An effective modeling method for multi-scale and multilayered power/ground plane structures
ABSTRACT As more packages or chips are integrated into a single system in recent electronic designs, power/ground planes tend to contain numerous and diverse features than ever, such as via holes, small apertures, plane gaps, etc. Although dimensions of these features are usually very small compared to the power/ground planes, their influence on the power delivery network cannot be neglected. However, power/ground plane analysis methods using square or rectangular meshes, such as multilayer-finite difference method, can generate large number of unit cells for small features in a large design. Multi-scale structures can be effectively meshed by using triangular mesh scheme, which can then be solved using finite element method (FEM). However, it is difficult to include external circuit elements to FEM, since the equivalent circuit model of FEM is not physically intuitive. In this article, a new modeling method, multilayer triangle element method, which is efficient for solving multi-scale and multilayer power/ground plane structures with external circuit elements, is presented.
An Effective Modeling Method for Multi-scale and Multilayered Power/Ground Plane Structures
Jae Young Choi and Madhavan Swaminathan
School of Electrical and Computer Engineering
Georgia Institute of Technology
Atlanta, GA 30332
As more packages or chips are integrated into a single
system in recent electronic designs, power/ground planes tend
to contain numerous and diverse features than ever, such as
via holes, small apertures, plane gaps, etc. Although
dimensions of these features are usually very small compared
to the power/ground planes, their influence on the power
delivery network cannot
power/ground plane analysis methods using square or
rectangular meshes, such as multilayer-finite difference
method, can generate large number of unit cells for small
features in a large design. Multi-scale structures can be
effectively meshed by using triangular mesh scheme, which
can then be solved using finite element method (FEM).
However, it is difficult to include external circuit elements to
FEM, since the equivalent circuit model of FEM is not
physically intuitive. In this article, a new modeling method,
multilayer triangle element method, which is efficient for
solving multi-scale and multilayer power/ground plane
structures with external circuit elements, is presented.
be neglected. However,
Design and analysis of power delivery network (PDN)
become more challenging as more dissimilar electronic
components are integrated in an electronic system. To meet
the different power supply requirements of the integrated
components, power/ground planes are usually stacked on each
other or separated by an aperture or a gap. These apertures or
gaps can provide paths for horizontal/vertical coupling of the
electromagnetic energy. Furthermore, the effect of smaller
discontinuities such as via anti-pads for signal routing is no
longer negligible in high-speed systems. This coupled energy
leads to unwanted power/ground noise, which can result in
large fluctuations in supply voltage. Therefore, a modeling
method that can effectively analyze the impedance of the
PDN is required.
3D Full-wave solvers can provide accurate solutions for
most of the problems. However, they tend to generate too
many unknowns for complex and multi-scale structures, and
thus require large memory and long simulation time.
Therefore, considerable research is being devoted to obtain an
accurate solution for power/ground structures with efficient
One of the approaches is multi-layer finite difference
method (MFDM) , which is based on finite difference
method using square mesh. MFDM can accurately solve for
multi-layer structures while generating a sparse system
matrix. However, the square mesh may generate too many
unknowns for small features in large designs. Furthermore,
square mesh suffers from the staircase approximation error.
Thus, a square mesh is not suitable for the modeling of a
typical PDN, which contains multi-scale and arbitrary
geometries. Another approach for power/ground modeling is
multi-layer finite element method (MFEM) . MFEM uses
surface triangular mesh, which generates far fewer unknowns
than hexahedral or tetrahedral meshes. However, the
equivalent circuit of MFEM is complex and not physically
intuitive. Thus, connecting external circuit elements (e.g. via,
decoupling capacitor) to the model can be complicated.
This paper proposes a modeling method, multilayer
triangle element method (MTEM), which applies a surface
triangular mesh along with its dual graph, for the analysis of
multilayer power/ground planes. The method employs the
orthogonal property between Delaunay triangulation and its
dual graph, Voronoi diagram. Figure 1 shows an example of
the dual meshes generated on a rectangular plane, and the
equivalent circuit of a triangle cell. On the mesh, Kirchhoff’s
current law and Maxwell-Ampere circuital law are applied to
obtain the values for the equivalent circuit. Authors in 
presented a similar approach for a single plane pair; however,
the unit cell is not a triangle but a polygon formed by the dual
graph of triangulation. Since the polygon has more node
values that are subject to calculations, the generated system
matrix will be dense. Furthermore, it is only applied to a
single plane pair. MTEM is extended to multiple plane-pair
structures, taking into account the coupling between plane
pairs . To implement the coupling, MTEM uses the
indefinite admittance matrix method  and assigns each
reference node to a common node.
