EECS 412 Electromagnetic Fields III Formula Sheet
Maxwell’s Equations (general differential)
B
 E  
t
˜
 D  
D
 H  J 
t
B  0
Maxwell’s Equations (time harmonic )
  E  j B
 D  
D
H  J
t
 B 0
Maxwell’s Equations (integral )
B
C E  dl   S t  ds
SD  ds  V dv
 H  dl  
 B  ds  0
C
S
[1.11a]
[1.11b]
[1.11c]
[1.11d]
[1.11a]
[1.11b]
[1.11c]
[1.11d]
[1.1]
[1.2]
D
 ds
S t
J  ds  
[1.3]
[1.4]
S
Electromagnetic Boundary Conditions
nˆ  E1 E2  0
[1.12]
[1.13]
[1.14]
[1.15]
[1.16]
ˆ  H1 H 2  0
n
nˆ  H 1  J
˜s
nˆ  D1 D2  
nˆ  B1 B2  0
˜

nˆ  J1  J2   s
t
2/6/16
[1.17]
-1-
5:03 AM
EECS 412 Electromagnetic Fields III Formula Sheet
Reflection & transmission (simple dielectric)
E
   n  n2
  r0  2 1  1
E i0 2  1 n1  n 2
E
22
2n1
T  t0 

E i 0 2  1 n1  n 2
Basic waves
c
c
n
   rr 

p

2
2f

o
[3.2]
[3.3]
p.135
  
p
1

f   p 


 ;



Uniform plane waves in arbitrary direction
1
H  r  kˆ  E r 

Er  kˆ  H r
[2.61]
Reflection & transmission (multiple dielectrics)
 2  1 3  2    2  1 3  2 e 2 j2 d

eff 
 2  1 3   2 2  1 3  2e2 j2d
42 3e  j 2 d
Teff 
2  13  2    2  13  2 e 2 j 2d
3  j 2 tan 2 d

2  j 3 tan 2 d 23
Z d   1 23  1
eff  eff e j  2

Z2 d   1  23  1
Z2 d    2
[3.8]
[3.9]
[3.10]
r
2/6/16
[3.7]
-2-
5:03 AM
EECS 412 Electromagnetic Fields III Formula Sheet
Plane waves in lossy materials
    j  j   j 
[2.19]
12
 

2

  
[2.20]
 
1     1
2 




12
2

 



 1

[2.21]
 
   1
2 


Lossy materials
Polarization currents:
[p.46]
c  ' j"
 eff "
tan  c 

[p.48]
 '  '
Conduction Currents:
 eff    "
[p.48]
  "
tan  c 
[p.48]
 '

Use tan  c instead of
in expression for complex impedance





c 

e j 1 2Tan
  
2 14
 eff
   
 j
 1    


1
  
[2.22]
Good conductor approximations
  

1

c 
2/6/16


2
2


 
[p.54]
1
f 

1
 
1 j

 
12

[2.26]
j 


 j 45
e

-3-
[p.54]
5:03 AM
EECS 412 Electromagnetic Fields III Formula Sheet
Poynting Theorem

 E  J dv   t 
V
V
Sav  ˆz
1
2  H
2
1
2 
  E dv  S E  H  ds
2
E 02
2
[2.31]
[2.32]
1
Sav  R eE  H *
2
[2.43]
Arbitrarily directed uniform plane waves
ˆ
Er   E0 e  jkr
[2.58]
xˆ  x  yˆ  y  ˆz z
kˆ =
[p.97]
 
1
Hr   kˆ  Er 
[2.61]

Reflection and refraction of oblique waves at planar dielectric interfaces
sin i  p1

n

 2 2  2
sin  t  p 2
11 n1
(known as Snell’s Law)
[3.19]
E
 cos i  1 cos t
  r0  2

E i0 2 cos i  1 cos t
cos i   2   sin 2  i
1
[3.24]

