Chapter 3 Transmission Lines and Waveguides
3.1 General solutions for TEM, TE and TM waves
procedures, ad
3.5 Coaxial line (TEM line)
TEM mode, higher order mode effect
3.7 Stripline (TEM line)
conformal mapping solution, electrostatic solution
3.8 Microstrip (quasi-TEM line)
eeff concept, conformal mapping solution, electrostatic
solution
3.11 Summary of transmission lines and waveguides
3-1
微波電路講義
3.1 General solutions for TEM, TE and TM waves
• Procedure to analyze a TEM (Ez, Hz=0) line
1) solve (x, y) from Laplace's equation t2(x, y)  0
2) apply B.C.
 ( x,y )
3) e(x, y)  t (x, y)  E  (x , y , z)  e(x , y)e  jz
h(x, y) 
1
Z TEM
zˆ  e(x, y), Z TEM
E x
     H  (x, y , z)  h(x, y)e  jz
Hy
2
4) V   1   2   E   dl , I  ∮H   dl
1
c

V
5) v p 
, 
  e , Z o   
vp
I
εr
L

C
LC
1

C
v pC
• Procedure to analyze a TE (Ez=0) or TM (Hz=0) line
2
2
1) solve hz ( x, y ) from Helmholtz equation ( 2  2  kC2 )hz ( x, y)  0 : TE case
x y
2
2
solve ez ( x, y) from Helmholtz equation (

 kc2 )ez ( x, y )  0 : TM case
x 2 y 2
微波電路講義
3-2
2) apply B.C. to find kc
3) H z ( x, y,z )  hz ( x, y )e  jz
 jβ H z
 jβ H z   j H z
j H z


H  2
, Hy  2
, Ex 
, Hy  2
: TE case
2
kc x
kc y
kc
y
kc x

x
Ez ( x, y,z )  ez ( x, y )e  jz
je Ez
 je Ez   j Ez
 j Ez


H  2
, Hy 
, Ex  2
, Hy  2
: TM case
2
kc y
kc
x
kc x
kc y

x
Ex
Ex
 
 k 
4)   k  k ,ZTM (  ) 

, ZTE (  ) 

Hy
e
k
Hy


2
2
c
• Dielectric loss
k 2 tan 
  a d  j  a d 
( NP / m) : TE or TM case
2
k tan 
ad 
( NP / m) : TEM case
2
3-3
微波電路講義
(derivation of the dielectric loss expression)
wave eq.  2 E  k 2 E  0  ( 2xy   2z ) E  k 2 E  0   2xy E  (  2  k 2 ) E  0
 2xy E  kc2 E  0, kc2   2  k 2
   a d  j  kc2  kd2
kd   e   (e ' je ")   e '(1  j tan )  k 1  j tan 
tan  
e"
, k   e '
e'
   kc2  k 2 (1  j tan )  kc2  k 2  jk 2 tan 
1 jk 2 tan 
1 k 2 tan 
 k k 
 j 
 j  a d
2
2
2 kc  k
2

2
c
2
k 2 tan 
 ad 
( NP / m) : TE or TM case
2
k tan 
ad 
( NP / m) : TEM case kc  0
2
3-4
微波電路講義
y
3.5 Coaxial line
• TEM mode
a
ln b 
(  , ) 
Vo
ln b a
E  (  ,  , z )  t  (  ,  )e  jkz 
H (  , , z ) 


b
V ( z) 
E

1
Z TEM

Vo
ˆ
e  jkz
ln b a

b

Vo
x
ˆ
Vo
 jkz 
zˆ  E (  ,  , z ) 
e
 ln b a


(  ,  , z )  dl  Vo e  jkz  Vo e  jkz
a
I  ( z ) ∮
H  (  ,  , z )  dl 
2 Vo
e  jkz  I o e  jkz
 ln b a
V  ( z )  ln b a
60
b
Zo  


ln 
I ( z)
2
a
er
L
L
,   k   e
C

b
2e
ln , C 
(Table 2.1, p.54)
2
a
ln b / a
3-5
微波電路講義
• Higher order mode effect
TE11 mode (p.133 Fig.3.17)
f max  f c 
1
 ( a  b ) e
Power
meter
output
power
f
3-6
微波電路講義
Discussion
1. Conformal mapping solution (Collin, Field theory of guided waves,
p.262)
y Z-plane
Z  x  jy
W  ln Z  ln e j
W-plane
v
Vo
 ln   j
Vo
0
er

a
 u  jv

2
er
x
b
ln a

ln b
ln b  u
Vo
ln b  ln a
2  ln b
1
We  e 
2 0
C 
eVo2
 2
 2
1
(
)

