TRANSMISSION MEDIA
MAXWELL’S EQUATIONS AND
TRANSMISSION MEDIA
CHARACTERISTICS
ENEE 482 Spring 2002
DR. KAWTHAR ZAKI
MICROWAVE CIRCUIT ELEMENTS AND
ANALYSIS
Dielectric
Two conductor
wire
Coaxial line
Shielded
Strip line
ENEE482
2
Common Hollow-pipe waveguides
Rectangular
guide
Circular
guide
ENEE482
Ridge guide
3
STRIP LINE CONFIGURATIONS
W
SINGLE STRIP LINE
COUPLED LINES
COUPLED STRIPS
TOP & BOTTOM
COUPLED ROUND BARS
ENEE482
4
MICROSTRIP LINE CONFIGURATIONS
TWO COUPLED MICROSTRIPS
TWO SUSPENDED
SUBSTRATE LINES
SINGLE MICROSTRIP
SUSPENDED SUBSTRATE
LINE
ENEE482
5
TRANSMISSION MEDIA
• TRANSVERSE ELECTROMAGNETIC (TEM):
– COAXIAL LINES
– MICROSTRIP LINES (Quasi TEM)
– STRIP LINES AND SUSPENDED SUBSTRATE
• METALLIC WAVEGUIDES:
– RECTANGULAR WAVEGUIDES
–CIRCULAR WAVEGUIDES
• DIELECTRIC LOADED WAVEGUIDES
ANALYSIS OF WAVE PROPAGATION ON THESE
TRANSMISSION MEDIA THROUGH MAXWELL’S
EQUATIONS
ENEE482
6
Electromagnetic Theory Maxwell’s Equations
B
 E  
-M
t
D  
D
;  H 
J
t
; B  0

Continuity Equation :
 J  
t
E  Electric Filed Intensity V/m
D  Electric Flux Density C/m 2
B  Magnetic Flux Density T (Telsa or V - Sec./m 2 )
H  Magnetic Field Intensity A/m
J  Electric current density A/m 2
M  Magnetic (fictitiou s) current density V/m 2
ENEE482
7
Auxiliary Relations:
1. F  qE  v  B 
Newton
q  Charge
; v  Velocity
2. J  E
(Ohm' s Law)
  Conductivi ty
; J  Conduction Current
3. J  v
; J  Convection Current
4. D  E   r  o E
;  o  8.854 10 12 F/m
 r  Relative Dielectric Constant
5. B  H   r  o H
;  o  4 10 12
H/m
 r  Relative Permeabili ty
ENEE482
8
Maxwell’s Equations in Large Scale Form
 D  d S   dv
S
V
 B  dS  0
S

l E  d l   t SB  d S  S M  dS

l H  d l  SJ  d S  t SD  d S
ENEE482
9
Maxwell’s Equations for the Time - Harmonic Case
Assume e jt variations , then :
E ( x, y, z , t )  Re[ E ( x, y, z )e jt ]
E ( x, y, z )  a x ( E xr  jExi )  a y ( E yr  jE yi )
 a z ( E zr  jEzi )
E x  Re[( E xr  jExi )e jt ]  Re[ E xr2  E xi2 e jt  j ]
 E 2 xr  E 2 xi cos(t   ) ,   tan 1 ( E xi / E xr )
D  
,  B  0
  E   j B  M
,   H  J  j D
ENEE482
10
Boundary Conditions at a General Material Interface
1,1
Et1  Et2   M s
h
Dn1  Dn2   s
 s  Surface Charge Density
Bn1  Bn2
E1t
;
2,2
E2t
H t1  H t2  J s
nˆ  ( D1  D2 )   s
Ds
nˆ  ( B1  B2 )  0
D1n
h
nˆ  ( E1  E2 )   M s
D2n
nˆ  ( H1  H 2 )  J s
ENEE482
11
Fields at a Dielectric Interface
Et1  Et2  0
Dn1  Dn2  0
Bn1  Bn2
;
H t1  H t2  0
nˆ  ( D1 )  nˆ  ( D2 )
nˆ  ( B1 )  nˆ  ( B2 )
nˆ  ( E1 )  nˆ  ( E2 )
nˆ  ( H1 )  nˆ  ( H 2 )
ENEE482
12
Boundary Conditions at a Perfect Conductor :
Et  o
n̂  E  0
D n  ρ s  nˆ  D
Bn  0
n̂  B  0
J s  H t  nˆ  H
Js
s
+++
n
ENEE482
Ht
13
The magnetic wall boundary
condition
nˆ  ( D )  0
nˆ  ( B )  0
nˆ  ( E )   M s
nˆ  ( H )  0
ENEE482
14
Wave Equation
    E   E  (  E )   j  H 
2
 j ( J  jE )
For a Source free medium :
2 E  k 2 E  0
 H k H 0
2
2
;
;
k 2   2 
k  /v 
ENEE482
2

15
Plane Waves
2
2
2

E

E

E
2
2
 E  k 0 E  0  2  2  2  k 02 E
x
y
z
 2 Ei  2 E i  2 Ei
2


 k 0 Ei  0
, i  x, y , z
2
2
2
x
y
z
Solve for E x ( x, y , z ), Using separation of variables 
k 2x  k y2  k z2  k 02
 jks x  jk y y  jkz z
E x  Ae
 


r  ax x  a y y  az z
 


, Let k  a x k x  a y k y  a z k z
ENEE482
16
E x  Ae  jk r ,
E  E 0 e  jk  r
Similarly E y  Be  jk r ,
E z  Ce  jk r
Since   E  0  k  E 0  0
The vector E 0 is perpendicu lar to the direction of propagatio n k.
The solution is called plane wave
  E   j 0 H
H 
1
j 0
  E 0 e  jk  r 
1
j 0
E 0   e  jk  r 
1
 0
k  E 0 e  jk  r
0
1

nE 
n  E  Y0 n  E 
nE
 0
 0
0
k0
 0 is the interensic impedance of free space  377
Y0 is the intrinsic admittance of free space.
ENEE482
17
z
E
H
n
y
x
H is perpendicular to E and to n. (TEM waves)
E  Re( E0 e  jk r  jt )  E0 cos( k  r  t ), if E0 is real
k 0 0  2 ,
*
1
1
1
*
*
P  Re E  H  n  Re Y0 E  ( n  E )  n  Y0 E0  E0
2
2
2
The time average energy densities in the electric and
magnetic fields of a TEM wave are :
*
1
1
*
U e   0 E0  E0
, U m  0 H 0  H 0  U e
2
2
ENEE482
18
Plane Wave in a Good Conductor
  
    j  j 
s 

1

j




 (1  j)
2
j
2

1

 (1  j )
 (1  j)
 s
2
ENEE482
19
Boundary Conditions at the Surface
of a Good Conductor
The field amplitude decays exponentially from its surface
According to e-u/s where u is the normal distance into the
Conductor, s is the skin depth
s 
2

