Discrete Optimization of Decoupling Capacitors for Power Integrity

7/27/2019 Discrete Optimization of Decoupling Capacitors for Power Integrity

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Power Integrity Analysis and Discrete Optimization of Decoupling Capacitors

on High Speed Power Planes by Particle Swarm Optimization

Jai Narayan Tripathi1, Raj Kumar Nagpal2, Nitin Kumar Chhabra2, Rakesh Malik2, Jayanta Mukherjee1,

Prakash R. Apte1

1Dept. of EE, IIT Bombay, Mumbai, INDIA.2TR&D, STMicroelectronics Pvt. Ltd., Greater Noida, INDIA.

E-mail1 : {jai,jayanta,apte}@ee.iitb.ac.inE-mail2 : {rajkumar.nagpal,nitin.chhabra,rakesh.malik}@st.com

Abstract Power Integrity problem for a high speed powerplane is discussed in context of selection and placement ofdecoupling capacitors. The s-parameters data of power plane

geometry and capacitors are used for the accurate analysisincluding bulk capacitors and VRM, for a real world problem.The optimal capacitors and their optimum locations on theboard are found using particle swarm optimization. A novel andaccurate methodology is presented which can be used for anyhigh speed Power delivery Network.

Index Terms Power Integrity, Power Delivery Networks, De-coupling Capacitors, S-parameters, Particle Swarm Optimization(PSO).

I. Introduction

In high speed systems, the power delivery network design

becomes critical in order to supply noise suppressed power

to the core and i/o circuits. Power Integrity (PI) is a highspeed issue concerned with Power Delivery Networks (PDNs),

which ensures sufficient and efficient power supply within a

system [1]. If power integrity is not maintained in a high

speed system, the power supply noise may exceed above the

specified allowable ripple and thus may affect the functionality

of the system which is designed to work for the predefined

noise margins. To, maintain power integrity in a PDN, the

maximum allowable impedance should be lesser than the target

impedance, which is ratio of allowable voltage ripple to the

maximum transient current of the system.

Ztarget =(voltage(pp))(%ripple/100)

Imaxtran(1)

A PDN consists of many components such as Voltage Reg-ulator Module (VRM), Bulk Capacitors, Board, Decoupling

Capacitors, Package. Power is supplied from VRM, through

board and package. There are capacitive and inductive effects

associated with all of these components of PDN which impacts

the system in different frequency ranges. The impedance

of a PDN is defined by the the cumulative effect of all

these impedances. These capacitive and inductive behaviors

of components cause resonance and anti-resonance patterns

in a PDN, which deteriorates the power supply quality and

increases the ripples[2]. To avoid this, the impedance of the

PDN should be lower than the target impedance.

This paper takes into account the decoupling network in a

PDN which consists of decoupling capacitors. The frequency

ranges in which the decoupling network affects a PDN is typi-cally hundreds of MHz. Decoupling capacitors are convention-

ally placed as near as possible, to the package pins [3]. Recent

studies have shown that the best position to place decoupling

capacitors are not always near to the package pins [4][5]. There

are various papers available in literature which use stochastic

methods such as Genetic Algorithm (GA), Particle Swarm

Optimization (PSO), Cuckoo Search etc., to find the optimum

positions and values of decoupling capacitors [6][7][8]. Unlike

the earlier work, this paper presents the practical solution of

the industrial problem of discrete optimization for finding the

suitable decoupling capacitors (provided with their s-parameter

files) and their best positions from the available positions on

the board. Novelties in this work are :

1) The methodology for decoupling optimization is based

on the s-parameters models of the decoupling capacitors

as practically available from the various manufacturers

instead of rlc models adopted in tandem.

2) Using s-parameters data for board and package to take

into account the effects of via, pad and anti-pads.

3) Solving the optimization problem for PDN, after taking

into effect of bulk capacitors, VRM, and package in

order to increase the efficiency and accuracy.

4) Finding the trajectory of the particles in PSO when the

s-parameters data is used, unlike the r,l,c ranges taken

in the earlier papers.5) Interpolating the data of capacitor bank before using it

for optimization, if the frequency range of the board is

not the same as that of the s-parameters file from the

capacitor bank.

6) Explaining the rationale behind choosing the frequency

range of decoupling network analysis.

