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ESE370: Circuit-Level Modeling, Design, and Optimization for Digital Systems
Lec 23: October 30, 2020Driving Large Capacitive Loads
Penn ESE 370 Fall 2020 - Khanna
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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
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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
3Penn ESE 370 Fall 2020 - Khanna
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Call back: Large Fanout Delay
! What is delay if must drive fanout=100?
Penn ESE 370 Fall 2020 - Khanna 4
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Call back: …and Again
! Delay here?
Penn ESE 370 Fall 2020 - Khanna 5
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Start Cdiff=0
6Penn ESE 370 Fall 2020 - Khanna
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One Stage (Preclass 1)
! How do we size to minimize delay?
7
WN
Penn ESE 370 Fall 2020 - Khanna
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One Stage (Preclass 1)
! Delay equation?
8
WN
Penn ESE 370 Fall 2020 - Khanna
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One Stage (Preclass 1)
! Delay equation?
9
delay = R022WN ⋅C0 +
R0WN
⋅Cload
WN
Penn ESE 370 Fall 2020 - Khanna
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Minimize (Preclass 1)
! Differentiate and set to zero
! What’s WN?
10
delay = R0WN ⋅C0 +R0WN
⋅Cload
R0C0 −R0WN
2 ⋅Cload = 0
Penn ESE 370 Fall 2020 - Khanna
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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
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Concrete?
! What is WN for Cload=4x104C0?
12
WN
WN =Cload
C0
Penn ESE 370 Fall 2020 - Khanna
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k-stage (Preclass 2)
13
WN1 WN2 WNk
Penn ESE 370 Fall 2020 - Khanna
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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)
"
#$$
%
&''+
R0WNk
×Cload
WN1 WN2 WNk
Penn ESE 370 Fall 2020 - Khanna
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Size WNi to minimize delay (Preclass 2)
! How do we minimize?
15
2τ WN1
2+WN 2
WN1
+WN 3
WN 2
+...+ WNi
WN (i−1)
+WN (i+1)
WNi
+...+ WNk
WN (k−1)
"
#$$
%
&''+
R0WNk
×Cload
Penn ESE 370 Fall 2020 - Khanna
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Size WNi to minimize delay (Preclass 2)
! 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
Penn ESE 370 Fall 2020 - Khanna
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Size WNi to minimize delay (Preclass 2)
! 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
Penn ESE 370 Fall 2020 - Khanna
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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
Penn ESE 370 Fall 2020 - Khanna
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Delay
! Call that ratio 𝞺
19
ρ =WNi
WN (i−1)
=WN (i+1)
WNi
Penn ESE 370 Fall 2020 - Khanna
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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)
"
#$$
%
&''+
R0WNk
×Cload
Penn ESE 370 Fall 2020 - Khanna
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Stage Delay
21
2τ WN1
2+WN 2
WN1
+WN 3
WN 2
+...+ WNi
WN (i−1)
+WN (i+1)
WNi
+...+ WNk
WN (k−1)
"
#$$
%
&''+
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
$
%&&
'
())
Penn ESE 370 Fall 2020 - Khanna
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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
$
%&&
'
())
Penn ESE 370 Fall 2020 - Khanna
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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( )
Penn ESE 370 Fall 2020 - Khanna
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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( )
Penn ESE 370 Fall 2020 - Khanna
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Stage Delay
ρ k+1 =Cload
4C0→ ρ =
Cload
4C0k+1
25
ρ =Cload4C0
⎛
⎝⎜⎜
⎞
⎠⎟⎟
1k+1⎛
⎝⎜
⎞
⎠⎟
Penn ESE 370 Fall 2020 - Khanna
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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( )
Penn ESE 370 Fall 2020 - Khanna
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Total Delay (Preclass 2)
27
TotalDelay = 2τ (k +1)ρ
ρ =WN1
2=
WNi
WN (i−1)
=WN (i+1)
WNi
=Cload
WNk 2C0( )
Penn ESE 370 Fall 2020 - Khanna
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
$
%&&
'
())
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Total Delay
28
TotalDelay = 2τ (k +1)ρ
TotalDelay = 2τ (k +1) Cload
4C0
!
