Power Bus Sensitivity Analysis Pkb

IEEE TRANSACTIONS ON ADVANCED PACKAGING, VOL. 33, NO. 2, MAY 2010 447

Power-Bus Sensitivity AnalysisLuca De Camillis, Giulio Antonini, Senior Member, IEEE, and Vikram Jandhyala, Senior Member, IEEE

AbstractDesign of modern printed circuit boards requiresrobust simulation techniques to model power-bus structures. Areliable method is the cavity model which assumes small spacingbetween power and ground planes. Designers often make theproper tradeoffs between conflicting design requirements usingoptimization techniques, to obtain the best possible performance.To this aim, sensitivity information with respect to power-bus pa-rameters are required by optimizers which employ gradient-basedtechniques and need the knowledge of sensitivities of the outputresponses. In this paper, the sensitivity analysis of power-busstructures is presented. Relying on the cavity model, voltagesensitivity is computed in a rational function form or rationalpolynomial form leading to a time-domain macromodel which canbe efficiently interfaced with linear and nonlinear terminations orSpice-like simulators. The simulation results show the accuracy ofthe proposed approach.

Index TermsPower-bus, sensitivity analysis, transient analysis.

I. INTRODUCTION

M ULTILAYER printed circuit boards (PCBs) usually em-ploy entire layers for power and ground planes deliv-ering dc power distribution for integrated circuits (ICs) oper-ating at high speeds. The power-ground planes may representan important noise coupling path leading to signal integrity (SI)and electromagnetic interference/compatibility (EMI/C) prob-lems [1], [2]. As a result, power-bus noise has become a majorconcern for EMI/C engineers in PCB designs.

Power-bus structures can be efficiently modeled by full-wavetechniques such as the finite-difference time-domain (FDTD)method [3] or the partial element equivalent circuit (PEEC) [4]approach which are usually time and memory consuming. Avaluable alternative is represented by the cavity-mode modelwhich has also been used to characterize the rectangularpower-bus structure as a multiport microwave circuit [5], [6].This model permits analytical computation of the impedancematrix of rectangular power-bus structures as a doubleinfinite series. The rational form or rational polynomial formof each term of the summation allows to easily generate a

Manuscript received January 29, 2009; revised March 20, 2009. This workwas supported by the Italian Ministry of University (MIUR) under a Program forthe Development of Research of National Interest (PRIN Grant 2006095890).First published June 05, 2009; current version published May 05, 2010. Thiswork was recommended for publication by Associate Editor M. Cases uponevaluation of the reviewers comments.

G. Antonini and L. De Camillis are with the UAq EMC Laboratory, Di-partimento di Ingegneria Elettrica e dellInformazione, Universit degli StudidellAquila, 67040 LAquila, Italy (e-mail: [email protected]).

V. Jandhyala is with the Department of Electrical Engineering, Paul AllenCenter, University of Washington, Seattle, WA 98185 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TADVP.2009.2022322

state-space macromodel which is well suited for time-domainsimulations [7]. For irregular shapes, the segmentation methodcan be employed [8][11]. A vast literature is available on theimprovements of the method, dealing with the convergencespeedup of the series [12][15], modeling of fringing fields[16][18], incorporation of radiation losses [19], [20].

Due to the inadequacies in state-of-the-art electronic designautomation tools, physical layout of most analog functions isstill done manually. Designers usually rely on experience inmaking layouts and detecting layout problems. Despite this ex-perience, they very often need to go through many iterations tobring the physical design within the specifications. This does notcoincide with the digital VLSI design automation trends. Thisresults in a dramatic reduction in productivity and an increasein the design cycle time.

A tool that is able to pinpoint the important layout-related pa-rameters and quantify their effect on the performance functionscan be of great help. Being aware of the relative importance ofthe parameters, a designer can then modify the layout in a wayto reduce the effect of critical parameters. This has the effectof reducing the physical design iterations. Also, this can be ofgreat help in building a more effective layout methodology foranalog circuits.

