Lumped Port Condition¶

A lumped port models an electromagnetic field source located within the computational domain. It typically represents an attached transmission line or a voltage or a current source applied between electrodes. Unlike the Waveguide Input Port, which provides a complete mode specification (including propagation constant and field profile) and can only be applied at the external boundary of the computational domain, a lumped port is applied on an internal interface within the solution space and excites a predefined single-mode pattern.

A lumped port can be positioned on any internal two-dimensional interface connecting two conducting objects. The separation between the conductors should be much smaller than the wavelength of the radiating field to ensure that the local quasi-static approximation remains valid.

Mathematically, the electric field amplitude \(\tilde{\mathbf{E}}\) at an internal interface were a lumped port is applied satisfies the following equation:

\[ \hat{\mathbf{n}} \times \left(\frac{1}{\mu} \nabla \times \tilde{\mathbf{E}}\right) = j \frac{\omega}{Z_s} \hat{\mathbf{n}} \times \left(\tilde{\mathbf{E}} \times \hat{\mathbf{n}}\right) -j \frac{2 \omega}{Z_s} \hat{\mathbf{n}} \times \left(\tilde{\mathbf{E}}^\text{in} \times \hat{\mathbf{n}}\right), \]

where \(\hat{\mathbf{n}}\) is the unit vector normal to the boundary, \(\omega\) is the angular frequency, \(Z_s\) is the surface impedance, and \(\tilde{\mathbf{E}}^\text{in}\) is the incident electric field.

As an example, consider a scenario where the user wishes to use a lumped port to model a connected transmission line between two conducting terminals. In this case, the expressions for the surface impedance \(Z_s\) and the incident electric field \(\tilde{\mathbf{E}}^\text{in}\) will depend on the shape of the lumped port.

For instance, for a rectangular lumped port geometry, \(Z_s\) and \(\tilde{\mathbf{E}}^\text{in}\) can be defined as

\[ \begin{align*} Z_s = Z \frac{w}{l}, \\ \tilde{\mathbf{E}}^\text{in} = E_0\, \hat{\mathbf{l}}, \end{align*} \]

where \(Z\) is the transmission line impedance, \(w\) and \(l\) are the width and length of the port, \(E_0\) is the amplitude of the incident electric field, and \(\hat{\mathbf{l}}\) is a unit vector defining the polarization of the field (typically oriented in the direction between the two terminals).

In contrast, for a coaxial lumped port geometry, \(Z_s\) and \(\tilde{\mathbf{E}}^\text{in}\) are given by

\[ \begin{align*} Z_s = Z \frac{2\pi}{\ln\frac{b}{a}}, \\ \tilde{\mathbf{E}}^\text{in} = E_0\frac{a}{r}\, \hat{\mathbf{r}}, \end{align*} \]

where \(a\) and \(b\) are the inner and outer radii of the corresponding coaxial cable, \(r\) is the distance from the port center, and \(\hat{\mathbf{r}}\) is a unit vector specifying the radial direction of the port.

Furthermore, the transmission line impedance \(Z\) can be expressed in terms of the resistance \(R\), inductance \(L\) and capacitance \(C\) of the corresponding RLC circuit as

\[ \frac{1}{Z} = \frac{1}{R} + \frac{1}{j\omega L} + j\omega C. \]

To create a LumpedPortCondition, the user must specify a custom name for the condition, provide the internal interface Marker, and include the port surface impedance as well as the incident electric field vector:

w = 0.10  # [m] port width
l = 0.04  # [m] port length
R = 50  # [Ohm] transmission line resistance
Z = R  # [Ohm] transmission line impedance: 1/Z = 1/R + 1/(j*w*L) + j*w*C
Zs = Z * w / l  # [Ohm] surface impedance

condition = LumpedPortCondition(
    name = "My Lumped Port Condition",
    marker = my_marker,
    surface_impedance = Zs,
    incident_electric_field_vector = (0, 0, 1),
)