Chapter 5 Transmission Lines

5-1 Characteristics of Transmission Lines

Transmission line: It has two conductors carrying current to support an EM wave, which is TEM or quasi-TEM mode. For the TEM mode, HaZE nTEM

ˆ ,

EaZ

H n

ˆ

1

TEM

, and TEMZ .

The current and the EM wave have different characteristics. An EM wave

propagates into different dielectric media, the partial reflection and the partial

transmission will occur. And it obeys the following rules.

Snell’s law: 2

1

2

1

1

2

1

2

2

1

2

1

sin

sin

r

r

p

p

i

t

v

v

n

n

and θi=θr

The reflection coefficient: Γ=0

0

i

r

E

E and the transmission coefficient: τ=

0

0

i

t

E

E

)sin(

sincos2

coscos

cos2

cos/cos/

cos/2

)sin(

)sin(

coscos

coscos

cos/cos/

cos/cos/

21

1

12

2

21

21

12

12

it

ti

ti

i

it

t

it

it

ti

ti

it

it

nn

n

nn

nn

for perpendicular polarization (TE)

)cos()sin(

sincos2

cos/cos/

cos/2

coscos

cos2

)tan(

)tan(

cos/cos/

cos/cos/

coscos

coscos

21

1

12

2||

21

21

12

12||

titi

ti

ti

t

it

i

it

it

ti

ti

it

it

nn

n

nn

nn

for parallel polarization (TM)

In case of normal incidence,

12

2//

12

12//

2

, where η1=1

1

and η2=2

2

.

Equivalent-circuit model of transmission line section:

)/( mR , , , )/( mHL )/( mSG )/( mFC

Transmission line equations: In higher-frequency range, the transmission line model

is utilized to analyze EM power flow.

t

tzvCtzGv

z

tzitzzit

tziLtzRi

z

tzvtzzv

),(),(

),(),(

),(),(

),(),(

t

vCGv

z

it

iLRi

z

v

Set v(z,t)=Re[V(z)ejωt], i(z,t)=Re[I(z)ejωt]

)()(

)()(

zVCjGdz

dI

zILjRdz

dV

)()())(()(

)()())(()(

22

2

22

2

zIzICjGLjRdz

zId

zVzVCjGLjRdz

zVd

where γ=α+jβ= ))(( CjGLjR zzzz eIeIzIeVeVzV 0000 )(,)(

Characteristic impedance: Z0=CjG

LjR

CjG

LjR

I

V

I

V

0

0

0

0

Note:

1. International Standard Impedance of a Transmission Line is Z0=50Ω.

2. In transmission-line equivalent-circuit model, G≠1/R.

Eg. The following characteristics have been measured on a lossy transmission

line at 100 MHz: Z0=50Ω, α=0.01dB/m=1.15×10-3Np/m, β=0.8π(rad/m). Determine

R, L, G, and C for the line.

(Sol.) CjG

LjR8

8

102

10250

, 1.15×10-3+j0.8π= )102(50))(( 8CjGCjGLjR

)/(8050102

8.08

mpFC

, )/(103.210

50

15.1 53 mSG ,

)/(0575.02500 mGR , )/(2.02500 mFCL

Eg. A d-c generator of voltage and internal resistance is connected to a lossy

transmission line characterized by a resistance per unit length R and a

conductance per unit length G. (a) Write the governing voltage and current

transmission-line equations. (b) Find the general solutions for V(z) and I(z).

(Sol.) (a) RGCjGLjR ))((0

)()(

),()(

2

2

2

2

zRGIdz

zIdzRGV

dz

zVd

(b) zRGzRGzRGzRG eIeIzIeVeVzV 0000 )(,)(

Lossless line (R=G=0):

0,,1

,,0 00000 XC

LRjXR

C

LZ

LCvLCLCjj p

Low-loss line (R<<ωL, G<<ωC):

)](2

11[

1,),(

2

1)](

2

11(

0 C

G

L

R

jC

LZ

LCvLC

C

LG

L

CR

C

G

L

R

jLCjj p

Distortionless line (R/L=G/C):

C

LZ

LCvLC

L

CRLjR

L

Cj p 0,

1,,)(

Large-loss line (ωL<<R, ωC<< G):

21

21

)1()1())((G

Cj

R

LjRGjCjGLjR

)](2

1[G

C

R

LjRG

)(2

,G

RC

R

GLRG

, )(2

1

G

RC

R

GLv p

21

21

0 )1()1(

G

Cj

R

Lj

G

R

CjG

LjRZ

)](2

1[G

C

R

Lj

G

R

Eg. A generator with an open-circuit voltage vg(t)=10sin(8000πt) and internal

impedance Zg=40+j30(Ω) is connected to a 50Ω distortionless line. The line has a

resistance of 0.5Ω/m, and its lossy dielectric medium has a loss tangent of 0.18%.

