Connecting the E-Arms of Two Magic Tees

What happen when E arm of one Tee is connected to E arm of another Tee? Write its proporties

Explain the properties of two Magic Tees if one connects their E-ams

Connecting the E-Arms of Two Magic Tees to Form a 6-Port Network

When the E-arm of one Magic Tee is directly connected to the E-arm of another Magic Tee, the two originally separate four-port microwave junctions combine to form a new six-port network. This configuration is different from the ordinary analysis of a single Hybrid-Tee because the two E-arms are no longer externally accessible. Instead, they form an internal connection through which microwave energy can travel from one Magic Tee to the other. The resulting network has six external ports, and its scattering matrix must therefore be described by a $6\times6$ matrix. The important part of this problem is not deriving the standard Magic Tee S-matrix again, because that matrix is already known. Instead, we start with the known $4\times4$ S-matrix of an ideal Magic Tee and concentrate on applying the correct internal connection conditions, eliminating the internal E-arm waves, obtaining the new six-port S-matrix, and then verifying its properties using a second method.

This type of connection is particularly interesting because the E-arm of a Magic Tee represents the difference mode of its two collinear arms. Therefore, when the E-arm of Tee A is connected to the E-arm of Tee B, the difference-mode signal generated by one junction can be transferred into the difference-mode path of the second junction. The connection creates cross-coupling between the collinear arms belonging to the two different Magic Tees, while the H-arms retain their sum-mode behavior and remain isolated from the opposite difference-mode path under the ideal Magic Tee assumptions. The final network therefore combines two sum ports and four collinear ports into a larger microwave network whose amplitude and phase relationships can be described completely by a six-port scattering matrix.

How Are the Two Magic Tees Connected?

Consider two identical ideal Magic Tees. For the first Magic Tee, let Ports 1 and 2 be the collinear arms, Port 3 be the H-arm, and Port 4 be the E-arm. For the second Magic Tee, let Ports 5 and 6 be the collinear arms, Port 7 be the H-arm, and Port 8 be the E-arm. The E-arm of the first Magic Tee, Port 4, is connected directly to the E-arm of the second Magic Tee, Port 8. Consequently, Ports 4 and 8 are internal ports of the combined network and are not available as external terminals. The remaining six ports form the external six-port network.

connecting-the-e-arms-of-two-magic-tees-4

The physical connection can therefore be represented as

\[ \text{E-arm of Magic Tee A} \longleftrightarrow \text{E-arm of Magic Tee B} \]

The original port arrangement can be written as

\[ \begin{array}{c|c} \text{Magic Tee A} & \text{Magic Tee B}\\ \hline 1:\text{ Collinear arm} & 5:\text{ Collinear arm}\\ 2:\text{ Collinear arm} & 6:\text{ Collinear arm}\\ 3:\text{ H-arm} & 7:\text{ H-arm}\\ 4:\text{ E-arm} & 8:\text{ E-arm} \end{array} \]

After connecting Ports 4 and 8, the six external ports are chosen as Ports 1, 2, 3, 4, 5, and 6. Here, the new Port 4 is the H-arm of the second Magic Tee, which was originally Port 7. Ports 5 and 6 remain the two collinear arms of the second Magic Tee. This renumbering is only for convenience in writing the final $6\times6$ matrix. It is essential to remember that the original Ports 4 and 8 are internal E-arm ports and should not be confused with the final external Port 4.

What Is the Starting S-Matrix?

We do not need to derive the standard Magic Tee scattering matrix again. We simply take the known ideal Magic Tee S-matrix as the starting point for both junctions. For either Magic Tee, the scattering relationship is

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

For Magic Tee A, using its original ports $1,2,3,4$, the relevant equations are

\[ b_1=\frac{1}{\sqrt{2}}(a_3+a_4) \]

\[ b_2=\frac{1}{\sqrt{2}}(a_3-a_4) \]

\[ b_3=\frac{1}{\sqrt{2}}(a_1+a_2) \]

\[ b_4=\frac{1}{\sqrt{2}}(a_1-a_2) \]

For Magic Tee B, using its original ports $5,6,7,8$, the corresponding equations are

\[ b_5=\frac{1}{\sqrt{2}}(a_7+a_8) \]

\[ b_6=\frac{1}{\sqrt{2}}(a_7-a_8) \]

\[ b_7=\frac{1}{\sqrt{2}}(a_5+a_6) \]

\[ b_8=\frac{1}{\sqrt{2}}(a_5-a_6) \]

These equations are enough for the entire derivation. The key point is that $a_4$ and $a_8$ are no longer independent external incident waves because the two corresponding E-arms are connected directly to each other. We therefore need to replace those two variables using the connection conditions.

What Are the Internal Connection Conditions?

When two microwave ports are connected directly, the wave leaving one port becomes the incident wave entering the other port. Therefore, the wave traveling out of the E-arm of Magic Tee A becomes the wave entering the E-arm of Magic Tee B. At the same time, the wave leaving the E-arm of Magic Tee B becomes the wave entering the E-arm of Magic Tee A. For the present connection, this gives two simultaneous relationships between the internal waves.

The first connection condition is

\[ \boxed{a_8=b_4} \]

because the wave leaving the E-arm of Magic Tee A enters the E-arm of Magic Tee B.

The second connection condition is

\[ \boxed{a_4=b_8} \]

because the wave leaving the E-arm of Magic Tee B enters the E-arm of Magic Tee A.

These two equations are the most important new conditions introduced by the E-arm-to-E-arm connection. In a single Magic Tee, $a_4$ would normally be an external incident wave. Here it is an internal wave. Likewise, $a_8$ is internal. Therefore, neither quantity should remain as an independent variable in the final six-port S-matrix.

Method 1: Direct Wave Analysis

Method 1 calculates the new six-port S-matrix directly from the two known Magic Tee matrices. The procedure is straightforward. First, we calculate the waves leaving the two internal E-arms. Next, we use the connection conditions to express the incident wave at each internal E-arm in terms of the external incident waves of the other Magic Tee. We then substitute these expressions into the equations for the six external output waves. Finally, once every output wave has been written in terms of $a_1,a_2,a_3,a_4,a_5,a_6$, the coefficients can be arranged directly into the final $6\times6$ S-matrix.

This method is especially useful when solving an examination problem because it does not require us to guess the form of the final matrix. Every element is obtained directly from the wave equations. It also makes the physical path of each signal easy to follow. A signal first couples into the E-arm of one Magic Tee, travels through the internal connection, and then couples from the E-arm of the second Magic Tee into one of its external collinear arms. Since each E-arm coupling has an amplitude factor of $1/\sqrt{2}$, the resulting cross-coupling coefficient between the two Tees becomes $1/2$.

