EMLAB

1

Transmission line

EMLAB

2

An apparatus to convey energy or signal from one place to another place.

• Transmitter to an antenna

• connections between computers in a network

• hydroelectric generating plant and a substation several miles away

• interconnect between components of a stereo system

• CATV service provider and your TV set

• Connections between devices on a circuit board

Transmission line

EMLAB

3Example – Cable TV

EMLAB

4Example – Computer network

EMLAB

5Example – Electric power transmission line

EMLAB

6Example – Printed circuit board

EMLAB

7Example – Printed circuit board

EMLAB

8Types of transmission lines

Microstrip line

Coaxial cable

Two-wire transmission line

EMLAB

9Distribution of electric field strengths of typical TEM lines

E/H distributions vary as the structures of transmission lines change. For electromagnetic compatibility, E/H should be confined to small area.

MicrostripTwo-wireParallel plate

EMLAB

10Field representation in waveguides

EMLAB

11

Wave Solution

0 zz HE

0ˆ

0ˆ

ttztt

ttztt

Ej

Hj

HzH

EzE

)ˆ

ˆ

ztttzztztt

ztztzztt

Hjω

Hjω

zHEEE

)z(H )E(E)E(E

(

)Eqs.Maxwell(, JEHHE jj

ttttz

ttttz

jωzz

jω

jωzz

jω

E HE H

H EH E

)ˆ(

)ˆ(

0,0 22

22

2

2

tt

tt k

zk

zH

HE

E0

0,0

0

2

tt

ttt

ztt z

E

H

E

EE

)(zftt E

)(00jkzjkz

tjkzjkz

t eVeVee EEE

TEM mode (Transverse electromagnetic mode)

EMLAB

12

z

y

xParallel plate waveguide

02

2

2

2

2

22

xyxt

With a wide enough line trace, varia-tion along y-axis can be ignored.

xd

VCxC s 21

)(ˆ)(ˆ jkzjkzjkzjkzsTEM eVeVeCeC

d

V xxE

d

)(1

ˆ)(ˆ1 jkzjkzjkzjkzx

TEM eVeVeVeVk

z

H

jj

yyE

H

EMLAB

13

i (z, t)

v (z, t)+

-

z

L zC z

i (z+z, t)

v (z+ z,t)+

-

i (z, t)

z

v (z, t)+

-

Transmission line 등가 회로

tC

z

it

iL

z

),(

),(),( tzz

t

tzizLtz

),(),(

),( tzzit

tzzzCtzi

EMLAB

14Transmission line eq. solution

tC

z

i

t

iL

z

, 0,02

2

2

2

2

2

2

2

t

iLC

z

i

tLC

z

)()(),(

)()(),(

tuzItuzItzi

tuzVtuzVtz

C

LZ0)()(),()( 00 tIZtVtIZtV

LCu

1

IC

LVI

C

LV ,

)(z

z

)( tuz

z

)( tuz

z

01

)(),(

,1

2

2

2

2

2

2

22

2

2

22

2

2

2

2

2

2

dX

Vd

dX

Vd

t

V

uz

V

dX

Vdu

t

X

t

V

Xt

Vu

dX

dV

t

X

dX

dV

t

V

dX

Vd

z

X

z

V

Xz

V

dX

dV

z

X

dX

dV

z

V

tuzX

EMLAB

15Transmission line 의 특징

H

E

Direction of propagation

H

1. For a magnetic field and an electric field propagating in the same direction, the ratio of E and H (E+/H+) is kept constant.

