EMLAB

Chapter 2. Transmission line theory

1

EMLAB

Types of transmission lines

Microstrip line

Coaxial cable

Two-wire transmission line

전기 신호를 손실이 적게 전송하기 위한 구조

2

EMLAB

Signal propagation in tx-line

+

V

-

+

전기장의 진행 속도는 빛의 속도 c임

LZ

전기장의 방향

3

EMLAB

i (z, t)

v (z, t) +

-

z

L z C z

z

i (z+z, t)

v (z+ z,t) +

-

i (z, t)

v (z, t)

),(),(

),( tzzVt

tzIzLtzV

),(),(

),( tzzIt

tzzVzCtzI

Transmission line circuits modeling 4

EMLAB

Transmission line eq. solution

t

VC

z

I

t

IL

z

V

, 0,0

2

2

2

2

2

2

2

2

t

ILC

z

I

t

VLC

z

V

)()(),(

)()(),(

tzItzItzI

tzVtzVtzV

01

)(),(

,1

2

2

2

2

2

2

22

2

2

22

2

2

2

2

2

2

dX

Vd

dX

Vd

t

V

z

V

dX

Vd

t

X

t

V

Xt

V

dX

dV

t

X

dX

dV

t

V

dX

Vd

z

X

z

V

Xz

V

dX

dV

z

X

dX

dV

z

V

C

LZ0)()(),()( tI

C

LtVtI

C

LtV

LC

1

tzX

I

C

LV

dX

dIL

dX

dV

t

IL

z

V

5

EMLAB

Coax

a

W

a

bZ

ab

C

a

bL

ln2

1

ln

2

ln2

0

Parallel Plate

d

W

dZ

d

WC

W

dL

0

Transmission line parameter - examples

b

6

EMLAB

+ -

Parallel wire

Coplanar waveguide

a

DZ

aD

C

a

DL

2cosh

1

2cosh

2cosh

1

0

1

1

a

D

7

EMLAB

LZ+

V

-

Transmission line의 특징

H

E

진행 방향

H

1. 한 방향으로 진행하는 전파의 E+/H+의 비율이 일정.

2. 한 방향으로 진행하는 V+/I+ wave의 amplitude 비율도 일정. → 특성 임피던스 Z0

3. 비율이 맞지 않는 경우 반사파 생김.

