Magic Tee (E-H Plane Tee) S-Matrix Derivation

Important Properties of an Ideal Magic Tee

The unique behavior of a Magic Tee arises from the simultaneous presence of an E-plane tee and an H-plane tee within the same waveguide junction. Because of this combination, the Magic Tee exhibits both sum and difference characteristics, making it one of the most important hybrid microwave junctions used in power division, power combining, mixers, duplexers, and microwave measurement systems.

1. Sum and Difference Operation

When two signals of equal magnitude and equal phase are applied simultaneously to the two collinear ports (Ports 1 and 2), the signals combine constructively at the H-arm and cancel at the E-arm.

Therefore, the output at the E-arm (Port 3) becomes zero, while the total combined output appears at the H-arm (Port 4).

This behavior can be expressed as

$ b_3 = 0 $

while the output at Port 4 becomes the sum of the two input signals.

Because Port 3 responds to the difference between the signals applied at Ports 1 and 2, it is called the Difference Port or E-arm. Similarly, Port 4 responds to the sum of the signals and is therefore called the Sum Port or H-arm.

2. Excitation of the E-Arm (Difference Port)

When a microwave signal is incident at the E-arm (Port 3), the power divides equally between the two collinear ports. However, the two resulting output waves are 180° out of phase with each other.

The E-arm is isolated from the H-arm, so no power appears at Port 4.

Therefore,

$ S_{13}=-S_{23} $

and

$ S_{43}=0 $

This property demonstrates that excitation of the E-arm produces equal-amplitude but opposite-phase signals at the two collinear ports.

3. Excitation of the H-Arm (Sum Port)

When a signal is applied at the H-arm (Port 4), the power again divides equally between the two collinear ports. In this case, however, the two output signals are in phase.

The H-arm is isolated from the E-arm, so no signal appears at Port 3.

For an ideal Magic Tee,

$ S_{14}=S_{24} $

$ S_{41}=S_{42} $

and

$ S_{34}=S_{43}=0 $

For the normalized ideal Magic Tee,

$ S_{14}=S_{24}=S_{41}=S_{42} = \frac{1}{\sqrt{2}} $

Thus, excitation of the H-arm produces equal-magnitude and equal-phase signals at the two collinear ports.

4. Isolation Between the E-Arm and H-Arm

One of the most important characteristics of a Magic Tee is the complete isolation between the E-arm and H-arm. A signal entering one side arm cannot directly couple to the other side arm.

Therefore,

$ S_{34}=S_{43}=0 $

This isolation is responsible for the excellent performance of the Magic Tee in duplexers, balanced mixers, and microwave bridge circuits.

5. Isolation Between the Two Collinear Ports

In an ideal Magic Tee, the two collinear ports are also isolated from each other. A signal incident at Port 1 does not directly appear at Port 2, and vice versa.

Consequently,

$ S_{12}=S_{21}=0 $

This isolation occurs because the signals propagating through the E-plane and H-plane sections reach the opposite collinear port with equal magnitude but opposite phase contributions, resulting in complete cancellation.

Ideal Magic Tee Properties

The defining characteristics of an ideal Magic Tee can therefore be summarized as follows:

• Port 3 (E-arm) is the Difference Port.
• Port 4 (H-arm) is the Sum Port.
• Excitation of the E-arm produces equal-magnitude, opposite-phase outputs at Ports 1 and 2.
• Excitation of the H-arm produces equal-magnitude, in-phase outputs at Ports 1 and 2.
• The E-arm and H-arm are completely isolated.
• The two collinear ports are completely isolated.
• The Magic Tee functions as both a power divider and a power combiner.
• The output phase relationship depends on whether excitation occurs at the E-arm or the H-arm.

Derivation of the Reduced S-Matrix

Starting from the general matrix, apply reciprocity first. The matrix becomes symmetric:

\[ [S]= \begin{bmatrix} S_{11} & S_{12} & S_{13} & S_{14}\\ S_{12} & S_{22} & S_{23} & S_{24}\\ S_{13} & S_{23} & S_{33} & S_{34}\\ S_{14} & S_{24} & S_{34} & S_{44} \end{bmatrix} \]

The H-arm phase relationship gives

\[ S_{23}=S_{13} \]

The E-arm phase relationship gives

\[ S_{24}=-S_{14} \]

The isolation between the two side arms gives

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

Finally, perfect matching of the H-arm and E-arm gives

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

Substituting these relationships into the symmetric S-matrix produces the reduced form

\[ \boxed{ [S]= \begin{bmatrix} S_{11} & S_{12} & S_{13} & S_{14}\\ S_{12} & S_{22} & S_{13} & -S_{14}\\ S_{13} & S_{13} & 0 & 0\\ S_{14} & -S_{14} & 0 & 0 \end{bmatrix} } \]

This is the reduced S-matrix of an ideal symmetric Magic Tee before the remaining unknown coefficients are determined. The original sixteen S-parameters have now been reduced to a small number of unknown quantities by using the physical properties of the junction.

