md putting F = E. For more concentrated
substances, the next approximation may be considered to be the classic
one of Clausius. Clausius assumed that the dipole could be thought of
as being within a small spherical ca~rity within the medium. In this
rase
l’=fi+3:o. (30)
The use of this expression for F, together with the equations already
obtained, to find c results in the relation
e
— – 1 = -.~T%+3@
to
l-~(a’+~)
or, as it is more usuall.v written,
(31)
‘1’his approximation for F is not a very good one, and more exact exprt>s-
sions have been given by onsagerl and Kirkwood. ~ We have, however,
established an important fact which is true regardless of the expression
for F, namely, that the polarization consists of two parts, one part that
is independent of temperature and depends on the shift of the charge
within the molecule and one part whose contribution decreases with
increasing temperature and is caused by the permanent electric moment
of the molecule. The dependence of the dielectric constant upon
temperature will be that of the temperature dependence of the polariz-
ability, in general, since the effect of the local field would not be expected
to be greatly dependent upon temperature.
The effect of frequency on the polarizability is again a twofold one.
Since, in the microwave region, the natural frequencies of the molecule
are large compared with the frequency of the radiation, the molecular
polarizability al is independent of frequency. When the frequency of
the radiation approaches a natural frequency of the molecule, then al
changes and gives the familiar anomalous-dispersion curve for the fre-
quency variation. The lowest natural frequencies of most molecules lie
in the infrared and do not influence the values of the dielectric constant
at microwave frequencies. The effect of the rotation of the electric
moment, however, will be strongly dependent upon frequency. This
1L. Onsager, J. Am. Cherrt. L$’oc., 68, 1486 (1936).
~J. G. Kirkwood. J. Chem. Phys., 7, 911 (1939).
SEC. 114] THE ,4-A
TIIRE OF DI ELECTRIC’ PHENO>IE,VA 379
effect was first explained by Debye, 1 who showed that the transient part
of the effect of collisions is similar in character to the effect of viscous
forces that impede rotation. At low frequencies, these viscous forces
\vould be small and the dielectric constant high. At high frequencies,
the forces would be so large that, effectively, the molecules would be
prevented from aligning themselves and the dielectric constant would be
lo\v. Thus, if the viscous forces are proportional to the rate of change of
moment,
~o
(32)
a= l+jcw’
where ~ is a “ relaxation time” that is characteristic of the material and
aOis the value of a at u = O, namel-y,
~2
(33)
a“ = m“
If the Clausius hypothesis is used for obtaining the value of F, then
2=4++9
A rearrangement of this expression, and separation
(34)
(35)
where
(36)
and c, and Emare the values of c at zero frequency (static value) and infinite
frequency (optical value), respectively. In terms of the molecular
constants,
wl+a+l (37)
and
“=’O1
-%”’+%)’
80 -
70
60 I_
50
40
,--
// \
30
/
/
20 -
/
10
0-
10’0 10”
Frequency in cps
FIG. 11.7. —Variation of the dielectric constant of water with frequency,
of 10–10 sec for liquids and 10–5 to 10–6 sec for a solid, ice, These val~~es
are entirely reasonable judged from cruclc estimates made f mm the
knowm values of the viscous forces. A viscous force of this kind IYould
be expected to bc strongly clcpendent on the temperature at all frequencies
at which the polarity of the molecule contributes to c, ancl indeed this is
the case. For a strongly polar liquid, such as water, the agreement is not
exact, but the general nature of the frequency variation is unaltered.
Figures 11.7 and 11.8 show some experimental valuesl for the dielectric
I The obser~,ations from lvhich the curve was drawn were taken from E. L.
Younker, “ Dielectric Properties of Ivater and Ice at K-band, ” RI, Report No. 644,
I)ecember 1944, .4. von Hippcl, “ Progress Report on L-ltrahigh-frequency Dielec-
trics,” OSRD Report No. 11{)7, December 1942, and the references cited in these
reports. Somewhat diffrrcnt conclusions have been reached by J. A. Saxton, ‘(The
Dielectric Properties of l~atcr at \Vavelengths from 2 Ilrn to 10 Cm, and over the
‘~emperature Range 0° to 400(;,” Paper N’o. ltRB/(’l 15, April 1945. Stxton con-
cludes that the experimental evidence indicates that the I)ebye thcory correctly
represents the facts for ~vater. If this m the CM(,) water is very exceptional, since
Iuost dielectric liquids seem to poswss a whole range of relrm,atlontimes.
SEC, 11.4] THE NA l’il RE OF IJIELECTRIC PHE,VOMENA 381
Substance
E’ ,’
— ,’
—
,0 en c,)
Steatite ceramic, .41simag 243 6.3 0.0015 6.2 0.0004 5.4 0.0002
Ruby mica, 5.43 0.005 5.40 0 0004 5.4 0.0003
Quartz, fused ,.. ““ 3.85 0.0009 3.82 0.0002 3.80 0 0001
Corning glass—702P. 4,75 0 009 4.55 0 002 4.40 0 006
Corning glass—705A0. 500 0.03 4.75 0.034 4.70 0 007
Corning glass—707DG. 400 0 0006 4,00 0.0008 3.90 0 001
Black Bakelite, . . . . . . . . 50 0 10 4.9 0.03 4.7 0 05
Lucite . . . . . . . . . . . 3.3 0.07 2.6 0.015 2.5 0.005
Plexiglas, . . . . . . . . . . . 3.4 0.06 2.7 0.015 2.5 0.005
Polystyrene, ,, . . . . . . 2.51 0.0002 2.51 0.0003 2.45 0.0005
Polyethylene . . . . . . . . . 2.25 0.0001 2.25 0.0001 2.25 0.0002
Apiezon W.........,.. 2.80 0.022 2.65 0.0025 2.62 0.002
Paraffin . . . . . . . . . . . 2.25 0.0002 2.25 0,0002 2.20 0.0002
Mahogany plywood, dry 2.4 0.01 2.4 0.02 2.0 0.02
Water, 25°C, , ..,,..,,.,,.. 79. (3000) 79. 0.03 59. 0.46
Likewise the expression for the propagation constant and wave imped-
ance in a ferromagnetic medium is more complicated;
y = k: – (&J’ +jkl’%p”,
~H =j(w’ +(W”, (39)
Y }
It is seen that y has a real part and Z~ an imaginary part, both of which
are representative of the energy loss from hysteresis. Since, however,
most ferromagnetic are metals or at least semiconductors, an imaginary
part of the dielectric constant must also be included. The result is
(41)
where
~,
R=l –“tan q tan ~ —
,A’p’
(42)
where
~2 +&=
()[2–
z~
Ao
1421P12
-2’’P’:.Y’(?)+(W”
M
384 1)1.W.ECTRICSIiV M“AVEGIJI1)lLS ~sklc.115
z. = ; +
J- (1’+ J’ – ~ Vl#l – k“)” (44)
6’
— P’
= 20, – = 3.2,
to /Jo
t!
e P“
– = 1.4, – 4.2,
~o /Jo –
tan P = 0.07, tan ~ = 1,3.
1.4 recent summary lms been prx,parwl hy J ‘~. .\lli~lmon. “ ‘~he Permeability OF
Ferromagnetic hlaterials at Frrquenries Grwter than 105cps, ” ( ‘rntral Rad]o Buiwa~t
2545, WR-1157, *Jk~I.44281, .~llr. 21, 1944.
2W. Arkadie!v, Physik Z., 14, 561 (1913).
3 N, Llohrning, Ifochf) eq({e?utccl~nikl{. Elek((fktts, 63, l!M (193!))
4The material in q[lesti{)])Iv,asan experimentalsample of polvlrc)nftlrnishwlhy
H. L, (’rowley and (’o., Inc , \Vmt Orange. X. J
These values were calculated from measurements of the impedance w
small pieces of the material placed in a coaxial transmission line, as
described earlier in this chapter for dielectric plates. The value of
c’/cOis high because of the presence of the conducting iron particles which
are polarized under the influence of the field. The point corresponding
to these observations is plotted on Fig. 11.9 for comparison with the
results for the solid material.
Itmaybe seen byreference totheexpression for Z,, inEq. (43) that
mixtures of this sort can be compounded to make ZM have any arbitrary
real value. If the imaginary part of Z~ is set equal to zero,
.
– = tan {.
;
From Eqs. (41), for k. = 0, is found the condition that p must equal ~.
If this is true, then
(47)
2T ,’p’
a+~p=% J-
~, (tan p + j). (48)
where Y’ is the admittance looking to the left, at the left boundary of the
1.6
— tl.— I I I 4:1.0
1.4
llL-Jz=-
1.2
1.0
0.6
0.4
u I
0.2
1!
.
n I
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1,0 1.1
>
A
FIG. IllO.-Variation of A,/A, with a/k, for various values of alla, when cz/c = 2.45.
Case I: Dielectric in center of guide.
“ “-n(K’l)d+)J
where Y$) and Y~2Jare the characteristic admittances of portions (1 )
and (2), respectively. Setting Yti = O, we have
SEC. 11.6] GUIDES PARTIALLY FILLED 387
(49)
1.4
1.2
1.0
0.8
~
b
0.6
0,4
0.2
00
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0,8 0.9 1.0 1,1
&
A
IJIG. 11.11.—Variation of k,/A, with a/),, for various values of d/a, when ez/e = 2.45.
Case II: Dielectric at edge of guide.
The results are given in Fig. 11.10, which shows Xl/Xu as a function of a/Xl
for a series of values of d/a for the case where c2/cl = 2.45. It is to be
noted that for small values of d/a there is a large change in k. whereas the
change in X. between d/a = 0.75 and d/a = 1.0 is very small. This is
obviously because, for small d, dielectric has been added where the field
is high and the effect is much larger than when the dielectric is added
where the field is weak.
A second simple case is shown in Fig. 11.11. The equation for k.
