This paper presents PULP, a MATLAB-based package for the determination of the per unit length parameters of lossless Multiconductor Transmission Lines. The correct computation of these parameters is important in the Electromagnetic Compatibility field, because they allow to model the parasitic interactions between interconnects of electronic systems, which are difficult to consider in a design process. PULP evaluates the per unit length parameters through the Finite Element Method (FEM), using the capabilities of the MATLAB PDE toolbox. An extended set of transmission lines can be analyzed, including wires with circular conductors and multilayered structures. The package is fully based on a Graphical User Interface (GUI) that allows the user to interactively change the geometry and the simulation parameters. Therefore, PULP can be used as a reliable computation tool, but it is also a useful educational tool for students of EMC courses willing to improve their knowledge on the dependence of the electromagnetic coupling on the geometry of the system.

The design of modern electronic systems has become a challenging task due to their increasing complexity. The usual procedure is to split the design in separate parts which are then interconnected. However, even if all the parts have been designed correctly under a functional standpoint, the nonideality of the interconnections may influence and seriously affect the overall performance of the system. The main reason is the spurious electromagnetic coupling between different wires of an interconnect, which leads to the degradation of the electric signals travelling from one part of the system to another.

If we think of a modern printed circuit board, the number of interconnections can be quite large. Therefore, it is important to be able to characterize with good accuracy the electromagnetic fields in the vicinity of the structure with fewest possible parameters. The idealized model of Multiconductor Transmission Lines, henceforth denoted with MTL, stems from this need. It is an approximate model which allows to compute the evolution of the signals through a loaded interconnect, and then to estimate how they are affected by parasitic couplings. The MTL model includes a set of parameters, the per unit length parameters, which fully characterize these couplings. For a lossless MTL, these parameters can be grouped in two matrices, the capacitance matrix C and the inductance matrix L. The former models the electric field interactions due to the vicinity of the conductors, while the latter models the magnetic field interaction due to flux linkage between different subcircuits. These matrices can be evaluated in a closed form in very few cases, which are not significative for the applications. Therefore, a tool is needed for the numerical computation of L and C for an extended set of structures of practical interest for the applications.

The package PULP allows this computation. A GUI has been designed to guide the user through the input of the structure, the simulation, and the output of the results. The structures that can be analyzed include circular and rectangular conductors and multilayered microstrips. Each structure can be in free space, enclosed by a metallic shield, or above a ground plane. In order to deal with open boundaries, second order Absorbing Boundary Conditions (ABC) have been implemented.

This paper is outlined as follows. Section 1 introduces the MTL model and defines the per unit length matrices L and C. Section 2 describes the main features of PULP, and Section 3 shows some numerical examples and results.

The electromagnetic fields are governed by Maxwell’s Equations. The numerical solution of these equations for complicated structures is very difficult. However, when restricting the types of structures to be investigated, the equations can sometimes be reduced to simpler forms. This holds true in the particular case of Multiconductor Transmission Lines.

The general form of a MTL is a set of parallel conductors of translation invariant cross-section (Fig. 1) along a fixed direction z. Even when the cross-section is arbitrary, in the low frequency limit the Maxwell’s Equations reduce to the MTL Equations (also known as Telegrapher’s Equations). When losses in the conductors and in the dielectrics are neglected, these equations read [1]

where v and i are vectors of length N, and represent voltages and currents for each conductor along the line. The parameters L and C are the per unit length inductance and capacitance matrices, respectively. We have assumed that the total number of conductors is N+1, one of them being the reference for the voltages and the return for the currents.

It can be shown [1] that the matrix L for nonmagnetic media can be linked to the capacitance matrix of the structure with the diectric replaced with free space according to

,

where and are the permettivity and permeability of free space. Therefore we only need to evaluate numerically the capacitance matrix.