"#
$
%&
1k+1!
"#
$
%&
Penn ESE 370 Fall 2020 - Khanna
ρ =Cload4C0
⎛
⎝⎜⎜
⎞
⎠⎟⎟
1k+1⎛
⎝⎜
⎞
⎠⎟
![Page 29: and Optimization for Digital Systemsese370/fall2020/handouts/lec... · 2020. 10. 30. · ESE370: Circuit-Level Modeling, Design, and Optimization for Digital Systems Lec23: October](https://reader033.vdocument.in/reader033/viewer/2022060914/60a83fc50d0085272a3c4beb/html5/thumbnails/29.jpg)
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
TotalDelay = 2τ (k +1) Cload
4C0
!
"#
$
%&
1k+1!
"#
$
%&
Penn ESE 370 Fall 2020 - Khanna
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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)
Delay vs. Number of Stages
TotalDelay = 2τ (k +1) Cload
4C0
!
"#
$
%&
1k+1!
"#
$
%&
Penn ESE 370 Fall 2020 - Khanna
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Minimize (Preclass 3)
31
0 = 2τ Cload
4C0
!
"#
$
%&
1k+1!
"#
$
%&
− (k +1) ⋅ ln Cload
4C0
!
"#
$
%&Cload
4C0
!
"#
$
%&
1k+1!
"#
$
%&
1k +1!
"#
$
%&2)
*
+++
,
-
.
.
.
TotalDelay = 2τ (k +1) Cload
4C0
!
"#
$
%&
1k+1!
"#
$
%&
Penn ESE 370 Fall 2020 - Khanna
d(bx )dx
= ln(b) ⋅bx
d(1 x)dx
= −1x2
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Minimize (Preclass 3)
32
0 = 2τ Cload
4C0
!
"#
$
%&
1k+1!
"#
$
%&
− (k +1) ⋅ ln Cload
4C0
!
"#
$
%&Cload
4C0
!
"#
$
%&
1k+1!
"#
$
%&
1k +1!
"#
$
%&2)
*
+++
,
-
.
.
.
TotalDelay = 2τ (k +1) Cload
4C0
!
"#
$
%&
1k+1!
"#
$
%&
0 =1− 1k +1"
#$
%
&'ln
Cload
4C0
"
#$
%
&'
k = ln Cload
4C0
!
"#
$
%&−1
Penn ESE 370 Fall 2020 - Khanna
d(bx )dx
= ln(b) ⋅bx
d(1 x)dx
= −1x2
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Concrete (Preclass 3)
! What is optimal k for Cload=4x104C0?
33
k = ln Cload
4C0
!
"#
$
%&−1
Penn ESE 370 Fall 2020 - Khanna
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Zoom: Plot Delay vs. k (Cload=4x104C0)
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)
Delay vs. Number of Stages
TotalDelay = 2τ (k +1) Cload
4C0
!
"#
$
%&
1k+1!
"#
$
%&
Penn ESE 370 Fall 2020 - Khanna
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Optimum Scale Up
! For optimum delay
! What is r?
35
k = ln Cload
4C0
!
"#
$
%&−1
ρ =Cload4C0
⎛
⎝⎜⎜
⎞
⎠⎟⎟
1k+1⎛
⎝⎜
⎞
⎠⎟
Penn ESE 370 Fall 2020 - Khanna
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Optimum Scale Up
36
ρ =Cload4C0
⎛
⎝⎜⎜
⎞
⎠⎟⎟
1
lnCload4C0
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟ = Y( )
1ln Y( )⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
ln(ρ) = 1ln Y( )
ln(Y ) =1
ρ = e
Penn ESE 370 Fall 2020 - Khanna
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Optimum Scale Up
37
ρ =Cload4C0
⎛
⎝⎜⎜
⎞
⎠⎟⎟
1
lnCload4C0
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟ = Y( )
1ln Y( )⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
ln(ρ) = 1ln Y( )
ln(Y ) =1
ρ = e
Penn ESE 370 Fall 2020 - Khanna
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Call Back: Total Delay
38
TotalDelay = 2τ (k +1)ρ
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( )
Penn ESE 370 Fall 2020 - Khanna
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Delay at Optimum
39
k = ln Cload
4C0
!
