Termination of TLs Load Reflection Coefficient.
In the last lecture, we considered for the first time TLs in the sinusoidal steady state. We will now consider the termination of TLs that are excited by sinusoidal steady state sources.
Your text places z = 0 at the source end of the TL. Here, we’re placing z = 0 at the load for convenience. While some intermediate formulas will be different by doing this, the end results will be the same.
Adding terminations produces reflections so that the total voltage and current anywhere on the TL are sums of forward and reverse propagating waves. From results in the previous lecture, the voltage and current on the TL will have the form
The “lumped load” ZL that terminates the TL is considered a boundary condition for the voltage and current in
V( z = 0 = I( z = 0)zL
Therefore, we can solve for V0- in terms of V<sub<0< sub="">+ by applying this boundary condition as:
Forming the ratio of these quantities gives
As with time domain TL analysis, we will solve for V0- / V0+, and define this ratio as the voltage reflection coefficient at the load (z = 0), we find
This expression for TL is very similar to what we derived for TL and time varying TL analysis. Here, however, TL is a complex number, as is ZL in general
Example N17.1: For an open circuit load on the TL shown above, compute the load reflection coefficient and sketch the voltage and current magnitude on the TL.
For an open circuit load ZL=(i.e., an extremely large impedance), so that from TL=+1 . With this value of TL, then from (1) and (2) the solutions for V(z) and I(z) are
These two equations (5) and (6) are not traveling waves. So, where has the traveling wave behavior in V (z) and I(z) gone? The interference between the incident and reflected waves produces standing waves, such as these. |V(z)| and |I (z)| are shown here for the open circuit load:
Phasor voltage and current magnitudes vary noticeably along TLs provided the TL length is greater than about 0.05 l or so. Remember, though, that we are plotting the magnitude of phasor voltages and currents. The voltage and current oscillate as functions of time with amplitudes equal to |V(z)| and |I(z)) , respectively, at each point along the TL.
Notice that the “distance” is measured in wavelengths. This is common in EM work. The quantity “bz” is called the electrical distance. (It’s not really distance since its units are radians.) One method for determining b of TLs in the lab is to connect a short circuit load and probe for the voltage nulls. Defining the distance between adjacent nulls as d then
But, from (17) in the previous lecture