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

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

Lec 21: October 27, 2021

Distributed RC Wire and Elmore Delay

Penn ESE 370 Fall 2021 - Khanna

Today

!

Estimate delay in RC Network

"

Elmore delay calculation

!

Apply to wire delay

!

Apply to pass transistor circuits

2 Penn ESE 370 Fall 2021 - Khanna

Previously: Equivalent RC

3 Penn ESE 370 Fall 2021 - Khanna

2C

+3C

2C

+2C

Previously: Distributed RC (preclass 1)

!

What are the time constants?

Penn ESE 370 Fall 2021 - Khanna 4

1 2 3 4 5 6 7 8 9 10 (ns)

Elmore Delay: Distributed RC network

!

The delay from source s to node 5

"

N = number of nodes in circuit

5 Penn ESE 370 Fall 2021 – Khanna

5 C₅

𝑅

= # 𝑅

⇒ (𝑅

∈ 𝑝𝑎𝑡ℎ 𝑠 → 5 ∩ 𝑝𝑎𝑡ℎ 𝑠 → 𝑘 )

𝜏

= #

"%&

'

𝐶

𝑅

What is response?

6 Penn ESE 370 Fall 2021 - Khanna

y

P

Q P

Q

Elmore Delay: Practice (preclass 1)

!

The delay from source s to node i

"

N = number of nodes in circuit

7

R

= ∑ R

^{⇒ (R}

∈ [ path(s → i)∩ path(s → k)])

Penn ESE 370 Fall 2021 - Khanna

τ _Di = C _k R _ik

k=1 N

∑

Apply (preclass 1)

!

What is Elmore delay?

"

S # Z

8 Penn ESE 370 Fall 2021 - Khanna

/z

y z

C1 C2

C3

s

SPICE Response

9 Penn ESE 370 Fall 2021 - Khanna

1 2 3 4 5 6 7 8 9 10 (ns)

C1 C2

C3

Elmore Delay: Special Ladder Case

!

For each resistor C _k in path

"

Compute R

= sum of all Rs upstream of C

10

τ _DN = C _k

k=1 N

∑ ^{R j}

j=1 k

∑ ^{= C k}

k=1 N

∑ ^{R kk}

Penn ESE 370 Fall 2021 - Khanna

Elmore Delay: Special Ladder Case

!

For each resistor C _k in path

"

Compute R

= sum of all Rs upstream of C

11 Penn ESE 370 Fall 2021 - Khanna

C

*(R

)+C

*(R

+R

)

τ _DN = C _k

k=1 N

∑ ^{R j}

j=1 k

∑ ^{= C k}

k=1 N

∑ ^{R kk}

Elmore Delay: Special Ladder Case

!

For each resistor C _k in path

"

Compute R

= sum of all Rs upstream of C

12 Penn ESE 370 Fall 2021 - Khanna

C

*(R

)+C

*(R

+R

)

=3RC

=3ns

τ _DN = C _k

k=1 N

∑ ^{R j}

j=1 k

∑ ^{= C k}

k=1 N

∑ ^{R kk}

!

Equations from KCL?

13

R1 R2

C1 C2

Compare KCL: Setup

V

−V

R

= V

−V

R

+ C

dV

dt V

−V

R

= C

dV

dt

V

V

V

Penn ESE 370 Fall 2021 - Khanna

@V ₁ :

@V ₂ :

Penn ESE 370 Fall 2021 - Khanna 14

V

−V

R

= V

−V

R

+ C

dV

dt V

R

= V

R

+ V

R

− V

R

+ C

dV

dt V

= V

1+ R

R

"

# $ %

&

' − R

V

R

+ R

C

dV

dt

V

−V

R

= C

dV

dt V

R

= V

R

+ C

dV

dt V

= V

+ R

C

dV

dt dV

dt = dV

dt + R

C

d

V

dt

Compare KCL: Math

@V ₁ : @V ₂ :

Penn ESE 370 Fall 2021 - Khanna 15

V

= V

1+ R

R

!