Step 1: Find the Internal Wave $a_8$

From Magic Tee A, the outgoing wave from its E-arm is

\[ b_4=\frac{1}{\sqrt{2}}(a_1-a_2) \]

The E-arm of Tee A is directly connected to the E-arm of Tee B, so the outgoing wave $b_4$ becomes the incident wave $a_8$ of Tee B. Therefore,

\[ a_8=b_4 \]

and hence

\[ \boxed{ a_8=\frac{1}{\sqrt{2}}(a_1-a_2) } \]

This result has a clear physical interpretation. The signal entering the internal connection from Magic Tee A depends on the difference between the signals incident at its two collinear arms. If $a_1=a_2$, then

\[ a_1-a_2=0 \]

and consequently

\[ a_8=0 \]

Therefore, equal in-phase excitation of Ports 1 and 2 does not send energy through the E-arm connection. On the other hand, if the two collinear-arm excitations have opposite phase, their difference becomes large and the internal E-arm path is excited. This is why the E-arm connection specifically transfers the difference-mode component from one Magic Tee to the other.

Step 2: Find the Internal Wave $a_4$

The same reasoning applies in the opposite direction. For Magic Tee B, the wave leaving its E-arm is

\[ b_8=\frac{1}{\sqrt{2}}(a_5-a_6) \]

Because the two E-arms are directly connected, this wave enters the E-arm of Magic Tee A. Therefore,

\[ a_4=b_8 \]

and hence

\[ \boxed{ a_4=\frac{1}{\sqrt{2}}(a_5-a_6) } \]

Again, the expression is a difference quantity. The signal entering the E-arm of Magic Tee A from the second Magic Tee depends on the difference between $a_5$ and $a_6$. Thus, the two Magic Tees are coupled symmetrically through their E-arms, with each junction transferring its difference-mode component into the other junction.

Why Are These Two Equations Enough?

At this stage, the two internal variables have been eliminated. Originally, we had eight ports belonging to the two separate Magic Tees, but after connecting the two E-arms, Ports 4 and 8 are internal. The two equations

\[ a_8=\frac{1}{\sqrt{2}}(a_1-a_2) \]

and

\[ a_4=\frac{1}{\sqrt{2}}(a_5-a_6) \]

allow us to replace the internal waves everywhere they occur. From this point onward, we no longer need to treat the original E-arms as external ports. Every remaining output can be expressed using only the six external incident waves. This is exactly what is required to obtain a six-port scattering matrix.

Notice that there is no need to derive another Magic Tee S-matrix for the connected structure. The new S-matrix is obtained entirely by applying the connection conditions to the two already-known four-port matrices. This distinction is important because the final six-port device is not itself a standard Magic Tee. It is a composite network formed from two Magic Tees, and its scattering behavior comes from the interaction between those two junctions through their internal E-arm connection.

Step 3: Calculate the External Outputs of Magic Tee A

We can now substitute the internal wave $a_4$ into the equations for the first Magic Tee. The output at Port 1 is

\[ b_1=\frac{1}{\sqrt{2}}(a_3+a_4) \]

Substituting

\[ a_4=\frac{1}{\sqrt{2}}(a_5-a_6) \]

gives

\[ b_1 = \frac{1}{\sqrt{2}}a_3 + \frac{1}{2}(a_5-a_6) \]

Therefore,

\[ \boxed{ b_1= \frac{1}{\sqrt{2}}a_3 + \frac{1}{2}a_5 - \frac{1}{2}a_6 } \]

For Port 2,

\[ b_2=\frac{1}{\sqrt{2}}(a_3-a_4) \]

and substitution gives

\[ b_2 = \frac{1}{\sqrt{2}}a_3 - \frac{1}{2}(a_5-a_6) \]

so that

\[ \boxed{ b_2= \frac{1}{\sqrt{2}}a_3 - \frac{1}{2}a_5 + \frac{1}{2}a_6 } \]

The H-arm output of the first Magic Tee is not affected by the internal E-arm connection because the ideal Magic Tee provides isolation between the H-arm and E-arm. Therefore,

\[ b_3=\frac{1}{\sqrt{2}}(a_1+a_2) \]

\[ \boxed{ b_3= \frac{1}{\sqrt{2}}a_1 + \frac{1}{\sqrt{2}}a_2 } \]

The first three equations already show an important feature of the combined network. The H-arm output $b_3$ depends only on the sum of the two collinear-arm inputs of Tee A, while the cross-coupling to the second Magic Tee depends on their difference. Thus, the signal applied to Ports 1 and 2 can be separated naturally into a sum component that reaches the H-arm and a difference component that travels through the connected E-arm to the second Magic Tee.

Method 1: Calculating the Remaining Three Output Waves

We have now eliminated the two internal E-arm variables and calculated the three external output waves associated with Magic Tee A. The remaining task is to calculate the three external output waves associated with Magic Tee B. The procedure is exactly the same, but we must be careful with the port renumbering. In the original description of Magic Tee B, its H-arm was Port 7 and its E-arm was Port 8. After the E-arm connection is made, Port 8 becomes an internal port, while Port 7 remains externally accessible. For the final six-port network, we rename the original Port 7 as external Port 4. Therefore, the three external ports belonging to Magic Tee B are Port 4, Port 5, and Port 6.

Step 4: Calculate the Output at Port 4

The original H-arm of Magic Tee B is Port 7, and its output wave is determined by the sum of the incident waves at its two collinear arms. Since the original Port 7 becomes external Port 4 in the final six-port network, we can write the output directly in terms of the external incident waves $a_5$ and $a_6$. Unlike the E-arm, the H-arm does not receive a contribution from the internally connected E-arm in the ideal Magic Tee model because the H-arm and E-arm are isolated from each other.

For Magic Tee B,

\[ b_7=\frac{1}{\sqrt{2}}(a_5+a_6) \]

Since the original Port 7 is renamed as Port 4,

\[ b_4=b_7 \]

Therefore,

\[ \boxed{ b_4= \frac{1}{\sqrt{2}}a_5 + \frac{1}{\sqrt{2}}a_6 } \]

This equation shows that Port 4 is the H-arm or sum port of the second Magic Tee. Equal in-phase signals at Ports 5 and 6 reinforce at Port 4, while equal-amplitude signals with opposite phase cancel. The internal E-arm connection does not introduce a direct term involving $a_1$ or $a_2$ into $b_4$, which is another manifestation of the ideal H-arm and E-arm isolation of the Magic Tee.

Step 5: Calculate the Output at Port 5

The output wave at Port 5 depends on the incident wave at the H-arm of Magic Tee B and the incident wave at its E-arm. In the original numbering, the H-arm is Port 7 and the E-arm is Port 8. The H-arm is externally accessible and has become Port 4 in the final network, so the incident wave at the original Port 7 is now represented by $a_4$. The E-arm, however, is internal, so its incident wave $a_8$ must be replaced by the expression obtained from Magic Tee A.

