and optimization for digital systemsese370/fall2020/handouts/lec... · 2020. 10. 30. · ese370:...

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

Lec 23: October 30, 2020Driving Large Capacitive Loads

Penn ESE 370 Fall 2020 - Khanna

Today

! Back to CMOS today! How do we drive a large capacitive load?

" Stages and buffer sizing" Minimum delay

2Penn ESE 370 Fall 2020 - Khanna

Message

! To drive large loads" Scale buffers geometrically" Exponential scale up in buffer size

! Scale factor: 3—4 typically" One origin of FO4 target

! Drains contribute capacitance too (Cdiff)! Can formulate sizing to optimize


Call back: Large Fanout Delay

! What is delay if must drive fanout=100?

Penn ESE 370 Fall 2020 - Khanna 4

Call back: …and Again

! Delay here?

Penn ESE 370 Fall 2020 - Khanna 5

Start Cdiff=0


One Stage (Preclass 1)

! How do we size to minimize delay?

7

WN



! Delay equation?

8

WN



! Delay equation?

9

delay = R022WN ⋅C0 +

R0WN

⋅Cload

WN


Minimize (Preclass 1)

! Differentiate and set to zero

! What’s WN?

10

delay = R0WN ⋅C0 +R0WN

⋅Cload

R0C0 −R0WN

2 ⋅Cload = 0


Minimize

! Differentiate and set to zero

! What’s WN?

11

delay = R0WN ⋅C0 +R0WN

⋅Cload

R0C0 −R0WN

2 ⋅Cload = 0

WN2 =

Cload

C0WN =

Cload

C0Penn ESE 370 Fall 2020 - Khanna

Concrete?

! What is WN for Cload=4x104C0?

12

WN

WN =Cload

C0


k-stage (Preclass 2)

13

WN1 WN2 WNk


k-stage Delay (Preclass 2)

14

2τ WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

"

#$$

%

&''+

R0WNk

×Cload

WN1 WN2 WNk


Size WNi to minimize delay (Preclass 2)

! How do we minimize?

15

2τ WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

"

#$$

%

&''+

R0WNk

×Cload



! Take partial derivative with respect to WNi and set = 0

16

2τ 0+ 0+ 0+...+ 1WN (i−1)

−WN (i+1)

WNi( )2+...+ 0

"

#$$

%

&''+ 0 = 0

2τ WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

"

#$$

%

&''+

R0WNk

×Cload



! Take partial derivative with respect to WNi and set = 0

17

2τ 0+ 0+ 0+...+ 1WN (i−1)

−WN (i+1)

WNi( )2+...+ 0

"

#$$

%

&''+ 0 = 0

2τ WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

"

#$$

%

&''+

R0WNk

×Cload

1WN (i−1)

=WN (i+1)

W 2Ni

→WNi

WN (i−1)

=WN (i+1)

WNi


Delay

! Conclude: at optimal sizing, ratio of stages is same:

18

1WN (i−1)

=WN (i+1)

W 2Ni

→WNi

WN (i−1)

=WN (i+1)

WNi


Delay

! Call that ratio 𝞺

19

ρ =WNi

WN (i−1)

=WN (i+1)

WNi


Stage Delay

20

2τ WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

"

#$$

%

&''+

R0WNk

×Cload


Stage Delay

21

2τ WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

"

#$$

%

&''+

R0WNk

×Cload

2τ WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

+R0

2τ ⋅WNk

×Cload

$

%&&

'

())


Stage Delay

22

ρ =WN1

2=

WNi

WN (i−1)

=WN (i+1)

WNi

=Cload

WNk 2C0( )

2τ WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

"

#$$

%

&''+

R0WNk

×Cload

2τ WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

+R0

2τ ⋅WNk

×Cload

$

%&&

'

())


Stage Delay

23

Two simplifications?1) in terms of r?

2) without r?

ρ =WN1

2=

WNi

WN (i−1)

=WN (i+1)

WNi

=Cload

WNk 2C0( )


Stage Delay

24

WN1

2!