This paper discusses the importance of using sensitivityanalysis as a response to the aforementioned need. Sensitivitiesquantify the performance degradation induced by changes indifferent design and process parameters. Hence, critical pa-rameters can be easily detected. In this way, the layout designcan be controlled in order to avoid excessive changes in theseparameters.

The paper is organized as follows. In Section II the com-putation of the voltage sensitivity in terms of the closed-formGreens function for voltages via planar-circuit approach ispresented. Section III discusses the different parameter (portpositions and geometrical parameters) used in this work andthe computation of the corresponding sensitivities is presentedin Section IV. The time-domain macromodel is developedin Section V. Several examples, in both the time and fre-quency-domain, are presented in Section VI, confirming thecapability of the proposed approach to provide a fast andreliable method to sensitivity of power-bus structures.

II. ELECTRICAL MODEL FOR MODAL PROPAGATION

In the setup of CAD techniques, the compromise betweenaccuracy and simplicity is a stringent requirement. Exactanalyses are often impractical because of the exceedingly highcomputer time required. From this perspective, the planar-cir-cuit approach is a very powerful technique, which has beenbasically developed for the analysis of microstrip circuits, butcan be extended to other microwave circuit configurations, such

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448 IEEE TRANSACTIONS ON ADVANCED PACKAGING, VOL. 33, NO. 2, MAY 2010

Fig. 1. Schematic of a two-layer power-bus structure.

as reduced-height waveguide, striplines, suspended microstrips,etc.

A planar circuit is defined as an electrical circuit having twodimensions comparable with the wavelength, while the thirddimension is a negligible fraction of the wavelength. Thoughthe planar circuit is an approximate model of microstrip com-ponents, it constitutes a substantial improvement over conven-tional transmission-line models, providing accurate descriptionsof their performances. On the other hand, planar-circuit modelsare simple enough to keep computer analysis reasonably inex-pensive. A sample of a power-bus structure is sketched in Fig. 1.

As will be shown, a 2-D circuit theory can be developed forplanar components by extending to the 2-D case the concepts ofvoltage and current usually defined in transmission-line theory.

Because of planarity and open-circuit boundaryconditions, Maxwells equations, in the Laplace domain, reduceto [9]

(1)(2)

where is the 2-D nabla operator, is the unit vector normalto the plane of the circuit, and are the permeability and per-mittivity of the filling substrate material. The -field has onlythe -component, while the -field lies in the plane.

A 2-D form of telegraphers equations can be obtained from(1) and (2) defining at each point of the planar circuit a voltage

and a surface current density flowing on the topconductor as

(3)(4)

Inserting (3) and (4) into (1) and (2), we get

(5)(6)

in which we have considered

(7)(8)

We can define the input current vector by multiplying the densitycurrent vector to . In this way, multiplying and dividing the

term into (6) by we obtain

(9)

Differentiating (5) and (9) with respect to a parameter (whererepresents any electrical or physical interconnect parameter

of interest) yields the following relationship:

(10)

(11)

Now, using Schwarzs theorem, we can commute the derivativewith the nabla operator such that

(12)

(13)

in which we have defined the sensitivity variables as

(14)

(15)

If we apply the divergence at (12) we obtain

(16)

and so, with the vector identity ,the following:

(17)

The terms and are equalto zero, because the impedance and the admittance doesnot depend on , . Thus, substituting (9) and (13) into (17), weobtain

(18)where

(19a)(19b)

(19c)

and is a bi-dimensional Dirac delta function.Equation (18) represents a SturmLiouville problem for

the sensitivity ; its solution can be computed in terms of theGreens function as

(20)


DE CAMILLIS et al.: POWER-BUS SENSITIVITY ANALYSIS 449

where

(21)

For a rectangular plane structure with dimension , separatedby a dielectric of thickness ( being the minimumwavelength in the frequency range of interest), the Greens func-tion can be computed as in [21] and shown in (22b)

(22a)

(22b)

where are the coordinates at the center of the -port,are the dimensions of the -port, ,

, if , if ,represents the th mode associated with the -dimension, and

represents the th mode associated with the -dimension.