,
J  E
The surface Impedance :
1 j
Zm 
,
Et  Z m J s  Z m n̂  H
 s
ENEE482
20
Reflection From A Dielectric Interface
Parallel Polarization
Er
0

x
n2
q2
n3
q3
q1
Et
z
n1
E ijk n r
Ei  E1e 0 1 1
, H i  Y0n1  Ei
Er  E2e  jk0n2 r1
, H r  Y0n2  Er
k0   0
0
, Y0 
0
ENEE482
21
E t  E 3e
 jk n3  r1
Y  nY0
, H t  Y n3  Et
, k  k0 , n  
k0n1 x  k0n2 x  kn3 x  nk0n3 x


n1  a x sin q1  a z cos q1


n2  a x sin q 2  a z cosq 2


n3  a x sin q 3  a z cos q 3
q1  q 2
sin q1  n sin q 3
E1 x  E1 cos q1
, E1z   E1 sin q1
E2 x  E2 cos q 2
, E1z  E2 sin q 2
E3 x  E3 cos q 3
, E1z   E3 sin q 3
ENEE482
22
Energy and Power
Under steady-state sinusoidal time-varying
Conditions, the time-average energy stored in the
Electric field is
1
1
*
We  Re  E  D dV    E  E * dV
4V
4V
If  is constant and real, then
We 

E E

4
*
dV
V
ENEE482
23
Time average energy stored in the magnetic field is :
1
*
Wm  Re  H  B dV
4V


4
*
H

H
dV if  is real and constant

V
The time average power tran smitted across a closed
surface S is given by :
1
P  Re  E  H *  dS
2
S
ENEE482
24
Poynting Theorem
  E  H *  (  E )  H *  (  H * )  E
 ( jB  M )  H *  jD *  E  E  J *
J  J s  E

1
1
*
*


E

H
d
V


E

H
 dS


2V
2S
 
1
 j  ( B  H *  E  D * )dV   ( E  J *  H *  M s )dV
2V
2V
 B  H * E  D* 
1
*
*


 2 j  

dV

(
E

J

H
 M s )dV


4
4 
2V
V
ENEE482
25
If the medium is characteri zed by :
    - j  and conductivi ty 
     j ,
1
1
*
-  ( E  J Ss  H  M S )dV   E  H *  dS 
2S
2V

1
*
*
*




E  E dV   (  H  H   E  E )dV 

2V
2V
j

*
*


)dV
E

E


H

H

(

2V
1
P0   E  H *  dS
ENEE482
2S
26
P 

*
*




(

H

H


E

E
)dV 

2V
1
*

E

E
dV

2V
 Time average power loss
1
*
Ps    ( E  J s  H s  M s )dV
2V
*
*


1
H

H
E

E
*



dV
 Im  E  H  dS  2   

2S
4
4 
V
 2 (Wm  We )
Ps  P0  P  2 j (Wm  We )
The power delivered by the sources (Ps ) is equal to the sum of
the power tran smitted through t he surface P0 , the power lost to heat
in the volume ( P ) and 2 times the reactive energy stored in the
volume.
ENEE482
27
Circuit Analogy
I
L
R
C
V
1 * 1
1
j
VI  ZII *  II * ( R  jL 
)
2
2
2
C
*
1
1
1
II
 RII *  2 j ( LII * 
)
2
2
4
4 C
 P  2 j (Wm  We )
P  2 j (Wm  We )
Z
1 *
II
2
General Definitio n of the impedance of a network
ENEE482
28
Potential Theory

Let B    A ,
  E  jA   0 ,

  E   jB   j  A
E  jA  
E   jA  
1
  H      A  jE  J   2A  j  J

    A    A   2 A  k 2 A  j  J ,
k 2   2 
, Let   A   j
or   A   j
(Lorentz condition)
 2 A  k 2 A   J Inhomogene ous Helmholtz equation.

 D  0     k   

2
2
ENEE482
29
Solution For Vector Potential
J (x’,y’, z’) R
(x,y,z)
r’
r

e  jkR
A ( x, y , z )  A ( r ) 
J ( r )
dV  for an infinitism al current
4
R
R  ( x  x ) 2  ( y  y ) 2  ( z  z ) 2  r  r 

A(r ) 
4

V
e  jkR
J ( r )
dV 
R
ENEE482
30
Waves on An Ideal Transmission Line
Rg
z
Lumped element circuit model for a transmission line
I(z,t)
V(z,t)
Ldz
I(z,t)+I/z dz
Cdz
V(z,t)+v/z dz
ENEE482
31
 2V ( z, t )
 2V ( z, t )
 LC
0
2
2
z
t
 2 I ( z, t )
 2 I ( z, t )
 LC
0
2
2
z
t
1
v
LC
z
z
V ( z, t )  V  f  (t  )  V  f  (t  )
v
v
z
z
 
 
I ( z , t )  I f (t  )  I f (t  )
v
v


V
V
L


I 
, I 
, Zc 
Zc
Zc
C
Z c : Characteri stic Impedance
ENEE482
32
Steady State Sinusoidal Waves
V g (t )  V g cos t
V ( z )
  jLI ( z )
z
I ( z )
  jCV ( z )
z
d 2V ( z )  2

 2 V ( z)  0 ,   
2
v
dz
v
V ( z )  V  e  jz  V  e jz
I ( z )  I  e  jz  I  e jz , I   YcV  , I   YcV 
1
L
Zc 

Yc
C
,    LC
ENEE482
33
Transmission Line Parameters
Let the voltage between th e conductors be :
V0 e  jz and the current be I 0 e  jz . The time - average
stored magnetic energy for 1 m section of line :

L
Wm   H H dS  I 0
4S
4
*

C
We   E E dS  V0
4S
4
*
2
C1
2
C2
S
Power loss per unit length due to finite conductivi ty of
the conductor is
Rs
Pc 
2

