Hybrid-Tee Termination Analysis

Hybrid-Tee Termination Analysis: E-Arm Terminated, H-Arm Terminated, and Co-Planar Arms Terminated in a Magic Tee

Note:

a) E-arm terminated: H-arm input divides equally between the two co-planar arms with the same phase.

b) H-arm terminated: E-arm input divides equally between the two co-planar arms with a 180° phase difference.

c) Co-planar arms terminated: E-arm and H-arm are isolated. H-arm gives the sum signal, while E-arm gives the difference signal.

Hybrid Tee with E-Arm Terminated

Consider an ideal Hybrid Tee (Magic Tee) consisting of four ports. Ports 1 and 2 are the collinear arms, Port 3 is the H-arm (Sum Port), and Port 4 is the E-arm (Difference Port). The ideal Magic Tee is a lossless, reciprocal, and perfectly matched microwave junction that combines the characteristics of both E-plane and H-plane tees. One of the most useful analyses of the Hybrid Tee is studying its behavior when one of its ports is terminated with a matched load.

 

When the E-arm (Port 4) is terminated with a matched load, the reflection coefficient at that port becomes zero.

\[ \Gamma_4 = 0 \]

Since the load is perfectly matched, no reflected wave returns from Port 4. As a result, the E-arm can be removed from the external network analysis, leaving a three-port microwave junction consisting of Ports 1, 2, and 3.

Reduced S-Matrix After E-Arm Termination

Starting from the ideal Magic Tee scattering matrix,

\[ [S] = \frac{1}{\sqrt{2}} \begin{bmatrix} 0 & 0 & 1 & 1\\ 0 & 0 & 1 & -1\\ 1 & 1 & 0 & 0\\ 1 & -1 & 0 & 0 \end{bmatrix} \]

removing the terminated E-arm produces the following three-port network:

\[ [S]_{E\text{-terminated}} = \frac{1}{\sqrt{2}} \begin{bmatrix} 0 & 0 & 1\\ 0 & 0 & 1\\ 1 & 1 & 0 \end{bmatrix} \]

Method 1: Direct Excitation Analysis

To understand the operation of the reduced network, assume a signal is applied only at the H-arm while all other ports are matched.

\[ a_3 \neq 0 \]

\[ a_1=a_2=0 \]

Using the scattering relation

\[ [b]=[S][a] \]

the output waves become

\[ b_1=\frac{a_3}{\sqrt{2}} \]

\[ b_2=\frac{a_3}{\sqrt{2}} \]

\[ b_3=0 \]

The signal entering the H-arm therefore divides equally between the two collinear arms. Since both coefficients are positive and identical, the outputs have exactly the same phase.

Therefore:

• Power divides equally between Ports 1 and 2.
• Each port receives 50% of the incident power.
• The outputs have equal magnitude.
• The outputs are in phase.
• No signal appears at the terminated E-arm.

Method 2: Using S-Matrix Properties

The same result can be obtained directly from the properties of the reduced scattering matrix.

Because the Hybrid Tee is reciprocal,

\[ S_{ij}=S_{ji} \]

and because it is lossless,

\[ [S][S]^{\dagger}=[I] \]

The diagonal terms are zero,

\[ S_{11}=S_{22}=S_{33}=0 \]

showing that all remaining ports are perfectly matched.

The isolation condition is

\[ S_{12}=S_{21}=0 \]

which proves that the two collinear arms are isolated from one another.

The coupling coefficients are

\[ S_{13}=S_{23} = \frac{1}{\sqrt{2}} \]

The equal magnitudes indicate equal power division, while the identical signs indicate zero phase difference between the outputs. Therefore, the H-arm acts as an equal in-phase power divider when the E-arm is terminated.

Properties of the E-Arm Terminated Hybrid Tee

• All three remaining ports are perfectly matched.
• The two collinear ports remain isolated.
• The H-arm divides power equally between Ports 1 and 2.
• The outputs at Ports 1 and 2 are equal in magnitude.
• The outputs at Ports 1 and 2 are in phase.
• Each collinear port receives 50% of the incident power.
• The network behaves as an in-phase three-port power divider.

H-Arm Terminated Hybrid-Tee Analysis

In this case, the H-arm (Port 3) of the Hybrid-Tee is terminated with a perfectly matched load. Since a matched load absorbs all incident power without reflection, the reflection coefficient at Port 3 becomes zero. As a result, the H-arm can be removed from the external network analysis, leaving a three-port junction consisting of the two collinear arms (Ports 1 and 2) and the E-arm (Port 4).

