wave” from load to the source, whereas the wave incident at the load V+e(ωt − γx) appears as the “backward‐traveling” wave. The voltage drop across the load is
(2.1.83)
Keeping in view that the origin of the distance (x = 0) is at the load, the line voltage and current at the load end are written, from equation (2.1.79), as follows:
(2.1.84)
The voltage reflection coefficient at the load end is defined as follows:
(2.1.85)
The expression to compute the reflection coefficient at the load end is obtained from the above equations (2.1.83) – (2.1.85): equations:
The mismatch of a load impedance ZL with the characteristic impedance Z0 of a line is the cause of the reflection at the load end. For the condition ZL = Z0, the matched load terminated line avoids the reflection on a line, as ΓL = 0. At any distance x, the reflection coefficient is a complex quantity with both the magnitude and phase expressed as follows:
(2.1.87)
A lossless line has |Γ(x)| = |ΓL|, i.e. on a lossless line magnitude of the reflection is the same at all locations on a line. However, the lagging phase ϕ changes with distance. It has 180° periodicity, i.e. for an inductive load, the range of phase is 0 < ϕ < π, and a capacitive load has the phase in the range −π < ϕ < 0. Using equation (2.1.79), the line voltage and line current are written in term of the load reflection coefficient:
For a lossless transmission line, the above equations are written as follows:
(2.1.89)
In the above equations, the origin is at the load end, i.e. x < 0. The maxima and minima of the voltage and current waves along the line occur due to the phase variation along the line. The voltage maximum occurs at ej(ϕ + 2βx) = + 1. In this case, both the forward and reflected waves are in‐phase. The voltage minimum occurs at ej(ϕ + 2βx) = − 1. In this case, both the forward and reflected waves are out of phase. Finally, the maxima and minima of the voltage on a line are given as follows:
(2.1.90)
The reflection coefficient Γ(x) at any location x from the load end is related to the reflection coefficient at the load ΓL by
(2.1.91)
The measurable quantity voltage standing wave ratio (VSWR) is defined as follows:
(2.1.92)
For a lossless line, the VSWR is constant along the length of a line. Likewise, the current standing wave ratio is also defined.
The wave reflection also takes place at the sending end when the source impedance Zg is not matched to the characteristic impedance of a line. The reflection coefficient at x = − ℓ, i.e. at the generator (source) end is defined as Γ(x = − ℓ) = Γg. The voltage and current at the generator end are obtained from equation (2.1.88),
The amplitude factor V+ is determined by the reflections at both the source and load ends.
Figure (2.8b) shows that the port voltage
On substitution of equation (2.1.93) in (2.1.94), the voltage wave amplitude V+ is obtained as follows:
(2.1.95)
However, the reflection coefficient at the source end is
Therefore, the amplitude of the voltage wave launched by the source is
Equation (2.1.88a and b) give the voltage and current waves on a transmission line with the amplitude factor V+. The amplitude factor V+ is given by equation (2.1.97).
2.1.8 Application of Thevenin's Theorem to Transmission Line
Thevenin's theorem is a very popular concept used in the analysis of the low‐frequency lumped element circuits. It is equally applicable to a transmission line network. At the output end of the line, the input source voltage
