ECE 598 JS Lecture 08 Lossy Transmission Linesjsa.ece.illinois.edu/ece598js/Lect_08.pdf · 2013-03-23 · skin o VDJD j Z jf IJww π μρ σδ + == = + 1 ρ σ = skin skin D R Xf

Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 1

ECE 598 JS Lecture ‐ 08Lossy Transmission Lines

Spring 2012

Jose E. Schutt-AineElectrical & Computer Engineering

University of [email protected]


RF SOURCE

Loss in Transmission Lines

Signal amplitude decreases with distance from the source.


δ

Low Frequency High Frequency Very High Frequency

Skin Effect in Lines


εr

H. A. Wheeler, "Formulas for the skin effect," Proc. IRE, vol. 30, pp. 412-424,1942

Skin Effect in Microstrip


/ /y jyoJ J e eδ δ− −=

/ /

0 1y jy o

oJ wI J we e dy

jδ δ δ∞

− −= =+∫

oo o o

JE J Eσσ

= ⇒ =

oo

J DV E Dσ

= =

Current density varies as

Note that the phase of the current density varies as a function of y

The voltage measured over a section of conductor of length D is:



( ) ( )11o

skino

jJ DV DZ j fI J w w

π μρσ δ

+= = = +

1ρσ

=

skin skinDR X fw

π μσ= =

The skin effect impedance is

where


is the bulk resistivity of the conductor

skin skin skinZ R jX= +

with

Skin effect has reactive (inductive) component


V I =RI+Lz t

∂ ∂−∂ ∂

I V= GV Cz t∂ ∂

− +∂ ∂

L

Δz

C

I

V

+

-

G

R

Telegraphers Equation: Time Domain

Lossy Transmission Line


−∂V∂z

= (R+ jωL)I = ZI

−∂I∂z

= (G+ jωC)V = YV

L

Δz

C

I

V

+

-

G

R

Telegraphers Equation: Frequency Domain



z

R, L, G, C,

forward wave

backward wave



( ) z j zV z Ae e− −= α β z j zBe e+ ++ α β

1( ) z j z

o

I z Ae eZ

− −⎡ ⎤= ⎣ ⎦α β z j zBe e+ +− α β

( )( )j R j L G j Cγ α β ω ω= + = + +( )( )o

R j LZ

G j Cωω

+=

+

z

Zo βZ1 Z2

Vsl

γ



‐ Signal attenuation

‐ Dispersion

‐ Rise time degradation-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Vol

ts

0 0.4 0.8 1.2 1.6 2Time (ns)

Far End Response

BoardVLSISubmicronDeep Submicron

( )( )( )( ) j R j L G j C= + = + +ωγ α ω β ω ω

Effects of Losses


l

Zin

R

C

R : series resistance per unit lengthC : shunt capacitance per unit length

in

coth (1 )2Z =

(1 )2

Rl CRl jR

Rl C jR

ω

ω

+

+

inarg(Z ) 45≈For very high ω,

RC Transmission Line


l

Zin

R

C

2

2 << RCl

ωIf then

in1 1Z + = +

2 2T

T

Rl RjCl jCω ω

=

RT = Rl : total resistanceCT = Cl : total capacitance



Line

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Vol

ts

0 0.4 0.8 1.2 1.6 2Time (ns)

Far End Response


-0.1

0.175

0.45

0.725

1

0 0.4

Vol

ts

0.8 1.2 1.6 2Time (ns)

Near End Response


Pulse Characteristics: rise time: 100 ps fall time: 100 ps pulse width: 4ns

Line Characteristics length : 3 mm near end termination: 50 Ω far end termination 65 Ω

LogicthresholdLogic

threshold



0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0 0.02 0.04 0.06 0.08 0.1

Category 5/ 100-meter

Simulation

Measurement

S11

Mag

nitu

de

Frequency (GHz)

-50

-40

-30

-20

-10

0

10

20

30

0 0.02 0.04 0.06 0.08 0.1


Simulation

Measurement

S11

Phas

e (d

eg)

Frequency (GHz)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.02 0.04 0.06 0.08 0.1


