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ESE370: Circuit-Level Modeling, Design, and Optimization for Digital Systems

Lec 32: Deceember 2, 2020Transmission Line

Penn ESE 370 Fall 2020 – Khanna

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Next few Lectures

! Saw in action in lab ! Where transmission lines arise?! General wire formulation! Lossless Transmission Line! End of Transmission Line?! Termination! Discuss Lossy! Implications

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Where Transmission Lines Arise

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Transmission Lines

! Cable: coaxial! PCB

" Strip line" Microstrip line

! Twisted Pair (Cat5)

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Transmission Lines

! How did the traces behave in lab?! How does this differ from

" Ideal equipotential?" RC-wire on chip?

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Transmission Lines

! This is what long wires/cables look like" Aren’t an ideal equipotential" Signals take time to propagate" Maintain shape of input signal

" Within limits

" Shape and topology of wiring effects how signals propagate

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Transmission Lines

! Need theory/model to support design" Reason about behavior" Understand what can cause noise" Engineer high performance/speed communication

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Wire Formulation

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From the Way Way Back

! Problem: A long cable – the trans-atlantic telephone cable –is laid out connecting NY to London. We would like analyze the electrical properties of this cable.

! For simplicity, assume the cable has a uniform cross-sectional configuration (shown as two wires here)

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Trans-Atlantic Cable

! Can we do it with circuit theory? ! Fundamental problem with circuit theory is that it

assumes that the speed of light is infinite. So all signals are in phase: V(z) = V(z + ℓ)

! Consequently, all variations in space are ignored: ∂/∂z → 0

! This allows the lumped circuit approximation

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Lumped Circuit Properties of Cable

! Shorted Line: The long loop has inductance since the magnetic flux ψ is not negligible (long cable) (ψ = LI)

! Open Line: The cable also has substantial capacitance (Q = CV)

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Wires

! In general, our “wires” have distributed R, L, C components

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RC Wire (Preclass 1)

! When R dominates L" We have the distributed RC Wires " Typical of on-chip wires in ICs" What is RC response to step?

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Transmission Line

! When resistance is negligible " Have LC wire = Lossless Transmission Line

" No energy dissipation (loss) through R’s

" More typical of printed circuit board (PCB) wires and bond wires

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Build Intuition from LC (Preclass 2)

! What did one LC do?! What will chain do?

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Build Intuition from LC (Preclass 2)

! What did one LC do?! What will chain do?

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Intuitive: Lossless

! Pulses travel as waves without distortion" (up to a characteristic frequency)

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SPICE Simulation

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Step Response SPICE

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Pulse Response SPICE

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Contrast RC Wire

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Contrast

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Model

! Now voltage is a function of time and position" Position along wire – distance from source

! Want to get V(x,t)" And I(x,t)

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Setup Relations (Preclass 4)

! i is a node, x is the distance from source, Δx is distance between nodes

! Position along wire: x=i×Δx! So Vi(t) = V(x=iΔx, t)

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Vi Vi+1Vi-1Ii Ii+1

Ici

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Setup Relations

! KCL @ Vi: Ii-Ii+1=! Ici=! Vi-Vi-1 =

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Vi Vi+1Vi-1Ii Ii+1

Ici

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Setup Relations

! KCL @ Vi: Ii-Ii+1=! Ici=! Vi-Vi-1 =

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Vi Vi+1Vi-1Ii Ii+1

Ici

Ii − Ii+1 =CdVidt

Vi −Vi−1 = −LdIidt

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Setup Relations

! KCL @ Vi: Ii-Ii+1=! Ici=! Vi-Vi-1 =

! i is a node, but has spatial dimension along the Tline" No longer agnostic of wire length

! Vi changes at different positions

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Vi Vi+1Vi-1Ii Ii+1

Ici

Ii − Ii+1 =CdVidt

Vi −Vi−1 = −LdIidt

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Setup Relations

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Vi Vi+1Vi-1Ii Ii+1

Ici

Ii − Ii+1 =CdVidt

⇒−dIidx

=CdVidt

Vi −Vi−1 = −LdIidt

⇒dVidx

= −LdIidt

Penn ESE 370 Fall 2020 – Khanna

! KCL @ Vi: Ii-Ii+1=! Ici=! Vi-Vi-1 =

! i is a node, but has spatial dimension along the Tline" No longer agnostic of wire length

! Vi changes at different positions

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Reduce to Single Equation

! Eliminate I? Differential equation in voltage only.

