Constants
Constants
Constants
A.1
Fundamental Constants
c
o
o
o
e
m
mp
h
k
No
R
A.2
velocity of light
permittivity of free space
permeability of free space
characteristic impedance of free space
charge of an electron, (-e.v./Joule)
mass of an electron
mass of a proton
Planck constant
Boltzmann constant
Avogadros constant
Universal gas constant
6.14 107
5.80 107
4.10 107
3.54 107
1.81 107
1.57 107
1.28 107
1.0 107
0.5 1.0 107
0.48 107
Monel
Mercury
Sea Water
Distilled Water
Bakelite
Glass
Mica
Petroleum
Fused Quartz
- A421 -
0.24 107
0.1 107
35
2 10-4
10-8 10-10
10-12
10-11 10-15
10-14
<2 10-17
A.3
A.4
1.00
1.03
1.8 2.0
2.0 3.0
2.1
2.1
2.1
2.16
2.3 4.0
2.55
2.6
2.6 3.5
3.78
Vycor glass
Low-loss glass
Ice
Pyrex glass
Muscovite (mica)
Mica
Magnesium silicate
Porcelain
Aluminum oxide
Diamond
Ethyl alcohol
Distilled water
Titanium dioxide
Relative Permeability /o
Vacuum
Biological tissue
Cold steel
Iron (99.91%)
Purified iron (99.95%)
mu metal (FeNiCrCu)
Supermalloy (FeNiMoMn)
- A422 -
1
1
2,000
5,000
180,000
100,000
800,000
3.8
4.1
4.15
5.1
5.4
5.6 6.0
5.7 6.4
5.7
8.8
16.5
24.5
81.1
100
0.5
(B.1)
(B.2)
Therefore:
e j = 1+ j 2 2! j3 3!+ 4 4! + j5 5! ...
= 1 2
2! + 4 4! ... + j j3 3!+ j5 5!...
e j = cos + jsin
(B.3)
(B.4)
(B.5)
where the real part is Ar Re{A} and the imaginary part is Ai Im{A}.
It is now easy to use (B.4) and (B.5) to show that76:
Acos(t + ) = R e Ae (
} = R {Ae
j t +)
j jt
(B.6)
where:
A = Ae j = Acos + jAsin = A r + jAi
(B.7)
The physics community differs and commonly defines Acos(t + ) = Re{Ae-j(t + ) } and Ai -Asin, where the
rotational direction of is reversed in Figure B.1. Because phase is reversed in this alternative notation, the
impedance of an inductor L becomes -jL, and that of a capacitor becomes j/C. In this notation j is commonly
replaced by -i.
76
- B423 -
A r Acos,
Ai Asin
(B.8)
The definition of A given in (B.8) has the useful geometric interpretation shown in Figure
B.1(a), where the magnitude of the phasor A is simply the given amplitude A of the sinusoid,
and the angle is its phase.
(a)
(b)
Im{A}
A = Ar + jAi = Ae
Asin
(c)
Im{A,B}
A + B
j
-1 0
-j
1 Acos
Re{A}
Im{A,B}
Re{A,B}
AB = AB e j( A +B )
A
B
C A
B
0
Re{A,B}
A = A 2r + Ai2
0.5
(B.9)
= tan 1(Ai A r )
(B.10)
It is also easy to see, for example, that ej = -1, and that A = jA corresponds to -Asint.
Examples of equivalent representations in the time and complex domains are:
Acost
A
Asint jA
Acos(t + ) Ae j
Asin(t + ) jAe j = Ae (
j 2)
Complex numbers behave as vectors in some respects, where addition and multiplication are
also illustrated in Figure B.1(b) and (c), respectively:
- B424 -
A + B = B + A = A r + Br + j(Ai + Bi )
(B.11)
AB = BA = (A r Br Ai Bi ) + j(A r Bi + Ai Br ) = AB e (
j A +B )
(B.12)
A* = A r jAi = A e jA
(B.13)
A r = A + A* 2,
Ai = A A* 2
(B.14)
(B.15)
= A1 n e j n
(B.16)
= A (1 n ) e( j n ) e( j2 m n )
(B.17)
for m = 0, 1, , n 1.
