E-H Arm Connected Magic Tees: 6-Port
What happen when E arm of one Tee is connected to H arm of another Tee? Write its proporties
ORExplain the properties of two Magic Tees if one E connects another H-ams of another Tee
E-H Arm Connected Magic Tee: 6-Port Scattering Matrix Derivation
Construction of the General \(6\times6\) Scattering Matrix
Since the configuration and internal connection of the two Magic Tees have already been established, we can start the analysis directly from the scattering-parameter method. In this method, we do not begin by assigning arbitrary coefficients such as \(a\), \(b\), or \(c\). Instead, every possible transmission and reflection coefficient is first written explicitly as \(S_{ij}\). We then use the known properties of the ideal Magic Tee, including reciprocity, matching, E-arm and H-arm isolation, and the appropriate phase relationships, to gradually simplify the general matrix. This approach is useful because it shows exactly where every term in the final \(6\times6\) matrix comes from and allows the unknown coefficients to be determined systematically from the lossless or unitary condition.
Starting with the General \(6\times6\) S-Matrix
A six-port microwave network has six incident waves and six corresponding reflected or outgoing waves. Therefore, its complete scattering matrix contains \(6\times6=36\) scattering parameters. Before applying any property of the E-arm-to-H-arm connected structure, the most general scattering matrix can be written by listing every coefficient from \(S_{11}\) through \(S_{66}\). At this stage, no coefficient is assumed to be zero, equal to another coefficient, or assigned a particular magnitude. This is important because the purpose of Method is to begin with the complete mathematical representation and then reduce it step by step using the physical properties of the network.
The general scattering relationship is
\[ [b]=[S][a] \]
where \([a]\) represents the vector of incident waves, \([b]\) represents the vector of outgoing waves, and \([S]\) represents the six-port scattering matrix. Therefore,
\[ \begin{bmatrix} b_1\\ b_2\\ b_3\\ b_4\\ b_5\\ b_6 \end{bmatrix} = \begin{bmatrix} S_{11} & S_{12} & S_{13} & S_{14} & S_{15} & S_{16}\\ S_{21} & S_{22} & S_{23} & S_{24} & S_{25} & S_{26}\\ S_{31} & S_{32} & S_{33} & S_{34} & S_{35} & S_{36}\\ S_{41} & S_{42} & S_{43} & S_{44} & S_{45} & S_{46}\\ S_{51} & S_{52} & S_{53} & S_{54} & S_{55} & S_{56}\\ S_{61} & S_{62} & S_{63} & S_{64} & S_{65} & S_{66} \end{bmatrix} \begin{bmatrix} a_1\\ a_2\\ a_3\\ a_4\\ a_5\\ a_6 \end{bmatrix} \]
The six ports are arranged so that Ports 1 and 2 are the collinear arms of Magic Tee A, Port 3 is its E-arm, Port 4 is the E-arm of Magic Tee B, and Ports 5 and 6 are the collinear arms of Magic Tee B. The internal connection is between the H-arm of Tee A and the H-arm of Tee B. Because those H-arms are not external ports, their effect appears through the cross-coupling terms between the two groups of external ports. The purpose of the following steps is to determine which of the 36 original coefficients must be equal, which must be zero, and which coefficients remain unknown until the unitary condition is applied.
Applying Reciprocity to the General Matrix
The first property used in Method is reciprocity. An ideal passive microwave junction constructed from reciprocal materials satisfies the reciprocal condition that transmission from Port \(i\) to Port \(j\) is the same as transmission from Port \(j\) to Port \(i\). Mathematically, this means that the scattering matrix is symmetric. Therefore, every coefficient \(S_{ij}\) can be compared with the corresponding coefficient \(S_{ji}\). Rather than introducing a new variable, we simply replace the second coefficient with the first one. This keeps the derivation entirely in terms of the original \(S_{ij}\) notation.
The reciprocity condition is
\[ S_{ij}=S_{ji} \]
Therefore,
\[ S_{21}=S_{12} \]
\[ S_{31}=S_{13} \]
\[ S_{41}=S_{14} \]
\[ S_{51}=S_{15} \]
\[ S_{61}=S_{16} \]
Similarly,
\[ S_{32}=S_{23} \]
\[ S_{42}=S_{24} \]
\[ S_{52}=S_{25} \]
\[ S_{62}=S_{26} \]
For the remaining pairs,
\[ S_{43}=S_{34} \]
\[ S_{53}=S_{35} \]
\[ S_{63}=S_{36} \]
\[ S_{54}=S_{45} \]
\[ S_{64}=S_{46} \]
\[ S_{65}=S_{56} \]
After applying reciprocity, the general matrix becomes symmetric, so it can be written as
\[ [S]= \begin{bmatrix} S_{11} & S_{12} & S_{13} & S_{14} & S_{15} & S_{16}\\ S_{12} & S_{22} & S_{23} & S_{24} & S_{25} & S_{26}\\ S_{13} & S_{23} & S_{33} & S_{34} & S_{35} & S_{36}\\ S_{14} & S_{24} & S_{34} & S_{44} & S_{45} & S_{46}\\ S_{15} & S_{25} & S_{35} & S_{45} & S_{55} & S_{56}\\ S_{16} & S_{26} & S_{36} & S_{46} & S_{56} & S_{66} \end{bmatrix} \]
This step reduces the number of independent coefficients, but it does not yet determine their values. The diagonal terms are still unknown, the coupling terms are still unknown, and the isolation and phase properties have not yet been imposed. We therefore continue with the physical properties of the two Magic Tees.
Applying the Matching Conditions
For the ideal Magic Tee model, each accessible port is assumed to be perfectly matched to the characteristic impedance of the network. A matched port does not produce a reflected wave when it is excited by itself, so the corresponding reflection coefficient is zero. In S-parameter notation, the reflection coefficient at Port \(i\) is represented by \(S_{ii}\). Since all six external ports are treated as matched in the ideal model, all six diagonal elements of the six-port scattering matrix become zero.