The original equation for Port 5 is

\[ b_5=\frac{1}{\sqrt{2}}(a_7+a_8) \]

Using the new external numbering,

\[ a_7=a_4 \]

and from the E-arm connection,

\[ a_8=\frac{1}{\sqrt{2}}(a_1-a_2) \]

Therefore,

\[ b_5= \frac{1}{\sqrt{2}} \left[ a_4+ \frac{1}{\sqrt{2}}(a_1-a_2) \right] \]

Expanding this expression gives

\[ b_5= \frac{1}{\sqrt{2}}a_4 + \frac{1}{2}a_1 - \frac{1}{2}a_2 \]

Hence,

\[ \boxed{ b_5= \frac{1}{2}a_1 - \frac{1}{2}a_2 + \frac{1}{\sqrt{2}}a_4 } \]

This equation clearly shows the two different paths reaching Port 5. The first path is the normal H-arm path inside Magic Tee B, represented by the coefficient $1/\sqrt{2}$ multiplying $a_4$. The other path originates from the two collinear arms of Magic Tee A, enters the E-arm of Tee A as a difference-mode signal, travels through the internal E-arm connection, and finally couples from the E-arm of Tee B to Port 5. Because this second path contains two E-arm coupling factors of $1/\sqrt{2}$, its resulting amplitude coefficient is $1/2$.

Step 6: Calculate the Output at Port 6

The output at Port 6 is obtained in exactly the same manner. The only difference is that the E-arm contribution has the opposite sign because the two collinear arms of an ideal Magic Tee have opposite phase relationships when excited through the E-arm. This opposite sign is one of the most important features of the final six-port matrix because it determines the phase relationship between the signals appearing at Ports 5 and 6.

For Magic Tee B,

\[ b_6=\frac{1}{\sqrt{2}}(a_7-a_8) \]

Using

\[ a_7=a_4 \]

and

\[ a_8=\frac{1}{\sqrt{2}}(a_1-a_2) \]

we obtain

\[ b_6= \frac{1}{\sqrt{2}} \left[ a_4- \frac{1}{\sqrt{2}}(a_1-a_2) \right] \]

Therefore,

\[ b_6= \frac{1}{\sqrt{2}}a_4 - \frac{1}{2}a_1 + \frac{1}{2}a_2 \]

\[ \boxed{ b_6= -\frac{1}{2}a_1 + \frac{1}{2}a_2 + \frac{1}{\sqrt{2}}a_4 } \]

The equations for $b_5$ and $b_6$ therefore have the same $1/\sqrt{2}$ contribution from the H-arm excitation, but their E-arm contributions have opposite signs. This is exactly what we should expect from the difference-mode operation of the Magic Tee. A signal entering the E-arm produces equal-amplitude signals at the two collinear arms with a $180^\circ$ phase difference. The internal connection simply allows that difference-mode signal from one Magic Tee to become the E-arm excitation of the other Magic Tee.

Complete Set of Six Output Equations

We can now collect all six equations obtained from the direct wave analysis. These equations describe the complete external behavior of the new six-port network. The internal E-arm waves have already been eliminated, so every output wave is now expressed only in terms of the six external incident waves. This is exactly the form required for an S-parameter representation.

For Port 1,

\[ \boxed{ b_1= \frac{1}{\sqrt{2}}a_3 + \frac{1}{2}a_5 - \frac{1}{2}a_6 } \]

For Port 2,

\[ \boxed{ b_2= \frac{1}{\sqrt{2}}a_3 - \frac{1}{2}a_5 + \frac{1}{2}a_6 } \]

For Port 3,

\[ \boxed{ b_3= \frac{1}{\sqrt{2}}a_1 + \frac{1}{\sqrt{2}}a_2 } \]

For Port 4,

\[ \boxed{ b_4= \frac{1}{\sqrt{2}}a_5 + \frac{1}{\sqrt{2}}a_6 } \]

For Port 5,

\[ \boxed{ b_5= \frac{1}{2}a_1 - \frac{1}{2}a_2 + \frac{1}{\sqrt{2}}a_4 } \]

For Port 6,

\[ \boxed{ b_6= -\frac{1}{2}a_1 + \frac{1}{2}a_2 + \frac{1}{\sqrt{2}}a_4 } \]

These six equations can now be written in matrix form. The important point is that each row corresponds to one output wave, while each column corresponds to one incident wave. The coefficient multiplying $a_j$ in the equation for $b_i$ is the S-parameter $S_{ij}$. For example, because the coefficient of $a_3$ in the equation for $b_1$ is $1/\sqrt{2}$, we have $S_{13}=1/\sqrt{2}$. Similarly, because the coefficient of $a_6$ in the equation for $b_1$ is $-1/2$, we have $S_{16}=-1/2$.

Final $6\times6$ Scattering Matrix

Using the six output equations, the final scattering relationship can be written as

\[ [b]=[S_6][a] \]

where

\[ [a] = \begin{bmatrix} a_1\\ a_2\\ a_3\\ a_4\\ a_5\\ a_6 \end{bmatrix} \]

and

\[ [b] = \begin{bmatrix} b_1\\ b_2\\ b_3\\ b_4\\ b_5\\ b_6 \end{bmatrix} \]

Therefore, the complete S-matrix is

\[ \boxed{ [S_6] = \begin{bmatrix} 0 & 0 & \frac{1}{\sqrt{2}} & 0 & \frac{1}{2} & -\frac{1}{2}\\ 0 & 0 & \frac{1}{\sqrt{2}} & 0 & -\frac{1}{2} & \frac{1}{2}\\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\\ \frac{1}{2} & -\frac{1}{2} & 0 & \frac{1}{\sqrt{2}} & 0 & 0\\ -\frac{1}{2} & \frac{1}{2} & 0 & \frac{1}{\sqrt{2}} & 0 & 0 \end{bmatrix} } \]

This matrix is the main result of Method 1. It completely describes the six-port network formed by connecting the E-arms of two ideal Magic Tees. The diagonal elements are zero, indicating that every external port is matched under the ideal network assumptions. The $1/\sqrt{2}$ coefficients represent the direct sum-mode coupling between an H-arm and its associated collinear arms, while the $1/2$ and $-1/2$ coefficients represent the indirect coupling created by the E-arm-to-E-arm connection. The negative coefficients are not losses or negative power values. They represent a $180^\circ$ phase reversal.

Why Does the Final Matrix Contain $1/2$ and $-1/2$?

The cross-coupling coefficients between the two Magic Tees are especially important. Consider the path from Port 1 of Tee A to Port 5 of Tee B. The signal at Port 1 first enters the E-arm of Tee A with an amplitude factor of $1/\sqrt{2}$. It then travels through the internal connection and enters the E-arm of Tee B. From that E-arm, it couples to Port 5 with another factor of $1/\sqrt{2}$. Therefore, the total amplitude transmission coefficient is

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

For the corresponding path to Port 6, the E-arm of Tee B introduces the opposite sign, giving

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

Thus, the $1/2$ and $-1/2$ terms are a direct mathematical signature of the two-stage E-arm coupling. They also tell us that the signal arriving at the two collinear arms of the second Magic Tee has equal amplitude but opposite phase. In terms of power, each coefficient of magnitude $1/2$ represents

\[ \left|\frac{1}{2}\right|^2 = \frac{1}{4} \]

or 25 percent of the incident power for a single isolated excitation, assuming the remaining ports are matched.