C1  C 2
H  H d  R I 0 / 2, Pd 
*
2
ENEE482
 
2
 E E
S
*
dS  G
V0
34
2
2
Terminated Transmission Line
To generator

Zc
ZL
Z

V  V  V  VL
VL
1
I  I  I  IL 

(V   V  )
ZL Z c


V
L   Reflecti on coefficien t
V
1  L Z L
Z L / Zc  1

, L 
1  L Z c
Z L / Zc  1
ENEE482
35
1
1 
*
 2
P  Re(VL I L )  Re Yc V (1  L )(1  L )* 

2
2 
1
2
 2
 Yc V (1  L )
2
V  V  e  jz  LV  e jz
 V  e  jz  e jq V  e jz
, L  e jq
q 

2
V  V  1     4  sin 2 ( l  )
2 

1 
S
1 

Z
Z  jZ c tan 
Z in  in  L
Z c Z c  jZ L tan 
ENEE482
1/ 2
36
Transmission Lines & Waveguides
Wave Propagation in the Positive z-Direction is Represented By:e-jz
E  x, y , z   E t  x, y , z   E z  x, y , z 
 et  x, y e  jz  ez  x, y e  jz
H  x, y , z   H t  x, y , z   H z  x, y , z 
 ht  x, y e  jz  hz  x, y e  jz
  E  ( t  ja z )  ( et  ez )e  jz   j ( ht  hz )e  jz
 t  e  ja z  e   t  ez  ja z  ez   j ( ht  hz )e  jz
 t  et   jhz
,  t  ht  jez
 t  ht  jhz
,  t  et  jez
ENEE482
37
Modes Classification:
1. Transverse Electromagnetic (TEM) Waves
Ez  H z  0
2. Transverse Electric (TE), or H Modes
Ez  0
but H z  0
,
3. Transverse Magnetic (TM), or E Modes
Hz  0
,
But Ez  0
,
Ez  0
4. Hybrid Modes
Hz  0
ENEE482
38
TEM WAVES
 t  ht  0
 t  et  0
,
,
 t  ht  0 ,
 t  et  0
â z  e t   0 h t
â z  h t   0 e t
e  x, y    t   x, y   0
  Scalar Pot ential
  t   x, y   0
2
Et  et e  jz   t  ( x, y )e  jz
H t   ht e  jz  Y0 aˆ z  e e  jz
ENEE482
39

1
Y0 


Z0
 0  Wave Impedance
,
Ey
Ex

  0
Hy
Hx
 for wave propagatio n in the  or - z direction
The field must satisfy Helmholtz equation :
 2 Et  k 02 Et  0
, but    t  j a z
 t2 Et  (k 02   ) Et  0
,
,  2   t2   2
 t [ t2  (k 02   ) ]  0
   k 0 for TEM waves
ENEE482
40
TE WAVES
2 H  k 2 H  0
( t   2 )hz ( x, y )  k 2 hz  0
2
 t  ( k 2   2 )hz  0 ,
let k c2  k 2   2
2
 t h z  k c2 hz  0
2
 t  et   j hz
,
a z  et  ht
 t  ht  0
,
a z   t hz  ja z  ht   je
 t  ht  jhz
,
 t  et  0
ENEE482
41
j
 ht   2  t hz
kc
 0
k
et  
aˆ z  ht   Z 0 aˆ z  ht ;


Zh 
k


0 

 0  Wave Impedance
ey
ex

 Zh
hy
hx
ENEE482
42
TM WAVES
2 E  k 2 E  0
( t   2 )e z ( x, y )  k 2 e z  0
2
 t e z  ( k 2   2 )e z  0 ,
2
let k c2  k 2   2
 t e z  k c2 ez  0
2
j
 et   2  t ez
kc
ht  Ye aˆ z  e
Ye 
k

Y0  Wave Admittance
ENEE482
43
TEM TRANSMISSION LINES
Parallel -plate
Two-wire
ENEE482
Coaxial
44
COAXIAL LINES
a

b
1  
1  2

(r
) 2

0
for
0
2
r r r
r 

  C1 ln r  C2
  V0 at r  a,   0 at r  0
ln( r / b)
  V0
ln( a / b)
V0
E
a r e - jkz
r ln( b / a )
Y0 


and
ENEE482
V0
H   Y0
a  e - jkz
r ln( b / a )
45
V0
J s  nˆ  H  aˆ r  H  Y0
aˆ z e - jkz
a ln( b / a )
2
V0
2V0 - jkz
- jkz
I  Y0
ade  Y0
e

a ln( b / a ) 0
ln( b / a )
b 2
2

Y
V
1
0 0
P  Re   E  H * aˆ z rdrd 
2
ln( b / a )
a 0
• THE CHARACTERISTIC IMPEDANCE OF A COAXIAL IS Z0
V0
1
b
Zc 

ln   Ohms
I 0 2Y0  a 
ENEE482
46
Zc OF COAXIAL LINE AS A FUNCTION OF b/a
10
X
ENEE482
=
260
240
220
200
180
160
140
120
100
80
60
40
20
1
0
b/a
100
 r Zo
47
Transmission line with small losses
k  k 0 ( r  j r)1 / 2
r 

 ( r  j r)
0
and
Y  Y0 ( r  j r)1 / 2
For small losses     
 rk 0
 r 1 / 2
jk    j  j  r k 0 (1  j )  j  r k 0 
 r
2  r
 rk 0

2  r
,
E   t  e  jkz
Y   r Y0
   r k 0
, H  Yaˆ z  E
, k  k 0  r
 r is equivalent to the conductivi ty 
J  E
ENEE482
48
The power loss per unit length is :
*
1
 
*
P 
J  J dS 
E  E dS


2 S
2 S
P  P0 e
z
P
,  P  2P
z
*
Y
P   E  E dS ,
2S
 r
 
d 

 k0
2Y 2Y0  r
2  r

The power loss due to the conductor loss :
Zm 
1 j
 s
Rm
1
*
, P  Re Z m  J s  J s d 
2
2
S1  S2
1
Re  E  H * dS
2
RmY b  a
c 
2 ln( b / a ) ab
H
 H s d
*
s
S1  S2
P
, Y  Y0

