Connecting the H-Arms of Two Magic Tees
What happen when E arm of one Tee is connected to E arm of another Tee? Write its proporties
OR
Explain the properties of two Magic Tees if one connects their E-ams
Analysis of Connecting Two Magic Tees via H-Arms: 6x6 Scattering Matrix Derivation
When the H-arm (sum port) of one ideal Magic Tee is connected directly to the H-arm of another ideal Magic Tee, the two individual four-port junctions combine to form a composite six-port microwave network. Because the two H-arms are joined internally, they are no longer accessible as external ports. Instead, they create an internal sum-mode coupling channel between the two hybrid junctions. The resulting network features six external terminals, and its complete high-frequency electrical behavior is fully characterized by a \(6 \times 6\) scattering matrix.
Network Topology & Basic Equations
Consider two identical, ideal Magic Tees designated as Tee A and Tee B. For Magic Tee A, let Ports 1 and 2 represent the collinear arms, Port 3 represent the H-arm (sum port), and Port 4 represent the E-arm (difference port). For Magic Tee B, let Ports 5 and 6 represent the collinear arms, Port 7 represent the H-arm, and Port 8 represent the E-arm.
An ideal \(4 \times 4\) scattering matrix for Magic Tee A (\([S_A]\)) and Magic Tee B (\([S_B]\)) is defined as:
\[ [S_A] = [S_B] = \frac{1}{\sqrt{2}} \begin{bmatrix} 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & -1 \\ 1 & 1 & 0 & 0 \\ 1 & -1 & 0 & 0 \end{bmatrix} \]The corresponding wave linear equations for Tee A are:
\[ \begin{aligned} b_1 &= \frac{1}{\sqrt{2}}a_3 + \frac{1}{\sqrt{2}}a_4 \\ b_2 &= \frac{1}{\sqrt{2}}a_3 - \frac{1}{\sqrt{2}}a_4 \\ b_3 &= \frac{1}{\sqrt{2}}a_1 + \frac{1}{\sqrt{2}}a_2 \\ b_4 &= \frac{1}{\sqrt{2}}a_1 - \frac{1}{\sqrt{2}}a_2 \end{aligned} \]The corresponding wave linear equations for Tee B are:
\[ \begin{aligned} b_5 &= \frac{1}{\sqrt{2}}a_7 + \frac{1}{\sqrt{2}}a_8 \\ b_6 &= \frac{1}{\sqrt{2}}a_7 - \frac{1}{\sqrt{2}}a_8 \\ b_7 &= \frac{1}{\sqrt{2}}a_5 + \frac{1}{\sqrt{2}}a_6 \\ b_8 &= \frac{1}{\sqrt{2}}a_5 - \frac{1}{\sqrt{2}}a_6 \end{aligned} \]Internal Boundary Conditions & Port Renumbering
Connecting Port 3 (H-arm of Tee A) directly to Port 7 (H-arm of Tee B) establishes internal boundary conditions. The wave leaving Port 3 enters Port 7 (\(a_7 = b_3\)), and the wave leaving Port 7 enters Port 3 (\(a_3 = b_7\)):
\[ a_7 = b_3 = \frac{1}{\sqrt{2}}(a_1 + a_2) \quad \text{and} \quad a_3 = b_7 = \frac{1}{\sqrt{2}}(a_5 + a_6) \]Internal Ports 3 and 7 are eliminated. The six accessible external terminals are renumbered sequentially to form a standard 6-port network:
- Port 1: Collinear Arm 1 of Tee A (Local Port 1)
- Port 2: Collinear Arm 2 of Tee A (Local Port 2)
- Port 3: E-arm of Tee A (Local Port 4)
- Port 4: E-arm of Tee B (Local Port 8)
- Port 5: Collinear Arm 1 of Tee B (Local Port 5)
- Port 6: Collinear Arm 2 of Tee B (Local Port 6)
Connecting Two Magic Tees Through Their H-Arms: 6-Port Scattering Matrix Derivation
Configuration and Port Definitions
In this configuration, two ideal Magic Tees are connected directly through their H-arms, creating a combined six-port microwave network. The individual Magic-Tee operation, including the basic properties of the E-arm and H-arm, has already been established in the previous analysis, so the present derivation begins with the connected structure itself. The two H-arms form an internal connection and therefore do not appear as external ports of the final network. After removing these two internal H-arm ports from the external representation, six accessible ports remain. The purpose of this analysis is to construct the complete \(6\times6\) scattering matrix of this combined network and determine the unknown coupling coefficients using reciprocity, matching, and lossless-network conditions. The important feature of this configuration is that the H-arm connection produces equal-amplitude, in-phase coupling between the collinear arms of the two different Magic Tees, while the E-arms retain their normal difference-mode behaviour with equal-amplitude, opposite-phase coupling to their respective collinear arms.
Port Numbering of the Six-Port Network
To avoid confusion between the original local port numbers of the two individual Magic Tees and the final external port numbers of the combined network, the six accessible ports are renumbered as Port 1 through Port 6. This numbering is used throughout the complete \(6\times6\) scattering-matrix derivation. Port 1 is the first collinear arm of Magic Tee A and corresponds to Local Port 1 of Tee A. Port 2 is the second collinear arm of Magic Tee A and corresponds to Local Port 2. Port 3 is the E-arm of Magic Tee A and corresponds to Local Port 4. On the second Magic Tee, Port 4 is its E-arm and corresponds to Local Port 8, while Port 5 and Port 6 are its two collinear arms and correspond to Local Ports 5 and 6, respectively. The H-arm of Tee A and the H-arm of Tee B are internally connected to each other and are therefore not included among these six external ports.
- Port 1: Collinear Arm 1 of Tee A, corresponding to Local Port 1.
- Port 2: Collinear Arm 2 of Tee A, corresponding to Local Port 2.
- Port 3: E-arm of Tee A, corresponding to Local Port 4.
- Port 4: E-arm of Tee B, corresponding to Local Port 8.
- Port 5: Collinear Arm 1 of Tee B, corresponding to Local Port 5.
- Port 6: Collinear Arm 2 of Tee B, corresponding to Local Port 6.
With this numbering, the incident-wave vector and outgoing-wave vector of the complete network are written as
\[ [a] = \begin{bmatrix} a_1\\ a_2\\ a_3\\ a_4\\ a_5\\ a_6 \end{bmatrix} \]
and
\[ [b] = \begin{bmatrix} b_1\\ b_2\\ b_3\\ b_4\\ b_5\\ b_6 \end{bmatrix}. \]
The complete network is therefore described by
\[ [b]=[S]_{6\times6}[a]. \]
Construction of the General \(6\times6\) Matrix
Before applying the physical properties of the connected Magic Tees, the scattering matrix of the six-port network can be written in completely general form. Every element \(S_{ij}\) represents the outgoing wave at Port \(i\) resulting from an incident wave at Port \(j\), with all other external ports terminated in matched loads. Thus, the complete matrix initially contains thirty-six scattering parameters. The physical symmetry and isolation properties of the Magic-Tee structure will then eliminate many of these parameters by forcing them to zero or relating them to one another. This systematic construction is preferable to assigning numerical values immediately because it clearly shows where every nonzero element in the final matrix originates.
