EFFECTIVE PEEC MODELING OF TRANSMISSION LINES STRUCTURES USING A SELECTIVE MESH APPROACH

Electrical Engineering: An International Journal (EEIJ), Vol. 1, No. 1, June 2014
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EFFECTIVE PEEC MODELING OF
TRANSMISSION LINES STRUCTURES USING A
SELECTIVE MESH APPROACH
A.B Gurulakshmi1
and Dr. N. Suresh Kumar2
1
Department of Electronics and Communication Engineering,Vickram College of
Engineering , Enathi, Tamilnadu
2
Velammal College of Engineering and Technology, Madurai, Tamilnadu
ABSTRACT
The transmission lines structures are quite common in the system of electromagnetic compatibility (EMC)
analysis. The increasing complexities of physical structures make electromagnetic modeling an
increasingly tough task, and computational efficiency is desirable. In this paper, a novel selective mesh
approach is presented for partial element equivalent circuit (PEEC) modeling where intense coupling parts
are meshed while the remaining parts are eliminated. With the proposed approach, the meshed ground
plane is dependent on the length and height of the above transmission lines. Relevant compact formulae for
determining mesh boundaries are deduced, and a procedure of general mesh generation is also given. A
numerical example is presented, and a validation check is accomplished, showing that the approach leads
to a significant reduction in unknowns and thus computation time and consumed memories, while
preserving the sufficient precision. This approach is especially useful for modeling the electromagnetic
coupling of transmission lines and reference ground, and it may also be beneficial for other equivalent
circuit modeling techniques.
KEYWORDS
Transmission Lines, PEEC Method, selective mesh approach
1. INTRODUCTION
The increasing complexities of physical structures signal features and electromagnetic (EM)
environment of modern electronic systems make EM modeling an increasingly tough task.
Despite significant advances in EM modeling methodologies, computational efficiency is
desirable especially for complex modeling problems. Currently, the partial element equivalent
circuit (PEEC) method [1] is one of the promising numerical methods for EM modelling of
various engineering problems, e.g., EM compatibility (EMC), EM interference (EMI), and signal
integrity (SI) of high-speed digital circuits [2]. The main advantage of PEEC is its ability to
provide a circuit interpretation of the electric field integral equation (EFIE) in terms of partial
elements, namely resistances, partial inductances and coefficients of potential. It especially has
great potentials for mixed electromagnetic-circuit problems because it is ease to integrate the field
solver with real circuit elements. Integration of a PEEC model directly into a circuit simulator is
computationally expensive for two main facts. One is that a large number of circuit elements are

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generated for complex structures at high frequencies; and the other is that the circuit matrices
based on modified transmission lines [3] are usually dense due to full inductive and capacitive
coupling. In order to model/simulate such problems efficiently, developing compact model
representation via meshing for the PEEC Model is necessary.
2. OVERVIEW OF PEEC MODELING
A numerous literature survey on the main numerical methods is carried out and a great attention
is addressed to those fulfil the following issues:
• Formulation in both time and frequency domain;
• Full-wave method: the mathematical formulation is based on the complete Maxwell's
equations;
• Full-spectrum method: model valid from DC to the maximum frequency determined by
the meshing;
• Low computational time.
On the basis of these requirements, the Partial Element Equivalent Circuit (PEEC) method is
considered the most suitable approach: it is based on the circuital interpretation of the mixed-
potential integral and it is characterized by a great versatility in its applications.
2.1. Partial Ele ment Equivalent Circuit Construction
The classical PEEC method is derived from the equation for the total electric field at a point[2].
An orthogonal metal strip with 3 nodes and 2 cells.

