H-Arm Connected to E-Arm in Two Magic Tees
H-Arm Connected to E-Arm in Two Magic Tees: Derivation of the 6×6 Scattering Matrix and Network Properties
When the H-arm of one Magic Tee is connected directly to the E-arm of another Magic Tee, the resulting structure forms a six-port microwave network. Unlike the conventional E-arm-to-E-arm or H-arm-to-H-arm interconnections, this configuration combines the in-phase power division characteristic of the H-arm with the out-of-phase power division characteristic of the E-arm. Consequently, the resulting six-port network exhibits unique coupling paths, phase relationships, and isolation properties that differ significantly from other Hybrid-Tee interconnection arrangements.
The objective of this derivation is to obtain the complete six-port scattering matrix by systematically applying the physical properties of the interconnected Magic Tees. Instead of immediately assuming numerical coupling coefficients, the derivation begins with the most general six-port scattering matrix and progressively reduces it using reciprocity, matching conditions, E-arm properties, H-arm properties, symmetry conditions, and finally the unitary property of a lossless microwave network.
Configuration and Port Definitions
The six external ports of the network are defined according to the physical arrangement of the two Magic Tees. The first Magic Tee contributes the two collinear arms and its H-arm as external ports, while its E-arm becomes part of the internal connection. The second Magic Tee contributes its E-arm and two collinear arms as external ports, while its H-arm participates in the internal connection.
- Port 1 = Collinear Arm 1 of Tee A
- Port 2 = Collinear Arm 2 of Tee A
- Port 3 = H-arm of Tee A
- Port 4 = E-arm of Tee B
- Port 5 = Collinear Arm 1 of Tee B
- Port 6 = Collinear Arm 2 of Tee B
The E-arm of Tee A is connected internally to the H-arm of Tee B. Since these two ports are no longer accessible externally, the original eight-port system is reduced to a six-port microwave network.
Construction of the General 6×6 Scattering Matrix
The scattering behavior of the six-port network is described by the standard scattering relationship
\[ [b]=[S][a] \]
where \([a]\) represents the vector of incident waves, \([b]\) represents the vector of outgoing waves, and \([S]\) represents the six-port scattering matrix. Therefore,
\[ \begin{bmatrix} b_1\\ b_2\\ b_3\\ b_4\\ b_5\\ b_6 \end{bmatrix} = \begin{bmatrix} S_{11}&S_{12}&S_{13}&S_{14}&S_{15}&S_{16}\\ S_{21}&S_{22}&S_{23}&S_{24}&S_{25}&S_{26}\\ S_{31}&S_{32}&S_{33}&S_{34}&S_{35}&S_{36}\\ S_{41}&S_{42}&S_{43}&S_{44}&S_{45}&S_{46}\\ S_{51}&S_{52}&S_{53}&S_{54}&S_{55}&S_{56}\\ S_{61}&S_{62}&S_{63}&S_{64}&S_{65}&S_{66} \end{bmatrix} \begin{bmatrix} a_1\\ a_2\\ a_3\\ a_4\\ a_5\\ a_6 \end{bmatrix} \]
At this stage, every element of the matrix is unknown. The next objective is to reduce the matrix using the physical properties of the interconnected Magic Tees.
Applying Reciprocity to the General Matrix
Since the network consists entirely of passive microwave junctions and contains no ferrite materials or non-reciprocal components, the reciprocity condition must be satisfied. Reciprocity requires that the transmission coefficient from Port i to Port j be identical to the transmission coefficient from Port j to Port i.
\[ S_{ij}=S_{ji} \]
Therefore,
\[ S_{12}=S_{21} \]
\[ S_{13}=S_{31} \]
\[ S_{14}=S_{41} \]
\[ S_{15}=S_{51} \]
\[ S_{16}=S_{61} \]
and similarly for all remaining off-diagonal terms. After applying reciprocity, the matrix becomes symmetric about its principal diagonal and the number of independent unknown quantities is significantly reduced.
Applying the Matching Conditions
The external ports of an ideal Magic Tee are perfectly matched. Therefore, an incident wave arriving at any external port does not experience reflection back toward the same port. Mathematically, this condition is expressed by setting all diagonal elements equal to zero.
\[ S_{11}=S_{22}=S_{33}=S_{44}=S_{55}=S_{66}=0 \]
Substituting these conditions into the general matrix eliminates all self-reflection coefficients. The matrix now contains only transmission coefficients between different ports.
