Sam PalermoAnalog & Mixed-Signal Center

Texas A&M University

ECEN689: Special Topics in High-Speed Links Circuits and Systems

Spring 2011

Lecture 3: Transmission Lines

Announcements

• HW1 due 1/28• One page summary of recent link design paper

• Lecture Reference Material• Dally, Chapter 3.1 – 3.4

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Agenda

• Transmission Lines• Propagation constant• Characteristic impedance• Loss• Reflections• Termination examples• Differential transmission lines

3

Wire Models

• Model Types• Ideal• Lumped C, R, L• RC transmission line• LC transmission line• RLGC transmission line

• Condition for LC or RLGC model (vs RC)

4

LRfπ20 ≥

Wire R L C >f (LC wire)

AWG24 Twisted Pair 0.08Ω/m 400nH/m 40pF/m 32kHz

PCB Trace 5Ω/m 300nH/m 100pF/m 2.7MHz

On-Chip Min. Width M6 (0.18µm CMOS node) 40kΩ/m 4µH/m 300pF/m 1.6GHz

RLGC Transmission Line Model

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( ) ( ) ( )t

txILtxRIx

txV∂

∂−−=

∂∂ ,,,

( ) ( ) ( )t

txVCtxGVx

txI∂

∂−−=

∂∂ ,,,

0 dx As →(1)

(2)

General Transmission Line Equations

Time-Harmonic Transmission Line Eqs.

• Assuming a traveling sinusoidal wave with angular frequency, ω

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( ) ( ) ( )xILjRdx

xdV ω+−=

( ) ( ) ( )xVCjGdx

xdI ω+−=

• Differentiating (3) and plugging in (4) (and vice versa)

( ) ( )xVdx

xVd 22

2

γ=

( ) ( )xIdx

xId 22

2

γ=

• where γ is the propagation constant

( )( ) ( )-1m CjGLjRj ωωβαγ ++=+=

(5)

(6)

Time-Harmonic Transmission Line Equations

(3)

(4)

Transmission Line Propagation Constant

• Solutions to the Time-Harmonic Line Equations:

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( ) ( ) ( ) xr

xfrf eVeVxVxVxV γγ

00 +=+= −

• What does the propagation constant tell us?• Real part (α) determines attenuation/distance (Np/m)• Imaginary part (β) determines phase shift/distance (rad/m)• Signal phase velocity

( ) ( ) ( ) xr

xfrf eIeIxIxIxI γγ

00 +=+= −

where ( )( ) ( )-1m CjGLjRj ωωβαγ ++=+=

(m/s) βωυ =

Transmission Line Impedance, Z0

• For an infinitely long line, the voltage/current ratio is Z0

• From time-harmonic transmission line eqs. (3) and (4)

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( )( ) ( )Ω

++

== 0 CjGLjR

xIxVZ

ωω

• Driving a line terminated by Z0 is the same as driving an infinitely long line

[Dally]

Lossless LC Transmission Lines

• If Rdx=Gdx=0

9

LC

LCjj

ωβ

αωβαγ

=

==+=

0

CLZ

LC

=

==

0

1βωυ

No Loss!

• Waves propagate w/o distortion• Velocity and impedance

independent of frequency• Impedance is purely real

[Johnson]

Low-Loss LRC Transmission Lines

• If R/ωL and G/ωC << 1

• Behave similar to ideal LC transmission line, but …• Experience resistive and

dielectric loss• Frequency dependent

propagation velocity results in dispersion

• Fast step, followed by slow DC tail

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( )( )

βαα

ωωω

ωω

ωωβαγ

j

CG

LRLCjGZ

ZR

LCGLRCjLCj

CjGLjRj

DR ++=

+

+++≅

+−≅

++=+=

220

0

21

81

811

22

1

2

2

0

0

GZZR

D

R

≅

≅

α

α

+

+≅

22

81

811

CG

LRLC

ωωωβ

122

81

811

−

+

+≅

CG

LRLC

ωωυ

Resistive Loss

Dielectric Loss

Skin Effect (Resistive Loss)

• High-frequency current density falls off exponentially from conductor surface

• Skin depth, δ, is where current falls by e-1 relative to full conductor• Decreases proportional to

sqrt(frequency)

• Relevant at critical frequency fswhere skin depth equals half conductor height (or radius)• Above fs resistance/loss increases

proportional to sqrt(frequency)

11

δd

eJ−

= ( ) 21

−= µσπδ f

2

2

=h

fs

πµ

ρ

( )21

=

sDC f

fRfR

21

02

=

s

DCR f

fZ

Rα

For rectangular conductor:

[Dally]

Skin Effect (Resistive Loss)

12

[Dally]

MHzfmR sDC 43 ,7 =Ω=

5-mil Stripguide

kHzfmR sDC 67 ,08.0 =Ω=

30 AWG Pair

21

02

=

s

DCR f

fZ

Rα

Dielectric Absorption (Loss)

• An alternating electric field causes dielectric atoms to rotate and absorb signal energy in the form of heat

• Dielectric loss is expressed in terms of the loss tangent

• Loss increases directly proportional to frequency

13

CG

D ωδ =tan

LCf

CLfCGZ

D

DD

δπ

δπα

tan2

tan22

0

=

==

[Dally]

Total Wire Loss

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[Dally]

Reflections & Telegrapher’s Eq.