This paper is organized as follows. Section II explains the
modeling method for a single plane-pair. Section III extends
the problem to multi-layer structures and describes inclusion
of external circuit elements. Section IV demonstrates results
for test cases with comparison to other simulation tools, and
Section V summarizes the conclusions.
Fig. 1. Triangulation (thick line) and dual graph (thin line) are applied to a
978-1-61284-498-5/11/$26.00 ©2011 IEEE477 2011 Electronic Components and Technology Conference
II. Modeling of Single Plane-Pair
A. Two Models: Equivalent Circuit and Electromagnetic
Since the thickness of typical power/ground planes is
electrically very small, the electric field along the z-axis can
be assumed as invariant. Hence, the equivalent circuit for
power/ground planes can be approximated as a planar circuit
. In Figure 2, an example of power/ground planes with
arbitrary shape is shown with its equivalent circuit, which is
composed of lumped circuit elements. The circuit model can
be described by Kirchhoff’s Circuit Law (KCL),
?jωC?? G??V?? ?
From the viewpoint of electromagnetics, the electric flux
leaving or entering the given structure can be assumed that it
exists only on x-y plane as shown in Figure 3 (a). According
to Maxwell-Ampere’s circuital law (Equation (2)), the time-
varying electric field will produce a magnetic field along the
contour of the given structure (Figure 3 (b)),
? ? H ???? J?? jωεE ? ??
Finally, the voltage solution of the given structure, Vi, can
be obtained by using an analogy between the two equations,
namely Equation (1) and (2). Detailed procedures are
explained in the following subsections.
B. Mesh and Formulation
A triangular mesh is built based on Delaunay
triangulation. It has an important property that it has a dual
graph, Voronoi diagram, which intersects the edges of the
meshed triangles orthogonally. This property of the mesh
geometry is important in the sense that electric and magnetic
fields are orthogonal to each other. Hence, the concept of
Maxwell-Ampere’s circuital law (Equation (2)) can be applied
to the triangular mesh with the dual graph.
In Figure 4, a triangle simplex with its neighboring
triangle cells is depicted with its duality, Voronoi diagram.
To apply Maxwell-Ampere’s circuital law to a finite object, it
is convenient to use an integral form ,
? H ???∙ dl ????? ??J?? jωϵE ? ??? ∙ ds ?
where C and S are the contour and surface of the unit triangle,
respectively. The E- and H-fields can be represented by two-
dimensional voltage distribution, V(x,y),
E ? ??? ?z ?V?x,y?
H ???? ?
jωμd??V?x,y? ? z ?? ∙ dl ???? (5)
where d is the dielectric thickness. The gradient of V(x,y) can
be approximated as a linear voltage difference between the
unit triangle and the neighboring triangles. Therefore, the
left-hand side of Equation (3) is derived as follows:
? H ???∙ dl ????? ?
???V?x,y? ? z ?? ∙ dl ????
jωμd??V?x,y? ∙ ?z ? ? dl ?????
Fig. 2. Equivalent circuit of an arbitrary power/ground plane.
Fig. 3. Two-dimensional representation of (a) electric flux and (b) magnetic
field in an arbitrary power/ground plane.
Fig. 4. A triangle unit cell with its neighboring triangle cells. Dotted lines
are dual graph of the triangulation.
u ????∙ ?u ????l??
Notice that the unit triangle has only three neighboring
nodes that are subject to calculation. On the contrary, 
used a polygon that is created by Voronoi diagram as a unit
cell. Because of the nature of Delaunay triangulation, each
polygon will face about six neighboring nodes that are subject
to calculation, as a result of which the system matrix can be
dense. Comparison of the memory consumption will be
discussed in section IV.