2
2
cos i 
1  sin i
E
2 2 cos i
2cos i
T  t 0 

[3.25]
E i 0 2 cos i  1 cos t

2
2
cos i 
1  sin i
|| 
E r0 1 cos i  2 cos t


Ei 0
1 cos i   2 cos t
E
22 cos i
T||  t 0 

E i0 1 cos i  2 cos t
2/6/16
 cos i  1   2   sin 2 i
2
1
cos i  1   2   sin 2 i
2
1
2 1  cos i
2
cos i  1  1 1  sin 2  i
2
2
-4-
[3.26]
[3.27]
5:03 AM
EECS 412 Electromagnetic Fields III Formula Sheet
Total internal reflection
sin  ic 
2 n 2

1 n1
[3.29]
for i  ic cos t   j
 
1 2
sin  i 1
2
cos  i  j sin 2 i   2 
1
cos  i  j sin  i  2 
1
[3.31]
 1e j 
[3.32]
2
tan

2

sin 2  i 
2
cos i
1
[p.192]
 2 cos  j sin 2    2
i
i

1
||   1
 1e j
 2 cos  j sin 2    2
i
i
1
1
tan
||
2

sin 2  i 
2
2
||
1
[3.33]
[p.192]
1 cos i
Normal incidence on a lossy medium
12
 2 2 
12
1 j
 2  j  eff  j 
  j2 2  
 j 

[3.39]
12
 2 j 2 

1 j
c  Zs  Rs  jXs 
 
 2 

 eff   2 
 2  2
[3.40]
 2 cos  j sin 2    2
i
i

1
||   1
 1e j
 2 cos  j sin 2    2
i
i
1
1
[3.33]
tan
2/6/16
||
2

sin 2  i 
2
2
||
1
[p.192]
1 cos i
-5-
5:03 AM
EECS 412 Electromagnetic Fields III Formula Sheet
Parallel plate waveguide
Parallel-plate TEm modes: m=0,±1,±2,…
m  z
E y  C1 sin
x e
a 
1 E y
m
m  z
Hz  

C1 cos
x e
 a 
j x
j a


m  z
Hx  
Ey  
C1 sin 
x e
j
j
a 
[4.12a]
[4.12b]
[4.12c]
Parallel-plate TMm modes: m=0,±1,±2,…
m   z
H y  C4 cos
x e
a 


m 
Ex 
Hy 
cos x e z
j
j
a
jm
m  z
Ez 
C4 sin 
x e
a
a 
[4.13a]
[4.13b]
[4.13c]
Parallel-plate TEM mode
Hy  C4 e z

Ex 
C4 e z
j
Ez  0
[4.14a]
[4.14b]
[4.14c]
Propagation constants
m p
m
f cm 

2a
2a 
[4.15]
2
 f c 
m 
  jm  j       j  1  m  , f  f c m
a
 f 
2
2
[4.16]
2
2
 f c 
m 
  m  j     2     m   1, f  f cm
a
 f 
m 
p 
m
2/6/16
2
 m



m

[4.14c]
[4.18]
2
f

1  c m f 


p
[4.18]
2
 f

1  c m 
f 

-6-
5:03 AM
EECS 412 Electromagnetic Fields III Formula Sheet
Conduction losses
R
1 o
 cTEM  s 
a  a 2
[4.22]
2
c

c

TEm
TM m
 f 
2Rs cm 
 f 
 f 
a 1  cm 
 f 
2Rs
[4.23]
2
[4.24]
2
 f 
a 1  c m 
 f 
Dielectric losses
  '  " '
d 
2
 c m 
2 1  
  
[4.27]
For parallel plate TE modes the total power through the guide is
b a
C12 2 m 
C12 ab
[p.277]
Pav   
sin 
x dxdy 
2
a 
4
0 0
For the parallel plate TEM (TM0) mode the total power through the guide is
1 2
Pav  C4 ba
[p.277]
2
2/6/16
-7-
5:03 AM

EECS 412 Electromagnetic Fields III Formula Sheet
Dielectric slab waveguide
TM Modes The non-zero field components are E z , E x ,and
For x≤-d/2
Ez0x  Co sin x x  Ce cosx x
where the transverse propagation constant is given by
x2   2d  d  2  hd2
For x≥-d/2
C e  x x  d 2 ,x  d
 a
2
E z0 x     x  d 
x
2
d