0
ENEE482
49
Qc OF COAXIAL LINE AS A FUNCTION OF Zo
3000
2800
2600
2400
2200
180
170
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
2000
10
Qc
b f GHz
3200
0
Q-Coppe r of Coa xia l Line
3400
r Z
c
ENEE482
50
TEM Modes
 t2  ( x, y )  0
Parallel Plate Waveguide
0  x  w,
y
0yd
 (x, y)  A  By
d
(x,0)  0,
 (x, d)  V0
x
w
V0 y
V0
 ( x, y ) 
, e ( x, y )   t    yˆ
d
d
V0  jkz
 jkz
E ( x, y )  e ( x, y ) e
  yˆ e
d
d
V0  jkz
H ( x, y )  zˆ  E  xˆ
e , V   E y dy  V0 e  jkz
0
d
w
w
wV0  jkz
I   J s  zˆdx   ( yˆ  H )  zˆdx 
e
0
0
d
ENEE482
51
TM modes
 
2
 2  k c e z ( x, y )  0
 y

e z ( x, y )  A sin k c y  B cos k c y
2
e z ( x, y )  0 at y  0, d
, k c d  n , n  0,1,2,3,....
B0
n 2
  K ( )
d
n
e z ( x, y )  An sin
y
d
n
E z ( x, y, z )  An sin
y e - jz
d
j
n
H x ( x, y , z ) 
An cos
y e - jz
kc
d
2
ENEE482
52
 j
n
E y ( x, y , z ) 
An cos
y e - jz , E x  H y  0
kc
d
fc 
kc
2 


2d 
The wave impedance of the TM modes is :


Z TM  


H x 
k

2
vp 
, g 


Ey
1 w d
1 w d
*
P0    E  H  zˆdydx     E y H x* dydx
2 x 0 y 0
2 x 0 y 0
2
 Re(  )d
An
for n  0
2

4k c

 Re(  )d
2

A
for n  0
n
2

2k c
ENEE482
53
Attenuatio n due to conductor loss
P
c 
2 P0
2
 2  2 Rs w
 Rs  w
P  2   J s dx 
An
2
x

0
kc
 2 
2Rs 2kRs
c 

Np/m for n  0
d
d
ENEE482
2
54
TE Modes
 
2
 2  k c hz ( x, y )  0
 y

hz ( x, y )  A sin k c y  B cos k c y
2
e x ( x, y )  0 at y  0, d
, k c d  n , n  1,2,3,....
A0
n 2
  k ( )
d
n
hz ( x, y )  Bn cos
y
d
n
H z ( x, y, z )  Bn cos
y e - jz
d
j
n
E x ( x, y , z ) 
Bn sin
y e - jz
kc
d
2
ENEE482
55
j
n
H y ( x, y , z ) 
Bn sin
y e - jz , E y  H x  0
kc
d
fc 
kc
2 


2d 
The wave impedance of the TM modes is :
E x  k
Z TE 


Hy



vp 

,
g 
2

1 w d
1 w d
*
P0    E  H  zˆdydx    E x H *y dydx
2 x 0 y 0
2 x 0 y 0
dw
2

B
Re(  )
For n  0
n
2
4k c
2k c2 Rs
c 
kd
Np/m
ENEE482
56
COUPLED LINES EVEN & ODD
MODES OF EXCITATIONS
AXIS OF EVEN SYMMETRY
AXIS OF ODD SYMMETRY
P.M.C.
P.E.C.
ODD MODE ELECTRIC
FIELD DISTRIBUTION
EVEN MODE ELECTRIC
FIELD DISTRUBUTION
Z 0o =ODD MODE CHAR.
Z 0e =EVEN MODE CHAR.
IMPEDANCE
IMPEDANCE
Equal &opposite currents are
flowing in the two lines
Equal currents are flowing
in the two lines
ENEE482
57
WAVEGUIDES
• HOLLOW CONDUCTORS RECTANGULAR OR
CIRCULAR.
• PROPAGATE ELECTROMAGNETIC ENERGY ABOVE
A CERTAIN FREQUENCY (CUT OFF)
• INFINITE NUMBER OF MODES CAN PROPAGATE,
EITHER TE OR TM MODES
• WHEN OPERATING IN A SINGLE MODE, WAVEGUIDE
CAN BE DESCRIBED AS A TRANSMISSION LINE WITH
C/C IMPEDANCE Zc & PROPAGATION CONSTANT 
ENEE482
58
WAVEGUIDE PROPERTIES
• FOR A W/G FILLED WITH DIELECTRIC r :
r 1 1
 2  2  2
2
1   g c
1
WHER E
1 IS WAVELENG TH IN DIELECTRIC
 IS WAVELENGT H IN FREE SPACE
 g IS GUIDE WAVELENG TH
C IS CUT OFF WAVELENGT H
WAVE IMPEDANCE Z w IS :
ZW 

377   g 
 
1
r 
FOR TE MODES

FOR TM MODES
377
r
1
g 
ENEE482
59
• PROPAGATION PHASE CONSTANT:
2

RADIANS/UN IT LENGTH
g
• FOR RECTANGULAR GUIDE a X b, CUTOFF
WAVELENGTH OF TE10 MODES ARE:
11.8
C  2a
,
fc 
r
c
fc
: CUT OFF FREQUENCY IN GHz (c INCHES):
• FOR CIRCULAR WAVEGUIDE OF DIAMETER D
CUTOFF WAVE LENGTH OF TE11 MODE IS:
c  1.706 D
• DOMINANT MODES ARE TE10 AND TE11 MODE
FOR RECTANGULAR & CIRCULAR WAVEGUIDES
ENEE482
60
RECTANGULAR WAVEGUIDE
MODE FIELDS
y
b
z
x
a
CONFIGURATION
ENEE482
61
TE modes
 2
2
2
 2  2  k c hz ( x, y )  0
y
 x

k c2  k 2   2
hz ( x, y )  X ( x)Y ( y )
1 d 2 X 1 d 2Y
2


k
c  0
2
2
X dx
Y dy
1 d2X
2


k
x
2
X dx
1 d 2Y
2
,


k
y
2
Y dy
k x2  k y2  k c2
hz ( x, y )  ( A cos k x x  B sin k x x)(C cos k y y  D sin k y y )
mx
ny  jz
H z ( x, y, z )  Amn cos
cos
e
a
b
ENEE482
62
TEmn MODES mx
ny
H z  cos(
) cos(
) e  jz , E z  0
a
b
Ex  Zh H y
,
E y  Z h H x
m
Hx  j
ak c2
Hy  j
Zh 
n
bk c2
mx
ny
sin(
) cos(
) e  jz
a
b
m
ny
cos(
) sin(
) e  jz
a
b
k Z0
m 2
n 2
) (
)
a
b
2ab
c 
( m 2 b 2  n 2 a 2 )1 / 2
k 2c  (
;

