Transcript ppt

ESE370:
Circuit-Level
Modeling, Design, and Optimization
for Digital Systems
Day 25: October 29, 2014
Driving Large Capacitive Loads
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Penn ESE370 Fall2014 -- DeHon
Today
• Back to CMOS today
• How do we drive a large load?
– Stages and buffer sizing
– Minimum delay
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Penn ESE370 Fall2014 -- DeHon
Message
• To drive large loads
– Scale buffers geometrically
– Exponential scale up in buffer size
• Scale factor: 3—4 typically
– One origin of fanout 4 target
• Drains contribute capacitance, too
• Can formulate math to optimize
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Penn ESE370 Fall2014 -- DeHon
Start Cdiff=0
(because it’s easier…do it first to see
what looks like)
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One Stage
• How size to minimize delay?
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Penn ESE370 Fall2014 -- DeHon
One Stage
• Delay equation?
R0
R0
2WN  C0 
 Cload
2
WN
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Penn ESE370 Fall2014 -- DeHon
Minimize
• Differentiate and set to zero.
R0
R0WN  C0 
 Cload
WN
R0
R0C0 
2  Cload  0
WN
• What’s WN?

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Penn ESE370 Fall2014 -- DeHon
Solving for size
Cload
WN 
C0
2

Cload
WN 
C0
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Concrete?
• What is WN for Cload=4x104C0?
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N-stage
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Penn ESE370 Fall2014 -- DeHon
N-stage Delay
WN1 WN2 WN 3
 R0
WNi
WNi 1
2


 ...

 ...
 Cload
WN1 WN2
WNi 1 WNi
 2
 WNN
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Penn ESE370 Fall2014 -- DeHon
Size WNi to minimize delay
• How minimize?
WN1 WN 2 WN 3
 R0
WNi
WNi 1
2


 ...

 ...
 Cload
 2
 WNN
WN1 WN2
WNi 1
WNi
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Penn ESE370 Fall2014 -- DeHon
Size WNi to minimize delay
• Take partial derivative wrt WNi
WN1 WN 2 WN 3
 R0
WNi
WNi 1
2


 ...

 ...
 Cload
 2
 WNN
WN1 WN2
WNi 1
WNi


1
WNi 1

2  0  ...
0  0  0  ... WNi 1 
 0  0
WNi


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Penn ESE370 Fall2014 -- DeHon
Solving for WNi


1
WNi 1

2  ...
0  0  0  ... WNi 1 
 0  0
WNi



WNi
WNi 1

WNi 1
WNi
WNi  WNi 1  WNi 1
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Penn ESE370 Fall2014 -- DeHon
Delay
• Conclude: at optimal sizing, ratio of
stages is same:
WNi
WNi 1

WNi 1
WNi
WN1 WN 2 WN 3
 R0
WNi
WNi 1
2


 ...

 ...
 Cload
 2
 WNN
WN1 WN2
WNi 1
WNi

Penn ESE370 Fall2014 -- DeHon
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Delay
• Call that ratio f
WNi
WNi 1
f 

WNi 1
WNi
WN1 WN 2 WN 3
 R0
WNi
WNi 1
2


 ...

 ...
 Cload
 2
 WNN
WN1 WN2
WNi 1
WNi

Penn ESE370 Fall2014 -- DeHon
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Stage Delay
WN1 WN 2 WN 3
 R0
WNi
WNi 1
2


 ...

 ...
 Cload
 2
 WNN
WN1 WN2
WNi 1
WNi
WN1
WNi
WNi 1
Cload
f 



2
WNi 1
WNi
WNN 2C0 
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Penn ESE370 Fall2014 -- DeHon
Stage Delay
WN1
WNi
WNi 1
Cload
f 



2
WNi 1
WNi
WNN 2C0 
WN1WN2 WN 3  WNi WNi 1



L 

L
 2 WN1 WN2  WNi 1 WNi 
 WNN  Cload


WN(N 1) WNN 2C0 
Two simplifications?
1) in terms of f?
2) without f?
Penn ESE370 Fall2014 -- DeHon
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Stage Delay
WN1
WNi
WNi 1
Cload
f 



2
WNi 1
WNi
WNN 2C0 
WN1WN2 WN 3  WNi WNi 1



L 

L
 2 WN1 WN2  WNi 1 WNi 
 WNN  Cload
 f N 1


WN(N 1) WNN 2C0 
WN1WN2 WN 3  WNi WNi 1  WNN  Cload
Cload




L 

L 

 2 WN1 WN2  WNi 1 WNi  WN(N 1) WNN 2C0  4C0
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Stage Delay
WN1WN2 WN 3  WNi WNi 1  WNN  Cload
 f N 1