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The corresponding PEEC circuit.
where is an incident electric field, is a current density, is the magnetic vector potential,
is the scalar electric potential, and the electrical conductivity all at observation point . In
the figures on the right, an orthogonal metal strip with 3 nodes and 2 cells, and the corresponding
PEEC circuit are shown. By using the definitions of the scalar and vector potentials, the current-
and charge-densities are discretized by defining pulse basis functions for the conductors and
dielectric materials. Pulse functions are also used for the weighting functions resulting in a
Galerkin type solution. By defining a suitable inner product, a weighted volume integral over the
cells, the field equation can be interpreted as Kirchhoff's voltage law over a PEEC cell consisting
of partial self inductances between the nodes and partial mutual inductances representing the
magnetic field coupling in the equivalent circuit. The partial inductances are defined as
for volume cell α and β. Then, the coefficients of potentials are computed as
2.2 PEEC model reduction
The rigorous full-wave version of the PEEC method is called (Lp,P,R,t) PEEC, where Lp is
partial inductance, P is potential coefficient (inverse of capacitance), R is resistance, and t is
delay. If available, reduced model of the full-wave version can be used. For example, if the EIP
structure is electrically small, the delay term t can be omitted and the model can be reduced to
(Lp,P,R) PEEC model. In addition, if frequency is sufficiently high so that w*Lp >> R, we can
omit R term and use approximate (Lp,P) PEEC model. According to various modeling situations,
(Lp) and (Lp,R) models are also useful.
2.3 Meshing for the PEEC Model
Meshing is an important issue in accurate and effective PEEC modeling, as other numerical
methods [2]. Two kinds of discretizations are constructed in this method. After the initial node
placement, surfaces are meshed using quadrilateral elements from which coefficients of potentials
are calculated using (1). Depending on the boundaries of the surface mesh, volume cells are
created as hexahedral cells from which partial inductances and resistances are calculated using (3)
and (4), respectively [5]. It facilitates the partial elements calculation using such quadrilateral and
hexahedral elements in the mesh [5, 6].

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Figure 1(a) shows the elementary surface and volume discretization in three dimensions, where
the numbers 0 » 6 denote the different surfaces, and I ~ VI are the volume cells. Fig. 1(b)
presents the node placement of a three dimensional conductor. Three kinds of nodes are set
including one inner node as (1), six surface nodes as (2), eight vertex nodes as (3), and 12 edge
nodes as (4). Totally, 27 nodes are obtained. With such node placement, 54 surface cells
(6x1+8x3+12x2 = 54) are formed as in Fig. 1(c) indicated with different patterns. Fig. 1(d) shows
two volume cells, and actually 18 such volumes cells are created in one current orientation by any
two adjacent nodes. This results in 54 volumes in total in an ordinary 3-D orthogonal coordinate
system [7]. The basic rule of thumb when carrying out the PEEC discretization is to use a fixed
number of cells per shortest wavelength λ min (corresponding to the highest frequency of
interest) to assure sufficient accuracy. Originally, approximately 10 cells per λmin was
Figure 1. Surface - and volume- mesh of a rectangular structure (a) elementary mesh;(b)
node placement; (c) surface cells; (d)volume cells.
3. A NOVEL SELECTIVE MESH APPROACH
3.1 Transmission Lines
If the lengths of traces are in the range of the signal's wavelength, then the user has to consider
the effects of transmission lines. The problems that a user must deal with are time delay,
reflections, and crosstalk. To get a better understanding of these problems and where and how
they arise, it is useful to know what transmission lines are. They are simply the traces on a PCB
and depend on the length and the frequency of the signals passing through them. Many different
structures of trace routing are possible on a PCB. Two common structures are shown in Figure 2.

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On the left, a microstrip structure is illustrated and on the right, a stripline technique. A microstrip
has one reference, often a ground plane, and these are separated by a dielectric. A stripline has
two references, often multiple ground planes, and are surrounded with the dielectric.
Figure 3. Structure and Dimension of Microstrip and Stripline
The following sections describe some important properties of transmission lines which are
significant for PCB design. Many software tools are available to calculate the properties of the
several transmission line structures. In this application report, the freeware AppCAD from
Agilent is used so that the reader can become familiar with these properties. Figure 4 shows the
two structures, microstrip (top) and stripline (bottom). The dimensions, the material, and the
frequency are in each case shown on the left. The results that are used in the following sections
can be seen on the right. Table 1 defines the symbols used in Figure 3 and Figure 4.
Figure 4. Calculation of Properties of Microstrip and Stripline (AppCAD)

