Parameterized Model Order Reduction of Power Distribution Planes

Majid Ahmadloo*, Sourajeet Roy, Anestis Dounavis

Department of ECE, University of Western Ontario London, Ontario, Canada

Email: adounavis@eng.uwo.ca; mahmadlo@uwo.ca

Abstract— This paper proposes an algorithm to obtain a parameterized reduced order model for system level representations of large power distribution planes over a wide frequency of interest. The key advantage of the proposed algorithm is that, the electromagnetic behaviour of the plane can be efficiently modeled for a wide variation of design parameters without the need to regenerate the reduced model for each parameter change.

I. INTRODUCTION

With increase in operating frequency and decrease in supply voltage, transient currents in the power planes lead to voltage fluctuations, ground bounce and electromagnetic interference [1], [2]. Thus, power planes forms a critical area for system performance and reliability in contemporary high speed digital systems. To accurately characterize the electrical performance of these power planes, accurate modeling of power planes over the entire bandwidth of operation (in the high GHz range) is required. In the past, various full wave numerical techniques like method of moments (MoM), finite element method (FEM) and finite difference time domain (FDTD) [2]-[4] have been developed for analysis of power planes. However on account of the computational expense involved by full wave techniques, representation of power planes using RLGC lumped elements and two dimensional grid of transmission lines have been used [1], [5]-[7]. Lumped circuit representations can be effectively used to model irregular plane geometries and multi plane layers [1], [5] including decoupling capacitors within a SPICE-like simulation environment. However, the high bandwidth of operation, number of planes and decoupling capacitors can lead to large system matrices, requiring high memory and computational time demands. This problem is further exacerbated when one considers the typical design process which includes optimization and design space exploration and thus requires repeated simulations of the same problem for different parameter values.

To address the computational complexity for power distribution planes, model order reduction based on Krylov subspace projection, like PRIMA [8] have been reported [9]. However, these algorithms require the regeneration of the reduced models each time a design parameter is modified. In this work, a parameterized reduced order model for system level representations of large power distribution systems is presented. The algorithm is based on multi dimensional subspace projection that matches the moments of the original system with respect to frequency as well as design parameters of interests. Such an approach is significantly more CPU efficient in optimization since a new reduced model is not required each time a design parameter is modified.

II. MODELING POWER DISTRIBUTION PLANES USING RLGC LUMPED ELEMENTS

In this section, a rectangular power/ground plane is considered which is subdivided into numerous unit cells as shown in Fig. 1. Representing each unit cell with lumped RLC elements using the quasi-static model, similar to [1] allows the plane to be modeled by a large RLC network. The RLC elements of each cell can be analytically derived from the electromagnetic properties of the metal and the geometry of the cell involved. Considering an unit cell of dimensions (a,b) with a dielectric separation of ‘d’ between planes, thickness of metal (t), metal conductivity (σ) and dielectric constant ( rε ), the equivalent RLC parameters are computed as [1]

tR

σ2= ,

dabC ro εε= , dL oμ= (1)

where oε and oμ are the permittivity and the permeability of free space. For this example skin effect losses were ignored. Once the plane has been discretized, the plane and the decoupling capacitors can be represented using RLC lumped elements in an MNA formulation as shown

BuXCG =+ ))()(( λλ s

⎥⎦

⎤⎢⎣

⎡=

0E-EN

G T)(λ , ⎥⎦

⎤⎢⎣

⎡=

Q00P

C )(λ , ⎥⎦

⎤⎢⎣

⎡=

IV

X (2)

where N , P and Q consist of the stamp of the resistive, capacitive and inductive elements respectively, V and I represent the nodal voltages and the inductance currents, E maps the contribution of the current through each inductor and the port voltage sources, B is a selector matrix to map the port voltages u and ],,,[ 21 nλλλλ …= are the design parameters of interest. The next section derives a parameterized reduced model to efficiently solve (2).

III. PARAMETERIZED MODEL ORDER REDUCTION

The computation of the parameterized reduced order model calculates the moments of (2) with respect to frequency and design parameters λ using a procedure similar to [10], [11] to obtain the multi dimensional moment matrix K as

[ ]( )Xkscolsp MMMMK λλ1= (3)

where sM contains the moments with respect to frequency, iλM contains the moments with respect to parameter iλ and XM contain the cross moments. The matrix K is generally ill-conditioned and is converted to an orthonormal matrix Q as described in [12]. Using the orthonormal matrix Q the parametric reduced order model is obtained by a change of variables as

),(ˆ),( λλ ss XQX = (4)

Substituting (4) into (2) and pre-multiplying by TQ yields

( ) uBXCG ˆ),(ˆ)(ˆ)(ˆ =⋅+ λλλ ss (5)

where

QGQG )()(ˆ λλ T= , QCQC )()(ˆ λλ T= , BuQuB Tˆ = (6)

It can be shown that the reduced system of (5) preserves the moments of the original system using techniques presented in [10]. Once (5) is calculated, it can be used to efficiently calculate the response of power distribution planes within a user defined range of frequency and design parameters.