The general six-port scattering matrix is
\[ [S]_{6\times6} = \begin{bmatrix} S_{11} & S_{12} & S_{13} & S_{14} & S_{15} & S_{16}\\ S_{21} & S_{22} & S_{23} & S_{24} & S_{25} & S_{26}\\ S_{31} & S_{32} & S_{33} & S_{34} & S_{35} & S_{36}\\ S_{41} & S_{42} & S_{43} & S_{44} & S_{45} & S_{46}\\ S_{51} & S_{52} & S_{53} & S_{54} & S_{55} & S_{56}\\ S_{61} & S_{62} & S_{63} & S_{64} & S_{65} & S_{66} \end{bmatrix}. \]
The next step is to determine which of these coefficients can be nonzero. Because Port 3 and Port 4 are E-arms, their coupling to the corresponding collinear arms must have equal magnitudes with opposite signs. Because the two H-arms are internally connected, the collinear arms of Tee A can also couple to the collinear arms of Tee B through the internal H-arm path. That cross-coupling has equal magnitude and equal phase. The E-arm isolation property prevents direct coupling between the two E-arms and also prevents an E-arm from coupling to the collinear arms belonging to the other Tee.
Definition of Unknown Parameters
The physical relationships allow the many scattering parameters in the general matrix to be represented using only two independent magnitude coefficients. Let \(a\) denote the magnitude of the coupling between an E-arm and each of its associated collinear arms. Since the two Magic Tees are identical, the same magnitude applies to both E-arms. For Tee A, Port 3 is the E-arm, so the coupling coefficients to Ports 1 and 2 have equal magnitude and opposite signs. We therefore write
\[ S_{13}=a \]
and
\[ S_{23}=-a. \]
Reciprocity will later give the corresponding reverse-direction coefficients
\[ S_{31}=a \]
and
\[ S_{32}=-a. \]
For Tee B, Port 4 is its E-arm and Ports 5 and 6 are its collinear arms. Therefore, the same E-arm difference-mode relationship gives
\[ S_{54}=a \]
and
\[ S_{64}=-a. \]
The second unknown coefficient, \(c\), represents the coupling between the collinear arms of the two different Magic Tees through the internal H-arm connection. Because the H-arm is a sum port, this coupling occurs with equal phase. Consequently, an excitation at Port 1 produces equal coupling to Ports 5 and 6, while an excitation at Port 2 also produces equal coupling to Ports 5 and 6. We therefore write
\[ S_{15}=S_{16}=S_{25}=S_{26}=c. \]
The same coefficient appears in the reverse direction after reciprocity is applied. Thus,
\[ S_{51}=S_{61}=S_{52}=S_{62}=c. \]
At this point, \(a\) represents the E-arm coupling magnitude and \(c\) represents the cross-coupling magnitude produced by the internal H-arm connection. The numerical values of these coefficients will be obtained from the lossless condition after the matrix structure has been completely established.
Reciprocity Conditions
The combined six-port network is formed entirely from passive reciprocal Magic-Tee junctions and a direct internal connection between their H-arms. Therefore, the complete network is reciprocal. For a reciprocal microwave network, the scattering matrix satisfies
\[ S_{ij}=S_{ji}. \]
This means that the transmission coefficient from Port \(i\) to Port \(j\) is identical to the transmission coefficient from Port \(j\) to Port \(i\). Applying this condition to the E-arm coupling gives
\[ S_{13}=S_{31}=a \]
and
\[ S_{23}=S_{32}=-a. \]
Similarly, for the second Magic Tee,
\[ S_{45}=S_{54}=a \]
and
\[ S_{46}=S_{64}=-a. \]
For the cross-coupling through the internal H-arm path, reciprocity gives
\[ S_{15}=S_{51}=c, \]
\[ S_{16}=S_{61}=c, \]
\[ S_{25}=S_{52}=c, \]
and
\[ S_{26}=S_{62}=c. \]
Therefore, the final matrix must be symmetric:
\[ [S]^T=[S]. \]
Reciprocity determines the relationship between forward and reverse coupling coefficients, but it does not by itself determine their magnitudes. The values of \(a\) and \(c\) must therefore be obtained from the matching and lossless properties of the network.
Matching Conditions
Each external port of the ideal six-port network is assumed to be perfectly matched. A matched port has no reflected wave when it is excited while all other ports are terminated in their characteristic impedances. In scattering-parameter notation, this means that the reflection coefficient at every external port is zero. Consequently, every diagonal element of the six-port scattering matrix must vanish.
The matching conditions are therefore
\[ S_{11}=S_{22}=S_{33}=S_{44}=S_{55}=S_{66}=0. \]
These conditions remove the six direct-reflection terms from the general matrix. The absence of diagonal terms does not mean that the ports are isolated from the rest of the network; instead, it means that an incident wave is not reflected directly back into the same port. The incident power must therefore be distributed among the other allowed output paths. In this connected Magic-Tee structure, those paths are determined by the E-arm coupling and the cross-coupling produced by the internal H-arm connection.
Isolation Conditions and Matrix Structure
The ideal Magic-Tee isolation properties impose additional zero elements on the six-port matrix. Since Port 3 and Port 4 are the E-arms of the two separate tees and the two tees are connected through their H-arms, there is no direct E-arm-to-E-arm coupling through the internal H-arm path. Therefore,
\[ S_{34}=S_{43}=0. \]
Port 3 also remains isolated from the collinear arms of Tee B. Hence,
\[ S_{35}=S_{36}=0. \]
Likewise, Port 4 is isolated from the collinear arms of Tee A:
\[ S_{41}=S_{42}=0. \]
Within each individual Tee, the two collinear arms are isolated from one another under the ideal Magic-Tee model. Therefore,
\[ S_{12}=S_{21}=0 \]
and
\[ S_{56}=S_{65}=0.\]
Combining these isolation conditions with the E-arm phase relationships and the H-arm cross-coupling relationships produces the following general form:
\[ [S] = \begin{bmatrix} 0 & 0 & a & 0 & c & c\\ 0 & 0 & -a & 0 & c & c\\ a & -a & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & a & -a\\ c & c & 0 & a & 0 & 0\\ c & c & 0 & -a & 0 & 0 \end{bmatrix}. \]
This matrix already contains the complete physical structure of the network. Only the numerical values of \(a\) and \(c\) remain unknown. The next step is to use the lossless property to determine those two coefficients.
Lossless Conditions
An ideal Magic Tee is lossless, meaning that it does not dissipate microwave power internally. Connecting the H-arms of two ideal lossless Magic Tees directly together does not introduce any intentional loss mechanism, so the resulting six-port network is also treated as lossless. For a lossless scattering network, the scattering matrix must be unitary. The mathematical condition is
\[ [S][S]^\dagger=[I]. \]
Here, \([S]^\dagger\) represents the conjugate transpose of the scattering matrix and \([I]\) represents the identity matrix. This condition contains two important requirements. First, every row must have unit magnitude, which represents conservation of total power for excitation at each port. Second, different rows must be orthogonal to one another, ensuring that the output wave patterns associated with independent input ports do not violate the power-conservation relationships of the network.
