微波電路講義3-1

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

procedures, d3.5 Coaxial line (TEM line)

TEM mode, higher order mode effect3.7 Stripline (TEM line)

conformal mapping solution, electrostatic solution3.8 Microstrip (quasi-TEM line)

eff concept, conformal mapping solution, electrostatic solution

3.11 Summary of transmission lines and waveguides

微波電路講義3-2

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 2 1

1) solve , from Laplace s equation , 02) apply B.C. x,y3) , , , ,

1 ˆ , , , , ,

4)

t

j zt

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)eZ H

V E dl

15) , , p op pr

, I H dl

c V L LCv Zv I C C v Cε

2 22

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

2 2

22 2

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

微波電路講義3-3

2 2 2 2

2 2

2) apply B.C. to find

3) ( ) ( )

, : TE case

( ) ( )

,

cj z

z z

z z z zx y x y

c c c cj 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 , Hk x k y k y k x

E x, y,z e x, y e

E Ej jH , H Ek y k x

2 2

2 2

: TM case

4) ( ) , ( )

z zx y

c c

x xc TM TE

y y

E Ej j, Hk x k y

E E kk k ,Z ZH k H

• Dielectric loss2 tan ( / ) : TE or TM case2

tan ( / ) : TEM case2

d d

d

kj NP m

k NP m

微波電路講義3-4

(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 tan2

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 kk

2

2 2

2

1 tan2

tan ( / ) : TE or TM case2

tan ( / ) : TEM case 02

d

d

d c

kj jk

k NP m

k NP m k

微波電路講義3-5

3.5 Coaxial line

• TEM modeln( , )ln

ˆ( , , ) ( , )

lnˆ1 ˆ( , , ) ( , , )

ln

( ) ( , , )

2( ) ( , , )ln

( ) ln( )

o

jk z jk zot

jk zo

T E Mb

jkz jk zo o

a

jk z jk zoo

o

b Vb a

VE z e eb a

VH z z E z eZ b a

V z E z d l V e V e

VI z H z d l e I eb a

V z b aZI z

6 0 ln ,

2

2ln , (T a b le 2 .1 , p .5 4 )2 ln /

r

b L ka C

bL Ca b a

x

y

ba

Vo

微波電路講義3-6

• Higher order mode effect

TE11 mode (p.133 Fig.3.17) )ba(1ff cmax

Power meter

outputpower

f

微波電路講義3-7

Discussion1. Conformal mapping solution (Collin, Field theory of guided waves,

p.262)

ln lnln

j

Z x jyW Z e

ju jv

2 ln 22 2 2

0 ln

lnln ln

1 1( ) ( )2 ln 2

2 1, ln ,ln 2

o

bo

e oa

op

b u Vb a

VW d u d v C Vu v b a

bC Zb a v C C a

x

y

a

b

r

Z-planeW-plane

r

ln a ln b u

vVo 0Vo

微波電路講義3-8

2. Reasons for selecting Zo=50Coaxial line has minimum attenuation as Zo=77 (Prob. 2-27), and maximum power capacity as Zo=30 (Prob. 3-28).

3. Types of coaxial connectorstype-N, SMA, APC3.5, APC2.4,….. (p.134, point of interest)

characteristic impedancefor coaxial airlines (ohms)

10 20 30 40 50 60 70 80 90100

1.0

0.8

0.7

0.6

0.5

0.9

1.5

1.4

1.3

1.2

1.1

norm

aliz

ed v

alue

s

50 ohm standard

attenuation is lowest at 77

ohms

power handling capacity peaks at 30

ohms

微波電路講義3-9

3.7 Stripline• Conformal mapping solution

(Collin’s book, p.265)

oo1B

)1Z)(xZ(ZdZA'W

CZ,

v4C

vC

oo

o

=Vo

=Vo 1

=Vo

=Vo

= 0

= 0

= 0

= 0

= 0

r

r

W-plane

W’-plane

Z-plane

W

b

b/2

vo

B)Z(Z

dZAW

1

-x 1

bw

o

2cosh 2

微波電路講義3-10

• Electrostatic solution2 ( ) 0 2 0

( ) 0 2 0

cos sinh 0 2( )

cos sinh ( ) 2

( ) cos cosh

t

nodd

nodd

n

y

x, y x a , y bB.C. x, y x a , y ,b

nπ nπA x y y ba a

x, ynπ nπA x b y b y ba a

nπ nπ nπA x y a a a

Ey

2

0

0 2

( ) cos cosh ( ) 2

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

0 2

( 0 ) 2 sinh4

odd

nodd

s y y n

b

y n

y b

nπ nπ nπA x b y b y ba a a

x Wρ x D x, y b D x, y b A

x W

nπV E x , y dy A b, Qa

2

2

( )

1...(3.192),

W

sodd W

ro

p

ρ x dx W

εQ L C ZV C v C cC

-a/2 a/2 x

ybW

r

微波電路講義3-11

Discussion1. analysis eq.(3.179)

synthesis eq.(3.180)

2.