IV. NUMERICAL RESULTS

In this section a numerical example is provided to illustrate the validity of the proposed parameterized model order reduction. A rectangular power/ground plane pair of geometry as provided in [1] is considered (Fig. 1). The

Power plane

Ground plane

Dielectric

b

ad

t

……….……….

……….

0.2 in

0.2 in

0.1 in

Location of decoupling capacitors

y

x

(a) (b)

Fig. 1: (a) Rectangular power plane. (b) Example of a rectangular power plane showing the positions of the decoupling capacitors

Frequency

Log

|Z11

|

Log

|Z11

|=1

Frequency

rε

=12

=1

=12

Reduced Model

Original Model

Reduced Model

Original Model

rε

rε

rε

(a) (b)

Frequency

Log

|Z11

|

Log

|Z11

|

Frequency

Reduced Model

Original Model

Reduced Model

Original Model

d=15mic

d=45mic

d=15mic

d=45mic

(c) (d) Fig.2. Frequency response comparison of Z11 using proposed model with SPICE for different parameter values. (a) Frequency response of Z11 with

1=rε and 12=rε and d = 15 micron (b) Frequency response of Z11 with 1=rε and 12=rε at d = 45 micron. (c) Frequency response of Z11 with d = 15 micron and d = 45 micron at 1=rε . (d) Frequency response of Z11 with d = 15 micron and d = 45 micron at 12=rε .

dimensions of the planes are 6.35 cm by 6.35 cm, thickness 3 microns and separated by a 25.4 micron thick FR4 with relative permittivity 4=rε . The electrical parameters of each unit cell are Ω= mR 131.1 , pHL 3.31= and pFC 98.8= . Decoupling capacitors are placed as shown in Fig. 1(b) and are represented using a series RLC model with constant parameters Ω= mRd 100 , nHLd 47.0= and nFCd 10= . The input port is located at (0.25 cm, 6.1 cm) while the output port is located at (6.1 cm, 0.25 cm). Using a unit cell of dimensions 0.25 cm by 0.25 cm, the plane was divided into 625 unit cells resulting in 4078 unknown variables. The parameters of interest are chosen to be frequency (s) varying from 0 to 15 GHz, the height of dielectric (d) varying from 15 microns to 45 microns and the relative permittivity of material used ( rε ) varying between 1 and 12. The original system is reduced using the parameterized model order reduction methodology outlined in the previous section using MATLAB 2008a on a Pentium 4 (2.8 GHz) PC with 2048 MB memory. The application of the proposed algorithm resulted in the reduction of the original system from 4078 unknown variables into 238 unknown variables. For this example the proposed algorithm required 20 moments for frequency, 15 moments for ε , 20 moments for d and 4 cross moments to capture the frequency domain response of the power plane for the given range of parameters. Fig.2 compares the magnitude of the driving point impedance ( 11Z ) using the parameterized model order reduction proposed with SPICE for height of dielectric (d) varying from 15 micron to 45 micron and the relative permittivity of material used ( rε ) varying from 1 to 12. Fig. 3 shows similar comparisons between the proposed algorithm and SPICE for impedance variable 12Z . In all the above cases, the reduced model was found to display good agreement with the original system. Simulation of the original system requires 4 hours and 24 minutes while the reduced model requires only 13 minutes thereby providing a speed up of more than 20.

=1

Frequency

Reduced Model

Original Model

=1

=12rε

rε

Frequency

Log

|Z12

|

=12

Reduced Model

Original Model

rε

Log

|Z12

|

=1rε

(a) (b)

Frequency

Reduced Model

Original Model

d=15micd=45mic

Frequency

Log

|Z12

|

Reduced Model

Original Model

d=15micd=45mic

Log

|Z12

|

(c) (d) Fig.3. Frequency response comparison of Z12 using proposed model with SPICE for different parameter values. (a) Frequency response of Z12 with

1=rε and 12=rε and d = 15 micron (b) Frequency response of Z12 with 1=rε and 12=rε at d = 45 micron. (c) Frequency response of Z12

with d = 15 micron and d = 45 micron at 1=rε . (d) Frequency response of Z12 with d = 15 micron and d = 45 micron at 12=rε .

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