here to go to our main page on Wilkinson power dividers
here to go to our main page on lumped elements
New for March 2007! Content
was donated by Dr. Antonije Djordjevic, a Professor at University
of Belgrade. The real
Belgrade, not Belgrade,
Maine, by the way. Thanks, Tony!
design below provides lumped-element equivalent of simple one-stage
Wilkinson power divider with nominal port impedance 50 ohms.
The center frequency was picked to be 1.592 GHz, so that angular
radians/second. You can apply scale factors to the capacitors
and inductors to change center frequency, both are inversely
proportional to frequency. Note that the isolation resistor
value is the same as in a "normal" Wilkinson, at 2xZ0.
The normalized inductor
reactance at center frequency:
The normalized capacitor
susceptances at center frequency:
Let's review the formulas for
capacitive and inductive reactance while we're on the subject:
Here's how it performs over frequency:
Now for the derivation...
One way to derive the lumped-element
equivalent of the Wilkinson power divider is to begin by deriving
a lumped-element equivalent of a quarter-wave transmission line.
Suppose that the characteristic impedance of the line is ZC=Z0,
where Z0 is the port nominal impedance. If the line length
the matrix of scattering parameters of the line is:
We consider the pi-network of
lumped elements, which consists of capacitors in shunt branches
and an inductor in the series branch. We want the network to have
the same scattering parameters as the transmission line. Due to
the symmetry, the two capacitors must be identical (C1=C2=C).
We place a generator of emf E=2V, angular frequency ,
and internal resistance R1=Z0 at one port.
We terminate the other port in R2=Z0.
Let V1 and V2 be the input and output voltages
(the lower nodes are assumed grounded). To obtain the required scattering
matrix, we should have complex voltages V1=1 Volt and
V2=-j Volt. The nodal equations for this circuit read:
Substituting V1= 1V,
V2=-jV, R1=R2=Z0, and
C1=C2=C into the second equation, we readily
so that C=1/Z0.
Hence, the reactance of the coil is XL=Z0
and the susceptance of the capacitor is BC=1/Z0.
In the transmission-line Wilkinson
power divider (for 50 Ohm nominal port impedances), we have two
quarter-wavelength transmission lines whose characteristic impedance
are 50xSQRT(2)=70.7 ohms. Each line can be replaced by a pi-network
with XL=70.7 Ohms and BC=1/70.7 Siemens, which
are the same values as in our scheme. (Two capacitors at port 1
are merged into an equivalent capacitor with two times larger capacitance.)
References for this work:
Djordjevic, A. R., Baždar, M.
B., Vitoševic, G. M., Sarkar, T. K., Harrington, R. F., Scattering
Parameters of Microwave Networks with Multiconductor Transmission
Lines (software and user's manual), Artech House, Boston, 1989.
Djordjevic, A. R., Baždar, M.
B., Harrington, R. F., Sarkar, T. K., LINPAR for Windows: Matrix
Parameters for Multiconductor Transmission Lines, Version 2.0
(software and user's manual), Artech House, Boston, 1999.
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