III. SENSITIVITY PARAMETERSLet us consider the sensitivity parameter . In order to com-

pute the sensitivity (20) we need to consider the forcing term(21) and the expressions of the terms therein. The derivative in(21) can assume different expression depending on the partic-ular selected sensitivity parameter. Let us consider two cases:

1) as the position of one or more ports on the power-bus;2) as a geometrical or electrical parameter.

In order to simplify the calculus, we can decompose the forcingterm (21) into three terms. Let us consider

, as

(23)

A. Position of PortsTo compute the sensitivity parameter with respect to the po-

sition of one or more ports we need to define variables if wehave ports that is .This is because variation of these ports is independent from theothers, and should be evaluated as the derivative with respect tothe correlated variable. For example, the voltage sensitivity atport with respect to the variation of port reads

(24)

For the sake of simplicity, in the following we consider the vari-ation of one port as a single sensitivity parameter, . Inthis way the sensitivity can be evaluated as in (24).

Now, let us consider the forcing term in (39). As stated before,and do not depend on and, thus, the and

terms are equal to zero. The term reads

(25)

in which we have defined

(26)

(27)

In order to compute (25) we require two properties of the Diracdelta function [22]

(28)

(29)

where the symbol stands for the derivative with respect toand for all continuous .

To compute the response in (26) we need (28). The resultreads

(30)where

(31a)(31b)

(31c)The integral of (27) reads

(32)

To compute (32), we apply (29). Although we cannot directlyuse this property, because the variable of differentiation and in-tegration can be the same, however, by means of a change of



variable, the problem can be circumvented. This is done by mul-tiplying and dividing by . Considering for simplicityonly the first term, we obtain

(33)

Thus, the sensitivity reads

(34)

where

(35)

B. Geometrical or Electrical Parameter

The voltage sensitivity with respect to a geometrical or elec-trical parameter can be treated in a different way. In this case,the and parameter could be dependent on the parameterand the values of and could be different from zero.

First we consider the term from (23). We can solve

(36)

where can be considered as the solution of the SturmLi-ouville problem, with the same equivalent current vector asin [9]. So

(37)

For the Dirac delta function property (28), if we consider, the integral of (37) becomes

(38)

Since the terms , and do not depend on , sub-stituting (38) into (36), we obtain

(39)

We also notice that the term, due to the property of the matrixproduct, can be placed on the right side of the integral. Now wecan consider the integral (39)

(40)

For the particular structure of in (31a), the result of theintegral will be a vector of the form

(41)

where

(42)

For the sake of simplicity we can consider only the case of thefirst term , because the procedure can be extended alsoto the remaining terms. Thus, recalling (22b), we can write

(43)

The eigenfunctions and constitute two or-thonormal bases and, therefore, the integral of their product isdifferent from zero only if . In this way, wecan consider only one double sum

(44)

where

(45)

Expanding (45), we obtain

(46)

where

(47)



We can distinguish the case of the pair from thecase . In the first case the argument of integral(46) will be unity and therefore the result can be written as

(48)

Otherwise, the solution of (46) will be

(49)

In the case is equal to zero, we can only consider thesecond (first) part of (49). Hence, the result reads

(50)

The second term reads

(51)

Again, due to the Dirac delta function property (28), the resultof the integral becomes

(52)

We can notice that, if does not depend on , the termcan be neglected.

For the third term , the derivative of is equal tozero because the function does not depend on unlike as (25).We can write

(53)

Applying the property (28) we obtain

(54)

Finally, the sensitivity reads

(55)

IV. COMPUTATION OF DERIVATIVESTo compute the voltage sensitivity in (34) and (55) we need

to know the expression of several derivatives depending on thesensitivity parameter.