Applying the E-Arm and H-Arm Properties
The next step is to incorporate the fundamental properties of the E-arm and H-arm. These properties determine how signals divide between the collinear arms and establish the phase relationships that ultimately define the final six-port scattering matrix.
Applying the E-Arm Difference-Port Relationship
Port 4 is the E-arm of Tee B. A signal entering an E-arm divides equally between the two collinear arms while producing a phase reversal of 180° between them. Consequently,
\[ S_{54}=-S_{64} \]
By reciprocity,
\[ S_{45}=-S_{46} \]
These relationships represent the characteristic difference-mode operation of the E-arm.
Applying the H-Arm In-Phase Coupling Relationship
Port 3 is the H-arm of Tee A. A signal entering an H-arm divides equally between the two collinear arms without producing a phase reversal. Therefore,
\[ S_{13}=S_{23} \]
and by reciprocity,
\[ S_{31}=S_{32} \]
These relationships represent the sum-mode operation of the H-arm.
Collinear-Arm Isolation and Symmetry Conditions
The two collinear arms of an ideal Magic Tee are mutually isolated. Consequently, the collinear arms of Tee A satisfy
\[ S_{12}=S_{21}=0 \]
Similarly, the collinear arms of Tee B satisfy
\[ S_{56}=S_{65}=0 \]
These conditions further reduce the number of independent unknown coefficients and simplify the matrix before the application of the unitary condition.
Reduced S-Matrix Before Applying the Unitary Condition
After applying reciprocity, matching conditions, H-arm properties, E-arm properties, and collinear-arm isolation, the original general matrix is reduced considerably. The remaining unknown terms correspond only to the physically allowed coupling paths of the interconnected Magic Tees.
The next stage of the derivation is to determine the numerical values of these remaining unknown coefficients using the lossless property of the network.
Preparing the Matrix for the Unitary Condition
Because the network is assumed to be ideal and lossless, total microwave power must be conserved. The scattering matrix must therefore satisfy the unitary condition
\[ [S][S]^\dagger=[I] \]
This condition provides a set of normalization and orthogonality equations that can be used to determine all remaining unknown S-parameters.
Therefore, the next stage is to explicitly multiply the reduced scattering matrix by its conjugate transpose and compare the result with the identity matrix.
Method: Applying the Unitary Condition to the Reduced 6×6 Matrix
In the next section, the reduced matrix will be multiplied according to
\[ [S][S]^\dagger=[I] \]
to obtain the row-normalization equations, orthogonality equations, and the numerical values of the remaining unknown scattering coefficients. Once these coefficients are solved, the complete six-port scattering matrix of the H-arm-to-E-arm connected Magic Tee network can be obtained.
Reduced 6×6 S-Matrix Before Numerical Evaluation
After applying reciprocity, matching conditions, collinear-arm isolation, H-arm sum-port behavior, and E-arm difference-port behavior, the general scattering matrix is significantly simplified. At this stage, many of the original thirty-six scattering coefficients have either become zero or have been related to other coefficients through symmetry and physical constraints. However, the exact numerical values of the remaining coefficients are still unknown and must be determined using the lossless property of the network.
The reduced matrix can therefore be written as
\[ [S] = \begin{bmatrix} 0 & 0 & S_{13} & S_{14} & S_{15} & S_{16} \\ 0 & 0 & S_{13} & S_{24} & S_{25} & S_{26} \\ S_{13} & S_{13} & 0 & S_{34} & S_{35} & S_{36} \\ S_{14} & S_{24} & S_{34} & 0 & S_{45} & -S_{45} \\ S_{15} & S_{25} & S_{35} & S_{45} & 0 & 0 \\ S_{16} & S_{26} & S_{36} & -S_{45} & 0 & 0 \end{bmatrix} \]
The negative sign associated with the E-arm coupling coefficients originates directly from the difference-port property of the E-arm. Whenever a signal enters an E-arm, equal-amplitude outputs are produced at the two collinear arms, but the outputs differ in phase by 180°. This property is represented mathematically through opposite signs in the corresponding scattering coefficients.
