Cavity (Waveguide) Resonators

A resonator is primarily an energy storage device. It forms an essential part of an oscillator system.
At low frequencies (below 100MHz) an RLC circuit or a quartz crystal is used as a resonant circuit
element. However at higher frequencies (microwave range) the dimensions of the circuits are
comparable with the operating resonant wavelength and hence it results in unwanted radiations.
To avoid this at high frequencies waveguide resonators are used.
A waveguide resonator is formed by shorting the ends of a waveguide. These are commonly called
Cavity Resonators. Two types of cavity (waveguide) resonators are shown in Fig.1.

(a) (b)

Fig. 1: Cavity Resonators (a) Rectangular (b) Cylindrical

z
y

(a,0,c)
(0,b,0)

(a, 0, 0) x
Fig.2: A rectangular cavity with dimensions (a,b,c)

Page | 1
 Cavity (Waveguide) Resonators

A cavity is a closed conducting box. A standing wave is expected in a cavity resonator. It can
have TE and TM modes of propagation. The wave inside the cavity may be in x, y or z direction.
There is actually no wave propagation but a standing wave is developed.
For a lossless medium (ε, µ) the three dimensional wave equations are written as
2 E    2   E (1)
2 H    2   H (2)
Each of these equations represents three Helmholtz equations.
Assume the +z-direction as the "direction of wave propagation". However, In fact, there is no wave propagation.
Rather, there are standing waves. A standing wave is a combination of two waves traveling in opposite directions.
For z-component of electric field E, the eq.(1) can be written as
 2 EZ    2   EZ (3)
2 2 2
Or E  E  EZ    2   EZ (4)
x y z
2 Z 2 Z 2

Where EZ  F ( x, y, z, t )
This is a second order partial differential equation and using variable separable method of
solution one can express Ez as
EZ  F ( x) F ( y ) F ( z ) e j  t (5)
2
Let EZ   k x2 (6A)
x 2

2
EZ   k y2 (6B)
y 2

2
EZ   k z2 (6C)
z 2
Where k x2  k y2  k z2   2   (7)
The general solutions of (6A), (6B) and (6C) are
F ( x)  A Cos k x x  B Sin k x x (8A)
F ( y)  C Cos k y y  D Sin k y y (8B)
F ( z )  E Cos k z z  F Sin k z z (8C)
Hence EZ  ( A Cos k x x  B Sin k x x) (C Cos k y y  D Sin k y y) ( E Cos k z z  F Sin k z z) e  j  t (9A)
Similarly proceeding along same steps one can obtain from eq. (2)
H Z  ( J Cos k x x  K Sin k x x) ( L Cos k y y  M Sin k y y) ( N Cos k z z  P Sin k z z ) e  j  t (9B)
The other field components Ex, Ey, Hx and Hy can be obtained in terms of Ez and Hz using
Maxwell Curl equations as follows

Page | 2
 Cavity (Waveguide) Resonators

ux uy uz
 H 

x

y

z

 j   Ex  E y  Ez  (10A)
Hx Hy HEz
ux uy uz
 E 

x

y

z

 j  Hx  Hy  Hz  (10B)
Ex Ey Ez
 
Or Hz  H y  j   Ex (11A)
y z
 
Hx  H z  j   Ey (11B)
y z
 
Hy  H x  j   Ez (11C)
y z
 
Ez  E y   j   Ex (11D)
y z
 
Ex  Ez   j   E y (11E)
y z
 
Ey  Ex   j   Ez (11F)
y z
Multiply (11A) by –jωµ we have
 
 j  Hz  ( j   H y )   2   E x
y z
Substituting (11E) we get
   
 j  Hz  ( Ex  Ez )   2   Ex
y z z x
 2  
 j  H z  2 Ex  Ez   2   Ex
y z x z
Since all field components are varying with respect to as {Cos kzz +Sin kzz}
2 
Ex   k z Ex Ez  j Ez
2
Therefore and
z 2
z
 
Hence we have  j   H z  k z2 E x  j k z Ez   2   Ex
y x
j   jk 
Or Ex   H z  2z Ez (12)
h y
2
h x
j   jk 
Similarly Ey  H z  2z Ez (13)
h x
2
h y

Page | 3
 Cavity (Waveguide) Resonators

j kz  j 
Hx  Hz  2 Ez (14)
h x
2
h y
j kz  j 
Hx  Hz  2 Ez (15)
h y
2
h x

Where 
h2  2    k z2  k x2  k y2  (16)

Case I: Ez = 0 and Hz = 0.
This corresponds to TEM mode of propagation. Under above situation all field components
vanish, hence TEM modes of operation are not supported by rectangular cavity resonator.
Case II: Ez ≠ 0 and Hz = 0
This corresponds to TM mode of operation. In this case Ez is given by eq. (9A)
Applying boundary conditions we have
Ez = 0 at x = 0; this yield from eq. (9A) A=0
Ez = 0 at y = 0; this yield from eq. (9A) C=0

Also Ex = 0 at y = 0 & Ey = 0 at x = 0; hence E z  0 ; this leads to F = 0
z
Hence EZ  B DE Sin k x x Sin k y y Cos k z z e  j  t
EZ  E0 z Sin k x x Sin k y y Cos k z z e  j  t
Again Ez = 0 at x = a; this yield from eq. (9A)
m
Sin k x a  0 or kxa  m or k x  where m  0,1, 2, 3 .......
a
Ez = 0 at y = b; this yield from eq.(9A)
n
Sin k y b  0 or k yb  n  or k y  where n  0,1, 2, 3 .......
b