 2  k 2  k c2 ;
f cmn 
kc
2


1
2

ENEE482
m 2
n 2
(
) (
)
a
b
63
The dominant mode is TE10
 x  j z
H z  A10 cos
e
a
 j a
 x  j z
Ey 
A10 sin
e

a
j a
 x  j z
Hx 
A10 sin
e

a
Ex  Ez  H y  0
kc   / a ,   k 2  ( / a) 2
 a A10 b
1
*
ˆ
P10  Re   E  H  zdydx

Re(  )
2
x

0
y

0
2
4
2
Rs
a  2 a3
2
P   J s d  Rs A10 (b  
)
2
2 C
2 2
a
b
Rs
c  3
(2b 2  a 3 k 2 ) Np/m
a b  k
ENEE482
3
2
64
TMmn MODES
mx
ny
E z  sin(
) sin(
) e  jz
a
b
Hz  0
H x   E y /Z e
Ex   j
Ey   j
Ze 

2
m
ak c2
 n
bk c2
Z 0
k k
2
mx
ny
cos(
) sin(
) e  jz
a
b
m
sin(
)
a
2
c
ny
cos(
) e  jz
b
m 2
n 2
k (
) (
)
a
b
2
c
;
k
H y  E x / Ze
,
;
2ab
c 
( m 2 b 2  n 2 a 2 )1 / 2
ENEE482
65
TE Modes of a Partially Loaded
Waveguide y
TE m0 have no y - variation and the structure is uniform in the y-direction
 2
2
 2  kd  hz  0 for 0  x  t
 x

 2
2
x
 2  ka  hz  0 for t  x  a
 x

kd , ka are the cutoff wavenumbers for dielectric and air regions
   r k02  kd2  k02  ka2
 A cos kd x  B sin kd x
hz  
C cos ka (a  x)  D sin k a (a  x)
ENEE482
for 0  x  t
for t  x  a
66
 j 0
 k [ A sin k d x  B cos k d x]
ey   d
j 0

[C sin k a (a  x)  D cos k a (a  x)
 k a
for 0  x  t
for t  x  a
To satisfy th e Boundary conditions that E y  0 at x  0 and x  a 
B  D  0 , (E y , H x ) are continuous at x  t 
A
C
sin k d t  sin k a (a  t )
kd
ka
A cos k d t  C cos k a (a  t )
k a tan k d t  k d tan k a (a  t )  0
This is the characteri stic equation t hat can yields to 
ENEE482
67
CIRCULAR WAVEGUIDE MODES
y
r

a
z
ENEE482
x
68
TE Modes
2 H z  k 2 H z  0
H z (  ,  , z )  hz (  ,  )e  jz
 2
1 
1 2
2
 2 
 2
 k c hz (  ,  )  0
2
   
 

hz (  ,  )  R (  ) ( )
1 d 2R
1 dR
1 d 2
2



k
c  0
2
2
2
R d
 R d    d
 1 d 2
2

k

 d 2
d 2
2
,

k
   0
2
d
2
d
R
dR
2 2
2
2



(

k

k
c
 )R  0
2
d
d
ENEE482
69
 ( )  A sin n  B cos n
, k2  n 2
2
d
R
dR
2
2 2
2




(

k

n
)R  0
c
2
d
d
Bessel' s Different ial equation. The solution is :
R(  )  CJ n ( k c  )  DY n ( k c  )
J n ( k c  ), Yn ( k c  ) are the Bessel function of first and second kinds.
Yn ( k c  ) is infinite at   0  D  0
h z (  ,  )  ( A sin n  B cos n ) J n ( k c  )
The boundary condition E (  ,  )  0 at   a
j
E (  ,  , z ) 
( A sin n  B cos n ) J n ( k c  )e  jz
kc
ENEE482
70
J n ( kc a )  0
 )0
, J n ( pnm
  mth root of J n
pnm

 2
pnm
pnm
2
2
2
kcnm 
,  nm  k  kc  k  (
)
a
a

kc
pnm
f cnm 

2  2a 
 jn
 jz
E 
(
A
cos
n


B
sin
n

)
J
(
k

)
e
n
c
2
kc 
E 
j
( A sin n  B cos n ) J n ( kc  )e  jz
kc
 j
H 
( A sin n  B cos n ) J n ( kc  )e  jz
kc
H 
 jn
 jz
(
A
cos
n


B
sin
n

)
J
(
k

)
e
n
c
kc2 
ENEE482
71
E
Z TE 
H

 E
H
k


Dominant Mode is TE 11
H z  A sin J1 (kc  )e  jz
 j
E   2 A cos J1 (kc  )e  jz
kc 
j
E 
A sin J1(kc  )e  jz
kc
H 
 j
A sin J1(kc  )e  jz
kc 
 j
H   2 A cos J1 (kc  )e  jz
kc 
Ez  0
ENEE482
72
TEnm MODES
 
 pnm
H z  Jn

 a 
Ez  0
e
 jz
cos( n ) 


sin(
n

)


H
 J n ( pnm
  / a )  jz
 jpnm

e
2
ak c
H
  / a )  jz
 jnJ n ( pnm

e
2
rk c
E  Z h H
cos( n ) 


sin(
n

)


 sin( n ) 


cos(
n

)


E   Z h H 
;
 is the m' th zeros of Jn ( x )
pnm
 /a
Z h  kZ0 / 
; k c  pnm
 2  k 2  k c2
;

c  2a / pnm
ENEE482
73
TMnmMODES
 pnm  
Ez  J n 

 a 
Hz  0
e
 jz
cos( n ) 


sin(
n

)


  / a )  jz
 jpnm J n ( pnm
E 
e
2
ak c
cos( n ) 


sin(
n

)


 sin( n ) 


cos(
n

)