Therefore,
\[ S_{11}=S_{22}=S_{33}=S_{44}=S_{55}=S_{66}=0 \]
Substituting these matching conditions into the reciprocal matrix gives
\[ [S]= \begin{bmatrix} 0 & S_{12} & S_{13} & S_{14} & S_{15} & S_{16}\\ S_{12} & 0 & S_{23} & S_{24} & S_{25} & S_{26}\\ S_{13} & S_{23} & 0 & S_{34} & S_{35} & S_{36}\\ S_{14} & S_{24} & S_{34} & 0 & S_{45} & S_{46}\\ S_{15} & S_{25} & S_{35} & S_{45} & 0 & S_{56}\\ S_{16} & S_{26} & S_{36} & S_{46} & S_{56} & 0 \end{bmatrix} \]
The matching condition removes the six diagonal reflection coefficients, but the remaining off-diagonal terms must still be determined. In particular, some terms represent coupling between the two collinear arms of the same Magic Tee, some represent E-arm coupling, and some represent coupling between the two different Magic Tees through the internally connected H-arms. These terms are determined next by applying isolation and phase relationships.
Applying the E-Arm and H-Arm Properties
The next step is to use the defining properties of the Hybrid Tee. Port 3 is the E-arm of Magic Tee A, while Port 4 is the E-arm of Magic Tee B. An E-arm is isolated from the H-arm of the same Magic Tee, and the H-arm connection is the path responsible for coupling the two Magic Tees. Therefore, an excitation at an E-arm does not enter the H-arm path and cannot produce a cross-coupled output at the other Magic Tee. This produces the required isolation conditions between the E-arm ports and the opposite group of collinear ports.
Since Port 3 is the E-arm of Tee A, it remains isolated from the H-arm connection leading to Ports 5 and 6. Therefore,
\[ S_{35}=S_{53}=0 \]
and
\[ S_{36}=S_{63}=0 \]
Similarly, Port 4 is the E-arm of Tee B and is isolated from the H-arm path leading toward Ports 1 and 2. Therefore,
\[ S_{14}=S_{41}=0 \]
and
\[ S_{24}=S_{42}=0 \]
The two E-arms also remain isolated from one another because the E-arm of each tee is isolated from its own H-arm, and the only internal connection between the two tees is through their H-arms. Thus, there is no direct E-arm-to-E-arm transmission path in this ideal configuration. Hence,
\[ S_{34}=S_{43}=0 \]
These isolation conditions are particularly important because they separate the E-arm difference-mode behavior from the H-arm cross-coupling behavior. Port 3 interacts with the collinear arms of Tee A, while Port 4 interacts with the collinear arms of Tee B. The H-arm connection, on the other hand, establishes coupling between the two groups of collinear arms.
Applying the E-Arm Difference-Port Relationship
The E-arm produces equal-amplitude signals with opposite phase at the two collinear arms of its Magic Tee. Therefore, the transmission coefficients from the E-arm of Tee A to Ports 1 and 2 must have equal magnitude and opposite sign. Since Port 3 is the E-arm of Tee A, the corresponding relationship is obtained by comparing its coupling to Port 1 and Port 2. Thus,
\[ S_{23}=-S_{13} \]
Because the network is reciprocal, the corresponding reverse transmission relationship follows automatically:
\[ S_{32}=S_{23}=-S_{13} \]
Similarly, Port 4 is the E-arm of Tee B. Its coupling to Ports 5 and 6 must also have equal magnitude and opposite phase. Therefore,
\[ S_{64}=-S_{54} \]
and by reciprocity,
\[ S_{46}=S_{64}=-S_{54} \]
These relationships do not yet specify the numerical magnitude of the coefficients. They only establish the relative phase. The actual magnitude will be obtained later from the unitary condition. This distinction is important because the phase relationships come from the physical E-arm property, whereas the numerical magnitudes follow from power conservation and the lossless nature of the ideal network.
Applying the H-Arm In-Phase Coupling Relationship
The internal connection is between the H-arm of Magic Tee A and the H-arm of Magic Tee B. Unlike the E-arm, the H-arm produces equal-amplitude outputs that are in phase. Consequently, the signals transferred through this internal H-arm path from one Magic Tee to the other appear with the same phase at the two collinear arms of the receiving tee. Therefore, the cross-coupling coefficients from Ports 1 and 2 toward Ports 5 and 6 must follow the in-phase H-arm relationship. For example, when a signal from Tee A reaches Tee B through the connected H-arms, the outputs at Ports 5 and 6 have equal phase, so
\[ S_{51}=S_{61} \]
Reciprocity gives
\[ S_{15}=S_{51} \]
and
\[ S_{16}=S_{61} \]
Therefore, the relationship can be written consistently as
\[ S_{15}=S_{16} \]
The same argument applies to excitation at Port 2. The signal travelling through the connected H-arm path again produces equal-phase outputs at Ports 5 and 6. Hence,
\[ S_{52}=S_{62} \]
and, using reciprocity,
\[ S_{25}=S_{26} \]
These equalities are the mathematical representation of the H-arm's sum-mode behavior. The coefficients are still not assigned numerical values at this stage. We only know that the corresponding cross-coupled terms must have the same magnitude and phase. Their actual values will be obtained by applying the unitary condition.
Reduced S-Matrix Before Applying the Unitary Condition
After applying reciprocity, matching, E-arm isolation, E-arm phase reversal, and H-arm in-phase coupling, the original 36-parameter matrix has been reduced substantially. The remaining unknown coefficients are still written entirely as \(S_{ij}\), with no arbitrary coefficient such as \(a\) or \(c\) introduced. The reduced matrix can therefore be arranged according to the physical relationships of the network.
The matrix takes the form
\[ [S]= \begin{bmatrix} 0 & S_{12} & S_{13} & 0 & S_{15} & S_{15}\\ S_{12} & 0 & -S_{13} & 0 & S_{25} & S_{25}\\ S_{13} & -S_{13} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & S_{54} & -S_{54}\\ S_{15} & S_{25} & 0 & S_{54} & 0 & S_{56}\\ S_{15} & S_{25} & 0 & -S_{54} & S_{56} & 0 \end{bmatrix} \]
At this point, several terms such as \(S_{12}\), \(S_{25}\), and \(S_{56}\) must still be examined using the physical structure and the orthogonality requirements of the ideal Magic Tee network. In particular, the collinear arms belonging to the same Magic Tee are isolated from one another in the ideal Hybrid Tee, and the symmetry of the two identical structures imposes corresponding relationships between the remaining coefficients. These conditions will further simplify the matrix before the unitary equation is expanded.
