서강대학교 전자공학과 윤상원 교수 * “RF Circuit Design: Theory and Applications”,...

서강대학교 전자공학과윤상원 교수

* “RF Circuit Design: Theory and Applications”, R. Ludwig & P. Bretchko

10. Transmission Line

Microwave & Millimeter-wave Lab. 2

차 례

10-1. Introduction ------------------------------------------------10-2. Transmission Lines -------------------------------------10-3. Equivalent Circuit ---------------------------------------10-4. General Transmission Line Equation ---------10-5. Lossless transmission line --------------------------10-6. Microstrip Transmission Lines --------------------10-7. Terminated Lossless line ----------------------------10-8. Standing Waves ------------------------------------------10-9. Special Termination Conditions -----------------10-10. Sourced and loaded line --------------------------10-11. Power considerations for a line ---------------

3 4 6 811142023283538


1-1. Introduction

At RF and microwave frequencies Physical size of circuit approaches to the wave-

length - the phase of ac signal must be considered At higher frequency range For larger size of the circuits

Voltage and Current must be treated as waves Phasor notation is very convenient On the circuit board one dimensional analysis is possible

Distributed circuit approach must be used Lumped element equivalent circuit approach enable us to

use Basic Circuit Theory Impedance is very important as in the Circuit Theory


1-2. Transmission Lines

Two wire lines

Coaxial line


Transmission lines(2)

Microstrip lines and Striplines

Parallel-plate transmission line


1-3. Equivalent Circuit(1)


Equivalent circuit (2)


1-4. General Transmission Line Equation

For a small segment of a transmission line Lumped element equivalent circuit

Apply KVL and KCL

+V(z)

-

I ( z)

+V(z+z)

-

I ( z+z)

z z+z

CzGz

Rz Lz

CjGY

LjRZ

)()()()()(

)()()()()(

zzVYzIzIzIzzI

zzIZzVzVzVzzV


General Transmission Line Equation (2)

leads to the differential form as

or

,ZIdz

dV YV

dz

dI

022

2

Ikdz

Id,02

2

2

2

2

Vkdz

VdZYV

dz

Vd

where propagation constant k given as

jCjGLjRZYk


General Transmission Line Equation (3)

Voltage and current waves

,)( kzkz eVeVzV kzkz eIeIzI )(

kzkzkzkz eVeVZ

eVeVZ

k

dz

dV

ZzI

0

11)(

where the characteristic impedance given as

CjG

LjR

Y

Z

I

V

I

VZ

0


1-5. Lossless transmission line

C

L

Y

ZZ 0

jLCjZYk

0R and 0G

Propagation constant becomes

Characteristic impedance becomes

Voltage and current waves become

,)( zjzj eVeVzV zjzj eIeIzI )(


1-6. Microstrip Transmission Lines

Microstrip Line geometry

Assume that ‘t’ is negligible compared to ‘h’ ; t/h < 0.005

→ depend only on ‘w’, ‘h’ and r.


Microstrip Transmission Lines (2)

For a narrow lines ; w/h < 1

h

w

w

hZZ

eff

f

48ln

20

: characteristic line

impedance

8.37600 fZ

221

104.01212

1

2

1

h

w

w

hrreff

: wave impedance in

free space

: effective dielectric constant



For a wide lines ; w/h > 1

Wavelength

444.1ln

32

393.10

hw

hw

ZZ

eff

f

: characteristic

impedance

21

1212

1

2

1

w

hrreff

: effective

dielectric constant

effeff

p c

ff

v

01



Z0 and εeff are plotted as w/h and εr



Assuming an infinitely thin line conductor,w/h ≤ 2 ;

w/h ≥ 2 ;

2

82

A

A

e

e

h

w

rr

rr

fZ

ZA

11.0

23.01

1

2

12 0

rr

r BBBh

w

61.0

39.01ln2

112ln1

2

r

f

Z

ZB

02



Corrections for nonzero strip thickness t ;

twπh/wx

hwhx

t

xtwweff 22 if 2

2t 2/ if

21


1-7. Terminated Lossless line

Voltage Reflection Coefficient ;

z=- d z=0z

Z0 Z L

Z in 0

zjzj

zjzj

zjzj

eIeI

eVeVZ

zI

eVeVzV

)0()0(

)0()0(1

)(

)0()0()(

0

djdj eZ

VdIeVdV 2

00

20 1)( , 1)(

Use standing wave concept)0(

)0(Γ0

zV

zV

0Γ


Terminated Transmission line (2)

dj

dj

ine

eZ

dI

dVZ

20

20

01

1

)(

)(

Input impedance ;

Input impedance at ;0z

0

00 1

1

)0(

)0()0(

ZI

VZZ Lin

1

1

0

00

L

L

L

L

Z

Z

ZZ

ZZ : Reflection coefficient

at load

dz


Terminated Transmission line (3)

Reflection coeff. for various terminations ; Open line : Short circuit : Impedance matched :

For a infinite transmission line ; Phase constant :