The line is 50m long and is terminated in a matched load. Find the instantaneous

expressions for the voltage and current at an arbitrary location on the line.

(Sol.) 0.18%= GCC

G 21021.2

, Vg=10j

∵ Distortionless, ∴ mHLG

C

R

L/1011.1 2 ,

0Z

R

L

CR =0.01Np/m,

mradZ

L

L

CLLC /58.5

0

, γ=α+jβ=0.01+j5.58

53

5

0

00 j

ZZ

VZV

g

g

, , 00 V zjz ejeVzV )58.501.0(

0 )53

5()(

)6.7158.58000cos(3

105])(Re[),( 01.08000 zteezVtzV ztj

)6.7158.58000cos(102

1),(),( 01.0

0

teZ

tzVtzI z

Relationship between transmission-line parameters:

21

21

)1()1())((

jj

Cj

GLCjCjGLjR G/C=σ/ε

and LC=με

Two-wire line: saJI 2 , )2

(2

1 2

a

RIP s

c

cs f

aa

RR

1

)2

(2

Coaxial-cable line: sosi bJaJI 22 , )2

(2

1 2

a

RIP s

i , )2

(2

1 2

b

RIP s

o

)11

(2

1)

11(

2 ba

f

ba

RR

c

cs

Eg. It is desired to construct uniform transmission lines using polyethylene

(εr=2.25) as the dielectric medium. Assume negligible losses. (a) Find the distance

of separation for a 300Ω two-wire line, where the radius of the conducting wires

is 0.6mm; and (b) find the inner radius of the outer conductor for a 75Ω coaxial

line, where the radius of the center conductor is 0.6mm.

(Sol.) Two-wire line: )2/(cosh 1 aD

C

, )2

(cosh 1

a

DL

, a=0.6mm, ε=2.25ε0

mmDa

D

C

LZ 5.25

1036

125.2

104)2

(cosh300

9

71

0

Coaxial line: )/ln(

2

abC

, )ln(

2 a

bL

a=0.6mm,

9

7

0

1036

125.2

104

2

)ln(75

a

b

C

LZ b=3.91mm

Parallel–plate transmission line:

xz

yz

HxeE

xH

EyeEyE

ˆˆ

ˆˆ

0

0

0

, ,jj

At y=0 and y=d, Ex=Ey=0, Hy=0

At y=0, , ya ˆˆ n

zjzslsl

zjyslsl

eE

zHzJJHy

eEEDy

0

0

ˆˆˆ

ˆ

At y=d, , ya ˆˆ n

zjxsusu

zjysusu

eE

zHzJJHy

eEEDy

0

0

ˆˆˆ

ˆ

HjE

, xy Hj

dz

dE

d

x

d

y dyHjdyEdz

d00

)/()(])()[()()(

mHw

dLzLIjwzJ

djdzJj

dz

zdVsusu

EjH

, yx Ej

dz

dH

w

y

w

x dxEjdxHdz

d00

)/()(])()[()()(

mFd

wCzCVjdzE

d

wjwzEj

dz

zdIyy

w

d

w

d

C

L

zI

zVZ

LCvLC p

)(

)(

11,

0

)()(

)()(

22

2

22

2

zLCIdz

zId

zLCVdz

ZVd

CVjdz

dI

LIjdz

dV

Lossy parallel–plate transmission line: d

wCG

Surface impedance: c

cssc

x

z

s

ts

fjjXR

H

E

J

EZ

)1(

RIw

RIRJZJP s

ssussu2222

2

1)(

2

1

2

1)Re(

2

1

)/(2

)(2 mf

ww

RR

c

cs

Eg. Consider a transmission line made of two parallel brass strips σc=1.6×107S/m

of width 20mm and separated by a lossy dielectric slab μ=μ0, εr=3, σ=10-3S/m of

thickness 2.5mm. The operating frequency is 500MHz. (a) Calculate the R, L, G,

and C per unit length. (b) Find γ and Z0.

(Sol.) (a) )/(11.12 0 m

f

wR

c

, )/(108 3 mSd

wG

)/(1057.1 70 mH

w

dL , )/(1012.2 10 mF

d

wC

(b) γ= ))(( CjGLjR =18.13∠-0.41°, ω=2π×500×106, Z0=CjG

LjR

=27.21∠0.3°

Eg. Consider lossless stripline design for a given characteristic impedance. (a)

How should the dielectric thickness d be changed for a given plate width w if the

dielectric constant εr is doubled? (b) How should w be changed for a given d if εr

is doubled? (c) How should w be changed for a given εr if d is doubled?