H  E  / Z e
 jnJ n ( pnm  / a )  jz
E 
e
2
rk c
H    E /Z e
;
pnm is the m' th zeros of J n ( x )
Z e  Z 0  /k
 2  k 2  k c2
k c  pnm / a
;
;
c  2a / pnm
ENEE482
74
Cutoff frequencies of the first few TE
And TM modes in circular waveguide
TE11
0
TE21 TE01 TE31
1
fc/fcTE11
TM01
TM11
ENEE482
TM21
75
ATTENUATION IN WAVEGUIDES
• ATTENUATION OF THE DOMINANT MODES (TEm0) IN
A COPPER RECTANGULAR WAVEGUIDE DIM. a X b, AND
(TE11) CIRCULAR WAVEGUIDE, DIA. D ARE:
 c (TE
 c (TE
m0)
11 )
1.9 x10 4  r

b
 2b  f  2 
 c  
1 
a  f  
f 
 dB/unit length
2
 fc 
1   
 f 
3.8 x10 4  r

D
 f  2

c
   0.42
f  f 

dB/unit length
2
f 
1   c 
 f 
WHERE f IS THE FREQUENCY IN GHz
ENEE482
76
ATTENUATION IN COPPER WAVEGUIDES
DUE TO CONDUCTOR LOSS
Alfa*a/Sqrt(epsr*f(GHz)) dB (GHz)^(-1/2)
0.0020
Alfa TE0m;b/a=.45
Alfa TEm0;b/a=.5
0.0018
Alfa Circ. TE11
0.0016
a
a
0.0014
b
0.0012
E
E
Rectangular
Guide
0.0010
Circular Guide
0.0008
0.0006
0.0004
0.0002
0.0000
1
1.2
1.4
1.6
1.8
2
2.2
(f/fc)
ENEE482
77
Higher Order Modes in Coaxial Line
TE Modes:
hz (  ,  )  ( A sin n  B cos n )( CJ n (kc  )   DYn ( kc  ))
Boundary conditions E ( kc  )  0 at   a, b
CJ n (kc a )  DYn(kc a )  0
CJ n (kcb)  DYn(kcb)  0
J n (kc a )Yn( kcb)  J n ( kcb)Yn(kc a )
This is the characteri stic equation to solve for kc
ENEE482
78
Grounded Dielectric Slab
x
d z
Dielectric
Ground plane
TM Modes
Assume e -jz variation
 2
2
2
 2   r k 0   ez ( x, y )  0 for 0  x  d
 x

 2
2
2
 2  k 0   ez ( x, y )  0 for d  x  
 x

E z ( x, y, z )  ez ( x, y )e - jz
Let kc2   r k02   2 , h 2   2  k02
ENEE482
79
ez ( x, y )  A sin kc x  B cos kc x for 0  x  d
ez ( x, y )  Ce hx  De  hx for d  x  
Boundary conditions are :
E z(x,y,z)  0
at x  0
E z(x,y,z)  
at x  
E z(x,y,z) continuous at x  d
H y(x,y,z) continuous at x  d
H x  Ey  H z  0
B0
, C 0
A sin kc d  De
 hd
kc tan kc d   r h
,
r A
D  hd
cos kc d  e
kc
h
kc2  h 2  ( r  1)k02
Solving the two equations  kc , h.
ENEE482
80
Stripline
y
w
b
x
z
Approximate Electrostatic Solution:
y
b/2
 t  ( x, y )  0
2
 ( x, y )  0
-a/2
0
a/2
at x   a / 2
& y  0, b
ENEE482
81

nx
ny

  An cos a sinh a for 0  y  b/2
 ( x, y )    n 1
 Bn cos nx sinh n (b  y ) for b/2  y  b
 n 1
a
a
Potential must be continuous at y  b/2  An  Bn

Ey  
y


nx
ny
 n 
A
cos
cosh
for 0  y  b/2
  n  a 
a
a

E y    n 1 
 An  n  cos nx cosh n (b  y ) for b/2  y  b
 n 1  a 
a
a
ENEE482
82
1
for x  w / 2
Let  s ( x)  
for x  w / 2
0
 s ( x )  D y ( x, y  b / 2  )  D y ( x, y  b / 2  )
n
n
nb
 2 0  r  An ( ) cos
cosh
a
a
2a
n 1

odd
2a sin( nw / 2a )
(n ) 2  0  r cosh( nb / 2a )
An 
V  
b/2
0
nb
E y ( x  0, y )dy   An sinh
2a
n 1

odd
Q
w/ 2
w / 2
 s dx  w
ENEE482
83
Q
C 
V
w
nb

2a

2
(
n

)
 0 r cosh( nb / 2a)
n 1
2a sin( nw / 2a ) sinh
odd
r
L
1
Z0 


C v p C cC
Z 0 is the characteri stic impedance
ENEE482
84
Microstrip
y
w
d
vp 
-a/2
,   k0  e
c
e
a/2
x
 e is the effective dielectric constant.
1  e  r
An Approximat e Electrosta tic solution
 t  ( x, y )  0
2
 ( x, y )  0
at x   a / 2 ,
 ( x, y )  0 at y  0, 
ENEE482
85

nx
ny
A
cos
sinh
for 0  y  d
 n
a
a
n 1
 ( x, y )   
ny

n

x
  Bn cos
e a for d  y  
 n 1
a

nx
Potential must be continuous at y  d  An sin
 Bn e
a

Ey  
y
nd
a


nx
ny
 n 
  An 
cosh
for 0  y  d
 cos

a
a
 a 
n 1
Ey   
 An  n  cos nx sinh n (d e- n (y - d)/a for d  y  
 n 1  a 
a
a
ENEE482
86
1
for x  w / 2
Let  s ( x)  
for x  w / 2
0
 s ( x )  D y ( x, y  d  )  D y ( x, y  d  )
n
n
 2 0  An ( ) cos
a
a
n 1

nd
nd 

sinh a   r cosh a 
odd
4a sin( nw / 2a )
An 
(n ) 2  0 [sinh( nd / a )   r cosh( nd / a )
V  
d
0
nd
E y ( x  0, y )dy   An sinh
a
n 1

odd
Q
w/ 2
w / 2
 s dx  w
ENEE482
87
Q
C 
V
1
nd
2a sin( nw / 2a ) sinh

a

2
(
n

)
w 0 r [sinh( nd / a )   r cosh( nd / a )
n 1
odd
C  Capacitanc e per unit length of the microstrip line with
a dielectric constant  r
C 0  Capacitanc e per unit length of the microstrip line with
an air dielectric ( r  1)
C
e 
C0
e
1
Z0 

v pC
cC
Z 0 is the characteri stic impedance
ENEE482
88
The Transverse Resonance Technique
For a resonant line, at anypoint on the line, The input
impedances seen looking to either side must be zero
Zinr ( y )  Zin ( y )  0
for all y
TM Modes for the parallel plate waveguide
y
y
d
d
0
w
Zinr ( y)
Zin ( y)
x
ENEE482
89
Z0  ZTM  k y / k