W,b,Z Z,b,W

o

o

eq.(3.181) cd ,2tank

roZ

bW /

roZ

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

微波電路講義3-12

3.8 Microstrip

• Characteristicsfabrication by printed circuitdevices can be bonded to stripcomponent are accessiblein-circuit characterization of devices is straightforward to implementdc as well as ac signals can be transmittedlarge variation in Zoc > dmonolithic applicationsstructure is rugged and can withstand high voltages and power levelspower handling is best with BeO substrate used up to 300GHz or morequasi-TEM mode for d

微波電路講義3-13

• eff concept

o p op

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

oaa a

effeff

o ae

oa aeff eff

L cZ , v ,β k εC ε

LZC cC

L , vv LC

C cZ C v

ff

reff

微波電路講義3-14

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

rr

eff

r

)1(21

r

dW /

微波電路講義3-15

• Conformal mapping solution (see Gupta, Garg and Bahl, Microstrip lines and slotlines, p.9)

'Z2'ZtanhdjZ

'gs'a'gq

"s's"ssr

x

yW/2d

r

Z-plane

Z’-plane

a’ g’ x’

y’

y’y’

a’ g’ x’

a’ g’ x’

s’

s’-s” s”

s

微波電路講義3-16

• Electrostatic solution2

( )

( ) 0 2 0( ) 0 2 0

cos sinh 0( )

cos sinh

( )

t

nodd

n y d an

odd

n

y

x, y x a , yB.C. x, y x a , y ,

nπ nπA x y y da a

x, ynπ nπA x de d ya anπAa

Ey

( )

0

cos cosh 0

( ) cos sinh

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

0 2

odd

n y d an

odd

s y y o y o r y n

d

y

nπ nπx y y da a

nπ nπ nπA x d e d ya 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

2

2

( 0 ) sinh ( )

1(3.211), ,

W

n sodd W

effeff o

a p

nπx , y dy A d, Q ρ x dx W a

εQ C L C ε ZV C C v C cC

r

W d

-a/2 a/2 x

y

微波電路講義3-17

Discussion1. analysis eq.(3.196, 195)

synthesis eq.(3.197, 195)

2.

effo

effo

,W,d,Z, Z,d,W

(3.199) (3.198)12

1WZ

Rα,)(

tan)(k

o

sc

reff

effrod

r

W/d0.2 0.5 1 2 5 10 20

16

10

5

2

150

120

100

80 50 30 20 10

微波電路講義3-18

(derivation of eq.(3.198))

tantanTEM line 2 2

microstrip line:1

(1) (1 ) ( 1) 11

(2)

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

tan "" "2 2 2 2

o rd

effeff r r

r

r eff

effeff 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

1 1tantan

1 12 2eff effo o r

rr reff eff

k k

微波電路講義3-19

4. substrate material

capacitor MIM :* g)(sputterin 6.5

phase)(vapor 7.6 NSi4 SiO

0.005 11.7 Si 0.025-0.018 4.6-4 10)-(G FR4

0.0003 6.8 BeO0.0027 0.156.15 6006 RT/duroid0.0009 0.02 2.2 5880 RT/duriod0.0001 3.8 quartz

0.0002 0.2510.1 (99.5%)OAl0.002 12.9 GaAs@10GHz tan ε

43*

2*

32

r

δ

conductor material

s thicknesum) (35 mil 1.4Cu oz 1

4.0 7.2 Ta 2.7 4.7 Cr 1.9 3.3 Al1.7 3.0 Au1.5 2.6 Cu1.4 2.5 Ag

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

s

ωμσ δ s

2depthskin

微波電路講義3-20

3.11 Summary of transmission lines and waveguides

• comparison of coaxial line, stripline and microstrip, (p.158, Table 3.6)

• CPW (coplanar waveguide)fabrication easyquasi-TEM operationradiation problem when gap width approaches /2monolithic applicationsless radiation than microstrip if well-balancedhigher order modes (coupled slot mode)

• www.appwave.com for computer assisted transmission line analysis

r

Ws s

微波電路講義3-21

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

capacity

3021

2377

2210

22

1capacity power maximum

103air ofstrength fieldbreakdown

22

2

6

ablnZ

abln

dadP

ablnEaP

ablnZ,

ZVP

ablnaEV

abln

Vm/VE

oo

max

o

dmax

oo

o

maxmax

dmaxo

d

ADS examples: Ch3_prj