(Sol.)

w

d

C

LZ 0

(a) dd 22 , (b) 2

2w

w

(c) wwdd 22

Attenuation constant of transmission line: α =)(2

)(

zP

zPL , where PL(z) is the

time-average power loss in an infinitesimal distance.

]))((Re[)Re( CjGLjRj

Suppose no reflection, , zjeVzV )(0)( zje

Z

VzI )(

0

0)(

zeRZ

VzIzVzP 2

02

0

20*

2)]()(Re[

2

1)( ze 2

)(2

)()(2)(

)(

zP

zPzPzP

z

zP LL

Microstrip lines: are usually used in the mm wave range.

ff

p

c

,

C

L

Cp

1,

ff

p

f

Assuming the quasi-TEM mode:

Case 1: t/h＜0.005, t is negligible.

Given h, W, and εr, obtain Z0 as follows:

For W/h1:

h

W

W

h

ff

25.08ln60

,

where

22/1

104.01212

1

2

1

h

W

W

hrrff

For W/h1: 444.1ln667.0393.1

120

hW

hW

ff

,

where 2/1

1212

1

2

1

W

hrrff

Given Z0, h, and εr, obtain W as follows:

For W/h2: 2

82

A

A

e

heW , where

rr

rr

11.0

23.01

1

2

1

600

For W/h>2:

rr

r BBBh

W

61.039.01ln

2

112ln1

2, where

r

02

377

Case 2: t/h＞0.005. In this case, we obtain Weff firstly.

For W/h 21 :

t

h

h

t

h

W

h

Weff 2ln1

For W/h 21 :

t

W

h

t

h

W

h

Weff

4ln1

And then we substitute Weff into W in the expressions in Case 1.

Assuming not the quasi-TEM mode:

ffeff fW

f

h377

, where 2

1

0

p

effeff

ff

WWWfW ,

hp 8f (h in cm)

and 00

0ff

effW

377h

, G=0.6+0.009Z0, 2

1

p

ffrrff

ffG

f

(f in GHz)

The frequency below which dispersion may be neglected is given by

1

3.00

rhzGf

, where h must be expressed in cm.

Attenuation constant: α=αd +αc

For a dielectric with low losses:

tan

1

13.27

r

ff

ff

rd (

cm

dB)

For a dielectric with high losses:

2/1

1

134.4

rff

ffd (

cm

dB)

For W/h → ∞: sc RW

68.8

, where

fRs

For W/h 21 :

W

t

t

W

W

h

W

h

h

PR

effeff

sc

4ln1

2

68.8

For 21 ＜W/h2: PQ

h

Rsc

2

68.8, where

2

41

h

WP eff

and

h

t

t

h

W

h

W

hQ

effeff

2ln1

For W/h2:

94.02

94.02

2ln268.8

2

hW

hW

h

W

h

We

h

W

h

QR

eff

effeffeffeffs

c

Eg. A high-frequency test circuit with microstrip lines.

5-2 Wave Characteristics of Finite Transmission Line

Eg. Show that the input impedance is Zi=

tanh

tanh)(

0

00

'0

L

L

zz ZZ

ZZZZ

.

(Proof) ,

)2.....()(

)1...()(

00

00

zz

zz

eIeIzI

eVeVzV

0

0

0

00

I

V

I

VZ

Let z=l, V(l)=VL, I(l)=IL

eZIVV

eZIVV

eZ

Ve

Z

VI

eVeVV

LL

LL

L

L

)(2

1

)(2

1

00

00

0

0

0

0

00

])()[(2

)(

])()[(2

)(

)(0

)(0

0

)(0

)(0

zL

zL

L

zL

zL

L

eZZeZZZ

IzI

eZZeZZI

zV

)'cosh'sinh()'(

)'sinh'cosh()'(

])()[(2

)'(

])()[(2

)(

00

0

'0

'0

0

'0

'0

'

zZzZZ

IzI

zZzZIzV

eZZeZZZ

IzI

eZZeZZI

zV

LL

LL

zL

zL

L

zL

zL

L

'tanh

'tanh)'(

0

00 zZZ

zZZZzZ

L

L

, Zi=

tanh

tanh)(

0

00

'0

L

L

zz ZZ

ZZZZ

Lossless case (α=0, γ=jβ, Z0=R0, tanh(γl)=jtanβl): Zi=ljZR

ljRZR

L

L

tan

tan

0

00

Note: In the high-frequency circuit, the input current Ii=Lg

g

ig

g

ZZ

V

ZZ

V

: the

value in the low-frequency case. And the high-frequency Ii is dependent on the length

l, the characteristic impedance Z0, the propagation constant γ of the transmission line,

and the load impedance ZL. But the low-frequency Ii is only dependent on Z0 and ZL.