"

# $

% & − R

R

V

+ R

C

dV

dt V

= V

+ R

C

dV

dt

!

"

# $

% & 1+ R

R

!

"

# $

% & − R

R

V

+ R

C

dV

dt + R

C

d

V

dt

!

"

# $

% &

V

= V

+ R

C

1+ R

R

!

"

# $

% & + R

C

!

"

## $

% && dV

dt + R

C

R

C

d

V

dt

V

= V

+ (R

C

+ R

C

+ R

C

) dV

dt + R

C

R

C

d

V

dt

Compare KCL: Math

Penn ESE 370 Fall 2021 - Khanna 16

V

= V

+ (R

C

+ R

C

+ R

C

) dV

dt + R

C

R

C

d

V

dt

V

= V

+ 3RC dV

dt + R

C

d

V

dt

R

= R

= R C

= C

= C

Compare KCL: Math

Penn ESE 370 Fall 2021 - Khanna 17

V

= V

+ (R

C

+ R

C

+ R

C

) dV

dt + R

C

R

C

d

V

dt

V

= V

+ 3RC dV

dt + R

C

d

V

dt

V

= A(1+ e

) →

V

= A(1+ e

) − 3RC ⋅ α ^Ae

+ R

C

⋅ α

^Ae

R

= R

= R

C

= C

= C

Compare KCL: Math

Penn ESE 370 Fall 2021 - Khanna 18

V

= A(1+ e

) →

V

= A(1+ e

) − 3RC ⋅ α ^Ae

+ R

C

⋅ α

^Ae

t = ∞ → V

= V

= A

V

= V

(1+ e

) − 3RC ⋅ α ^V

^e

+ R

C

⋅ α

^V

^e

0 = e

− 3RC ⋅ α ^e

+ R

C

⋅ α

^e

0 = e

(1− 3RC α + R

C

⋅ α

⁾ 0 = 1− 3RC α + R

C

⋅ α

Compare KCL: Math

Penn ESE 370 Fall 2021 - Khanna 19

V

= A(1+ e

) →

V

= A(1+ e

) − 3RC ⋅ α ^Ae

+ R

C

⋅ α

^Ae

t = ∞ → V

= V

= A

V

= V

(1+ e

) − 3RC ⋅ α ^V

^e

+ R

C

⋅ α

^V

^e

0 = e

− 3RC ⋅ α ^e

+ R

C

⋅ α

^e

0 = e

(1− 3RC α + R

C

⋅ α

⁾ 0 = 1− 3RC α + R

C

⋅ α

Compare KCL: Math

Penn ESE 370 Fall 2021 - Khanna 20

V

= V

(1+ e

)

0 = 1− 3RC α + R

C

⋅ α

α = 3RC ± 9(RC)

− 4(RC)

2(RC)

= 3RC ± 5RC

2(RC)

= 3± 5 2RC

→ τ = 2

3± 5 RC ≈ 2.6RC

Compare KCL: Math

Penn ESE 370 Fall 2021 - Khanna 21

V

= V

(1+ e

)

0 = 1− 3RC α + R

C

⋅ α

α = 3RC ± 9(RC)

− 4(RC)

2(RC)

= 3RC ± 5RC

2(RC)

= 3± 5 2RC

→ τ = 2

3± 5 RC ≈ 2.6RC

Compare KCL: Math

=3RC

=3ns

Elmore Delay: Distributed RC network

!

The delay from source to node i

"

N = number of nodes in circuit

!

Special ladder case

22

R

= ∑ R

^{⇒ (R}

∈ [ path(s → i)∩ path(s → k)])

Penn ESE 370 Fall 2021 - Khanna

τ _Di = C _k R _ik

k=1 N

∑

τ _DN = C _k

k=1 N

∑ ^{R j}

j=1 k

∑ ^{= C k}

k=1 N

∑ ^{R kk}

Wire Delay

Penn ESE 370 Fall 2021 - Khanna 23

Wire Resistance

100um

1um

1um A

B

R = ρ ^L A

R = 10

Ω⋅ m ⋅100 µ ^m

1 µ m ⋅1 µ ^{m = 10Ω}

Penn ESE370 Fall2021 – Khanna 24

Wire Capacitance

4 m

0.2 mm

0.1 mm A

B

0.1 mm

metal plate

dielectric material

C = ε _d ^A d

Penn ESE370 Fall2021 – Khanna 25

Wire as Distributed RC Ladder

!