"#

$

%&WN 2

WN1

!

"#

$

%&WN 3

WN 2

!

"#

$

%&!

WNi

WN (i−1)

!

"##

$

%&&WN (i+1)

WNi

!

"#

$

%&!

WNk

WN (k−1)

!

"##

$

%&&

Cload

WNk 2C0( )= ρ k+1

ρ =WN1

2=

WNi

WN (i−1)

=WN (i+1)

WNi

=Cload

WNk 2C0( )


Stage Delay

ρ k+1 =Cload

4C0→ ρ =

Cload

4C0k+1

25

ρ =Cload4C0

⎛

⎝⎜⎜

⎞

⎠⎟⎟

1k+1⎛

⎝⎜

⎞

⎠⎟


Total Delay

26

2τ WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

+R0

2τ ⋅WNk

×Cload

$

%&&

'

())

ρ =WN1

2=

WNi

WN (i−1)

=WN (i+1)

WNi

=Cload

WNk 2C0( )


Total Delay (Preclass 2)

27

TotalDelay = 2τ (k +1)ρ

ρ =WN1

2=

WNi

WN (i−1)

=WN (i+1)

WNi

=Cload

WNk 2C0( )


2τ WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

+R0

2τ ⋅WNk

×Cload

$

%&&

'

())

Total Delay

28


TotalDelay = 2τ (k +1) Cload

4C0

!

"#

$

%&

1k+1!

"#

$

%&


ρ =Cload4C0

⎛

⎝⎜⎜

⎞

⎠⎟⎟

1k+1⎛

⎝⎜

⎞

⎠⎟

Plot Delay vs. k (Cload=4x104C0)

29

0

50

100

150

200

250

300

350

400

450

1 2 3 4 5 6 7 8 9 10 11 12

Del

ay (t

units

)

Stages (k)

Delay vs. Number of Stages


4C0

!

"#

$

%&

1k+1!

"#

$

%&


Zoom: Plot Delay vs. k (Cload=4x104C0)

30

30

40

50

60

70

80

90

100

1 2 3 4 5 6 7 8 9 10 11 12

Del

ay (t

units

)

Stages (k)



4C0

!

"#

$

%&

1k+1!

"#

$

%&



31

0 = 2τ Cload

4C0

!

"#

$

%&

1k+1!

"#

$

%&

− (k +1) ⋅ ln Cload

4C0

!

"#

$

%&Cload

4C0

!

"#

$

%&

1k+1!

"#

$

%&

1k +1!

"#

$

%&2)

*

+++

,

-

.

.

.


4C0

!

"#

$

%&

1k+1!

"#

$

%&


d(bx )dx

= ln(b) ⋅bx

d(1 x)dx

= −1x2


32

0 = 2τ Cload

4C0

!

"#

$

%&

1k+1!

"#

$

%&


4C0

!

"#

$

%&Cload

4C0

!

"#

$

%&

1k+1!

"#

$

%&

1k +1!

"#

$

%&2)

*

+++

,

-

.

.

.


4C0

!

"#

$

%&

1k+1!

"#

$

%&

0 =1− 1k +1"

#$

%

&'ln

Cload

4C0

"

#$

%

&'

k = ln Cload

4C0

!

"#

$

%&−1


d(bx )dx

= ln(b) ⋅bx

d(1 x)dx

= −1x2

Concrete (Preclass 3)

! What is optimal k for Cload=4x104C0?

33

k = ln Cload

4C0

!

"#

$

%&−1



34

30

40

50

60

70

80

90

100

1 2 3 4 5 6 7 8 9 10 11 12

Del

ay (t

units

)

Stages (k)



4C0

!

"#

$

%&

1k+1!

"#

$

%&


Optimum Scale Up

! For optimum delay

! What is r?

35

k = ln Cload

4C0

!