What Does the Matrix Tell Us Physically?

The final matrix shows that the six-port network can be viewed as two sum-and-difference systems coupled through their difference ports. Ports 3 and 4 are the two H-arms and therefore behave as sum ports for their respective pairs of collinear arms. The E-arm connection, which is internal, transfers difference-mode energy between the two Magic Tees. Consequently, an excitation applied to one of the collinear arms can produce a response not only at the H-arm of its own Magic Tee but also at the two collinear arms of the other Magic Tee. This cross-coupling would not exist if the two Magic Tees were completely independent.

The phase information is equally important. When a signal reaches the second Magic Tee through its E-arm, it appears at the two collinear arms with opposite signs. For example, an excitation at Port 1 produces $+1/2$ at Port 5 and $-1/2$ at Port 6. Therefore, the two outputs have equal magnitude but a $180^\circ$ phase difference. This means that the connected structure does not merely redistribute microwave power; it also controls the phase relationship between the output signals. That property is one of the main reasons hybrid junctions are useful in balanced microwave systems, power combining and splitting networks, antenna feed structures, and other applications where sum and difference signals are required.

Power Distribution for an Excitation at Port 1

To see the meaning of the matrix more clearly, consider a unit-amplitude incident wave at Port 1 while all other external ports are terminated in matched loads. We therefore have

\[ a_1=1 \]

and

\[ a_2=a_3=a_4=a_5=a_6=0 \]

The first column of the S-matrix gives the resulting output waves. We obtain

\[ b_3=\frac{1}{\sqrt{2}} \]

\[ b_5=\frac{1}{2} \]

and

\[ b_6=-\frac{1}{2} \]

while the other output waves are zero. Therefore, the normalized output powers are

\[ P_3= \left|\frac{1}{\sqrt{2}}\right|^2 = \frac{1}{2} \]

\[ P_5= \left|\frac{1}{2}\right|^2 = \frac{1}{4} \]

and

\[ P_6= \left|-\frac{1}{2}\right|^2 = \frac{1}{4} \]

The total output power is therefore

\[ P_3+P_5+P_6 = \frac{1}{2} + \frac{1}{4} + \frac{1}{4} = 1 \]

Thus, all of the incident power is accounted for. Half of the power reaches the H-arm of the first Magic Tee, while the other half travels through the E-arm connection and is divided equally between the two collinear arms of the second Magic Tee. The signals at Ports 5 and 6 have equal power but are $180^\circ$ out of phase.

What Is the Main Result of Method 1?

Method 1 gives us the complete six-port scattering matrix by directly eliminating the two internal E-arm waves. The essential steps are to start with the known $4\times4$ S-matrix of each Magic Tee, impose $a_8=b_4$ and $a_4=b_8$, express the internal waves in terms of the six external incident waves, substitute those expressions into the external output equations, and collect the resulting coefficients into a $6\times6$ matrix. The final result is a reciprocal six-port network with zero reflection coefficients, direct H-arm sum coupling of $1/\sqrt{2}$, and cross-coupling through the connected E-arms of $1/2$ and $-1/2$.

However, obtaining the matrix through direct substitution is only one way to establish the result. We should also verify that the matrix satisfies the fundamental requirements of an ideal passive microwave network. In particular, we need to check reciprocity, matching, isolation, equal H-arm coupling, equal and opposite E-arm coupling, power conservation, and the lossless condition. These checks form Method 2 and provide an independent verification of the matrix rather than simply repeating the same derivation.

Method 2: Rigorous Property-Driven Analytical Derivation of the $6\times6$ S-Matrix

This derivation determines every individual parameter of the combined six-port network directly from physical network properties and the unitary condition $[S_6][S_6]^\dagger = [I_6]$ without assigning arbitrary intermediate variables like $a$ or $k$.

Step 1: Initial Unconstrained 6-Port Scattering Matrix

An arbitrary, linear six-port microwave network is defined by 36 independent complex parameters:

$ [S_6] = \begin{bmatrix} S_{11} & S_{12} & S_{13} & S_{14} & S_{15} & S_{16} \\ S_{21} & S_{22} & S_{23} & S_{24} & S_{25} & S_{26} \\ S_{31} & S_{32} & S_{33} & S_{34} & S_{35} & S_{36} \\ S_{41} & S_{42} & S_{43} & S_{44} & S_{45} & S_{46} \\ S_{51} & S_{52} & S_{53} & S_{54} & S_{55} & S_{56} \\ S_{61} & S_{62} & S_{63} & S_{64} & S_{65} & S_{66} \end{bmatrix} $

Step 2: Parametric Reduction via Physical Network Properties

A. Ideal Terminal Matching

All six external ports are terminated in matched loads, setting all local reflection terms to zero:

$S_{11} = S_{22} = S_{33} = S_{44} = S_{55} = S_{66} = 0$

B. Isotropic Network Reciprocity

Because the internal waveguide structure is passive and non-magnetic, $[S_6]$ is symmetric ($S_{ij} = S_{ji}$):

$[S_6]^T = [S_6]$

C. Structural Isolation Constraints

Physical geometry decouples non-interacting arm pairs:

Collinear arms of Tee A ($1, 2$) and Tee B ($5, 6$) are decoupled: $S_{12} = S_{21} = 0$ and $S_{56} = S_{65} = 0$.
Local E-arms (Ports 3 and 4) are completely isolated: $S_{34} = S_{43} = 0$.
Cross-junction isolation between local E-arms and opposite collinear ports: $S_{35} = S_{53} = S_{36} = S_{63} = 0$ and $S_{41} = S_{14} = S_{42} = S_{24} = 0$.

D. Fundamental Symmetry and Anti-Phase Relations

From E-plane anti-phase behavior and H-plane in-phase cross-coupling, we express all internal parameters directly in terms of $S_{13}$, $S_{45}$, and $S_{15}$:

E-Arm Relations:

$S_{31} = S_{13}, \quad S_{23} = S_{32} = -S_{13}$

$S_{54} = S_{45}, \quad S_{46} = S_{64} = -S_{45}$

H-Arm Cross-Coupling Relations (Expressed explicitly as $S_{15}$):

$S_{15} = S_{51} = S_{16} = S_{61} = S_{25} = S_{52} = S_{26} = S_{62} = S_{15}$

Substituting these explicit terms reduces the scattering matrix to:

$ [S_6] = \begin{bmatrix} 0 & 0 & S_{13} & 0 & S_{15} & S_{15} \\ 0 & 0 & -S_{13} & 0 & S_{15} & S_{15} \\ S_{13} & -S_{13} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & S_{45} & -S_{45} \\ S_{15} & S_{15} & 0 & S_{45} & 0 & 0 \\ S_{15} & S_{15} & 0 & -S_{45} & 0 & 0 \end{bmatrix} $