"#
$
%&−1 ρ = e
TotalDelay = 2τ (k +1)ρ
Penn ESE 370 Fall 2020 - Khanna
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Delay at Optimum (Preclass 3)
40
k = ln Cload
4C0
!
"#
$
%&−1 ρ = e
TotalDelay = 2τ (k +1)ρ
TotalDelay = 2τ ⋅ ln Cload
4C0
"
#$
%
&'⋅e
! What is optimal delay for Cload=4x104C0 in tau units?
Penn ESE 370 Fall 2020 - Khanna
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Zoom: Plot Delay vs. k (Cload=4x104C0)
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)
Delay vs. Number of Stages
TotalDelay = 2τ (k +1) Cload
4C0
!
"#
$
%&
1k+1!
"#
$
%&
Penn ESE 370 Fall 2020 - Khanna
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Cdiff=gCgate
42Penn ESE 370 Fall 2020 - Khanna
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Diffusion Capacitance (Preclass 4)
! What does this do to t model?" Delay of middle stage cascade?
43Penn ESE 370 Fall 2020 - Khanna
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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
"
#$
%
&'
Penn ESE 370 Fall 2020 - Khanna
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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)
"
#$$
%
&''+
R0WNk
×Cload
delayW1→W 2 = 2τ γ +W2
W1
"
#$
%
&'
Penn ESE 370 Fall 2020 - Khanna
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k-stage Delay (Preclass 4)
46
2τ WN1
2+WN 2
WN1
+WN 3
WN 2
+...+ WNi
WN (i−1)
+WN (i+1)
WNi
+...+ WNk
WN (k−1)
"
#$$
%
&''+
R0WNk
×Cload
delayW1→W 2 = 2τ γ +W2
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)
"
#$$
%
&''
+R0WNk
(Cload + 2γWNkC0 )Penn ESE 370 Fall 2020 - Khanna
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k-stage Delay
47
2τ γk +WN1
2+WN 2
WN1
+WN 3
WN 2
+...+ WNi
WN (i−1)
+WN (i+1)
WNi
+...+ WNk
WN (k−1)
"
#$$
%
&''+
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)
"
#$$
%
&''+
R0WNk
Cload + 2τγ
2τ γk +WN1
2+WN 2
WN1
+WN 3
WN 2
+...+ WNi
WN (i−1)
+WN (i+1)
WNi
+...+ WNk
WN (k−1)
+R0
2τWNk
Cload +γ"
#$$
%
&''
Penn ESE 370 Fall 2020 - Khanna
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)
"
#$$
%
&''
+R0WNk
(Cload + 2γWNkC0 )
![Page 48: and Optimization for Digital Systemsese370/fall2020/handouts/lec... · 2020. 10. 30. · ESE370: Circuit-Level Modeling, Design, and Optimization for Digital Systems Lec23: October](https://reader033.vdocument.in/reader033/viewer/2022060914/60a83fc50d0085272a3c4beb/html5/thumbnails/48.jpg)
k-stage Delay
48
2τ γk +WN1
2+WN 2
WN1
+WN 3
WN 2
+...+ WNi
WN (i−1)
+WN (i+1)
WNi
+...+ WNk
WN (k−1)
"
#$$
%
&''+
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)
"
#$$
%
&''+
R0WNk
Cload + 2τγ
2τ γk +WN1
2+WN 2
WN1
+WN 3
WN 2
+...+ WNi
WN (i−1)
+WN (i+1)
WNi
+...+ WNk
WN (k−1)
+R0
2τWNk
Cload +γ"
#$$
%
&''
Penn ESE 370 Fall 2020 - Khanna
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)
"
#$$
%
&''
+R0WNk
(Cload + 2γWNkC0 )
