Compact Wilkinsons make up
another class of Wilkinson power splitters. When would you want
to use a compact Wilkinson? Suppose you wanted to use a Wilkinson
power splitter at 1 GHz. The quarter-wave sections, laid out in
microstrip on a soft substrate such as Duroid, would be a couple
of inches long. You might not have room for them. Engineers have
been trying different ways to shrink the Wilkinson since it was
invented in 1960, but there are tradeoffs to each approach.
1: Webb splitter
Update December 2005:
this 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 to find
a copy of the original article, but we have it now!
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
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:
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.
Example 2: Scardelletti splitter
This device was pointed out
to us by Microwaves101 knifewinner Emily. It was described in
an January 2002 IEEE article entitled Miniaturized Wilkinson
Power Dividers using Capacitive Loading, by Scardelletti et
The technique used is to increase
the line impedance of the arms, while loading the structure with
capacitors C1 at the split ports and C2 at the common port (C2=2xC1).
This is actually a pretty basic concept that we will eventually
describe in our page on quarterwave
tricks. Here' s a schematic we generated in Eagleware's Genesys
(once again we ignored proper capacitor designators!)
In the referenced paper, equations
are given for line impedance as a function of electrical length.
Guess what? There's a mistake in the equation given for capacitance!
We've fixed the mistake here:
The plot below
shows the line impedances and shunt capacitor value as functions
of line length in degrees, for this type of splitter in a fifty
ohm system. When the line length is 90 degrees, the design reverts
to a conventional Wilkinson coupler, with impedances 70.7 ohms
and capacitors at zero value. The capacitor values given are for
1 GHz center frequency, if you want to scale them to other frequencies,
merely divide them by the frequency you need (in GHz). Line impedances
are NOT a function of frequency.
Below is the frequency
response of the Scardelletti splitter given a 50% shrink (arms
are 45 degrees long). Looks like it doesn't achieve the same bandwidth
as the Webb splitter, it rolls off quickly above the center frequency.
3: Kang splitter
This technique was described
by In-Ho Kang and Jin-San Park in their July 2003 Microwave Journal
article entitled A Reduced-size Power Divider Using the Coupled
Line Equivalent to a Lumped Inductor. It was pointed out to
us on the Microwaves101 message board.
Update November 2006...
we finally simulated the Kang splitter. Thanks to David from down
under for sending us the article!
What Kang and Park figured
out is that you can fake the quarterwave section using a coupled
line that is shorted on two ports, with a pair of capacitors.
Before we describe Kang's splitter,
let's look at a conventional Wilkinson realized using microstrip
on 10 mil alumina. The next two images show the ADS schematic
and its response:
Notice that a quarterwave section
at 10 GHz is 120 mils.
The layout below represents
about 10 minutes worth of work, trying to get the Kang splitter
to behave at X-band, using 10 mil alumina. As you can see, the
coupled line section is far less than a quarter wavelength, in
this case it is closer to 1/16 wavelength. Tradeoffs that can
be made include the coupling factor (spacing between the arms),
length and impedance of the coupled section, and the capacitor
values. Note that C1 and C2 could be combined into one capacitor
of double the value, but we like the symmetry as it is shown.
Here's the response, it has
NOT been fully optimized.
More to come!