The approximate rlc models of the decoupling capacitor do

not take into account the nonlinear behavior of the capacitors

at higher frequencies while s-parameters models do. The rlc

models of the capacitors are defined at one spot frequency or

a narrow band, instead of broader range.

978-1-4673-4953-6/13/$31.00 2013 IEEE 670 14th Int'l Symposium on Quality Electronic Design


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II . Power Delivery Network

The main components of the power delivery network are

the die, package, the PCB planes, decoupling capacitors, bulk

capacitors, and the voltage regulator module. The decoupling

capacitors help the voltage regulator supply current when there

is high demand of current into the chip by supplying the charge

stored in them. In a typical PDN, the electrolytic bulk capaci-

tors are effective only at very low frequencies (


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Fig. 4. Power delivery plane and nets in a typical package

s-parameters and rlc values. This is the profile of capacitor

LLR185C70G105ME01 from Murata Manufacturing Co. The

r,l,c values (ESR of 100 m and ESL of 120 pH) are takenfrom the data sheet and the s-parameters file is obtained from

vendors website [9][10]. There is clearly a difference betweenthe two profiles. The maximum difference is in hundreds of

ohms at lower frequency range, even at high frequencies,

second order effects will be visible only in s-parameter models

and not in linear rlc models as generally adopted in analysis.To

avoid the inaccuracies involved in rlc models, we have used

s-parameters data in this analysis.

Fig. 5. Impedance profiles of a cap obtained from s-parameters and rlcmodel

III. Particle Swarm Optimization

Particle Swarm Optimization (PSO) is a metaheuristic op-

timization technique belonging to the group of algorithms

inspired by the nature. Introduced by Kennedy and Eberhart

in 1995, PSO is inspired by the movement of fishes while

schooling and the same of the birds while flocking. The

fishes or birds follow the group behavior and the collective

intelligence is used for the movement of the entire swarm. In

this optimization technique, a basic entity is called particle,

which follows a trajectory based on the past memory, to

find the solution for the given problem. In PSO, there are

P particles generated or defined randomly in a design space.

Each particle is assigned a position and an initial velocity.

Each particle claims to be a solution of the problem within

the search space, intending to attain optimal position. After

generating particles, the fitness of all the particles is calculated.

Fitness is the output parameter of optimization problem, which

we want to optimize. Particles move towards the fittest particle

and by this process they reach to the optimal solution. After

calculating fitness of all the particles, the one with the best

fitness is dened as globally best particle gbest. The movement

of all the particles is decided by the surrounding particles.

After calculating the fitness of all the particles, particles are

assigned new velocities and positions according to a set of

equations, which are following :

vi(t + t) = (t)vi(t) +p1r1(xl xi) +p2r2(xg xi) (1)

xi(t + t) = xi(t) + vi(t + t)t (2)

(t) = (i f).tmax t

tmax+ f (3)

In above equations, vi(t) is the velocity of a particle, xi(t)is the position of the particle, and (t) is weighing factor, all attime t. The velocity and position of a particle may be vector

depending on the dimension of the particle. Equation 1 shows

the calculation for new velocity of a particle for next iteration

and equation 2 shows the same for the position of the particle

in next iteration. Weighing factor is updated (reduced) after

each iteration according equation 3. The bold characters in the

above equations stand for vectors. For more details of PSO,

readers are requested to refer [7], [11]-[14]. The dynamics of

convergence of particles is elborated mathematically in [15].

IV. Optimization Methodology

A real world problem (industrial case study) is presented

with the proposed methodology. The methodology aims to

find the optimal number of capacitors, their values and their

positions on the board, in order to meet the target impedance

of the system. The methodology is based on impedance profile

generation from s-parameters data for the complex geometry

of the multilayer board and the given bank of capacitors. The

steps followed in this methodology are :

1) Extraction of s-parameters of multilayer board and pack-

age from the design database.2) Including the effects of VRM, bulk capacitor, board and

package to find composite system impedance profile.

3) Generation of decoupling capacitor bank (z-profiles)

from their given s-parameters profiles provided by dif-

ferent vendors.

4) Indexing and ordering the capacitor bank, according to

the resonance points in their impedance profiles.

5) Indexing the coordinates of the available ports on the

board, based on their relative positioning.

6) Applying PSO in iterative loop to achieve the target

impedance.


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The cumulative z-parameter matrix of a board loaded with

decaps can be given by the following formula [4].