Collinear-Arm Isolation and Symmetry
For an ideal Magic Tee, the two collinear arms are isolated from one another. A signal entering one collinear arm does not appear directly at the other collinear arm because the junction separates the sum and difference modes through the H-arm and E-arm. Therefore, for Tee A, the coupling between Ports 1 and 2 is zero:
\[ S_{12}=S_{21}=0 \]
Similarly, for Tee B, the two collinear arms are isolated, giving
\[ S_{56}=S_{65}=0 \]
Because the two Magic Tees are identical, the E-arm coupling magnitudes of the two structures must be equal. Their phase conventions are chosen consistently with the E-arm difference-port property, so the corresponding coefficients satisfy
\[ S_{54}=S_{13} \]
The cross-coupling terms through the connected H-arms likewise have corresponding values because the two Magic Tees are identical and the internal connection is reciprocal. Thus, the cross-coupling relationships can be represented by the corresponding \(S_{ij}\) terms rather than introducing a separate arbitrary coefficient.
The scattering matrix is therefore reduced to
\[ [S]= \begin{bmatrix} 0 & 0 & S_{13} & 0 & S_{15} & S_{15}\\ 0 & 0 & -S_{13} & 0 & S_{25} & S_{25}\\ S_{13} & -S_{13} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & S_{13} & -S_{13}\\ S_{15} & S_{25} & 0 & S_{13} & 0 & 0\\ S_{15} & S_{25} & 0 & -S_{13} & 0 & 0 \end{bmatrix} \]
The remaining unknown quantities are now contained in the E-arm coupling coefficient \(S_{13}\) and the cross-coupling coefficients \(S_{15}\) and \(S_{25}\). The exact values and relative signs of these remaining terms cannot simply be assumed. They must be obtained from the lossless condition. This is where the unitary property of the scattering matrix becomes essential.
Preparing the Matrix for the Unitary Condition
An ideal Magic Tee network is lossless, meaning that the total incident power must equal the total outgoing power. In scattering-parameter form, a lossless network satisfies the unitary condition. This condition provides the mathematical equations required to determine the remaining unknown coefficients. Instead of guessing the magnitude of \(S_{13}\), \(S_{15}\), or \(S_{25}\), we will obtain them by multiplying the reduced matrix by its conjugate transpose and comparing the result with the identity matrix.
The required condition is
\[ [S][S]^\dagger=[I] \]
where \([S]^\dagger\) denotes the conjugate transpose of the scattering matrix and \([I]\) is the \(6\times6\) identity matrix. The diagonal elements of the resulting product must be equal to \(1\), while all off-diagonal elements must be equal to \(0\). The diagonal equations provide the power-normalization relationships, whereas the off-diagonal equations provide the orthogonality relationships needed to determine the remaining unknown terms and their relative signs.
Therefore, the next stage is to explicitly multiply
\[ \begin{bmatrix} 0 & 0 & S_{13} & 0 & S_{15} & S_{15}\\ 0 & 0 & -S_{13} & 0 & S_{25} & S_{25}\\ S_{13} & -S_{13} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & S_{13} & -S_{13}\\ S_{15} & S_{25} & 0 & S_{13} & 0 & 0\\ S_{15} & S_{25} & 0 & -S_{13} & 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 0 & S_{13}^{*} & 0 & S_{15}^{*} & S_{15}^{*}\\ 0 & 0 & -S_{13}^{*} & 0 & S_{25}^{*} & S_{25}^{*}\\ S_{13}^{*} & -S_{13}^{*} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & S_{13}^{*} & -S_{13}^{*}\\ S_{15}^{*} & S_{25}^{*} & 0 & S_{13}^{*} & 0 & 0\\ S_{15}^{*} & S_{25}^{*} & 0 & -S_{13}^{*} & 0 & 0 \end{bmatrix} = [I] \]
The complete diagonal and off-diagonal equations obtained from this multiplication, followed by the systematic solution of the remaining \(S_{ij}\) coefficients and the final completed \(6\times6\) scattering matrix, form the next stage of the derivation.
Method: Applying the Unitary Condition to the Reduced \(6\times6\) Matrix
We now continue the Method derivation from the reduced \(6\times6\) scattering matrix obtained after applying the structural properties of the E-arm-to-H-arm connected configuration. At this stage, the remaining coefficients are deliberately kept in their original \(S_{ij}\) form. We do not introduce new symbols such as \(a\), \(b\), or \(c\). The purpose of this method is to determine the unknown scattering parameters directly from the mathematical conditions imposed on a reciprocal, matched, lossless six-port network. The most important condition is the unitary property, which states that the scattering matrix multiplied by its conjugate transpose must produce the identity matrix. The diagonal terms of this multiplication give power-normalization equations, while the off-diagonal terms give orthogonality equations that determine the relationships between the remaining unknown coefficients.
Reduced \(6\times6\) S-Matrix Before Numerical Evaluation
After applying reciprocity, matching, E-arm isolation, E-arm difference-port behavior, H-arm in-phase coupling, and collinear-arm isolation, the scattering matrix can be written entirely in terms of the remaining unknown \(S\)-parameters. The E-arm of Tee A is represented by Port 3, while the E-arm of Tee B is represented by Port 4. Ports 1 and 2 belong to Tee A, and Ports 5 and 6 belong to Tee B. The remaining unknown coefficients are therefore retained explicitly so that their values can be obtained from the unitary equations rather than assumed in advance.