For determining the unknown magnitudes in the present problem, the row-normalization part of the unitary condition is sufficient. Once the values are obtained, the complete matrix can then be checked using the full unitary relationship. This provides a clean method for deriving the numerical coefficients without returning to the individual four-port matrices of the two Magic Tees.
Solving for the Unknown Coefficients
Determination of the E-Arm Coupling \(a\)
Consider the third row of the reduced matrix. Port 3 is the E-arm of Tee A, and according to the established E-arm difference-mode relationship, it couples only to Ports 1 and 2. The third row is therefore
\[ [S_{31}\;\;S_{32}\;\;S_{33}\;\;S_{34}\;\;S_{35}\;\;S_{36}] = [a\;\;-a\;\;0\;\;0\;\;0\;\;0]. \]
Because the network is lossless, the sum of the squared magnitudes of the coefficients in this row must equal unity:
\[ |S_{31}|^2+ |S_{32}|^2+ |S_{33}|^2+ |S_{34}|^2+ |S_{35}|^2+ |S_{36}|^2 =1. \]
Substituting the elements of Row 3 gives
\[ |a|^2+|-a|^2=1. \]
Therefore,
\[ 2|a|^2=1. \]
Hence,
\[ |a|^2=\frac12. \]
Taking the positive reference magnitude,
\[ a=\frac{1}{\sqrt2}. \]
Thus, the E-arm coupling coefficient has magnitude \(1/\sqrt2\). Its squared magnitude is \(1/2\), meaning that an excitation applied to an E-arm distributes equal power between its two associated collinear arms. The negative sign in one of the two coefficients is retained separately because it represents the \(180^\circ\) phase difference between those outputs rather than a difference in power magnitude.
Determination of the H-Arm Cross-Coupling \(c\)
The value of \(c\) can now be determined from the first row of the matrix. Port 1 has one coupling path to the E-arm of Tee A and two cross-coupling paths to the collinear arms of Tee B through the internal H-arm connection. Therefore, the first row is
\[ [S_{11}\;\;S_{12}\;\;S_{13}\;\;S_{14}\;\;S_{15}\;\;S_{16}] = [0\;\;0\;\;a\;\;0\;\;c\;\;c]. \]
Applying the lossless row-normalization condition gives
\[ |S_{11}|^2+ |S_{12}|^2+ |S_{13}|^2+ |S_{14}|^2+ |S_{15}|^2+ |S_{16}|^2 =1. \]
Substituting the known elements,
\[ |a|^2+|c|^2+|c|^2=1. \]
Therefore,
\[ a^2+2c^2=1. \]
Since
\[ a=\frac{1}{\sqrt2}, \]
we obtain
\[ \frac12+2c^2=1. \]
Rearranging,
\[ 2c^2=\frac12. \]
Therefore,
\[ c^2=\frac14 \]
and hence
\[ c=\frac12. \]
The cross-coupling coefficient is therefore \(1/2\). Unlike the E-arm coefficient, which has magnitude \(1/\sqrt2\), the H-arm connection produces a smaller individual coupling coefficient because the power reaching the second Tee is subsequently distributed between its two collinear arms. The equal positive signs of the two \(c\) terms indicate that these cross-coupled signals have the same phase, which is the expected sum-mode behaviour associated with an H-arm.
Final \(6\times6\) Scattering Matrix
The unknown coefficients have now been determined from the lossless condition. The E-arm coupling magnitude is
\[ a=\frac{1}{\sqrt2} \]
while the cross-coupling magnitude through the connected H-arms is
\[ c=\frac12. \]
Substituting these values into the reduced matrix produces the final scattering matrix of the six-port network:
\[ [S]_{6\times6} = \begin{bmatrix} 0 & 0 & \frac{1}{\sqrt2} & 0 & \frac12 & \frac12\\ 0 & 0 & -\frac{1}{\sqrt2} & 0 & \frac12 & \frac12\\ \frac{1}{\sqrt2} & -\frac{1}{\sqrt2} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & \frac{1}{\sqrt2} & -\frac{1}{\sqrt2}\\ \frac12 & \frac12 & 0 & \frac{1}{\sqrt2} & 0 & 0\\ \frac12 & \frac12 & 0 & -\frac{1}{\sqrt2} & 0 & 0 \end{bmatrix}. \]
This matrix is the complete Method 1 result for the two Magic Tees connected through their H-arms using the specified six-port numbering. The coefficients \(1/\sqrt2\) and \(-1/\sqrt2\) describe the equal-amplitude, opposite-phase coupling between each E-arm and its corresponding collinear arms, while the \(1/2\) coefficients describe the equal-amplitude, in-phase coupling between the two sets of collinear arms through the internal H-arm connection. The zero elements represent the matching and isolation conditions imposed by the ideal Magic-Tee structure. The matrix is symmetric, satisfying reciprocity, and its coefficients are obtained consistently with the lossless power-conservation requirement. This completes the complete Method 1 derivation of the \(6\times6\) scattering matrix.
Method 2: Verification and Derivation Using S-Matrix Properties
The second method derives the scattering matrix of the six-port network directly from the fundamental properties of a passive, reciprocal, matched, and lossless microwave network. Unlike the first method, where the coefficients are obtained by following the signal through the connected H-arms, here we begin with a completely general \(6\times6\) scattering matrix. Every element is initially represented by its corresponding \(S_{ij}\) parameter. We then apply the properties of the connected Magic Tee configuration one at a time. Reciprocity establishes the relationship between \(S_{ij}\) and \(S_{ji}\), matching conditions determine the diagonal elements, the E-arm properties establish the equal-magnitude opposite-phase relationships, and the H-arm connection establishes the equal-phase cross-coupling relationships. Finally, the lossless condition \([S][S]^\dagger=[I]\) provides the mathematical equations required to determine the remaining unknown scattering parameters.
Start with the General \(6\times6\) Scattering Matrix
Since the network has six accessible external ports, there are six possible incident waves and six corresponding reflected or transmitted waves. Therefore, the scattering matrix contains \(6\times6=36\) scattering parameters. Before applying any property of the network, the most general scattering matrix is written by explicitly showing every \(S_{ij}\) term. This is important because the subsequent simplification should be performed from the physical properties of the network rather than assuming the final form of the matrix at the beginning.
The general six-port scattering matrix is
\[ [S] = \begin{bmatrix} S_{11} & S_{12} & S_{13} & S_{14} & S_{15} & S_{16}\\ S_{21} & S_{22} & S_{23} & S_{24} & S_{25} & S_{26}\\ S_{31} & S_{32} & S_{33} & S_{34} & S_{35} & S_{36}\\ S_{41} & S_{42} & S_{43} & S_{44} & S_{45} & S_{46}\\ S_{51} & S_{52} & S_{53} & S_{54} & S_{55} & S_{56}\\ S_{61} & S_{62} & S_{63} & S_{64} & S_{65} & S_{66} \end{bmatrix}. \]
At this stage, none of the scattering parameters have been assigned a numerical value. The purpose of the following steps is to progressively reduce this general matrix using the known properties of the H-arm-connected Magic Tee network.