- B425 -
- B426 -
A B = det A x
Bx
Ay
Az
By
Bz
= x ( A y Bz A z B y ) + y ( A z Bx A x Bz ) + z ( A x B y A y Bx )
= a b A B sin
Ai( B C ) = Bi( C A ) =
Ci( A B )
A ( B C ) = ( AiC ) B ( AiB )
C
( A B )i( C D ) = ( AiC ) ( Bi
D ) ( AiD ) ( BiC )
= 0
i( A ) = 0
( A ) = ( iA ) 2
A
A ( A ) = ( Ai ) A 1 ( Ai
A )
2
( ) =
+
i( A ) = Ai +
iA
( A
) = A
+ A
2
= i
( AiB ) = ( Ai ) B + ( Bi ) A + A ( B ) + B (
A )
i( A
B ) = Bi( A ) Ai( B )
( A B ) = A ( iB ) B ( iA ) + ( Bi ) A (
Ai ) B
- C427 -
x
y
z
A x A y A z
iA =
+
+
x
y
z
A y
A
A A y A x
A
A = x z
+ y x z + z
z
x x
y
z
y
2
2
2
2 = + +
x 2 y2 z 2
r
r y
z
( rA r ) 1 A A
z
iA = 1
+
+
r r
r
z
r
r
z
rA
1 A z A
(
)
A
A
A
1
1
r
z
r
+
A = r
+ z
= det r z
z
r
r r
r
z
r
A r rA A z
( )
2
2
2 = 1 r + 1 +
r r r
r 2 2 z 2
r
r
r sin
A
r 2
A r
( sin A )
1
iA =
+ 1
+ 1
r
r sin
r sin
r2
1 A r 1 ( rA )
( r sin A ) A
1 ( rA ) A r
A = r 1
+
+
r sin
r r
r sin r r
r
r
r sin
1 det r
=
2
r sin
A r rA r sin A
sin +
1
2
2 = 1 r 2 + 1
r
r 2 r
r 2 sin 2 2
r 2 sin
- C428 -
V iG dv = A Gin da
Stokes Theorem:
A ( G )in da = C Gid
Fourier Transforms for pulse signals h(t):
H( f ) =
h( t ) =
h(t)e
j2 ft
H(f )e
dt
+ j2 ft
df
- C429 -
- C430 -
p = ( Ne2 mo )
Fundamentals
f = q ( E + v o H ) [ N ]
0.5
eff = (1 j )
E = B t
c E ds = dt A
B da
E1// E 2 // = 0
H = J + D t
H1// H 2 // = J s n
B1 B2 = 0
c H ds = A J da + dt A D da
( D1 D2 ) = s
A D da = V dv
B = 0 A B da = 0
D =
0 = if =
J = t
Electromagnetic Quasistatics
2 = 0
KCL : i Ii (t) = 0 at node
o = 4 10-7 Hm-1
e = -1.60 10-19 C
o 377 ohms = (o/o)
( 2 2
i(t) = C dv(t)/dt
f = q ( E + v o H ) [ N ]
0.5
t 2 ) E = 0 [Wave Eqn.]
f z = dw T dz
F = I o H [ Nm
E e = v o H inside wire
A ( E H ) da + ( d dt ) V ( E 2 + H
= V E J dv (Poynting Theorem)
2
-1
P = T = WTdVolume/dt [W]
2 dv
vi =
dw T
+ f dz
dt
dt
D = o E + P
Electromagnetic Waves
D = f , =
( 2 2
t 2 ) E = 0 [Wave Eqn.]