The reduced matrix is
\[ [S]= \begin{bmatrix} 0 & 0 & S_{13} & 0 & S_{15} & S_{15}\\ 0 & 0 & -S_{13} & 0 & S_{25} & S_{25}\\ S_{13} & -S_{13} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & S_{54} & -S_{54}\\ S_{15} & S_{25} & 0 & S_{54} & 0 & 0\\ S_{15} & S_{25} & 0 & -S_{54} & 0 & 0 \end{bmatrix} \]
Because the two Magic Tees are identical, the E-arm coupling magnitude of Tee A and Tee B must be the same. With the same port-reference convention used for both tees, the corresponding E-arm coefficients can therefore be related through
\[ S_{54}=S_{13} \]
Substituting this relationship gives the matrix that will be used for the unitary analysis:
\[ [S]= \begin{bmatrix} 0 & 0 & S_{13} & 0 & S_{15} & S_{15}\\ 0 & 0 & -S_{13} & 0 & S_{25} & S_{25}\\ S_{13} & -S_{13} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & S_{13} & -S_{13}\\ S_{15} & S_{25} & 0 & S_{13} & 0 & 0\\ S_{15} & S_{25} & 0 & -S_{13} & 0 & 0 \end{bmatrix} \]
Notice that no numerical value has yet been assigned to \(S_{13}\), \(S_{15}\), or \(S_{25}\). These are the coefficients that must now be determined from the lossless condition. The negative signs associated with the E-arm coupling have already been established from the difference-port property of the E-arm, whereas the equality of the cross-coupling terms follows from the in-phase behavior of the internally connected H-arm path.
Applying the Lossless Condition
Since the ideal six-port network is lossless, no incident power can disappear inside the junction. Every unit of incident power must appear at one or more of the external output ports. In scattering-parameter form, a lossless network satisfies the unitary condition. Therefore, the reduced scattering matrix must satisfy
\[ [S][S]^{\dagger}=[I] \]
where \([S]^{\dagger}\) is the conjugate transpose of \([S]\), and \([I]\) is the \(6\times6\) identity matrix. This means that every diagonal element of \([S][S]^{\dagger}\) must be equal to \(1\), while every off-diagonal element must be equal to \(0\). We will use these equations one by one rather than immediately jumping to the final numerical matrix.
Row 3 Normalization
We first consider the third row because Port 3 is an E-arm and its row contains only the two E-arm coupling terms associated with the collinear arms of Tee A. The third row is
\[ \begin{bmatrix} S_{13} & -S_{13} & 0 & 0 & 0 & 0 \end{bmatrix} \]
The diagonal element corresponding to this row must be equal to \(1\). Therefore,
\[ |S_{13}|^2+|-S_{13}|^2=1 \]
Since the magnitude of a quantity is unchanged by a negative sign,
\[ |S_{13}|^2+|S_{13}|^2=1 \]
Hence,
\[ 2|S_{13}|^2=1 \]
giving
\[ |S_{13}|^2=\frac{1}{2} \]
and therefore
\[ |S_{13}|=\frac{1}{\sqrt{2}} \]
This result has a direct physical meaning. The E-arm signal is divided equally between the two collinear arms of its Magic Tee. Since the two output powers must add to the total incident power, each coupling coefficient must have a power magnitude of \(1/2\), corresponding to an amplitude magnitude of \(1/\sqrt{2}\).
Row 4 Normalization
The same calculation can be performed for Row 4. Port 4 is the E-arm of Magic Tee B, so its row contains the two coefficients associated with the collinear arms of Tee B:
\[ \begin{bmatrix} 0 & 0 & 0 & 0 & S_{13} & -S_{13} \end{bmatrix} \]
The corresponding diagonal element of \([S][S]^{\dagger}\) must again be equal to \(1\). Therefore,
\[ |S_{13}|^2+|-S_{13}|^2=1 \]
which gives
\[ 2|S_{13}|^2=1 \]
and hence
\[ |S_{13}|=\frac{1}{\sqrt{2}} \]
Thus, both E-arms have the same coupling magnitude, as expected for two identical Magic Tees. The result also confirms that the E-arm of Tee B divides its incident signal equally between Ports 5 and 6, with the two outputs having opposite phase because of the negative sign in the corresponding scattering coefficient.
Row 1 Normalization
We now consider Row 1. Port 1 receives contributions from the E-arm of Tee A and from the two collinear arms of Tee B through the internally connected H-arm path. The first row of the matrix is
\[ \begin{bmatrix} 0 & 0 & S_{13} & 0 & S_{15} & S_{15} \end{bmatrix} \]
Applying the diagonal condition of the unitary matrix gives
\[ |S_{13}|^2+|S_{15}|^2+|S_{15}|^2=1 \]
Therefore,
\[ |S_{13}|^2+2|S_{15}|^2=1 \]
We already obtained
\[ |S_{13}|^2=\frac{1}{2} \]
Substituting this result gives
\[ \frac{1}{2}+2|S_{15}|^2=1 \]
Therefore,
\[ 2|S_{15}|^2=\frac{1}{2} \]
and
\[ |S_{15}|^2=\frac{1}{4} \]
Hence,
\[ |S_{15}|=\frac{1}{2} \]
This result is important because the signal entering Port 1 has two different paths available in the ideal six-port structure. One part couples to the E-arm of Tee A, while the remaining power is distributed through the H-arm connection toward Ports 5 and 6. The E-arm coupling has an amplitude magnitude of \(1/\sqrt{2}\), while each H-arm cross-coupling coefficient has an amplitude magnitude of \(1/2\).
Row 2 Normalization
The second row provides the corresponding condition for Port 2. Its non-zero coefficients are \(S_{23}=-S_{13}\), \(S_{25}\), and \(S_{26}=S_{25}\). Therefore, the row is
\[ \begin{bmatrix} 0 & 0 & -S_{13} & 0 & S_{25} & S_{25} \end{bmatrix} \]
Applying the row-normalization condition gives
\[ |-S_{13}|^2+|S_{25}|^2+|S_{25}|^2=1 \]
or
\[ |S_{13}|^2+2|S_{25}|^2=1 \]
Substituting
\[ |S_{13}|^2=\frac{1}{2} \]
gives
\[ \frac{1}{2}+2|S_{25}|^2=1 \]
Hence,
\[ |S_{25}|^2=\frac{1}{4} \]
and therefore
\[ |S_{25}|=\frac{1}{2} \]
Thus, the cross-coupling from Port 2 toward the two collinear arms of Tee B has the same amplitude magnitude as the corresponding cross-coupling from Port 1. The remaining task is to determine whether \(S_{25}\) and \(S_{15}\) have the same or opposite phase. That information comes from the off-diagonal orthogonality equations.