L 

L 

 2 WN1 WN2  WNi 1 WNi  WN(N 1) WNN 2C0 
WN1WN2 WN 3  WNi WNi 1



L 

L
 2 WN1 WN2  WNi 1 WNi 
f
Penn ESE370 Fall2014 -- DeHon
N 1
Cload

4C0
 WNN  Cload
Cload



WN(N
1)

WNN 2C0  4C0
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Stage Delay
f

N 1
Cload

4C0
Cload
f  N 1
4C0
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Penn ESE370 Fall2014 -- DeHon
Math
Cload
f  N 1
4C0
 C   1  C 
load
load
N
1
ln f  ln 
 
ln 

 4C0  N 1 4C0 
Penn ESE370 Fall2014 -- DeHon
f e
 1  C load 

ln

N 1  4C 0 
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Total Delay
WNi
WNi 1
f 

WNi 1
WNi

WN1 WN 2 WN 3
 R0
WNi
WNi 1
2


 ...

 ...
 Cload
 2
 WNN
WN1 WN2
WNi 1
WNi
TotalDelay  2(N 1) f
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Penn ESE370 Fall2014 -- DeHon
Total Delay
TotalDelay  2(N 1) f
f e
C lo a d  1 
ln


 4C 0 N 1
TotalDelay  2(N 1)e
 1  C load 

ln

N 1  4C 0 
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Penn ESE370 Fall2014 -- DeHon
How many stages?
• How does this trend with N?
TotalDelay  2(N 1)e
 1  C load 

ln

N 1  4C 0 
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Penn ESE370 Fall2014 -- DeHon
Plot Delay vs. N
450
400
350
300
250
Delay
( units)
Series1
200
150
100
50
0
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Penn ESE370 Fall2014 -- DeHon
2
3
N
4
5
6
7
8
9
10
11
12
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Zoom Delay vs. N
100
90
80
70
Series1
60
50
40
30
1
2
3
4
5
6
7
8
9
10
11
12
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Penn ESE370 Fall2014 -- DeHon
Minimize
TotalDelay  2(N 1)e
0e
 1  C load 

ln

N 1  4C 0 
 1  C load 

ln

N 1  4C 0 
2  1 lnC load 

 

N 1  4C 0 
Cload  1 
 (N 1)ln
 e

4C0 N 1
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Penn ESE370 Fall2014 -- DeHon
Solve
0e
 1  C load 

ln

N 1  4C 0 
2  1 lnC load 

 

N 1  4C 0 
Cload  1 
 (N 1)ln
 e

4C0 N 1
 1  Cload 
0  1  
ln

N 1 4C0 

Cload 
N 1  ln

4C0 
Penn ESE370 Fall2014 -- DeHon
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Concrete
• What is optimal N for Cload=4x104C0?
Cload 
N 1  ln

4C0 
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Penn ESE370 Fall2014 -- DeHon
Zoom Delay vs. N
100
90
80
70
Series1
60
50
40
30
1
2
3
4
5
6
7
8
9
10
11
12
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Penn ESE370 Fall2014 -- DeHon
Optimum Scale Up
f e

 1  C load 

ln

N 1  4C 0 
Cload 
N 1  ln

4C0 
Penn ESE370 Fall2014 -- DeHon
What is f?
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Optimum Scale Up
f e
 1  C load 

ln

N 1  4C 0 
Cload 
N 1  ln

4C0 
f e

Penn ESE370 Fall2014 -- DeHon
Deep result – take time to digest.
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Delay at Optimum
Cload 
N 1  ln

4C0 
f e
TotalDelay  2(N 1)f
Cload 
TotalDelay  2e  ln

 4C0 
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Penn ESE370 Fall2014 -- DeHon
Cdiff=gCgate
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Penn ESE370 Fall2014 -- DeHon
Diffusion Capacitance
• What does this do to  model?
– Delay of middle stage?
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Penn ESE370 Fall2014 -- DeHon
Stage Delay
 R0 
  2W 1  Cdiff 0  2W 2  C0
W 1
Cdiff 0  gC0

 R0 
2 W 1  gC0  W 2  C0 
W 1

Penn ESE370 Fall2014 -- DeHon

37
Stage Delay
 R0 
2 W 1  gC0  W 2  C0 
W 1
 W 2 
2g 

 W 1 
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Penn ESE370 Fall2014 -- DeHon
N-stage Delay
 WN1
 R0
WN1 WN2
WN2 WN 3
WNi
WNi WNi 1
2g 
g

g

 ...
g

 ...
 Cload

 WNN
2
WN1 WN1
WN2 WN2
WNi 1
WNi
WNi
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Penn ESE370 Fall2014 -- DeHon
N-stage Delay
 WN1
 R
WN1 WN2
WN2 WN 3
WNi
WNi WNi 1
2g 
g

g

 ...
g

 ... 0  Cloa

 WNN
2
WN1 WN1
WN2 WN2
WNi 1
WNi
WNi

 R0
WN1 WN2 WN 3
WNi
WNi 1
2g N 1 


 ...