,x   2
Ca e
where the transverse attenuation constant is given by
x2   2   2 0 0  h02 or x  x2   2 0 0
The cutoff frequencies are given by
m  1,3,5,...odd
m 1
fCTM m 
2d d  d  0 0 m  2,4,6...even
Hy
[4.34]
[4.35]
[4.36]
[4.37]
[4.45]
TE Modes The non-zero field components are Hz , Hx ,and E y
For x≤-d/2
[4.46]
Hz0x  Co sin x x  Ce cosx x
where the transverse propagation constant
x2   2d  d  2  hd2
For x≥-d/2
C e  x x  d 2 ,x  d
 a
0
2
Hz x     x  d 
x
2
d

,x   2
Ca e
where the transverse propagation constant
x2   2   2 0 0  h02 or x  x2   2 0 0
[4.37]
For Odd TM Modes:
x  0  x d
 tan
 x  d  2 
For Even TM Modes:
x

 d 
  0 cot  x 
x
d
2
For ALL TM Modes:
x   2 d  d  0 0  x2
The cutoff frequencies are given by
m  1,3,5,...odd
m  1
fCTEm 
2d d  d   0 0 m  2,4,6...even
[4.40]
[4.44]
[4.42]
[4.49]

2/6/16
-8-
5:03 AM
EECS 412 Electromagnetic Fields III Formula Sheet
Dielectric covered ground plane
fCTEm 
m  1,3,5,...odd _ TMm
2d d  d   0 0 m  2,4,6...even _TE m
m 1
[4.50]
Dielectric slab waveguide ray theory
tan  i 

x
[p.306]
Detailed example of odd TM Modes for slab dielectric waveguide: free space above the
guide
For x≥d/2
 d   x  d 
E z0 x   C0 sin  x e x 2
[4.39a]
2
j 
 d   x d 2
E x0 x   
C0 sin x e x
[4.39b]
 2 
x
j  0
 d   x d 
[4.39c]
Hy0 x   
C0 sin  x e x 2
x
2
For |x|≤d/2
Ez0x  C0 sin x x
j 
E x0 x   
C0 cosx x 
[4.39d]
[4.39e]
x
j  d
H y0  x  
C cos x x 
x 0
[4.39f]
For x≤-d/2
 d   x  d 
E z0 x   C0 sin  x e x 2
2
j 
 d   x d 2
E x0 x   
C0 sin x e x
 2 
x
j  0
 d   x d 
Hy0 x   
C0 sin  x e x 2
x
2
2/6/16
[4.39g]
[4.39h]
[4.39i]
-9-
5:03 AM
EECS 412 Electromagnetic Fields III Formula Sheet
Rectangular waveguides: TM modes
m  n   jmn z
E z  C sin
x sin
y e
a   b 
j C m
m  n   jmn z
E x   mn2
cos
x sin
y e
h
a
a   b 
j C n m  n   jmnz
E y   mn2
sin 
x cos
y e
h
b
a   b 
jC n m  n   jmnz
Hx 
sin 
x cos
y e
h2 b
a   b 
jC m
m  n   jmnz
Hy   2
cos
x sin
y e
h
a
a   b 
[5.13a]
[5.13b]
[5.13c]
[5.13d]
[5.13e]
2
ZTM mn
0
 f c 
Ex
 0   1  mn 
Hy
 f 
[5.16]
Rectangular waveguides: TE modes
m  n   jmn z
Hz  C cos
x cos
y e
a   b 
j
m m  n   jmn z
Hx  mn
sin 
x cos
y e
2 C
h
a
a   b 
j
n
m  n 
Hy  mn
cos x sin  ye  jmnz
2 C
h
b
a
b
j n
m  n   jmn z
Ex  2 C
cos
x sin
y e
h
b
a   b 
j m m  n   jmnz
Ey   2 C
sin 
x cos
y e
h
a
a   b 
[5.21a]
[5.21b]
[5.21c]
[5.21d]
[5.21e]