Orthogonality of Rows 1 and 2
The diagonal equations have now determined the magnitudes of the remaining unknown coefficients, but magnitudes alone are not enough to construct the complete scattering matrix. We must also determine the relative phase of the coefficients. For a unitary matrix, different rows must be orthogonal. Therefore, the inner product of Row 1 with Row 2 must be zero.
Row 1 is
\[ \begin{bmatrix} 0 & 0 & S_{13} & 0 & S_{15} & S_{15} \end{bmatrix} \]
while Row 2 is
\[ \begin{bmatrix} 0 & 0 & -S_{13} & 0 & S_{25} & S_{25} \end{bmatrix} \]
Their inner product must satisfy
\[ [\text{Row 1}][\text{Row 2}]^{\dagger}=0 \]
Therefore,
\[ S_{13}(-S_{13})^{*} + S_{15}S_{25}^{*} + S_{15}S_{25}^{*} =0 \]
Since
\[ (-S_{13})^{*}=-S_{13}^{*} \]
the equation becomes
\[ -|S_{13}|^2 + S_{15}S_{25}^{*} + S_{15}S_{25}^{*} =0 \]
Hence,
\[ -|S_{13}|^2+2S_{15}S_{25}^{*}=0 \]
or
\[ 2S_{15}S_{25}^{*}=|S_{13}|^2 \]
Since
\[ |S_{13}|^2=\frac{1}{2} \]
we obtain
\[ S_{15}S_{25}^{*}=\frac{1}{4} \]
We already know that
\[ |S_{15}|=|S_{25}|=\frac{1}{2} \]
Therefore, the product \(S_{15}S_{25}^{*}\) has its maximum possible magnitude of \(1/4\), and the equation requires this product to be positive real. Consequently, \(S_{15}\) and \(S_{25}\) must have the same phase. With the usual reference convention for the ideal symmetric structure, they can be taken as equal:
\[ S_{25}=S_{15} \]
This is exactly the mathematical expression of the in-phase behavior produced by the H-arm connection. The signal transferred from one Magic Tee through the connected H-arms reaches the two collinear arms of the other Tee with equal phase.
Orthogonality of Rows 3 and 5
The same relationship can be confirmed using another pair of rows. Row 3 is
\[ \begin{bmatrix} S_{13} & -S_{13} & 0 & 0 & 0 & 0 \end{bmatrix} \]
and Row 5 is
\[ \begin{bmatrix} S_{15} & S_{25} & 0 & S_{13} & 0 & 0 \end{bmatrix} \]
Their inner product must be zero:
\[ S_{13}S_{15}^{*} - S_{13}S_{25}^{*} =0 \]
Therefore,
\[ S_{13} \left( S_{15}^{*}-S_{25}^{*} \right) =0 \]
Since \(S_{13}\neq0\), it follows that
\[ S_{15}^{*}=S_{25}^{*} \]
and therefore,
\[ S_{15}=S_{25} \]
This independently verifies the result obtained from the orthogonality of Rows 1 and 2. The cross-coupling coefficients associated with the H-arm path must therefore have equal magnitude and equal phase.
Determining the Numerical Magnitudes
We can now collect the results obtained from the diagonal and off-diagonal unitary equations. The E-arm coupling coefficient satisfies
\[ |S_{13}|=\frac{1}{\sqrt{2}} \]
The corresponding E-arm coefficient of the second Magic Tee has the same magnitude:
\[ |S_{54}|=\frac{1}{\sqrt{2}} \]
The H-arm cross-coupling coefficients satisfy
\[ |S_{15}|=\frac{1}{2} \]
and
\[ |S_{25}|=\frac{1}{2} \]
with the orthogonality condition giving
\[ S_{25}=S_{15} \]
The remaining phase reference can be selected according to the conventional reference-plane definition of the ideal Magic Tee. Choosing the E-arm coupling coefficient as positive real gives
\[ S_{13}=\frac{1}{\sqrt{2}} \]
and therefore
\[ S_{23}=-\frac{1}{\sqrt{2}} \]
Similarly, for the second Magic Tee,
\[ S_{54}=\frac{1}{\sqrt{2}} \]
and
\[ S_{64}=-\frac{1}{\sqrt{2}} \]
The H-arm cross-coupling terms can then be selected as
\[ S_{15}=S_{16}=S_{25}=S_{26}=\frac{1}{2} \]
The positive signs indicate that these cross-coupled outputs are in phase, which is the characteristic behavior associated with the H-arm connection.
Final \(6\times6\) Scattering Matrix
Substituting all the determined coefficients into the reduced matrix produces the complete scattering matrix for the ideal six-port network formed by connecting the H-arm of one Magic Tee directly to the H-arm of another Magic Tee. The matrix contains both characteristic behaviors of the Hybrid Tee: the E-arms produce equal-amplitude outputs with a \(180^\circ\) phase difference, while the internally connected H-arms produce equal-amplitude cross-coupled outputs with the same phase.
The final scattering matrix is
\[ [S]_{6\times6} = \begin{bmatrix} 0 & 0 & \frac{1}{\sqrt{2}} & 0 & \frac{1}{2} & \frac{1}{2}\\ 0 & 0 & -\frac{1}{\sqrt{2}} & 0 & \frac{1}{2} & \frac{1}{2}\\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}\\ \frac{1}{2} & \frac{1}{2} & 0 & \frac{1}{\sqrt{2}} & 0 & 0\\ \frac{1}{2} & \frac{1}{2} & 0 & -\frac{1}{\sqrt{2}} & 0 & 0 \end{bmatrix} \]
This matrix is reciprocal because it is symmetric, with \(S_{ij}=S_{ji}\). It is matched because every diagonal element is zero. It also contains the expected E-arm difference-port relationships, such as \(S_{23}=-S_{13}\) and \(S_{64}=-S_{54}\), while the H-arm cross-coupling terms appear with equal signs, such as \(S_{15}=S_{16}\) and \(S_{25}=S_{26}\). These relationships are not merely numerical coincidences; they directly represent the physical behavior of the two different modes supported by the Hybrid Tee structure.
