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.A3Tsinghua University, Beijing 100084, China

4San Diego State University, U.S.A

2

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

3

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.

4

Power Network Model with Controlled-ESR

VDD

VDD

Current Source

Inductor

Decap

Resistor

Controlled-ESR

5

Voltage Variation Analysiswith Circuit State Equation

From KCL and KVL, we have the circuit state equation:

which is denoted to be:

0

0 T

C v G E vBU

L i E R i

Cx Ax Bu

(1)

(2)

If add extra decap ⊿C and controlled-ESR ⊿A, solution x will be updated by ⊿x, so (2) becomes:

( )( ) ( )( )C C x x A A x x Bu (3)

By subtracting (2) from (3): ( ) ( ) ( )C C x A A x Ax Cx (4)

The solution of (4) is:1 1

0

0

( ) ( ) 10 ( )

tC A t t C A t

t

x e x e C U d

where: , , C C C A A A U Ax Cx 2 3

...2! 3!

A A Ae I A

(5)

6

Effect of Controlled-ESR on reducing the noise

0 2 4 6 8 10 12 14 16 18 200.8

0.85

0.9

0.95

1

1.05

1.1

Time [ns]

Vol

tage

[V

]

Controlled-ESR=10 mOhm

Controlled-ESR=100 mOhmControlled-ESR=1 Ohm

Controlled-ESR=10 Ohm

Decap

Controlled-ESR

VDD

Time (ns)

Vol

tage

of

V0 (

V)

V0

7

Power Network Noise Considering Overshoot

Vdd

Vmin

Violation Area

ts

V

te T

Vdd

Vmin

Violation Area

ts1

V

te1 T

Vmax

ts2 te2

Violation Area

min

0

max( ( ), 0)T

j jg V v t dt min max

0

max[max( ( ), 0), max( ( ) , 0)]T

j j jg V v t v t V dt

8

Problem Formulation

Objective function: Min

Constraints: (1) Voltage response satisfies the circuit equation with

given stimulus; (2) Total decap budget: (3) Space constraint for each decap location: (4) Space constraint for each controlled-ESR location:

1

N

jj

g

1

M

ii

c Q

max0 i ic c

max0 i iCtrlESR CtrlESR

9

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:

jij

i

gs

c

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

( ) ( )s eu t t u t t

Merged adjoint sensitivity is calculated with

,1 0

( ( )) ( ) , ( 1, 2,..., )TN

i ij i all ij

s s v T t v t dt i M

10

Revised Sensitivity Computation Considering Overshoot

We denote the port currents and voltages by vectors Ip and Vp. Denote the non-source branch currents and voltages by vectors Ib and Vb. From Tellegen’s theorem, we have

0

0

ˆ ˆ[ ( ) ( ) ( ) ( )]

ˆ ˆ[ ( ) ( ) ( ) ( )]

T

p p p p T t

T

b b b b T t

i v t v i t dt

i v t v i t dt

We set all voltage sources in the adjoint network to zero and apply a current source for each violation node: 1

[ ( ) ( )]vN

s k sk ekk

I D u t t u t t

1, if ( ) Vdd

1, if ( ) Vddsk

ksk

v tD

v t

10

[ ( ) ( )] ( )vT N

k sk ek pk T t

g D u t t u t t v t dt

Left hand:

0

ˆ[ ( ) ( )]T

C c c T t

gs v v t dt

C

0

ˆ[ ( ) ( )]T

R R R T t

gs i i t dt

R

Right hand:

11

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.

12

A Simple Case

Initial values: 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.

Constraints: 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:

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: 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%.

+-

CtrlESR 1

Decap 1 Decap 2 Decap 3

CtrlESR 2 CtrlESR 3

P1 P2 P3

13

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.

Experimental ResultsTable 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.

15

Voltage Waveforms with Different Optimizations

0 5 10 15 200.94

0.96

0.98

1

1.02

1.04

1.06

Time [ns]

Voltage [

V]

Without optimization

Optimization with evenly distributed decap

SQP result with decap only

SQP result with decap and Controlled-ESR

Time (ns)

Vol

tage

(V

)

16

Relationship between Decap Budget and Noise

5 10 15 20 25 30 35 40 45 500

50

100

150

200

250

300

350

Decap [nF]

Nois

e [

V*p

s]

Larger decap budget leads to smaller noise

Tradeoff between the noise reduction and the decap investment

Decap (nF)

Noi

se (

V*p

s)

17

Conclusions

Optimize power network with both decap and controlled-ESR.

Revised sensitivity computation considering voltage overshoot.

SQP based optimization

18

Thank You!

Q & A