 ...
 Cload


2
WN1 WN2
WNi 1
WNi
WNN
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Penn ESE370 Fall2014 -- DeHon
Impact on Min Wni ?
• Partial Derivative unchanged

 R0
WN1 WN2 WN 3
WNi
WNi 1
2g N 1 


 ...

 ...
 Cload

 WNN
2
WN1 WN2
WNi 1
WNi
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Penn ESE370 Fall2014 -- DeHon
Stage Delay:
f unchanged (fixed N)
WN1 WN 2 WN 3
 R0
WNi
WNi 1
2


 ...

 ...
 Cload
 2
 WNN
WN1 WN2
WNi 1
WNi
WN1
WNi
WNi 1
Cload
f 



2
WNi 1
WNi
WNN 2C0
f
Penn ESE370 Fall2014 -- DeHon
N 1
Cload

4C0
42
Total Delay
TotalDelay  2(N 1) ( f  g )
C lo a d  1 

ln


 4C 0 N 1

2(N 1) g  e




Penn ESE370 Fall2014 -- DeHon
43
Impact of Gamma
450
400
350
300
Series1
250
Series2
Series3
200
Series4
150
100
50
g1.5
g1.0
g0.5
g0
0
1
2
3
4
5
6
7
8
9
10
11
12
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Penn ESE370 Fall2014 -- DeHon
Impact of Gamma
100
90
g1.5
80
Series1
Series2
70
Series3
Series4
60
50
g1.0
g0.5
g0
40
1
2
3
4
5
6
7
8
9
10
11
12
45
Penn ESE370 Fall2014 -- DeHon
Minimize
TotalDelay  2(N 1)e
 1  C load 

ln

N 1  4C 0 
0 g e
 1  C load 

ln

N 1  4C 0 
 1  C load 

ln

N 1  4C 0 
 1  Cload 
 (N 1)
 ln
e
N 1  4C0 
2
46
Penn ESE370 Fall2014 -- DeHon
Solve
 1  C load 

ln

N 1  4C 0 
0 g e
 1  C load 

ln

N 1  4C 0 
 1  Cload 
 (N 1)
 ln
e
N 1  4C0 
2
  1  C 
load
0  g  f 1  
ln 

 N 1  4C0 
 1  Cload 
g  f  
ln
f
N 1 4C0 
47
Penn ESE370 Fall2014 -- DeHon
Solve
Cload  1 
g  f  ln
f

4C0 N 1


C
1 
g 1  ln load 


f
4C0 N 1



Cload  1
N 1  ln 
 g
4C0 1

Penn ESE370 Fall2014 -- DeHon




f 
48
Optimum Scale Up
f e

C lo a d  1 
ln


 4C 0 N 1

Cload  1
N 1  ln 
 g
4C0 1

Penn ESE370 Fall2014 -- DeHon
f e

1 g 

f 





f 
49
Optimal Staging Any g
1g / f 
f e
50
Penn ESE370 Fall2014 -- DeHon
F and gamma?
• f=4 is optimal for what g?
• f=3 is optimal for what g?
1g / f 
f e
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Penn ESE370 Fall2014 -- DeHon
Optimal Fanout
• Clearer why we use f=4
as our benchmark?
– Remember HW4.3
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Penn ESE370 Fall2014 -- DeHon
Idea
• To drive large loads
– Scale buffers geometrically
– Exponential scale up in buffer size
• Scale factor: 3—4 typically
– One origin of fanout 4 target
• Drains contribute capacitance, too
• Can formulate math to optimize
53
Penn ESE370 Fall2014 -- DeHon
Admin
• Project 1 due tomorrow
• Class on Friday
• Midterm 2 on Monday
– Previous midterms linked in
• Start w/ versions w/out answers
– Everything on those is fair game
– Cases at end of Monday are fair game
– Explicit roundup on Friday
54
Penn ESE370 Fall2014 -- DeHon