Similarly, the repeated terms involving the H-arm arise from the sum-port behavior of the H-arm. A signal entering the H-arm divides equally between the collinear arms with identical phase. Consequently, the associated scattering coefficients must have equal magnitudes and identical signs.
Additional Physical Constraints Before Applying the Unitary Condition
The internal connection between the H-arm of Tee A and the E-arm of Tee B introduces additional restrictions on the signal flow within the network. Certain transmission paths become physically impossible because the H-arm and E-arm are isolated ports inside an ideal Magic Tee. As a result, several coefficients must be zero.
Since Port 3 is associated with the H-arm behavior of Tee A and Port 4 is associated with the E-arm behavior of Tee B, direct coupling between these ports does not occur. Therefore,
\[ S_{34}=S_{43}=0 \]
Likewise, excitation entering the H-arm path does not directly excite the collinear outputs associated solely with the E-arm coupling mechanism. Hence,
\[ S_{35}=S_{36}=0 \]
Similarly,
\[ S_{14}=S_{24}=0 \]
because the E-arm excitation of Tee B remains isolated from the collinear arms of Tee A.
Substituting these physical constraints into the reduced matrix gives
\[ [S] = \begin{bmatrix} 0 & 0 & S_{13} & 0 & S_{15} & S_{16} \\ 0 & 0 & S_{13} & 0 & S_{25} & S_{26} \\ S_{13} & S_{13} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & S_{45} & -S_{45} \\ S_{15} & S_{25} & 0 & S_{45} & 0 & 0 \\ S_{16} & S_{26} & 0 & -S_{45} & 0 & 0 \end{bmatrix} \]
This matrix now contains only the coefficients that correspond to physically realizable transmission paths in the six-port network.
Applying Additional Symmetry Relationships
Because the two Magic Tees are assumed to be identical and the internal connection does not favor one collinear arm over another, symmetry conditions can be applied to the remaining unknown coefficients.
The coupling from Port 1 into Tee B must be identical to the coupling from Port 2 into Tee B. Therefore,
\[ S_{15}=S_{25} \]
Similarly,
\[ S_{16}=S_{26} \]
The H-arm coupling path divides power equally into the two collinear arms of Tee B. Consequently,
\[ S_{15}=S_{16} \]
Combining these symmetry relationships yields
\[ S_{15}=S_{16}=S_{25}=S_{26} \]
Substituting these equalities into the matrix produces
\[ [S] = \begin{bmatrix} 0 & 0 & S_{13} & 0 & S_{15} & S_{15} \\ 0 & 0 & S_{13} & 0 & S_{15} & S_{15} \\ S_{13} & S_{13} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & S_{45} & -S_{45} \\ S_{15} & S_{15} & 0 & S_{45} & 0 & 0 \\ S_{15} & S_{15} & 0 & -S_{45} & 0 & 0 \end{bmatrix} \]
The matrix now contains only three independent unknown quantities:
\[ S_{13} \]
\[ S_{15} \]
\[ S_{45} \]
The next step is to apply the lossless condition
\[ [S][S]^\dagger=[I] \]
to obtain normalization equations and solve these remaining unknown scattering coefficients. Once those values are determined, the complete numerical six-port scattering matrix can be written directly.
Applying the Unitary Condition to the Reduced 6×6 Matrix
The remaining unknown scattering coefficients cannot be determined solely from reciprocity, matching conditions, symmetry, and port-isolation properties. The final numerical values must be obtained from the lossless condition of the network. Since the interconnected Magic-Tee structure is assumed to be ideal and contains no dissipative elements, all incident microwave power must be conserved. Therefore, the scattering matrix must satisfy the unitary condition
\[ [S][S]^\dagger=[I] \]
where \([S]^\dagger\) represents the conjugate transpose of the scattering matrix and \([I]\) represents the identity matrix. Physically, this condition ensures that the total output power is exactly equal to the total input power and that different rows of the matrix remain orthogonal to one another.
The reduced six-port scattering matrix obtained previously is
\[ [S] = \begin{bmatrix} 0 & 0 & S_{13} & 0 & S_{15} & S_{15} \\ 0 & 0 & S_{13} & 0 & S_{15} & S_{15} \\ S_{13} & S_{13} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & S_{45} & -S_{45} \\ S_{15} & S_{15} & 0 & S_{45} & 0 & 0 \\ S_{15} & S_{15} & 0 & -S_{45} & 0 & 0 \end{bmatrix} \]
The unknown quantities that remain to be determined are
\[ S_{13} \]
\[ S_{15} \]
\[ S_{45} \]
Row Normalization Equations
For a lossless network, the inner product of each row with its own complex conjugate must be equal to unity. This requirement produces the row-normalization equations.