0ZI

V

SZI

V

0ZI

V

I

V

8

EMLAB

LZ+

V

-

SZI

V

I

V

VVVL

IIIL

LLL IZV

)tcoefficien reflection;(, VV

0Z

II

VV

L

L

)1(

)1(

0

0

01

1

1

1

ZZ

ZZ

ZZI

V

I

V

L

L

L

L

L

Reflection coefficient 9

EMLAB

LZ+

V

-

SZ

20 40 60 800 100

0

1

-1

2

time, nsec

Vin

, V

Vo

ut, V

20 40 60 800 100

0

1

-1

2

time, nsec

Vin

, V

Vo

ut, V

20 40 60 800 100

0

1

-1

2

time, nsec

Vin

, V

Vo

ut,

V

20 40 60 800 100

0

1

-1

2

time, nsec

Vin

, V

Vo

ut, V

20 40 60 800 100

0

1

-1

2

time, nsec

Vin

, V

Vo

ut,

V

LZ+

V

-

SZ

LZ+

V

-

SZ

LZ+

V

-

SZ

LZ+

V

-

SZ

Zs = 20

Z0= 50

ZL= 1k

0.5m

Line길이에 따른 반사파 영향

rc

LT

/delay

[ns]250 T

[ns]6d T

[ns]3d T

[ns]5.1

[ns]375.0

[ns]75.0

Impedance mismatched

Vin Vout R

R2

R=1k Ohm

MLIN R

R1

R=20 Ohm

VtPulse

SRC1

t

Z0= 50

43.05020

50200

ZZ

ZZ

s

ss

9.05020

50200

ZZ

ZZ

L

LL

10

EMLAB

20 40 60 800 100

0

1

-1

2

time, nsecV

in,

VV

ou

t, V

20 40 60 800 100

0

1

-1

2

time, nsec

Vin

, V

Vo

ut,

V

20 40 60 800 100

0

1

-1

2

time, nsec

Vin

, V

Vo

ut, V

20 40 60 800 100

0

1

-1

2

time, nsec

Vin

, V

Vo

ut, V

20 40 60 800 100

0

1

-1

2

time, nsec

Vin

, V

Vo

ut,

V

LZ+

V

-

SZ

LZ+

V

-

SZ

LZ+

V

-

SZ

LZ+

V

-

SZ

Zs = 1

Z0= 50

ZL= 50

0.5m

rc

LT

/delay

[ns]6d T

[ns]3d T

[ns]5.1

[ns]375.0

[ns]75.0

LZ+

V

-

SZ

Impedance matched

Line 길이에 따른 수신 신호

Vin Vout R

R2

R=50 Ohm

MLIN R

R1

R=50 Ohm

VtPulse

SRC1

t

Z0= 50

11

EMLAB

Narrow band signal 12

EMLAB

0,02

2

2

2

2

2

2

2

t

iLC

z

i

tLC

z

Frequency domain solution

0,0 2

2

22

2

2

LCI

z

ILCV

z

V

tjtj ezItziezVtz ),(),(,),(),(

zjzj

zjzj

eIeIzI

eVeVzV

),(

),(

VCjz

IILj

z

V

,

)()(),()( IC

LVI

C

LV

LCLC p

p

1,

β : propagation constant, vp : speed of light

13

EMLAB

dezItzi

dezVtz

tj

tj

),(2

1),(

),(2

1),(

0,02

2

2

2

2

2

2

2

t

iLC

z

i

tLC

z

0),(),(

2

1

0),(),(

2

1

2

2

2

2

2

2

dezILCz

zI

dezLCVez

zV

tj

tjtj

0,0 2

2

22

2

2

ILC

z

IVLC

z

V

zjzj

zjzj

eIeIzI

eVeVzV

),(

),(

14

EMLAB

]}[Re{

}Re{

)/()/(),(

)/()/(

zjzjtj

cztjcztj

eVeVe

eVeV

cztVcztVtz

Phasor representation

zjzj eVeVzV ),(

}Re{)/( )/( cztjeVcztV

LZ+

V

-

SZI

V

I

V

0Z

tj

SS eVtlz ),(

}),(Re{),( tjezVtz lz 0z

)cos(),( ztVtzV

)cos(),( ztVtzV

15

EMLAB

Transmission line terminated with short, open

Zs = Zo

Vrefl V

inc

For reflection, a transmission line terminated in a short or open reflects all power

back to source

In phase (0 ) for open o

Out of phase (180 ) for short

Vrefl

o

16

EMLAB

Transmission Line Terminated with 25 Ω

Zs = Zo

ZL = 25

Vrefl V

inc

Standing wave pattern does not go to zero as with short or open

3

1

5025

5025

17

EMLAB

)(][1

),(

)(),(

00

ljljljlj

ljljljlj

eeZ

VeVeV

ZlI

eeVeVeVlV

0

0

ZZ

ZZ

L

L

ljZZ

ljZZZ

ee

eeZ

lI

lVZ

L

L

ljlj

ljlj

in

tan

tan

),(

),(

0

000

Equivalent input impedance 18

EMLAB

Input impedance of short

ljZZin tan0

19

EMLAB

Input impedance of open

ljZZin cot0

20

EMLAB

Input impedance of ¼ wavelength line

L

inZ

ZZ

2

0

4/

Quarter wavelength transformer

21

EMLAB

Reflection/Transmission

IV

IV TV

TI

TT

TT

IIVV

IIIVVV

)1(,)1(

,

101

1

)1(

)1(

ZZI

V

III

TVVV

T

T

TT

T

22

EMLAB

Reflection measurement – slotted line

||1

||1SWR

min

max

V

V

Standing wave ratio

23

EMLAB

Smith chart

j

L

L

L

L eZZ

ZZ

ZZ

ZZ||

1/

1/

0

0

0

0

각각의 반사계수에 해당하는 부하 임피던스를 표시한 그림

real

imag

0/ ZZz L

Normalized impedance

je||

반사계수 측정을 위해 사용된

transmission line의 특성 임피던스 = Z0

24

EMLAB

25

EMLAB

Network analyzer 26

EMLAB

Smith chart review

.

