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}.

Read more about this topic:  Partial Element Equivalent Circuit

Famous quotes containing the word theory:

    “It is not enough for theory to describe and analyse, it must itself be an event in the universe it describes. In order to do this theory must partake of and become the acceleration of this logic. It must tear itself from all referents and take pride only in the future. Theory must operate on time at the cost of a deliberate distortion of present reality.”
    —Jean Baudrillard (b. 1929)

    “Lucretius
    Sings his great theory of natural origins and of wise conduct; Plato
    smiling carves dreams, bright cells
    Of incorruptible wax to hive the Greek honey.”
    —Robinson Jeffers (1887–1962)

    “The things that will destroy America are prosperity-at-any- price, peace-at-any-price, safety-first instead of duty-first, the love of soft living, and the get-rich-quick theory of life.”
    —Theodore Roosevelt (1858–1919)