Theory for the Transmission Line, Transient Boundary Conditions
The frequency-domain form of the Terminating Impedance boundary condition is derived from the current flowing through the load impedance, as is shown in Equation 4-9. The time derivative of Equation 4-9 reads
(4-15)
Inserting the expression for the time derivative of the current I from Equation 4-11 into Equation 4-15 yields to the time-domain Terminating Impedance boundary condition equation:
(4-16)
To derive the Absorbing Boundary boundary condition, the load impedance in Equation 4-10 is replaced by the characteristic impedance. This results in the frequency-domain form of the Absorbing Boundary condition:
(4-17)
Equation 4-17 can be expressed in terms of the propagation constant as
(4-18)
where the propagation constant is . In order to derive the time-domain form of Equation 4-18, it is required to perform the Taylor series expansion of the propagation constant. The resulting Taylor series of the corresponding products of γ are given by
In the case when R and G are very small, the higher-order terms can be neglected and the propagation constant simplifies to
(4-19)
Substituting in Equation 4-19, the Absorbing Boundary condition equation yields
(4-20)
To ensure the continuity condition and current conservation, COMSOL Multiphysics solves the following time-domain Absorbing Boundary equation (instead of Equation 4-20)
(4-21)
Following the approaches detailed in Theory for the Transmission Line Boundary Conditions, the Incoming Wave boundary condition is written in terms of the propagation constant as
(4-22)
Inserting Equation 4-19 into Equation 4-22, gives the time-domain Incoming Wave boundary condition:
(4-23)
In the The Transmission Line Interface, the Lumped Port boundary condition (for a Cable lumped port) reads as
(4-24)
Substituting in Equation 4-24, gives the time-domain form of the Lumped Port boundary condition:
(4-25)
In the case when the wave excitation at this port is turned off and the port impedance is replaced by the load impedance, Equation 4-25 results in the Terminating Impedance boundary condition equation.
In the case of current Lumped Port, the time-domain boundary condition can be evaluated from Equation 4-11 as
For a current type lumped port, Iin is the terminal current and is an input parameter.