! Take derivative with respect to time

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dIidt−dIi+1dt

=C d2Vidt2

Ii − Ii+1 =CdVidt

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Reduce to Single Equation

! Eliminate Is ?

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dIidt−dIi+1dt

=C d2Vidt2

dVidx

= −LdIidt

dVi+1dx

= −LdIi+1dt

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Reduce to Single Equation

! Eliminate Is ?

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dIidt−dIi+1dt

=C d2Vidt2

dVidx

= −LdIidt

dVi+1dx

= −LdIi+1dt

dIidt

−dIi+1dt

=Cd 2Vidt2

−1LdVidx

− −1LdVi+1dx

⎛

⎝⎜

⎞

⎠⎟=C

d 2Vidt2

dVi+1dx

−dVidx

= LCd 2Vidt2

d 2Vidx2

= LCd 2Vidt2

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Implication

" Wave equation:

" Solution:

" What is w?

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∂ 2V∂x2

= LC ∂2V∂t 2

V (x, t) = A+Bex−wt

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Implication

" Wave equation:

" Solution:

" What is w?

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V (x, t) = A+Bex−wt

Bex−wt = LCw2Bex−wt

w = 1LC

w is the rate of propagation

Penn ESE 370 Fall 2020 – Khanna

∂ 2V∂x2

= LC ∂2V∂t 2

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Propagation Rate in Example

! L=1uH! C=1pF! What is w?

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€

w = 1LC

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Signal Propagation

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Signal Propagation

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Delay linear in length

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Contrast RC Wire

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RC wire delay quadratic in length

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Propagation

" Solution:

" Rate of propagation, w:

" Previously:

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V (x, t) = A+Bex−wt

w = 1LC

CL = εµ

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Propagation

" Solution:

" Rate of propagation, w:

" Previously:

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V (x, t) = A+Bex−wt

w = 1LC

CL = εµ

w = 1εµ

=1

ε0εrµ0µr

c2 = 1ε0µ0

⇒

w = cεrµr

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Longer LC (open)

! 40 Stages! L=100nH! C=1pF

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€

w = 1LC

=c0εrµr

Penn ESE 370 Fall 2020 – Khanna

Stage delay? How long to propagate?

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Reflection Teaser: Pulse Travel RC

! V1,V3,V4,V5,V6 about 10 stages apart

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Wire “Resistance”

• What is the resistance at Vi ?" Q needed to charge C to Vi?" Ii given velocity w? " R = Vi/Ii?

Vi Vi+1Vi-1Ii Ii+1

Ici

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Wire “Resistance”

• Q=CVi• Ii = dQ/dt• Moving at rate w• Ii=wCVi• R=Vi/Ii=1/(wC)

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€

w = 1LC

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R = LCC

Vi Vi+1Vi-1Ii Ii+1

Ici

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Characteristic Impedance

! Z0 = R

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R = LCC

=LC= Z0

Vi Vi+1Vi-1Ii Ii+1

Ici

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Impedance

! Assuming infinitely long wire, how does the impedance look at Vi, Vi+1, Vi+2 ?

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Vi Vi+1Vi-1Ii Ii+1

Ici

Penn ESE 370 Fall 2020 – Khanna

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Impedance

! Assuming infinitely long wire, how does the impedance look at Vi, Vi+1, Vi+2 ?

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Z0 =LC

Vi Vi+1Vi-1Ii Ii+1

Ici

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Impedance

! Transmission lines have a characteristic impedance

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Z0 =LC

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Infinite Lossless Transmission Line

! Transmission line looks likeresistive load

! Input waveform travels down line at velocity" Without distortion

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Z0

€

Z0 =LC

€

w = 1LC

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Idea

! Signal propagates as wave down transmission line" Delay linear in wire length, if resistance negligible " Rate of propagation" Characteristic impedance

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w = 1LC

=cεrµr

Penn ESE 370 Fall 2020 – Khanna

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Z0 =LC

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Admin

! Project 2 due Friday 12/4! HW 7 out on Friday

" Due 12/10

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