ZTE mn 
[5.22]
2
 f 
1  cmn 
 f 
For both TM and TE modes:
m  n 
  h            2 
a
b
2
2
2
2
[5.14]
m  n 
where h      
a
b
2
2
2
mn
c
m  n 


  a   b 
1
mn
2
2
[5.15]
2
 f c 
m  n 
mn            1  mn 
a
b
 f 
2
2
2
2/6/16
-10-
[5.16]
5:03 AM
EECS 412 Electromagnetic Fields III Formula Sheet
Dominant TE10 mode:
 x 
Hz  C cos e  j1 0 z
a
j aC x 
Hx  10 sin  e  j10z

a
jaC  x   j10 z
Ey  
sin  e

a
Ex  H y  0
[5.23a]
[5.23b]
[5.23c]
[5.23d]
 
2   
10            
a

a
2
2
2
2
c
TE10
10 
f c10 
 2b  f c 2 
1  10  
a  f  
R 

 s 
2
b
 f c 
1  10 
 f 
2
10

[5.29]

[5.23f]
  
1
2a 
2
1
[5.24]
2a 
ZTE1 0 
2/6/16
[5.23e]

2
 f 
1  c1 0 
 f 

2af
4a 2f 2  1
[5.24]
-11-
5:03 AM
EECS 412 Electromagnetic Fields III Formula Sheet
Circular waveguides: TM modes where Hz  0
t 
E z  Cn J n  nl rcosn e  jn l z
a
jaTM nl
t 
Cn J n '  nl rcosn e  jnl z
tnl
a
2
ja nTM nl
t 
E  
Cn J n  nl rsin n e  jnl z
2
tnl
a
2
jna
t 
Hr   2
Cn J n nl r sin n e  jnl z
 a 
tnlr
ja
t 
H  
Cn J n '  nl rcosn e  jnl z
tnl
a
where
Er  
t 
a
[5.46a]
[5.46b]
[5.46c]
[5.46d]
[5.46e]
2
TM   2    nl 
[5.44]
t nl
2a 
Circular waveguides: TE modes where E z  0
s 
Hz  Cn J n  nl rcosn e  jnl z
a
[5.45]
nl
f cTMnl 
jaTE nl
s 
Cn J n '  nl rcosn e  jnl z
snl
a
2
jna TE nl
s 
H 
Cn J n  nl rsin n e  jnl z
2
snlr
a
2
ja n
s 
Er  
Cn J n nl r sin n e  jnl z
2
 a 
snlr
ja
s 
E  
Cn J n '  nl rcosn e  jnl z
snl
a
Hr  
[5.48a]
[5.48b]
[5.48c]
[5.48d]
[5.48e]
where
2
 TE

nl
f cTEnl
2/6/16
s 
    nl
 a 
snl

2a 
2
[p.359]
[p.359]
-12-
5:03 AM
EECS 412 Electromagnetic Fields III Formula Sheet
Table 5.2 lth roots (tnl) of Jn(.)=0
l=
1
2
3
4
0
2.405
5.520
8.654
11.792
1
3.832
7.016
10.173
13.323
2
5.136
8.417
11.620
14.796
3
6.380
9.761
13.015
16.223
n=
4
7.588
11.065
14.372
17.616
5
8.771
12.339
15.700
18.980
6
9.936
13.589
17.004
20.321
7
11.086
14.821
18.288
21.642
3
4.201
8.015
11.346
14.586
n=
4
5.317
9.282
12.682
15.964
5
6.416
10.520
13.987
17.313
6
7.501
11.735
15.268
18.637
7
8.578
12.932
16.529
19.942
Table 5.3 lth roots (snl) of Jn’(.)=0
l=
1
2
3
4
2/6/16
0
3.832
7.016
10.173
13.324
1
1.841
5.331
8.536
11.706
2
3.054
6.706
9.969
13.170
-13-
5:03 AM