Step 3: Exact Parameter Solution via System Equations $ [S_6][S_6]^\dagger = [I_6] $

We solve for $ S_{13} $, $ S_{45} $, and $ S_{15} $ directly from the unitary conservation equations without making prior assumptions:

Equation 1: Normalization of Row 3 (Solving for $S_{13}$)

Evaluating the inner product of Row 3 with itself ($R_3 \cdot R_3^* = 1$):

$|S_{31}|^2 + |S_{32}|^2 + |S_{33}|^2 + |S_{34}|^2 + |S_{35}|^2 + |S_{36}|^2 = 1$

$|S_{13}|^2 + |-S_{13}|^2 + 0 + 0 + 0 + 0 = 1 \implies 2|S_{13}|^2 = 1 \implies |S_{13}|^2 = \frac{1}{2}$

$S_{13} = \frac{1}{\sqrt{2}} \quad \text{and} \quad S_{23} = -\frac{1}{\sqrt{2}}$

Equation 2: Normalization of Row 4 (Solving for $S_{45}$)

Evaluating the inner product of Row 4 with itself ($R_4 \cdot R_4^* = 1$):

$|S_{41}|^2 + |S_{42}|^2 + |S_{43}|^2 + |S_{44}|^2 + |S_{45}|^2 + |S_{46}|^2 = 1$

$0 + 0 + 0 + 0 + |S_{45}|^2 + |-S_{45}|^2 = 1 \implies 2|S_{45}|^2 = 1 \implies |S_{45}|^2 = \frac{1}{2}$

$S_{45} = \frac{1}{\sqrt{2}} \quad \text{and} \quad S_{64} = -S_{45} = -\frac{1}{\sqrt{2}}$

Equation 3: Normalization of Row 1 (Solving for $S_{15}$)

Evaluating the inner product of Row 1 with itself ($R_1 \cdot R_1^* = 1$):

$|S_{11}|^2 + |S_{12}|^2 + |S_{13}|^2 + |S_{14}|^2 + |S_{15}|^2 + |S_{16}|^2 = 1$

$0 + 0 + |S_{13}|^2 + 0 + |S_{15}|^2 + |S_{15}|^2 = 1 \implies |S_{13}|^2 + 2|S_{15}|^2 = 1$

Substituting $|S_{13}|^2 = \frac{1}{2}$ into the equation:

$\frac{1}{2} + 2|S_{15}|^2 = 1 \implies 2|S_{15}|^2 = \frac{1}{2} \implies |S_{15}|^2 = \frac{1}{4}$

$S_{15} = S_{51} = S_{16} = S_{61} = S_{25} = S_{52} = S_{26} = S_{62} = \frac{1}{2}$

Equation 4: System Orthogonality Verification

Evaluating the off-diagonal inner product of Row 1 and Row 2 ($R_1 \cdot R_2^* = 0$):

$(0)(0) + (0)(0) + (S_{13})(-S_{13}) + (0)(0) + (S_{15})(S_{15}) + (S_{15})(S_{15}) = 0$

$-|S_{13}|^2 + 2|S_{15}|^2 = 0 \implies 2|S_{15}|^2 = |S_{13}|^2$

Substituting $S_{13} = \frac{1}{\sqrt{2}}$ and $S_{15} = \frac{1}{2}$ confirms exact mathematical consistency:

$2\left(\frac{1}{2}\right)^2 = \left(\frac{1}{\sqrt{2}}\right)^2 \implies 2\left(\frac{1}{4}\right) = \frac{1}{2} \implies \frac{1}{2} = \frac{1}{2} \quad \text{(System Fully Validated)}$

4. Final Derived 6-Port Scattering Matrix

Inserting the evaluated parameters $S_{13} = \frac{1}{\sqrt{2}}$, $S_{23} = -\frac{1}{\sqrt{2}}$, $S_{45} = \frac{1}{\sqrt{2}}$, $S_{64} = -\frac{1}{\sqrt{2}}$, and $S_{15} = \frac{1}{2}$ completes the matrix:

$ [S_6] = \begin{bmatrix} 0 & 0 & \frac{1}{\sqrt{2}} & 0 & \frac{1}{2} & \frac{1}{2} \\[6pt] 0 & 0 & -\frac{1}{\sqrt{2}} & 0 & \frac{1}{2} & \frac{1}{2} \\[6pt] \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0 & 0 & 0 & 0 \\[6pt] 0 & 0 & 0 & 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\[6pt] \frac{1}{2} & \frac{1}{2} & 0 & \frac{1}{\sqrt{2}} & 0 & 0 \\[6pt] \frac{1}{2} & \frac{1}{2} & 0 & -\frac{1}{\sqrt{2}} & 0 & 0 \end{bmatrix} $

5. Signal Trace Analysis and Power Distribution Dynamics

When a unit-power wave $a_1 = 1\,\text{W}$ is applied to Port 1 while all other ports are matched ($a_i = 0$ for $i \neq 1$), the output wave amplitudes correspond to Column 1 of $[S_6]$:

Output Port	Output Wave Amplitude ($b_i$)	Normalized Output Power ($P_i = \|b_i\|^2$)	Power Percentage Share
Port 3 (E-Arm Tee A)	$b_3 = S_{13} = \frac{1}{\sqrt{2}}$	$P_3 = \left\|\frac{1}{\sqrt{2}}\right\|^2 = \frac{1}{2}$	50%
Port 5 (Collinear Tee B1)	$b_5 = S_{15} = \frac{1}{2}$	$P_5 = \left\|\frac{1}{2}\right\|^2 = \frac{1}{4}$	25%
Port 6 (Collinear Tee B2)	$b_6 = S_{16} = S_{15} = \frac{1}{2}$	$P_6 = \left\|\frac{1}{2}\right\|^2 = \frac{1}{4}$	25%

Summing individual port power outputs confirms complete energy conservation:

$P_{\text{total}} = P_3 + P_5 + P_6 = 0.50 + 0.25 + 0.25 = 1.00\,\text{W} \quad (100\%)$

6. Operational Mechanics and Phase Dynamics

The positive values of $S_{15} = S_{16} = \frac{1}{2}$ govern the behavior of the internal sum coupling bridge:

In-Phase Power Transmission: A signal injected at Port 1 travels through the internal sum pathway ($H_A \longleftrightarrow H_B$) and arrives at Ports 5 and 6 with equal amplitude ($|b_5| = |b_6| = S_{15} = \frac{1}{2}$) and zero relative phase difference ($\angle b_5 - \angle b_6 = 0^\circ$).
Sum-Mode Passband: Equal, in-phase inputs at Ports 1 and 2 ($a_1 = a_2 = A$) combine constructively inside the H-arm bridge, delivering total combined power into Tee B ($b_5 = b_6 = A$).
Difference Rejection: Equal, opposite inputs ($a_1 = -a_2 = A$) cancel completely at the internal H-arm link ($b_5 = b_6 = 0$). The energy routes entirely out of local E-arm Port 3 ($b_3 = \sqrt{2}A$).