The next step is to apply the lossless or unitary condition,

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

to determine the magnitudes and remaining relationships among \(S_{11}\), \(S_{12}\), \(S_{13}\), and \(S_{14}\). This ultimately leads to the complete ideal Magic Tee S-matrix.

Unitary and Lossless Derivation of the Magic Tee S-Matrix

The reduced S-matrix obtained in the previous section contains the remaining unknown coefficients of the ideal Magic Tee. These coefficients can be determined by applying the lossless condition. Since an ideal Magic Tee is assumed to have no internal power dissipation, the total incident power must equal the total outgoing power. In S-parameter theory, this requirement is expressed by the unitary condition

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

where \([S]^\dagger\) is the conjugate transpose of the scattering matrix and \([I]\) is the identity matrix. This condition ensures both power conservation and orthogonality of the rows and columns of the S-matrix.

Lossless Condition of the Magic Tee

For a lossless network, no incident microwave power is dissipated inside the junction. Therefore, if a wave is applied to any one port, the total power carried by all outgoing waves must equal the incident power. In terms of S-parameters, the squared magnitudes of the elements in each row or column must satisfy the appropriate power-conservation relationship.

For the reduced Magic Tee matrix

\[ [S]= \begin{bmatrix} S_{11} & S_{12} & S_{13} & S_{14}\\ S_{12} & S_{22} & S_{13} & -S_{14}\\ S_{13} & S_{13} & 0 & 0\\ S_{14} & -S_{14} & 0 & 0 \end{bmatrix} \]

the unitary condition requires the rows and columns to have unit norm and to be mutually orthogonal. We can therefore determine the unknown coefficients systematically rather than assuming their values.

Applying the Unitary Condition

The conjugate transpose of the reduced matrix is

\[ [S]^\dagger= \begin{bmatrix} S_{11}^{*} & S_{12}^{*} & S_{13}^{*} & S_{14}^{*}\\ S_{12}^{*} & S_{22}^{*} & S_{13}^{*} & -S_{14}^{*}\\ S_{13}^{*} & S_{13}^{*} & 0 & 0\\ S_{14}^{*} & -S_{14}^{*} & 0 & 0 \end{bmatrix} \]

Therefore,

\[ \boxed{ \begin{bmatrix} S_{11} & S_{12} & S_{13} & S_{14}\\ S_{12} & S_{22} & S_{13} & -S_{14}\\ S_{13} & S_{13} & 0 & 0\\ S_{14} & -S_{14} & 0 & 0 \end{bmatrix} \begin{bmatrix} S_{11}^{*} & S_{12}^{*} & S_{13}^{*} & S_{14}^{*}\\ S_{12}^{*} & S_{22}^{*} & S_{13}^{*} & -S_{14}^{*}\\ S_{13}^{*} & S_{13}^{*} & 0 & 0\\ S_{14}^{*} & -S_{14}^{*} & 0 & 0 \end{bmatrix} =[I] } \]

Diagonal Equations from Power Conservation

The diagonal elements of \([S][S]^\dagger\) must each be equal to one. The first diagonal element is obtained from the inner product of the first row with itself:

\[ S_{11}S_{11}^{*} +S_{12}S_{12}^{*} +S_{13}S_{13}^{*} +S_{14}S_{14}^{*}=1 \]

Therefore, the correct first row power equation is

\[ \boxed{ |S_{11}|^2+|S_{12}|^2+|S_{13}|^2+|S_{14}|^2=1 } \]

Notice that both \( |S_{13}|^2 \) and \( |S_{14}|^2 \) must appear. The H-arm and E-arm both contribute to the total outgoing power from Port 1.

Similarly, the second row gives

\[ S_{12}S_{12}^{*} +S_{22}S_{22}^{*} +S_{13}S_{13}^{*} +(-S_{14})(-S_{14}^{*})=1 \]

and hence

\[ \boxed{ |S_{12}|^2+|S_{22}|^2+|S_{13}|^2+|S_{14}|^2=1 } \]

The third row contains only the two H-arm coupling coefficients. Therefore, its power equation is

\[ |S_{13}|^2+|S_{13}|^2=1 \]

which gives

\[ 2|S_{13}|^2=1 \]

Therefore, the magnitude of the H-arm coupling coefficient is

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

The fourth row gives an equivalent equation for the E-arm:

\[ |S_{14}|^2+|-S_{14}|^2=1 \]