388 1)11{1.ECTRICS I:V ll-A J’EG1‘IDE,Y [SW, 116
now represents the condition that the impedance looking to the left
vanish on the right-hand boundary of the wa~’eg~lide. We have
Z(2)
~~, tan K~2)d = –
0
z~2)
~=
For cz/el = 2.45 the results of the calculation are as shown. It is seen
from the curves that for small values of d the effect is small. As the
dielectric interface approaches the center of the waveguide, the effect
becomes much larger and then decreases again as the region of weak
1.6
I
I
1.2
0.4
n
-o
ml=’’’” 0.1 02 0.3 0.4 0.5 0.6 0.7
+
of X)A, with b/X. Case III: Dielectricat bottomof guide.
FIG. 11.12.—Variation
fields near the right-hand wall of the guide is approached. The circles
on the curve indicate the values of a/A for which the next H,mode can
propagate. The losses have again been neglected.
Figure 11.12 shows a somewhat more interesting example. The pre-
ceding cases involved a mode of transmission that was transverse-electric
both in the direction normal to the dielectric interface and in the direction
of propagation. In the present case, if the interface normal is chosen as
the reference direction, the field configuration may be considered to be
that of an E-mode. The field has components E=, E,, EZ, Hg, and HZ.
Hence, with respect to the z-axis, the mode is neither a pure E-mode
nor a pure H-mode but must be a combination of the two. The
impedance method of calculation is still valid, but now the E-mode
impedance [Eq. (11)] is used as the characteristic impedance of the lines,
sm. l17] DIELECTRIC POST IN WA VEGUIDE 389
The results for a particular case are shown in Fig. 11.12, for the values
Q/~1 = 2.45 and b/a = 0.45, for a guide half full of dielectric. Figure
11.13 shows the variation with d/b 1.6
for two values of b/A. This case
has been treated in a more general /
1.4
fashion by Pincherle, 1 who dis-
cusses other modes in rectangular
1.2 —
guide. Pincherle also examines a
waveguide of circular cross section
\vith a dielectric rod down the 1.0
center. This case can be con-
sidered from an impedance point of
view by the methods of Chap. 8.
11.7. Dielectric Post in Wave- 0.6
h = 0,45
a
guide.—If there is a cylindrical ke= 2,45
dielectric post of circular cross
0.40
section in rectangular guide 0.2 0.4 0.6 0.8 1)
operating in the Hlo-mode, which _g
extends in the direction of E at FIG. 11,13.—Variation of X/Xc with d/b for
the center of the guide, the rela- two values of b/X.
where R is the radius of the post and a the width of the waveguide. This
expression was derived for the case for which lc/cOl (2ml?/A)z <<1 and
the series arms of the equivalent T-network have a negligibly small
impedance. The expression is valid to within 3 per cent for the range of
wavelengths given by ~ < a/X < 1,provided that the radius of the post
is small enough.
The expression holds for a complex c as well as for real values. It is
possible to solve for c in terms of a measured Y and in this way measure
dielectric constants, For example, it was found for a = 0.424) in. and
A = 1.25 cm, a column of water for which R was 0.009 in. had a measured
admittance of 1 – j. The value of c/co deduced from this was
c
— = 39 – j17.
eo
where nisan integer. The information given in these figures does not
make it possible to find the Q of the cavity. We notice that Ql, the
dielectric Q, will depend on the mean value of E2 in the dielectric, relative
to the mean value of E2 in the whole cavity, and hence Ql, as well as
the resonant frequency, depends on the position of the dielectric within
the cavity.
To find an expression for Q1 it is convenient to regard as the direction
of propagation the direction of the
normal to the dielectric interface as
before and let this be the z-axis.
The metal losses will be neglected. L 1 1
Substitution for B from Eq. (52) and the use of Eq. (54) reduces this to
(56)
where
1.
M = a – E “n “a Cos “a
N = (1 – a) sinz ~,a + (1 – a) ~ az~,a + ~ sin ~,a cos ~la.
(2 Y:! Y)’
and the voltage across Y will be
(2 YOZ+”Yj”
This quantity is proportional to the ~
dominant-mode field; therefore its ~
square is proportional to the dielec- ~
tric loss. It is evident that the loss =
decreases monotonically as Y is
increased. To this must be added
the loss produced by the hi.gher- Admittance of aDerture Y
mode fields. If the aperture is com- ~10. 11.16.—L0ss vs. admittance for a
dielectric-filled
couplingaperture.
pletely open, -no higher-mode fields
are excited. This is also true when the aperture has no opening, Th\w
the dielectric loss caused by the higher modes will be zero ~vhen Y is zero;
and as Y increases, the loss will increase, pass through a maximum, anti
then decrease again, approaching zero as Y approaches infinity, The
total loss may then be represented as in Fig. 11.16.
1E. Feenberg, “ Use of Cylindrical Resonatorto }Ieasure Dir.lcrtritPropertiesat
UltrahighFrequencies,” Sperry Gyroscope Co,, July 1942.
SEC. 119] PIWPAGA Y’IOATIN IOiVIZED GASES 393
(57)
when the medium filling the waveguide has a conductivity u. lfu isa
complex quantity, it may be written
~ = ~t — jcff, (58)
The propagation constant is then
2
2=! +jupu’–wzu ,–: (59)
7
() a ()
Thus u“/a is the contribution to the dielectric constant of the con-
ductivity of the medium. It will be assumed that c and p are real. If
it is remembered that ~ = a + j~, then
If the substitution of
2T 2
U%p = —
()
and
It should be pointed out that h and h, are the wavelengths in the medium
and in the wave guide, respective y, when the conductivity u is zero.
The expression for P may be written as
Thus if
2/32 = UPC”
-(w- [a~’’-(l’l
27r 2
> w.w”,
(-)h,
and if
/32=
() :2–
Q
UW’’>O;
(-)
2U 2
A,
< W&J”,
& = 0.
In the latter case, the waveguide is beyond cutoff. The cutoff wave-
length in the waveguide is
%
(l,)..,.,, = -. (63)
<cop.”
The following approximate expressions are useful. If
Zr 2
l.qlur’ — >> wlu’,
() ~
-(?J
‘+%’-;:*15T
.=+ (64)
and
““
and
where 1 is the meanf ree path, n the density of charged particles of charge
e and mass m, k is Boltzmann’s constant, and T’ is an effective tempera-
ture defined bv
(69)
P=l>%.
()
2k T
(70)
If only small fields are considered, the difference between T and T’ may
be neglected. It is seen immediately that the effects of the positive ions
may be neglected compared with those of the electrons because of the
occurrence of m in the denominator of the expression for a. This expres-
sion reduces to simple form when the frequency is very low or very high.
For low frequencies, that is, for
mlz
“ ‘< ~T’
4 et in . wez12n
(71)
“ = ~ (2~mkT)~ – 3 3kT “
1 H. Margenau, Phys. Rev.,69, 508 (1946); RL Report No. S36,Oct. 26, 1945.
396 DIELECTRICS IN IVAVEGUIDES [SEC.11.10
The real part of u is the Langevin formula usually written in terms of the
mobility. The imaginary part corresponds to a change in dielectric
constant that is independent of CO. For high frequencies, co’ >> m12/2kT,
(73)
I
o 0.5 1.0 1,5
Conductance.
FIG. 1119. -Impedrmce chart for the absorbing sheet of Fig, 11.17. The v,’avele”gths
are indicated on the curve.
and depends, therefore, for loss angles that are not too large, essentially
15
10
0
2345678 9 10 11 12 13 14 15 16
Wavelength in cm
FIG. 11,20.—Reflection coefficient of absorbing sheet,
which depends upon the sum of the loss angles. An efficient ferro-
magnetic absorber should have large and equal values of q and { and large
values of c’ and p’ subject to the condition that ~’/c’ be nearly equal to
Po/eo.
The construction of an absorber utilizing these principles is shown in
Fig. 11.21. The synthetic rubber is impregnated with iron powder,
prepared from iron carbonyl, of particle size less than 10Y. The material
has a specific gravity of about 4. At a wavelength of 10 cm the dielectric
constant C’/COis approximately 25, ~’/AO varies from 3 to 4, tan p is
approximately equal to tan ~, and tan ~ and tan p lie between 0.3 to 0.4.
The intrinsic impedance is therefore real but rather small, and a resonant
construction has been adopted to match into the absorbing material.
The waffle construction, at wavelengths long compared with the grid
400 DIELECTRICS IN WA VEGUIDES [sm. 1110
-IL-IL
R
Spacer
Metal
IDE ron-impregn ated
rubber
lrlr
,4mm/mm~
1mm
FIG. 11.21 ,—.+bsorbing sheet with resonant construction.
L2 L1
——
::
Short
Grid - c1 circuit
Yo= 1 Ml -
1 ! 1 I 1 1 1 1
7 8 9 10 11 12 13 14
Wavelength in cm
FIG. 11.23.—Reflection coefficient of three samples of absorbing sheet of Fig. 1121
BY R. H. DICKE
-
(a) (b)
Symmetry about a single plane
>
e ..--—--
K$?!3 Q
,.
,, ;
,’ .“ .- : ._
Y
,--6--- ‘,
. . \’\-
<, ,,,
(d) (6)” \ (f) (9)
&a=3- (h)
Complete symmetry
FIG. 12 1.—Junctions having reflection symmetries only. Symmetry of one or more planes.
I
12.1. Classes of Symmet~.—A number of symmetrical junctions are
illustrated in Figs. 12.1 to 12.3, inclusive. It is evident that this col- 1
lection is by no means complete and that there is an unlimited number
I
of possible symmetrical junctions. Nearly all junctions of transmission
401 I
402 THE S’YM.!l ETR Y OF WA VEGIJIDE J 7JNCTIONS [SEC. 121
+
(a)
One axis Three axes
t (d) (e)
Symmetry about a point, bn axis, Symmetry about a point
and a plane
FIG. 12,2.—Junctions hating reflection symmetries only, continuation.
operation, any given solution need not be. For instance, a wave moving
to the right can be transformed under a reflection into a wave moving to
the left. Ho\vever, a standing \vave with the symmetry plane at a node
or loop is left unchanged by the reflection. Suclr a solution is said to be
invariant under the symmetry operation.
\
Symmetrical junctions will be investigated by looking for symmetrical
\__—
solutions of Maxwell’s equations that satisfy the boundary conditions of
..
