chapter 3 transmission lines and waveguides

微波電路講義3-1

Chapter 3 Transmission Lines and Waveguides3.1 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 General solutions for TEM, TE and TM waves

• Procedure to analyze a TEM (Ez, Hz=0) line

• Procedure to analyze a TE (Ez=0) or TM (Hz=0) line

2

2

1 21

1) solve , from Laplace s equation , 0

2) apply B.C. x,y

3) , , , ,

1ˆ , , , , ,

4)

t

j z

t

j zxTEM

TEM y

(x y) ' (x y)

( )

e(x y) (x, y) E (x y z) e(x y)e

Eh(x, y) z e(x y) Z H (x y z) h(x y)e

Z H

V E dl

15) , , p o

p pr

, I H dl

c V L LCv Z

v I C C v Cε

e

∮

2 22

2 21) solve ( ) from Helmholtz equation ( ) ( ) 0 : TE casez C zh x, y k h x, y

x y

2 2

22 2

solve ( ) from Helmholtz equation ( ) ( ) 0 : TM casez zce x,y k e x,yx y


2 2 2 2

2 2

2) apply B.C. to find

3) ( ) ( )

, : TE case

( ) ( )

,

c

j z

z z

z z z zx y x y

c c c c

j z

z z

z zx y

c c

k

H x, y,z h x, y e

H H H Hjβ jβ j jH , H E , H

k x k y k y k x

E x, y,z e x, y e

E Ej jH , H E

k y k x

e e

2 2

2 2

: TM case

4) ( ) , ( )

z zx y

c c

x xc TM TE

y y

E Ej j, H

k x k y

E E kk k ,Z Z

H k H

e

• Dielectric loss2 tan

( / ) : TE or TM case2

tan( / ) : TEM case

2

d d

d

kj NP m

kNP m

a a

a


(derivation of the dielectric loss expression)

2 2 2 2 2 2 2 2

2 2 2 2 2

2 2

2 2 2 2 2

22 2

. 0 ( ) 0 ( ) 0

0,

( ' ") '(1 tan ) 1 tan

"tan , '

'

(1 tan ) tan

1 tan

2

xy z xy

xy c c

d c d

d

c c

c

c

wave eq E k E E k E E k E

E k E k k

j k k

k j j k j

k

k k j k k jk

jkk k

k

a

e e e e

e e

e

2

2 2

2

1 tan

2

tan( / ) : TE or TM case

2

tan( / ) : TEM case 0

2

d

d

d c

kj j

k

kNP m

kNP m k

a

a

a


3.5 Coaxial line

• TEM modeln

( , )ln

ˆ( , , ) ( , )

ln

ˆ1ˆ( , , ) ( , , )

ln

( ) ( , , )

2( ) ( , , )

ln

( ) ln

( ) 2

o

jkz jkzot

jkzo

TEM

b

jkz jkz

o o

a

jkz jkzoo

o

bV

b a

VE z e e

b a

VH z z E z e

Z b a

V z E z dl V e V e

VI z H z dl e I e

b a

V z b aZ

I z

∮

60ln ,

2ln , (Table 2.1, p.54)

2 ln /

r

b Lk

a C

bL C

a b a

e e

e

x

y

b

a

Vo


• Higher order mode effect

TE11 mode (p.133 Fig.3.17) e )ba(

1ff cmax

Power

meter

output

power

f


Discussion

1. Conformal mapping solution (Collin, Field theory of guided waves,

p.262)

ln ln

ln

j

Z x jy

W Z e

j

u jv

2 ln 22 2 2

0 ln

ln

ln ln

1 1( ) ( )

2 ln 2

2 1, ln ,

ln 2

o

b

oe o

a

o

p

b uV

b a

VW dudv CV

u v b a

bC Z

b a v C C a

e e

ee

e

x

y

a

b

er

Z-plane

W-plane

2

er

ln a ln b u

v

Vo 0Vo


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).

3. Types of coaxial connectors

type-N, SMA, APC3.5, APC2.4,….. (p.134, point of interest)

characteristic impedance

for coaxial airlines (ohms)

10 20 30 40 50 60 70 80 90 100

1.0

0.8

0.7

0.6

0.5

0.9

1.5

1.4

1.3

1.2

1.1no

rmal

ized

val

ues

50 ohm standard

attenuation is lowest

at 77 ohms

power handling capacity

peaks at 30 ohms


3.7 Stripline

• Conformal mapping solution

(Collin’s book, p.265)

o

o

1 B)1Z)(xZ(Z

dZA'W

CZ,

v

4C

vC

o

o

o

ee

e

=Vo

=Vo 1

=Vo

=Vo

= 0

= 0

= 0

= 0

= 0

er

er

W-plane

W’-plane

Z-plane

W

b

b/2

vo

B)Z(Z

dZAW

1

-x 1

b

w

o

2cosh

2


• Electrostatic solution

2 ( ) 0 2 0

( ) 0 2 0

cos sinh 0 2

( )

cos sinh ( ) 2

( )cos cosh

t

n

odd

n

odd

n

y

x, y x a , y b

B.C. x, y x a , y ,b

nπ nπA x y y b

a ax, y

nπ nπA x b y b y b

a a

nπ nπ nπA x y

a a aE

y

2

0

0 2

( )cos cosh ( ) 2

1 2( ) ( 2) ( 2) ...(3.189)

0 2

( 0 ) 2 sinh4

odd

n

odd

s y y n

b

y n

y b

nπ nπ nπA x b y b y b

a a a

x Wρ x D x, y b D x, y b A

x W

nπV E x , y dy A b, Q

a

2

2

( )

1...(3.192),

W

s

odd W

r

o

p

ρ x dx W

εQ L C Z

V C v C cC

-a/2 a/2 x

yb

Wer


Discussion

1. analysis eq.(3.179)

synthesis eq.(3.180)

2.