Since the magnitude of \(-S_{14}\) is the same as the magnitude of \(S_{14}\),

\[ 2|S_{14}|^2=1 \]

and therefore

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

Deriving the Collinear-Port Coefficients

We now substitute the two coupling magnitudes into the first diagonal equation:

\[ |S_{11}|^2+|S_{12}|^2 +\frac{1}{2} +\frac{1}{2} =1 \]

Therefore,

\[ \boxed{|S_{11}|^2+|S_{12}|^2=0} \]

Since the squared magnitudes of complex quantities are nonnegative, the only possible solution is

\[ \boxed{S_{11}=0} \]

and

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

The second diagonal equation similarly gives

\[ |S_{12}|^2+|S_{22}|^2 +\frac{1}{2} +\frac{1}{2} =1 \]

so that

\[ |S_{22}|^2=0 \]

and therefore

\[ \boxed{S_{22}=0} \]

Thus, the two collinear ports are also perfectly matched in the ideal Magic Tee:

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

This result is important because it shows that the ideal Magic Tee is a four-port junction in which all four ports can be perfectly matched under the ideal symmetric conditions.

Determining the Required Signs

The magnitudes obtained from power conservation do not by themselves determine the phase signs of the coupling coefficients. The relative signs must be consistent with the physical phase behavior of the E-arm and H-arm.

For H-arm excitation, the outputs at Ports 1 and 2 must be equal and in phase. Therefore,

\[ \boxed{S_{13}=S_{23}} \]

For E-arm excitation, the outputs at Ports 1 and 2 must be equal in magnitude but opposite in phase. Therefore,

\[ \boxed{S_{24}=-S_{14}} \]

Using the conventional reference-phase choice, the coupling coefficients may be selected as

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

and

\[ \boxed{ S_{14}=\frac{1}{\sqrt{2}}, \qquad S_{24}=-\frac{1}{\sqrt{2}} } \]

Reciprocity then gives

\[ S_{31}=S_{13}=\frac{1}{\sqrt{2}} \]

\[ S_{32}=S_{23}=\frac{1}{\sqrt{2}} \]

\[ S_{41}=S_{14}=\frac{1}{\sqrt{2}} \]

\[ S_{42}=S_{24}=-\frac{1}{\sqrt{2}} \]

The remaining coefficients are zero:

\[ \boxed{ S_{11}=S_{12}=S_{21}=S_{22}=S_{33}=S_{34}=S_{43}=S_{44}=0 } \]

Complete Ideal Magic Tee S-Matrix

Substituting all the derived coefficients into the reduced matrix gives the complete ideal Magic Tee S-matrix:

\[ \boxed{ [S]= \begin{bmatrix} 0 & 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\\[4pt] 0 & 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}\\[4pt] \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 & 0\\[4pt] \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0 & 0 \end{bmatrix} } \]

This matrix clearly shows the defining properties of the ideal Magic Tee. The H-arm, represented by Port 3, couples equally and in phase to Ports 1 and 2. The E-arm, represented by Port 4, couples equally but with opposite phase to the two collinear ports. The H-arm and E-arm are isolated, as indicated by \(S_{34}=S_{43}=0\), while the two collinear ports are also isolated, as indicated by \(S_{12}=S_{21}=0\).

Verification of the Resulting Matrix

The derived matrix should satisfy reciprocity, power conservation, and the required phase and isolation properties. First, the matrix is symmetric:

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

Therefore, the network is reciprocal.

Next, consider the norm of the first row:

\[ \left|0\right|^2+ \left|0\right|^2+ \left|\frac{1}{\sqrt{2}}\right|^2+ \left|\frac{1}{\sqrt{2}}\right|^2 = 0+0+\frac{1}{2}+\frac{1}{2}=1 \]

The same result is obtained for every row, confirming power conservation.

The orthogonality of the first two rows provides an especially useful check:

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

Thus, the two rows are orthogonal, as required by the unitary condition. Similarly, the third and fourth rows are orthogonal because

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

Therefore, the complete matrix satisfies

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

The derived result is therefore consistent with the ideal assumptions of a reciprocal, lossless, symmetric, perfectly matched Magic Tee. The most important mathematical result is that each side arm couples one-half of its incident power to each collinear port, while the relative sign determines whether the resulting collinear waves are in phase or opposite in phase.

Complete Ideal Magic Tee S-Matrix

The previous sections established the physical properties and the unitary conditions of the ideal Magic Tee. We can now assemble these results into the complete four-port scattering matrix and verify that every matrix element satisfies the required physical and mathematical conditions. The standard port arrangement used here is Port 1 and Port 2 as the collinear ports, Port 3 as the H-arm or sum port, and Port 4 as the E-arm or difference port.