/,\—_
—.——
the junction. Any solution can then be expressed as a linear combination
—
8!=&/
\ /
/
/’
I
~+~+-q
‘4 I
X1
\ 4
Terminals Iris Terminals
(1) (2)
el = ZIIil + Z12i2,
(1)
ez = Z21i1 + Z22i2. }
where the e’s and i’s are the currents and volt ages at the junction.
The first observation that can be made from symmetry is that
Z,, = Z,,. This is evident because a reflection of the waveguide through
the plane of symmetry leaves the guide unchanged. However, this
reflection interchanges the field quantities at terminals (1) and (2),
Thus an interchange of (1) and (2) in the elements of the impedance matrix
should leave it unchanged. This is possible only if
Z]2 = Z21,
(2)
Z22 = z,,. }
el = il(Z1l — Z12),
ez = i2(Z11 — 212), (3)
e2 = —el. 1
For this particular antisymmetrical solution the impedance seen iooking
into terminals (1) or (2) is
from Eq. (4), Z1l = ZIZ. This is just the condition that the iris be a
shunt susceptance at the symmetry plane. Perhaps the easiest way to
see this is from the T-equivalent of the impedance Eq. (1) (see Chap. 4).
If Z,, = Z,, = Z,*, the circuit becomes a pure shunt element of imped-
ance ZI* across the junction.
The odd distribution in electric field of Fig. 12.5 is one symmetrical
solution of Maxwell’s equations. It is evident that the even solution is
another,
el = ez,
(5)
i, = iz,
These are the only two solutions which are symmetrical about the sym-
metry plane. It is evident that any other solution of Eq. (1) can be
obtained as a linear combination of the solutions given in Eqs. (5) and (3).
MATRIX ALGEBRA
12.3. The Eigenvalue Problem.—The problem just considered was so
simple that it could be seen by inspection that an odd or even distribution
of fields about the symmetry plane was a symmetrical solution of Max-
well’s equations. In more complicated cases involving many waveguides
in complicated configurations the intuitive approach may not be sufficient
to obtain a correct solution. It is the purpose of the next sections to
develop formal methods that are applicable to these more complicated
cases.
A formalism that is useful in the discussion of symmetrical junctions
is that provided by the theory of the eigenvalue equations. This theory
is developed here only to the point actually needed in the subsequent
problems. A more complete treatment can be found in any of the stand-
ard works on matrix algebra. There are also introductory treatments
t’or the reader unfamiliar with this field. 1
The Eigenvulue Equations .-For a square matrix P, a column vector” a,
md a number p, the equation
Pa = pa, (6)
Zi = ~i, (7)
where
1
i=i, (8)
() –1 “
(P – pl)a = O, (9)
z3
c~aj = O (11)
2 c,a, = O. (13)
All other terms in the sum vanish. Equation (15) can be satisfied only
by c, = O. In a similar way each of the other c’s can be shown to vanish,
and the n eigenvectors are linearily independent.
1 Jlrirgenatl and llllrphy, op. cit., Chap. 10, p. 299.
SEC. 12.4] SYMMETRICAL MATRICES 407
z
,=(J
cja, =0 (16)
a. =
2
j= [
b,a,. (17)
P=P. (20)
Multiplying Eq. (23) on the right by a,, Eq. (22) on the left by ak, and
subtracting,
(p, – p,)~,aj = 0. (24)
Ilultiplying Eq. (27) on the right by a, and Eq. (25) on the left by 5:
and subtracting,
(P; – PJ)~J*aj = o (28)
Thus all eigenval(les are pure re:tl. If Eq, (29) is substituted in Ml (26) ,
it is seen that both a, and a; tire eigenvwtors of the same eigenval[le.
SEC.12.5] RA TIO.VAL MATRIX F[l,Y(:TIONS, DEFINITIONS 409
Zai = z,a,,
\vhere
and
a]=c)’ az=E)
z, = o, Zz = 2Z12.
It should be noted that z, and .z2are pure imaginary and a, and a. are
real and orthogonal.
12.5. Rational Matrix Functions, Definitions.-Any expression of the
form
may be evaluated by taking the product of ak by the last factor, then the
product by the second last factor and so on. The result is, using Eqs. (33)
and (34),
j(P)a, = j(pi)ao
Sal = siai,
Zj—1
—
‘7=%+1
The general result has been proved that the impedance, admittance and
scattering matrices have common eigenvectors. It should be noted that
since z; = —z~,
Zj—1z:—1
)S,p =——=1.
z~+lz:+l
p“+c,p”-’+. .”=c)
SEC. 127] C.41’1,B1’-HA,Jl ILl’OiV’S THEOREM 411
and let the 7Lroots of this equation be pk. Form the matrix
M = P;’ + c,p”-’ + + C,,.
a — d~a~.
2
k
Then
, Ma — 7.d~.(pl +
y
C,P;-l . . “ + c.)a~.
*
The expression in parentheses on the right of this equation vanishes.
Therefore
Ma = 0,
The sum of the roots of the polynomial is equal to c,. The characteristic
equation is the expansion of det (P – pl ). In the expansion of the
Z“nn = Zpk n
n k
I
412 THE SYMMETRY OF WA VEGUIDE JUNCTIONS [SEC. 128
F.. x=–xj
(38)
F= ~E.(z,y,z,u) = –E.(–x,g,z,co), I
SEC,12.8] REFLECTION’ IV A PLAA’E 413
It is evident that
F. ~ (cE=) = C(FZ ~E.), (39)
An operator is said to be linear when conditions (39) and (40) are satisfied.
Information concerning general properties of the junction can often
be obtained by searching for solutions of’ Maxwell’s equations that are
invariant under the symmetry transformation. It is desired to find
solutions of Maxwell’s equations that are left unchanged (except for a
possible change in phase) by the symmetry operator. A change in phase
can be compensated by a change in time zero, and such a change in solu-
tion is not significant.
If E. is a symmetrical solution, then
H the solution is continuous across the symmetry plane, that is, if the
414 THE SI”MMETR Y OF WA VEGUIDE JUNCTIONS [SEC. 12.9
symmetry plane does not contain a metallic sheet at the point in question,
then setting z = O in Table 12.1 yields
Even Odd
E= = O, E. = O,
H, = O, E, = O, (45)
H, = O, Hz = 0,
I
Conditions (45) for the odd case are just the ones that must be satisfied
by the field quantities at the surface of an electric wall. In other words
the field distribution is the same as though the symmetry plane were
replaced by a perfectly conducting metallic film. In a similar way the
even solutions correspond to a magnetic wall at the symmetry plane.
12.9. Symmetry Operators.—Operators F. and F. can be introduced
in a similar manner to repreqent reflections in the *Y- and the zz-planes.
All the above results follow exactly as for the operator F=. These reflec-
tion operators may be applied in combinations to the various coordinates.
For instance a reflection in the yz-plane followed by a reflection in the
zz-plane is equivalent to a reflection in the z-axis (or a rotation of 180”
about the z-axis). A new operator R, may be introduced to represent
this reflection in the z-axis. Formally,
R, = F=FY == FVFZ. (46)
In a similar way
R= = FVFZ= FZFU,
(47)
R, = FZF= = FJZ.
1
Another symmetry operator is
P = FZFVF,. (48)
‘rABLElzz.-~~ULTIF’LICAmON
TABLEFORTHEREF1.ECTION
GROUP
I R. Ru R, ‘:F. F. F. P
T I R. R, R,!Fz F, F. P
R. R, I R. RV~P F, F. F.
R, R, Rz I R,!F. Pz F. Fv
R. R. R. R, I ~FU F, P F.
.
F. F. P F. FV:I R. R, R.
F, F. F, P F.!R. I R, R,
F, F. Fy F. P R. R. I R.
P P F. F, F,:Rz R, R, I
I
It can be verified from the table that a 180° rotation about the z-axis
follc~\ved bytireflection intheyz-plane isecluivalent toa reflection in the
,rz-plane. Symbolically,
F,R, = R, F,.
,.
...:
problem again is to introduce with a simple illustration the formal
-.
,,,
methods of solution. The junction is shown in Fig. 12.6. The iris may
have an aperture of any shape in a Y
metal plate of uniform thickness d.
&
The junction has a symmetry h
e=[::1i=k]
of the junction, then el is a measure of E, at junction (1) and ZI is a
measure of H. at junction (1). Under F=, which symbolizes the trans-
f{]..matiou (37),
The sign convention on the junction currents (into the network) results
in no sign reversal in Eq. (53). The reflection operator F= takes the form
Thus,
u
10
(54)
Fi = i, Fe = e’. (55)
9
The transformation given in Eqs. (52) and (53) may be made by per-
forming the matrix operation of Eq. (55). The matrix F is said to repre-
sent the operator F..
418 THE S1’.11 M E1’RF OF WA VEGUIDE JUNCTIONS [SEC.12.11
-., . . , ,.. .. . ...= .
11 the Impedance matrix 01 the junction 1s Z, then
e = Zi. (56)
But the transformed voltages and currents also satisfy Eq. (56)
e’ = Zi’,
Fe = ZFi,
FZi = ZFi, (57)
(FZ – ZF)i = O. 1
Even Odd
el = e2, el = —ez,
(59)
‘il = iz, il = —iz. )
det
[) f
If
1 =jz–l=().
11
“=72 H 1’ a’=$ H-:”
Note that a, and az are linearly independent (Theorem 1), orthogonal
(Theorem 3), and pure real (Theorem 4) and are normalized to unity.
From Eq. (58) and Theorem 6, a, and a, are also eigenvectors of Z.
Thus the eigenvalue equations of Z can be written as
Zak = z~ah; (62)
As was discussed previously (Sec. 12.8) the even and odd solutions
●
are those for which the symmetry plane becomes a magnetic and an
. ----- . .
electric wall respectively. Figure 12- 7 m
“ a cross section ot llg. 12% Ior
these two cases, showing one side 1
only.
