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On-Chip Power Network
Optimization with Decoupling
Capacitors and Controlled-ESRs
Wanping Zhang1,2, Ling Zhang2, Amirali Shayan2, Wenjian
Yu3, Xiang Hu2, Zhi Zhu1, Ege Engin4, Chung-Kuan Cheng2
1Qualcomm
Inc. 5775 Morehouse Dr., San Diego, U.S.A
2UC San Diego, U.S.A
3Tsinghua University, Beijing 100084, China
4San Diego State University, U.S.A
Outline of Optimization with
Decap and Controlled-ESR







Introduction
 Existing works add decap to reduce noise
 Controlled-ESR is shown to be effective to suppress the
resonance
Power network model with controlled-ESR
Problem statement
 Power network noise considering overshoot
 Formulation
Revised sensitivity computation
 Sensitivity computation with merged adjoint network
 Revised sensitivity computation considering voltage overshoot
SQP based optimization
Experimental results
Conclusions
2
Our Contributions

We propose to allocate decaps and controlled-ESRs
simultaneously to suppress the resonance and
reduce SSN of power network.

We consider both voltage drop and overshoot for
voltage violation. Derive revised sensitivity.

An optimization formulation with the objective
function of minimizing the voltage violation area and
a constraint of decap budget is presented, and
solved with an efficient SQP algorithm.
3
Power Network Model with
Controlled-ESR
VDD
Current
Source
Inductor
VDD
ControlledESR
Decap
Resistor
4
Voltage Variation Analysis
with Circuit State Equation
If add extra decap ⊿C and
controlled-ESR ⊿A, solution x will
be updated by ⊿x, so (2) becomes:
The solution of (4) is:
(1)
(2)
Cx  Ax  Bu
which is denoted to be:
By subtracting (2) from (3):
0  v   G  E  v 
 T
 BU






L  i   E
R   i 
C
0

From KCL and KVL, we have
the circuit state equation:
(C  C )( x  x)  ( A  A)( x  x)  Bu
(3)
(C  C )x  ( A  A)x  (Ax  Cx)
(4)
x  e
1
C A( t t0 )
t
x0   e
C 1 A( t  )
C 1U ( )d
(5)
t0
where: C
 C  C, A  A  A, U  Ax  Cx
2
3
A
A
eA  I  A 

 ...
5
2! 3!
Effect of Controlled-ESR on
reducing the noise
C ontrolled-E S R = 10 m O hm
C ontrolled-E S R = 100 m O hm
1.1
C ontrolled-E S R = 1 O hm
C ontrolled-E S R = 10 O hm
VDD
ControlledESR
Decap
of V[V0] (V)
Voltage
V oltage
V0
1.05
1
0.95
0.9
0.85
0.8
0
2
4
6
8
10
12
14
16
18
20
Tim e [ns ]
Time (ns)
6
Power Network Noise
Considering Overshoot
V
V
V io la tio n A re a
Vm ax
Vdd
Vdd
Vmin
V m in
Violation Area
ts
T
V io la tio n A re a
te
g j   max(Vmin  v j (t ), 0)dt
0
T
t s1
te 1 ts 2
te2
T
T
g j   max[max(Vmin  v j (t ), 0), max(v j (t )  Vmax , 0)]dt
0
7
Problem Formulation

Objective function:
N


Min  g
j 1
j
Constraints:




(1) Voltage response satisfies the circuit equation with
given stimulus;
M
(2) Total decap budget:  ci  Q
i 1
(3) Space constraint for each decap location: 0  ci  cmax i
(4) Space constraint for each controlled-ESR location:
0  CtrlESRi  CtrlESRmax i
8
Sensitivity Computation with
Merged Adjoint Network
The sensitivity sij is defined to be the contribution of
decap added at node i to remove violation at node j:
sij 
g j
ci
The merged adjoint sensitivity is defined to be the contribution
of decap added at node i to remove the violation for all nodes.
The merged adjoint network has a current source
applied at every node j
u (t  ts )  u (t  te )
Merged adjoint sensitivity is calculated with
N
T
j 1
0
si   sij   (vi ,all (T  t ))  vi (t )dt , (i  1, 2,..., M )
9
Revised Sensitivity Computation
Considering Overshoot
T
We denote the port currents and voltages
by vectors Ip and Vp. Denote the nonsource branch currents and voltages by
vectors Ib and Vb. From Tellegen’s
theorem, we have
 [iˆ ( )v
p
p
(t )  vˆ p ( )i p (t )] T t dt
0
T
  [iˆb ( )vb (t )  vˆb ( ) ib (t )] T t dt
0
We set all voltage sources in the adjoint
network to zero and apply a current source
for each violation node:
I s   Dk [u(t  tsk )  u (t  tek )]
k 1
1, if v(tsk )  Vdd
Dk  
 1, if v(tsk )  Vdd
 Nv