Complete Ideal 4 × 4 S-Matrix

From the previous derivation, the important relationships are

\[ S_{11}=S_{12}=S_{21}=S_{22}=0 \]

\[ S_{33}=S_{34}=S_{43}=S_{44}=0 \]

\[ S_{13}=S_{23}=S_{31}=S_{32}=\frac{1}{\sqrt{2}} \]

and

\[ S_{14}=S_{41}=\frac{1}{\sqrt{2}}, \qquad S_{24}=S_{42}=-\frac{1}{\sqrt{2}}. \]

Substituting these values into the general four-port S-matrix gives the complete ideal Magic Tee matrix:

\[ \boxed{ [S]= \begin{bmatrix} 0 & 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\\[4pt] 0 & 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}\\[4pt] \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 & 0\\[4pt] \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0 & 0 \end{bmatrix} } \]

This matrix completely describes the ideal scattering behavior of the Magic Tee under the assumed reference-plane and phase conventions.

Derivation of Every Matrix Element

The diagonal elements \(S_{11}\) and \(S_{22}\) describe reflection at the two collinear ports. For an ideal Magic Tee, the collinear ports are perfectly matched. Therefore, there is no reflected wave when either collinear port is individually excited:

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

The coefficients \(S_{12}\) and \(S_{21}\) describe direct transmission between the two collinear ports. The ideal Magic Tee isolates these two ports, so an excitation at either collinear port does not produce an output at the other collinear port:

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

The coefficients \(S_{13}\) and \(S_{23}\) describe coupling from the H-arm to Ports 1 and 2. An excitation at the H-arm divides equally between the two collinear ports and produces outputs with the same phase. Therefore,

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

By reciprocity,

\[ \boxed{ S_{31}=S_{32}=\frac{1}{\sqrt{2}} } \]

The coefficients \(S_{14}\) and \(S_{24}\) describe coupling from the E-arm to the two collinear ports. An excitation at the E-arm produces equal-magnitude but opposite-phase outputs at Ports 1 and 2. Thus,

\[ \boxed{ S_{14}=\frac{1}{\sqrt{2}}, \qquad S_{24}=-\frac{1}{\sqrt{2}} } \]

Reciprocity gives

\[ \boxed{ S_{41}=\frac{1}{\sqrt{2}}, \qquad S_{42}=-\frac{1}{\sqrt{2}} } \]

Finally, Ports 3 and 4 are isolated from one another, and both side arms are matched. Therefore,

\[ \boxed{ S_{33}=S_{34}=S_{43}=S_{44}=0 } \]

These individual results account for all sixteen elements of the four-port S-matrix.

Physical Meaning of Every Row

Each row of an S-matrix describes the outgoing wave at one particular port produced by incident waves at all four ports. Therefore, the rows of the Magic Tee matrix can be interpreted directly in terms of the physical junction.

First row:

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

This row describes the outgoing wave \(b_1\) at Port 1. There is no reflection from Port 1 because \(S_{11}=0\), and there is no direct transmission from Port 2 because \(S_{12}=0\). The H-arm produces a positive \(1/\sqrt{2}\) contribution, while the E-arm produces another contribution of \(1/\sqrt{2}\) under the selected reference convention.

Second row:

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

This row describes the outgoing wave \(b_2\) at Port 2. The H-arm contribution has the same sign as the contribution at Port 1, whereas the E-arm contribution has the opposite sign. This is the mathematical representation of the sum and difference behavior of the Magic Tee.

Third row:

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

This row describes the outgoing wave \(b_3\) at the H-arm. An excitation at either collinear port couples to the H-arm with equal magnitude and the same phase. There is no reflection at the H-arm and no coupling from the E-arm to the H-arm.

Fourth row:

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

This row describes the outgoing wave \(b_4\) at the E-arm. Excitation at Ports 1 and 2 produces equal-magnitude contributions with opposite signs. This is the difference-port behavior of the Magic Tee.

Physical Meaning of Every Column

Each column describes the response of the entire network when only the corresponding port is excited while all other ports are terminated in their characteristic reference impedances.

Column 1: Port 1 excitation

\[ \boxed{ \begin{bmatrix} 0\\ 0\\ \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}} \end{bmatrix} } \]

An excitation at Port 1 produces no reflection at Port 1 and no output at Port 2. The incident signal divides between the H-arm and E-arm with equal power.

Column 2: Port 2 excitation

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

An excitation at Port 2 similarly produces no reflection at Port 2 and no output at Port 1. Equal power is delivered to the H-arm and E-arm, but their relative phase is represented by the opposite signs in the column.