 

The termination condition is

\[ \Gamma_3 = 0 \]

For a matched termination,

\[ b_3 = 0 \]

The ideal Hybrid-Tee scattering matrix is

\[ [S] = \frac{1}{\sqrt{2}} \begin{bmatrix} 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & -1 \\ 1 & 1 & 0 & 0 \\ 1 & -1 & 0 & 0 \end{bmatrix} \]

After terminating Port 3, the remaining accessible ports are Port 1, Port 2, and Port 4. The reduced three-port scattering matrix becomes

\[ [S]_{H\text{-terminated}} = \frac{1}{\sqrt{2}} \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & -1 \\ 1 & -1 & 0 \end{bmatrix} \]

Method 1: Direct Excitation Analysis

To understand the behavior of the reduced network, assume a signal is applied only at the E-arm while all other ports are terminated in matched loads.

\[ a_4 \neq 0 \]

\[ a_1=a_2=a_3=0 \]

Using the scattering matrix relationship

\[ [b]=[S][a] \]

the output waves become

\[ b_1=\frac{a_4}{\sqrt{2}} \]

\[ b_2=-\frac{a_4}{\sqrt{2}} \]

\[ b_3=0 \]

These equations immediately reveal the fundamental operating principle of the E-arm. The signal entering Port 4 is divided equally between the two collinear arms. Since both coefficients have the same magnitude, each output receives exactly half of the input power. However, the negative sign associated with \(b_2\) indicates a phase reversal of \(180^\circ\).

Therefore,

\[ |b_1|=|b_2| \]

and

\[ \angle b_1-\angle b_2=180^\circ \]

This proves that the E-arm behaves as a difference port. Any signal entering the E-arm produces equal-amplitude outputs that are opposite in phase at the two collinear arms.

Resulting Properties

• Equal power division between Port 1 and Port 2.
• Outputs have equal magnitude.
• Outputs are 180° out of phase.
• H-arm remains isolated.
• No power is coupled into Port 3.
• E-arm acts as the Difference Arm.

Method 2: Verification Using S-Matrix Properties

The same conclusions can be obtained directly from the properties of the ideal Hybrid-Tee scattering matrix without explicitly exciting the network.

First, the matrix is reciprocal:

\[ S_{ij}=S_{ji} \]

which confirms that the transmission characteristics are identical in both directions.

Second, the network is lossless and therefore satisfies the unitary condition

\[ [S][S]^{\dagger}=[I] \]

This condition guarantees conservation of power. Since

\[ |S_{14}|^2 = |S_{24}|^2 = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]

the total power leaving Ports 1 and 2 becomes

\[ \frac{1}{2}+\frac{1}{2}=1 \]

which proves that all incident power entering the E-arm is delivered to the collinear arms.

The phase relationship is determined directly from the coupling coefficients:

\[ S_{14}=+\frac{1}{\sqrt{2}} \]

\[ S_{24}=-\frac{1}{\sqrt{2}} \]

The opposite signs indicate a phase difference of

\[ 180^\circ \]

between the two outputs.

The isolation property is obtained from

\[ S_{34}=S_{43}=0 \]

which proves that no signal can be transferred directly between the E-arm and the H-arm.

Similarly,

\[ S_{12}=S_{21}=0 \]

showing that the two collinear arms remain isolated from one another even though they receive power from the same excitation source.

Key Conclusion

When the H-arm is terminated and the E-arm is excited, the Hybrid-Tee operates as an equal power divider with opposite-phase outputs. The signal entering the E-arm splits equally between the two collinear ports, each receiving 50% of the input power, while maintaining a phase difference of \(180^\circ\). This unique phase property is the reason Port 4 is called the Difference Arm and forms the foundation of balanced mixers, microwave comparators, and difference-channel radar systems.

Co-Planar Arms Terminated Hybrid-Tee Analysis

A third important operating condition occurs when both collinear arms of the Hybrid-Tee are terminated with perfectly matched loads. In this configuration, Ports 1 and 2 are connected to matched terminations so that all incident energy reaching these ports is completely absorbed without reflection. The only remaining accessible ports are the H-arm (Port 3) and the E-arm (Port 4). Studying this condition clearly demonstrates one of the most important characteristics of the Magic Tee: the complete isolation between the E-arm and the H-arm.

