# Compact Wilkinson Example 1: the Webb power divider

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New for August 2024.  We split the content below from this page which was getting too long.  Now each compact Wilkinson has its own page:

Example 1: Webb power divider

Example 2: Scardelletti power divider

Example 3: Kang power dividers

The Webb Wilkinson variation was described in a 1981 issue of MicroWaves magazine,which later evolved into Microwaves and RF. The article was titled Power Divider/Combiners: Small Size, Big Specs, authored by Richard C. Webb. Don't bother searching on the MW&RF web site, they haven't made great stuff like this accessible to the teeming masses like we have. It took us some time (and some connections) to find a copy of the original article, but we have it now!

Update July 2024: we posted the Webb article in our download area.  If anyone with a legitimate copyright claim has any objections, let us know!

Webb's technique involves splitting each quarterwave arm into two smaller transmission lines, of unequal impedance. In the circuit shown below, the Wilkinson's quarter-wave sections of impedance 1.414xZ0 are replaced with two transmission lines in series, each of length "L1", measured in electrical degrees (90 degrees is one quarter wavelength). When L1 is 45 degrees, the Webb splitter is identical to the "classic" Wilkinson splitter (all impedances are 1.414x Z0). When L1 is shrunk to less than 45 degrees, the impedances of the lines are not equal, the impedance of the line closest to port 1 (Z0A) is lower than 1.414xZ0, while the other line impedance (Z0B) is higher than 1.414xZ0. A capacitor is used in the isolation network, this is the trick that makes it work.

Webb offered two variants of the splitter: one uses a parallel RC isolation network, and one uses a series RC network. Below are schematics we generated in Eagleware's Genesys, unfortunately we didn't use the "proper" nomenclature for the line impedances and lumped elements (maybe we'll fix this later).

Series isolation network (10 GHz capacitor value):

Parallel isolation network (10 GHz capacitor value):

Webb provided closed-form equations for line impedances and RC values for both variants, demonstrating math skills far beyond anything we possess! Below are the two solutions, we modified them so that you solve directly for capacitance in picofarads, instead of ohms of capacitive reactance. The angle "theta" is the electrical length in degrees (or radians if you use Excel). There was only one typo in the equations in the article, hopefully there are no typos here! (Oops, there was a typo, but it's fixed now thanks to Ramya).

Here are Webb's equations if you want to cut and paste them into Excel. Note that we replaced "theta" in the equations with "t" because we are too lazy to insert Greek characters in this case... Update December 18, 2006... Dave pointed out that the Excel equations for Cp and Cs were off by a factor of (-1). We think that they are all fixed now, but with that many parentheses it's still "buyer beware".

Z01=(Z0/tan(t))*((1+8*(tan(t)^4))^0.5-1)^0.5

Z02=(2*(Z0)^2/Z01)

Rp=2*Z0

Cp=((1000/(2*pi*f))*((Z01+Z02)*(cos(t)^2)-
Z02*(1+(Z02/Z01))*(sin(t))^2))/((Z01+Z02)^2*sin(t)*cos(t))

Rs=((Z01+Z02)^2/Z0)*((sin(t)^2)*(cos(t)^2))

Cs=(1000/(2*pi*f))*1/(2*(Z01+Z02)*(cos(t)^3)*sin(t)-
(Z02*(1+(Z02/Z01))*(sin(t)^3)*cos(t)))

Three things to note:

1. The line impedances Z01 and Z02 are the same for both series and parallel RC solutions.
2. Only the capacitor value is a function of design frequency.
3. In the parallel RC solution, the resistor value is unchanged from the classic Wilkinson at 2Z0.

Webb further offered a solution that would take into account the series inductance of the mounting pads of the lumped element RC network (we suggest that you model the isolation network an use an optimizer to finalize your design).

The two plots below shows the impedances of the lines, as well as the capacitor and resistor values (the first plot is for series RC, the second plot is for parallel RC). The compact Wilkinson has limitations: for L1=20 (length of arms compacted from 90 degrees to 40 total), the line impedance Z02 value of 140 ohms may be unattainably high. Note that when L1=45 degrees, the capacitor reaches zero value, and the line impedances are all 70.7 ohms, which is the solution for a regular fifty-ohm Wilkinson.

Below we show frequency response plots for the series and parallel variations, given a 50% shrink (L1 is 22.5 degrees). They are quite similar, but the series RC version has better isolation bandwidth.

Webb splitter, L1=22.5 degrees, series isolation network, 10 GHz design:

Webb splitter, L1=22.5 degrees, parallel isolation network, 10 GHz design:

The final thing to consider when using this compact Wilkinson is that your circuit will be at the mercy of the tolerance and temperature stability of capacitor C1. However, you could consider using open-circuit stubs in place of the capacitors for better uniformity.

Author : Unknown Editor