Introduction to Transmission Line Theory
Figure 4-9 is an illustration of a transmission line of length l and characteristic impedance Zc. The distributed resistance R, inductance L, conductance G, and capacitance C characterize the properties of the transmission line.
Figure 4-9: Schematic of a transmission line with a load impedance.
The distribution of the electric potential V and the current I describe the propagation of the signal wave along the line. The following equations relate the current and the electric potential:
(4-2)
(4-3)
Equation 4-2 and Equation 4-3 can be combined to the second-order partial differential equation
(4-4)
where
Here γ, α, and β are called the complex propagation constant, the attenuation constant, and the (real) propagation constant, respectively.
The attenuation constant, α, is zero if R and G are zero.
The solution to Equation 4-4 represents a forward- and a backward-propagating wave
(4-5)
By inserting Equation 4-5 in Equation 4-2 the current distribution is obtained:
If only a forward-propagating wave is present in the transmission line (that is, there are no reflections), dividing the voltage by the current gives the characteristic impedance of the transmission line:
To make sure that the current is conserved across interior boundaries, COMSOL Multiphysics solves the following wave equation (instead of Equation 4-4):
(4-6)