Row 3 Normalization
The third row of the reduced matrix is
\[ \begin{bmatrix} S_{13} & S_{13} & 0 & 0 & 0 & 0 \end{bmatrix} \]
Applying the normalization condition gives
\[ |S_{13}|^2 + |S_{13}|^2 =1 \]
\[ 2|S_{13}|^2=1 \]
Therefore,
\[ |S_{13}| = \frac{1}{\sqrt2} \]
Row 4 Normalization
The fourth row of the matrix is
\[ \begin{bmatrix} 0 & 0 & 0 & 0 & S_{45} & -S_{45} \end{bmatrix} \]
Applying the normalization condition,
\[ |S_{45}|^2 + |-S_{45}|^2 =1 \]
\[ 2|S_{45}|^2=1 \]
Therefore,
\[ |S_{45}| = \frac{1}{\sqrt2} \]
Row 1 Normalization
The first row of the matrix is
\[ \begin{bmatrix} 0 & 0 & S_{13} & 0 & S_{15} & S_{15} \end{bmatrix} \]
Applying the normalization condition,
\[ |S_{13}|^2 + |S_{15}|^2 + |S_{15}|^2 =1 \]
\[ |S_{13}|^2 + 2|S_{15}|^2 =1 \]
Substituting
\[ |S_{13}|^2 = \frac12 \]
gives
\[ \frac12 + 2|S_{15}|^2 = 1 \]
\[ 2|S_{15}|^2 = \frac12 \]
\[ |S_{15}|^2 = \frac14 \]
Therefore,
\[ |S_{15}| = \frac12 \]
Summary of the Solved Coefficients
The normalization equations have now determined all remaining unknown magnitudes in the scattering matrix. The results are
\[ S_{13} = \frac{1}{\sqrt2} \]
\[ S_{45} = \frac{1}{\sqrt2} \]
\[ S_{15} = \frac12 \]
The signs associated with these coefficients are obtained directly from the H-arm and E-arm phase relationships already incorporated into the matrix. The H-arm produces equal in-phase outputs, while the E-arm produces equal outputs with a phase difference of 180°. Therefore, the magnitudes obtained above are sufficient to determine every remaining scattering coefficient.
Final Numerical Values of the Remaining S-Parameters
Substituting the solved coefficient values into the symmetry relationships gives
\[ S_{13} = S_{23} = S_{31} = S_{32} = \frac{1}{\sqrt2} \]
\[ S_{45} = S_{54} = \frac{1}{\sqrt2} \]
\[ S_{46} = S_{64} = -\frac{1}{\sqrt2} \]
\[ S_{15} = S_{16} = S_{25} = S_{26} = S_{51} = S_{52} = S_{61} = S_{62} = \frac12 \]
All remaining coefficients are either zero due to matching and isolation conditions or are determined through reciprocity.
Final 6×6 Scattering Matrix
Substituting all solved coefficients into the reduced matrix yields the complete six-port scattering matrix for the network formed by connecting the H-arm of one Magic Tee to the E-arm of another Magic Tee.
\[ [S]_{6\times6} = \begin{bmatrix} 0 & 0 & \frac{1}{\sqrt2} & 0 & \frac12 & \frac12 \\ 0 & 0 & \frac{1}{\sqrt2} & 0 & \frac12 & \frac12 \\ \frac{1}{\sqrt2} & \frac{1}{\sqrt2} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{\sqrt2} & -\frac{1}{\sqrt2} \\ \frac12 & \frac12 & 0 & \frac{1}{\sqrt2} & 0 & 0 \\ \frac12 & \frac12 & 0 & -\frac{1}{\sqrt2} & 0 & 0 \end{bmatrix} \]
This matrix completely characterizes the microwave behavior of the six-port network. Every power division ratio, phase relationship, isolation property, and transmission path can now be determined directly from the matrix elements. The next section examines the physical meaning of these coefficients and develops the practical properties and applications of the network.