In the right-hand side of Equation (2), current density, J, is
the density of conduction current, whose field meets the
integral path orthogonally. Thus, taking surface integral of
the equation eliminates the J-term. Moreover, when the size
of the unit triangle is electrically small enough, the electric
potential can be assumed to be uniform. Therefore, the
equation can be derived as follows:
??J?? jωϵE ? ??? ∙ ds ?
d∙ ds ?
where Ai is the area of the unit triangle.
Finally, the values of the lumped elements in the
equivalent circuit can be obtained by merging Equations (6)
and (7) into Equation (1):
For clarity, loss terms are not included for though their
incorporation is straightforward.
III. Modeling of Multiple Plane-Pairs
A. Shifting of Reference Nodes
Multiple plane-pairs are nothing but a set of single plane-
pairs. The equivalent circuits for each single plane-pair can
be obtained by the procedure explained in the previous
section. However, the equivalent circuits for different plane
pairs do not reference to the same layer. Therefore, stacking
the each equivalent circuit on top of each other without
modifying its reference node will result in an incorrect.
One way to correctly model multiple plane-pairs is to shift
reference nodes using indefinite admittance matrix .
Consider the unit cell model shown in Figure 5. This three-
layered structure can be decomposed into two single plane-
pairs as shown in Figure 5 (b). The inductance and
capacitance models are shown in Figure 5 (b) and (c). L12 and
L34 are per unit cell inductances for each plane pair that can be
obtained from Equation (8). Assuming the plane 3 is the
common ground, the indefinite admittance matrices for the
top and bottom plane-pairs can be derived as follows:
? ? ?
I?? ? ?Y?
Similarly, an admittance matrix for capacitance between
plane-pairs is obtained as follows:
? ? ?
Loss terms are omitted in both models for simplification.
Finally, interposing all the indefinite admittance matrices,
Equation (9), (10), and (12), completes the total admittance
matrix for the given three-layered structure:
Fig. 5. (a) Cross-section of a three-layer structure, (b) inductance model, and
(c) capacitance model.
Y??? Y??? Y?? Y?
Y??? Y??? Y?? Y??
Since the total admittance matrix is a sparse matrix, a
direct linear equation solver can effectively solve the problem.
For example, using nested dissection method , the
computation time for the matrix of size N×N can be achieved
to O(N1.5), and required storage can be O(0.5Nlog2N), where
N=kM, and k and M are the number of layers and unit
B. Meshing and Sub-domains of Apertures
To apply the indefinite admittance matrix method for
multiple plane-pairs as described in the previous section,
mesh nodes on each layer should be in the same (x, y)-
location. This can be done by meshing a layer on which all
the features from each layer are put together . The Features
can include any physical structures such as apertures and
plane boundaries, which need to be meshed. Each feature is
assigned to its unique sub-domain.
When a problem structure contains apertures, the node
within a sub-domain of apertures needs to be considered in a
different way. If a calculation node is inside of a sub-domain
of a plain-conductor, the values of lumped elements
connected to the node can be directly obtained by Equation
(8). However, if a calculation node is inside of a sub-domain
of an aperture, it should be treated with three different cases:
1) Apertures are neither on the current layer nor on the
layer below: The node calculation is executed
equivalently when the node is in the sub-domain of a
2) The node and an aperture are on the same layer:
Since the node has no contribution to the system, no
calculation is conducted.
3) An aperture or apertures are on the layer(s) below
(consecutively): modifications of permittivity (εr),
permeability (µr), dielectric thickness of Equation (8),
and the reference node are required.
To further explain case three, Figure 6 (a) shows a three-
layer structure with an aperture in the middle layer. The
equivalent circuit for m-th calculation node is shown in Figure
6 (b). Since the conductor plane on the second layer is
missing, the thickness of the dielectric between m-th node and
(2N+m)-th node is d1+d2. If dielectric material is
homogeneous on both layers, modified values of the lumped
elements for the m-th node are,
L??? μ?d?? d??h??
where k=1,2, and 3, representing the indices of neighboring
Since a conductor is missing on (m+N)-th node on the
second layer, the reference node of the m-th node has to be
shifted to (m+2N)-th node on the third layer, where N is the
number of the nodes on a single plane-pair. Shifting reference
node is done using an indefinite admittance matrix,
?2? ? ????