Eg. A 2m lossless air-spaced transmission line having a characteristic impedance

50Ω is terminated with an impedance 40+j30(Ω) at an operating frequency of

200MHz. Find the input impedance.

(Sol.) 3

4

pv, , 500R 3040 jZ L , m2

87.93.26)2

3

4tan()3040(50

)23

4tan(50)3040(

50 jjj

jjZi

Eg. A transmission line of characteristic impedance 50Ω is to be matched to a

load ZL=40+j10(Ω) through a length l’ of another transmission line of

characteristic impedance R0’. Find the required l’ and R0’ for matching.

(Sol.)

105.0'),(7.381500''tan)1040(

'tan104050 0'

0

'0'

0

R

jjR

jRjR

Eg. Prove that a maximum power is transferred from a voltage source with an

internal impedance Zg to a load impedance ZL over a lossless transmission line

when Zi=Zg*, where Zi is the impedance looking into the loaded line. What is the

maximum power transfer efficiency?

(Proof) gi

i ZZ

VI

, V

ZZ

ZV

gi

ii

])()[(2*]Re[

2

1)(

22

2

gigi

iiiout XXRR

VRIVPower

When and , ∴ gi RR gi XX MaxPower out )( , *gi ZZ

In this case, g

out R

VPower

4)(

2

, g

is R

VVIP

2*]Re[

2

12

, s

out

P

Powere

)(

2

1

Transmission lines as circuit elements:

Consider a general case: Zi=

tanh

tanh

0

00

L

L

ZZ

ZZZ

1. Open-circuit termination (ZL→∞): Zi=Zio=Z0coth(γl)

2. Short-circuit termination (ZL =0): Zi=Zis=Z0tanh(γl)

∴ Z0= isi ZZ 0 , γ=0

1tanh1

i

is

Z

Z

3. Quarter-wave section in a lossless case (l=λ/4, βl=π/2): L

i Z

RZ

20

4. Half-wave section in a lossless case (l=λ/2, βl=π): Li ZZ

Eg. The open-circuit and short-circuit impedances measured at the input

terminals of an air-spaced transmission line 4m long are 250∠-50°Ω and

360∠20°Ω, respectively. (a) Determine Z0, α, and β of the line. (b) Determine R,

L, G, and C.

(Sol.) (a) 6.778.289360250 20500 jeeZ jj ,

jj

235.0139.050250

20360tanh

4

1 1

(b) 3.575.580 jZLjR , )/(812.03.573.57

mHc

L

45

0

1076.8105.24 jZ

CjG , )/(4.12

1076.8 4

mpFc

C

g. Measurements on a 0.6m lossless coaxial cable at 100kHz show a capacitance E

of 54pF when the cable is open-circuited and an inductance of 0.30μH when it is

short-circuited. Determine Z0 and the dielectric constant of its insulating

medium.

(Sol.) (a) )/(1096.0

1054 1112

mFC

, )/(1056.0

103.0 76

mHL

C

LRZ 00 =74.5Ω, 05.400 rrr LC Lossless

General expressions for V(z) and I(z) on the transmission lines:

Let Γ= j

L

L eZZ

ZZ

0

0 , z’=l-z

]1[)(2

)'(

]1[)(2

)'(

'2'0

0

'2'0

zzL

L

zzL

L

eeZZZ

IzI

eeZZI

zV

]1[)(2

)'(

]1[)(2

)'(

)'2('0

0

)'2('0

zjzL

L

zjzL

L

eeZZZ

IzI

eeZZI

zV

For a lossless line, V(z)= ]1[ )(2

0

0 zjzj

g

g eeZZ

VZ

Eg. A 100MHz generator with Vg=10∠0° (V) and internal resistance 50Ω is

connected to a lossless 50Ω air line that is 3.6m long and terminated in a

25+j25(Ω) load. Find (a) V(z) at a location z from the generator, (b) Vi at the

input terminals and VL at the load, (c) the voltage standing-wave radio on the

line, and (d) the average power delivered to the load.