k   
Z inr ( y )  jZTM tan k y (d  y )
Z in ( y )  jZTM tan k y y
Condition for transv erse resonance 
jZTM [tan k y (d  y )  tan k y y ]  0
jZTM
sin k y y
cos k y (d  y ) cos k y y
n
kc  k y 
d
0
for n  0,1,2,..
ENEE482
90
MODES IN DIELTECTRIC LOADED WAVEGUIDE
b
er1
a
er
2
CATEGORIES OF FIELD SOLUTIONS:
• TE0m MODES
• TM0m MODES
• HYBRID HEnm MODES
ENEE482
91
BOUNDARY CONDITIONS
FIELDS SATISFY THE WAVE EQUATION,
SUBJECT TO THE BOUNDARY CONDITIONS
Ez , E , Hz , H ARE CONTINUOUS AT r=b
Ez , E VANISH AT r=a
E z  AJ n (1r ) cos n
for 0  r  a
j

H z  AJ n (1r ) sin n
 Er 
1
  J n (1r ) / 1r 
 jH     A cos n n
n  k 2 /  2   J  ( r ) 



n
1
1



1
 

 E 
   J n (1r ) / 1r 
 jH   A sin n  n
r
nk 2 /  2     J  ( r ) 



n
1
 1


1



ENEE482
92
for a  r  b
E z  An Rn ( 2 r ) cos n
j
H z  APn ( 2 r ) sin n

 Er 
P
(

r
)
/

r

n
1




A
cos
n

n
2
1
 jH    


2
2 



 2 n  k 2 /    Rn ( 2 r ) 
  
 E 
   Rn ( 2 r ) / 1r 
 jH    A sin n  n
r




2
2



P
(

r
)
nk
/



2
 2
 n 2




WHERE A IS AN ARBITRARY CONSTANT
ENEE482
93
Characteristic equation
Gn  U n2  k02 a 2VnWn  0
Where z=1a is the radial wave number in 
12  k12   2
;
k12   r1k 02
;
 22  ( k 22   2 )
k 22   r 2 k 02
; k 02   0 0
 1
1 
U n  nJ n (1a )  2 2  2 2 
2a 
 1 a
 J  ( a ) P  ( a ) 
Vn   n 1  n 2 
( 2 a ) 
 (1a )

ENEE482
94
 J n (1a )
Rn ( 2 a ) 
  r2
Wn   r1

a

a

2
1


 K n ( 2 r ) I n ( 2 b)  I n ( 2 r ) K n ( 2 b) 
Pn( 2 r )  J n (1a ) 

 K n ( 2 a ) I n ( 2 b)  I n ( 2 a ) K n ( 2 b) 
 K n ( 2 r ) I n ( 2 b)  I n ( 2 r ) K n ( 2 b) 
Rn ( 2 r )  J n (1a ) 

 K n ( 2 a ) I n ( 2 b)  I n ( 2 a ) K n ( 2 b) 
ENEE482
95
For n = 0, the Characteristic Equation Degenerates in two
Separate Independent Equations for TE and TM Modes:
 J  ( a ) P  ( a ) 
Vn   n 1  n 2   0
( 2 a ) 
 (1a )

For TE Modes
And:
 J n (1a )
Rn ( 2 a ) 
Wn   r1
  r2
0

1 a
 2a 

For TM Modes
ENEE482
96
COMPLEX MODES
• COMPLEX PROPAGATION CONSTANT :
   j
• ONLY HE MODE CAN SUPPORT COMPLEX WAVES
• PROPAGATION CONSTANT OF COMPLEX MODES
ARE CONJUGATE :
    j
• COMPLEX MODES DON’T CARRY REAL POWER
• COMPLEX MODES CONSTITUTE PART OF THE
COMPLETE SET OF ELECTROMAGNETIC FIELD
SPACE
• COMPLEX MODES HAVE TO BE INCLUDED IN THE
FIELD EXPANSIONS FOR CONVERGENCE TO
CORRECT SOLUTIONS IN MODE MATCHING
TECHNIQUES.
ENEE482
97
OPTICAL FIBER
2a
IN CIRCULAR CYLINDRICAL COORDINATES:
1
Step-index fiber
 2 E z 1 E z 1  2 E z
2
2



(
k


) E z  0 ; i  1 for r  a,
i
2
2
2
r
r r
r 
i  0 for r  a
ra
ra
cos n 
E z1  AJ n ( k c1r ) 

 sin n 
cos n 
 AK n ( k c 2 r ) 

 sin n 
;
E z2
 sin n 
H z1  BJ n ( k c1r ) 

cos
n



;
H z2
k c1  ( k 12   2 )1 / 2
;
k c 2  (  k 02   2 )1 / 2
ENEE482
 sin n 
 BK n ( k c 2 r ) 

cos
n



98
For the symmetric case n=0, the solution break into Separate
TE and TM sets. The continuity condition for Ez1= Ez2
and H1= H2 at r=a gives for the TM set:
J 1 ( k c1 a )
 0 k c1 a K 1 ( k c 2 a )

J 0 ( k c1 a )
 1k c 2 a K 0 k c 2 a 
The continuity condition for Hz1= Hz2
and E1= E2 at r=a gives for the TE set:
J 1 ( k c1a )
k c1 K1 ( k c 2 a )

J 0 ( k c1a )
k c 2 K 0 k c 2 a 
If n is different from 0, the fields do not separate into TM
and TE types, but all the fields become coupled through
continuity conditions.
ENEE482
99
Parallel Plate Transmission Line
y
c
a
Assume e  jz

b
x
Partially loaded parallel
Plate waveguide
variation , no variation with x
2
2

k

in the air region
 0
2
2
2
TM modes :  t ez  k c ez  0 , k c   2
2
k

in the dielectric region

for air region
p
Let k c  
 for the dielectric region
ENEE482
100
 2  p 2  ( r  1)k 02
d 2 ez
2


ez  0
2
dy
for 0  y  a
d 2 ez
2

p
ez  0
for a  y  b
2
dy
ez ( y )  0
at y  0, b , e z ( y ) is continuous at y  a
H x is continuous at y  a 


ey ( y)  