-90 o

0 o

180 o +

- .2

.4

.6

.8

1.0

90 o

0

0 +R

+jX

-jX

Smith Chart maps rectilinear

impedance plane onto polar plane

Rectilinear impedance plane

Polar plane

Z = Zo L

= 0

Constant X

Constant R

Z = L

= 0 O

1

Smith Chart

(open)

L

Z = 0

= ±180 O

1

(short)

0

0

ZZ

ZZ

Z-plane

Γ-plane Z-to-Γ transform

27

EMLAB

Smith chart :

• graphical representation in the reflection coefficient plan plane

• passive impedance plane Re(z)

• z one-to-one correspondance

)2()1(

2

)1()1(

1

)1(

})1}{()1{(

)1(

)1(

1

1

1

1

impedance normalized:/ 1

1

22

22

22

22

00

0

0

0

xr

xv

xr

xru

xr

jxrjxr

jxr

jxr

z

zivulet

z

Z

Xj

Z

Rjxrz

ZZzz

z

ZZ

ZZ

Constant resistance, reactance circles 28

EMLAB

22

2

2

2

2

2222

22

11)1( )2()6(

)1(

1

1 )1()5(

)6()1(

)5(1

1

222

2 )4()3(

)4(12)1(2

)2(

)3(12 ))2/()1((2

xxvu

ru

r

ru

v

vuxr

vu

rx

rv

x

v

ux

xrrxrv

x

xrv

uxx

29

EMLAB

x

R 0 0.5 1 2

r=2

r=1

r=0.5

r=0

x

R

2

1

0.5

0.5

1

2

x=2

x=0.5 x=1

x=-0.5

x=-1

x=-2

real

imag

real

imag

Constant resistance, reactance circles 30

EMLAB

Constant admittance circles

1/

1/

/1

/1

/1

/1

0

0

0

0

0

0

0

0

YY

YY

YY

YY

ZZ

ZZ

ZZ

ZZ

1

1

1

1

y

y

z

z

0

+jX

-jX

Z-plane

0

+jB

Y-plane

+jG 1

1)(

y

y

Impedance chart를 원점에 대칭이동하면 admittance chart가 됨.

31

EMLAB

Basic smith chart operation

ljzj

zj

zj

L

L

ezezeV

eV

lV

lVlz

ZZ

ZZ

V

Vz

22

0

0

)0()0()(

)()(

)0(

)0( z)( lz

ljezlz 2)0()(

Wavelength toward generator

32

EMLAB

Example 2.2

ZL = 40 + j 70Ω 의 부하가 특성 임피던스 100 Ω이고 길이가 0.3λ인

transmission line 에 연결되어 있다.

)0( z)( lz

부하단에서 반사계수, transmission line의 입력단에서 반사계수, 입력임피던스, SWR, return loss를 구하라.

7.04.0100

7040j

jzL

1045890.00.5714 + -0.1429

1

1j

z

z

L

LL

1125890.0

21611045890.02 lj

Lin e

dB6.4||log20RL

87.359.01

59.01SWR

33

EMLAB

34

EMLAB

Example 2.4

)2(2)( lj

L

lj

L eelz

)2(1 lje

)2(1 lje

0.2, 2.2, 4.2cm

0.72, 2.72, 4.72cm

특성 임피던스 50Ω 인 slotted

line을 이용한 측정 결과이다.

부하 임피던스를 구하라.

SWR =1.5

7.193.471

150

1996.00126.0*2.0

2.0

4

1.481.48cm0.52cm2cm

48.1*4/4

48.1*4/42

min

min

jZ

je

ee

l

L

LL

j

L

j

L

lj

L

35

EMLAB

Example 2.5

10050

71.7050*1001Z

36

EMLAB

2.7 Lossy transmission line 37

EMLAB

0I(z)dz

I(z)d

0V(z)dz

V(z)d

2

2

2

2

2

2

)()(dz

dI(z)

)()(dz

dV(z)

zVCjG

zILjR

C)jL)(Gj(R

j

C

Lossy transmission line 38

EMLAB

zz

zz

ee

ee

00

00

III(z)

VVV(z)

Wave Solution

CjG

LjRZ

LjR0

]VV[LjR

I(z) 00

zz ee

: Characteristic Impedance

0

00

0

00

I

VZ

I

VZ

2 fvp

For Lossless Line

LCv

LC

p

1

22

0

LC

LCjj

C

LZ 0

zjzj

zjzj

eZ

e

ee

0

0

0

0

00

I

Z

VI(z)

VVV(z)

39