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Symbol Description
H Height of the dielectric
W Width of the trace
L Length of the trace
Z0 Characteristic Impedance of the trace
λ Wavelength of the trace at the given frequency
W/H Ratio between trace width and trace length
Table 1. Description of the Symbols Used in Figures 3 and 4
3.2 A Selective Mesh Approach
The magnetic field H indicates the inductive coupling, while the electric field E represents the
capacitive coupling. Since most of coupling energy is concentrated in a limited region, only
intense coupling parts are meshed in the proposed selective mesh approach, while the other parts
are eliminated. This can reduce the computational time and consumed memory. Here we set
mesh boundaries in both sides of a filament, and conventional mesh will be carried out only
within the boundaries. The following problem is to quantitatively determine the mesh
boundaries.
…(4)
A threshold value k which means the occupation of field within the boundaries over the total
field in free space is defined in (4). The threshold value is user-defined which determines the
approximation accuracy, and normally a larger value of k results in a better approximation.
Numerical validations indicate that a value larger than 0.9 is suitable for most practical
problems. It is not difficult to solve the value of xo in (4), and it is a compact function of the
height hand the threshold value k, as shown in (5).
…(5)
It is evident in (5) that, with the selective mesh approach, the necessary mesh boundaries of
ground plane are dependent on height and the length of the above wires. In some wire-ground
structures where the ground conductor is relatively much larger than the mesh region defined by
(5), meshing the entire structure in a conventional manner with an identical discretization size is
not effective. Figure 5 illustrates a framework of the proposed mesh generation.
1) Count the number of straight wire segments (Sw) and flat planes (Sp). The notation w(i)
x , w(j)
y
, and w(k)
z are used to identify wire segments, and P(l)
XY , P(m)
XZ , P(n)
YZ to name different planes.
The counters i, j, k, l, m, n all start with one. So, Sw = i + j + k and Sp = l + m + n..
2) Calculate the distances of wire-plane pair indicated in Table 1.
3) Calculate the mesh boundaries of each non-perpendicular wire- plane pair using (5).
4) Overlapping process. Once the selective mesh surfaces are overlapped, the overlapping
subdivisions are merged as

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Figure 5. Frame work of the proposed mesh generation.
4. NUMERICAL RESULTS
The configuration of coupled microstrip lines shown in Fig. 7 is typical in printed circuit board
(PCB) structures. The common ground provides a possibility of interference due to crosstalk. The
two transmission conductors are 50mm long and 3mm apart. They have identical rectangular
cross sections with dimensions of 1mm in width and 0.2mm in thickness. A voltage excitation
source, VS (1 volt) consisting of a source resistance RS (50 ohms) is connected at a terminal S,
and another end of the generator conductor is terminated with a 50 ohms resistor. A receptor
conductor connects two terminations NE and FE, represented by resistors RNE (50 ohms) and RFE
(50 ohms).Excellent agreements are achieved with a suitable threshold value k = 0:95.
Figure 7. A simulation structure of coupled microstrip lines .

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Since our PEEC modelling code at present is mainly based on the quasi-static PEEC model,
some difference between measurements and calculations presents at high frequencies.
Figure 8. Measurement and nume rical results by full mesh and selective mesh.
Table 2 Computation consumption of the full model and reduced model
Table 2 shows the numbers of unknowns and the consumed time of the full model and the
reduced model. It is evident that the consumed time of the element calculation and circuit
solving of the reduced model in this example is only 1/5 and 1/25 of the counterparts of the full
model. Numerous practical applications indicate that both the number of unknowns and the
consumed time are reduced using the proposed selective mesh approach.
5. CONCLUSION
A selective mesh approach which is consistent with coupled field distribution is proposed for
PEEC modeling in this paper. The intense coupling parts are meshed while the remaining parts
are eliminated. The resulting meshed region of ground plane is dependent on the length and
height of the above conductive path of transmission line. Numerical results show that the mesh
approach can greatly reduce the unknowns while preserving the sufficient precision, which
follows the reduction of consumed modeling time. This approach especially has potentials for
transmission analysis in aspect of system EMC and can be extend to other equivalent circuit
modeling techniques.
REFERENCES
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ructures Using A Selective Mesh Approach “, Progress In Electro magnetics Research, Vo l. 123, 355
- 370, 2012

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[2] Z. F. Song and D. L. Su “ Model Order Reduction For Peec Modeling Based On Moment Matching ”
Progress In Electro magnetics Research, Vo l. 114, 285 -299, 2011
[3] Paul, C. R., Introduction to Electro magnetic Co mpatibility, 2nd edition, John Wiley & Sons,
Hoboken, New Jersey, 2006.
[4] Kong, L. and M. Luo, “Co -frequency interference suppression algorithm via maximu m signal minus
interference level," Progress In Electro magnetics Research, Vo l. 104, 183-199, 2010.
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suppression of simultaneous switching noise with the time-do main impedance function for h igh-
speed circu it design," Progress In Electro magnetics Research, Vo l. 112, 4-62, 2011.
[7] Paul, C. R., Analysis of Mult iconductor Transmission Lines, 2nd edition, Wiley-IEEE Press, New
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[8] Roy, A., S. Ghosh, and A. Chakraborty, “Simp le crosstalk model o f three wires to predict near-end
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