To derive a system of integral equations, we enforce the boundary conditions on the integral definitions of the '''E''' and '''H''' fields as follows:
:<math>L_E(E) = L_E \bigg\{ E^{inc} + \iiint\limits_V \overline{\overline{G}}_{EJ}(r|r') \cdot J(r') \, dv' + \iiint\limits_V \overline{\overline{G}}_{EM}(r|r') \cdot M(r') \, dv' \bigg\} </math>Â :<math>L_H(H) = L_H \bigg\{ H^{inc} + \iiint\limits_V \overline{\overline{G}}_{HJ}(r|r') \cdot J(r') \, dv' + \iiint\limits_V \overline{\overline{G}}_{HM}(r|r') \cdot M(r') \, dv' \bigg\} </math><!--[[File:PMOM4(2).png]]-->
where '''L<sub>E</sub>''' is the boundary value operator for the electric field and '''L<sub>H</sub>''' is the boundary value operator for the magnetic field. For example, '''L<sub>E</sub>''' may require that the tangential components of the '''E'''field vanish on perfect conductors:
:<math> \hat{n} \times \hat{n} \times \mathbf{E} = 0, \quad \mathbf{r} \in PEC </math><!--[[File:PMOM65.png]]-->
Or '''L<sub>E</sub>''' and '''L<sub>H</sub>''' may require that the tangential components of the '''E''' and '''H''' fields be continuous across an aperture in a perfect ground plane:
:<math>\begin{cases}\hat{n} \times \hat{n} \times (\mathbf{E}^+ - \mathbf{E}^-) = 0 \\\hat{n} \times \hat{n} \times (\mathbf{H}^+ - \mathbf{H}^-) = 0\end{cases} \quad \Rightarrow \quad\mathbf{M}^+(r) = \mathbf{M}^-(r), \quad r \in PMC</math><!--[[File:PMOM66(1).png]]-->
Given the fact that the dyadic Greenâs functions and the incident or impressed fields are all known, one can solve the above system of integral equations to find the unknown currents '''J''' and '''M'''.
In EM.CUBE's [[Planar Module|Planar module]], magnetic currents are always surface current with units of V/m. Electric currents, however, can be surface currents with units of A/m as in the case of metallic traces like microstrip lines, or they can be volume currents with units of A/m<sup>2</sup> as in the case of perfectly conducting vias. Dielectric inserts are modeled as volume polarization currents that are related to the electric field '''E''' in the following manner:
:<math>\mathbf{J}_p(r) = jk_0 Y_0(\varepsilon_r - \varepsilon_b)\mathbf{E}(r)</math><!--[[File:PMOM5.png]]-->
where k<sub>0</sub> is the free space propagation constant, Y<sub>0</submath>Y_0 =\tfrac{1/Z<sub>0}{Z_0} = \tfrac{1}{120\pi}</submath> =1/(120p) is the free space intrinsic admittance, eε<sub>r</sub> is the permittivity of the dielectric insert, and eε<sub>b</sub> is the permittivity of its background layer. In a 2.5-D formulation, it is assumed that the volume currents have only a vertical component along the Z direction, and their circumferential components are negligible.
=== Numerical Solution Of Integral Equations ===