15

T

iT ZZ

VI+

=0

2

+−

=

+−=

−=

0

0

0

00

2

ZZZZ

ZVI

ZZV

ZVI

III

T

Tir

T

iir

Tfr

0

0

ZZZZ

VV

IIk

T

T

i

r

i

rr +

−===

Termination Current:

• With a Thevenin-equivalent mode of the line:

• KCL at Termination:Telegrapher’s Equation or Reflection Coefficient

[Dally]

Termination Examples - Ideal

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RS = 50ΩZ0 = 50Ω, td = 1nsRT = 50Ω

050505050

050505050

5.05050

501

=+−

=

=+−

=

=

+=

rS

rT

i

k

k

VVV

in (step begins at 1ns)

source

termination

Termination Examples - Open

17

RS = 50ΩZ0 = 50Ω, td = 1nsRT ~ ∞ (1MΩ)

050505050

15050

5.05050

501

=+−

=

+=+∞−∞

=

=

+=

rS

rT

i

k

k

VVV

in (step begins at 1ns)

source

termination

Termination Examples - Short

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RS = 50ΩZ0 = 50Ω, td = 1nsRT = 0Ω

050505050

1500500

5.05050

501

=+−

=

−=+−

=

=

+=

rS

rT

i

k

k

VVV

in (step begins at 1ns)

source

termination

Arbitrary Termination Example

19

RS = 400ΩZ0 = 50Ω, td = 1nsRT = 600Ω

778.05040050400

846.05060050600

111.050400

501

=+−

=

=+−

=

=

+=

rS

rT

i

k

k

VVV

in (step begins at 1ns)

sourcetermination

0.111V0.205V

0.278V0.340

Lattice Diagram

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RS = 400Ω

RT = 600ΩZ0 = 50Ω, td = 1ns

in (step begins at 1ns)

Rings up to 0.6V(DC voltage division)

Termination Reflection Patterns

21

RS = 25Ω, RT = 25Ω

krS & krT < 0

Voltages Converge

RS = 25Ω, RT = 100Ω

krS < 0 & krT > 0

Voltages Oscillate

RS = 100Ω, RT = 25Ω

krS > 0 & krT < 0

Voltages Oscillate

RS = 100Ω, RT = 100Ω

krS > 0 & krT > 0

Voltages Ring Up

source

termination

sourcetermination

source

termination

source

termination

Termination Schemes

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• No Termination• Little to absorb line energy• Can generate oscillating

waveform• Line must be very short

relative to signal transition time• n = 4 - 6

• Limited off-chip use

• Source Termination• Source output takes 2 steps up• Used in moderate speed point-

to-point connections

LCnlnTt triproundr 2=> −

LClt porch 2≅

Termination Schemes

23

• Receiver Termination• No reflection from receiver• Watch out for intermediate

impedance discontinuities• Little to absorb reflections at driver

• Double Termination• Best configuration for min

reflections• Reflections absorbed at both driver

and receiver

• Get half the swing relative to single termination

• Most common termination scheme for high performance serial links

Differential Transmission Lines

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• Differential signaling advantages• Self-referenced • Common-mode noise rejection• Increased signal swing• Reduced self-induced power-

supply noise

• Requires 2x the number of signaling pins relative to single-ended signaling• But, smaller ratio of

supply/signal (return) pins• Total pin overhead is typically

1.3-1.8x (vs 2x)

[Hall]

• Even mode• When equal voltages drive both

lines, only one mode propagates called even more

• Odd mode• When equal in magnitude, but out

of phase, voltages drive both lines, only one mode propagates called odd mode

Balanced Transmission Lines

• Even (common) mode excitation• Effective C = CC

• Effective L = L + M

• Odd (differential) mode excitation• Effective C = CC + 2Cd

• Effective L = L – M

25

21

21

2

+−

=

+=

dcodd

ceven

CCMLZ

CMLZ

[Dally]

2 ,2 even

CModdDIFFZZZZ ==

PI-Termination

26

1RZeven =

2||2|| 221 RZRRZ evenodd ==

−

=oddeven

evenodd

ZZZZR 22

T-Termination

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12 2RRZeven +=

( )oddeven

odd

ZZR

RZ

−=

=

21

1

2

Next Time

• Channel modeling• Time domain reflectometer (TDR)• Network analysis

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