Simulation

Measurement

S21

Mag

nitu

de

Frequency (GHz)-200

-150

-100

-50

0

50

100

150

200

0 0.02 0.04 0.06 0.08 0.1


Simulation

Measurement

S21

Phas

e (d

eg)

Frequency (GHz)

100m Category‐5 Cable

Long Cable


0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.05 0.1 0.15 0.2

Category 5/ 1-meter

Simulation

Measurement

S11

Mag

nitu

de

Frequency (GHz)-200

-150

-100

-50

0

50

100

150

0 0.05 0.1 0.15 0.2

Category 5/ 1-meter

Simulation

Measurement

S11

Phas

e (d

eg)

Frequency (GHz)

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 0.05 0.1 0.15 0.2

Category 5/ 1-meter

Simulation

Measurement

S21

mag

nitu

de

Frequency (GHz)-200

-150

-100

-50

0

50

100

150

200

0 0.05 0.1 0.15 0.2

Category 5/ 1-meter

Simulation

Measurement

S21

phas

e (d

eg)

Frequency (GHz)

Short Cable1m Category‐5 Cable


0

1

2

3

4

5

6

0 0.02 0.04 0.06 0.08 0.1


Res

ista

nce

(Ohm

s/m

)

Frequency (GHz)

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.02 0.04 0.06 0.08 0.1


Vel

ocity

Rat

io

Frequency (GHz)

Resistance and velocity

Category 5 Cable


( ) * psR f R f=

*r ro rsv v v f= +

( ) ( )skin skinZ R f j L R j R Lω ω= + = + +

Cable Loss Model

Category 5 100 0.724 -0.165 15.38 0.482 0.224-Ga 100 0.678 1.157 29.03 0.593 0.1Category 3 100 0.705 11.06 12.31 0.473 0.01 SMA 50 0.700 0.113 7.94 0.415 0.2

Zo vro vrs Rs p fmax(Ω) (m/ns) (m/ns-GHz) (Ω/m-GHzp) (GHz)


Lossy TL Simulation

{ }( , ) z j z z j zv t z IFFT Ae e Be eα β α β− − + += +

1( , ) + z j z z j z

o

i t z IFFT Ae e Ae eZ

α β α β− − + +⎧ ⎫⎡ ⎤= ⎨ ⎬⎣ ⎦

⎩ ⎭

( )( )j R j L G j Cγ α β ω ω= + = + +( )( )o

R j LZ

G j Cωω

+=

+

• To simulate lossy TL with resistive loadsNo closed form solutionSimplest method is to use IFFT

21 2

( ) 1

sl

TVAe γ

ω−=

−Γ Γ2

2 lB e Aγ−= Γ1

o

o

ZTZ Z

=+

11

1

o

o

Z ZZ Z

=−

Γ+

22

2

= o

o

Z ZZ Z

−Γ

+


cableZs = 50 Ω

Vs

open

near end far end

Time‐Domain Simulations


-0.5

0

0.5

1

1.5

2

0 500 1000 1500 2000

22GA/Cu/4-cond Near End

volts

Time (ns)

-0.5

0

0.5

1

1.5

2

2.5

3

0 500 1000 1500 2000

22GA/Cu/4-cond Far End

volts

Time (ns)

Pulse Propagation (CAT‐5)


-0.5

0

0.5

1

1.5

2

0 500 1000 1500 2000 2500 3000 3500

MP/CM Shielded Near Endvo

lts

Time (ns)

-0.5

0

0.5

1

1.5

0 500 1000 1500 2000 2500 3000 3500

MP/CM Shielded Far End

volts

Time (ns)

Pulse Propagation (MP/CM)


-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 500 1000 1500 2000

RG174 Near End

volts

Time (ns)

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 500 1000 1500 2000

RG174

volts

Time (ns)

Pulse Propagation (RG174)

Documents

ECE 598 JS Lecture 08 Lossy Transmission Linesjsa.ece.illinois.edu/ece598js/Lect_08.pdf · 2013-03-23 · skin o VDJD j Z jf IJww π μρ σδ + == = + 1 ρ σ = skin skin D R Xf