2. For a voltage and a current current propagating in the same di-rection, V+/I+ ratio is equal to Z0 .

3. When the ratio is disturbed, reflected waves are generated.

0ZI

V

I

V

0ZI

V

I

V)(tV

SZLZ

EMLAB

16

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

LL

L

Reflection coefficient

}Re{)1(2

1)1()1(Re

2

1}Re{

2

102

0

2

2

*0

*** Z

Z

V

Z

VVIVP LLL

EMLAB

17

LZ+V-

SZ

20 40 60 800 100

0

1

-1

2

time, nsecV

in, V

Vou

t, V

20 40 60 800 100

0

1

-1

2

time, nsec

Vin

, VV

out,

V

LZ+V-

SZ

LZ+V-

SZ

LZ+V-

SZ

LZ+V-

SZ

Zs = 20

Z0= 50

ZL= 1k

0.5m

Influence of line length on load voltage

rc

LT

/delay

[ns]250 T

[ns]6d T

[ns]3d T

[ns]5.1

[ns]375.0

[ns]75.0

Impedance mismatched

Vin VoutRR2R=1k Ohm

MLINRR1R=20 Ohm

VtPulseSRC1

t

Z0= 50

20 40 60 800 100

0

1

-1

2

time, nsec

Vin

, VV

out,

V

20 40 60 800 100

0

1

-1

2

time, nsec

Vin

, VV

out,

V

20 40 60 800 100

0

1

-1

2

time, nsec

Vin

, VV

out,

V

EMLAB

18

EMLAB

19Ringing : Time domain

EMLAB

20

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.91.0 2.0

-1

0

1

2

-2

3

time, usec

Vin

, V

Vout, V

Vs Vin Vout

RR1R=10 Ohm

VtPulseSRC1

Period=50 nsecWidth=25 nsecFall=1 nsecRise=1 nsecEdge=cosineDelay=0 nsecVhigh=1 VVlow=0 V

t

RR2R=1000 Ohm

MLINTL1

L=5 meterW=0.242 mmSubst="MSub1"

Signal source

Load

~ 10SZ

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.91.0 2.0

-0.0

0.2

0.4

0.6

0.8

-0.2

1.0

time, usec

Vin

, V

Vout, V

kZL 1500Z

Mismatched load

Ringing

EMLAB

21Impedance matching – Digital logic

~

10SZ

kZL 1500Z

R1

1k

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.950.00 1.00

0.0

0.5

-0.5

1.0

time, usec

Vin

, V

Vout, V

40

~ 10SZ

kZL 1

500Z 53

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.950.00 1.00

-0.0

0.2

0.4

0.6

0.8

-0.2

1.0

time, usec

Vin

, V

Vout, V

Source matching

Load matching

EMLAB

22Impedance matching topologies

EMLAB

23

0,02

2

2

2

2

2

2

2

t

iLC

z

i

tLC

z

Frequency domain solution

0,0 22

22

2

2

LCIz

ILCV

z

V

dezItzi

dezVtz

tj

tj

),(2

1),(

),(2

1),(

zjzj

zjzj

eIeIzI

eVeVzV

),(

),(

VCjz

IILj

z

V

,

)()(),()( IC

LVI

C

LV

LCLC p

p

1,

β : propagation constant, vp : speed of light

dezVLC

dz

zVd

tLC

ztj),(

),(

2

1 22

2

2

2

2

2

EMLAB

24

]}[Re{

}Re{

)()(),(

zjzjtj

zjtjzjtj

eVeVe

eeVeeV

tzVtzVtz

Phasor representation

zjzj eVeVzV ),(

LZ+V-

SZI

V

I

V

0Z

tjSS eVtlz ),(

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

)cos(}Re{),( ztVeeVtzV zjtj

)cos(}Re{),( ztVeeVtzV zjtj

EMLAB

25Transmission line terminated with short, open

Zs = Zo

VreflV inc

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

In phase (0 ) for openo

Out of phase (180 ) for short

Vrefl

o

EMLAB

26Transmission Line Terminated with 25 Ω

Zs = Zo

ZL = 25 W

VreflV inc

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

3

1

5025

5025

EMLAB

27

)(][1

),(

)(),(

00

ljljljlj

ljljljlj

eeZ

VeVeV

ZlI

eeVeVeVlV

0

0

ZZ

ZZ

L

L

ljZZ

ljZZZ

ee

eeZ

lI

lVZ

L

Lljlj

ljlj

in

tan

tan

),(

),(

0

000

Equivalent input impedance

EMLAB

28Input impedance of short

ljZZ in tan0

EMLAB

29Input impedance of open

ljZZ in cot0

EMLAB

30Some transmission line examples

300

][513.2105.2

101002 18

6

m

rad][6.1288180)/(28.0 l

300tan

tan

0

000 ljZZ

ljZZZ

ee

eeZZ

L

Lljlj

ljlj

in

]W[5.1}1.030Re{2

1}Re{

2

1 6.16.1* eeVIPL

300

V][)6.1102cos(30}30Re{ 86.1102 8

teeV tL

A][)6.1102cos(1.0}1.0Re{ 86.1102 8

teeI tL

case 1) matched load

EMLAB

31

150

206466288tan150300

288tan3001500 j

j

jZZin

300

][150756.0206466300

60A

jI in

]W[333.1466)0756.0(2

1}Re{

2

1 22 inL ZIP

case 2) unmatched load

EMLAB

32Ringing : Time/frequency domain