o E = f + p
( 2 + k 2 ) E = 0, E = E o e jk ir
P = p , J = E
k = ()0.5 = /c = 2/
B = H = o ( H + M )
= o 1 p 2
vp = /k, vg = (k/)-1
- D431 -
1
2
r = i
sin t sin i = k i k t = n i n t
Cparallel = C1 + C2
c = sin 1 ( n t n i )
> c E t = Ei
Te+x jk z z
we = Cv2(t)/2; wm = Li2(t)/2
k = k ' jk ''
Lsolenoid = N2A/W
= RC, = L/R
= B da (per turn)
= T 1
T TE = 2 (1 + [ o cos t t cos i ])
T TM = 2 (1 + [ t cos t i cos i ])
B = tan
( t
i )
0.5
Zseries = R + jL + 1/jC
for TM
Pd J S
2 [ Wm
2
-2
Ypar = G + jC + 1/jL
Q = owT/Pdiss = o/
E = A t , B = A
( ( r ) e
A(r) = ( J ( r ) e
(r) =
jk r
'r
V'
V'
o = (LC)-0.5
)
4 r ' r ) dv '
v 2 ( t ) R = kT
4 o r ' r dv '
jk
r 'r
+ o o = o
dv(z)/dz = -Ldi(z)/dt
A + o o A = o J
di(z)/dz = -Cdv(z)/dt
d2v/dz2 = LC d2v/dt2
v(z,t) = f+(t z/c) + f-(t + z/c)
f z = dw T dz
F = I o H [ Nm
c = (LC)-0.5 = 1
-1
E e = v o H inside wire
P = T = WTdVolume/dt [W]
Max f/A = B2/2o, D2/2o [Nm-2]
dw T
+ f dz
vi =
dt
dt
f = ma = d(mv)/dt
x = xo + vot + at2/2
P = fv [W] = T
wk = mv2/2
T = I d/dt
2
I = i mi ri
Zo = Yo-1 = (L/C)0.5
Power Transmission
(d2/dz2 + 2LC)V(z) = 0
k = 2/ = /c = ()0.5
Z(z) = V(z) I(z) = Zo Zn (z)
Circuits
KCL : i Ii (t) = 0 at node
L = /I
i(t) = C dv(t)/dt
PR = Pr ( , , r ) r 2 sin d d
- D432 -
Acoustics
Prec = Pr(,)Ae(,)
P = Po + p, U = U o + u
Ae
(, ) = G(, ) 2
4
(U
= 0 here )
p = o u t
R r = PR i 2
(t)
E ff ( 0 ) = ( je
r ) A E t (x, y)e
jkr
jk x x + jk y y
u = (1 Po ) p t
dxdy
Prec = PR ( G 4r 2 ) s 4
( 2 k 2 2 t 2 ) p = 0
E = i a i E i e jkri
= (element factor)(array f)
k 2 = 2 cs = 2 o Po
cs = v p = vg = ( Po o )
(d2/dz2 + 2LC)V(z) = 0
p, u continuous at boundaries
p = p+e-jkz + p-e+jkz
uz = s-1(p+e-jkz p-e+jkz)
A up da + ( d dt )V ( o
0.5
or ( K o )
0.5
u 2 + p2 2Po dV
Zn (z)
= [1 + (z) ] [1 (z) ] = R n + jX n
Mathematical Identities
sin2 + cos2 =1
r = i
H(f ) =
sin t sin i = k i k t = n i n t
e x = 1 + x + x 2
2! + x 3 3! + ...