The matched termination conditions are

\[ \Gamma_1=\Gamma_2=0 \]

Therefore,

\[ b_1=b_2=0 \]

The ideal Hybrid-Tee scattering matrix is

\[ [S] = \frac{1}{\sqrt{2}} \begin{bmatrix} 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & -1 \\ 1 & 1 & 0 & 0 \\ 1 & -1 & 0 & 0 \end{bmatrix} \]

Removing the two terminated collinear arms leaves only Ports 3 and 4. The reduced network becomes

\[ [S]_{\text{Co-Planar Terminated}} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \]

where the remaining ports are

\[ \text{Port 3 (H-arm)} \]

and

\[ \text{Port 4 (E-arm)} \]

Method 1: Direct Excitation Analysis

Assume a signal is applied only to the H-arm while the E-arm is left terminated.

\[ a_3 \neq 0 \]

\[ a_1=a_2=a_4=0 \]

Using

\[ [b]=[S][a] \]

the output at Port 4 becomes

\[ b_4=S_{43}a_3 \]

Since

\[ S_{43}=0 \]

we obtain

\[ b_4=0 \]

Thus, no power reaches the E-arm when the H-arm is excited.

Similarly, if the E-arm is excited

\[ a_4 \neq 0 \]

\[ a_1=a_2=a_3=0 \]

then

\[ b_3=S_{34}a_4 \]

and since

\[ S_{34}=0 \]

we obtain

\[ b_3=0 \]

Therefore, no signal can propagate from the E-arm to the H-arm. The two side arms remain completely isolated from one another.

Resulting Properties

• E-arm and H-arm are completely isolated.
• No power is transferred between Ports 3 and 4.
• Both ports remain perfectly matched.
• No reflections occur at either side arm.
• All incident power is absorbed by the matched loads connected to Ports 1 and 2.
• The network behaves as two isolated matched ports.

Method 2: Verification Using S-Matrix Properties

The same conclusion can be obtained directly from the mathematical properties of the ideal Hybrid-Tee S-matrix.

The defining isolation parameters of a Magic Tee are

\[ S_{34}=0 \]

and

\[ S_{43}=0 \]

These two elements indicate that no signal transmission exists between the H-arm and E-arm under any operating condition.

The matching condition is obtained from

\[ S_{33}=0 \]

and

\[ S_{44}=0 \]

Since the reflection coefficients are zero, both side arms are perfectly matched.

The lossless property is verified through the unitary condition

\[ [S][S]^{\dagger}=[I] \]

Because the columns corresponding to Ports 3 and 4 are orthogonal,

\[ S_{13}S_{14}^{*} + S_{23}S_{24}^{*} + S_{33}S_{34}^{*} + S_{43}S_{44}^{*} = 0 \]

Substituting the ideal values,

\[ \left(\frac{1}{\sqrt{2}}\right) \left(\frac{1}{\sqrt{2}}\right) + \left(\frac{1}{\sqrt{2}}\right) \left(-\frac{1}{\sqrt{2}}\right) = 0 \]

which confirms mathematically that the H-arm and E-arm remain isolated.

This orthogonality condition is one of the most important characteristics of the Hybrid-Tee and is responsible for its use in duplexers, balanced mixers, microwave bridges, and radar systems.

Physical Interpretation

The H-arm distributes energy equally to the two collinear arms with the same phase, whereas the E-arm distributes energy equally with opposite phase. When the waves generated by these two excitation modes are compared, their vector sums cancel at the opposite side arm. Because of this cancellation mechanism, energy entering the H-arm cannot appear at the E-arm, and energy entering the E-arm cannot appear at the H-arm. This natural cancellation creates an electrical isolation that does not require any additional components and is the reason the Magic Tee is widely used in microwave systems requiring high isolation between transmit and receive paths.

Key Conclusion

When both collinear arms are terminated with matched loads, the remaining H-arm and E-arm behave as two perfectly matched but completely isolated ports. No signal transmission occurs between them because

\[ S_{34}=S_{43}=0 \]

while

\[ S_{33}=S_{44}=0 \]

ensures perfect matching. This isolation property is one of the fundamental reasons why the Hybrid-Tee is extensively used in duplexers, balanced microwave mixers, bridge circuits, and high-frequency measurement systems.