Verification of the Final Matrix
The final matrix can now be checked against the original unitary requirement. For example, the first row has a total power coefficient equal to
\[ \left|\frac{1}{\sqrt{2}}\right|^2 + \left|\frac{1}{2}\right|^2 + \left|\frac{1}{2}\right|^2 = \frac{1}{2} + \frac{1}{4} + \frac{1}{4} = 1 \]
Likewise, the third row satisfies
\[ \left|\frac{1}{\sqrt{2}}\right|^2 + \left|-\frac{1}{\sqrt{2}}\right|^2 = \frac{1}{2}+\frac{1}{2}=1 \]
The same normalization occurs for the other rows, while the off-diagonal inner products cancel because the positive and negative E-arm contributions and the equal-phase H-arm contributions occur in the required combinations. Therefore,
\[ [S][S]^{\dagger}=[I] \]
The final matrix is consequently consistent with the ideal assumptions of reciprocity, perfect matching, losslessness, E-arm difference-mode operation, and H-arm in-phase cross-coupling.
Physical Interpretation of the E-Arm-to-H-Arm Connected Six-Port Network
The final \(6\times6\) scattering matrix gives the mathematical description of the six-port network, but the matrix becomes much easier to understand when we look at what actually happens to a signal as it travels through the two connected Magic Tees. In this configuration, the H-arm of Magic Tee A is directly connected to the H-arm of Magic Tee B, while the E-arms remain available as external Ports 3 and 4. This creates an interesting combination of two different Hybrid-Tee behaviors inside one six-port structure. The E-arms continue to produce equal-amplitude signals with opposite phase at their respective collinear arms, whereas the internal H-arm connection allows signals from one Magic Tee to reach the collinear arms of the other Magic Tee with equal amplitude and the same phase. The resulting network therefore combines difference-mode behavior at Ports 3 and 4 with sum-mode cross-coupling between the two groups of collinear arms.
Signal Flow Interpretation
The easiest way to understand the network is to divide the six external ports into two groups. Ports 1, 2, and 3 belong to Magic Tee A, while Ports 4, 5, and 6 belong to Magic Tee B. Port 3 is the E-arm of Tee A and Port 4 is the E-arm of Tee B. The two H-arms are not externally accessible because they are directly connected to one another. Consequently, an excitation at a collinear arm of one tee can have two components: one component reaches its own E-arm, while another component travels through the internally connected H-arm and appears at the collinear arms of the second tee. This internal path is responsible for the cross-coupling terms such as \(S_{15}\), \(S_{16}\), \(S_{25}\), and \(S_{26}\). Because the connection is through the H-arms, the cross-coupled signals at the receiving collinear arms have the same phase.
The final matrix is
\[ [S]_{6\times6} = \begin{bmatrix} 0 & 0 & \frac{1}{\sqrt{2}} & 0 & \frac{1}{2} & \frac{1}{2}\\ 0 & 0 & -\frac{1}{\sqrt{2}} & 0 & \frac{1}{2} & \frac{1}{2}\\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}\\ \frac{1}{2} & \frac{1}{2} & 0 & \frac{1}{\sqrt{2}} & 0 & 0\\ \frac{1}{2} & \frac{1}{2} & 0 & -\frac{1}{\sqrt{2}} & 0 & 0 \end{bmatrix} \]
Each column of this matrix describes the complete response of the network when one particular port is excited while the remaining ports are terminated in matched loads. This makes the matrix especially useful for understanding the physical signal paths. Rather than looking at all 36 entries individually, we can examine the four important excitation cases: excitation at Port 1, excitation at Port 2, excitation at Port 3, and excitation at Port 4.
Excitation at Port 1
Suppose a signal is applied only to Port 1. In this case, \(a_1\) is non-zero while all other incident waves are zero. From the first column of the scattering matrix, the output waves are
\[ b_1=0 \]
\[ b_2=0 \]
\[ b_3=\frac{1}{\sqrt{2}}a_1 \]
\[ b_4=0 \]
\[ b_5=\frac{1}{2}a_1 \]
\[ b_6=\frac{1}{2}a_1 \]
Therefore, the input at Port 1 produces one output at the E-arm of its own Magic Tee and two outputs at the collinear arms of the second Magic Tee. The E-arm output has amplitude \(1/\sqrt{2}\), while each cross-coupled output has amplitude \(1/2\). Most importantly, the signals at Ports 5 and 6 have the same sign:
\[ b_5=b_6 \]
This means that the two cross-coupled signals are in phase. Their powers are
\[ |b_5|^2=\frac{1}{4}|a_1|^2 \]
and
\[ |b_6|^2=\frac{1}{4}|a_1|^2 \]
while the E-arm receives
\[ |b_3|^2=\frac{1}{2}|a_1|^2 \]
Thus, the total output power is
\[ \frac{1}{2}|a_1|^2 + \frac{1}{4}|a_1|^2 + \frac{1}{4}|a_1|^2 = |a_1|^2 \]
which confirms power conservation. Physically, half of the input power reaches the E-arm of Tee A, while the remaining half is transferred through the internal H-arm connection and divided equally between Ports 5 and 6.
