Transcript ppt

ESE370:
Circuit-Level
Modeling, Design, and Optimization
for Digital Systems
Day 23: November 3, 2010
Driving Large Capacitive Loads
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Penn ESE370 Fall2010 -- DeHon
Today
• How do we drive a large load?
– Stages and buffer sizing
– Minimum delay
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Penn ESE370 Fall2010 -- DeHon
Start Cdiff=0
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Penn ESE370 Fall2010 -- DeHon
One Stage
• How size to minimize delay?
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Penn ESE370 Fall2010 -- DeHon
One Stage
• Delay equation?
R0
R0
3.5WN  C0 
 Cload
2
WN
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Penn ESE370 Fall2010 -- DeHon
Minimize
• Differentiate and set to zero.
R0
R0
3.5WN  C0 
 Cload
2
WN
R0
R0
 3.5C0 
2  Cload  0
2
WN
 •
What’s WN?
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Penn ESE370 Fall2010 -- DeHon
Solving for size
Cload  2C0
WN 
3.5
2

Cload  2C0
WN 
3.5
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Penn ESE370 Fall2010 -- DeHon
N-stage
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Penn ESE370 Fall2010 -- DeHon
N-stage Delay
WN1 WN2 WN 3
 R0
WNi
WNi 1
3.5


 ...

 ...
 Cload
WN1 WN2
WNi 1 WNi
 2
 WNN
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Penn ESE370 Fall2010 -- DeHon
Size WNi to minimize delay
• Take partial derivative wrt WNi
WN1 WN 2 WN 3
 R0
WNi
WNi 1
3.5


 ...

 ...
 Cload
 2
 WNN
WN1 WN2
WNi 1
WNi


1
WNi 1

2  ...
0  0  0  ... WNi 1 
 0  0
WNi


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Penn ESE370 Fall2010 -- 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 Fall2010 -- DeHon
Delay
WN1 WN 2 WN 3
 R0
WNi
WNi 1
3.5


 ...

 ...
 Cload
 2
 WNN
WN1 WN2
WNi 1
WNi
WNi
WNi 1

WNi 1
WNi
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Penn ESE370 Fall2010 -- DeHon
Stage Delay
WN1 WN 2 WN 3
 R0
WNi
WNi 1
3.5


 ...

 ...
 Cload
 2
 WNN
WN1 WN2
WNi 1
WNi
WNi
WNi 1
f 

WNi 1
WNi
2f
Penn ESE370 Fall2010 -- DeHon
N 1
Cload

3.5C0
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Stage Delay
2f

N 1
Cload

3.5C0
Cload
f  N 1
7C0
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Penn ESE370 Fall2010 -- DeHon
Stage Delay
WNi
WNi 1
f 

WNi 1
WNi

WN1 WN 2 WN 3
 R0
WNi
WNi 1
3.5


 ...

 ...
 Cload
 2
 WNN
WN1 WN2
WNi 1
WNi
TotalDelay  3.5(N 1) f
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Penn ESE370 Fall2010 -- DeHon
Cload
f  N 1
7C0 Total Delay
Cload
TotalDelay  3.5(N 1)N 1
7C0
TotalDelay  3.5(N 1)e
C load  1 
ln


 7C 0 N 1
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Penn ESE370 Fall2010 -- DeHon
Minimize
TotalDelay  3.5(N 1)e
0e
C load  1 
ln




7C
N
1
 0 
C load  1 
ln


 7C 0 N 1
2 lnC load  1 





7C
N
1
 0 
Cload  1 
 (N 1)ln 
 e

 7C0 N 1
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Penn ESE370 Fall2010 -- DeHon
Solve
0e
C load  1 
ln



 7C 0  N 1
2 lnC load  1 




 7C 0  N 1
Cload  1 
 (N 1)ln 
 e

 7C0 N 1
Cload  1 
0  1  ln


 7C0 N 1

Cload 
N 1  ln

 7C0 
Penn ESE370 Fall2010 -- DeHon
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Optimum Scale Up
f e

C lo a d  1 
ln


 7C 0 N 1
Cload 
N 1  ln

 7C0 
Penn ESE370 Fall2010 -- DeHon
f e
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Delay at Optimum
Cload 
N 1  ln

 7C0 
f e
TotalDelay  3.5(N 1) f
Cload 
TotalDelay  3.5e  ln

 7C0 
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Penn ESE370 Fall2010 -- DeHon
Cdiff=gCgate
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Penn ESE370 Fall2010 -- DeHon
N-stage Delay
WN1
 R0
WN1 WN2
WN2 WN 3
WNi
WNi WNi 1
3.5
g

g

 ...
g

 ...
 Cload
 2
 WNN
WN1 WN1
WN2 WN2
WNi 1
WNi
WNi
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Penn ESE370 Fall2010 -- DeHon
N-stage Delay
WN1
 R
WN1 WN2
WN2 WN 3
WNi
WNi WNi 1
3.5
g

g

 ...
g

 ... 0  Cload
 2
 WNN
WN1 WN1
WN2 WN2
WNi 1
WNi
WNi

 R0
WN1 WN2 WN 3
WNi
WNi 1
3.5gN 


 ...

 ...
 Cload


2
WN1 WN2
WNi 1
WNi
WNN
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Penn ESE370 Fall2010 -- DeHon
Impact on Min Wni ?
• Partial Derivative unchanged

 R0
WN1 WN2 WN 3
WNi
WNi 1
3.5gN 


 ...

 ...
 Cload

 WNN
2
WN1 WN2
WNi 1
WNi
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Penn ESE370 Fall2010 -- DeHon
Total Delay
TotalDelay  3.5(N 1) f  3.5gN
3.5(N 1)e
C load  1 
ln


 7C 0 N 1
 3.5gN
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Penn ESE370 Fall2010 -- DeHon
Stage Delay: N=7
3.5(N 1)e
C load  1 
ln


 7C 0 N 1
Cload 
N 1  ln

 7C0 
 3.5gN
N+1=8
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Penn ESE370 Fall2010 -- DeHon
Stage Delay, g=0.5
3.5(N 1)e
C load  1 
ln


 7C 0 N 1
 3.5gN
N+1=8
2

10
3.5  8  8
 3.5  7  0.5  88
7
4
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Penn ESE370 Fall2010 -- DeHon
Stage Delay: N=6, g=0.5
3.5(N 1)e
C load  1 
ln


 7C 0 N 1
 3.5gN
N+1=7
2

10
3.5  7  7
 3.5  6  0.5  87
7
4
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Penn ESE370 Fall2010 -- DeHon
Stage Delay: N=5, g=0.5
3.5(N 1)e
C load  1 
ln


 7C 0 N 1
 3.5gN
N+1=6
2

10
3.5  6  6
 3.5  5  0.5  88
7
4
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Penn ESE370 Fall2010 -- DeHon
Stage Delay: N=4, g=0.5
3.5(N 1)e
C load  1 
ln


 7C 0 N 1
 3.5gN
N+1=5
2

10
3.5  5  5
 3.5  4  0.5  93
7
4
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Penn ESE370 Fall2010 -- DeHon
Optimal Staging g0
1g / f 
f e
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Penn ESE370 Fall2010 -- DeHon
Admin
• Friday
– Lecture
– HW5 due
– Proj2 out
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Penn ESE370 Fall2010 -- DeHon
Idea
• Drive large loads with geometrically
sized buffer chain
• There is an optimal buffer ratio
– Technology/gate dependent
– Depends on self-loading
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Penn ESE370 Fall2010 -- DeHon