Partial Element Equivalent Circuit - Theory

Theory

The classical PEEC method is derived from the equation for the total electric field at a point written as

\vec{E}^i(\vec{r},t) = \frac{\vec{J}(\vec{r},t)}{\sigma} + \frac {\partial \vec{A}(\vec{r},t)}{\partial t} + \nabla \phi (\vec{r},t)

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

L_{p_{\alpha \beta}} = \frac {\mu}{4 \pi}\frac{1}{a_{\alpha} a_{\beta}} \int_{v_{\alpha}} \int_{v_{\beta}} \frac {1} {| \vec{r}_{\alpha} - \vec{r}_{\beta}|} d v_{\alpha} dv_{\beta}

for volume cell and . Then, the coefficients of potentials are computed as

P_{ij} = \frac{1}{S_i S_j} \frac{1}{4 \pi \epsilon_0} \int_{S_i} \int_{S_j} \frac{1}{|\vec{r}_i - \vec{r}_j|} \; dS_j \; dS_i

and a resistive term between the nodes, defined as

R_\gamma = \frac{l_\gamma}{a_\gamma \sigma_\gamma}.

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