Designing and Modeling Connectors in Power Circuits

Increasing communication speed within an electronic system requires minimizing interconnection inductance. Many high-power applications get lower inductance by using busbar structures with the different power modules. Even if the designer can ignore the busbar contribution on total loop inductance, he still needs to characterize system connectors. Busbar design is tricky, and validating a correct geometry is often based on a long and expensive prototype realization. A challenge is in choosing the proper choice of CAD tools to design such structures.

Simulation allows the designer to compute the parasitic inductance of a given structure. With simulation, the designer also gets the current distribution in the conductor in order to investigate coupled thermal and electromechanical effects. With proper design tools and models, it is possible to obtain an accurate model of the system's interconnect structures.

We first choose a model and compare several methods of computing its characteristics. Then we review the chosen method, PEEC (Partial Element Equivalent Circuit), to show the assumptions made by a designer and how it is possible to use PEEC for modeling power-electronics components. The paper then presents some simple structures and busbars. Even if the chosen geometry is complicated, due to a large width/thickness ratio, small spacing between the copper sheets on a printed-circuit board, and no preferential direction for the current in large copper sheets, the proposed modeling is still accurate.

The next section deals with the use of the electrical equivalent model of connectors. With this model, you can calculate equivalent impedances to estimate over-voltage, coupling between power and control parts, and current flowing into each conductor. The current flow information lets a designer treat some coupled aspects of the design. It is possible to estimate conductor losses, temperature, the surrounding magnetic field, and electrodynamic effects between conductors.

Which Model for Connectors?
The aim of connector modeling is to determine electrical values of the power electronics structure. You need an accurate electrical-equivalent circuit model that must be easy to use in a simulation environment.

Due to the frequency, voltage ranges, and the technologies power electronic components experience, we need to evaluate only resistive and inductive characteristics of the connectors. The model for any conductor could be a resistance in series with an inductance. All inductances of the model are coupled.

Which Method?
There are three methods to use to solve Maxwell's equations:

Analytical Methods

Analytical methods are based on simplifications of the equations that are valid only with some assumptions. Thus it is possible to use analytical methods only to design structures similar to some existing structures—you cannot use these methods to design new structures.

Numerical Methods

You can do an electromagnetic analysis of any kind of structure using a finite-element analysis. This method gives good results in all cases provided that meshing and boundary conditions are correctly done. The meshing state is one of the biggest problems—for busbars, it is especially difficult. This method results in large file sizes and long computation times, since you also need to mesh the area surrounding the busbar. Finally, applying the results is tricky, since only global information is available. This prevents us from knowing which part of the busbar is the major contributor to loop inductance.

The PEEC Method

Another method is preferred—the PEEC method. This method is useful for inductance calculation for any kind of connector. PEEC consists in replacing each straight part of a conductor by an L-R equivalent circuit and mutual coupling M (Figure 1). One of the big advantages of the PEEC method is that you can calculate the global inductance of a closed loop and the contribution of each part of this loop. PEEC is based on analytical formulations to obtain resistance, inductance and mutual-inductance values. The assumptions PEEC uses are well known: uniform current density in the cross section of the conductor, no magnetic material, parallel or perpendicular conductors, and a rectangular cross section for each conductor.

Figure 1:  PEEC Model for two parallel conductors

Note that you have to describe only conductors. A meshing of the conductor allows you to take into account proximity and frequency effects. This meshing will be different according the shape of the conductors you are modeling. For massive bars, the assumption of current in one direction inside the conductor is valid, so you only need to mesh cross section (Figure 2a). If this hypothesis is not valid, for example, to describe a busbar, adopt a 2D meshing (Figure 2b).

Figure 2:  Conductor meshing for PEEC analysis: (a) 1D meshing, (b) 2D meshing

If there is a ground plane under the structure, you can use an image method to determine the electrical characteristics for modification.

The electrical characteristics of each equivalent circuit, corresponding to all sub-conductors, leads to the frequency-dependent equivalent impedance of the interconnection.

The following two examples show how you obtain equivalent inductance from the electrical equivalent circuit given by PEEC. This value is important when studying power-electronic structures to evaluate over-voltage, CEM-conducted performance, and other parameters.

1D Problem

Figure 3:  1D modeling example that describes the study and shows the frequency influence on electrical characteristics

You use the obtained equivalent circuit to determine electrical waveforms for the structure in a circuit simulation program such as PSpice.

2D Study
For a busbar study, the current path is unknown, requiring a 2D analysis. Current propagation is separated into two orthogonal directions; an equivalent electrical circuit represents each one. The result is an electrical equivalent network comprising several R, L, and M values. You can reduce this large electrical network using PSpice for simple structures, or by solving circuit equations.

Figure 4 shows a 2D modeling example of busbar, analysis of the busbar, and how resistance and inductance varies with frequency

The evaluation of equivalent inductance does not produce a unique inductance value. A simple treatment of the problem and use of physical laws produces other values.

Current Evaluation
Using Kirchoff's equations lets you determine current in each subdivision of the problem if you know the voltage sources. This is a complex linear system of electrical equations (Equation 1) to solve, but you can easily compute the currents. You can then evaluate the current density.

Figure 5 shows the analysis of three-phase power-distribution bars using FEM and PEEC methods. The current density in three bars of one phase is drawn, showing proximity effects between conductors.

Knowing the currents, it is then possible to calculate the losses due to the connections in the structure. This information helps the designer to choose a cooling system, for example, or to make a thermal analysis using a dedicated EDA tool to obtain the temperature at any point of the structure.

Figure 5:  Current density evaluation for three-phase power-distribution bars

Magnetic Field Determination
Knowing the current in each conductor, it is then possible to evaluate the magnetic field at any point of the system using Biot and Savart's law (assuming thin conductors) (Equation 2).

Figure 6 shows the simple case of two parallel conductors. The magnetic field is evaluated on the x-axis and compared with the results obtained using FEM techniques.

Figure 6:  Calculating the magnetic field for two parallel conductors

Electrodynamics Evaluation
Following the previous approach for calculating magnetic fields, it is then possible to evaluate the electrodynamic efforts using Laplace's law, again assuming thin conductors (Equation 3). This is not difficult and once again the calculated results are very close to those obtained using FEM analysis.

Then, as you do for current, it is possible to use these results in a mechanical software program in order obtain electrodynamic parameters.

Conclusions
In this section, we have seen that the electrical equivalent circuit of connections using PEEC analysis is very useful to obtain a global analysis of a structure and to treat coupled phenomenon such as electrothermal and electromechanical parameters. In addition, all these evaluations are based upon analytical formulations, so they can be easily computed and can made part of a real CAD tool.

Since each evaluation is based on analytical equations, it is then possible to transform the previous analysis process into an optimization process. The variables are geometrical parameters of the connections—constraints can be geometrical and mechanical but also may include cost, weight, and other parameters. The objective function can be the minimization of equivalent inductance, of losses, the equal sharing of current into parallel conductors, and so on.

Figure 7 shows an example of minimization of losses for system comprising three-phase power-distribution bars. The optimized solution results in reduced losses of 19%.