Column 3: H-arm excitation

\[ \boxed{ \begin{bmatrix} \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}\\ 0\\ 0 \end{bmatrix} } \]

The H-arm signal divides equally between Ports 1 and 2. Since the two coefficients have the same sign, the two collinear-port waves are in phase.

Column 4: E-arm excitation

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

The E-arm signal also divides equally between Ports 1 and 2, but the two outputs have opposite phase because the corresponding coefficients have opposite signs.

Verification of Reciprocity

A reciprocal network satisfies

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

The derived matrix is symmetric about its main diagonal. For example,

\[ S_{13}=S_{31}=\frac{1}{\sqrt{2}} \]

\[ S_{23}=S_{32}=\frac{1}{\sqrt{2}} \]

\[ S_{14}=S_{41}=\frac{1}{\sqrt{2}} \]

and

\[ S_{24}=S_{42}=-\frac{1}{\sqrt{2}}. \]

The zero elements also occur in reciprocal pairs. Therefore,

\[ \boxed{[S]^T=[S]} \]

and the ideal Magic Tee is reciprocal.

Verification of Matching

A port is perfectly matched when its reflection coefficient is zero. The reflection coefficients are the diagonal elements of the S-matrix. From the complete matrix,

\[ \boxed{ S_{11}=S_{22}=S_{33}=S_{44}=0 } \]

Therefore, all four ports are perfectly matched in the ideal Magic Tee.

Verification of Isolation

The H-arm and E-arm are isolated from one another. This is represented by

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

Thus, an ideal signal applied to the H-arm does not directly appear at the E-arm, and an ideal signal applied to the E-arm does not directly appear at the H-arm.

The two collinear ports are also isolated:

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

Therefore, direct transmission between the two collinear ports is zero under the ideal Magic Tee conditions.

Verification of Power Conservation

For a lossless network, the sum of the squared magnitudes of the elements in each row must equal unity.

For the first row,

\[ \left|0\right|^2+ \left|0\right|^2+ \left|\frac{1}{\sqrt{2}}\right|^2+ \left|\frac{1}{\sqrt{2}}\right|^2 = \frac12+\frac12=1 \]

For the second row,

\[ \left|0\right|^2+ \left|0\right|^2+ \left|\frac{1}{\sqrt{2}}\right|^2+ \left|-\frac{1}{\sqrt{2}}\right|^2 =1 \]

The third and fourth rows similarly satisfy

\[ \frac12+\frac12=1. \]

Thus, every row has unit power norm, which is consistent with a lossless junction.

Verification of Orthogonality

Power conservation alone is not sufficient to establish the complete unitary property. Different rows must also be orthogonal. Consider the first and second rows:

\[ \begin{aligned} &\left(0\right)\left(0\right) +\left(0\right)\left(0\right) +\left(\frac{1}{\sqrt{2}}\right) \left(\frac{1}{\sqrt{2}}\right) +\left(\frac{1}{\sqrt{2}}\right) \left(-\frac{1}{\sqrt{2}}\right)\\ &=\frac12-\frac12=0 \end{aligned} \]

Therefore, the first and second rows are orthogonal.

The third and fourth rows give

\[ \left(\frac{1}{\sqrt{2}}\right) \left(\frac{1}{\sqrt{2}}\right) + \left(\frac{1}{\sqrt{2}}\right) \left(-\frac{1}{\sqrt{2}}\right) = \frac12-\frac12=0. \]

Thus, the third and fourth rows are also orthogonal. The remaining row pairs are orthogonal because their nonzero elements occur in separate positions or cancel in equal and opposite terms.

Consequently, the complete matrix satisfies the required unitary condition:

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

The final matrix therefore satisfies all the defining ideal properties of the Magic Tee: reciprocity, perfect matching at all four ports, isolation between the E-arm and H-arm, isolation between the two collinear ports, equal power division, the sum and difference phase relationships, power conservation, and row and column orthogonality.

Physical Meaning of Every Matrix Element

The ideal Magic Tee is a four-port microwave junction, so its scattering matrix contains sixteen S-parameters. Each element has a specific physical meaning based on its row and column position. The first subscript identifies the port where the outgoing wave is observed, while the second subscript identifies the port where the incident wave is applied. Therefore, \(S_{ij}\) represents the ratio of the outgoing wave at Port \(i\) to the incident wave at Port \(j\), with all other ports terminated in their characteristic reference impedances.

