180^o RF Hybrid
by Michael Ellis
Copyright 2000, All Rights Reserved

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 R₁ is defined as I₁, then the voltage on the top of R₁ is V₁-I₁R₁.

2. The current through the secondary of T₁ has to be the same as the primary current, I₁.

3. The current through R₄ is defined as I₂.

4. The voltage on the top of R₄ can be written in terms of I₂ and is V₄-R₄I₂.

5. The voltage across the secondary of T₁ is the same as the difference in voltage

across the primary. The voltage is V₁-I₁R₁-V₄+R₄I₂.

6. The current through the secondary of T₂ has to be I₁+I₂.

7. The voltage on the top of R₃ can be written as the difference in voltage across the

secondary of T₂ and is V₁-I₁R₁-V₄+R₄I₂-V₄+R₄I₂.

Figure 2. 180^o hybrid analysis

From figure 2, the current through R₂ (which is 2I₁+I₂) can also be written in terms of the difference in the voltage across R₂, or

(1)

Also the current through R₃ (which is I₁+I₂) can be written in terms of the difference in voltage across R₃, divided by R₃, or

(2)

Rearranging equations (1) and (2) for solution of I₁ and I₂ yields

(3)

and

(4)

For analysis, only one of the voltages V₁, V₂, V₃, or V₄, will be non-zero at any given time. To calculate the impedance into port 1, let V₁ = 1 and V₂=V₃=V₄=0. The port 1 input impedance becomes

(5)

The other input impedances are

(6)

(7)

and

(8)

If R₁, R₂, R₃, and R₄ are set to 75 ohms, then Z₁ = Z₃ = 150 ohms and Z₂ = Z₄ = 37.5 ohms.