Measure wire length in units

"

Say l

"

Each l unit has C

, R

"

Unit capacitance and resistance of wire of length l

26 Penn ESE 370 Fall 2021 - Khanna

Wire Delay

!

Delay of Wire N units long:

27 Penn ESE 370 Fall 2021 - Khanna

Wire Delay

!

Delay of Wire N units long:

R

C

28 Penn ESE 370 Fall 2021 - Khanna

Wire Delay

!

Delay of Wire N units long:

R

C

+ 2R

C

29 Penn ESE 370 Fall 2021 - Khanna

Wire Delay

!

Delay of Wire N units long:

R

C

+ 2R

C

+ 3R

*C

+ … +NR

*C

=(R

*C

)*(N+(N-1)+(N-2)+….1)

30 Penn ESE 370 Fall 2021 - Khanna

Sum of integers (preclass 2)

!

What’s the sum of the integer 1 to N?

31

k

k=0 N

∑ ⁼

Penn ESE 370 Fall 2021 - Khanna

Sum of integers

!

What’s the sum of the integer 1 to N?

32

k

k=0 N

∑ ^{= N(N +1)} ₂ ^{≈ N}

2

2

Penn ESE 370 Fall 2021 - Khanna

Wire Delay (preclass 3)

!

Delay of Wire N units long:

R

*(N*C

)

+ R

((N-1)*C

+ R

*(N-2)*C

+ … +R

*C

=(R

*C

)*(N+N-1+N-2+….1)

~=R

C

*N

/2

33 Penn ESE 370 Fall 2021 - Khanna

Lumped RC Wire? (preclass 4)

!

What would the delay be if we treated the wire as lumped R and C?

34

R

= N×R

C

= N×C

Penn ESE 370 Fall 2021 - Khanna

Wire Delay

!

R _wire = N*R _unit

!

C _wire =N*C _unit

!

Lumped RC wire delay = R _unit *C _unit *N ²

!

Distributed RC Wire delay = R _unit *C _unit *N ² /2

!

Distributed has half the delay of lumped RC product

!

Delay is quadratic in length of wire in both cases

35 Penn ESE 370 Fall 2021 - Khanna

Apply to Gates

Pass Transistor and CMOS

Penn ESE 370 Fall 2021 - Khanna 36

Pass Transistor XOR

!

Delay when A=B=1?

37 Penn ESE 370 Fall 2021 - Khanna

2C

+3C

2C

+2C

Pass Transistor XOR (preclass 5)

!

Delay when A=1, B=0?

"

Start with equivalent

RC circuit

38 Penn ESE 370 Fall 2021 - Khanna

Unbuffered (preclass 6)

!

Circuit # Delay?

"

2 stages

39 Penn ESE 370 Fall 2021 - Khanna

Unbuffered (preclass 7)

!

Circuit # Delay?

"

3 stages

40 Penn ESE 370 Fall 2021 - Khanna

Unbuffered (preclass 8)

!

Delay as a function of number of stages, k?

41 Penn ESE 370 Fall 2021 - Khanna

…

CMOS XOR (preclass 9)

!

Delay with Cdiff>0?

42 Penn ESE 370 Fall 2021 - Khanna

Idea

!

Elmore delay calculation allows us to estimate delay for distributed RC network

"

Necessary for pass transistors

!

Wires are distributed RC

"

Half delay lumped calculation

"

Still quadratic in length

43 Penn ESE 370 Fall 2021 - Khanna

Admin

!

Project 1

"

Final report due Friday 10/29 midnight

44 Penn ESE 370 Fall 2021 - Khanna