Zeff = (Z1 + Z1decap)

1 (4)

Where the z-parameters matrix of the board is Z and Zdecapis the diagonal matrix in which the diagonal elements are

associated with the impedance of the decoupling capacitors

on the ports corresponding to the diagonal elements while all

other elements are zero. The problem with the above formula

appears when the decoupling capacitors on the board are lesser

than the available number of ports. In that case, one or more

number of diagonal elements of matrix Zdecap are zero, which

will not allow the inverse of it. To avoid this problem, we

used y-parameters for that step particularly. The analysis is

carried out for a high speed serial link board. The extracted s-

parameters file was having 39 ports. There are 39 ports in the

board from which 4 ports are reserved, one for VRM and 3 for

bulk capacitors, while one of the ports is the observation port

where the package pin is connected. Thus, there are 34 ports

available while defining or initializing the ports for decoupling

capacitors. There are thousands of capacitors (3702 for this

study) used for creating the bank. For defining the particles,

the s-parameters data and the port numbers were used. Each

particle has two dimensions corresponding to one capacitor,

one for the capacitor number from the capacitor bank and

one for the port number on the board. If the particle is using

multiple capacitors, the dimension of it will be multiple of

two to the number of capacitors.

Suppose if each available port has one capacitor from the

capacitor bank for meeting the target impedance of the system,

the total possible combinations will be 370234 10121 whichwill be practically impossible for available computing systems.

A. Movements of particlesThe particles in this case study are multidimensional and

can have discrete values only. If one of the dimensions of

a particle is some capacitor number or a port number, after

the the next iteration it should avail only the feasible values

which should be some port number (integer values between 5

and 39 except 33) or capacitor number (integer value between

1 and 3702) depending on the dimension. In PSO, in order

to move the particles for iterations, there must be a varying

directional vector depending upon locally best and globally

best particles. Here, in the case of capacitors, they are sorted

according to their resonance points. Each decoupling capacitor

available in the market, has ESR and ESL values associated

with it, which cause the resonance behavior. Fig. 6 shows theimpedance profiles of three capacitors with different resonance

points. We have used the resonance points of the decoupling

capacitors for the movement of the dimensions of the particle

with which the capacitor numbers are associated. For example,

if a capacitor as one of the dimensions of the globally best

particle in one iteration is having resonance at 40 MHz and one

of the dimensions of another particle is having its resonance

as 180 MHz, the later one will move randomly towards the

capacitors having value between 40 MHz and 180 MHz.

In the case of particle dimensions concerned with the port

numbers, the movement is decided by the distance between the

Fig. 6. Resonance points of various capacitors defined for guiding themovement of particles

ports which are found by the coordinates of them. A particle

will move to the global best particle and it can avail any value

or port number (based on random variables defined by PSO)

which is having lesser distance than that from the globally

best particle.

B. Interpolation and other issues

The S-parameters files of the capacitor provided by a vendor

were having a set of frequency points, while the same was

different in case of a different vendor. Additionally, the S-

parameters files of the system (VRM, bulk, package, board)

was having a different points of frequency. Thus to calculate

the impedance of the system after placing capacitor became

difficult. To solve this problem, interpolation was used and thefrequency points for the S-parameters for all the capacitors

were defined according to the frequency points given in the

S-parameters file of system.

The combined impedance matrix Zeff, after placing the

decoupling capacitor on a board can be found by the formula

shown in the previous section. But if the number of capacitors

are not equal to the number of ports or the dimension of the

Z matrix, then all the diagonal elements in the matrix Zdecapwill not be non-zero. When taken inverse, this will provide

the value of determinant zero resulting in a singular matrix.

Thus this formula cant be used in its present form so Y-

parameters based addition was used. The conversion from S-

parameters to Y-parameters (for Zdecap) was also a problemin case of capacitor files though there are formulae available

for the same. The capacitors S-parameters were given in shunt

mode as shown in fig. 7. In shunt configuration all the four

values of the Z-parameters for a capacitor are same thus the

inverse is not possible if used as a matrix and thus the Y-

parameters cant be found in that case. In that case, the value

of the z11 was inversed.