For the standard Magic Tee numbering used in this article, Ports 1 and 2 are the two collinear ports, Port 3 is the H-arm or sum port, and Port 4 is the E-arm or difference port. The ideal S-matrix is

\[ \boxed{ [S]= \begin{bmatrix} 0 & 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\\[4pt] 0 & 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}\\[4pt] \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 & 0\\[4pt] \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0 & 0 \end{bmatrix} } \]

Note:

\(S_{11}\) and \(S_{22}\): Reflection at the Collinear Ports

The parameters \(S_{11}\) and \(S_{22}\) are the reflection coefficients at the two collinear ports. \(S_{11}\) describes the wave reflected back toward Port 1 when Port 1 is excited, while \(S_{22}\) describes the wave reflected back toward Port 2 when Port 2 is excited.

For the ideal Magic Tee, both collinear ports are perfectly matched. Therefore, no reflected wave appears at the excited collinear port:

\[ \boxed{S_{11}=0} \]

and

\[ \boxed{S_{22}=0} \]

Thus, an incident wave at either collinear port does not return to the same port under the ideal matched condition.

\(S_{33}\) and \(S_{44}\): Reflection at the H-arm and E-arm

The parameters \(S_{33}\) and \(S_{44}\) represent the reflection coefficients at the two side arms. \(S_{33}\) describes reflection at the H-arm when Port 3 is excited, while \(S_{44}\) describes reflection at the E-arm when Port 4 is excited.

Both side arms are perfectly matched in the ideal Magic Tee. Hence,

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

and

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

This means that an incident wave at either side arm is completely transferred to the appropriate output ports without reflection back toward the excited side arm.

\(S_{12}\) and \(S_{21}\): Isolation Between the Collinear Ports

The parameter \(S_{12}\) describes the wave emerging from Port 1 when Port 2 is excited. Conversely, \(S_{21}\) describes the wave emerging from Port 2 when Port 1 is excited.

For the ideal Magic Tee, the two collinear ports are isolated from each other. Therefore,

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

Thus, an ideal signal applied to Port 1 does not directly appear at Port 2, and an ideal signal applied to Port 2 does not directly appear at Port 1. This isolation is one of the important characteristics that allows the Magic Tee to perform independent sum and difference operations.

\(S_{13}\) and \(S_{23}\): H-arm to Collinear-Port Coupling

The parameters \(S_{13}\) and \(S_{23}\) describe the waves appearing at Ports 1 and 2 when the H-arm, Port 3, is excited.

From the ideal S-matrix,

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

The equal magnitudes indicate equal power division between the two collinear ports. Since the two coefficients have the same sign under the selected reference convention, the resulting waves at Ports 1 and 2 are in phase.

The power delivered to each collinear port is therefore

\[ |S_{13}|^2=|S_{23}|^2=\frac{1}{2}. \]

Thus, excitation of the H-arm produces equal-magnitude, equal-phase outputs at the two collinear ports.

\(S_{14}\) and \(S_{24}\): E-arm to Collinear-Port Coupling

The parameters \(S_{14}\) and \(S_{24}\) describe the waves appearing at Ports 1 and 2 when the E-arm, Port 4, is excited.

The ideal matrix gives

\[ \boxed{ S_{14}=\frac{1}{\sqrt{2}}, \qquad S_{24}=-\frac{1}{\sqrt{2}} } \]

The magnitudes are equal, so the incident power is divided equally between Ports 1 and 2. However, the negative sign indicates a \(180^\circ\) phase difference between the two output waves.

Therefore,

\[ |S_{14}|^2=|S_{24}|^2=\frac{1}{2}. \]

The E-arm consequently produces equal-magnitude, opposite-phase waves at the two collinear ports.

\(S_{31}\) and \(S_{32}\): Collinear Ports to H-arm Coupling

The parameters \(S_{31}\) and \(S_{32}\) describe the waves emerging at the H-arm when Ports 1 and 2 are excited, respectively.

For the ideal reciprocal Magic Tee,

\[ \boxed{ S_{31}=S_{13}=\frac{1}{\sqrt{2}} } \]

and

\[ \boxed{ S_{32}=S_{23}=\frac{1}{\sqrt{2}} } \]

Therefore, an excitation at either collinear port couples equally to the H-arm. The equal signs show that the H-arm responds equally to the two collinear ports when they are excited individually under the selected reference convention.

This property is the basis of the sum-port operation. When equal-amplitude, in-phase signals are applied to Ports 1 and 2, their contributions at the H-arm add constructively.

\(S_{41}\) and \(S_{42}\): Collinear Ports to E-arm Coupling

The parameters \(S_{41}\) and \(S_{42}\) describe the waves emerging at the E-arm when Ports 1 and 2 are excited.

From the ideal matrix,

\[ \boxed{ S_{41}=\frac{1}{\sqrt{2}} } \]

and

\[ \boxed{ S_{42}=-\frac{1}{\sqrt{2}} } \]

The equal magnitudes and opposite signs mean that the E-arm responds to the difference between the signals at the two collinear ports.