Electric wall
Nothing very much can be said Even a
about the eigenvalue zI for the even Magnetic wall
//////////// ~
case without a solution of the bound- (1)
Terminals 2
ary-value problem. However, it is
( clear that the obstacle for the odd
I case will reflect in such a way that an
I
effective short circuit lies somewhere odd a
between the symmetry plane and the
left side of the iris. The eigenvalue
Z2 will be capacitive, because the of short circuit
short circuit lies between one-quarter FIG, 12.7.—Boundary conditions for
and one-half guide wavelength from symmetrical and antisymmetrical solu-
tions for the thick iris.
the terminals. If the thickness of
the iris, d, is small compared with the guide wavelength x,, then
I
I 0<jz2<7r$
Q
#
The impedance matrix can be written down directly in terms of zl and zZ.
However, it may be obtained formally by the following procedure.
Equation (62) may be combined to form the single equation
(63)
where
Note that A has al and az as columns. The matrix A has columns that
are orthogonal and normalized to unity. Such a matrix is said to be
orthogonal. It has the property that
~ = A-’. a
“=-
OejZ2e~ $
FIG. 1Z.S.—Equivalent circuit of a thick iris.
The symmetries illustrated in Fig. 12.10 are not the only ones. The
plane containing the axes of @he three guides is a symmetry plane, and
the intersection of this plane with the other planes (that is, the z-axes of
the guides) are symmetry axes. However, these symmetries do not play
an important role in the properties of the Y-junction. In fact, these
extraneous symmetries will later be removed by placing a post, along the
three-fold axis, that does not extend completely across the guide.
1
SEC. 12.12] THE SYMMETRICAL Y-JUNCTION 421
The unit operator together with R,, R,, F,, F2, and Fs form a group
4
whose multiplication table is Table 12.6. It should be emphasized that
TARL~12.6.—SYMM~T~Y
GROUP
I R, R,;F, F, F,
T I R, R,!F, F, F,
R, R, R, I !F, F, F,
R, Rz I R,jF, F, F,
1 .....................
F, F, F, F,! I R, R,
F, F, F, F,!R, I R,
FZF3 F, F,~R2 R, I
100
F,=oo
[)
010
I.
Let
TI= !,
b’=i%~(a’ I
- a’)”
424 THE LS’YMME7’R Y OF WA VEGUIDE JUNCTIONS [SEC. 1212
F,b, = b,,
F,b, = b,, (81)
F,b, = – b,. )
It can be seen by inspection that the b’s are real, orthogonal, and linearly
independent.
By Theorem 5 (see example at end) the a’s and b’s are also eigenvec-
tors of S and Y. Let
Magnetic
walls
I
y!= [~’’”!, ~?/,
‘ (3)
I
the symmetry planes. Thus arm (1) is terminated in a V-shaped
magnetic wall. The solutions bz and b~ are eigensolutions of Fl, even
and odd respectively, and are therefore characterized by a magnetic
and an electric wall in the symmetry plane (see Fig. 12.11).
1 The Scattering Matrix .—The eigenvalue equations
Sb: = s,b~,
SB = BSd,
[see Eq. (63)] where
B=
—— ——
in v% @
and
S,oo
s.= o .s, o.
[10 0 .s~
1’
S = BSfi. (83)
If S is multiplied out, there is obtained
[1
ffPP
S= pap, (84)
pp.
where
a = +(s1 + 2s2),
(85)
B = +(s1 – 5’2). )
Equation (84) also gives the impedance and admittance matrices Z and Y,
provided that in Eqs. (85) Sj is replaced by Zi and yj respectively, where
l+sj l–sj
(86)
“=l-S: y~=l+$j”
Note that the sum of the diagonal elements, or spur, of the matrix given
by Eq. (84) is equal to the sum of the eigenvalues (Theorem 8).
Power Division.—A junction is said to be matched when all the
diagonal elements in the scattering matrix are zero. Clearly a necessary
condition for a matched junction is that the sum of all eigenvalues of S be
zero. In Eqs. (85) the phases of s, and S2 can never be of such values
that a = O. Hence the symmetrical Y-junction can never be matched.
This property is much more general than appears above. In fact
no junction of three transmission lines can be matched. To show this,
assume that such a junction has been matched. Its scattering matrix is
[J
~M ~13
S,Is:, = o.
In a similar way the other two products give
~
S12SY = o, s,,% = o.
These three equations cannot be satisfied unless two of the three
quantities S12, S23, S13 are zero. But in this case there is a column of
the matrix that is zero. This is impossible, since the product of every
column by the complex conjugate of itself is unity. Thus it is impossible
to match a T-junction or any other j unction of three guides.
T
I It is to be noted that the best match which can be obtained with the
6 symmetrical Y-junction is when
s, = —s2.
I (88)
I
I ‘=4-:
-1-i
With two of the waveguides terminated -in their characteristic imped-
’89)
ances, eight-ninths of the power entering the third arm goes into these
terminating loads. The remaining ninth is reflected back to the genera-
tor. In order to satisfy the condition (88), it is necessary to adjust the
phases of sl and .sZrelative to each other. One way in which this can be
done is to insert a pin in the guide along the symmetry axis. It will be
remembered that the electric field is zero along the axis for the modes
whose eigenvalue is sZ. The pin does not affect these modes at all. The
electric field is a maximum at this point for the eigenvector al with the
eigenvalue s,. Hence as the pin is extended across the guide, the phase
of SI would be expected to change without altering Si. It does not neces-
sarily follow that by such an adjustment, the phase of S1can be made to
have any desired value, but it is to be expected that a sizable variation
can be obtained in this way.
12.13. Experimental Determination of SI and st.—If plungers are
inserted in two of the arms in symmetrical positions, then power entering
the third arm will set up standing waves in the system. The plungers
are adjusted until the nodal points come in the same place in each of the
three arms; then the position of the nodal point is measured. The
phase determined in this way is the phase of one of the eigenvalues.
The procedure just outlined is correct in principle but would be very
difficult in practice. A procedure that involves a single plunger is much
better in many ways. It is to be noticed that the eigenvector b%has no
fields in one of the arms. If there are no fields, the plunger can be omitted
in this arm. In fact, the condition of no power in this arm is a convenient
test for determining when the remaining plunger is in the correct position.
To see this algebraically, let a plunger be placed in arm (3) of the Y-junc-
tion and let a matched load terminate arm (1). Then the scattering
equation is
Sa = b,
where
al = O,
aa = b~e–idI ‘ (90)
428 THE SYMMETRY OF WA VEGUIDE JUNCTIONS [SEC.12.13
(a – 5’2) f?
. 0. (94)
B (a – ~io)
Thus if the reflection coefficient in arm (2) is sZ, the plunger is in such a
position that Eq. (95) holds. The ratio of az to as is the ratio of the
minors of a column of the determinant in Eq. (94).
a2 _a_ei+
— — – 1,
a3— —~—
a2+a: =0.
Substituting in Eq. (92),
b, = O.
Under these conditions, therefore, no power enters the load on arm (l).
Conversely, if the plunger is adjusted until no power enters arm (l),
then s, is given by relation (95).
The determination of s, is the next step. One way in which it can be
determined is to measure a by measuring the reflection coefficient atone
of the arms with the other arms matched. Then, making use of Eqs.
(85), s, can be determined. Another method, which is capable of greater
accuracy, will now be outlined.
It is to be noted that the eigenvector ba is odd with respect to F but
that b, and b~ are even. The eigenvectors b, and bz have clifferent eigen-
values with respect to S, and the positions of nodes are therefore clifferent
for bl and b~. A linear combination of b, and b, with the same time phase
is a new standing-wave solution. However, the nodes occur at different
places. In particular, by taking the right combination, the nodes in
arms (2) and (3) can be made to occur *XOaway from the nodes of b3. A
linear combination of this standing-wave solution with ba, since the time
phase of ba is in quadrature and its amplitude is equal to the other solu-
~
(
SEC. 12.13] EXPERIMENTAL DETERMINATION 429
tion, can be made such that there are pure running waves in arms (2)
and (3).
To recapitulate, there is a solution that corresponds to running waves
in arms (2) and (3) and a standing \~ave in (I). This solution can be set
up by a plunger in arm (1). The positionof this plunger.can be adjuster’
until there is no reflection by the Y-junction. The position of the plunger
is then an accurate measure of a combination of sl and sZ. Since S2
I is known accurately, S1can be determined.
It will now be shown that the linear combination
I
~=-L - b, + & b, + ~S:/S:+s:2) b, (96)
S1 + S2
s a set of incident waves which results in pure running waves in arms (2)
and (3). The components of Eq. (96) are, from Eqs. (S2),
2s2 + s,
‘1 = S2(S1+ S2)’
gz . ——S2 — sl
S2(S1 + S2)’
Thus this solution corresponds to waves incident upon the junction from
arms (1) and (2). After scattering by the junction, the waves are given
by
h = Sg,
where
since b~, bt, and bt are eigenvectors with eigenvahles s1, s2, and s2. The
components of h are
2s, + S2 2s, + St
h,=—=
SI + S2 2s2 + s~ ’29”
hz = O,
sl — S2
h,=—= —Szgz.
S1 + S2
Ihd = lgd.
Thus there is a pure standing wave in arm (1). If this standing wave is
( set up by a plunger in the correct position, the position of this plunger is a
430 THE SYh4METR Y OF WA VEGUIDE JUNCTIONS [SEC, 1214
(97)
1
SEC. 1214] S1’MMETRICAL T-JUNCTIONS 431
If Eq. (98) is substituted in Eq. (99) and the matrix product performed,
the elements of S must satisfy
S1l = S22,
(loo)
S,, = sZ&
I
‘=1 ~ -~1”
The rotation operator applied to the axial T-junction takes the form
‘=! : -~1
Thus the rotation operator is quite equivalent to the reflection
operator applied to the series T-junction, and the properties of the axial
, $..,>* ~ ...~.,. \
These relations, except for the change of sign, are the same as Eqs. (100).
For this reason the properties of all the T-junctions will be very similar.
In order to avoid duplication, only the shunt T-junction will be discussed
in detail.
432 THE SYMME1’R Y OF WA VEGUIDE JUNCTIO1h-S [SEC. 1215
det (F –fl) = O,
Thus there are two positive eigenvalues and one negative. The negative
eigenvalue is nondegenerate; its eigenvector is therefore unique, except
for the usual multiplicative constant. The eigenvectors of the degenerate
positive eigenvalue are not unique, but the following set of eigenvectors
are orthogonal and real:
“=M“=l-~l’
‘3=[-!