g    Dk [u (t  tsk )  u (t  tek )]v p (t ) 
dt
 T t
0  k 1
T
Left hand:
g
  [vˆc ( )vc (t )] T t dt
C 0
T
Right hand:
Nv
sC 
g
sR 
  [iˆR ( )iR (t )] T 10t dt
R 0
T
SQP Based Optimization
Algorithm for the SQP based optimization:
1. Select the intrinsic capacitance and controlled-ESR to be the initial
solution X(0).
2. Simulate the power network circuit, and compute the sensitivity as
gradient using the revised method.
3. Use the gradient to approximate the problem with a linearly
constrained QP subproblem at X(t).
4. Solve for the step size d(t) to move.
5. If meet with termination condition, stop;
Else, let X(t+1) = X(t) + d(t).
6. Increase t and return to step 2.
11
A Simple Case
P1
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CtrlESR 2
CtrlESR 3
Decap 1
Decap 2
Decap 3
+
-
R = 1 ohm and L=1 nH. Decap=0.01 nF, controlled-ESR = 1.0e-4 ohm.
Vdd is 1V, and the allowable voltage drop is 0.05V.
Maximum allowable decap at each node to be 0.1 nF,
Total decap should not exceed 0.2 nF,
Maximum allowable controlled-ESR at each node is 0.2 Ohm
Without optimization: The overall noise is 193.7 V*ps.
Optimize with decap only:
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CtrlESR 1
Constraints:
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P3
Initial values:
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P2
Decap at each node are: 0.1 nF, 0.09 nF, and 0.01 nF.
The noise after optimization is 6.3 V*ps.
Optimize with both decap and controlled-ESR:
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Controlled-ESR values at each node are: 0.2 Ohm, 2.77e-2 Ohm, and 1.0e-4 Ohm.
The noise is further reduced to be 5.3 V*ps.
The controlled-ESR improves the noise by 15.9%.
12
Experimental Results
Table I. Effect of considering voltage overshoot
 The noise is the voltage violation area and the number of violation nodes.
 Total noise is on average underestimated by 4.8% due to neglecting the
voltage overshoot.
 The number of violation nodes is almost the same for both cases.
13
Experimental Results
Table II. Comparison among three methods for the minimization of power
network noise
 The noise (column 2, 4, 6) and the number of violation nodes (column 3, 5, 7)
are reduced.
 The improvement brought by considering the controlled-ESRs is 25% on
average.
With the third method, the average allocated controlled-ESR ranges from
0.038 Ohm to 0.083 Ohm for different cases.
Voltage Waveforms with
Different Optimizations
1.06
Without optimization
Optimization with evenly distributed decap
SQP result with decap only
SQP result with decap and Controlled-ESR
(V)
Voltage
Voltage [V]
1.04
1.02
1
0.98
0.96
0.94
0
5
10
Time [ns]
15
20
Time (ns)
15
Relationship between Decap
Budget and Noise
350
(V*ps)
Noise
Noise [V*ps]
300
250
200
150
100
50
0
5
10
15
20
25
30
Decap [nF]
35
40
45
50
Decap (nF)
Larger decap budget leads to smaller noise
Tradeoff between the noise reduction and the decap investment
16
Conclusions

Optimize power network with both decap
and controlled-ESR.

Revised sensitivity computation considering
voltage overshoot.

SQP based optimization
17
Thank You!
Q&A
18