RF Electronics Chapter 3: Transformers and Hybrids Page 62
2022, C. J. Kikkert, James Cook University, ISBN 978-0-6486803-9-0.
The equations for this hybrid are:
V
a
-V
d
=V
1
V
c
-V
e
=V
1
Eqn. 3.6
V
c
-V
d
=V
2
V
b
-V
e
=V
2
Subtracting the first two equations and the last two equations and adding the resulting
equations gives:
2
b a
c
V V
V
Eqn. 3.7
Now consider the conditions required for isolation. If V
b
=0. For isolation between ports
A and B, the current through the bottom transformer is zero, since with a load connected
to B the voltage must be zero. That means V
2
=0 and thus V
e
=0 and V
d
=½V
a
.
For the input impedance at A to be Z
0
then the load resistor must be ½Z
0
. Under those
conditions, the current into A goes through the load and out of C. The voltages are thus:
At input A V
a
= Z
0
I Eqn. 3.8
Across the Load ½V
a
= ½Z
0
I Eqn. 3.9
At the output C ½V
a
= ½Z
0
I Eqn. 3.10
The equations are thus consistent and the assumption that V
b
=0 is justified and we have
full isolation when R=½Z
0
. The circuit works thus as a proper hybrid.
Note: Like the Wilkinson transformer, the impedance at the summing port C has an
impedance of ½Z
0
and a transformer needs to be used to change that to Z
0
if required,
similar to that for the Wilkinson transformer of figure 3.14.
Many Way Hybrids
Figure 3.21. A four-way combiner.
RF Electronics: Design and Simulation
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