Let the eigenvalue equation for the scattering matrix of a shunt
T-junction be
Sbj = sibi.
b, =
>2
—— “
[1 c1
By Theorem 6, b~ = a~. The form of the b’s is not particularly simple,
because the positions of the reference planes in the arms 1, 2, and 3 have
been chosen in an arbitrary way. We shall now indicate how these planes
may be chosen so that a = 1.
SEC,1215] THE SH uNT T-JIJ,VCTIOA- 433
or the same thing with the opposite sign for the third element. It is
always possible to choose the original linear combination in such a way
that al is the correct eigenvector. A comparison with b, shows that
a=l. Thlls for these new reference planes,
Sai = s,aj,
ad-r
s=ha~, (103)
[177/3
434 THE S 1“,11.lfETR 1- OF 1!”.4VEGUIDE J1;.VCTIONS [SEC. 1215
where
a = +(1 + sz + %s),
B = +(1 + s,), )
(104)
1
@
7= T(l– S2),
6 = +(1 + s, – 2s3).
h’ote that
%+@= l+ S2+Ss
a=p=o.
o S,, S31
S= S210 S3,
[1~31 s32 s33
Since S is unitary, the product of the first column by the complex conjll-
gate of the second vanishes;
S31S;2 = o.
Is,,l = 1,
since the square of each column must be unity. Then
S31 = S32 = O,
Is,,l = 1s331= 1.
I
(106)
where
S1l = +(s1 + 52),
With the iris present, S is given by Eq. (106) which, it is to be noted, can
be obtained from Eq. (107) by adding ~(.sl + sZ) to every element of
Eq. (107). Thus a pure susceptance generates waves of equal amplitude
in either direction. Another way of expressing this relation is
TIP- ‘rA
+ 111% = ‘lyP
1
FIG. 12.13.—Operation of a T-junction as a stub tuner.
in (3) but with traveling waves in the other guides. To be more explicit,
let a solution with power entering guide (1) be combined with a solution
with power flowing into guide (3). This is illustrated in Fig. 12.13, in
which the small Greek letters have the same meaning as in Eq. (103).
436 THE SY.U,IIEI’R1’ OF lf’A VEGUIDE JC.VC1lIO,VS [SW. 1216
It is apparent from Eq. (104) that this relation is correct. Hence the
junction acts as a shunt susceptance. From Eq, (103),
a—L$=s’~.
Thus the electrical length of line between the two terminals is sS. This
electrical line length may be deter-
mined experimentally by inserting a
plunger in arm (2) and adjusting until
J ~Ma’neticwa” there is no coupling between (1) and
(3). Under these conditions the plung-
1 er is electrically an integral number of
EigenVectors a, and a2 half wavelengths from the effective
FIG. 12. 14.—Boundary conditions for susceptance. Another method that can
eigenvectom a, and al.
be used to determine this length will be
taken up in the next section.
It should be noted that the reflection coefficient of the junction is
It may be verified, by substituting from Eq. (103), that this equation has a
solution with real #.
The eigenvectors a, and az are also eigenvectors of F with eigenvalue + 1.
This is the usual even solution which is equivalent to the field distribution
when the symmetry plane is replaced by a magnetic wall (see Fig. 12.14).
The magnetic wall reduces the junction to a two-terminal junction for
which there are two eigenvectors al and az. The eigenvector at is odd
ahmlt the symmetry plane. The fields are those with the symmetry
%x. 12.18] THE SINGLE-HOLE I>I1tlWTIONAL L’OUPIiER 437
plane replaced byametd wall (see Fig. 12.15). Tomeasure the proper-
ties of aright-angle bend with a magnetic wall, it would be necessary to
apply equal incident waves to the symmetrical arms of the ‘I’-junction.
The right-angle bend of Fig. 12”15, how-
ever, can be constructed, and its prop-
Electtic wall
erties measured directly. NTote that the
eigenvalue of S for as (Fig. 12”15) is SS
which is also the distance along the line to 41
the effective position of the susceptance
o0- Eigenvector as
when the junction is used as a tuner. FIG. 12.15.—Boundary condi -
This, then, is another way in which this tions for field,ine,,
~,gnetic eigenvectora~ showing
lineJength can be determined. The series
T-junction is analyzed in a similar way. If a plunger is inserted in guide
(3), it acts as a series reactor in the line.
12.17. Directional Couplers. —Most of the directional couplers in use
have a symmetry such that their scattering matrices have the form
ap’.yti
fia;tiy
s = . .. . .. . ... (109)
-y~’afl
[1c$~;fla
It will now be shown that if this junction is matched (a = O), it is also a
directional coupler; and conversely if it is a directional coupler (say
~ = O), then it is matched. First let a = O. Then, since S is unitary,
In Fig. 12.16 the symmetry axes are designated by El, Rz, and Ra.
The terminal planes are numbered (1), (2), (3), and (4), and the direction
of positive electric field in each plane is indicated by an arrow. Figure
12.17 is a representation of the common metal wall between the two
R3
(l)t 1
N’
R2
FIG. 12.16,—Directional coupler.
0 oil o
0 o~o 1
RI = (112)
1 0/0 o
0 1;0 o
1--------------1”
The operator Rl, operating on the terminal quantities
al
~= U2 (113)
Ua
IIa4
SEC. 12.18] THE SINGLE-HOLE DIRECTIONAL COUPLER 439
interchanges the first and third as well as the second and fourth com- I
ponents. Inasimilar waytheoperator RZtakes the form
o 010 1
0 0!1 o
R, = ---------------- (114)
0 l~o o
1 0;0 o
!1
Both RI and Rz have doubly degenerate eigenvalues. However, it is
possible to take a linear combination of R 1 and Rz whose eigenvalues are
nondegenerate. Let
M = c,RI + c,R2, (115)
where
Note that
cl = +(1 +j),
q = +(1 —j). ) (116)
(117)
al = (cIRl + czRJal
. ~lRlal + czRZal
= clrla, + czr2al
> = (w, + c,rz)al,
where r, and i-z are eigenvalues of RI and RZ and must be either f 1.
1 From Eqs. (116), t, + c, = 1; thus for the eigenvalue ml = 1, the
.
(120)
a! 1) = ~(,1),
(121) I
a~l) = a$). }
az=tl
‘3=’[4
Note that M is symmetric, and by Theorem 3 the a’s are all mutually
orthogonal. They have been normalized to unity. The eigenvalues of
a4=i+l “24)
the three symmetry operators Rl, Ra, and Rj for the four eigenvectors
al . . . aAare given in Table 12.7.
~4
R, 11–1–1
Symmetry ~ ~_l_l,
I;igcmvalues
operators ~
I !3, 1–11–1 1
Since the scattering matrix must commute with R, and R,, it also
commutes with M. By Theorem 6, the a’s are also eigenvectors of S,
In. Eq. (124), the eigenvectors of S are pure real, a condition required bv
SEC. 1218] THE SINGLE-HOLE DIRECTIONAL COUPLER 441
1 –1 –1
–1 –1 1
S,o[o o
0 S,jo o
.: ........
o 0;s30
I.0 o~o S4
Since T is symmetrical and orthogonal, Eq. (126) may be written as
S = TS,T. (127)
The product on the right-hand side of Eq. (127) maybe expanded to give
s= (128)
where
~ = +(sl + S2 + 53 + w),
p == *(S, – s, + s, – s,),
(129)
7 = +(s1 + 52 – 53 – S4),
a = +(s1 – S2 – S3 + SJ. [
spur S=sl+s2+s3+s4.
(131)
442 THE SYMMETRY OF WA VEGUIDE JUNCTIONS [SEC.12.18
s!:; : :;,
S$ —
(133)
~io) = +1!
Sjo)= —1. ‘1
The conditions of Eq. (135) can be shown formally to follow from Eq.
(134). Let us introduce the operator R, which rotates o through 180°.
Thk has the effect of interchanging the terminals (1) and (2). This
transformation, in acting on the terminals, takes the form of the matrix
o l\o o
1 o~o o
0 011 0
0------------1
o~o 1
444 THE S1’MMETRY OF WA VEGUIDE JUNCTIONIS [SEC. 12.18
R&R;lRpak = skRdak,
S’(R~a,) = sJR~aJ,
\vhere S’ = R&3R;1 is the scattering matrix of the transformed junction. 8
But
s’ak = S;ak, (136) $
where
Sj(f?) = sk(~ + 6). (137)
‘rhus
Sl(e + T) = SI(e),
S2(L9+ T) = s4(e),
(138)
s3(e + 7r) = S3(19),
s4(e + m) = s2(e). I
Also it may be seen that
Sk(e) = Sk(–e),
I I
(a) (b)
Fm. 12,1 S.—Perturbations of eigenvalue of S as a function of a.
‘l=[$!
‘2=
N-I
Note that although the symmetries of Figs. 12.19 and 12.16 are quite
different, the generators of each of the symmetry groups take exactly
the same form when written as matrices. A comparison may be made
with Eqs. (111) and (113). FI and RI are quite equivalent to each other,
and F, is equivalent to R,. As a result of this fact, except for those
results connected with the symmetries of the field, namely, the field
dktributions for the various eigenvectors, all the results obtained for the
single-hole directional coupler also apply to Fig. 12.19. In particular,
the eigenvectors of Eqs. (123) and (124) are eigenvectors of the scattering
matrix that takes the form of Eq. (128). Table 12.7 is still valid if
R, and R, are replaced by F, and F, and R, by the rotation R.
Fields Associated with the Four Eigenvectors.—From Table 12.7 it
can be seen that the eigenvalues of FI and Fa for the eigenvect or a ~ are
+1. From Eq. (45) it is seen that the fields at the two symmetry planes
satisfy the conditions imposed by magnetic walls along the symmetry
planes. In Fig. 12.20, these boundary conditions are shown. The
v figure also shows the boundary conditions that the remaining eigenvectors
must’ satisfy. Note that if a thin metal plate is inserted in the plane F,
in such a way as not to destroy the symmetry, the phases of S2 and SI
446 THE 8YMMETR Y OF WA VEGUIDE JUNCTIONS [SEC. 1219
will be changed but not those of SSor s4. Also SI or S4can be changed by a
symmetrical metal wall in Fz. Moreover, it can be seen that a metal pin
insert ed in the guide along the symmetry axis will change sI but leave
the other eigenvalues of S unchanged.