Excitation at Port 2
Now consider an excitation at Port 2, with \(a_2\) non-zero and all other incident waves equal to zero. From the second column of the scattering matrix,
\[ b_1=0 \]
\[ b_2=0 \]
\[ b_3=-\frac{1}{\sqrt{2}}a_2 \]
\[ b_4=0 \]
\[ b_5=\frac{1}{2}a_2 \]
\[ b_6=\frac{1}{2}a_2 \]
The most important difference between Port 1 and Port 2 excitation appears at Port 3. For Port 1 excitation,
\[ b_3=\frac{1}{\sqrt{2}}a_1 \]
whereas for Port 2 excitation,
\[ b_3=-\frac{1}{\sqrt{2}}a_2 \]
Thus, the E-arm responds with opposite phase to the two collinear arms. This is the characteristic difference-mode behavior of the E-arm. At the same time, the cross-coupled outputs at Ports 5 and 6 remain in phase because
\[ b_5=b_6=\frac{1}{2}a_2 \]
Again, the total output power is conserved:
\[ \frac{1}{2}|a_2|^2 + \frac{1}{4}|a_2|^2 + \frac{1}{4}|a_2|^2 = |a_2|^2 \]
Therefore, Ports 1 and 2 behave differently at their local E-arm but identically along the H-arm cross-coupling path. This combination of opposite-phase local coupling and equal-phase remote coupling is one of the defining characteristics of this connected configuration.
Excitation at Port 3
Port 3 is the E-arm of Magic Tee A. When a signal is applied to this port, the signal is divided equally between Ports 1 and 2, but the two output signals have opposite phase. From the third column of the scattering matrix,
\[ b_1=\frac{1}{\sqrt{2}}a_3 \]
\[ b_2=-\frac{1}{\sqrt{2}}a_3 \]
and
\[ b_4=b_5=b_6=0 \]
Therefore, Port 3 does not transfer power into the second Magic Tee through the H-arm connection. The reason is that the E-arm and H-arm modes are isolated. Since the internal connection is specifically between the H-arms, an E-arm excitation does not launch the required H-arm mode for propagation through that internal path. The E-arm therefore behaves as a local difference port, producing equal-amplitude but \(180^\circ\) out-of-phase signals at Ports 1 and 2.
The power division can be verified directly:
\[ |b_1|^2 = \frac{1}{2}|a_3|^2 \]
\[ |b_2|^2 = \frac{1}{2}|a_3|^2 \]
Hence,
\[ |b_1|^2+|b_2|^2=|a_3|^2 \]
All of the incident power at the E-arm is therefore delivered to the two collinear arms of the same Magic Tee.
Excitation at Port 4
Port 4 is the E-arm of Magic Tee B, so its behavior is the corresponding counterpart of Port 3. When \(a_4\) is the only non-zero incident wave, the fourth column of the matrix gives
\[ b_1=0 \]
\[ b_2=0 \]
\[ b_3=0 \]
\[ b_4=0 \]
\[ b_5=\frac{1}{\sqrt{2}}a_4 \]
\[ b_6=-\frac{1}{\sqrt{2}}a_4 \]
Thus, Port 4 produces equal-amplitude outputs at Ports 5 and 6 with a \(180^\circ\) phase difference. It does not transfer power to Ports 1 and 2 through the internally connected H-arms. This is again a direct consequence of E-arm and H-arm isolation.
The power delivered to Ports 5 and 6 is
\[ |b_5|^2+|b_6|^2 = \frac{1}{2}|a_4|^2 + \frac{1}{2}|a_4|^2 = |a_4|^2 \]
Therefore, Port 4 acts as the difference port of the second Magic Tee, exactly as Port 3 acts for the first Magic Tee.
Sum-Mode Behaviour Through the Connected H-Arms
The most important new feature introduced by connecting the two H-arms is the cross-coupling between the two groups of collinear ports. When a signal enters Port 1 or Port 2, part of that signal reaches the H-arm of Tee A and then travels through the direct internal connection to the H-arm of Tee B. At Tee B, the H-arm distributes the signal equally between its two collinear arms, Ports 5 and 6. Since an H-arm produces equal-phase outputs, the two cross-coupled signals satisfy
\[ S_{15}=S_{16} \]
and
\[ S_{25}=S_{26} \]
This is fundamentally different from the E-arm behavior. The E-arm produces
\[ S_{23}=-S_{13} \]
while the H-arm cross-coupling produces equal signs at the receiving collinear arms. Therefore, the same six-port network contains both difference-mode and sum-mode behavior. The E-arm controls the opposite-phase component, while the internally connected H-arms establish the in-phase cross-coupled component.
Why the H-Arm Connection Produces In-Phase Coupling
The reason for the equal-phase cross-coupling can be understood from the fundamental operation of an H-plane tee. When a signal enters the H-arm, the junction distributes the signal between the two collinear arms with equal phase. In the present six-port configuration, the H-arm of Tee A is directly connected to the H-arm of Tee B, so the signal arriving at the second tee enters its H-arm and is again divided equally and in phase between Ports 5 and 6. The internal connection therefore preserves the sum-mode nature of the H-arm excitation. This is why the cross-coupling terms appear with the same sign rather than opposite signs.
Mathematically, for excitation at Port 1,
\[ S_{51}=S_{61}=\frac{1}{2} \]
and for excitation at Port 2,
\[ S_{52}=S_{62}=\frac{1}{2} \]
The equal signs are the key indication that the H-arm path produces in-phase coupling. If the internal connection had instead been made between the two E-arms, the cross-coupling relationships would have been different because the E-arm produces a difference-mode output with a \(180^\circ\) phase reversal.
Comparison with E-Arm-to-E-Arm Connection
Connecting the H-arms of two Magic Tees and connecting the E-arms of two Magic Tees are not equivalent configurations. The difference comes directly from the modes associated with the two arms. In an E-arm-to-E-arm connection, the internal path is associated with difference-mode behavior, so the corresponding cross-coupled outputs acquire opposite phase relationships. In an H-arm-to-H-arm connection, the internal path is associated with sum-mode behavior, so the corresponding cross-coupled outputs have equal phase. This makes the H-arm connection particularly useful when an in-phase combination or distribution between two groups of microwave ports is required.
In the present configuration, Ports 3 and 4 remain E-arm difference ports. They are isolated from one another and do not transfer energy through the H-arm connection. The cross-coupling occurs primarily between the collinear-port groups, where the H-arm connection creates equal-phase outputs. Thus, the E-arm-to-H-arm connected structure should be viewed as a combination of local difference-mode operation and remote sum-mode coupling rather than as a simple six-port power divider.
