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. The Waveguide Input Port gives a complete mode specification, which includes the propagation constant and field profile. You can apply it only at the external boundary of the computational domain. A lumped port is different: you apply it on an internal interface within the solution space, and it 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, so the local quasi-static approximation stays valid.

Mathematically, the electric field amplitude \(\tilde{\mathbf{E}}\) at an internal interface where 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.

For example, use a lumped port to model a 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.

You can also express the transmission line impedance \(Z\) 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, give a name, the internal interface Marker, the port surface impedance, and 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*omega*L) + j*omega*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),
)