The insertion of irises in the two symmetry planes and a pin along
the symmetry axis are three adjustments that can be used to obtain any
scattering matrix consist ent with this symmetry, for example, that for a
dkectional coupler. It must be assumed, however, that these adjust-
ments have sufficient range to obtain the desired results.
In order to be a directional coupler, the junction must be matched.
The condition for a match is the vanishing of the sum of the eigenvalues
q
- F2
y
‘x/
S2
a2
Magnetic
Reflected planes S4
amptitude = S1
a, a,
FIG. 12.20.—Boundary conditions for eigenvector solutions al, a~, aj, a~.
S1 = —s3,
(141)
St = —s4. 1
Figure 12”21 shows the form that the obstacles might take. The power
1 distribution is that given by Eqs. (140). Notice that the dk.tribution is
I
just opposite that which one might expect from simple optics.
If in Fig. 12.19 the angle @is very small, the two eigenvalues ss and S4
are very nearly equal. This can be seen by reference to Fig. 12.20.
In this case the wall F1 together with the remainder of the guide forms a
tapered waveguide that at some position reaches the cutoff width. The
amplitude of the wave dies down quickly after this. As a result there is
very little electromagnetic field at
/ the wall Fz. Hence, it matters –Y2(S3+S4) Y2(S3-S4)
little whether Fz is electric or mag- Pin
netic. Thus it is seen that for (3)\ / (2)
small o the wave incident in guide
(1) goes only into guide (3) when
the junction is matched as in Fig.
12.21. This result is even more
x
general. In fact it can be seen by
the inspection of Eq. (129) that *
if SS= s4, P = 6. If in addition FIG, 1221.-One method of matching the
a=O, then@ =6= Oand]~l=l. junctionof Fig. 12.19.
<
Thus independently of the mechanism used to match the junction, y
is the only nonvanishing matrix element.
12.20. The Magic T.—The “ magic T” may be defined as a direc-
tional coupler with equal power division. Clearly the scattering matrix
of a magic T may always be written in the form
Cj
gh
s= (142)
00
00
The elements e, ~, O, and h are not completely independent but must
1
satisfy the unitary conditions
e~ + gh* = O,
eg’ +~h* = O,
(143)
lel’ + lg)’ = ljl’ + lhl’ = 1,
Iel’ + Ifl’ = Igl’ + Ihl’ = 1. 1
By the correct choice of the positions of the terminals in three of the four
guides, three of the four parameters ~,j, g, and h can be made to have any
448 1’HE S YMMh’TR Y OF WA VEGUIDE J UNC’TIONS [%c> 12.21
i
-4 I
o
(o
O~j
0;; –i
~:............ ,
1
~
(144)
I
s =
{\o o
–];0 o
0 Oil j’
o o~j 1
(145)
‘=+2
1 j~o o
(j 1;0 o
S1 = —s3,
S2 = —s4,
sl = js2.
If these values are substituted in Eq. (129), the elements of the scattering
matrix are found to be
a=o,
/3=0,
(146)
T = ~sl(l – j),
6 = ~sl(l +j). I
1 Except for a phase factor, y and 6 are the same as in Eq, (145). In fact
by choosing the terminals in a new symmetrical set of positions Eqs.
1
(146) become
I
(YII YI*
yzl y22 .
Y= . . (147)
. . .
. . .
I
Thk may be written as
. . . . . . .I
. . . . . . .
J \
‘o 0 0..
o !/22 0 . . .
o 0 ,..
+ 0 . . . . +
. . . . (148)
. . . .
\“
. . . .
Each of these submatrices may be examined individually. The first
matrix has zeros in all rows and columns except the first. “ This implies
that if voltages are applied to all the terminals, the only terminal influ-
enced by the first sub matrix is terminal (1). At this terminal a current
proportional to yl, flows. In other words, the first matrix in Eq. (148)
450 THE SYMMETRY OF WA VEGUIDE JUNCTIONS [SEC. 12.21
(149)
S’+l=o. (150)
Y = (1 – S)(1 + s)-’.
t
Substituting from Eq. (150),
I
SEC. 12.22] COUPLING-HOLE MAGIC T’S 451
o 1; o–&
Oj–a
Y = j ........! ............ ~....................O . (152)
o–@~ o
[ -w @ 1 :I
The synthesis, by coaxial lines, of a junction whose admittance matrix is
given by Eq. (152) follows in the same way as before. Note that the
lines with a characteristic conductance of unity are one-quarter wave-
length long. Those whose character-
istic conductance is @ are three-
quarters wavelength long.
From a practical point of view
a three-quarter-wavelength line is
more frequency-sensitive than a
quarter-wavelength line, and it is
desirable, if possible, to choose ref- (1) (2) 4
~ti, , ,,.=1
erence planes in such a way that all
: elements in the matrix of Eq. (152)
are positive. If the reference planes FIG. 12.23.—Synthesis of a magic T in
I
!
are moved back one-auarter wave-
coaxial line.
length, S in Eq. (151j changes sign and, as may be verified, the admit-
tance matrix of this new junction is
o 1! O+fi
oi+@
Y = j ........! ............~..... ................. .
o+w~ o 1
[ +@ 0: 1 0 I
r
This junction may be synthesized by the circuit of Fig. 1223. It should
be noticed that power entering arm (1) is split equally between arms (3)
and (4) and no power leaves arm (2).
12.22. Coupling-hole Magic T’s.—As another example of a magic T
with a scattering matrix of the form of Eq. (145), consider a directional
coupler of the type shown in Figs. 12.16 and 12.17 with o = O. The
coupling hole or holes will be assumed to be large enough to produce
equal power division. One possible arrangement is a set of two holes
about a quarter-wavelength apart. Another possibility is a large slot
or oblong hole in the direction of the guide axis. As was pointed out
previously, the symmetry of the junction becomes complete for 8 = O.
The symmetry plane containing the coupling holes becomes effectively
an electric wall for the eigenvectors as and ad. Consequently the coupling
452 THE ,9 YM JIIil’Ii 1’ OF IVA VEGIJIIIE ,J(JNCTION,q [SEC. 12.23
hole has no effect upon these standing waves, and the field distributions
are identical with those of two independent waveguides. As a result,
because of the location of reference planes, the eigenvalues of the scatter-
ing matrix for these eigenvectors are (see paragraph entitled Symm~try
with 0 = O, Sec. 12.18).
53 = 1, 54 = –1.
‘~he eigenvalues SI and 52 are dependent upon the size and shape of the
coupling hole or holes. However, it is to be noted that these two eigen-
values are the only parameters of the junction left open. Thus it requires
only two adjustments to convert the junction into a magic T. For
example, if the coupling is produced by two circular coupling holes, there
may exist a particular diameter for the holes and a distance between the
holes for which the device is a magic T.
With reference to Eqs. (129), it is evident that for the matched junc-
tion for which a = O, the cond~tions
=
F of Eq. (130) require that ~ = O.
Thus, if the junction is a magic T
~. t (for which a = O), it must be one for
“. -+ 4 which there is no coupling between
1 guides (1) and (3) or between (2) and
(4). It is to be noted that this is
just the opposite of the behavior of
1 the small-single-hole coupler for
plane 3 which the coupling between (1) and
(4) is the least (see last paragraph
FIG. 12.24.—Magic T of Sec. 12.18). Referring again to
Eqs. (129), it is evident that the conditions to be satisfied in order that
the junction be a magic T are
s, = —s2 = kj.
It should be noted that only one condition must ho satisfied in order that
the junction be a directional coupler, namely,
!s1= —s2.
o l~o o
F = :-------”~-o-------o .
0 0!1 o
[;l0 0!0 –1
S = FSF.
! Multiplying out the right-hand side and setting the product equal to the
left-hand side,
s,, = s,,, Y
S31 = S32,
(153)
S41 = ‘S42,
s,, = o. I
It can be seen also from the field distributions in the junction that the
above conditions are correct. The junction is not yet a magic T, as
there is still coupling between (1) and (2); moreover it is not matched.
Assume that matching transformers are inserted in such a way as to
make
S33 = S,4 = o.
6~7 ~
;
s = ---------:~-----:: (154)
‘r T!o o
6 –~jo o
II
The scattering matrix is unitary, and the absolute square of each column
must be unity; that is,
la12 + M’ + M’ + 1612
= 1,
21712
= 1, (155)
21al’ . 1. 1
From Eqs. (155),
Ial’ + pl’ = o.
This is ~ossible only if both a and B vanish. Thus with a = 6 = O the
scattering matnix of a magic T is given by Eq. (154). If the positions of
454 THE SYMMETRY OF WA VEGUIDE JUNCTIONS ]SEC. 12.24
the reference planes in guides (3) and (4) are chosen correctly, the scat-
tering matrix can be made to take the form
‘o oil 1’
s=~ !.......”.i 1. ..-.l (1 56)
4 1 1;0 o
1 –1!0 o
and (2) and (3) into each other. It will be this type of symmetry which
will appear in the coaxial magic T.
Notice that the matrix of Eq. (154) is pure imaginary. Therefore,
since S is unitary,
s,= ._I. (157)
Y = (1 – S)(I + s)-’
= (1 – S) ’(I – S)-’(1 + s)-’ (158)
= (1 – 2s + S’)(1 – s’)-’.
Y = –s. (159)
~
+ Symmetry
I plane 4
3A~
C@+ (3)
H
Fa
“--- = * \
///// ___ .,
~go=l (n? ,/, I ~;$ (4)
----- —-. J f ! ; ;e======= FI
1/4A
F4
FI...F6.
These ten symmetry operators form a group with the following multipli-
cation table:
456 THE SYMMETRY OF WA VEGUIDE JIJNCTIONS [SEC. 1225
F, F, F, F, F, F3~I R, R, R, R,
F, F, F, F, F, F,:R, I R, R, R,
F, F, FL F, F, F,~R, R, I R, R,
F, F, F, F, F, F,:R, R, R, I R,
F, F, F, F, F, F,:R, R, R, R, I
00001
10000
R,=o1ooo, (160)
00100
[100010
10000)
00001
F,= OOO1O. (161)
00100
I
[101000
By referring to Table L2.8 it may be seen that R, satisfies the equation
R! = 1.
and that the eigenvalues of RI are the five fifth roots of 1 (see Theorem 7).