"#

$

%&−1

ρ =Cload4C0

⎛

⎝⎜⎜

⎞

⎠⎟⎟

1k+1⎛

⎝⎜

⎞

⎠⎟


Optimum Scale Up

36

ρ =Cload4C0

⎛

⎝⎜⎜

⎞

⎠⎟⎟

1

lnCload4C0

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟ = Y( )

1ln Y( )⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

ln(ρ) = 1ln Y( )

ln(Y ) =1

ρ = e


Optimum Scale Up

37

ρ =Cload4C0

⎛

⎝⎜⎜

⎞

⎠⎟⎟

1

lnCload4C0

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟ = Y( )

1ln Y( )⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

ln(ρ) = 1ln Y( )

ln(Y ) =1

ρ = e


Call Back: Total Delay

38


2τ WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

"

#$$

%

&''+

R0WNk

×Cload

2τ WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

+R0

2τ ⋅WNk

×Cload

$

%&&

'

())

ρ =WN1

2=

WNi

WN (i−1)

=WN (i+1)

WNi

=Cload

WNk 2C0( )


Delay at Optimum

39

k = ln Cload

4C0

!

"#

$

%&−1 ρ = e



Delay at Optimum (Preclass 3)

40

k = ln Cload

4C0

!

"#

$

%&−1 ρ = e


TotalDelay = 2τ ⋅ ln Cload

4C0

"

#$

%

&'⋅e

! What is optimal delay for Cload=4x104C0 in tau units?



41

30

40

50

60

70

80

90

100

1 2 3 4 5 6 7 8 9 10 11 12

Del

ay (t

units

)

Stages (k)



4C0

!

"#

$

%&

1k+1!

"#

$

%&


Cdiff=gCgate


Diffusion Capacitance (Preclass 4)

! What does this do to t model?" Delay of middle stage cascade?


Diffusion Capacitance (Preclass 4)

! What does this do to t model?" Delay of middle stage cascade?

44

delayW1→W 2 =R0W1

"

#$

%

&' 2W1 ⋅Cdiff 0 + 2W2 ⋅C0( )

delayW1→W 2 = 2R0W1

"

#$

%

&' W1 ⋅γC0 +W2 ⋅C0( )

delayW1→W 2 = 2τ γ +W2

W1

"

#$

%

&'


k-stage Delay

45

2τ WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

"

#$$

%

&''+

R0WNk

×Cload


W1

"

#$

%

&'


k-stage Delay (Preclass 4)

46

2τ WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

"

#$$

%

&''+

R0WNk

×Cload


W1

"

#$

%

&'

2τ γ +WN1

2+γ +

WN 2

WN1

+γ +WN 3

WN 2

+...+γ + WNi

WN (i−1)

+γ +WN (i+1)

WNi

+...+γ + WNk

WN (k−1)

"

#$$

%

&''

+R0WNk

(Cload + 2γWNkC0 )Penn ESE 370 Fall 2020 - Khanna

k-stage Delay

47

2τ γk +WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

"

#$$

%

&''+

R0WNk

(Cload + 2γWNkC0 )

2τ γk +WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

"

#$$

%

&''+

R0WNk

Cload + 2τγ

2τ γk +WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

+R0

2τWNk

Cload +γ"

#$$

%

&''


2τ γ +WN1

2+γ +

WN 2

WN1

+γ +WN 3

WN 2

+...+γ + WNi

WN (i−1)

+γ +WN (i+1)

WNi

+...+γ + WNk

WN (k−1)

"

#$$

%

&''

+R0WNk

(Cload + 2γWNkC0 )

k-stage Delay

48

2τ γk +WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

"

#$$

%

&''+

R0WNk

(Cload + 2γWNkC0 )

2τ γk +WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

"

#$$

%

&''+

R0WNk

Cload + 2τγ

2τ γk +WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

+R0

2τWNk

Cload +γ"

#$$

%

&''


2τ γ +WN1

2+γ +

WN 2

WN1

+γ +WN 3

WN 2

+...+γ + WNi

WN (i−1)

