Download Parameterized Model Order Reduction of Power Distribution Planes Majid Ahmadloo

2010 14th International Symposium on Antenna Technology and Applied Electromagnetics [ANTEM] and the American Electromagnetics Conference [AMEREM]
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: [email protected]; [email protected]
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 ( ε r ), the equivalent RLC parameters are computed as [1]
R =
2
σt
, C = ε oε r
ab
d
, L = μod
(1)
where ε o and μ o 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
978-1-4244-5050-3/10/$26.00 ©2010 IEEE
2010 14th International Symposium on Antenna Technology and Applied Electromagnetics [ANTEM] and the American Electromagnetics Conference [AMEREM]
Location of decoupling capacitors
b
0.2 in
d
0.2 in
a
……….
……….
Power plane
y
……….
t
x
Dielectric
Ground plane
0.1 in
(a)
(b)
Fig. 1: (a) Rectangular power plane. (b) Example of a rectangular power plane showing the positions of the decoupling capacitors
(G(λ ) + sC (λ )) X = Bu
⎡ N
G(λ ) = ⎢ T
⎣- E
E⎤
⎡P 0 ⎤
⎡V ⎤
⎥ , C (λ ) = ⎢
⎥, X =⎢ ⎥
0⎦
⎣ 0 Q⎦
⎣I ⎦
(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 λ = [λ1 , λ2 , …, λn ] 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
([
K = colsp M s
M λ1
M λk
MX
])
(3)
where M s contains the moments with respect to frequency, M λi contains the moments with respect to parameter λi
and M X 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
X ( s, λ ) = QXˆ ( s, λ )
(4)
Substituting (4) into (2) and pre-multiplying by Q T yields
(Gˆ (λ ) + sCˆ (λ ))⋅ Xˆ (s, λ ) = Bˆ u
(5)
where
Gˆ (λ ) = Q T G (λ )Q ,
Cˆ (λ ) = Q T C (λ )Q ,
Bˆ u = Q T Bu
(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
2010 14th International Symposium on Antenna Technology and Applied Electromagnetics [ANTEM] and the American Electromagnetics Conference [AMEREM]
Reduced Model
Original Model
Original Model
Log |Z11|
ε r =1
Log |Z11|
Reduced Model
ε r =1
ε r =12
ε r =12
Frequency
Frequency
(b)
Reduced Model
Reduced Model
Original Model
Original Model
Log |Z11|
Log |Z11|
(a)
d=45mic
d=45mic
d=15mic
d=15mic
Frequency
(c)
Frequency
(d)
Fig.2. Frequency response comparison of Z11 using proposed model with SPICE for different parameter values. (a) Frequency response of Z11 with
ε r = 1 and ε r = 12 and d = 15 micron (b) Frequency response of Z11 with ε r = 1 and ε r = 12 at d = 45 micron. (c) Frequency response of Z11 with d =
15 micron and d = 45 micron at ε r = 1 . (d) Frequency response of Z11 with d = 15 micron and d = 45 micron at ε r = 12 .
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 ε r = 4 . The electrical parameters of each unit cell are R = 1.131mΩ , L = 31.3 pH
and C = 8.98 pF . Decoupling capacitors are placed as shown in Fig. 1(b) and are represented using a series RLC
model with constant parameters Rd = 100mΩ , Ld = 0.47 nH and C d = 10nF . 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 ( ε r ) 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 ( Z11 ) 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 ( ε r ) varying from 1 to 12. Fig. 3 shows similar comparisons between the proposed
algorithm and SPICE for impedance variable Z 12 . 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.
Log |Z12|
Log |Z12|
2010 14th International Symposium on Antenna Technology and Applied Electromagnetics [ANTEM] and the American Electromagnetics Conference [AMEREM]
ε r =12
Original Model
Original Model
Frequency
Frequency
(a)
(b)
d=45mic
d=15mic
Log |Z12|
Log |Z12|
ε r =1
Reduced Model
=1
ε r =1
Reduced Model
ε r =12
d=15mic
d=45mic
Reduced Model
Reduced Model
Original Model
Original Model
Frequency
Frequency
(c)
(d)
Fig.3. Frequency response comparison of Z12 using proposed model with SPICE for different parameter values. (a) Frequency response of Z12 with
ε r = 1 and ε r = 12 and d = 15 micron (b) Frequency response of Z12 with ε r = 1 and ε r = 12 at d = 45 micron. (c) Frequency response of Z12
with d = 15 micron and d = 45 micron at ε r = 1 . (d) Frequency response of Z12 with d = 15 micron and d = 45 micron at ε r = 12 .
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