Let
rl = 1,
r2 = ei~’, ~, = *,
TS= e~~’, 42 = a,
.
.
Flal = al,
F,az = as,
FlaS = a’, (164)
F,a’ = aa,
F,a, = aj. I
b’=+3
a“
b, = *O (a, + a,),
b, = ~ (a~ – a,),
fio
b, = *O (a, – a,)
I
0s,00 0
Sd=oos, o 0.
000s3 0
[
0000 S.2
Since B is orthogonal,
S = BS,B.
If this equation is multiplied out, it is found that
( 166)
where
Id = 171, ...
I and P and ~ are at an angle of 120”
with respect to each other. Thus
the matched junction distributes the
3
I power equally among the remaining
four guides. Thus if the five star is
matched, power entering one of the
arms is split equally among the other FIG..12.27.—Turn8tilejunction.
four arms.
12,26. The Turnstile Junction.—The turnstile junction shown in
Fig. 12.27 is a six-terminal-pair device. The two polarizations in the
round guide furnish two of the terminal pairs. F@re 12.28 shows the
numbering scheme of the terminal planes.
RI
I // F3
~~t~,
//
———— — —+——— F
1
4
3 //’ , ‘.,
3yrnm@fy axis
:--- 4
[ Reference plane ,
+ ‘<
FIG. 12.2S.—Symmetry propertie~ of the turnstile junction. The symmetry planes
are F,, Fz, F~, F4; the terminal Planes in the rectangular waveguide are 1, 2, 3, 4; the
terminals in the round wavoguide for the two Polarizations are 5 and 6. A rotation of
90° about the symmetry axis is R,.
The junction has a fourfold symmetry axis and four symmetry planes.
The symmetry planes in Fig. 1228 are designated by F,, F,, Fs, and F,.
Let a counterclockwise rotation of the fields by 90° be designated by
R,. Let RZ, R,, and I represent rotations of 180°, 270°, and 0° respec-
‘1(M) THli ,V}’Jlhl ETltl’ OF \VAVIWIIIDE .ll INCTIONS [SEC. 1226
tively. Terminal numbers are kept in a fixed position under any sym-
metry operation.
The rotations and reflections form a group \vith the follo\ving multipli-
cation table.
[ R, R, R,:F, F, F, F,
T I R, R, R,:F, F, F, F,
R, R, R, R, I jF, F, F, F,
R, R, R, I R,:F, F, F, F,
R, R, I R, R,F, F, F, F,
....................... ... ..
F, F, F, F, F,{ I R, R, R,
F, Fz F4 F1 F~~R2 I R, R,
F, F, F, F, F,~R, R, I R,
F, F, F, F3 F,:R, R~ R? I
o 0 0 1!0 o~
1 0 0 0!0 o
0 1 0 0!0 o
R, = 0 0 1 0;0 o ,
000 0!0 –1
o 0 0 o\l o
1 0 0 0;0 o
D O 0 1;0 O
......................
0 0 1 0/0 o
F, = D 1 0 o~o o.I
I000
000
oil
o~o
R,a~ = Tkak,
SEC. 12.26] THE TURNSTILE JUNCTION 461
rl = 1,
rz = j (doubly degenerate),
r3 = –1,
r4 = – j (doubly degenerate).
A set of eigenvectors is
1’
–1
+1
–1 ‘
o
0,
respectively.
a4=lii1
as=i!l
a=!
The eigenvectors as and ‘6 are eigenvectors of the eigenvalues r2 and
It may be seen by inspection that
Flal = al,
T4
Flaz = a’,
Fla3 = as,
(168)
Fla’ = az, I
Flas = ae,
Flae = a,. )
Since the scattering matrix S commutes with R 1, the eigenvectors a 1
and ai are also eigenvectors of S (Theorem 6). Linearly independent
eigenvectors of S can be formed by taking linear combinations of at and
a, and of a’ and a,. Let
b, = a,,
b, = i(a2 + a4) + $a(a, + a6),
bs = aa,
bl = *(az + a4) + *6(a5 + a6),
[1
1 0 –1
1 –1 1
b, = b, = b, =
1 o ‘ –1 ‘
0 o
0 ;, 0
1’ o 0
0 1 1
–1 o 0
b, = b, = b,= _l
o –1 ‘
o 0
~ [1 a !: P
The eigenvectors bi satisfy the eigenvalue equations
S bi = sjbi)
where
S5 = Sz,
se = s,.
a~ = —2.
0000 SZ o
0000 0 s,,
1 11 1! o 0
1 0 –1 0/ 1 1
l–11–l~o o
B=; 1 0 –1 0:–1 –1
(-”””””---------”---
o
0
%6
00
O–WI
B is orthogonal.
Solving Eq. (170) for S, —
S = BSd~,
or
s=
..... ..........
o —e O! /3
c o —c; o
where
~ = *(S I + S2 + S3 + S4),
IS = +(s2 + s,),
7 = +(s1 – S3),
b = +(s1 – S2 + .93 – S4),
The spur of S is
&z+zj3= s,+%z+s3+zs4,
464 7’HE ,$’I’.4IMETR Y OF W.1 VEGUIDE .JCINCTIONS [SRC. 1226
d~k J L
__ —-.
The conditions for this are
r ,,~i],](,
‘:::}
(171)
~%
Elgenvector
b,
1- +
Eigenvector bz
# % ,hereforei!rnstileis
Elgenvectorb~ E!genvectorsb. and b: matched, power entering guide
(1) leaves by guides (2), (4), and
Elecwcwall (5). One-half of the power leaves
--- ‘-- Magneticwall by the round guide; the remainder
~1~.12.29.—Eigenvector solutions b, and be’, divides equally between guides (2)
b,, b,, and b,.
and (4).
Field Dislribu~ions.-The eigenvalues of the four reflection opw-at ors
are indicated in Table 12.10 for the six eigenvectors b~.
b, b, b, b, bi b,
K +1 +1 +1 +1 –1 –1
F, +1 –1 +1 – 1 + 1 + 1 eigrmvalues
F, +1 -–1 - - -
F, +l––l --–
Note that only b, and b, are eigenvectors of F, and F4. As the boundary
conditions on the symmetry planes are determined by the eigenvalues of
the symmetry operators, Table 12.10 can be used to verif y the correctness
of the diagrams in Fig. 12.29. The remainder of the eigenvectors satisfy
boundary conditions similar to those of b~.
As bl and b, are eigenvectors of S with the same eigenvalue s,, any
linear combination is also an eigenvector. Let
b, — b, + b,, (172)
i
t
S1.r. 1226] 7’1{1< 7’L:ILV,$’TII>I<
JIT.V(:’I’lO.V
i
.ilsotllc
\“ect,lJr’
b{, = b, + b,
I is an eigenvector of S ~vith the eigenvalue ,S4.
->
[‘1-42
I The eigenvector given in Eel. (172) is introduced because it satisfies
different boundary conditions from those of either b, or bs (bi or b,).
Electric wall-
Lowest mode
propagates
I B
I
,
(a) Eigenvector b, (~) EigenvaluesS2 and .%
Reflection coefficientS, Eigenvectorsb2 and ba
Single propagating
R
No p:o~eating Magnetic wall mode
(c) Eigenvector b3
(o!) Eigenvectorsb, and b’,
Reflection coefficientS3
EigenvaluesS2 and S4
FIG. 12.30.—Junction partitioned by electric and magnetic walls for the various eigenvector
solutions.
Note that bOand b; are eigenvectors of F~and F, but not of F, or F,. The
boundary conditions satisfied by bOor b{ are illustrated also in Fig. 12.29.
Since the magnetic and electric walls shown in Fig. 1229 divide the
466 THE SYMMETRY OF WA VEGUIDE JUNCTIONS [SEC. 12.27
After this the pin is inserted. It will be remembered from the previous
discussion that this affects only s,. Its position is adjusted until
S1 = —s3. Then a = O, and the junction is completely matched.
12.27. Purcell’s Junction.-The device shown in Fig. 12”32 is a junc-
tion of six rectangular guides. It is completely symmetrical in the sense
that all the waveguides are equivalent. The junction may be regarded
as one generated by a cube, each guide being mounted on a face of a cube.
The terminal planes are shown as dotted lines in Fig. 12.32 and are
numbered according to a rule that allows a regular progression from one
number to the next. Each terminal plane has an arrow assigned to it
that represents the direction of positive electric field. The way in
which the arrows are assigned is evident from the figure.
SEC. 12.27] PURCELLS JUNCTION 467
M,
IM2
of terminal (1) goes into that of terminal (6) without a change in sign.
This, together with the other permutations of the terminals, are
(1) + (6)
(2) 4 (5)
(3) + (4) Symmetry operator
(4) + (3) R1.
(5) + (2)
(6) -+ (1)
\o o 1’
0 \olo
\loo
R, = ......... .... (174)
ool~
olo~o
,1 0 o~ 1
operating on a current or voltage column vector.
In a similar way the 180° rotations about the axes R, and R8 are
symmetry operators and have, as matrices representing the corresponding
permutations of the terminal pairs,
4(i8 THE SYMMETRY OF H-.4 VEGUIDE JUNCTIOi%-S [SEC. 1227
‘o o o~l o o’
0 () l~o () o
() 1 o~o o 0
R, = ...... ..... ..... ,
100:000
Oooiool
\o o Oio 1 0,
[010 000’
1100 000
000 001
R, =
000 010
000 100
,001 000
At first glance it might be thought that these three rotations are the
only symmetry operators, except for the identity operator 1. If this were
true, however, the product of any two ought to yield the third matrix.
It is found that, on the contrary, the product of two of them, RI and R!