Consequently, equal-amplitude, in-phase signals applied to Ports 1 and 2 cancel at the E-arm, while equal-amplitude signals with opposite phase reinforce at the E-arm. This is why Port 4 is called the difference port.

\(S_{34}\) and \(S_{43}\): Isolation Between H-arm and E-arm

The parameter \(S_{34}\) describes the wave emerging at the H-arm when the E-arm is excited. Conversely, \(S_{43}\) describes the wave emerging at the E-arm when the H-arm is excited.

For the ideal Magic Tee, the two side arms are isolated:

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

Therefore, an excitation applied directly to the H-arm does not appear at the E-arm, and an excitation applied directly to the E-arm does not appear at the H-arm.

Complete Physical Interpretation

The sixteen elements of the ideal Magic Tee matrix can now be grouped according to their physical functions. The four diagonal elements are zero because all four ports are matched. The two coefficients \(S_{12}\) and \(S_{21}\) are zero because the collinear ports are isolated. The two coefficients \(S_{34}\) and \(S_{43}\) are zero because the E-arm and H-arm are isolated.

The four coefficients associated with H-arm coupling are

\[ \boxed{ S_{13}=S_{23}=S_{31}=S_{32}=\frac{1}{\sqrt{2}} } \]

These represent equal-magnitude, equal-phase coupling between the H-arm and the two collinear ports.

The four coefficients associated with E-arm coupling are

\[ \boxed{ S_{14}=S_{41}=\frac{1}{\sqrt{2}}, \qquad S_{24}=S_{42}=-\frac{1}{\sqrt{2}} } \]

These represent equal-magnitude, opposite-phase coupling between the E-arm and the two collinear ports.

Therefore, the physical behavior of the ideal Magic Tee can be summarized by two fundamental relationships:

\[ \boxed{ \text{H-arm: equal magnitude and equal phase} } \]

and

\[ \boxed{ \text{E-arm: equal magnitude and opposite phase} } \]

These relationships explain why the H-arm functions as the sum port and the E-arm functions as the difference port. They also provide a practical method for interpreting an unknown Magic Tee S-matrix without memorizing every individual matrix element.

Sum and Difference Operation of a Magic Tee

The most important practical function of a Magic Tee is its ability to perform sum and difference operations. Ports 1 and 2 are the two collinear ports, Port 3 is the H-arm or sum port, and Port 4 is the E-arm or difference port. The ideal Magic Tee combines the characteristics of an H-plane tee and an E-plane tee so that signals applied to the two collinear ports can be combined according to their relative amplitude and phase.

Using the standard ideal S-matrix established above, the relationship between the incident waves and outgoing waves is

\[ \begin{bmatrix} b_1\\ b_2\\ b_3\\ b_4 \end{bmatrix} = \begin{bmatrix} 0 & 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\\[4pt] 0 & 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}\\[4pt] \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 & 0\\[4pt] \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0 & 0 \end{bmatrix} \begin{bmatrix} a_1\\ a_2\\ a_3\\ a_4 \end{bmatrix}. \]

For the sum and difference operation, the two side-arm inputs are normally absent, so \(a_3=a_4=0\). The two relevant output equations then reduce to

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

and

\[ \boxed{ b_4=\frac{a_1-a_2}{\sqrt{2}} } \]

These two equations provide the mathematical basis of the sum and difference behavior of the Magic Tee.

Equal In-Phase Inputs at Ports 1 and 2

Consider two signals of equal magnitude and identical phase applied simultaneously to Ports 1 and 2. Let

\[ a_1=a_2=A. \]

Substituting these values into the output equations gives

\[ b_3=\frac{A+A}{\sqrt{2}} =\sqrt{2}A \]

and

\[ b_4=\frac{A-A}{\sqrt{2}}=0. \]

Therefore, equal in-phase signals applied to the two collinear ports combine constructively at the H-arm and cancel at the E-arm.

Hence,

\[ \boxed{b_3=\sqrt{2}A,\qquad b_4=0} \]

The H-arm therefore acts as the sum port, while the E-arm is isolated from this particular excitation.

Equal Opposite-Phase Inputs at Ports 1 and 2

Now consider two equal-magnitude signals that are \(180^\circ\) out of phase. Under the chosen reference convention, this can be represented as

\[ a_1=A,\qquad a_2=-A. \]

The H-arm output becomes

\[ b_3= \frac{A+(-A)}{\sqrt{2}} =0. \]

The E-arm output becomes

\[ b_4= \frac{A-(-A)}{\sqrt{2}} =\sqrt{2}A. \]

Therefore, equal-amplitude, opposite-phase signals applied to Ports 1 and 2 combine constructively at the E-arm and cancel at the H-arm.