(Sol.) , )(010 VVg )(50 gZ

)

, , ,

,

)(108 Hzf )(500 Z

(4525 Z L 36.3525 j

)(6.3 m , )/(3

2

103

1028

8

mradc

, )/(4.2 mrad

648.0447.050)2525(

50)2525(

0

0

j

j

ZZ

ZZ

L

L , 0g

(a) ]447.0[5]1[)( )152.03/2(3/2)(2

0

0

zjzjzjzj

g

g eeeeZZ

VZzV

(b) )(43.806.7)447.01(5)0( 152.0 VeVV ji

(c) )(5.4547.4]447.0[5)6.3( 248.04.0 VeeVV jjL

(d) 62.2447.01

447.01

1

1

S , )(200.025)

36.35

47.4(

2

1

2

1 2

2

WRZ

VP L

L

Lav

Eg. A sinusoidal voltage generator Vg=110sin(ωt) and internal impedance

Zg=50Ω is connected to a quarter-wave lossless line having a characteristic

impedance Z0=50Ω that is terminated in a purely reactive load ZL=j50Ω. (a)

Obtain the voltage and current phasor expressions V(z’) and I(z’). (b) Write the

instantaneous voltage and current expressions V(z’,t) and I(z’,t).

(Sol.) (a) 4

,0,5050

5050,110

0

0

0

0

ZZ

ZZj

j

j

ZZ

ZZjV

g

gg

L

Lg

(c) )(55)1()'( '''2'

0

0 zjzjzjzj

g

g jeejeeZZ

VZzV

,

)(1.1)1()'( '''2'

0

zjzjzjzj

g

g jeejeeZZ

VzI

),0'(),0'( tzItzVP )2cos(5.60 t , 01

0

T

av PdtT

P

(b) )]'cos()'[sin(55])'(Im[),'( ztztezVtzV tj )]'cos()'[sin(1.1])'(Im[),'( ztztezItzI tj

Eg. A sinusoidal voltage generator with Vg=0.1∠0° (V) and internal impedance

Zg=Z0 is connected to a lossless transmission line having a characteristic

impedance Z0=50Ω. The line is l meters long and is terminated in a load

resistance ZL=25Ω. Find (a) Vi, Ii, VL錯誤! 尚未定義書籤。, and I ; (b) the

standing-wave radio on the line; and (c) the average power delivered to the load.

L

(Sol.) (a) 3

1

0

0

ZZ

ZZ

L

L , 00

0

ZZ

ZZ

g

gg , )

3

11(

2

1.0)'( 2 j

i ezVV

)3

11(

100

1.0)'( 2 j

i ezII

jjL eezVV

30

1)

3

11(

2

1.0)0'(

eezII jL 750

1)

3

11(

100

1.0)0'(

(b) 21

1

S , (c) WIVP LLav

51022.2*]Re[2

1

Eg. Consider a lossless transmission line of characteristic impedance R0. A

time-harmonic voltage source of an amplitude Vg and an internal impedance

Rg=R0 is connected to the input terminals of the line, which is terminated with a

load impedance ZL=RL+jXL. Let Pinc be the average incident power associated

with the wave traveling in the +z direction. (a) Find the expression for Pinc in

terms of Vg and R0. (b) Find the expression for the average power PL delivered

to the load in terms of Vg and the reflection coefficient Γ. (c) Express the ratio

PL/Pinc in terms of the standing-wave ratio S.

(Sol.) )(1

)(,)( 000

00zjzjzjzj eVeV

RzIeVeVzV ,

2)0( 0

ginc

VVzV ,

00 2

)0(R

VIzI g

inc

(a) 0

2

0

2

0

00 82)*](Re[

2

1

R

V

R

VIVP g

inc

(b) 2

1)])(Re[(

2

1)](*)(Re[

2

1 2

0

2

00

*0

*000

0

VVR

eVeVeVeVR

zIzVP zjzjzjzjL

18

12

2

0

22

0

2

0

R

V

R

Vg

(c) 2

22

)1(

4)

1

1(11

S

S

S

S

P

P

inc

L

Case 1 For a pure resistive load: ZL=RL

'sin)/('cos)'(

'sin)/('cos)'(

'sin'cos)'(

'sin'cos)'(

220

2

220

2

0

0

zRRzIzI

zRRzVzV

zR

VjzIzI

zRjIzVzV

LL

LL

LL

LL

1

1S ,

1

1

S

S1. Γ=0S=1 when ZL= Z0 (matched load)

2. Γ=-1 S=∞ when Z L=0 (short-circuit), 3. Γ=1S=-∞ when ZL=∞ (open-circuit)

minmax & IV occurs at nz 2'2 max

maxmin & IV occurs at )12('2 min nz

If ....3,2,1,0,2

',00 max0 nn

zRRL

If 2

',0 min0

nzRRL

If 2

'max

nzRL

Eg. The standing-wave radio S on a transmission line is an easily measurable

quality. Show how the value of a terminating resistance on a lossless line of

known characteristic impedance R0 can be determined by measuring S.

(Sol.) If , 0RRL 0 , maxV occurs at 0'z and minV occurs at 2' z .