Also Ex = 0 at y = b & Ey = 0 at x = a; hence E z  0 at z=0;
z
p
Sin k z c  0 or kzc  p  or k z where p  0,1, 2, 3.......
c
Thus for TM mode of operation for rectangular cavity we have
m n p
Hence EZ  E0 Sin x Sin y Cos z e j t (17A)
a b c
HZ  0
1 p  m  m n p
Ex  2 E0Cos x Sin y Sin z e j t (17B)
h c a a b c
 p m 2 m n p
Ex  2
E0Cos x Sin y Sin z e  j  t (17C)
h ca a b c

Page | 4
 Cavity (Waveguide) Resonators

 p n 2 m n p
Ey  2
E0 Sin x Cos y Sin z e j t (17D)
h cb a b c
 j   n m n p
Hx  2
E0 Sin x Cos y Cos z e j t (17E)
h b a b c
 j   m m n p
Hx  2
E0Cos x Sin y Cos z e j t (17F)
h a a b c

Also  2    k x2  k y2  k z2   2 (18)
If fr is the resonant frequency of the cavity then
r2    k x2  k y2  k z2   2
 m   n   p 
2 2 2
1
Or fr       
2    a   b   a 
2 2 2
1 m  n   p 
Or fr        (19)
2   a  b a
2 2 2
uP  m   n  p
Or fr        (20)
2  a  b a
u 2
Resonant wavelength r  P  (21)
fr m
2
 n   p 
2 2

     
 a  b a
Again at resonance r2     2
 1
fr   uP  (22)
2   2
Where up is the phase velocity of wave in an unbounded lossless medium.
Various TM modes are designated by TMmnp.
TM000, TM00p, TM0np, TMm0p modes cannot exist.
The Lowest TM which exists is a Rectangular Cavity Resonator is TM110. This is known as
dominant mode.
Case III: Ez = 0 ; Hz ≠ 0.
This condition represents TE mode of operation.
Under this case the other field components are given by
j  
Ex   2 Hz (23A)
h y
j  
Similarly Ey  Hz (23B)
h2  x
j kz 
Hx  Hz (23C)
h2  x

Page | 5
 Cavity (Waveguide) Resonators

j kz 
Hx  Hz (23D)
h2  y
Eq. (9B) defines the Hz. Applying boundary conditions which state that
Ex = 0 at y = 0 and y = b and Ey = 0 at x = 0 and x = a
This implies that

H z  0 at x = 0 and x = a
y

H z  0 at y = 0 and y = a
x
Also Hz = 0 at z = 0 and z = c.
Applying these boundary conditions we get
m n p   j t
H Z  H 0 Cos x Cos y Sin ze
a b c
Where m, n, p are positive integers.
In this case the Resonant frequency is given by
2 2 2
1 m  n   p 
fr        (24)
2   a  b a
2 2 2
u m  n   p 
fr  P         (25)
2  a  b a
2
And Resonant wavelength r  (26)
2 2 2
m  n   p 
     
 a  b a
Thus for TE mode of operation we have
Ez  0 (27A)
m n p   j t
H Z  H 0 Cos x Cos y Sin ze (27B)
a b c
j   n m n p   j t
Ex  2
H 0 Cos x Sin y Sin ze (27C)
h b a b c
 j   m m n p   j t
Ey  2
H 0 Sin x Cos y Sin ze (27D)
h a a b c
 m p 2 m n p   j t
Hx  2
H 0 Sin x Cos y Cos ze (27E)
h ac a b c
 n p 2 m n p   j t
Hy  2
H 0 Cos x Sin y Cos ze (27F)
h bc a b c
Different TE modes
(i) If m = n = p = 0; this refers to TE000 mode of operation. In this case all field components
vanish, hence TE000 mode does not exist.

Page | 6
 Cavity (Waveguide) Resonators

(ii) Similarly all those modes for which p = 0, Hz = 0 and hence all other field components
become zero. Thus TEmn0 modes do not exist.
(iii) If m = 0, n = 0 and p ≠ 0. In this case also no field component exists except Hz, hence
TM00p modes do not exist.
(iv) The lowest TE modes which exist in a Rectangular Cavity Resonator are TE101 and
TE011. One of them can qualify as dominant mode.

Dominant Mode: The modes with lowest resonant frequency are termed as dominant mode.
Degenerate Modes: In cavity resonators there can be more than one mode having same resonant
frequency. These modes are called Degenerate Modes.
Quality factor Q of a cavity
A practical resonant cavity is made of a material having finite conductivity, hence it suffers loss
in the stored energy. The means of finding the loss is the quality factor of the cavity.
The Q-factor (Quality factor) of a resonator is defined as the ratio of the amount of energy stored
in the cavity and the amount of energy lost per cycle through the walls of the cavity. The quality
factor Q also gives the measure of the bandwidth of the cavity resonator. Mathematically it is
defined as
Time average Energy Stored
Q  2
Energy loss per cycle of Oscillation
W W
Q  2  (28)
PL T PL
Where T = 1/f ; Time period of oscillation
PL is the time average power loss in the cavity
W is the time average energy stored in the cavity in the electric and magnetic fields.
For TE101 mode of operation the quality factor Q is
(a 2  c 2 ) a b c
QTE101  (29)
 [2 b (a 3  c 3 )  a c ( a 2  c 2 ) ]
1
Where  (30)
 f101 0  c
And it is the skin depth of the walls of cavity.

Page | 7