A. Port Positions

The derivatives involved in (34) are the derivative of the portcurrent vector and the derivative of the Greens function

function with respect to the parameter.The derivative with respect to of the function

can be analytically computed. In particular, recalling (22b), wecan distinguish the case of the self and the mutual Greens func-tion in this way

(56)

Otherwise, if the Greens function does not depend on, its derivative is equal to zero

(57)

The port voltages can be expressed in terms of the port currentsas

(58)

where represents the vector of the port voltages

(59)



and the impedance matrix can be expressed in terms of theGreens functions

.

.

.

.

.

.

(60)

In addition, boundary condition have to be enforced at the elec-trical ports. If linear terminations are considered, they can bewritten in a matrix form as

(61)where the input voltage source vector and the terminationimpedance matrix are

.

.

.

.

.

.

(62)

After trivial matrix manipulations, port currents can be ex-pressed directly in terms of the source voltages as

(63)Now we are ready to compute the derivative of the port currentvector (63) as

(64)

Such a computation can be more efficiently performed throughthe use of the following theorem

Theorem 1: Suppose is a square matrix depending on areal parameter taking values in an open set . Further,suppose all component functions in are differentiable, and

is differentiable for all . Then, in , we have

where is the derivative. Hence, (64) becomes

(65)

The derivative of matrix can now be computed from (60)taking (57) and (56) into account and by recalling that isindependent of .

B. Geometrical or Electrical Parameter

Equation (55) shows the derivative for the sensitivity appliedto geometrical or electrical parameter. The derivative of and

depends on the choice of the parameter. Otherwise thederivative of can be computed as in (65). The only differenceis that the derivative of the Greens function has to be com-puted in different form as in (56) and (57) and also in this casedepends on the choice of the parameter.

V. TIME-DOMAIN MODEL

From (58), the port voltage can be expressed in terms of theport current, and the sensitivity applied at (58) reads

(66)

Impedance matrix entries can be rewritten as [7]

(67)

The rational structure of (67) is well suited for a pole/residueidentification, leading to

(68)

which can be easily converted to a state-space model [23]

(69)

Furthermore, the impedance matrix sensitivity can bewritten as

(70)

where matrices and are obtained by computing (41) and(31a) at the electrical ports.

Since both the two contributions to the impedance sensitivityhave a rational form, they admit a state-space realization

(71)

where , . In addi-tion, (58) admits a space-state realization , , which coin-cides with , , (69)

(72)

The voltage sensitivity can be finally expressed as

(73)

and a global state-space model can be generated as

(74)

(75)



A. Terminations

Port terminations are assumed to be described by currentsources and voltage driven lumped elements such that thefollowing equation holds:

(76)

where and are the vectors of the unknown port voltagesand currents, represents the current source vector, matrices

, describe linear resistive and capacitive lumped elements,respectively. The sensitivity of the terminations with respect toa parameter can be obtained using (76) as

(77)

Substituting (73) into (77), we obtain

(78)In order to compute (76) and (78), we need to evaluate theport voltages and currents, and respectively. They canbe computed from (58) which admits a space-state realization.Once and are computed through (72) and the corre-sponding terminal conditions (76), we can solve the system ofordinary differential equations (ODEs) (69) and (71) along withthe terminations (78) where the input vector is represented bythe port currents . To this aim, we can substitute (78) into(71)

(79)The expression of can be obtained directly from (69). Sub-stituting (69) into (79), we obtain

(80)The solution of (80) requires the vector of the unknown ,already computed in (69), and the port voltage and current,and , respectively, already evaluated by (72) and (76) and,therefore, it can be easily carried out.

Remark 2: The presented technique is based on the closed-form Greens function for voltages via planar-circuit approachfor a single power-bus pair [5], [6]. The extension of such anapproach to multiple plains can be obtained as described in[24]. The sensitivity analysis of multiple plains structures canbe based on the proposed technique and will be investigated inforthcoming reports.

VI. NUMERICAL RESULTS

In this section we propose both frequency and time-domainanalysis in order to verify the validity and strength of theproposed approach. The frequency-domain results are validatedthrough comparison with the perturbative approach computingthe voltage sensitivity as

(81)

Fig. 2. Test board geometry.