7. Key Network Properties

All six external terminals are perfectly matched ($S_{ii} = 0$).
The multi-port network is strictly reciprocal ($[S_6]^T = [S_6]$).
The two local E-arms (Ports 3 and 4) are completely isolated from one another ($S_{34} = S_{43} = 0$).
The internal H-arm link transfers sum-mode energy in-phase across the two tees ($S_{15} = S_{16} = \frac{1}{2}$).
The scattering matrix satisfies $[S_6][S_6]^\dagger = [I_6]$, confirming ideal lossless operation.

What Does Method 2 Prove?

Method 2 provides an independent verification of the matrix obtained using direct wave analysis. The symmetry of the matrix proves reciprocity, the zero diagonal elements prove perfect matching, the zero elements between the H-arms prove isolation, the equal $1/\sqrt{2}$ coefficients verify sum-mode H-arm coupling, and the $+1/2$ and $-1/2$ coefficients verify the equal-amplitude opposite-phase difference-mode coupling produced by the internal E-arm connection. Finally, the unitary condition confirms that the complete network conserves power and is therefore lossless under ideal assumptions.

The two methods complement each other. Method 1 is the better method when the actual S-matrix must be calculated from the physical connection because it explicitly eliminates the internal E-arm waves. Method 2 is the better method for checking the answer and explaining why the resulting coefficients and zero elements make physical sense. Using both methods gives a much stronger analysis than relying on either method alone.

Physical Meaning of Connecting the E-Arms of Two Magic Tees

The mathematical result becomes much easier to understand when we look at what the internal E-arm connection is actually doing. Each Magic Tee separates microwave signals into two basic modes. The H-arm represents the sum mode of its two collinear arms, while the E-arm represents the difference mode. When the E-arm of one Magic Tee is connected directly to the E-arm of another Magic Tee, the difference-mode signal generated by the first junction becomes the difference-mode excitation of the second junction. In the opposite direction, the second Magic Tee can also send its difference-mode component back through the same internal connection. Therefore, the connection creates a bidirectional difference-mode coupling path between the two Magic Tees. The resulting six-port network can be understood as two sum and difference junctions linked together through their difference ports.

This interpretation also explains why the H-arms do not become directly connected to each other. The H-arm and E-arm of an ideal Magic Tee are isolated. A signal entering the H-arm produces equal in-phase signals at the two collinear arms, whereas a signal entering the E-arm produces equal-amplitude signals with opposite phase. Since the connection is made only between the E-arms, the internal path carries the difference-mode component rather than the sum-mode component. Consequently, the H-arm of the first Magic Tee remains isolated from the H-arm of the second Magic Tee, even though the two junctions are physically connected through their E-arms.

What Happens When Port 1 Is Excited?

Consider the most useful example in which Port 1 is excited while all other external ports are terminated with matched loads. We can write the incident waves as

\[ a_1\neq0 \]

and

\[ a_2=a_3=a_4=a_5=a_6=0 \]

From the first column of the final S-matrix, the nonzero output waves are

\[ b_3=\frac{1}{\sqrt{2}}a_1 \]

\[ b_5=\frac{1}{2}a_1 \]

and

\[ b_6=-\frac{1}{2}a_1 \]

This means that the input signal at Port 1 is divided into two physically different paths. One portion travels directly to the H-arm of the first Magic Tee, producing the $1/\sqrt{2}$ term at Port 3. The difference-mode portion enters the E-arm of the first Magic Tee and travels through the internal E-arm connection into the second Magic Tee. It is then divided between Ports 5 and 6 with equal magnitude and opposite phase. Therefore, the signal at Port 1 simultaneously produces a sum-mode response at Port 3 and a difference-mode response at the second Magic Tee.

The normalized powers are

\[ P_3= \left| \frac{1}{\sqrt{2}} \right|^2 = \frac{1}{2} \]

\[ P_5= \left| \frac{1}{2} \right|^2 = \frac{1}{4} \]

and

\[ P_6= \left| -\frac{1}{2} \right|^2 = \frac{1}{4} \]

Thus,

\[ P_3+P_5+P_6 = \frac{1}{2} + \frac{1}{4} + \frac{1}{4} = 1 \]

So, if the input power is normalized to $1$, half of the power reaches the H-arm of the first Tee and the remaining half reaches the second Tee through its two collinear arms. The two signals arriving at Ports 5 and 6 have equal power but opposite phase.

What Happens When Port 2 Is Excited?

The same analysis can be performed for Port 2. If only Port 2 is excited, then

\[ a_2\neq0 \]

while

\[ a_1=a_3=a_4=a_5=a_6=0 \]

The relevant output waves become

\[ b_3=\frac{1}{\sqrt{2}}a_2 \]

\[ b_5=-\frac{1}{2}a_2 \]

and

\[ b_6=\frac{1}{2}a_2 \]

Notice that the signals at Ports 5 and 6 have again equal magnitude but opposite signs. However, the signs are reversed compared with the Port 1 excitation case. This happens because the E-arm responds to the difference $a_1-a_2$. When Port 1 is excited, the difference is positive, while when Port 2 is excited, the difference is negative. The internal E-arm connection therefore preserves the phase information associated with the original difference-mode excitation.

What Happens When the Two Collinear Arms Are Excited Together?

The sum and difference behavior becomes even clearer when Ports 1 and 2 are excited simultaneously. Suppose the two incident waves are equal and in phase:

\[ a_1=a_2=a \]

The internal E-arm excitation generated by Magic Tee A becomes

\[ a_8 = \frac{1}{\sqrt{2}}(a_1-a_2) \]

Therefore,

\[ a_8 = \frac{1}{\sqrt{2}}(a-a) = 0 \]

The E-arm connection is therefore not excited. The second Magic Tee receives no difference-mode signal from the first Tee, and consequently there is no cross-coupled output at Ports 5 and 6. At the same time, the H-arm output becomes

\[ b_3 = \frac{1}{\sqrt{2}}(a_1+a_2) \]

which gives

\[ b_3 = \frac{1}{\sqrt{2}}(2a) = \sqrt{2}a \]

This demonstrates the fundamental sum-mode property. Equal in-phase excitation of the two collinear arms produces a strong response at the H-arm while cancelling at the E-arm. Therefore, the E-arm connection does not respond to the common-mode or sum component of the input. It responds specifically to the difference component.

What Happens When the Two Collinear Arms Are Excited in Opposite Phase?

Now consider the opposite case. Suppose the two collinear arms of the first Magic Tee are excited with equal magnitude but opposite phase:

\[ a_1=a \]

and

\[ a_2=-a \]

The H-arm output becomes

\[ b_3 = \frac{1}{\sqrt{2}}(a_1+a_2) \]

so that

\[ b_3 = \frac{1}{\sqrt{2}}(a-a) = 0 \]

However, the E-arm excitation becomes

\[ a_8 = \frac{1}{\sqrt{2}}(a_1-a_2) \]

and therefore

\[ a_8 = \frac{1}{\sqrt{2}}(a-(-a)) = \sqrt{2}a \]

Thus, the H-arm is cancelled while the E-arm is strongly excited. The difference-mode signal then travels through the internal connection and reaches the second Magic Tee. This is one of the clearest ways to understand the purpose of the E-arm connection: it provides a path for the difference component of the signal between the two Magic Tees.