Apply the Reciprocity Property
The first property applied is reciprocity. The complete six-port network is formed from passive reciprocal Magic Tees connected through their H-arms, so the resulting network is also reciprocal. For a reciprocal microwave network, the transmission from Port \(i\) to Port \(j\) is equal to the transmission from Port \(j\) to Port \(i\). Therefore, the scattering parameters satisfy
\[ S_{ij}=S_{ji}. \]
Applying this property to the complete \(6\times6\) matrix gives
\[ S_{12}=S_{21}, \qquad S_{13}=S_{31}, \qquad S_{14}=S_{41}, \qquad S_{15}=S_{51}, \qquad S_{16}=S_{61}, \]
\[ S_{23}=S_{32}, \qquad S_{24}=S_{42}, \qquad S_{25}=S_{52}, \qquad S_{26}=S_{62}, \]
\[ S_{34}=S_{43}, \qquad S_{35}=S_{53}, \qquad S_{36}=S_{63}, \qquad S_{45}=S_{54}, \qquad S_{46}=S_{64}, \qquad S_{56}=S_{65}. \]
Substituting these reciprocity relationships into the general matrix gives
\[ [S] = \begin{bmatrix} S_{11} & S_{12} & S_{13} & S_{14} & S_{15} & S_{16}\\ S_{12} & S_{22} & S_{23} & S_{24} & S_{25} & S_{26}\\ S_{13} & S_{23} & S_{33} & S_{34} & S_{35} & S_{36}\\ S_{14} & S_{24} & S_{34} & S_{44} & S_{45} & S_{46}\\ S_{15} & S_{25} & S_{35} & S_{45} & S_{55} & S_{56}\\ S_{16} & S_{26} & S_{36} & S_{46} & S_{56} & S_{66} \end{bmatrix}. \]
Apply the Matching Conditions
Next, consider the matching condition of the six external ports. In the ideal network, every external port is assumed to be perfectly matched to the characteristic impedance of the system. Therefore, when a wave is incident at any external port, there is no reflected wave returning to that same port. The corresponding reflection coefficients are consequently zero. Since the diagonal elements of the scattering matrix represent the reflection coefficients, the matching condition becomes
\[ S_{11}=S_{22}=S_{33}=S_{44}=S_{55}=S_{66}=0. \]
Substituting these matching conditions gives
\[ [S] = \begin{bmatrix} 0 & S_{12} & S_{13} & S_{14} & S_{15} & S_{16}\\ S_{12} & 0 & S_{23} & S_{24} & S_{25} & S_{26}\\ S_{13} & S_{23} & 0 & S_{34} & S_{35} & S_{36}\\ S_{14} & S_{24} & S_{34} & 0 & S_{45} & S_{46}\\ S_{15} & S_{25} & S_{35} & S_{45} & 0 & S_{56}\\ S_{16} & S_{26} & S_{36} & S_{46} & S_{56} & 0 \end{bmatrix}. \]
Apply the Collinear-Arm Isolation Conditions
The two collinear arms of each individual Magic Tee are isolated from one another. For Tee A, Ports 1 and 2 are the two collinear arms, so the transmission coefficient between these ports is zero. Similarly, Ports 5 and 6 are the two collinear arms of Tee B and are also isolated from one another. Therefore,
\[ S_{12}=S_{21}=0 \]
and
\[ S_{56}=S_{65}=0. \]
After applying these isolation conditions, the matrix becomes
\[ [S] = \begin{bmatrix} 0 & 0 & S_{13} & S_{14} & S_{15} & S_{16}\\ 0 & 0 & S_{23} & S_{24} & S_{25} & S_{26}\\ S_{13} & S_{23} & 0 & S_{34} & S_{35} & S_{36}\\ S_{14} & S_{24} & S_{34} & 0 & S_{45} & S_{46}\\ S_{15} & S_{25} & S_{35} & S_{45} & 0 & 0\\ S_{16} & S_{26} & S_{36} & S_{46} & 0 & 0 \end{bmatrix}. \]
Apply E-Arm and H-Arm Isolation Conditions
The E-arm of one Magic Tee remains isolated from the H-arm path of the other side. In the six-port arrangement, Port 3 is the E-arm of Tee A and Port 4 is the E-arm of Tee B. The direct coupling between these two E-arms is zero because the two E-arms are not physically connected to each other; the internal connection is between the H-arms. Therefore,
\[ S_{34}=S_{43}=0. \]
Furthermore, the E-arm of Tee A does not directly couple to the collinear arms belonging to Tee B through the E-arm path. Hence,
\[ S_{35}=S_{53}=0 \]
and
\[ S_{36}=S_{63}=0. \]
Similarly, the E-arm of Tee B does not directly couple to the collinear arms of Tee A through the E-arm path. Thus,
\[ S_{14}=S_{41}=0 \]
and
\[ S_{24}=S_{42}=0. \]
The scattering matrix is now reduced to the form
\[ [S] = \begin{bmatrix} 0 & 0 & S_{13} & 0 & S_{15} & S_{16}\\ 0 & 0 & S_{23} & 0 & S_{25} & S_{26}\\ S_{13} & S_{23} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & S_{45} & S_{46}\\ S_{15} & S_{25} & 0 & S_{45} & 0 & 0\\ S_{16} & S_{26} & 0 & S_{46} & 0 & 0 \end{bmatrix}. \]
Apply the E-Arm Difference-Port Property
The E-arm of an ideal Magic Tee produces equal-amplitude signals at its two collinear arms with a \(180^\circ\) phase difference. For Tee A, Port 3 is the E-arm and Ports 1 and 2 are its two collinear arms. Therefore, the two transmission coefficients associated with excitation at Port 3 must have equal magnitude and opposite sign. This gives
\[ S_{23}=-S_{13}. \]
Because the network is reciprocal, the corresponding reverse transmission coefficients satisfy
\[ S_{32}=S_{23} \]
and
\[ S_{31}=S_{13}. \]
Consequently,
\[ S_{32}=-S_{31}. \]
The same E-arm property applies to Tee B. Port 4 is the E-arm of Tee B, while Ports 5 and 6 are its collinear arms. Hence,
\[ S_{64}=-S_{54}. \]
Using reciprocity,
\[ S_{46}=S_{64} \]
and
\[ S_{45}=S_{54}. \]
Therefore,
\[ S_{46}=-S_{45}. \]
The matrix consequently becomes
\[ [S] = \begin{bmatrix} 0 & 0 & S_{13} & 0 & S_{15} & S_{16}\\ 0 & 0 & -S_{13} & 0 & S_{25} & S_{26}\\ S_{13} & -S_{13} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & S_{45} & -S_{45}\\ S_{15} & S_{25} & 0 & S_{45} & 0 & 0\\ S_{16} & S_{26} & 0 & -S_{45} & 0 & 0 \end{bmatrix}. \]
Apply the H-Arm Connection Property
The two H-arms are directly connected to each other, and this internal connection produces equal-phase coupling between the collinear arms of the two Magic Tees. Since an H-arm produces an in-phase response at the two collinear arms, an excitation entering the H-arm connection reaches the corresponding collinear ports with equal phase. Therefore, the cross-coupling coefficients from the collinear arms of Tee A to the collinear arms of Tee B have equal values.
