Lumped elements are conceptualized in a similar way as gap or probe sources. They are indeed considered as infinitesimally narrow gaps placed in the path of current flow, across which Ohm's law is enforced. If a lumped element is placed on a PEC or conductive sheet trace, it is treated as a series connection. The boundary condition at the location of the lumped element is:
:<math> V_{gap} = Z_L I_{in} \quad\quad \int_{\delta} \hat{x}\cdot \mathbf{E_{gap}} \, dx = Z_L \int_W \hat{y} \cdot \mathbf{J_s} \, dy</math><!--[[File:PMOM67.png]]-->
where Z<sub>L</sub> is the total impedance across the two terminals of the series element. If the lumped element is placed on a slot trace, it is treated as a shunt connection that creates a current discontinuity. In this case, the magnetic current across the gap is continuous, and the boundary condition at the location of the lumped element is:
:<math> I_{gap} = Y_L V_{in} \quad\quad \int_{\delta} J_Y^{fila} \, dx = Y_L \int_W E_y \, dy </math>Â :<math> \int_{\delta} \hat{x}\cdot\hat{n} \times (\mathbf{H_{gap}^+ - H_{gap}^-}) \, dx = Y_L \int_W \hat{y}\cdot\mathbf{M_s} \, dy </math><!--[[File:PMOM70(1).png]]-->
where Y<sub>L</sub> is the total admittance across the two terminals of the shunt element. If a lumped element is placed on a PEC via that is connected to a metal strip from one side and to a PEC ground plane from the other end, it is indeed as a series connection across a gap discontinuity at the middle plane of the via. If the via is short, it is meshed using a single prismatic element. In that case, the lumped element in effect shunts the metal strip to the ground. The boundary condition at the location of the lumped element across the PEC via is:
:<math>V_{gap} = Z_L I_{in} \quad\quad \int_{\delta} \hat{z}\cdot \mathbf{E_{gap}} \, dz = Z_L \int_S \hat{z} \cdot \mathbf{J_p} \, ds</math><!--[[File:PMOM68.png]]-->
[[File:PMOM69.png]]