### Section 3: Sets with a Wilkinson with Extra Line Length

The circuit in Figure 5.4 features a Wilkinson with some additional transmission lines on the outputs. In a previous lesson, we saw that having two quarter-wave lines improved the bandwidth compared to having just one quarter-wave line. We will extend that concept with the Wilkinson divider by adding quarter-wave lines on the output ports (TL3 and TL4), and see if that helps improve the performance. We will start with TL3 and TL4 being equal to 50Ώ, such that the magnitude of the match and isolation will not be effected.

TL5 is also a 50Ώ line running into a 50Ώ port, and as such acts as an extension- it should not affect the magnitude of the match or the isolation. It will make one of the arms electrically longer than the other. If the line length is chosen correctly, the two arms will be 180° out of phase at some frequency, making the outputs equal in magnitude but opposite in phase (known as a balun).

We have several parameters we would like to optimize: the line lengths and widths of the quarter-wave transformers, and the line length (but not the width) of the 180° extension. We really have two different optimizations we want to run: improving the bandwidth by adjusting the quarter-wave lines, and adjusting the relative phase of the outputs by adjusting the length of TL5. We would like to decouple these two goals- in particular we do not want the circuit to try to sacrifice the bandwidth to improve the 180° phasing, and vice versa.

Mason allows you to associate certain optimization goals, and certain variables, with particular optimization sets. Each of these can have a “set” parameter, where each optimization step is discretely numbered. Each goal and variable has a “set” parameter that defines the association. In the example shown in Figure 5.4 and Figure 5.5: “set 1” consists of 100 random iterations, plus the log based calculations, plus the variables associated with the quarter-wave lines; “set 2” consists of 2 iterations of simplex, plus the log based calculations, plus the variables associated with the quarter-wave lines; and “set 3” consists of 3 iterations of simplex, plus the arg based calculations, plus the variable associated with the half-wavelength line.

The first two sets will optimize the performance of the match and isolation, and the variable associated with the third set does not impact the match or isolation. This is a very important point to keep in mind. The trick works best when the variables and the goals can be separated into sets that do not interact, or are weakly coupled.

Run the optimization (the 'M' button), and plot the match (the 'P' button). Because the first optimizer uses a random search to break out of local minimums, the first run may not get the -20dB performance as shown in Figure 5.6. Try increasing the number of random iterations to 10000, and rerunning as needed. The plot should automatically update at the end of each run. When a good performance is seen in the plot, update the schematic (the 'U' button). If the values appear to be okay, save the new values over the old values (the 'S' button).

Questions:

5.2.1: Over what bandwidth does the original Wilkinson divider have a return less than -20dB? Over what bandwidth does the new Wilkinson have a return loss less than -20dB?

5.2.2: Change the log-based variables (shown in Figure 5.5) to belong to all three sets: “1 2 3”, but do not change the goals. The variables and the goals are now cleanly separated. How does this affect the optimization, and why?

5.2.3: Using
Mason Plot's “Log” plot type, plot the phase of S_{21}
using “arg(S_2_1)” and the button “Autogen Y-Limits”.
Using the “Arg” plot type, plot the phase of S_{21}.
How do these plots differ? Are they really showing different data?

5.2.4: Plot the following three functions in the same plot (using the “Arg” plot type) and comment on the similarities and differences:

- arg(S_2_1/S_3_1) - 180
- arg(-S_2_1/S_3_1)
- arg(-S_3_1/S_2_1)

5.2.5: Over
what bandwidth is the phase within the +/- 20 degree limits in the
goal?