![Page 49: and Optimization for Digital Systemsese370/fall2020/handouts/lec... · 2020. 10. 30. · ESE370: Circuit-Level Modeling, Design, and Optimization for Digital Systems Lec23: October](https://reader033.vdocument.in/reader033/viewer/2022060914/60a83fc50d0085272a3c4beb/html5/thumbnails/49.jpg)
k-stage Delay
49
2τ γk +WN1
2+WN 2
WN1
+WN 3
WN 2
+...+ WNi
WN (i−1)
+WN (i+1)
WNi
+...+ WNk
WN (k−1)
"
#$$
%
&''+
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)
"
#$$
%
&''+
R0WNk
Cload + 2τγ
2τ γk +WN1
2+WN 2
WN1
+WN 3
WN 2
+...+ WNi
WN (i−1)
+WN (i+1)
WNi
+...+ WNk
WN (k−1)
+R0
2τWNk
Cload +γ"
#$$
%
&''
Penn ESE 370 Fall 2020 - Khanna
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)
"
#$$
%
&''
+R0WNk
(Cload + 2γWNkC0 )
![Page 50: and Optimization for Digital Systemsese370/fall2020/handouts/lec... · 2020. 10. 30. · ESE370: Circuit-Level Modeling, Design, and Optimization for Digital Systems Lec23: October](https://reader033.vdocument.in/reader033/viewer/2022060914/60a83fc50d0085272a3c4beb/html5/thumbnails/50.jpg)
k-stage Delay
50
2τ γk +WN1
2+WN 2
WN1
+WN 3
WN 2
+...+ WNi
WN (i−1)
+WN (i+1)
WNi
+...+ WNk
WN (k−1)
"
#$$
%
&''+
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)
"
#$$
%
&''+
R0WNk
Cload + 2τγ
2τ γk +WN1
2+WN 2
WN1
+WN 3
WN 2
+...+ WNi
WN (i−1)
+WN (i+1)
WNi
+...+ WNk
WN (k−1)
+R0
2τWNk
Cload +γ"
#$$
%
&''
Penn ESE 370 Fall 2020 - Khanna
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)
"
#$$
%
&''
+R0WNk
(Cload + 2γWNkC0 )
![Page 51: and Optimization for Digital Systemsese370/fall2020/handouts/lec... · 2020. 10. 30. · ESE370: Circuit-Level Modeling, Design, and Optimization for Digital Systems Lec23: October](https://reader033.vdocument.in/reader033/viewer/2022060914/60a83fc50d0085272a3c4beb/html5/thumbnails/51.jpg)
Size WNi to minimize delay
! Take partial derivative with respect to WNi = 0
51
2τ 0+ 0+ 0+...+ 1WN (i−1)
−WN (i+1)
WNi( )2+...+ 0
"
#$$
%
&''+ 0 = 0
1WN (i−1)
=WN (i+1)
W 2Ni
→WNi
WN (i−1)
=WN (i+1)
WNi
Penn ESE 370 Fall 2020 - Khanna
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
#
$%%
&
'((
![Page 52: and Optimization for Digital Systemsese370/fall2020/handouts/lec... · 2020. 10. 30. · ESE370: Circuit-Level Modeling, Design, and Optimization for Digital Systems Lec23: October](https://reader033.vdocument.in/reader033/viewer/2022060914/60a83fc50d0085272a3c4beb/html5/thumbnails/52.jpg)
Impact on Minimum WNi ?
! Partial derivative unchanged
52Penn ESE 370 Fall 2020 - Khanna
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
#
$%%
&
'((
![Page 53: and Optimization for Digital Systemsese370/fall2020/handouts/lec... · 2020. 10. 30. · ESE370: Circuit-Level Modeling, Design, and Optimization for Digital Systems Lec23: October](https://reader033.vdocument.in/reader033/viewer/2022060914/60a83fc50d0085272a3c4beb/html5/thumbnails/53.jpg)
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⎛
⎝⎜
⎞
⎠⎟
Penn ESE 370 Fall 2020 - Khanna
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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 )
TotalDelay = 2τ (k +1) Cload
4C0
!
