# Lesson 5: Power Dividers and Advanced Optimization Techniques

### Motivation

This Lesson will also transition to circuits using actual microstrip components, rather than ideal transmission lines. These models are less intuitive to use, but map into the real world more readily. For one, these models are defined by physical geometries that can readily be built. For another, these models include loss and other non-ideal characteristics which help with design and measure/ model agreement.

Mason provides a number of optimization techniques. While the optimizer is no substitute for good design, proper management of the optimizers can mean huge improvements in the performance of the circuit. This lesson will show some the advanced features of the optimizer, and provide some insights on how to effectively use the optimizer.

### Section 1: Dividers

A Wilkinson divider is one many circuits which can divide the power incident at one port into two output ports. Dividers are generally reciprocal- they can also be used to combine the power from two input port into one output port. A number of concepts are important when dealing with power dividers/ combiners.

Match (S11, S22, S33): In general, one would like to have most of the power going through the divider, and not reflected back to the source.

Isolation (S23): When combining power, most of the power should go from the input port to the output port. However, some amount of power may go from one of the input ports to the other input port. The amount that the circuit suppresses energy going to the wrong port is the isolation.

Insertion Loss (S21): The amount of loss within the circuit, not caused by return loss. When the match is pretty good this is often approximated by S21; when the match is poor, or when being pedantic: Equation 5.1

We will compare and contrast a few dividers: the T-junction, the Wilkinson, and the lossy divider.

T-Junction divider

The T-Junction divider is about as simple as it gets, three lines come together to form a junction with perhaps a matching element if needed. It only uses transmission lines (no resistors), requires less area than a Wilkinson, and has a low insertion loss. Unfortunately, some of the ports are not matched, and the isolation is poor.

In the circuit in Figure 5.1, the 50Ώ impedances of ports 2 and 3 are transformed into 100Ώ impedances by TL1 and TL2. These two 100Ώ impedances are in parallel with each other at port 1, which means that the impedance looking into the junction of TL1 and TL2 are 50Ώ, therefore port 1 is matched. However, the impedance that TL2 sees looking towards Port 1 and TL1 (which are both in parallel from TL2's perspective) is 33Ώ... which means port 2 (and port 3) are not matched. Figure 5.1: Basic T-Junction

The return loss at Port 2 can be calculated using the following formula: Equation 5.2

Questions:

5.1: Using Mason Plot, what is the bandwidth for S11 less than -20dB?

5.2: Using Mason Plot, how does the modeled S22 compare to the calculated Γ? Why might these two differ? What happens if the microstrip parameters of cu_sigma is changed to 10e9, and the loss_tan changed to 0?

5.3: Is there a relationship between the S22 (return loss) and the S23 (isolation)?

Wilkinson divider

The Wilkinson divider can be matched and provide good isolation over a narrow bandwidth. Specifically, a Wilkinson with one section has relatively low bandwidth, however more arms can be added to the Wilkinson to increase the bandwidth. The insertion loss is generally low. The circuit does take up more area than a T-junction. Figure 5.2: Basic Wilkinson combiner

Questions:

5.4: Using Mason Plot, what is the bandwidth for S11 less than -20dB?

5.5: How does the S22 compare between this circuit and the T-Junction?

5.6: How does the S23 compare between this circuit and the T-Junction?

Lossy divider

The lossy divider is matched, has okay isolation, and operates over a wide bandwidth. It can be fairly small, made up of just a few resistors and no quarter-wave transmission lines. Unfortunately, as the name implies, this circuit has a relatively high amount of insertion loss.

For all ports having a common impedance, the resistor values are Z0/3. Because this circuit has no frequency dependent components, the S-Parameters do not change over frequency- this is what causes the circuit to have so broad of a band. Figure 5.3: Lossy divider

Questions:

5.7: Using Mason Plot, what is the bandwidth for S11 less than -20dB?

5.8: How does the S23 compare between this circuit and the Wilkinson?

5.9: How does the S21 compare between this circuit and the Wilkinson?

### Section 2: 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). Figure 5.4: WIlkinson with 180° arm

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. Figure 5.5: Variables set up for three sets of optimizations

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). Figure 5.6: Match of the optimized circuit

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 S21 using “arg(S_2_1)” and the button “Autogen Y-Limits”. Using the “Arg” plot type, plot the phase of S21. 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:

1. arg(S_2_1/S_3_1) - 180

2. arg(-S_2_1/S_3_1)

3. arg(-S_3_1/S_2_1)

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