j
2
j
p2
ez
y
e z
y
 r ez
 2 y
y a
1 ez
 2
p y
y a
For dielectric region
for air region
ENEE482
101
 j r k 0Y0 e z
  2
y
hx ( y )  
jk Y ez
 02
 p y
e z ( y )  C1 sin y
e z ( y )  C 2 sin p (b  y )
For dielectric region
for air region
0 ya
a yb
C1 sin a  C 2 sin pc
r
1
C1 cos a   C 2 cos pc

p
tan a   r p tan pc Transceden tal equation must be solved
simultaneo usly with  2  p 2  ( r  1) k 02
ENEE482
 , p
102
  k 02  p 2  k 2  l 2
Most of the modes will be nonpropaga ting if  is imaginary
The variation is e
 z
and the field decays exponentia lly.
The value of  between k 0 and k can occur if p  jp0
Let  0 to be the correspond ing value of  then :
 0 tan  0 a   r p0 tan p0 c
 20  p02  ( r  1)k 02
ENEE482
103
Low Frequency Solution
When the frequency is low,
k 02 is very small number ,  0 and p0 are very small 
 0 a   r p02 c
2
( r  1) k 02  p02 
 r p02 c
a
2
(


1
)
k
2
r
0a
or p0 
The solution for  is
a   rc
  k  p0
2
0
2
 rb

k0   e k0
a   rc
 e is the effective dielectric constant
ENEE482
104
y
c
b

a
x
-W
 e   LC
W
L, C are the static inductance and capacitanc e
per meter.
The time average stored magnetic energy is Wm
0
0
1 2
Wm 
H dxdy 
WbJ  LI z


0

w
4
2
4
b
I z  2WJ z
L 0
2W
b
w
2
x
2
z
ENEE482
105
C2  C1 sin  0 a / j sinh p0 c   jC1  0 a / p0 c
0 ya
ez  C1 0 y
j
b
ey  
C1   jC1
0
( r  1)c
j r k 0Y0
( r c  a ) r
hx 
C1  jY0C1
0
( r  1)c
Ca Cd
 0 2W
 0 r 2W
The capacitanc e C 
, Ca 
, Cd 
Ca  Cd
c
a
 0 r 2W
C
 rc  a
 0 r  0 b
LC 
 rc  a
   LC
ENEE482
106
The capacitanc e C 
 0 r 2W
C
 rc  a
 0 r  0 b
LC 
 rc  a
Ca Cd
Ca  Cd
, Ca 
 0 2W
c
, Cd 
 0 r 2W
a
   LC
Field expression s :
C2  C1 sin  0 a / j sinh p0 c   jC1  0 a / p0 c
0 ya
ez  C1 0 y
ey  
j
b
C1   jC1
0
( r  1)c
ENEE482
107
j r k 0Y0
( r c  a ) r
hx 
C1  jY0C1
0
( r  1)c
In the air region :
(b  y )
e z  C1 0 a
c
j 0 a
b
e y   2 C1   jC1 r
( r  1)c
p0 c
hx 
j k 0Y0
0
C1  jY0C1
b
V    e y dy  jC1
0
( r c  a ) r
( r  1)c
b
(a   r c)
( r  1)c
I z  2WJ z  2WH x
ENEE482
108
The characteri stic impedance is :
Z0
V
Zc  
I 2W
( a   r c )b
r
L

C
In the low frequency limit, the dominant mode of
propagatio n becomed a TEM mode (quasi - TEM mode)
At hight frequency the mode of propagatio n is an E mod
ENEE482
109
High Frequency Solution:
At high frequency k 0 and  0 , p0 are large.
tanh p0 c  1
 0 tanh  0 a   r p0   r ( r  1)k 02   20
 20  p02  ( r  1)k 02 The solution is independen t of b
e z ( y )  C1 sin  0 y
for 0  y  a
sinh p0 (b  y )
e z ( y )  C 2 j sinh p0 (b  y )  C1 sin  0 a
sinh p0 (b  a )
e p0 ( b y )
 C1 sin  0 a p0 ( ba )  C1 sin  0 ae p0 ( y a ) for a  y  b
e
The field decays exponentia lly away from the air - dielectric surface
and does not depend on b as long as p0 c  p0 (b  a ) is large. The field
is guided by the dielectric sheet. This type of mode is called surface
ENEE482
110
wave mode.
Microstrip Transmission Line
w
y
H
x
 ( x, y , z )   s ( x, z ) ( y  H )
J ( x, y , z )  J s ( x, z ) ( y  H )
B    A,
  E   jB   j  A
E   jA  

  H  jD  J
For anisotropi c dielectric :
D   0 r ( E x aˆ x  E z aˆ z )   0 y E y aˆ y
ENEE482
111
    A    A   2 A  j 0 D   0 J


D   0 r [ j ( Ax aˆ x  Az aˆ z )  aˆ x
 aˆ z
]
x
z

  0 y ( jAy aˆ y  aˆ y
)
y
let
  A   j 0 r  0  (Lorentz condition)
-  2 A  j 0 [  j 0 r A   0   r

 
  0 ( y   r ) jaˆ y Ay  aˆ y
]   0 J
y 

J does not have a y component 
 2 Ax   r ( y )k 02 Ax    0 J x
 2 Az   r ( y )k 02 Az    0 J z
ENEE482
112
  r
 
 Ay   y ( y )k Ay  j 0 0 
 ( y   r )


r

y


2
2
0



 j 0 0 ( y   r )
  ( H ) r  1) ( y  H ) 
y


 D  

  2   2      
 r  2  2     y
   r2 k 02 
y  y  y 
 x
Ay

   j ( y  1) Ay ( H ) ( y  H )  j ( y   r )
0
y
ENEE482
113
Boundary conditions:
lim  0 
H 
H 
 Ax
Ax
dy 
2
y
y
2
H
H
 lim  0 
H 
H 
  0 J sx ( x, z ) ( y  H )dy
   0 J sx
Ax
y
H
   0 J sx
Ay
y

y
Az
y
,
H
Ax
 0 H z
y
H
,
   0 J sz
H
Az
 0 H x
y
H
 j 0 0 ( r  1) ( H )
H
H

y
y
H
s

 j ( y  1) Ay ( H )
0
ENEE482
114
In the substrate region away from the interface we have :
( 2   r k 02 ) Ax  0
( 2   r k 02 ) Az  0
( 2   y k 02 ) Ay  j 0 0 ( y   r )

y
Ay
 2  2  y  2
2
 2 
  r k 0    j ( y   r )
2
2
x
z
 r y
y
In the air region  r   y  1
For an isotropic substrate  r   y
ENEE482
115