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Penn ESE370 Fall DeHon 1 ESE370: Circuit-Level Modeling, Design, and Optimization for Digital Systems Day 24: October 27, 2014 Distributed RC Wire and Elmore Delay

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Previously: Equivalent RC Penn ESE370 Fall DeHon 2

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Previously: Chain without Inverters Penn ESE370 Fall DeHon 3

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Today Estimate delay in RC Network –Elmore delay calculation Wire Delay Apply to pass transistor circuits Penn ESE370 Fall DeHon 4

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Distributed RC Penn ESE370 Fall DeHon 5

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What is response? Penn ESE370 Fall DeHon 6

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SPICE Response Penn ESE370 Fall DeHon 7

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Intuition Look at series of R’s on path –Must move Q=V( C) across each R Not as much as if both R’s precede C’s Penn ESE370 Fall DeHon 8

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Elmore Delay For each resistor Ri in path –Compute C Ri = sum of all C’s downstream of Ri –Delay through Ri is Ri×C Ri Penn ESE370 Fall DeHon 9

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Superposition Penn ESE370 Fall DeHon 10

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Superposition Penn ESE370 Fall DeHon 11 R1 C1 R1 R2 C2 R1 R2 C1 C2

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Superposition Penn ESE370 Fall DeHon 12 R1 C1 R1 R2 C2 R1 R2 C1 C2 R1*C1(R1+R2)*C2 R1*(C1+C2)+R2*C2

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Superposition not concurrent Don’t happen concurrently since must divide current Penn ESE370 Fall DeHon 13

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Superposition For R1=R2=R, C1=C2=C –Delay = 3RC Penn ESE370 Fall DeHon 14 R1 R2 C1 C2 R1*(C1+C2)+R2*C2

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Compare KCL: Setup Penn ESE370 Fall DeHon 15 R1 R2 C1 C2 R1*(C1+C2)+R2*C2 Equations from KCL? (V1-V2)/R1 = (V2-V3)/R2 + C1(dV2/dt) (V2-V3)/R2=C2(dV3/dt) V2 V3

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Rearrange Penn ESE370 Fall DeHon 16 R1 R2 C1 C2 R1*(C1+C2)+R2*C2 (V1-V2)/R1 = (V2-V3)/R2 + C1(dV2/dt) V1/R1=V2/R1+V2/R2-V3/R2+C1(dV2/dt) (V2-V3)/R2=C2(dV3/dt) V2=V3+R2*C2(dV3/dt) V2 V3 ALGEBRA (will skip)

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Substitute Penn ESE370 Fall DeHon 17 R1 R2 C1 C2 R1*(C1+C2)+R2*C2 V1/R1=V2/R1+V2/R2-V3/R2+C1(dV2/dt) = V2(1/R1+1/R2)-V3/R2+C1(dV2/dt) V2=V3+R2*C2(dV3/dt) V1/R1=(V3+R2*C2(dV3/dt))(1/R1+1/R2)- V3/R2+C1(dV3/dt+R2*C2*d 2 V3/dt) V2 3 ALGEBRA (will skip)

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Multiply by R1 Penn ESE370 Fall DeHon 18 R1 R2 C1 C2 R1*(C1+C2)+R2*C2 V1/R1=(V3+R2*C2(dV3/dt))(1/R1+1/R2) -V3/R2+C1(dV3/dt+R2*C2*d 2 V3/dt) V1=V3+(R1*C2+R2*C2+R1*C1)(dV3/dt )+R2*C2*R1*C1*(d 2 V3/dt) V2 V3 ALGEBRA (will skip)

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Simplify Problem Penn ESE370 Fall DeHon 19 R1 R2 C1 C2 R1*(C1+C2)+R2*C2 V1=V3+(R1*C2+R2*C2+R1*C1)(dV3/dt )+R2*C2*R1*C1*(d 2 V3/dt) Simplify R1=R2=R, C1=C2=C V1=V3+(3RC)dV3/dt+(RC) 2 dV3/dt V2 V3

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Exponential Solution Penn ESE370 Fall DeHon 20 R1 R2 C1 C2 R1*(C1+C2)+R2*C2 V1=V3+(3RC)dV3/dt+(RC) 2 dV3/dt V3=A(1+e xt ) V1=A+Ae xt +(3RC)x Ae xt +(RC) 2 x 2 Ae xt A=V1 0=1+(3RC)x+(RC)x 2 V2 V3