+γ +WN (i+1)

WNi

+...+γ + WNk

WN (k−1)

"

#$$

%

&''

+R0WNk

(Cload + 2γWNkC0 )

k-stage Delay

49

2τ γk +WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

"

#$$

%

&''+

R0WNk

(Cload + 2γWNkC0 )

2τ γk +WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

"

#$$

%

&''+

R0WNk

Cload + 2τγ

2τ γk +WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

+R0

2τWNk

Cload +γ"

#$$

%

&''


2τ γ +WN1

2+γ +

WN 2

WN1

+γ +WN 3

WN 2

+...+γ + WNi

WN (i−1)

+γ +WN (i+1)

WNi

+...+γ + WNk

WN (k−1)

"

#$$

%

&''

+R0WNk

(Cload + 2γWNkC0 )

k-stage Delay

50

2τ γk +WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

"

#$$

%

&''+

R0WNk

(Cload + 2γWNkC0 )

2τ γk +WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

"

#$$

%

&''+

R0WNk

Cload + 2τγ

2τ γk +WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

+R0

2τWNk

Cload +γ"

#$$

%

&''


2τ γ (k +1)+WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

+R0

2τ ⋅WNk

⋅Cload

#

$%%

&

'((

2τ γ +WN1

2+γ +

WN 2

WN1

+γ +WN 3

WN 2

+...+γ + WNi

WN (i−1)

+γ +WN (i+1)

WNi

+...+γ + WNk

WN (k−1)

"

#$$

%

&''

+R0WNk

(Cload + 2γWNkC0 )

Size WNi to minimize delay

! Take partial derivative with respect to WNi = 0

51

2τ 0+ 0+ 0+...+ 1WN (i−1)

−WN (i+1)

WNi( )2+...+ 0

"

#$$

%

&''+ 0 = 0

1WN (i−1)

=WN (i+1)

W 2Ni

→WNi

WN (i−1)

=WN (i+1)

WNi


2τ γ (k +1)+WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

+R0

2τ ⋅WNk

⋅Cload

#

$%%

&

'((

Impact on Minimum WNi ?

! Partial derivative unchanged


2τ γ (k +1)+WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

+R0

2τ ⋅WNk

⋅Cload

#

$%%

&

'((

Stage Delay: r unchanged (for fixed k)

53

ρ =WN1

2=

WNi

WN (i−1)

=WN (i+1)

WNi

=Cload

WNk 2C0( )

ρ =Cload4C0

⎛

⎝⎜⎜

⎞

⎠⎟⎟

1k+1⎛

⎝⎜

⎞

⎠⎟


Stage Delay: r unchanged (for fixed k)

54

ρ =WN1

2=

WNi

WN (i−1)

=WN (i+1)

WNi

=Cload

WNk 2C0( )

TotalDelay = 2τ (k +1)(ρ +γ )


4C0

!

"#

$

%&

1k+1!

"#

$

%&

+γ

!

"

###

$

%

&&&

ρ =Cload4C0

⎛

⎝⎜⎜

⎞

⎠⎟⎟

1k+1⎛

⎝⎜

⎞

⎠⎟


2τ γ (k +1)+WN1

2+WN 2

WN1

+WN 3

WN 2

+...+ WNi

WN (i−1)

+WN (i+1)

WNi

+...+ WNk

WN (k−1)

+R0

2τ ⋅WNk

⋅Cload

#

$%%

&

'((

Impact of Gamma

55

0

50

100

150

200

250

300

350

400

450

1 2 3 4 5 6 7 8 9 10 11 12

Del

ay (t

uni

ts)

Stages (k)


g=0g=0.5g=1.0

g=1.5


Impact of Gamma

56

40

50

60

70

80

90

100

1 2 3 4 5 6 7 8 9 10 11 12

Del

ay (t

uni

ts)

Stages (k)


g=0g=0.5g=1.0

g=1.5


Minimize

57

TotalDelay = 2τ (k +1)(ρ +γ )


4C0

!