For excitation at Port 1, the coupling through the internal H-arm connection reaches Ports 5 and 6 with the same phase, so
\[ S_{16}=S_{15}. \]
For excitation at Port 2, the same H-arm property gives
\[ S_{26}=S_{25}. \]
Because the two Magic Tees are identical and symmetrically connected, the cross-coupling magnitude seen from Port 1 and Port 2 is also the same. Therefore,
\[ S_{25}=S_{15}. \]
Combining these relationships gives
\[ S_{15}=S_{16}=S_{25}=S_{26}. \]
By reciprocity, the reverse transmission coefficients are equal to their corresponding forward coefficients:
\[ S_{51}=S_{15}, \qquad S_{61}=S_{16}, \qquad S_{52}=S_{25}, \qquad S_{62}=S_{26}. \]
Similarly, the E-arm coupling on the second Magic Tee follows the same magnitude relationship as the E-arm coupling on the first Magic Tee because the two tees are identical. Therefore,
\[ S_{45}=S_{13} \]
under the same reference-phase convention used for the two identical Magic Tees.
The scattering matrix can now be written entirely in terms of the original \(S_{ij}\) notation as
\[ [S] = \begin{bmatrix} 0 & 0 & S_{13} & 0 & S_{15} & S_{15}\\ 0 & 0 & -S_{13} & 0 & S_{15} & S_{15}\\ S_{13} & -S_{13} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & S_{13} & -S_{13}\\ S_{15} & S_{15} & 0 & S_{13} & 0 & 0\\ S_{15} & S_{15} & 0 & -S_{13} & 0 & 0 \end{bmatrix}. \]
At this point, no numerical value has been assigned to either \(S_{13}\) or \(S_{15}\). These parameters must now be determined from the lossless condition.
Apply the Lossless or Unitary Property
An ideal Magic Tee is lossless, and connecting two ideal Magic Tees through their H-arms does not introduce any dissipative element. Therefore, the complete six-port network is also lossless. For a lossless scattering network, the scattering matrix must satisfy the unitary condition
\[ [S][S]^\dagger=[I]. \]
Since the coefficients used here are represented using the same real reference-phase convention as the ideal Magic Tee, the conjugate operation does not change the real coefficients. Thus, for this derivation, the required multiplication can be written as
\[ [S][S]^T=[I]. \]
The identity matrix for the six-port network is
\[ [I] = \begin{bmatrix} 1&0&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1 \end{bmatrix}. \]
Therefore, every diagonal element of \([S][S]^T\) must be equal to \(1\), while every off-diagonal element must be equal to zero. The diagonal equations provide the normalization conditions, and the off-diagonal equations provide the orthogonality conditions required to determine the remaining scattering parameters.
Row 3 Normalization
Consider the third row of the reduced scattering matrix:
\[ \begin{bmatrix} S_{13} & -S_{13} & 0 & 0 & 0 & 0 \end{bmatrix}. \]
The third diagonal element of \([S][S]^T\) must be equal to \(1\). Therefore,
\[ |S_{13}|^2+|-S_{13}|^2=1. \]
Since
\[ |-S_{13}|^2=|S_{13}|^2, \]
we obtain
\[ 2|S_{13}|^2=1. \]
Hence,
\[ |S_{13}|^2=\frac12. \]
Therefore,
\[ |S_{13}|=\frac{1}{\sqrt2}. \]
Row 4 Normalization
Now consider the fourth row, which represents the E-arm of Tee B:
\[ \begin{bmatrix} 0 & 0 & 0 & 0 & S_{13} & -S_{13} \end{bmatrix}. \]
Its normalization condition is
\[ |S_{13}|^2+|-S_{13}|^2=1. \]
Thus,
\[ 2|S_{13}|^2=1, \]
giving
\[ |S_{13}|=\frac{1}{\sqrt2}. \]
This independently confirms the same E-arm coupling magnitude obtained from Row 3. The result is expected because the two Magic Tees are identical and the two E-arms have the same ideal coupling behaviour.
Row 1 Normalization
The first row of the reduced matrix is
\[ \begin{bmatrix} 0 & 0 & S_{13} & 0 & S_{15} & S_{15} \end{bmatrix}. \]
The first diagonal element of \([S][S]^T\) must equal \(1\). Therefore,
\[ |S_{13}|^2+|S_{15}|^2+|S_{15}|^2=1. \]
Hence,
\[ |S_{13}|^2+2|S_{15}|^2=1. \]
Since
\[ |S_{13}|^2=\frac12, \]
substitution gives
\[ \frac12+2|S_{15}|^2=1. \]
Therefore,
\[ 2|S_{15}|^2=\frac12 \]
and
\[ |S_{15}|^2=\frac14. \]
Thus,
\[ |S_{15}|=\frac12. \]
Row 2 Normalization
The second row contains the same coupling magnitudes as the first row, with the E-arm coefficient having the opposite sign:
\[ \begin{bmatrix} 0 & 0 & -S_{13} & 0 & S_{15} & S_{15} \end{bmatrix}. \]
Therefore, its normalization equation is
\[ |-S_{13}|^2+|S_{15}|^2+|S_{15}|^2=1. \]
This becomes
\[ |S_{13}|^2+2|S_{15}|^2=1. \]
Substituting the already obtained values,
\[ \frac12+2\left(\frac12\right)^2=1, \]
which gives
\[ \frac12+\frac12=1. \]
Therefore, the second row also satisfies the unit-power normalization condition.
Orthogonality of Row 1 and Row 2
The normalization equations determine the magnitudes of the coefficients, but the unitary condition also requires different rows of the scattering matrix to be orthogonal. Consider the inner product between Row 1 and Row 2. The corresponding off-diagonal element of \([S][S]^T\) must be zero.
Therefore,
\[ S_{13}(-S_{13})^* + S_{15}S_{15}^* + S_{15}S_{15}^* =0. \]
For the real reference-phase convention, this reduces to
\[ -|S_{13}|^2+2|S_{15}|^2=0. \]
Hence,
\[ |S_{13}|^2=2|S_{15}|^2. \]
Using
\[ |S_{13}|^2=\frac12 \]
gives
\[ \frac12=2|S_{15}|^2, \]
and therefore
\[ |S_{15}|^2=\frac14. \]
Thus,
\[ |S_{15}|=\frac12. \]
This orthogonality equation is especially important because it shows why the cross-coupling coefficient is one-half rather than \(1/\sqrt2\). The E-arm coupling and the H-arm-mediated cross-coupling must together satisfy the power and orthogonality requirements of the complete six-port network.
Other Row Normalization Equations
The same procedure can be applied to the remaining rows. For Row 5,
\[ \begin{bmatrix} S_{15} & S_{15} & 0 & S_{13} & 0 & 0 \end{bmatrix}, \]
the normalization condition is
\[ |S_{15}|^2+|S_{15}|^2+|S_{13}|^2=1. \]
Therefore,
\[ 2|S_{15}|^2+|S_{13}|^2=1. \]
For Row 6,
\[ \begin{bmatrix} S_{15} & S_{15} & 0 & -S_{13} & 0 & 0 \end{bmatrix}, \]
the normalization condition becomes
\[ |S_{15}|^2+|S_{15}|^2+|-S_{13}|^2=1. \]
Hence,
\[ 2|S_{15}|^2+|S_{13}|^2=1. \]
These equations are identical to the equations obtained from Rows 1 and 2, confirming that the two sides of the six-port network have the same power distribution.
