Introduction to Transmission Line Theory
Figure 4-9 is an illustration of a transmission line of length L. 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 describes the propagation of the signal wave along the line. The following equations relate the current and the electric potential
(4-1)
(4-2)
Equation 4-1 and Equation 4-2 can be combined to the second-order partial differential equation
(4-3)
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 4-3 represents a forward- and a backward-propagating wave
(4-4)
By inserting Equation 4-4 in Equation 4-1 the current distribution is obtained.
If only a forward-propagating wave is present in the transmission line (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 4-3)
(4-5)