Sum and Difference Interpretation of the Six-Port Network

The behavior can therefore be summarized using the two basic combinations of the collinear-arm signals. For the first Magic Tee, the sum component is proportional to $a_1+a_2$, while the difference component is proportional to $a_1-a_2$. The sum component appears at the H-arm, while the difference component travels through the internally connected E-arm. The same process occurs in the second Magic Tee, where $a_5+a_6$ controls the H-arm output and $a_5-a_6$ controls the internal E-arm connection. This gives the entire six-port structure a very natural modal interpretation.

Mathematically, the two important combinations for the first Magic Tee are

\[ \boxed{ \text{Sum component} \propto a_1+a_2 } \]

and

\[ \boxed{ \text{Difference component} \propto a_1-a_2 } \]

For the second Magic Tee, the corresponding combinations are

\[ \boxed{ \text{Sum component} \propto a_5+a_6 } \]

and

\[ \boxed{ \text{Difference component} \propto a_5-a_6 } \]

This is why the connected structure can be viewed as a network that transfers difference-mode information between two separate sum and difference junctions. It is not simply a six-port power divider. The phase relationship between the ports is an essential part of its operation.

Why Is This Configuration Useful?

Connecting the E-arms of two Magic Tees becomes useful when a microwave system needs to process sum and difference signals in more than one stage. Instead of using one Magic Tee independently, the internal E-arm connection allows the difference output of one junction to become the difference input of another junction. This creates a controlled coupling mechanism between two pairs of collinear ports while keeping the H-arms isolated. Such an arrangement can be useful in balanced microwave networks, signal comparison systems, beam-forming structures, antenna feed networks, monopulse radar architectures, and microwave combining or splitting arrangements where phase relationships are important.

In a balanced microwave system, signals are often intentionally combined or subtracted rather than simply added as ordinary power. The E-arm is particularly useful for this purpose because it naturally produces equal-amplitude opposite-phase outputs. Connecting two E-arms allows that difference-mode information to be transferred to another hybrid junction. The second Magic Tee can then redistribute the signal into another pair of ports. This gives the designer a way to create more complex amplitude and phase relationships without requiring a completely new microwave junction.

Application in Balanced Microwave Networks

One important application is balanced signal processing. In a balanced system, two signals are often treated according to their common-mode and differential-mode components. The H-arm of a Magic Tee naturally responds to the common or sum component, while the E-arm responds to the differential or difference component. By connecting the E-arms of two hybrid junctions, the differential component from one part of the network can be transferred to another part while the common-mode paths remain separated. This can be useful in microwave circuits where unwanted common-mode signals must be separated from useful differential information.

The same principle can be extended to antenna feed systems. Hybrid junctions are frequently used where two antenna signals must be combined or separated according to amplitude and phase. Because the E-arm provides a controlled $180^\circ$ phase relationship between its collinear outputs, a connected pair of hybrid junctions can be incorporated into feed networks that require precise phase control. The exact implementation depends on the antenna geometry and the required radiation pattern, but the underlying microwave principle is the same: the network manipulates sum and difference modes rather than treating all signals as simple independent power paths.

Application in Monopulse and Comparison Systems

Another important area is microwave comparison and monopulse-type systems. These systems often need to form sum and difference signals from multiple antenna channels. The sum signal represents the combined response of the channels, while the difference signal represents the imbalance between them. A Magic Tee is naturally suited to this operation because its H-arm provides the sum response and its E-arm provides the difference response. Connecting hybrid junctions through their difference ports can therefore be useful when several comparison stages must be interconnected while maintaining the required phase relationships.

For example, if two signals applied to the collinear arms are equal and in phase, their difference component is zero and the E-arm is not excited. If the signals differ in amplitude or phase, a nonzero difference component appears at the E-arm. The connected second Magic Tee can then redistribute that difference information to another pair of ports. This makes the configuration useful conceptually for systems that need to detect or process differences between microwave signals rather than only measuring their total power.

Application in Microwave Power Combining and Splitting

The connected structure can also be used as part of microwave power combining and splitting networks. A Magic Tee already provides controlled power division with specific phase relationships. When two such junctions are connected through their E-arms, the resulting six-port network can distribute an input signal among multiple output ports while maintaining predictable phase relationships. For example, an excitation at Port 1 produces half the normalized power at Port 3 and one quarter at each of Ports 5 and 6. The signals at Ports 5 and 6 have opposite phase, which can be useful when the next stage of a microwave network requires balanced or differential excitation.

The important advantage is not simply the number of ports. The real advantage is the controlled phase relationship between them. In microwave engineering, two outputs having equal power but different phase are not equivalent. A $180^\circ$ phase difference can cause cancellation in one location and reinforcement in another. Therefore, the $+1/2$ and $-1/2$ coefficients in the six-port S-matrix provide useful information for designing larger networks.

Important Assumptions in This Analysis

The derived $6\times6$ matrix represents an idealized microwave network. We have assumed that both Magic Tees are ideal, all external ports are properly referenced to the same characteristic impedance, the individual Magic Tee ports are perfectly matched, and the direct E-arm connection introduces no additional reflection or discontinuity. We have also assumed that the connection has no appreciable transmission loss and that there are no additional phase shifts associated with the physical length of the connection. Under these assumptions, the S-matrix contains only the ideal amplitude and phase coefficients discussed above.

In a practical microwave structure, the physical connection between the two E-arms may have a finite length. If that length is electrically significant, the signal travelling between the two E-arms acquires a phase factor. In that situation, the simple connection conditions used here would need to be modified to include the propagation phase. For example, instead of treating the connection as an ideal zero-length connection, a phase factor such as

\[ e^{-j\beta l} \]

would be introduced for a transmission path of length $l$. The resulting six-port S-matrix would then contain phase terms in the cross-coupling elements. Therefore, the matrix derived in this analysis should be understood as the ideal direct-connection result.

Advantages of Connecting the E-Arms

The main advantage of the configuration is that it provides controlled coupling between two Magic Tees through their difference-mode ports. The H-arms remain isolated, while the E-arm connection transfers the differential component from one junction to the other. This gives the designer control over both amplitude and phase. Another advantage is that the resulting network can be analyzed systematically using standard S-parameter techniques. Once the internal ports are eliminated, the entire structure can be represented by a single $6\times6$ matrix, making it easier to include the network in a larger microwave system analysis.