c = sin 1 ( n t n i )
sin = ( e j e j ) 2 j
> c E t = Ei Te+x jk z z
cos = ( e j + e j ) 2
k = k '
jk ''
h(t)e jt dt
Vector Algebra
= x x + y y + z z
= T
1
A B = A x B x + A y B y + A z Bz
At o ,
w e = w m
(
)
= ( H 4 ) dv
( A ) = 0
( A ) = ( A ) 2
A
Q n = n w Tn Pn = n 2 n
sn = jn - n
2 0.5
f mnp = ( c 2 ) [ m a ] + [ n b] + [ p d ]
2
2 = 2 x 2 + 2
y2 + 2 z 2
w e = V
E 4 dv
wm
A ( G ) da = c G
ds
Optical Communications
e jt = cos t + jsin t
dn 2 dt = An
2 + B ( n 2 n1 )
Spherical Trigonometry
4 r
- D433 -
sin dd = 4
- D434 -
Appendix E:
Expressions
sin = a/c
cos = b/c
tan = a/b
a
b
a2 + b2 = c2
sin2 + cos2 = 1
ej = cos + jsin
(d/d)sin = cos
(d/d)cos = -sin
ax = (eln a)x
(d/dx)xn = nxn-1
(d/dx)f1[f2()] = [df1/df2][df2()/d]d/dx
(d/dx)sin[f()] = cos[f()][df()/d]d/dx
sin d = -cos
cos d = sin
eax dx = eax/a
xn dx = xn+1/(n+1)
- E435 -
- E436 -
Index
acceptor atoms, 245, 391
Ampere, 14
Amperes Law, 40
angular frequency, 30
atmosphere, 355
beam-splitter, 394
biaxial, 291
birefringence, 293
bit, 181
branch currents, 90
branches, 88
bridge circuit, 91
capacitance, 68
capacitors, 68
coercivity, 49
commutator, 164
complex frequency, 93
- i437
complex notation, 32
conduction, 42
conductivity, 25
conservation of energy, 12
conservation of momentum, 14
conservation of power, 13
constitutive relations, 41
Coulomb, 14
coupling, 71
critical temperature, 43
critically matched, 99
curl, 23
cylindrical capacitor, 71
del operator, 23
demagnetize, 49
diamagnetic, 47
dielectric constant, 68
diffraction, 338
divergence, 23
dot product, 23
duality, 274
dynode, 389
electric charge, 14
electric dipoles, 44
electric field, 15
electron, 14
external Q, 223
Farad, 68
Faradays Law, 52
integral form, 40
ferromagnetic, 48
flux density, 60
395
- i438
Gausss Law
for B , 40, 50, 51, 52, 53, 55, 56, 57, 58,
generator, 167
362
half-power bandwidth, 98
Henry, 134
holes, 42
hysteresis curve, 49
inductance, 73
inductors, 72
internal Q, 222
ionosphere, 355
Joule, 13
kinetic energy, 13
lasers, 380
LC resonant frequency, 94
linewidth, 388
loop currents, 90
loudspeaker, 415
lumped elements, 88
magnetic domains, 48
magnetic flux, 73
magnetic saturation, 48
magnetic susceptibility, 47
magnetization, 47
magnetization curve, 48
magnetoquasistatics, 85
masers, 380
matched load, 91
Maxwell equations
time-harmonic, 33
Maxwell's equations, 24
mechanical power, 14
MEMS, 154
metals, 42
microphone, 414
- i439
momentum, 11, 14
nodes, 88
ohm, 66
parallel-plate capacitor, 69
paramagnetic, 47
permanent magnet, 49
permeability, 25, 47
permittivity, 25, 44
phasor, 32
photodiodes, 390
photonics, 368
photons, 14
phototube, 389
planar resistor, 67
plasma, 296
polarization, 28, 35
polarization vector, 45
polysilicon, 250
position vector, 27
potential energy, 13
power radiated, 60
Poynting theorem, 56
complex, 59
Poynting vector, 57
external, 99
internal, 99
loaded, 99
quadratic equation, 94
RC time constant, 93
- i440
relays, 171
resistors, 65
resonator bandwidth, 98
RL time constant, 94
RLC resonators, 92
rotor, 159
shielding, 270
sidelobes, 360
spatial frequency, 31
Stokes' theorem, 39
susceptibility, 45
lossy, 248
toroid, 75
with a gap, 77
toroidal inductor, 76
torque, 159
transformers, 80
transistors, 240
uniaxial, 291
velocity of light, 28
Volts, 300
VSWR, 204
wave amplitude, 30
wavelength, 31
wavenumber, 31
- i441
- i442
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