**180**^{o}**
RF Hybrid**

by Michael Ellis

Copyright 2000, All Rights Reserved

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The 180^{o} hybrid
functions as a splitter/combiner for both 0^{o} and 180^{o}
signals. You may download an analysis program that performs all the calculations described in this article. The 180^{o} hybrid is usually represented as in figure 1 with the angles
labeled along the sides. A signal into port 1 appears in-phase, 3
dB attenuated, at port 2 and port 4. No signal appears at port 3
therefore port 3 is said to be isolated relative to port 1.

A signal into port 2
appears in-phase at both ports 1 and 3, and isolated at port 4.
Similarlly, a signal into port 3 appears in-phase at port 2 and
180^{o} out-of-phase at port 4.

Figure 1. 180^{o }hybrid

The 180^{o} hybrid
can be analyzed using the following steps.

1. If the current through R_{1}
is defined as I_{1}, then the voltage on the top of R_{1}
is V_{1}-I_{1}R_{1}.

2. The current through the
secondary of T_{1} has to be the same as the primary
current, I_{1}.

3. The current through R_{4}
is defined as I_{2}.

4. The voltage on the top
of R_{4} can be written in terms of I_{2} and is
V_{4}-R_{4}I_{2}.

5. The voltage across the
secondary of T_{1} is the same as the difference in
voltage

across the primary. The
voltage is V_{1}-I_{1}R_{1}-V_{4}+R_{4}I_{2}.

6. The current through the
secondary of T_{2} has to be I_{1}+I_{2}.

7. The voltage on the top
of R_{3} can be written as the difference in voltage
across the

secondary of T_{2}
and is V_{1}-I_{1}R_{1}-V_{4}+R_{4}I_{2}-V_{4}+R_{4}I_{2}.

Figure 2. 180^{o}
hybrid analysis

From figure 2, the current
through R_{2} (which is 2I_{1}+I_{2}) can
also be written in terms of the difference in the voltage across
R_{2}, or

(1) |

Also the current through R_{3}
(which is I_{1}+I_{2}) can be written in terms of
the difference in voltage across R_{3}, divided by R_{3},
or

(2) |

Rearranging equations (1)
and (2) for solution of I_{1} and I_{2} yields

(3) |

and

(4) |

For analysis, only one of
the voltages V_{1}, V_{2}, V_{3}, or V_{4},
will be non-zero at any given time. To calculate the impedance
into port 1, let V_{1} = 1 and V_{2}=V_{3}=V_{4}=0.
The port 1 input impedance becomes

(5) |

The other input impedances are

(6) |

(7) |

and

(8) |

If R_{1}, R_{2},
R_{3}, and R_{4} are set to 75 ohms, then Z_{1}
= Z_{3} = 150 ohms and Z_{2} = Z_{4} =
37.5 ohms.