LVV max , L

L R

RVV 0

min , LII min , 0

max R

RII L

L , 0min

max

min

max

R

RS

I

I

V

VL or

. 0SRRL

If , 0RRL , minV occurs at 0'z , and maxV occurs at 2' z .

LVV min , L

L R

RVV 0

max , LII max , 0

min R

RII L

L .LR

RS

I

I

V

V0

min

max

min

max or

S

RRL

0

Case 2 For a lossless transmission line, and arbitrary load:

ZL=mm

mm

jRR

jRRR

tan

tan

0

00

, zm’+lm=λ/2

Find ZL=?

1. 1

1

S

S, 2. At θΓ=2βzm’-π, V(z’) is a minimum.

3. ZL=RL+jXL =

1

1

1

100 R

e

eR j

j

Eg. Consider a lossless transmission line. (a) Determine the line’s characteristic

resistance so that it will have a minimum possible standing-wave ratio for a load

impedance 40+j30(Ω). (b) Find this minimum standing-wave radio and the

corresponding voltage reflection coefficient. (c) Find the location of the voltage

minimum nearest to the load.

(Sol.)

3

1500,

1

1,]

30)40(

30)40([

3040

30400

0

2122

0

220

0

0

0

0

RdR

dSS

R

R

jR

jR

RZ

RZ

L

L

, 23

190

3

1

3090

3010

0

0

j

j

RZ

RZ

L

L , 2

2 S

8)

2(

2

1'min

z , 8

3

82

m

Eg. SWR on a lossless 50Ω terminated line terminated in an unknown load

impedance is 3. The distance between successive minimum is 20cm. And the first

minimum is located at 5cm from the load. Determine Γ, ZL, lm, and Rm.

(Sol.)

52

,4.02.02

m

'2

05.0',5.013

13mmm zz

=0.15m

5.0'2 mz , 2

5.0 5.0 jee jj

mm

mmL jR

jRj

j

j

ZR

tan50

tan50504030

)2

(1

)2

(150,500

)(7.163

50 mR

Eg. A lossy transmission line with characteristic impedance Z0 is terminated in

an arbitrary load impedance ZL. (a) Express the standing-wave radio S on the

line in terms of Z0 and ZL. (b) Find the impedance looking toward the load at the

location of a voltage maximum. (c) Find the impedance looking toward the load

at a location of a voltage minimum.

(Sol.) (a) '2

00

'200'2

0

0

1

1,

zLL

zLLz

L

L

eZZZZ

eZZZZSe

ZZ

ZZ

(b) )'(1

1)'(,2'2

max

0

'2

'2

0max'2'2

maxmax

max

maxmax

zS

Z

e

eZzZeeenz

z

zzzj

(c) '2'2min

minmin)12('2 zzj eeenz

)'(1

1)'(

min

0'2

'2

0minmin

min

zS

Z

e

eZzZ

z

z

5-3 Introduction to Smith Chart

1

1

1/

1/

0

0

0

0

L

L

L

Lj

L

L

z

z

RZ

RZe

RZ

RZ ir j

j

j

Le

ez

1

1

1

1jxr

, 22)1(

2

ir

ix

22

22

)1(

1

ir

irr

222 )1

1()

1(

rr

rir

: r-circle, 222 )

1

xe ()

1()1(

xir : x-circl

Several salient properties of the r-circles:

1. The centers of all r-circles lie on the Γr-axis.

d centered at the origin, is the largest.

as r increases from 0 toward ∞,

se for x>0 (inductive reactance)

ose for x<0 (capacitive reactance) lie below the

cle becomes progressively smaller as |x| increases from 0 toward ∞,

pen-circuit.

r radii vary uniformly from 0 to

ough the point representing zL equals θΓ.

dio S.

2. The r=0 circle, having a unity radius an

3. The r-circles become progressively smaller

ending at the (Γr=1, Γi=0) point for open-circuit.

4. All r-circles pass through the (Γr=1, Γi=0) point.

Salient properties of the x-circles:

1. The centers of all x-circles lie on the Γr=1 line, tho

lie above the Γr–axis, and th

Γr–axis.

2. The x=0 circle becomes the Γr–axis.

3. The x-cir

ending at the (Γr=1, Γi=0) point for o

4. All x-circles pass through the (Γr=1, Γi=0) point.

Summary

1. All |Γ|–circles are centered at the origin, and thei

1.

2. The angle, measured from the positive real axis, of the line drawn from the origin

thr

3. The value of the r-circle passing through the intersection of the |Γ|–circle and the

positive-real axis equals the standing-wave ra

Application of Smith Chart in lossless transmission line:

]1

1[

)'(

)'()'(

'2

'2

0 zj

zj

i e

ez

zI

zVzZ

,

j

j

zj

zji

i e

e

e

e

Z

zZzz

1

1

1

1)()'(

'2

'2

0

when

'2 z

keep |Γ| constant and subtract (rotate in the clockwise direction) an angle

'4

'2z

z from θΓ. This will locate the point for |Γ|ejφ, which determine Zi.