Advantages of the E-Arm-to-H-Arm Connected Configuration
One major advantage of this configuration is that it combines two useful Hybrid-Tee properties within a single multiport network. The external E-arms provide controlled difference-mode access, while the internal H-arm connection creates an in-phase coupling path between the two Magic Tees. This gives the designer separate access to phase-opposed and phase-aligned signal paths without requiring an additional external junction. The ideal network is also reciprocal and lossless, meaning that the same coupling relationships are available in either direction and the total microwave power is conserved.
- The two E-arms remain isolated from each other.
- The E-arms provide equal-amplitude, \(180^\circ\) out-of-phase outputs.
- The connected H-arms provide equal-amplitude, in-phase cross-coupling.
- The network is reciprocal under the ideal passive-network assumption.
- The ideal six-port network is matched at all external ports.
- The unitary scattering matrix ensures power conservation.
- The structure provides both difference-mode and sum-mode signal paths.
Limitations of the Ideal Six-Port Model
The derived matrix represents an ideal mathematical model, so practical hardware will not reproduce every coefficient exactly. Real Magic Tees have conductor losses, dielectric losses, manufacturing tolerances, finite isolation, impedance mismatch, and frequency-dependent phase characteristics. The direct connection between the two internal H-arms can also introduce additional phase shift depending on the physical length and waveguide structure of the connection. As a result, practical measurements may show small non-zero reflection coefficients, imperfect isolation, unequal output amplitudes, and phase errors. Nevertheless, the ideal \(6\times6\) matrix remains extremely useful because it provides the reference behavior against which a practical device can be analyzed.
Practical Applications
A six-port network formed by connecting the H-arms of two Hybrid Tees can be useful wherever controlled in-phase and out-of-phase microwave signals need to coexist in the same network. The H-arm connection can be used to distribute a signal between two groups of ports with equal phase, while the external E-arms provide difference-mode access. Such structures are relevant to microwave signal combining and splitting, balanced microwave networks, phase-sensitive measurement systems, antenna-feed networks, radar front ends, and other RF systems in which sum and difference signal components must be controlled separately. The exact application depends on the physical implementation, operating frequency, required bandwidth, and desired port characteristics.
From a network-analysis perspective, the configuration is also useful as an example of how internal connections between microwave junctions can be converted into a larger multiport scattering matrix. Instead of treating the two Magic Tees as completely independent components, the internal H-arm connection allows their individual scattering properties to be combined into one six-port representation. This approach is valuable when analyzing larger microwave networks because it provides a systematic way to replace interconnected junctions with a single equivalent S-matrix.
Frequently Asked Questions
What happens when the H-arms of two Magic Tees are connected?
Connecting the H-arms creates an internal sum-mode coupling path between the two Magic Tees. A signal transferred through this path is divided equally and in phase at the collinear arms of the receiving Magic Tee. The result is cross-coupling between the two groups of collinear ports while the E-arms retain their difference-mode behavior.
Are the two E-arms coupled to each other?
In the ideal configuration considered here, the two E-arms are isolated from each other. Therefore,
\[ S_{34}=S_{43}=0 \]
This occurs because the internal connection is made through the H-arms, while the E-arm and H-arm modes are isolated within each Hybrid Tee.
Why are the cross-coupled signals in phase?
The internal connection is between two H-arms. An H-arm produces equal-amplitude, in-phase signals at the two collinear arms. Consequently, the cross-coupling terms at the receiving tee have equal signs, such as
\[ S_{15}=S_{16} \]
and
\[ S_{25}=S_{26} \]
Why do the E-arm outputs have opposite phase?
The E-arm represents the difference mode of the Hybrid Tee. Therefore, excitation at an E-arm produces equal-amplitude signals at the two collinear arms with a \(180^\circ\) phase difference. For Port 3, for example,
\[ S_{13}=\frac{1}{\sqrt{2}} \]
while
\[ S_{23}=-\frac{1}{\sqrt{2}} \]
The opposite signs represent the \(180^\circ\) phase difference.
What is the meaning of \(1/\sqrt{2}\) in the final matrix?
The magnitude \(1/\sqrt{2}\) represents equal power division between two output paths. Since
\[ \left|\frac{1}{\sqrt{2}}\right|^2=\frac{1}{2} \]
each output receives one-half of the available input power when an E-arm is excited.
What is the meaning of \(1/2\) in the final matrix?
The magnitude \(1/2\) appears in the cross-coupling path through the connected H-arms. Its corresponding power fraction is
\[ \left|\frac{1}{2}\right|^2=\frac{1}{4} \]
Therefore, each of the two cross-coupled collinear ports receives one-quarter of the incident power for the relevant collinear-port excitation, while the remaining one-half of the power reaches the local E-arm.
Is the final six-port network lossless?
Yes, under the ideal assumptions used in the derivation. The scattering matrix satisfies the unitary condition
\[ [S][S]^\dagger=[I] \]
which means that the total outgoing power equals the total incident power.
Is the six-port network reciprocal?
Yes. The final matrix is symmetric, so
\[ S_{ij}=S_{ji} \]
This confirms reciprocity for the ideal passive network.
Final Understanding of the E-Arm-to-H-Arm Connected Network
The main idea behind this six-port configuration is the combination of two different Hybrid-Tee modes in one interconnected structure. The E-arms, represented by Ports 3 and 4, operate as difference ports and generate equal-amplitude outputs with a \(180^\circ\) phase difference at their respective collinear arms. The internally connected H-arms provide a separate sum-mode path between the two Magic Tees, producing equal-amplitude and in-phase cross-coupled signals at the receiving collinear arms. By starting with the complete \(6\times6\) S-matrix, applying reciprocity and matching, imposing the E-arm and H-arm relationships, and finally using the unitary condition, the unknown coefficients can be determined systematically rather than assumed. The resulting matrix therefore provides a complete ideal representation of the E-arm-to-H-arm connected six-port network and clearly shows how phase, isolation, power division, and cross-coupling are controlled by the underlying Hybrid-Tee structure.