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Broadband Wilkinson


Wilkinson as matching network
Most microwave engineering texts will present the Wilkinson as a matching network for even- and odd- mode analysis (perhaps I've had time to add this to my pages).  This page will assume you already understand this analysis (if not, go back and review).  In general, one can add more elements and increase the bandwidth, within the laws of diminishing returns.  A caveat and a tip are presented in these pages.

Please assume a 50-ohm system for the following analysis.  It is entirely reasonable to use a Wilkinson for a matching network, but for now assume a vanilla circuit.

Isolation versus Match
Consider the Broadband Wilkinson Design below.  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. 
Broadband Wilkinson Design (Click to enlarge)

In the odd-mode analysis, Za and Zx are both looking into a short.  That means for our isolation, L1 does not figure into our analysis at all.  Therefore, we can get a broader band match than we can get isolation performance.  Since the L4 line does help improve the match from Zc to the virtual short, L4 actually can improve the isolation despite the lack of a terminating resistor.  Two observations:
  1. The match can be made over a slightly better frequency range than the isolation.
  2. To exploit the first point, care must be taken in the optimizer not to sabotage the isolation and/or match.
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.  It's on my list.

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.

Implementing with Mason
The design file discussed on this page has been attached at the bottom for your convenience.

ċ
broadband_double_wilkinson.dsn
(87k)
Gregory Kiesel,
Jun 27, 2010, 8:39 PM
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