C. Inclusion of Decoupling Capacitors and Vias
The addition of a decoupling capacitor to the
power/ground plane model can be conducted in the concept of
equivalent circuit. The equivalent circuit for a typical
decoupling-capacitor can be modeled connecting capacitors,
equivalent series inductance (ESL), and resistance (ESR) in
series. With the values of the circuit elements, the two-port
admittance matrix for the decoupling-capacitor network can
be created. Finally, the two-port admittance matrix is
stamped on to the admittance matrix of the power/ground
plane in accordance with the node connectivity as similarly
done in Equation (16). For example, if a decoupling capacitor
is connected between node-n on the upper plane and node-
(n+N) on the lower plane, the admittance matrix of the
decoupling-capacitor network (Ydecap) is added to the
power/ground system as follows:
Fig. 6. (a) Cross-section of a three-layer structure, (b) inductance model, and
(c) capacitance model.
?? ? ????
A. Single Plane-Pair
To compare memory requirements, and to provide the
validity of MTEM, the first example is a simple single plane-
pair with rectangular planes. It consists of two solid metal
planes and a dielectric layer between them. The metal planes
are size of 40mm × 30mm, and a 200µm dielectric layer is
placed between them. Dielectric constant of the dielectric is
4.5, and conductivity of the metal is 5.8×107S/m. Two ports
are located at (10, 15) mm and (20, 15) mm. Unit cells are
created by Delaunay triangulation and Voronoi diagram as
shown in Figure 7. For better comparison of the simulation
accuracy, dielectric loss is ignored.
Since the structure is completely rectangular without an
aperture, cavity resonator method  can provide accurate
impedance responses at the port locations. The structure was
also simulated with MFDM and MFEM, as well as with
MTEM. Figure 8 shows the transfer impedance between port
1 and 2, Z21, and a good agreement among MTEM, MFEM,
MFDM, and cavity resonator method.
Results from all of the four methods converge to the exact
solution within a percent error of 0.04. With this accuracy,
MTEM created the fewest non-zero elements in a system
matrix. Fewer non-zeros in a matrix mean less memory is
required for the matrix calculation. Note that the method in
 uses the same mesh nodes as MFEM; triangle vortexes. In
addition, a node in both methods is subject to account for the
interactions between the neighboring nodes and itself. Hence,
the number of unknowns and non-zeros are equivalent as
MFEM for the same mesh. Memory comparison of MTEM
and the other methods is summarized in Table I.
B. Multiple Plane-Pair with Aperture
To validate MTEM of handling more complex problems, a
three-layer power/ground plane with small apertures in the
middle layer is designed as shown in Figure 9. The
dimensions of the conductor planes are 40mm × 30mm, and
the apertures are 1mm × 1mm. Planes are separated by a
dielectric layer whose thickness is 200µm, with εr=4.5. Loss
tangent of the dielectric is 0.02, and the conductivity of the
metal is 5.8×107S/m. Two ports are located at (1.25, 1.25)mm
and (39.25, 15.25)mm, connecting plane layer 2 and 3, and 1
and 2, respectively. Meshes are generated as shown in Figure
The simulation results from MFDM, MFEM, and MTEM
agree well as shown in Figure 11 and 12. Notice that
considerable amount of coupling is observed in the transfer
impedance. This coupling is due to the presence of the small
apertures in the middle layer, and the electromagnetic energy
gets coupled through the holes. MTEM created a system
matrix with 2,226 unknowns, which is smaller than that of
MFEM (5,712) and MFDM (9,600). The number of nonzero
Fig. 9. Magnitude of the transfer impedance (Z21)
COMPARISON OF MEMORY CONSUMPTION
Method Non-Zeros Unknowns
Z21 at 1GHz
- - 0 -j0.657609
4,434 1,126 0.027 -j0.657433
MFEM 7,925 1,153 0.037 -j0.657854
MFDM 23,720 4,800 0.035 -j0.657036
Fig. 7. Top view of a single plane-pair structure with triangulation (solid)
and dual graph (dotted) mesh
Fig. 8. Magnitude of the transfer impedance (Z21)
Analyitic (Cavity Resonator Method)