Increasing z’ wavelength toward generator in the clockwise direction

A change of half a wavelength in the line length 2

'

z A change of

2)'(2 z in φ.

Eg. Use the Smith chart to find the input impedance of a section of a 50Ω lossless

transmission line that is 0.1 wavelength long and is terminated in a short-circuit. (Sol.) Given , ) , 0Lz (500 R 1.0'z

1. Enter the S tion ofmith chart at the intersec r=0 and x=0 (point on the

e chart

scP

extreme left of chart; see Fig.)

2. Move along the perimeter of th )1( by 0.1 “wavelengths toward

generator’’ in a clockwise direction to P1.

At P1, read r=0 and 725.0x , or 72.0jzi , )(3.36)725.0(50 jjZi . 5

g. A lossless transmission line of length 0.434λ and characteristic impedance

j180

j1.8 (point P2 in Fig.)

E

100Ω is terminated in an impedance 260+j180(Ω). Find (a) the voltage reflection

coefficient, (b) the standing-wave radio, (c) the input impedance, and (d) the

location of a voltage maximum on the line.

(Sol.) (a) Given l=0.434λ, R0=100Ω, ZL=260+

1. Enter the Smith chart at zL=ZL/R0=2.6+

60.02 OP2. With the center at the origin, draw a circle of radius .

( scOP =1) traight line and extend it to P2’ on the periphery. Read 3. Draw the s 2OP

0.22 on “wavelengths toward generator” scale. = 21 ,

2160.0je .

circle intersects with the positive-real axis at r=S=4. ocOP(b) The 60.0

(c) To find the input impedance:

1. Move P2’ at 0.220 by a total of 0.434 “wavelengths toward generator,” first to

0.500 and then further to 0.154 to P3’.

2. Join O and P3’ by a straight line which intersects the 60.0 circle at P3.

3. Read r=0.69 and x=1.2 at P3. )(12069)2.169.0(1000 jjzRZ ii .

(d) In going from P2 to P3, the 60.0 circle intersects the positive-real axis

ocOP at PM, where the voltage is a maximum. Thus a voltage maximum

appears at (0.250-0.220) or 0.030 from the load.

Application of Smith Chart in lossy transmission line

jz

jz

zjz

zjz

i ee

ee

ee

eez

'2

'2

'2'2

'2'2

1

1

1

1

∴ We can not simply move close the |Γ|-circle; auxiliary calculation is necessary for

the e-2αz’ factor.

Eg. The input impedance of a short-circuited lossy transmission line of length 2m

and characteristic impedance 75Ω (approximately real) is 45+j225(Ω). (a) Find α

and β of the line. (b) Determine the input impedance if the short-circuit is

replaced by a load impedance ZL= 67.5-j45(Ω). (Sol.) (a) Enter 0.360.075/)22545(1 jjzi in the chart as P1 in Fig.

Draw a straight line from the origin O through P1 to P1’.

Measure 211 89.0'/ eOPOP , )/(029.0)124.1ln(

4

1)

89.0

1ln(

2

1mNp

Record that the arc is 0.20 “wavelengths toward generator”. '1PPsc 20.0/ ,

8.0/42 . )/(2.04

8.0

2

8.0

mrad

.

(b) To find the input impedance for:

1. Enter 6.09.075/)455.67(/ 0 jjZZz LL on the Smith chart as P2.

2. Draw a straight line from O through P2 to P2’ where the “wavelengths toward

generator” reading is 0.364.

3. Draw a –circle centered at O with radius 2OP .

4. Move P2’ along the perimeter by 0.2 “wavelengths toward generator” to P3’ at

0.364+0.20=0.564 or 0.064.

5. Joint P3’ and O by a straight line, intersecting the –circle at P3.

6. Mark on line a point P3OP i such that 89.0/ 23 eOPOPi .

7. At Pi, read . 27.064.0 jzi )(3.200.48)27.064.0(75 jjZi

http://faculty.pccu.edu.tw/~meng/SmithChart-Lin.swf

5-4 Transmission-line Impedance Matching

Impedance matching by λ/4-transformer: R0’= LRR0

Eg. A signal generator is to feed equal power through a lossless air transmission

line of characteristic impedance 50Ω to two separate resistive loads, 64Ω and

25Ω. Quarter-wave transformers are used to match the loads to the 50Ω line. (a)

Determine the required characteristic impedances of the quarter-wave lines. (b)

Find the standing-wave radios on the matching line sections.

