Theory for the Transmission Line Boundary Conditions
The Transmission Line Interface has these boundary conditions:
(4-6)
and
(4-7)
In Equation 4-6 and Equation 4-7, the indices 1 and 2 denote the domains on the two sides of the boundary. The currents flowing out of a boundary are given by
where ni are the normals pointing out of the domain.
Because V is solved for, the electric potential is always continuous, and thus Equation 4-6 is automatically fulfilled. Equation 4-7 is equivalent to the natural boundary condition
which is fulfilled with the wave-equation formulation in Equation 4-5.
When the transmission line is terminated by a load impedance, as Figure 4-9 shows, the current through the load impedance is given by
(4-8)
Inserting Equation 4-1into Equation 4-8, results in the Terminating Impedance boundary condition
(4-9)
If the arbitrary load impedance ZL is replaced by the characteristic impedance of the transmission line Z0 you get the Absorbing Boundary condition. By inserting the voltage, defined in Equation 4-4, in Equation 4-9 you can verify that the boundary condition does not allow any reflected wave (that is, V is zero).
The Open Circuit boundary condition is obtained by letting the load impedance become infinitely large, that is, no current flows through the load impedance.
On the other hand, the Short Circuit boundary condition specifies that the voltage at the load is zero. In COMSOL Multiphysics this is implemented as a constraint on the electric potential.
To excite the transmission line, use the Incoming Wave boundary condition. Referring to the left (input) end of the transmission line in Figure 4-9, the forward propagating wave has a voltage amplitude of V0. Thus, the total voltage at this boundary is given by
Thereby, the current can be written as
resulting in the boundary condition
For the Lumped Port boundary condition, the port current (positive when entering the transmission line) defines the boundary condition as
where the port current Iport is given by
for a Cable lumped port (see the Lumped Port section for a description of the lumped port settings).
For a Current-controlled lumped port, you provide Iport as an input parameter, whereas it is part of an electrical circuit equation for a Circuit-based lumped port.