Fig. 3. Port positionmagnitude and phase sensitivity spectra at port 1.

where represents the perturbation. Time-domain values arecompared with those obtained by using the frequency-domaindata via the inverse fast Fourier transform (IFFT).

A. Frequency-Domain

We applied the proposed approaches to the power-bus struc-ture shown in Fig. 2. The board dimensions are mm(length), mm (width), mm thickness. On the boardare placed two ports with coordinates: port 1 (20 mm, 20 mm),port 2 (70 mm, 40 mm). The dielectric relative permittivity is

. The number of the modes used for computation of theGreens function is set to 10 along both the and axes and thefrequency range is .

1) Port Positions: First we consider the case of port variation.In particular, we compute the sensitivity on port 1 and port 2when a variation along of port 2 is affected. The input to thiscircuit through the port 1 is a 1 ns voltage pulse with 300 psrise and fall times. We compare the sensitivity computed as (34)and the perturbative approach. Both the ports are terminated in50 resistances. The magnitude and phase sensitivity spectraare shown in Figs. 3 and 4 for port 1 and 2, respectively. As it



Fig. 4. Port positionmagnitude and phase spectra at port 2.

clearly seen, a satisfactory agreement is obtained over the entirefrequency range.

2) Thickness of the Dielectric Substrate: In the second test,the voltage sensitivity is computed with respect to the thicknessof the dielectric substrate, . It is to be noted that, sincethe Greens function does not depend on , the derivativeof the functions are equal to zero for each .The input is the same of the previous test as well as the termina-tions. The comparison of the sensitivities at the port 1 and 2 arecomputed by (55) and the perturbative approach and are shownin Figs. 5 and 6.

In both the tests a satisfactory agreement is achieved.

B. Time-Domain Results

The time-domain analysis of the same board in Fig. 2 hasbeen carried out. As in the frequency-domain analysis, boththickness of dielectric substrate and port position have beenconsidered for the sensitivity analysis. The input at the port1 is represented by a 1 ns voltage pulse with amplitude of1 V, initial delay of 1 ns and 200 ps rise and fall times.The relative dielectric permittivity is . The seriesform has been limited to 10 modes along both the andaxes. As before, the ports are terminated in 50 resistances.The analysis is extended to 60 ns and the integration step is2 ps. We compare the integration of the space state modelproposed and the inverse fast Fourier transform (IFFT) of thefrequency sensitivity computed as explained in Section VI-A.

The voltage sensitivity with respect to the port position isshown in Fig. 7, while the voltage sensitivity with respect to

Fig. 5. Thickness of the dielectric substratemagnitude and phase spectra atport 1.

Fig. 6. Thickness of the dielectric substratemagnitude and phase spectra atport 2.

the thickness of dielectric substrate is shown, respectively, inFig. 8. Table I shows the a comparison of computational cost



Fig. 7. Port positiontime sensitivity analysis at port 1 and 2.

Fig. 8. Thickness of the dielectric substratetime sensitivity analysis at port1 and 2.

using the proposed approach versus the IFFT-based method, as-suming 3 time samples and 3600 frequency samples. The

TABLE ICOMPUTATIONAL COST: PROPOSED VERSUS IFFT

cpu-time saving is about 74% for a single parameter run and canbe even larger when multiparameter sensitivity is considered inthe framework of an optimization process.

VII. CONCLUSION

In this paper, a new methodology for both time and frequencydomain sensitivity analysis of power-bus has been presented.Relying on the fact the propagation of voltages and the corre-sponding sensitivities with respect to a generic parameter obeyto the same differential equation, the 2-D TM Greens functionis used to compute both voltage and sensitivities as convolu-tion integral between the Greens function itself and a forcingterm which can be analytically computed. Furthermore, sincethe model inherits the rational form from the Greens function,the time-domain macromodel in space-state form is easily gen-erated by using standard realization techniques. Hence, the pro-posed method provides a systematic approach to frequency andtime-domain sensitivity analysis of power-bus structures. Thevalidation of the proposed method is carried out by comparingthe numerical results with those obtained using standard ap-proaches in both the frequency and time-domain, confirming itsreliability and robustness.