W,b,Z

Z,b,W

o

o

e

e

eq.(3.181) cd ,2

tanka

a

roZ e

bW /

roZ e

180

160

140

120

100

80

70

60

50

40

30

20

0.2 0.3 0.4 0.5 0.6 0.71.0 2.0 3.0 4.0


3.8 Microstrip

• Characteristics

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

W

d er


• eeff concept

o p o

p

2og

p

TEM line ,

1air filled microstrip (TEM line)

1 1 cmicrostrip (quasi-TEM line) Z , k ,

C C

Z 1 C , ( )

C

oo p o r

r r

oa

a a

eff

eff

o ae

oa aeff eff

L cZ , v , β k ε

C ε

LZ

C cC

L, v

v LC

C c

Z C v

e

e

ee

e

e eff

er

eeff


1 ( 1)

2

1 ( 1)

2

1filling factor (1 )1 1 ( 1) , 1

2

eff r eff r

r eff r

eff r rq q q q q

e e e e

e e e

e e e

erer

effe

re

)1(2

1

re

dW /


• Conformal mapping solution (Gupta, Garg and Bahl, Microstrip

lines and slotlines, p.9)

'Z2

'ZtanhdjZ

'g

s'a'gq

"s's"ss

r

e

x

yW/2

d

er

Z-plane

Z’-plane

a’ g’ x’

y’

y’y’

a’ g’ x’

a’ g’ x’

s’

s’-s” s”

s


• Electrostatic solution

2

( )

( ) 0 2 0

( ) 0 2 0

cos sinh 0

( )

cos sinh

( )

t

n

odd

n y d a

n

odd

n

y

x, y x a , y

B.C. x, y x a , y ,

nπ nπA x y y d

a ax, y

nπ nπA x de d y

a a

nπA

aE

y

( )

0

cos cosh 0

( )cos sinh

1 2( ) ( ) ( ) ( ) ( )

0 2

odd

n y d a

n

odd

s y y o y o r y n

d

y

nπ nπx y y d

a a

nπ nπ nπA x d e d y

a a a

x Wρ x D x, y d D x, y d E x, y d E x, y d A

x W

V E

e e e

2

2

( 0 ) sinh ( )

1, ,

W

n s

odd W

eff

eff o

a p

nπx , y dy A d, Q ρ x dx W

a

εQ C L C ε Z

V C C v C cC

er

Wd

-a/2 a/2 x

y


Discussion

1. analysis eq.(3.196, 195)

synthesis eq.(3.197, 195)

2.

effo

effo

,W,d,Z

, Z,d,W

ee

ee

(3.199) (3.198)12

1

WZ

Rα,

)(

tan)(k

o

sc

reff

effro

d ee

eea

er

W/d0.2 0.5 1 2 5 10 20

16

10

5

2

150

120

100

80 50 30 20 10


(derivation of eq.(3.198))

tantanTEM line

2 2

microstrip line:

1(1) (1 ) ( 1) 1

1

(2)

"(3)tan tan , " " 0(1 ) "

tan "" "

2 2 2 2

o r

d

eff

eff r r

r

r eff

eff

eff eff

eff

o eff eff o eff o od r

eff reff eff

kk

q q q q

q q q

k k k q k qq

e a

e e e e

e

e e

e e e e

e

e e ee ea e

e ee e

1 1tan

tan1 12 2

eff effo o rr

r reff eff

k ke e e e

e e e e


4. substrate material

r

2 3

ε tanδ@10GHz

GaAs 12.9 0.002

Al O (99.5%) 10.1 0.25 0.0002

quartz 3.8

0.0001

RT/duroid 5880 2.2 0.02 0.0009

RT/duroid 6006 6.15 0.15 0.0027

BeO 6.8 0.0003

FR4 (G-10) 4-4.6

*2

*3 4

0.018-0.025

Si 11.7 0.005

SiO 4

Si N 7.6 (vapor phase)

6.5 (sputtering)

*: MIM capacitor

conductor material

s thicknesum) (35 mil 1.4Cu oz 1

4.0 7.2 Ta

2.7 4.7 Cr

1.9 3.3 Al

1.7 3.0 Au

1.5 2.6 Cu

1.4 2.5 Ag

(um)@2GHz )f10/sq( R -7

s

s

ωμσ δs

2depthskin


3.11 Summary of transmission lines and waveguides

• comparison of coaxial line, stripline and microstrip, (p.158,

Table 3.6)

• CPW (coplanar waveguide)

fabrication easy

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

er

Ws s


Solved Problems

Prob. 3.28 Find Zo of a coaxial line to have maximum power

capacity

6max

2 2 2max

max max

max

breakdown field strength of air 3 10 / ln

ln

1maximum power capacity , ln ln

2 2

1 377 10 ln ln 30

2 2 2 2

od d

o do

o o

oo

V bE V m V E a

b a

a

V b a E bP Z P

Z a a

dP b bZ

da a a

ADS examples: Ch3_prj

chapter 3 transmission lines and waveguides

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