This can be accomplished by noting that the entries of this matrix are

,

where is the static potential of the j-th conductor and is the static per unit length charge on the i-th conductor. The static potential is the solution of the Laplace equation

(1)

in the cross-sectional plane with Dirichlet boundary conditions on the conductors. The charge on each conductor can be evaluated, once is known, as the integral along the conductor surface of the electrix flux

, (2)

where is the outward unit vector normal to the surface.

From these expressions we can see that the matrix C can be evaluated through N solutions of (1), each time setting the voltage of one conductor to 1 and the voltage of the others to 0. For each solution of the potential , a column of C is obtained by evaluating the N integrals (2) along the boundaries of the N conductors.

Many common types of transmission lines, e.g. microstrips or power lines, are characterized by unbounded domains. Therefore, as the FEM based solutions for the Laplace equation (1) can only be applied to bounded domains, a procedure must be devised to create an artificial boundary that will affect as little as possible the solution. Stadard boundary conditions like Dirichlet or Neumann could be used. However, it is not difficult to provide examples where these conditions seriously affect the solution as well as the values of the per unit length parameters. For this reason, we implemented static Absorbing Boundary Conditions in PULP. While the Dirichlet and Neumann conditions require, respectively, to fix the solution and the normal derivative of the solution on the boundary, ABC determine a link between solution, normal derivative and tangential derivative on the boundary. This leads to far better approximations of the solution within the integration domain, and therefore to much more accurate values for the per unit length parameters. Section 3 will show this fact through a numerical example. A full description of the ABC, together with their analytical derivation, can be found in Ref. [2]

The package PULP (Per Unit Length Parameters) is a set of M-files. The main numerical processor of the package is the PDE toolbox, which provides the FEM engine. The package is built as an interface to the PDE toolbox. A GUI has been designed to guide the user through the three main blocks: input, simulation, and output. The following sections will detail each of these blocks separately.

We describe here the different structures that can be currently analyzed by PULP. The structure can be divided in three categories, itemized below:

• circular conductors with or without dielectric insulation (Fig. 2a);
• rectangular conductors with or without dielectric insulation (Fig. 2b);
• multilayered structures, with an arbitrary number of rectangular conductors above each layer (Fig. 2c).

(a)

(b)

(c)

Each of the structures above can be:

• in free space,
• within a perfectly conducting shield,
• above a ground plane.

These three ways to close the domain lead to different conditions on the outer boundary. The first forces ABC on all the outer boundary, the second uses Dirichlet conditions, while the third uses a combination of the ABC and Dirichlet conditions.

The shape of the outer boundary can be either rectangular (the only available when the geometry is a multilayered structure) or circular. When a rectangular boundary is selected, each of the four edges can be conducting (Dirichlet conditions) or absorbing (ABC conditions).

As mentioned in section 1, one of the conductors must be chosen as a reference. If any part of the outer boundary is conducting, PULP automatically selects it as the reference. If ABC are on the whole outer boundary, instead, the user can choose which of the conductors is the reference. This is a particularly useful feature, because sometimes the parasitic couplings can be reduced with a careful choice of the reference conductor.

The geometry can be fully edited through the GUI, loaded or saved to disk to MAT files. An automatic check for errors in the input parameters is included, and also a set of context-sensitive help windows are available, to guide the user to a correct interpretation of the parameters describing the geometry.

The calculation scheme that leads to the evaluation of the capacitance matrix C has already been described in Section 1. The dialog requiring the simulation parameters depends on the structure. When no dielectric insulations are present, both C and L are determined, while when the dielectric is not homogeneous the user must select which matrix to compute.

We show here how the user can improve the accuracy of the calculations by selecting different mesh refinement algorithms. The PDE toolbox capabilities have been exploited to provide three mesh generation options:

1. global mesh refinement (Fig. 3A),
2. adaptive mesh refinement (Fig. 3B),
3. selective refinement on conductors and outer boundary (Fig. 3C).