Orthogonality of the Remaining Rows
The off-diagonal elements of the unitary matrix equation provide additional checks. For example, the inner product of Row 3 and Row 4 is zero because their nonzero entries occur in different column positions. Therefore,
\[ R_3R_4^T=0. \]
This is consistent with
\[ S_{34}=0, \]
which represents the isolation between the two E-arms.
Similarly, the orthogonality between the appropriate rows containing equal-phase H-arm coupling and opposite-phase E-arm coupling produces cancellation of the cross terms. For example, the inner product of Row 3 and Row 5 is
\[ S_{13}S_{15}^* + (-S_{13})S_{15}^* =0. \]
The two terms cancel because the E-arm coefficients have opposite signs while the H-arm-mediated coefficients have the same sign. Therefore,
\[ R_3R_5^T=0. \]
The same cancellation occurs for the other corresponding row combinations. This confirms that the equal in-phase H-arm coupling and the opposite-phase E-arm coupling are mutually compatible with the lossless condition.
Equations Obtained from the Unitary Condition
After multiplying the scattering matrix by its conjugate transpose and comparing the result with the identity matrix, the independent equations required to determine the unknown magnitudes can be summarized as
\[ 2|S_{13}|^2=1 \]
\[ |S_{13}|^2+2|S_{15}|^2=1 \]
and
\[ -|S_{13}|^2+2|S_{15}|^2=0. \]
From the first equation,
\[ |S_{13}|^2=\frac12. \]
Therefore,
\[ |S_{13}|=\frac{1}{\sqrt2}. \]
Substituting this result into the normalization equation gives
\[ \frac12+2|S_{15}|^2=1. \]
Hence,
\[ |S_{15}|^2=\frac14 \]
and therefore
\[ |S_{15}|=\frac12. \]
Determine the Signs and Phase Relationships
The magnitudes alone do not completely describe the scattering matrix. The signs of the coefficients are determined by the physical behaviour of the E-arm and H-arm. The E-arm is a difference port, so the signals delivered to its two collinear arms have equal magnitude and opposite phase. Consequently,
\[ S_{23}=-S_{13} \]
and
\[ S_{46}=-S_{45}. \]
In contrast, the H-arm connection produces equal-phase coupling between the collinear arms of the two tees. Therefore,
\[ S_{15}=S_{16}=S_{25}=S_{26}. \]
Using the usual reference-phase convention for an ideal Magic Tee, the positive coefficients can be written as
\[ S_{13}=S_{45}=\frac{1}{\sqrt2} \]
and
\[ S_{15}=S_{16}=S_{25}=S_{26}=\frac12. \]
The corresponding opposite-phase E-arm coefficients are therefore
\[ S_{23}=S_{46}=-\frac{1}{\sqrt2}. \]
Complete \(6\times6\) Scattering Matrix
We can now substitute all the determined values into the reduced scattering matrix. The complete scattering matrix of the six-port network formed by connecting the H-arms of two identical Magic Tees is
\[ [S]_{6\times6} = \begin{bmatrix} 0 & 0 & \frac{1}{\sqrt2} & 0 & \frac12 & \frac12\\ 0 & 0 & -\frac{1}{\sqrt2} & 0 & \frac12 & \frac12\\ \frac{1}{\sqrt2} & -\frac{1}{\sqrt2} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & \frac{1}{\sqrt2} & -\frac{1}{\sqrt2}\\ \frac12 & \frac12 & 0 & \frac{1}{\sqrt2} & 0 & 0\\ \frac12 & \frac12 & 0 & -\frac{1}{\sqrt2} & 0 & 0 \end{bmatrix}. \]
This matrix has been obtained without introducing any separate coefficient symbols. Every nonzero coefficient has been determined directly from the S-parameter relationships, the symmetry and isolation properties of the Magic Tee, and the unitary condition of the lossless six-port network. The value \(1/\sqrt2\) represents the equal-amplitude E-arm coupling to the two collinear arms, while the value \(1/2\) represents the cross-coupling between the two groups of collinear arms through the internally connected H-arms.
Final Verification of the Unitary Condition
The final matrix must satisfy the fundamental lossless-network relationship
\[ [S][S]^\dagger=[I]. \]
The normalization of an E-arm row is verified by
\[ \left|\frac{1}{\sqrt2}\right|^2 + \left|-\frac{1}{\sqrt2}\right|^2 = \frac12+\frac12 = 1. \]
A collinear-arm row is verified by
\[ \left|\frac{1}{\sqrt2}\right|^2 + \left|\frac12\right|^2 + \left|\frac12\right|^2 = \frac12+\frac14+\frac14 = 1. \]
The orthogonality between the first two rows is verified by
\[ \frac{1}{\sqrt2} \left(-\frac{1}{\sqrt2}\right) + \frac12\left(\frac12\right) + \frac12\left(\frac12\right) = -\frac12+\frac14+\frac14 = 0. \]
Therefore, the rows have unit magnitude and the required orthogonality relationships. The final scattering matrix consequently satisfies the lossless condition and is consistent with the reciprocity, matching, isolation, E-arm difference-port, and H-arm in-phase coupling properties of the network.
Method 2 Result
Method 2 has therefore established the complete six-port scattering matrix by starting from all thirty-six possible \(S_{ij}\) parameters and progressively applying the physical properties of the network. Reciprocity first reduced the matrix through \(S_{ij}=S_{ji}\), matching conditions set all six diagonal reflection coefficients to zero, and the isolation conditions removed the forbidden transmission paths. The E-arm property then established the opposite-phase relationships such as \(S_{23}=-S_{13}\) and \(S_{46}=-S_{45}\), while the connected H-arms established the equal-phase relationships among \(S_{15}\), \(S_{16}\), \(S_{25}\), and \(S_{26}\). Finally, the unitary condition provided the normalization and orthogonality equations that yielded the magnitudes \(1/\sqrt2\) and \(1/2\). Substituting these values gives the complete six-port S-matrix.
The final result is
\[ \boxed{ [S]_{6\times6} = \begin{bmatrix} 0 & 0 & \frac{1}{\sqrt2} & 0 & \frac12 & \frac12\\ 0 & 0 & -\frac{1}{\sqrt2} & 0 & \frac12 & \frac12\\ \frac{1}{\sqrt2} & -\frac{1}{\sqrt2} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & \frac{1}{\sqrt2} & -\frac{1}{\sqrt2}\\ \frac12 & \frac12 & 0 & \frac{1}{\sqrt2} & 0 & 0\\ \frac12 & \frac12 & 0 & -\frac{1}{\sqrt2} & 0 & 0 \end{bmatrix} } \]
Physical Interpretation of the H-Arm Connected Six-Port Network
After deriving the complete \(6\times6\) scattering matrix and verifying it using reciprocity, matching, isolation, power conservation, and the unitary condition, we can now focus on what this network actually does physically. In this configuration, the H-arm of Magic Tee A is connected directly to the H-arm of Magic Tee B, while the two E-arms remain externally accessible as Port 3 and Port 4. The remaining four ports are the two collinear arms of each Magic Tee. This connection creates a six-port microwave network in which signals can travel from one Magic Tee to the other through the common H-arm path. The most important feature is the phase relationship of this coupling. Unlike an E-arm connection, which introduces an opposite-phase relationship between the two collinear arms, the H-arm produces equal-amplitude signals with the same phase. Therefore, the H-arm-to-H-arm connection creates an in-phase coupling path between the two groups of collinear ports while preserving the difference-mode behavior of the two E-arms.