"#
$
%&
1k+1!
"#
$
%&
+γ
!
"
###
$
%
&&&
ρ =Cload4C0
⎛
⎝⎜⎜
⎞
⎠⎟⎟
1k+1⎛
⎝⎜
⎞
⎠⎟
Penn ESE 370 Fall 2020 - Khanna
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
#
$%%
&
'((
![Page 55: and Optimization for Digital Systemsese370/fall2020/handouts/lec... · 2020. 10. 30. · ESE370: Circuit-Level Modeling, Design, and Optimization for Digital Systems Lec23: October](https://reader033.vdocument.in/reader033/viewer/2022060914/60a83fc50d0085272a3c4beb/html5/thumbnails/55.jpg)
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)
Delay vs. Number of Stages
g=0g=0.5g=1.0
g=1.5
Penn ESE 370 Fall 2020 - Khanna
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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)
Delay vs. Number of Stages
g=0g=0.5g=1.0
g=1.5
Penn ESE 370 Fall 2020 - Khanna
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Minimize
57
TotalDelay = 2τ (k +1)(ρ +γ )
TotalDelay = 2τ (k +1) Cload
4C0
!
"#
$
%&
1k+1!
"#
$
%&
+γ
!
"
###
$
%
&&&
0 = 2τ γ +Cload
4C0
!
"#
$
%&
1k+1!
"#
$
%&
− (k +1) ⋅ ln Cload
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
Penn ESE 370 Fall 2020 - Khanna
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Solve
58
γ + ρ = lnCload4C0
⎛
⎝⎜⎜
⎞
⎠⎟⎟1k +1⎛
⎝⎜
⎞
⎠⎟ρ
γρ+1= ln
Cload4C0
⎛
⎝⎜⎜
⎞
⎠⎟⎟1k +1⎛
⎝⎜
⎞
⎠⎟
Penn ESE 370 Fall 2020 - Khanna
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Solve
59
γ + ρ = lnCload4C0
⎛
⎝⎜⎜
⎞
⎠⎟⎟1k +1⎛
⎝⎜
⎞
⎠⎟ρ
γρ+1= ln
Cload4C0
⎛
⎝⎜⎜
⎞
⎠⎟⎟1k +1⎛
⎝⎜
⎞
⎠⎟
Penn ESE 370 Fall 2020 - Khanna
γρ+1= ln
Cload4C0
⎛
⎝⎜⎜
⎞
⎠⎟⎟
1k+1
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
γρ+1= ln ρ( )
![Page 60: and Optimization for Digital Systemsese370/fall2020/handouts/lec... · 2020. 10. 30. · ESE370: Circuit-Level Modeling, Design, and Optimization for Digital Systems Lec23: October](https://reader033.vdocument.in/reader033/viewer/2022060914/60a83fc50d0085272a3c4beb/html5/thumbnails/60.jpg)
Optimal Staging Any g
60Penn ESE 370 Fall 2020 - Khanna
𝜌 = 𝑒!"#$
![Page 61: and Optimization for Digital Systemsese370/fall2020/handouts/lec... · 2020. 10. 30. · ESE370: Circuit-Level Modeling, Design, and Optimization for Digital Systems Lec23: October](https://reader033.vdocument.in/reader033/viewer/2022060914/60a83fc50d0085272a3c4beb/html5/thumbnails/61.jpg)
r and g? (Preclass 4)
! r=4 is optimal for what g?! r=3 is optimal for what g?
61Penn ESE 370 Fall 2020 - Khanna
𝜌 = 𝑒!"#$
ln 𝜌 =𝛾𝜌+ 1
𝜌 ln 𝜌 − 𝜌 = 𝛾
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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)
Delay vs. Number of Stages
g=0g=0.5g=1.0
g=1.5
Penn ESE 370 Fall 2020 - Khanna
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Optimal Fanout
! Clearer why we use r=4 as our benchmark?
63Penn ESE 370 Fall 2020 - Khanna
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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
64Penn ESE 370 Fall 2020 - Khanna
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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
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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
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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