1 for example, yields new permutations
M, = R,R2
and
M, == R, RI,
where
(O o 0;0 1 o’
ooo~ool
100000
M, = ~~~------------------
I olo~ooo
001! 000
,0 0 oil o 0,
and
‘o o qo o o’
ooo~loo
ooo~olo
M, = -------------------------
ooo~ool
100:000
,() 1 o~o o 0, I
\vise (looking out along the arrow) of 120°. The operator M 2 is a rota-
tion of 240°.
The four symmetry axes of Fig. 1232 are also symmetry axes of the
cube upon which the junction is built. R 1, Rz, and R~ are symmetry
axes of the cube passing through midpoints of two opposite edges. The
fourth axis is a threefold axis of the cube which passes through diagonally
opposite corners of the cube.
It is evident from an inspection of the figure that there are no sym-
metry planes in the junction. Hence the symmetry operations are the
six rotations I, RI, Rt, Rz, Ml, and Mt. The multiplication properties
of these operators are summarized in Table 12.11. Note that the group
whose multiplication table is Table 12.11 is of order 6 and has three sub-
groups of order 2 and one of order 3. It is a subgroup of the symmetry
group of the cube.
As may be seen from Table 1211, the group may be generated by
the elements RI and M,. Since M! = 1, the eigenvalues of M, are the
three cube roots of 1. Each of these roots is a doubly degenerate eigen-
value. Thus
where
1 + V%j
ml, z = 1, al=a; =–—
2’
Let
1 1’
–1 0
1 ffz
a, ~ a2 = a4 =
–1 o
1 al
–1 o
l—
m2=l mA = m4 = a,
o’
1
0
a6 = (176)
ffl
o
ffz
I
M,S = SM,.
M,(Sa,) = m,(Sa~).
and (179) results, because of the linear independence of al and al, in the
t\vo equations
qukl + gnkz = sk~,
gnkl + gnkz = sk~.
g,, – s 912 = ~
gzl gzz — s
I,et these roots be s, and S2. There are t~vo linearly independent solu-
tions for kl and kz corresponding to each root. l,et these solutions be
k,l and klzfors = SIand kQ1and kzzfors = SC. Then
Sb, = s,b,, (180)
\vhere
bl = k,la, + klXa2. (181)
Thus both the sum and the difference of the terms in a, and az are eigen-
vectors of S with eigenvalue sl, and each of the terms kllal and kl~a~
must be independently an eigenvector of S. Therefore, because kl I
and k 12may not both vanish, either al or a~ is an eigenvector. If neither
k,, nor k,Q vanishes, both a, and az are eigenvectors, which implies that
s.Iis degenerate or that .sl = sZ. It will be assumed, without information
to the contrary, that SI and m. are nondegenerate, in which case k,, or
k,, must vanish. Without loss in generality it may be assumed that
k,z in Eq. (181) vanishes; thus
Sla, = slal.
In a similar way,
S,a, = s,a,.
b~ = ySSa3 + g~~a~,
b~ = g43a3 + g44a4,
(182)
b, = gssa, + 956%,
b6 = g6sa;+ g66a6,1
where
Sb, = .s,b,,. (18:3)
.
<
Then
J
S(R,b,) = sJR,bJ. (184)
g44 g66 !
However,
and
-(:6+%-SO
‘1b3=
b=da+:a) Either a6 and a6 are
are eigenvect ors of S with the same eigenvalue.
independently eigenvectors of S, which implies that
S,= S4=S5=S6,
or else
956 _ 933
——— (187)
g55 g34 ~
As usual the minimum condition on the eigenvalues is the one that !
will be assumed. Combining Eqs. (186) and (187),
1
SEC. 1227] I’( ‘R CELL’,V .111.VC7’1<).Y 473
In a similar way,
943 = T g4’1,
(189)
q65 = ~ g66. }
By the utilization of Eqs. (188) and ( 189), the b’s may be redefined,
without loss in generality, as
The eigenvectors given in Eqs. (190) are not pure real. However a
pure real set may be chosen. Let
e~=~(b~—b,)=~-(a~ +a4—a5—a6),
V% &
e~=~(bd–bJ=~.(aS –at–a~+a~).
V’% %/3
From Eq. (176) the e’s may be seen to be
1’ 2’ 2
[1
–1 2 –2
1 –1 –1
e4 =
–1 ‘ –1 ‘ 1’
1 –1 –1
–1 , –1 , 1
0 0
0 0
1 1
1 ‘ –1
–1 –1
–1/ 1/
474 7’1{1< ,Yl”.\[.llEl’Rl” <)F 11”.4j’EG(”IDE ,J(”.J-C’TIO.Y,q [SF(, 1227
It may be seen by inspection that the e’s are mutually orthogonal. Tfie
eigenvallle equation
Sek = skek
ST = TS,,,
\\llt’r(~
10 0 s,;
s, =
orthogonal.
o
S’j
e 6 (normalized
0
o
s
to unity)
T is an orthogonal
as columns.
matrix
and
T-1 = ~.
Therefore,
~a p 716 -f 8’
paa!’ycly
-f6a:/3y6
S = TS,,~ =
‘37B!a6y
yby!bab
dya, ypa
\vhere
a = &(S1 + SZ + %3 + zsA),
~ = %(s1 – sz + zs3 – 2s,),
~ = +(s, + S2 — S3 — S4),
d = +(s1 – S! – S3 + s,).
Then
7=0,
p = 1.
D= s,,
1. s, = –s’2 ~=o
S3 = —s4 7=0.
}
2. s, = s, 6=0
S2 = S4 7=0.
}
3. s, = S4 8 = –213
s? = S3 } 7=0.
c
Case 5: fi = 0.
‘l’his rcxluires that
. . – .s, – .s, +s, = ()
where
1 l+sk
(195) ‘
‘~=~=l–sk
and a~ are pure real, orthogonal, and normalized to unity. It should be
— —.. . .
SEC. 1228] THE EIGENVALUE FORMULATION 477
noted that a~ is a permissible wave column vector for Eq. (192), since Eq.
(193) is satisfied by a, with
@ = Sk.
hlany of the junctions that have been considered have enough symmetry
so that ak is completely determined by symmetry. In this case a~ is
independent of frequency and Eq. (197) becomes
(s% – Ss’)ak = 0. I
Since the ak’s form a complete set, Eq. (199) is satisfied for any vector
and hence
s’s – Ss’ = o.
If Ec~. ( 19g) is substituted in Eq. (196), it is seen that
(200)
This equation is important. It can be \vritten
or
(201 )
where
Sk = el+k.
_w,
dtw
dw=– () P
.
(202)
SA = AS.,
(203)
S’A = AS:, I
where A as usual has the vectors a~ as columns and
S,o o.......’
o
0.
s, = ,
.s,,\
S;o o.....
o
0
q =
w, ==
\ It’.
SEC. 12.29] WIDEBAND SYMMETRICAL JUNCTIONS 479
.9A = – 2jAW&
s’ = – 2jAwdAAsdA (205)
— – 2jws,
\\,here
W = AWd~.
It should be noted that S’, S, and W all have the same eigenvectors
and hence commute with each other.
In a completely similar way Eq. (191) becomes
Z; = @“k, (207)
where zj is the eigenvalue of Z’. Note that –jzj is always positive.
Z’ is obtained in a way completely analogous to that of Eq. (205) as
w = Wo(u)l, (209]
then the modulus of each element of S is independent of frequency, or
the junction is wideband. The necessary and sufficient condition for W
to have the form of Eq. (209) is for W,, of Eq, (196) to satisfy
Tvk = W,(Q).
Stated in words, this requires that the electromagnetic energy stored in
the junction be the same for all the various eigensolutions.
;.
Index
I’ Reciprocity theorem, 90
Reflection, from change in dielectric con-
II-network, 100 stant, 369–374
Phaae angle of dielectric constant, 365 ~ Plane, symmetry of, 412414
Phaae constant, 18 Reflection coefficient, 63, 137
Phase shifter, rotary, 355-358 current, 67, 266
Physical realizability, 142 in radial lines, 265-267
Pickering, W., 202 and stored energy, 138
Pincherle, L., 389 voltage, 67, 266
Pipes, half-wave, 355 Reflection group, multiplication table for,
quarter-wave, 354 414
rectangular, normal modes in, 33–38 symmetry types under, 415
round, normal modes in, 38–40 Refraction, index of, 366
Plane waves, nonuniform, 19-21 Reich, H. J,, 335
uniform, 17–19 Relaxation time, 379
Plates, parallel, Z’EAf-waves between, 22 Resistance, 84
Plungers for rectangular waveguide, 198 Resonance in closed circular waveguide,
Polarizations, basic, 350 361-364
transformation of, by rotation, 351 Resonant cavities, 207–239
Post, thick, 199 Resonant circuits, 127–129
Power, available, 95 Ring circuits, 30&31 1
Power flow in waveguide, 50-54 Robbins, T. E,, 172
Power transfer, maximum, 94 Roberts, S., 374
Poynting energy theorems, 14-16, 132- Row vector, 89
134
Poynting vector, complex, 16 s
Poynting vector theorems, 14-16
i
Propagation constant, 18 Sarbacher, R. I., 335
iterative, 113 Saxon, David S., 163, 174
Purcell’s junction, 466476 Saxton, J. .4., 380
Scattering, by antennas, 317-333
Q &attering matrix, 14&149
of directional coupler, 301–303
Q, 48, 127,230 of free space, 324
loaded, 228 of simple electric dipole, 325
unloaded, 228 transformation of, 149
Q,, 48 Schelkunoff, S. A., 19, 43, 202, 209, 211,
216, 252
R Schwinger, J., 45, 82, 163, 174, 318, 403
Schwinger reverse-coupling directional
Radar camouflage, 396 coupler, 312
Radial cotangents, large, 259 Screw tuners, 181
small, 259 Series elements, discontinuities with,
Radial lines, equivalent circuit in, 267- 19S206
271 Shunt elements, discontinuities with,
impedance description of, 25&265 19!3206
Radial transmission lines, 240-282 Shunt reactance, 163
Radiation, by antennas, 317-333 cavity formed by, 18>186
from thick holes, 201 diaphragms as, 163-179
Ramo, S., 252 Skin depth, 46
.( Reactance, 84 of various metals, 47
shunt (see Shunt reactance) Slater, J. C., 170, 190, 219, 315, 397
.——,
INDEX 485