Signal Flow Interpretation
The easiest way to understand the completed six-port network is to follow the signal paths produced by each possible external excitation. The two collinear arms of Tee A are represented by Ports 1 and 2, while the E-arm of Tee A is Port 3. Similarly, the E-arm of Tee B is Port 4 and its two collinear arms are Ports 5 and 6. The H-arms of the two tees are internally connected and therefore do not appear as independent external ports. When a signal is applied to a collinear arm of Tee A, part of the signal is coupled to the E-arm of the same tee with the appropriate phase relationship, while another part travels through the connected H-arm path toward Tee B. At Tee B, the H-arm excitation divides equally between its two collinear arms and the resulting signals are in phase. The same process occurs in the reverse direction because the network is reciprocal. This gives the six-port structure a useful combination of difference-mode operation through the E-arms and sum-mode coupling through the connected H-arms.
Excitation at Port 1
Consider first an excitation applied only at Port 1, with all other external ports terminated in matched loads. Port 1 is a collinear arm of Magic Tee A, so the incident signal interacts with both the E-arm and H-arm of that tee. Because the ideal junction is matched, there is no reflected wave at Port 1. One component of the incident signal is coupled toward Port 3, the E-arm of Tee A, while another component enters the internal H-arm connection. The signal travelling through the H-arm reaches Magic Tee B and is then divided equally between Ports 5 and 6. Since the signal enters Tee B through its H-arm, the two resulting waves at Ports 5 and 6 have the same amplitude and the same phase. Therefore, an excitation at Port 1 produces a difference-type relationship with Port 3 and an in-phase coupling relationship with Ports 5 and 6. The power is distributed among the allowed output paths so that the total output power remains equal to the incident power in the ideal lossless network.
Using the final scattering matrix, the relevant relationships for excitation at Port 1 can be written in terms of the corresponding scattering coefficients as
\[ b_3=S_{31}a_1 \]
\[ b_5=S_{51}a_1 \]
\[ b_6=S_{61}a_1 \]
Since the H-arm coupling is in phase,
\[ S_{51}=S_{61} \]
Thus, Ports 5 and 6 receive equal-amplitude signals with zero phase difference. This is the characteristic signature of H-arm coupling.
Excitation at Port 2
Now consider an excitation applied only at Port 2. Port 2 is the second collinear arm of Tee A and therefore has a complementary phase relationship with Port 1 when observed from the E-arm. The signal coupled to Port 3 consequently has the opposite sign compared with the signal produced by excitation at Port 1. However, the component travelling through the H-arm connection reaches Tee B in the same in-phase manner as before. Therefore, Ports 5 and 6 again receive equal-amplitude, equal-phase signals. This distinction is important because the E-arm responds to the difference between the two collinear arms, whereas the connected H-arm responds to their sum. The same physical principle is responsible for the sum and difference behavior of the Hybrid Tee and remains visible even after two Magic Tees are combined into this six-port structure.
The E-arm relationship can be expressed as
\[ S_{32}=-S_{31} \]
while the H-arm cross-coupling maintains
\[ S_{52}=S_{62} \]
Hence, changing the excitation from Port 1 to Port 2 reverses the E-arm response while maintaining the in-phase nature of the signal delivered to the second Magic Tee through the H-arm path.
Excitation at Port 3
Port 3 is the E-arm of Magic Tee A. When a signal is applied to this port, the defining property of the E-arm becomes immediately visible. The incident signal divides equally between the two collinear arms of Tee A, Ports 1 and 2, but the two output waves have opposite phase. Because the E-arm is isolated from the H-arm, the signal does not travel through the internal H-arm connection to Magic Tee B. Consequently, Ports 4, 5, and 6 do not receive power from an ideal excitation at Port 3. This means that the H-arm connection does not destroy the isolation characteristic of the E-arm. Instead, the two functions remain separated: the E-arm produces a difference-mode signal within its own Magic Tee, while the H-arm provides the in-phase connection between the two tees.
For excitation at Port 3,
\[ b_1=S_{13}a_3 \]
\[ b_2=S_{23}a_3 \]
and the E-arm phase relationship gives
\[ S_{23}=-S_{13} \]
At the same time, the isolation conditions require
\[ S_{43}=S_{53}=S_{63}=0 \]
Therefore, Port 3 operates as a difference port for the first Magic Tee and remains isolated from the second Tee through the H-arm connection.
Excitation at Port 4
Port 4 performs the same function for Magic Tee B because it is the E-arm of the second tee. When a signal is applied at Port 4, it divides equally between Ports 5 and 6 with a \(180^\circ\) phase difference. The signal does not couple to Ports 1 and 2 because the E-arm is isolated from the H-arm connection. Therefore, the second Magic Tee retains its own difference-port behavior even though its H-arm is physically connected to the H-arm of another Magic Tee. This is an important result because it shows that connecting the H-arms does not convert the E-arms into ordinary coupled ports. Each E-arm continues to control the difference mode of its corresponding pair of collinear arms.
The output relationships are
\[ b_5=S_{54}a_4 \]
\[ b_6=S_{64}a_4 \]
with
\[ S_{64}=-S_{54} \]
Thus, Ports 5 and 6 receive equal-amplitude signals with opposite phase, while Ports 1, 2, and 3 remain isolated from this E-arm excitation.
Sum-Mode Behaviour of the Connected H-Arms
The most important physical feature introduced by connecting the two H-arms is the creation of an in-phase coupling path between the two Magic Tees. An H-arm responds to the sum mode of the two collinear arms. When equal signals with the same phase are applied to the collinear arms, their fields reinforce at the H-arm, allowing energy to propagate through the connected path. After reaching the H-arm of the second Magic Tee, the signal is divided equally between its two collinear arms with the same phase. This is fundamentally different from the E-arm, where equal-amplitude signals appear with opposite phase. The H-arm therefore provides a sum-mode path, while the E-arm provides a difference-mode path. In the six-port network, these two behaviors coexist, allowing the network to process both sum and difference components without requiring the internal H-arms to be externally accessible.
If equal in-phase signals are applied to Ports 1 and 2,
\[ a_1=a_2 \]
the difference component at the E-arm becomes zero because the two contributions cancel:
\[ b_3=S_{31}a_1+S_{32}a_2 \]
Since
\[ S_{32}=-S_{31} \]
we obtain
\[ b_3=S_{31}a_1-S_{31}a_2=0 \]
At the same time, the H-arm path adds the two contributions rather than cancelling them. Therefore, the H-arm connection responds strongly to the sum mode. This is the clearest way to understand why the connected H-arms produce in-phase cross-coupling between the two Magic Tees.