The first option guarantees an almost uniform distribution of triangles on the whole integration domain. The second option is needed when the solution developes regions of fast variations opposed to more regular regions. An adaptive mesh is the best way to limit the number of unknowns in the final system. The third option has been added because the post-processing section requires a good discretization of the boundaries of the conductors. Indeed, the total charge on each conductor is obtained through a path integration along these boundaries, and the results are more accurate when a fine discretization is used. A fine discretization on the outer boundary is needed for a good performance of the ABC.

The mesh refinement algorithm can be determined by the user through a dialog as any combination of the three options above. A mesh control string is employed. For example, with the string like "AAB5000C", two global refinements are performed, then a sequence of adaptive refinements up to 5000 total triangles, and finally an additional refinement on the conductors and outer boundary.

During the simulation, the user can also enable partial plotting to visualize the mesh and the solution at any step of the loop described in Section 1.

Initial mesh

(A)

(B)

(C)

At the end of the simulation the per unit length matrices L and C are available to the main MATLAB workspace. In addition, they can be graphically displayed in a window mapping their entries to a colormap (Fig. 4).

This enables the user to quickly visualize which are the couplings between each pair of conductors. Mouse-clicking on the different blocks in the visualization window will show the exact numerical value of the corresponding entry of the matrix. To reduce the computational errors the matrices can be forced to be symmetric through C=(C+C’)/2 and similarly for L.

The deviation of the off-diagonal terms from the symmetry condition can be a measure of the accuracy of the computation. However, systematic errors and biases due to non optimal mesh generation cannot be avoided.

Figure 4. Graphical display of the capacitance matrix

This section includes few examples to validate the results obtained by PULP through comparisons with analytical expressions and with the existing literature.

This section presents numerical results for a two-conductor wire line in free space, depicted in Fig. 5(a). This geometry was chosen because there is a closed-form expression for its per-unit-length capacitance [1],

,

where D is the distance between the two conductors and d is the diameter of each conductor. In the following we will set and in normalized units, obtaining a nominal per-unit-length capacitance .

The domain was artificially limited with a circle of diameter , where H is the diagonal of the smallest box including the two conductors, and ABC were forced on the outer boundary to simulate the behavior of the potential at the infinity. Figure 5(b) reports the relative error on the computed per-unit-length capacitance obtained for different choices of k and for different number of triangles in the mesh. The final mesh was obtained through a sequence of adaptive refinements followed by two refinements on the conductors surface. Figure 5(c) compares the relative error as a function of k when the outer boundary is either an absorbing surface or a conductive shield. The number of triangles in the mesh ranges between 6000 and 8000 for all the points in panel (c). This illustrates that the ABC are quite accurate even when the outer boundary is close to the conductors.

The coupled microstrip shown in Fig. 3 has also been analyzed. The width of the strips is set to 3 (in normalized units), their separation to 2, and the thickness to 1. The strips are placed above a dielectric layer with relative permittivity and thickness equal to 1. The whole structure is enclosed by a box (width=18, height=7) with a conducting bottom side and ABC on the other sides.

(a)

The results obtained by PULP (with mesh control string "ACCB6000") are reported in Table 1, where they are compared to the existing literature [2,3] and to the output of LINPAR [4], a commercial software package based on the Method of Moments. Only the top row of C is reported, because and . The table shows that PULP produces accurate results with respect to the available references.

Table 1. Per unit length capacitance matrix in pF/m for the coupled microstrip of Fig. 6

1. Paul C. R., Analysis of Multiconductor Lines, Wiley & Sons, Inc., 1994.
2. Khebir A., Kouki A. B., and Mittra R., An Absorbing Boundary Condition for Quasi-TEM Analysis of Microwave Transmission Lines via the Finite Element Method, Journal of Electromagnetics Waves and Applications, Vol. 4, 1990, 145-157.
3. Weeks W. T., Calculation of coefficients of capacitance of multiconductor transmission lines in the presence of a dielectric interface, IEEE Trans. Microwave Theory Tech., Vol MTT-18, 1970, 35-43.
4. Djordjevic A. R., Miodrag B. B., Harrington R. F., Sarkar T. K., LINPAR, Copyright 1995, Artech House, Inc.