(Sol.) (a) )(1002 021 RRR ii .

)(806410011'01 Li RRR , )(502510022

'02 Li RRR

(b) Matching section No. 1:

11.08064

8064'011

'011

1

RR

RR

L

L, 25.1

11.01

11.01

1

1

1

11

S

Matching section No. 2:

33.05025

5025'022

'022

2

RR

RR

L

L , 99.133.01

33.01

1

1

2

22

S

Application of Smith Chart in obtaining admittance:

LL ZY /1 , LL

LL yYRR

Zz

11

00

, where jbyYRGYYYy LLL 0000 //

Eg. Find the input admittance of an open-circuited line of characteristic

impedance 300Ω and length 0.04λ.

(Sol.) 1. For an open-circuited line we start from the point Poc on the extreme right of

the impedance Smith chart, at 0.25 in Fig.

2. Move along the perimeter of the chart by 0.04 “wavelengths toward generator” to

P3 (at 0.29).

3. Draw a straight line from P3 through O, intersecting at P3’ on the opposite side.

4. Read at P3’: 26.00 jyi , mSjjYi 87.0)26.00(300

1 .

Application of Smith Chart in single-stub matching:

00

1

RYYYY SBi SB yy 1 , where yB=R0YB, ys=R0Ys

∵ 1+jbs= yB, ∴ ys=-jbs and lB is required to cancel the imaginary part.

Using the Smith chart as an admittance chart, we proceed as yL follows for

single-stub matching:

1. Enter the point representing the normalized load admittance.

2. Draw the |Γ|-circle for yL, which will intersect the g=1 circle at two points. At

these points, yB1=1+jbB1 and yB2=1+jbB2. Both are possible solutions.

3. Determine load-section lengths d1 and d2 from the angles between the point

representing yL and the points representing yB1 and yB2.

Determine stub length lB1 and lB2 from the angles between the short-circuit point on

the extreme right of the chart to the points representing –jbB1 and –jbB2, respectively.

Eg. Single-Stub Matching:

Eg. A 50Ω transmission line is connected to a load impedance ZL= 35-j47.5(Ω).

Find the position and length of a short-circuited stub required to match the line. )(500 R , )(5.4735 jZ L , 95.070.0/ 0 jRZz LL (Sol.) Given

–circle centered at O with radius 1OP . 1. Enter on the Smith chart as . Draw a Lz 1P

2. Draw a straight line from through O to on the perimeter, intersecting the 1P 2'P –circle at

which represents . No .109 at “wavelengths toward generator” scale.

3. T io

2P , Ly te 0 2'P on the

wo points of intersect n of the –c with the g=1 circle. ircle

3P : 11 12.11 BB jbjy At . At : 4P 22 12.11 BB jbjy ;

4. Solutions for th sition of the stubs:

For (f to ):

e po

rom3P 2'P 3'P 059.0)109.0168.0(1 d

For 4P (from 2'P to 4'P ): .0(2 d 223.0)109.0332

For (from to , which represents 3P scP 3''P 2.11 jjbB ):

111.0)250.0361.0(1 B

For (from to , w4P scP 4"P hich represents 2.12 jjbB ):

389.0)250.0139.0( 2B

5-5 Introduction to S-parameters

S-parameters: for analyzing the high-frequency circuits.

22

12

21

11

S

S

S

SS

Define xIZxVZ

xa 0

02

1 , xIZxV

Zxb 0

02

1

2212111111 laSlaSlb , 2222112122 laSlaSlb

, where

22

11

22

12

21

11

22

11

la

la

S

S

S

S

lb

lb 0

11

1111 22 lala

lbS ,

0

11

2221 22 lala

lbS ,

0

22

2222S

11 lala

lb, and

0

22

1112 11 lala

lbS .

New S-parameters obtained by shifting reference planes:

10111

jeblb , 10111jeala , 20222

jeblb , 20222jeala

0

0

0

0

2

1

222

12

21

211

2

1

2

21

21

1

a

a

eS

eS

eS

eS

b

bj

j

j

j

, where

=

22

12

21

11

'

'

'

'

S

S

S

S

2

21

21

1

222

12

21

211

j

j

j

j

eS

eS

eS

eS and =

22

12

21

11

S

S

S

S

2

21

21

1

222

12

21

211

'

'

'

'

j

j

j

j

eS

eS

eS

eS

21

11

21

22

12

21

11

1

S

S

S

T

T

T

T

21

221112

21

22

S

SSS

S

S

T-parameters:

lbla, where

22

22

22

12

21

1111

laTT 11

TT

lb

and

111211 TSS

11

21

2221 1

T

T

SS

1122 T

11

12

1221

T

T

TTT