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KCL-Based Solution Penn ESE370 Fall DeHon 21 R1 R2 C1 C2 R1*(C1+C2)+R2*C2 V3=A(1+e xt ) 0=1+(3RC)x+(RC)x 2 Quadratic Equation: V2 V3

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SPICE Response Penn ESE370 Fall DeHon 22

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Elmore Delay For each resistor Ri in path –Compute C Ri = sum of all C’s downstream of Ri –Delay through Ri is Ri×C Ri Penn ESE370 Fall DeHon 23

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Apply Y What is Elmore delay? Penn ESE370 Fall DeHon 24

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Apply to Y Imagine shorting A and B Penn ESE370 Fall DeHon 25 A B

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Apply Y 1000 ×3pF ×1pF =4ns Penn ESE370 Fall DeHon 26

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SPICE Response Penn ESE370 Fall DeHon 27

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Elmore Delay For each resistor Ri in path –Compute C Ri = sum of all C’s downstream of Ri –Delay through Ri is Ri×C Ri Penn ESE370 Fall DeHon 28

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Wire Penn ESE370 Fall DeHon 29

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Wire Capacitance Penn ESE370 Fall DeHon 30

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Wire Resistance Penn ESE370 Fall DeHon 31

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Wire as RC Ladder Penn ESE370 Fall DeHon 32

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Wire Delay as f(L) Measure wire length in units –Say –Each lambda have Cunit, Runit Capacitance and resistance of wire of length Penn ESE370 Fall DeHon 33

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Wire Delay Delay of Wire N units long: Runit*(N*Cunit) +Runit((N-1)*Cunit +Runit*(N-2)*Cunit+… +Runit*Cunit =(Runit*Cunit)*(N+N-1+N-2+….1) Penn ESE370 Fall DeHon 34

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Sum of integers What’s the sum of the integer 1 to N? N+N-1+N-2+…1 Penn ESE370 Fall DeHon 35

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Sum of integers What’s the sum of the integer 1 to N? N+N-1+N-2+…1 Penn ESE370 Fall DeHon 36

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Wire Delay Wire N units long: Runit*(N*Cunit)+Runit((N-1)*Cunit +Runit*(N-2)*Cunit+…+Runit*Cunit =(Runit*Cunit)*(N+N-1+N-2+….1) =Runit*Cunit*N 2 /2 Penn ESE370 Fall DeHon 37

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Lumped RC Wire? What would the delay be if we treated the wire as lumped R and C? Penn ESE370 Fall DeHon 38 Rwire = N×RunitCwire = N×Cunit

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Wire Delay Rwire = N*Runit Cwire=N*Cunit Wire delay = Runit*Cunit*N 2 /2 Wire delay = 0.5 * Rwire*Cwire Half the delay of lumped RC product Quadratic in length of wire Penn ESE370 Fall DeHon 39

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Wire Delay Wire N units long: =Runit*Cunit*N 2 /2 With –Runit=1000 –Cunit=1pF Penn ESE370 Fall DeHon 40

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RC Ladder Penn ESE370 Fall DeHon 41 Runit=1000 Cunit=1pF

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Animation estle/Animations/DAC01top.html Penn ESE370 Fall DeHon 42

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Apply to Pass Transistor (and CMOS) Penn ESE370 Fall DeHon 43

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Pass TR xor Delay when B=1? Penn ESE370 Fall DeHon 44

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Pass transistor xor Delay when B=0? –Start with equivalent RC circuit Penn ESE370 Fall DeHon 45

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Unbuffered Circuit Delay? Penn ESE370 Fall DeHon 46

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Unbuffered Circuit Delay? Penn ESE370 Fall DeHon 47

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Unbuffered Delay as a function of number of stages? Penn ESE370 Fall DeHon 48

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CMOS xor Delay with Cdiff>0? Penn ESE370 Fall DeHon 49

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Idea Lumped wiring calculation is pessimistic –Not all capacitance at end of wire Elmore delay calculation allows us to estimate Wires are distributed RC –Half delay lumped calculation –Still quadratic in length Penn ESE370 Fall DeHon 50

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Admin Project –Should have feedback from milestone on canvas –Due Thursday Andre office hour start 4:30pm this Tue. 2 nd midterm next Monday (11/3) Penn ESE370 Fall DeHon 51

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