### Motivation

Wilkinson combiners are made up of quarter-wave elements which provide a match from the two outputs to the input. We have already shown that by using multiple quarter-wave elements, we can improve the bandwidth of the circuit. We will use that philosophy to increase the bandwidth of the Wilkinson combiner.

## Section 1: Even and odd mode analysis

A simple model for a Wilkinson is shown in Figure 10.1. We are going to bisect this model, splitting Port 1's 50Ώ impedance into two parallel 100Ώ resistors, and splitting SER1 into two series 50Ώ resistors in order to keep the impedances the same (Figure 10.2). These two are equivalent from the output port's perspective (Port 3 is equivalent to Port 4, Port 2 is equivalent to Port 5).

### Even Mode:

First we consider the symmetric circuit, and what happens if we drive Port 4 and Port 5 with the same voltage in phase (+1V / +1V): this is called the even mode. Because the voltage is equal on either side of SER2 and SER3, then no current flows through either resistor. It is as if the node between SER2 and SER3 were open. Similarly, the voltage adds constructively on SER4 and SER5, again, one can consider the junction between them open. This is shown in Figure 10.3.

### Odd Mode:

Next we consider the symmetric circuit, and what happens if we drive Port 4 and Port 5 with an equal magnitude but opposite phase (+1V / -1V): this is called the odd mode. As long as the voltages are equal by opposite, the voltage divider of SER2 and SER3 will maintain a 0V potential between the resistors... the node is “virtually” grounded. Similarly, the voltages cancel each other at SER4 and SER5- again, the node is “virtually” grounded. This is shown in Figure 10.3.

Using the even and odd mode analysis, we have separated the circuit from a three-port network to two two-port networks made up of a quarter-wave transmission line. We already know we can design a multi-sectional quarter wave network... but before we do that, let's verify our math.

Enter the circuit shown in Illustration 58; Port 1 is the Even mode equivalent, Port 2 is the Odd mode equivalent, and Ports 3-5 are attached to a normal Wilkinson circuit. Care must be taken to properly compare the Even mode to the original circuit, and the odd mode to the original circuit.

The Even mode port is equivalent to having Port 3 and Port 4 driven in phase. The equivalent S-Parameter to the Even mode's S_1_1 is to sum S_3_3 and S_4_3, the voltage reflected off of Port 3 and the voltage transmitted from Port 4 to Port 3.

The Odd mode port is equivalent to having Port 3 and Port 4 driven out of phase. The equivalent S-Parameter to the Odd mode's S_2_2 is to take the difference between S_3_3 and S_4_3, the voltage reflected off of Port 3 and the voltage transmitted from Port 4 to Port 3.

Questions:

10.1.1: Compare the plots of the Even mode and comment:

- 20*log10(abs(S_1_1))
- 20*log10(abs(S_3_3+S_4_3))

10.1.2: Compare the plots of the Odd mode and comment:

- 20*log10(abs(S_2_2))
- 20*log10(abs(S_3_3-S_4_3))

Hopefully now we are comfortable that the Even and Odd mode analysis are correct, and we can focus on trying to exploit this relationship for a broader band design. Consider the Broadband Wilkinson Design in Figure 10.5. In Even-Mode analysis, Za and Zx are both looking into the 2*Z0 impedance (for 50-ohms, the translates into 100-ohms). We would like Zc and Zz to both be 50-ohms looking into the output ports. Therefore, we have three "proper" microstrip lengths to use for our matching: L2, L3, and L4. It is possible to use L1 to improve our match, since we are technically matching from the L1 impedance to the Zc impedance. The caveat relates to the odd-mode analysis.

In the odd-mode analysis, Za and Zx are both looking into a short. The odd mode corresponds to the power that is burned up in the resistor, so is associated with the isolation. Because of that short, that means for our isolation, L1 does not figure into our analysis at all. If we do use L1 as an optimized parameter, we can get a broader band impedance match than we can get isolation performance.

The match can be made over a slightly better frequency range than the isolation.

To exploit this, care must be taken in the optimizer not to sabotage the isolation and/or match; with L1 in play, one can optimize the match over a wider band than the isolation.

In Mason, optimization goals can be made frequency dependent
exploiting normal muParser rules, as
follows:

calculation="if(frequency>3e9,if(frequency<15e9,20*log10(abs(S_2_1)),0),0)"

Please note that units don't work in calculations, just as
units don't work in calculation blocks.

### Optimization technique

A second optimization tip is in
setting up the microstrip width and length variables. One could, of
course, allow the variables to have the same degrees of freedom- for
example, each width vary from 8mils to 22 mils. However, adding
intelligent constraints can be very valuable in finding optimal
solutions. Therefore, we will add a common constraint: every
subsequent width will be wider than the preceding one (therefore,
impedances will decrease monotonically from the even-mode 100-ohms
junction (Za, Zx) to the 50-ohms output port (Zc, Zx).

This is
done by having variables of the following pattern:

W_Wilk2
=8 mil; <opt min="8 mil" max="40 mil"
step="0.1 mil" set="1"/>

MW_Wilk3 =1.49581
; <opt min="1" max="2" step="0.01 mil"
set="1"/>

And calculations of the
following pattern:

W_Wilk3
= MW_Wilk3 * W_Wilk2

### Resistor choices

Optimal designs for the broadband
Wilkinson have, of course, been worked out in papers. For those who
choose to let the optimizers do the heavy lifting, all of Mason's
optimizers, including the gradient-like simplex, can optimize over
discrete resistor models and continuous microstrip values at the same
time. This means, given a rule of thumb, one can use downloaded
"real" resistor models and have the optimizer pick a good
result, assuming a fairly good starting place.

My rule of
thumb is: the outermost resistor (R2, in the above circuit) should
start the same as the single Wilkinson (100-ohms for a "normal"
Wilkinson combiner). As you go inwards, double the resistance
(200-ohms, 400-ohms, 800-ohms). The optimizer can work with
this.

Since the papers generally haven't had optimization
goals that exactly aligned with my goals, I find this a reasonable
approach.

### Circuit Design

Questions:

Model the circuit in Figure 10.6.

10.1: Run the optimizer and compare and comment on the differences between:

- 20*log10(abs(S_1_1))
- 20*log10(abs(S_4_4))

10.2: Try increasing the random iterations to 1000 or more. Looking at the console (the “C” button), does the optimizer find a lower FOM (figure of merit)?

10.3: Create new variables to optimize the lengths of the transmission lines.

- Create a new variable, L1, with a minimum of 45 and a maximum of 135
- Create a new calculation, L2 = 180-L1
- Assign L1 to the theta value of T3 and T4
- Assign L2 to the theta value of T5 and T6
- Run the optimizer again and compare the results to the previously optimized simulation.

10.4Reset the values for Z1_mult, Z2, R1, L1, and L2 to the original values above, and replace the random optimizer with the following genetic optimizer:

- genetic iterations=100 population=100 first_generation_multiplier=4 selection="tourney" tourney_size=5
- How does the answer compare? How does the time to solve compare?