Provides controlled coupling between two Magic Tees.
Transfers difference-mode signals between the two junctions.
Maintains isolation between the two H-arms in the ideal case.
Provides equal-amplitude opposite-phase outputs at the receiving collinear arms.
Allows a larger microwave network to be represented by a single $6\times6$ S-matrix.
Can be useful in balanced, comparison, antenna-feed, and power-distribution networks.
Provides predictable amplitude and phase relationships between multiple external ports.

Limitations of the Ideal Six-Port Model

Although the ideal S-matrix is very useful for analysis, a practical implementation will not behave perfectly. Waveguide losses, conductor losses, dielectric losses, junction discontinuities, manufacturing tolerances, impedance mismatches, and the physical length of the E-arm connection can all modify the actual S-parameters. In particular, a finite connection length introduces an additional phase shift, while losses reduce the magnitude of the transmission coefficients. Therefore, the ideal matrix should be treated as the theoretical reference model. Practical measurements or electromagnetic simulation may produce slightly different values.

Another important point is that the internal E-arm connection must be treated consistently when defining the reference planes. If the reference planes are moved away from the physical junctions, the S-parameters acquire additional phase factors because of transmission-line propagation. The simple real-valued matrix derived here assumes that the reference planes are positioned so that the direct E-arm connection does not introduce an additional propagation phase. This assumption is common when deriving an ideal network model for theoretical microwave engineering problems.

Final Result of the Six-Port Network Analysis

Connecting the E-arm of one ideal Magic Tee directly to the E-arm of another ideal Magic Tee produces a six-port composite microwave network. The two original E-arms become internal ports, while the two pairs of collinear arms and the two H-arms form the six external ports. Starting with the known $4\times4$ Magic Tee S-matrix, the internal connection conditions are $a_8=b_4$ and $a_4=b_8$. Eliminating the internal waves gives the final scattering matrix

The matrix is symmetric, so the network is reciprocal. All diagonal elements are zero, so all six external ports are matched in the ideal model. The two H-arms are isolated from each other, while each H-arm couples equally to its own pair of collinear arms. Most importantly, the internal E-arm connection creates cross-coupling between the two Magic Tees with coefficients $+1/2$ and $-1/2$, representing equal-amplitude signals with a $180^\circ$ phase difference. The matrix also satisfies the lossless condition

\[ \boxed{ [S_6][S_6]^\dagger=[I_6] } \]

which confirms conservation of power for the ideal network.

Frequently Asked Questions

What happens when the E-arm of one Magic Tee is connected to the E-arm of another?

The two E-arms become internal ports, and the two four-port Magic Tees combine into a six-port external network. The connection transfers difference-mode signals between the two junctions. The final external ports are the four collinear arms and the two H-arms.

Why does the final network have six ports instead of eight?

The two original Magic Tees contain eight ports in total. However, two of those ports are the E-arms that are directly connected to each other. Because those ports are internal to the combined network, they are not counted as external ports. Therefore, the number of accessible ports is

\[ 8-2=6 \]

What are the internal connection equations?

For a direct connection between the two E-arms, the outgoing wave from one E-arm becomes the incident wave at the other E-arm. Therefore,

\[ \boxed{ a_8=b_4 } \]

and

\[ \boxed{ a_4=b_8 } \]

These equations allow the internal waves to be eliminated from the final network description.

Why are the cross-coupling coefficients $1/2$?

A signal travelling from one Magic Tee to the other through the E-arm experiences two coupling factors of $1/\sqrt{2}$. One occurs when the signal enters the E-arm of the first Tee, and the second occurs when the signal leaves the E-arm of the second Tee. Therefore,

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

The opposite collinear arm has the opposite sign, giving $-1/2$.

Are the $1/2$ and $-1/2$ values power coefficients?

No. They are amplitude S-parameters. The corresponding power ratio is obtained by taking the squared magnitude. Therefore,

\[ \left|\frac{1}{2}\right|^2 = \frac{1}{4} \]

so each such path carries 25 percent of the incident power under a single-port excitation with all other ports matched.

Why are the two cross-coupled outputs 180 degrees out of phase?

The E-arm is a difference port. Its two collinear outputs have equal magnitude but opposite signs. Therefore, if one output is $+1/2$, the other is $-1/2$. The negative sign represents a phase difference of $180^\circ$.

Are the two H-arms isolated?

Yes, under the ideal Magic Tee assumptions. In the final matrix,

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

so there is no direct transmission between the two H-arms. The internal connection is through the E-arms, and the H-arm and E-arm of an ideal Magic Tee are isolated.

Is the six-port network reciprocal?

Yes. The final matrix is symmetric:

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

Therefore, the ideal six-port network is reciprocal.

Is the six-port network lossless?

Yes, under the ideal assumptions. Its S-matrix satisfies the unitary condition

\[ \boxed{ [S_6][S_6]^\dagger=[I_6] } \]

which means that the total output power equals the total incident power.

What is the main application of connecting the E-arms?

The main purpose is to transfer and process difference-mode microwave signals between two hybrid junctions while keeping the H-arm sum paths isolated. This can be useful in balanced microwave networks, signal comparison systems, antenna feed networks, monopulse and beam-forming structures, and microwave power distribution networks where controlled amplitude and phase relationships are required.

Note:

The E-arm-to-E-arm connection of two Magic Tees is a useful example of how internal microwave-port connections can be used to create a larger multiport network. The analysis begins with two known $4\times4$ Magic Tee S-matrices rather than deriving the individual Magic Tee matrix again. The two E-arms are then treated as internal ports, and the connection conditions $a_8=b_4$ and $a_4=b_8$ are applied. Eliminating these internal variables produces a $6\times6$ S-matrix for the combined network. Method 1 obtains this matrix directly from the wave equations, while Method 2 verifies it using reciprocity, matching, isolation, equal H-arm coupling, difference-mode phase relationships, and the lossless condition. Physically, the configuration creates a controlled difference-mode coupling path between two Magic Tees. This allows amplitude and phase information to be distributed among six external ports in a predictable way, making the configuration useful as a building block for more complex microwave networks.

Output Port	Output Wave Amplitude (\(b_i\))	Normalized Output Power (\(P_i = \|b_i\|^2\))	Power Percentage Share
Port 3 (E-Arm Tee A)	\(b_3 = S_{13} = \frac{1}{\sqrt{2}}\)	\(P_3 = \left\|\frac{1}{\sqrt{2}}\right\|^2 = \frac{1}{2}\)	50%
Port 5 (Collinear Tee B1)	\(b_5 = S_{15} = \frac{1}{2}\)	\(P_5 = \left\|\frac{1}{2}\right\|^2 = \frac{1}{4}\)	25%
Port 6 (Collinear Tee B2)	\(b_6 = S_{16} = S_{15} = \frac{1}{2}\)	\(P_6 = \left\|\frac{1}{2}\right\|^2 = \frac{1}{4}\)	25%

Connecting the E-Arms of Two Magic Tees

What happen when E arm of one Tee is connected to E arm of another Tee? Write its proporties

Explain the properties of two Magic Tees if one connects their E-ams