C. Experiments

The maximum current of the system is 20 mA, supply

voltage is 1.2 V and the tolerance is 3% so the target


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Fig. 7. Finding S-parameters of a capacitor in shunt configuraion

impedance for the system is 1.8 . The frequency rangeof interest for the analysis is 200 MHz as discussed in the

previous section. Thus the impedance after placing decoupling

capacitors should be lesser than 1.8 at all the frequencieslesser than 200 MHz. The optimal number of capacitors (N)

needed to achieve this, their name from the capacitor bank and

their optimum locations are to be found.

For applying PSO to this problem, 50 particles were initialized

with maximum number of iterations as 20. The coefficients

p1 and p2 were taken as 1.49 each, from [11] and similarly

i and f as 0.9 and 0.4. Each capacitor has two dimensions

associated with it, one for its number in capacitor bank and the

other one for its port number. Each particle was representing

N capacitors s-parameters files and their corresponding port

numbers. If the target is not met by N capacitors, PSO is

repeated by expanding the dimensions of each particle (i.e.

adding one more capacitor) till the target is met. The steps

used for applying PSO are as following :

Pseudo-Code for Optimization using PSO :

1) Importing the cumulative S-parameters file of the board,

VRM, bulk capacitor, and package; and arranging them

in a matrix.2) Importing the S-parameters file from capacitor bank and

arranging them in matrix form.

3) Interpolating the S-parameters of the capacitors, ac-

cording to the frequency points in the cumulative S-

parameters file of the board, VRM, package and bulk

capacitors.

4) Define maximum number of iterations Tmax, dimension

D = 2, number of capacitors N = D/2.

5) Loop 1 : While Zgbest ZT.6) For D dimensions , generate P particles, their respective

positions x(i, j) and velocities v(i, j); i {1, 2, . , P }and j {1, 2, ....D}, within the lower and upper bounds.

7) Calculate the impedance profiles for all the particles asZeffi = (Z

1i +Z

1decapi

)1 where each particle containsdecoupling capacitors and their respective port numbers.

8) Calculate the maximum impedance peak max imp(i)corresponding to all the Zeffi of all i particles.

9) Loop 2 : while Number of iterations t = t Tmax.10) Loop 3 : for i = 1 to P

11) Loop 4 : for j = 1 to D

12) Update inertia, velocities and positions

= (i f)(Tmax)t

Tmax+ f

v(i,j,t) = (t)v(i,j,t 1) + p1r1(lbest(i, j) x(i,j,t 1) + p2r2(gbest x(i, j)

x(i,j,t) = x(i,j,t 1) + v(i,j,t)13) Limit the positions and velocities within the lower and

upper bounds.

14) Update lbest(i,j) for all the particles.

15) END : Loop 4

16) End : Loop 3

17) Update gbest for each iteration.

18) Increment the counter for next iteration t = t + 119) END : Loop 2

20) Increase the Dimension of the particles D = D + 2, by

adding one more capcitor N = N + 1.21) END:Loop 1

22) Final Solution = gbest.

D. Results

The number of capacitors N needed to meet the target

impedance were 4. The particles converged to a solution which

is shown in table I. In the table, four capacitor and their

best positions which were found by the algorithm, and their

details like their serial numbers in capacitor bank and their

manufacturers are given. When used these capacitors at thegiven ports, the profile of the PDN was below the target

impedance 1.75 . The maximum impedance observed fromthe port 33 was 1.716 at 200 MHz.

Thus, the optimization problem is solved by PSO. Fig.9 shows

the flow used for this optimization process, which is generic

methodology and can be used for high speed systems. The

PSO based engine in fig. 9, is explained in the pseudo-code in

previous section. For 4 capacitors, the possible combinations

were > 1014.

TABLE I

The capacitors and their locations

Serial No.S.No. in Capacitor Capacitor Manufa Port

Bank Name -cturer Number

1. 1 GJ821BB31H105KA12 Murata 16

2. 3702 TWAE687M050 AVX 36

3. 2658 TPSB336K006R0450 AVX 16

4. 65 GRM033B31C332KA87 Murata 26

Fig. 8. PDN impedance with and without decaps


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V. Conclusion

A real world discrete optimization problem for power in-

tegrity has been solved by particle swarm optimization. Opti-

mum capacitors and their ports on the board are found. The

analysis was done using by s-parameter files for more accuracy

and to attain realistic approach. A generic methodology is

developed for similar power integrity analysis and decoupling

network design for any high speed system.

Fig. 9. Generic Methodology for PDN Optimization

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Discrete Optimization of Decoupling Capacitors for Power Integrity