Dispersion-free transmission line

)( LZ 10 )0( LZ 10

)( 0ZZL 00

LC

pv

f

vpSince


1-8. Standing Waves

z=- d z=0z

Z 0

Z in 0

Shorted transmission line ;

dZ

V

eeZ

VdI

dVj

eeVdV

djdj

djdj

cos2

)(

sin2

)(

0

0

in the time domain ;

2/cossin2

sin2Re Re ),(

tdV

deVjVetdV tjtj


Standing Waves(2)


Standing Waves(3)

z=- d z=0z

Z 0

Z L

Z in 0

V+

V -

V+ e - jd

0V+ e - jd0V+ e - j2d

Standing wave expressions ;

dj

dj

e

eV

VVdV

2

0

20

(d)

1

)(

djeVdA

dZ

dAdI

ddAdV

)(

)(1)(

)(

)(1)()(

0

Standing wave ratio(SWR) ;

11

1SWR

0

0

min

max

min

max

I

I

V

V


Standing Waves(4)


Standing Waves(5)

Graphical interpretation

Voltage standing wave ratio(VSWR) or return loss used: 0log20)(log20RL d

V+

0V+

2d

resulting standing w ave

O

| |


1-9. Special Termination Conditions

Input impedance of terminated line ;

z=- d z=0z

Z 0

Z L

Z in 0

V+

V -

V+ e - jd

0V+ e - jd0V+ e - j2d

dj

dj

ineV

eVZ

dI

dVdZ

2

0

20

01

1

)(

)()(

or

)(1

)(1)( 0 d

dZdZin

djZZ

djZZZ

eZZ

ZZ

eZZ

ZZ

ZdZL

L

dj

L

L

dj

L

L

in

tan

tan

1

1

)(0

00

2

0

0

2

0

0

0


Special Termination Conditions(2)

Short Circuit Transmission Line

z=- d z=0z

Z 0

Z in 0=- 1

djZdZin tan)( 0

dZ

V

eeZ

VdI

dVj

eeVdV

djdj

djdj

cos2

)(

sin2

)(

0

0

00

00 tan

tan)(

LZL

Lin djZZ

djZZZdZ



Open-circuit transmission line

z=- d z=0z

Z 0

Z in 0 =1

LZL

Lin djZZ

djZZZdZ

tan

tan)(

0

00

djZdZin cot)( 0

dZ

Vj

eeZ

VdI

dV

eeVdV

djdj

djdj

sin2

)(

cos2

)(

0

0



Quarter-wave transmission line in case

In case

d

Z0

Z LZ in

),2,1 ,2/2/(2/ mmd

LL

L

L

Lin

ZjZZ

jZZZ

djZZ

djZZZdZ

tan

tan

tan

tan)

2(

0

00

0

00

),2,1 ,2/4/(4/ mmd

LL

L

L

Lin Z

Z

jZZ

jZZZ

djZZ

djZZZdZ

20

0

00

0

00 2/tan

2/tan

tan

tan)

4(



Quarter-wave transformer

impedance matching condition ;

d

Z in ; desired Z L ; given

Z LZ 0 = Z L Z in

Lin Z

ZdZ

20 )4/(

LinZZZ 0


1-10. Sourced and loaded line

Phasor representation of source

Input voltage at plane AA’ ;

Z0 Z LVG

L=0

ZG

s

in out

A

A'

B

B'

Gin

inGininininin ZZ

ZVVVVV 1


Sourced and loaded line(2)

The input reflection coeff. at plane AA’ ;

The source reflection coeff. at plane AA’ ;

The source reflection coeff. at plane BB’ ;

d

0

00

0)(ZZ

Z

ZZ

ZZd

in

in

in

inin

0

0

ZZ

ZZ

G

Gs

2jsout e


Sourced and loaded line(3)

Transmission coefficient at plane AA’ ;

At the load end (at plane BB’) ;

0

21

ZZ

ZT

in

ininin

000

21

ZZ

ZT

L

L


1-11. Power considerations for a line

Time averaged power

The total power at plane AA’ ; the complex input voltage : input current :

Z0 Z LVG

L=0

ZG

s

in out

A

A'

B

B'

* Re 2

1IVPav

ininin VV 1

ininin ZVI 1)/( 0

2

0

2

12

1in

inininin Z

VPPP


Power considerations for a line(2)

In terms of generator voltage ;

The input and the generator impedances ;

The generator voltage in terms of

Gin

in

in

G

in

inin ZZ

ZVVV

11

s

sG

in

inin ZZZZ

1

1 ,

1

100

sin and

2

2

2

0

2

11

1

8

1in

ins

sGin Z

VP


Power considerations for a line(3)

or

When the impedances are matched ;

: available power

When the source is not matched ;

22022

0

2

0

2

11

1

8

1

j

js

sGin e

eZ

VP

G

GGin Z

V

Z

VP

2

0

2

8

1

8

1

2

0

2

18

1s

Gin Z

VP : available power at AA’

서강대학교 전자공학과 윤상원 교수 * “RF Circuit Design: Theory and Applications”,...

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Transcript of 서강대학교 전자공학과 윤상원 교수 * “RF Circuit Design: Theory and Applications”,...