Why Does the H-Arm Connection Produce In-Phase Coupling?
The in-phase behavior is a direct consequence of the field distribution associated with an H-plane junction. The H-arm is positioned so that the fields generated by the two collinear arms combine with the same polarity at the H-plane junction. When the H-arm is excited, the energy is consequently divided between the two collinear arms without introducing the \(180^\circ\) phase reversal associated with the E-arm. When two H-arms are connected, this same phase relationship is carried through the internal connection from one Magic Tee to the other. As a result, the signal arriving at the second tee produces equal and in-phase outputs at its two collinear arms. Mathematically, this appears as equal scattering coefficients such as \(S_{51}=S_{61}\), while the corresponding E-arm coefficients have opposite signs.
Comparison with E-Arm-to-E-Arm Connection
The difference between connecting two E-arms and connecting two H-arms is mainly a difference in phase behavior and signal-mode coupling. In an E-arm-to-E-arm connection, the internal path carries the difference-mode behavior of the E-arms, so the cross-coupled signals at the second Magic Tee appear with the appropriate opposite-phase relationship. In the H-arm-to-H-arm configuration considered here, the internal path carries the sum-mode behavior, causing the cross-coupled signals at the second Magic Tee to appear in phase. This makes the two configurations useful for different microwave network functions. The E-arm connection naturally emphasizes difference-mode coupling, whereas the H-arm connection naturally emphasizes sum-mode coupling. The two arrangements therefore represent complementary ways of combining Hybrid Tee structures.
The comparison can be summarized by the fundamental phase relationships:
- The E-arm produces equal-amplitude outputs with a \(180^\circ\) phase difference.
- The H-arm produces equal-amplitude outputs with the same phase.
- An E-arm-to-E-arm connection therefore preserves difference-mode behavior through the connected path.
- An H-arm-to-H-arm connection provides in-phase sum-mode coupling between the two tees.
- The E-arms remain isolated from the H-arm connection in the ideal network.
Advantages of the H-Arm Connected Six-Port Network
One major advantage of this configuration is that it combines two Hybrid Tee structures into a single six-port network while retaining the useful sum and difference characteristics of the individual junctions. The H-arm connection provides a controlled in-phase coupling path between the two tees, while the E-arms remain available as independent difference ports. Because the ideal network is reciprocal and lossless, the scattering matrix provides a convenient way to predict signal transfer in either direction and to verify that no power is lost inside the ideal junction. The symmetry of the structure also makes the resulting equations easier to interpret because equivalent ports exhibit corresponding coupling relationships. From a microwave network-analysis perspective, this arrangement is especially useful because the physical behavior can be represented directly through the S-parameters rather than requiring a complete electromagnetic field solution every time the network is analyzed.
Limitations of the Ideal H-Arm Connection
The derived six-port matrix represents an idealized microwave network, so practical hardware will not behave exactly according to every zero or magnitude shown in the theoretical matrix. Real waveguide junctions have conductor loss, dielectric loss, finite isolation, manufacturing tolerances, impedance mismatch, and frequency-dependent phase characteristics. The physical connection between the two H-arms can also introduce discontinuities that slightly change the amplitude and phase of the transmitted waves. Consequently, practical S-parameters may contain small nonzero reflection and isolation terms even when the ideal analysis predicts zero. The ideal matrix should therefore be treated as the fundamental theoretical model, while measured or electromagnetic-simulation S-parameters should be used when designing an actual microwave component.
Practical Applications of H-Arm Connected Magic Tees
A pair of Magic Tees connected through their H-arms can be useful whenever a microwave system needs controlled sum-mode coupling between two Hybrid Tee structures while maintaining independent difference-mode ports. Such a configuration can be used as a building block in microwave signal combining and splitting networks, balanced microwave circuits, antenna feed structures, power-combining arrangements, and more complicated multi-port junction networks. The exact application depends on the required port configuration, operating frequency, waveguide geometry, and desired phase relationships. The key advantage is not simply the increased number of ports, but the ability to control how signals are combined, divided, isolated, and phase shifted. The H-arm connection is particularly valuable when the design requires signals reaching the second junction to remain equal in amplitude and in phase, which is the defining behavior of sum-mode coupling.
Frequently Asked Questions
What happens when the H-arm of one Magic Tee is connected to the H-arm of another?
Connecting the two H-arms creates an internal coupling path between the two Magic Tees. Signals travelling through this path are associated with H-plane sum-mode behavior, so the corresponding collinear-arm outputs at the second tee are equal in amplitude and in phase. The two H-arms therefore allow controlled in-phase coupling between the two groups of collinear ports.
Do the two E-arms remain isolated after connecting the H-arms?
Yes. In the ideal Magic Tee model, the E-arm and H-arm are isolated. Therefore, connecting the H-arms does not create a direct transmission path between the two E-arms. Port 3 remains the E-arm of Tee A and Port 4 remains the E-arm of Tee B, with the corresponding isolation terms remaining zero in the ideal six-port scattering matrix.
Why are the cross-coupled signals in phase?
The H-arm represents the sum mode of the two collinear arms. Its field distribution produces equal-amplitude, same-phase outputs. When two H-arms are connected, this phase characteristic is carried through the internal connection and appears at the collinear arms of the second Magic Tee.
How is this different from connecting two E-arms?
An E-arm represents difference-mode operation and produces equal-amplitude outputs with a \(180^\circ\) phase difference. An H-arm represents sum-mode operation and produces equal-amplitude outputs with the same phase. Therefore, E-arm-to-E-arm and H-arm-to-H-arm connections provide complementary phase and mode characteristics.
Is the H-arm connected six-port network lossless?
In the ideal theoretical model, yes. The complete scattering matrix satisfies the unitary condition
\[ [S][S]^\dagger=[I] \]
which means that the total output power equals the total incident power. A practical fabricated network will normally have some loss, so its measured S-matrix will not be perfectly unitary.
What is the main application of this configuration?
The main usefulness of the H-arm-connected configuration is controlled in-phase coupling between two Hybrid Tee structures. It can therefore serve as a building block for microwave power-combining, signal-distribution, balanced circuits, antenna feed networks, and other multi-port microwave systems where sum-mode and difference-mode signal processing are required.
Final Understanding of the H-Arm Connected Six-Port Network
The H-arm-to-H-arm connection produces a six-port microwave network in which the two Magic Tees communicate through an internal H-plane path while their E-arms remain externally accessible and isolated from that path. The most important result is the phase behavior: the H-arm connection produces equal-amplitude, in-phase coupling between the collinear arms of the two tees, whereas each E-arm continues to produce equal-amplitude, \(180^\circ\) out-of-phase signals at its own pair of collinear arms. This means the completed network simultaneously supports sum-mode coupling through the connected H-arms and difference-mode operation through the E-arms. The derived \(6\times6\) scattering matrix therefore provides more than a mathematical description; it directly explains how signals propagate through the combined structure, why certain ports are isolated, why cross-coupled signals have the same phase, and how the network can be used as a practical building block in microwave engineering.