"#

$

%&

1k+1!

"#

$

%&

+γ

!

"

###

$

%

&&&

0 = 2τ γ +Cload

4C0

!

"#

$

%&

1k+1!

"#

$

%&


4C0

!

"#

$

%&Cload

4C0

!

"#

$

%&

1k+1!

"#

$

%&

1k +1!

"#

$

%&2)

*

+++

,

-

.

.

.

0 = γ + ρ − (k +1) ⋅ ln Cload

4C0

!

"#

$

%&ρ

1k +1!

"#

$

%&2

γ + ρ = ln Cload

4C0

!

"#

$

%&

1k +1!

"#

$

%&ρ


Solve

58

γ + ρ = lnCload4C0

⎛

⎝⎜⎜

⎞

⎠⎟⎟1k +1⎛

⎝⎜

⎞

⎠⎟ρ

γρ+1= ln

Cload4C0

⎛

⎝⎜⎜

⎞

⎠⎟⎟1k +1⎛

⎝⎜

⎞

⎠⎟


Solve

59

γ + ρ = lnCload4C0

⎛

⎝⎜⎜

⎞

⎠⎟⎟1k +1⎛

⎝⎜

⎞

⎠⎟ρ

γρ+1= ln

Cload4C0

⎛

⎝⎜⎜

⎞

⎠⎟⎟1k +1⎛

⎝⎜

⎞

⎠⎟


γρ+1= ln

Cload4C0

⎛

⎝⎜⎜

⎞

⎠⎟⎟

1k+1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

γρ+1= ln ρ( )

Optimal Staging Any g


𝜌 = 𝑒!"#$

r and g? (Preclass 4)

! r=4 is optimal for what g?! r=3 is optimal for what g?


𝜌 = 𝑒!"#$

ln 𝜌 =𝛾𝜌+ 1

𝜌 ln 𝜌 − 𝜌 = 𝛾

Impact of Gamma

62

40

50

60

70

80

90

100

1 2 3 4 5 6 7 8 9 10 11 12

Del

ay (t

uni

ts)

Stages (k)


g=0g=0.5g=1.0

g=1.5


Optimal Fanout

! Clearer why we use r=4 as our benchmark?


Idea

! To drive large loads" Scale buffers geometrically" Exponential scale up in buffer size (r = e)

! Scale factor: 3—4 typically" One origin of fanout 4 target

! Drains contribute capacitance too (Cdiff)! Can formulate sizing to optimize


Admin

! P/F deadline today! Daylight savings on Sunday (fall back)! Project 2 due Monday 11/2

" Don’t turn it in late. Get it off your plate Monday night.

! Monday 11/2 lecture" Review, old exam problems, no new material

! Wednesday 11/4 lecture cancelled" Will be office hours instead

! Friday 11/6 Midterm (no lecture)! HW 6 posted 11/6

65Penn ESE 370 Fall 2020 – Khanna

Midterm

! Friday 11/6 Midterm " 2 hr window between 12pm EST – 12am EST

" If this window doesn’t work for you, let me know ASAP

" Lectures 1-22" Different exams, random set of questions" Open note, open book" Can’t communicate with anyone else about the exam until

after exam period is over" All old exams online (Look at Midterm 2)

" Study them!

" Zoe Review Session" See Piazza

66Penn ESE 370 Fall 2020 – Khanna

Admin: Midterm 2 Topics (up to Lec 22)

! Sizing! Tau-model

" Estimation and optimization

! Elmore-delay" Estimation and

optimization

! Energy and power" Estimation and

optimization" Dynamic and static

! Logic" CMOS" Ratioed" Pass transistor

! Transistor" Regions of operation" Parasitic Capacitance

Model

67

“idea” slides

Penn ESE 370 Fall 2020 – Khanna

and optimization for digital systemsese370/fall2020/handouts/lec... · 2020. 10. 30. · ese370:...

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