(
)
dudv


CVo2

u
v
ln b a
2
ln a
2e
1
, Zo 

ln b a
v pC
3-7
e

b

ln ,  
C
2
a

e
微波電路講義
u
2. Reasons for selecting Zo=50
Coaxial line has minimum attenuation as Zo=77 (Prob. 2-27),
and maximum power capacity as Zo=30 (Prob. 3-28).
1.5
attenuation is lowest
at 77 ohms
1.4
1.3
normalized values
1.2
50 ohm standard
1.1
1.0
0.9
0.8
power handling capacity
peaks at 30 ohms
0.7
0.6
0.5
10
20
30
40
50
60 70 80 90 100
characteristic impedance
for coaxial airlines (ohms)
3. Types of coaxial connectors
type-N, SMA, APC3.5, APC2.4,….. (p.134, point of interest)
3-8
微波電路講義
3.7 Stripline
• Conformal mapping solution
(Collin’s book, p.265)
=0
b/2
= 0
 =Vo
W-plane
dZ
B
Z( Z  1 )
=0
 =Vo 1
C
W '  A1 
dZ
 Bo
Z ( Z  xo )( Z  1 )
-1
w
2b
W’-plane
er
Z-plane
  cosh 2
er
=0
vo
W  A
 xo
W
=0
 =Vo
 =Vo
b
3-9
e
vo
C 
e
4e
, Zo 
vo
C
微波電路講義
• Electrostatic solution
t2  ( x, y )  0
b
W
x  a 2, 0  y  b
 ( x, y )  0 x   a 2, y  0,b
B.C.
nπ
nπ

An cos
x sinh
y


a
a
 odd
 ( x, y )  
 A cos nπ x sinh nπ (b  y )
n

a
a
 odd
0 y b 2
0
 C
a/2 x
b 2 yb

1
ρs ( x)  Dy ( x, y  b  2)  Dy ( x, y  b  2)  

0
V 
er
-a/2
nπ
nπ
nπ


A
(
)
cos
x
cosh
y

n
a
a
a
 
 odd
Ey  

nπ
nπ
nπ
y 
An ( ) cos
x cosh
(b  y )


a
a
a
 odd
b 2
y
0 y b 2
b 2 y b
x W 2
x W 2
 An ...(3.189)
W 2
nπ
E y ( x  0, y )dy  2 An sinh
b, Q   ρs ( x)dx  W
4
a
odd
W 2
Q
...(3.192), Z o 
V
εr
L
1


C v pC
cC
3-10
微波電路講義
Discussion
1. analysis eq.(3.179) W , b ,e  Z o
synthesis eq.(3.180) Z , b ,e  W
o
ad 
k tan 
, a c eq.(3.181)
2
2.
70
180
60
160
50
Zo e r
140
40
120
Zo e r
30
100
20
80
0.2 0.3 0.4 0.5 0.6 0.71.0 2.0
W /b
3-11
3.0 4.0
微波電路講義
3.8 Microstrip
W
• Characteristics
d
er
fabrication by printed circuit
devices can be bonded to strip
component are accessible
in-circuit characterization of devices is straightforward to implement
dc as well as ac signals can be transmitted
large variation in Zo
ac > ad
monolithic applications
structure is rugged and can withstand high voltages and power levels
power handling is best with BeO substrate
used up to 300GHz or more
quasi-TEM mode for d<<λ
3-12
微波電路講義
• eeff concept
eeff
er
TEM line Z o 

L
c
, vp 
, β   e  ko εr ,   o
C
εr
er
air filled microstrip (TEM line) Z oa 
microstrip (quasi-TEM line) Zo 
g 
o
eeff

L
1

Ca cCa
L
1

, vp 
C vp C
Zo
Ca
1


,
Z oa
C
eeff
3-13
1
c

,   k o e eff ,
LC
eeff
C
c
 ( ) 2  e eff
Ca
vp
微波電路講義
er
er
1
eeff  (e r  1)
2
1
(e r  1)  eeff  e r
2
filling factor q
eeff  e r
eeff  qe r  (1  q)1  1  q(e r  1) ,
e eff
1
 q 1
2
er
1
(e r  1)
2
W /d
3-14
微波電路講義
• Conformal mapping solution (Gupta, Garg and Bahl, Microstrip
lines and slotlines, p.9)
y
y’ s’-s” s”
W/2
d
Z-plane
er
x
a’
g’
x’
s  s" 
Z'
Z  j  d tanh  Z'
2
q
y’
s’
Z’-plane
a’
g’
y’
x’
s
a’
3-15
g’
s'  s"
er
g' a'  s
g'
x’
微波電路講義
y
• Electrostatic solution
t2  ( x, y )  0
W
x  a 2, 0  y  
 ( x, y )  0 x   a 2, y  0, 
B.C.
er
-a/2
nπ
nπ