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Luca De Camillis was born in Teramo, Italy, in 1982.He received the Laurea degree in computer scienceengineering from the University of LAquila in 2007.

From June to December 2008, he was a ResearchScientist at the ACE Laboratory, University ofWashington, Seattle, focusing on macromodelingtechniques and sensitivity analysis applied tohigh-speed digital boards. He is now with MHT,a software-house working on enterprise resourceplanning.

Giulio Antonini (M94SM05) received the Laureadegree (summa cum laude) in electrical engineeringfrom the Universit degli Studi dellAquila, in 1994,and the Ph.D. degree in electrical engineering fromthe University of Rome La Sapienza, in 1998.

Since 1998, he has been with the UAq EMC Lab-oratory, Department of Electrical Engineering, Uni-versity of LAquila, where he is currently AssociateProfessor. His research interests focus on EMC anal-ysis, numerical modeling, and in the field of signalintegrity for high-speed digital systems. He has au-

thored or coauthored more than 180 technical papers and two book chapters.Furthermore, he has given keynote lectures and chaired several special sessionsat international conferences. He holds one European patent.

Dr. Antonini was the recipient of the IEEE TRANSACTIONS ONELECTROMAGNETIC COMPATIBILITY Best Paper Award in 1997, the CSTUniversity Publication Award in 2004, the IBM Shared University ResearchAward in 2004, 2005, and 2006. In 2006, he received a Technical AchievementAward from the IEEE EMC Society for innovative contributions tocomputational electromagnetic on the Partial Element Equivalent Circuit(PEEC) technique for EMC applications. He also received the IET-SMTBest Paper Award in 2008. He is vice-chairman of the dellIEEE EMC ItalyChapter, member of the TC-9 committee, and vice-chairman of the TC-10Committee of the IEEE EMC Society. He serves as member of the editorialboard of IET Science, Measurements, and Technology. He serves as reviewerin a number of IEEE journals.

Vikram Jandhyala (S96M00SM03) receivedthe B.Tech. degree in electrical engineering from theIndian Institute of Technology (IIT), Delhi, India,in 1993 and the M.S. and Ph.D. degrees from theUniversity of Illinois, Urbana-Champaign, in 1995and 1998, respectively.

From 1998 to 2000, he was a Research and Devel-opment Engineer at Ansoft Corporation, Pittsburgh,PA. Since 2000 he has been a faculty member inthe Electrical Engineering Department, as AssistantProfessor from 2000 to 2005, and then as tenured

Associate Professor. He directs the Applied Computational Engineering (ACE)Lab which has received research funding from DARPA, NSF, SRC, Intel,IBM, NASA, Air Force, Navy, Lawrence Livermore, the SBIR program, andthe state of Washington. His research interests include computational electro-magnetics, integral equations, fast multilevel algorithms, parallel computationtechniques, circuit and system applications of field solvers, signal and powerintegrity, EMI/EMC, multiphysics and hierarchical simulation methods, RFIDsystems, statistical and design techniques for field solvers, and graph theoryapplications of field techniques. He is currently on partial leave from theUniversity of Washington while serving as founder and CEO of Physware,Inc., a venture-funded startup producing electronic design automation toolsfor microelectronic simulation and design. He has served as a consultant andreviewer for NASA, DoD, NSF, Science Foundation Ireland, Cadence DesignSystems, McGraw-Hill, Prentice-Hall, Kluwer, Cambridge University Press,and Calypso Medical. He is a full-elected member of URSI Commission B. Hehas published approximately 150 papers in journals and refereed conferenceproceedings.

Dr. Jandhyala is a recipient of a 2008 NASA patent Inventor Award, a 2004Outstanding Graduate Research Advisor Award from UW EE, a 2001 NSF CA-REER award, a Raj Mittra Oustanding Graduate Research Award from UIUCin 1998, and an IEEE Microwave Society Graduate Fellowship in 1997.


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