A
cos
x
sinh
y
0 yd

n

a
a
 odd
 ( x, y )  
 A cos nπ x sinh nπ de  n ( y  d ) a
d  y
n

a
a
 odd
nπ
nπ
nπ


A
(
)
cos
x
cosh
y
0 yd

n

a
a
a
  odd
Ey  

nπ
nπ
nπ
y 
An ( ) cos
x sinh
d e  n ( y  d ) a
d  y


a
a
a
 odd
d
a/2

1
ρs ( x)  Dy ( x, y  d  )  Dy ( x, y  d  )  eo E y ( x, y  d  )  e o e r E y ( x, y  d  )  

0
x W 2
 An
x W 2
W 2
d
nπ
V    E y ( x  0, y )dy   An sinh
d, Q   ρs ( x) dx  W
a
odd
0
W 2
 C
Q
C
, εeff 
,
V
Ca
Zo 
εeff
L
1


C v pC
cC
3-16
x
微波電路講義
Discussion
1. analysis eq.(3.196, 195) W , d ,e  Z o ,e eff
synthesis eq.(3.197, 195) Z o , d ,e  W ,e eff
ad 
ko e r ( e eff  1 ) tan 
2 e eff ( e r  1 )
(3.198), αc 
Rs
(3.199)
Z oW
2.
16
10
er 5
80 50
30
20
10
100
120
150
2
0.2
0.5
1
3-17
2
5
10
20
W/d
微波電路講義
(derivation of eq.(3.198))
TEM line a d 
k tan  ko e r tan 

2
2
microstrip line:
(1)eeff  qe r  (1  q )  q (e r  1)  1  q 
er 
(2)

ko e eff tan eff
ko
2 eeff
er  1
e eff
(3)tan  tan eff 
ad 
e eff  1
2
e "eff
e eff
, e "eff  qe " 0(1  q )  qe "
ko e eff qe "
k qe "
ko q e "

 o

er
2
eeff
2 e eff
2 e eff e r
e eff  1
ko e r tan  e eff  1
tan e r 
er  1
er  1
2 e eff
3-18
微波電路講義
4. substrate material
εr
conductor material
R s (/sq  10-7 f )
tanδ@10GHz
0.0002
Ag
Cu
2.5
2.6
3.8
2.2  0.02
0.0001
0.0009
Au
Al
3.0
3.3
RT/duroid 6006
6.15  0.15
0.0027
BeO
6.8
0.0003
Cr
Ta
4.7
7.2
FR4 (G-10)
4-4.6
0.018-0.025
Si
11.7
GaAs
12.9
0.002
Al 2 O 3 (99.5%)
10.1  0.25
quartz
RT/duroid 5880
0.005
*
SiO 2
4
*
Si 3 N 4
7.6 (vapor phase)
 s (um)@2GHz
1.4
1.5
1.7
1.9
2.7
4.0
1 oz Cu  1.4 mil (35 um) thickness
skin depth δs 
2
ωμσ
6.5 (sputtering)
*: MIM capacitor
3-19
微波電路講義
3.11 Summary of transmission lines and waveguides
• comparison of coaxial line, stripline and microstrip, (p.158,
Table 3.6)
W
• CPW (coplanar waveguide)
s s
fabrication easy
er
quasi-TEM operation
radiation problem when gap width approaches /2
monolithic applications
less radiation than microstrip if well-balanced
higher order modes (coupled slot mode)
• www.appwave.com for computer assisted transmission line
analysis
3-20
微波電路講義
Solved Problems
Prob. 3.28 Find Zo of a coaxial line to have maximum power
capacity
Vo
b
breakdown field strength of air E d  3 10 6 V / m 
 V max  E d a ln
b
a
 ln
a
2
1 V max
o b
 a 2 E d2 b
maximum power capacity Pmax 
,Zo 
ln  Pmax 
ln
2 Zo
2 a
o
a
dPmax
b 1
 o b 377 1
 0  ln   Z o 
ln 
 30
da
a 2
2 a 2 2
ADS examples: Ch3_prj
3-21
微波電路講義