Thus,

\[ \boxed{b_3=0,\qquad b_4=\sqrt{2}A} \]

The E-arm therefore acts as the difference port.

Output at the H-arm

The H-arm output is determined by the sum of the two collinear-port incident waves:

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

If the two inputs have the same phase, their contributions add at Port 3. If they have opposite phase and equal magnitude, their contributions cancel. Therefore, the H-arm responds to the sum component of the two input signals.

For general unequal inputs, the H-arm does not necessarily become the only output port. Instead, it receives the normalized sum component while the E-arm receives the normalized difference component.

Output at the E-arm

The E-arm output is determined by the difference between the two collinear-port incident waves:

\[ \boxed{ b_4=\frac{a_1-a_2}{\sqrt{2}} } \]

If the two inputs are equal and in phase, their contributions cancel at Port 4. If they are equal in magnitude and opposite in phase, their contributions add at Port 4. Therefore, the E-arm responds to the difference component of the two input signals.

Why the Other Side Arm Becomes Isolated

The isolation of one side arm is a direct consequence of constructive and destructive interference. The two signals arriving at a side arm have equal coupling magnitude but their relative signs determine whether they add or cancel.

For equal in-phase inputs,

\[ a_1=a_2. \]

Therefore,

\[ b_4=\frac{a_1-a_2}{\sqrt{2}}=0. \]

The E-arm is isolated because the two contributions cancel exactly.

For equal opposite-phase inputs,

\[ a_1=-a_2. \]

Therefore,

\[ b_3=\frac{a_1+a_2}{\sqrt{2}}=0. \]

The H-arm is isolated because the two contributions cancel exactly.

Mathematical Demonstration Using the S-Matrix

The sum and difference behavior can be obtained directly from the third and fourth rows of the ideal S-matrix. With \(a_3=a_4=0\), the third row gives

\[ b_3 = S_{31}a_1+S_{32}a_2 = \frac{1}{\sqrt{2}}a_1+ \frac{1}{\sqrt{2}}a_2. \]

Therefore,

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

Similarly, the fourth row gives

\[ b_4 = S_{41}a_1+S_{42}a_2 = \frac{1}{\sqrt{2}}a_1- \frac{1}{\sqrt{2}}a_2. \]

Therefore,

\[ \boxed{ b_4=\frac{a_1-a_2}{\sqrt{2}} } \]

These equations show that the Magic Tee performs a normalized sum and difference transformation. Port 3 extracts the sum component of the two collinear-port signals, whereas Port 4 extracts their difference component.

The operation can therefore be summarized as

\[ \boxed{ \text{H-arm} \propto a_1+a_2 } \]

and

\[ \boxed{ \text{E-arm} \propto a_1-a_2 } \]

The factor \(1/\sqrt{2}\) ensures proper normalization and power conservation for the ideal lossless four-port network. Thus, the Magic Tee can function both as a power combiner and as a power divider, with the H-arm providing the sum response and the E-arm providing the difference response.

Exam Hint

Port Excitation Cases of a Magic Tee

The operation of a Magic Tee becomes much easier to understand when each port is considered separately rather than treating the scattering matrix as a collection of numbers to memorize. Using the standard port convention, Ports 1 and 2 are the collinear ports, Port 3 is the H-arm or sum port, and Port 4 is the E-arm or difference port. The ideal Magic Tee is represented by

\[ [S]= \begin{bmatrix} 0 & 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\\[4pt] 0 & 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}\\[4pt] \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 & 0\\[4pt] \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0 & 0 \end{bmatrix}. \]

The following excitation cases show exactly how the incident wave at each port is distributed among the other ports.

Port 3 Excited: H-Arm Excitation

When Port 3, the H-arm or sum port, is excited, let \(a_3=A\) while all other incident waves are zero:

\[ a_1=a_2=a_4=0,\qquad a_3=A. \]

From the S-matrix,

\[ b_1=\frac{A}{\sqrt{2}}, \qquad b_2=\frac{A}{\sqrt{2}}, \qquad b_3=0, \qquad b_4=0. \]

Therefore, the input at the H-arm divides equally between the two collinear ports, and the two output waves have the same phase.

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

At the same time, Port 3 is matched and Port 4 is isolated from Port 3:

\[ \boxed{S_{33}=0,\qquad S_{43}=0} \]

Thus, H-arm excitation produces equal-amplitude, in-phase outputs at the two collinear ports.

Port 4 Excited: E-Arm Excitation

When Port 4, the E-arm or difference port, is excited, let \(a_4=A\) while all other incident waves are zero:

\[ a_1=a_2=a_3=0,\qquad a_4=A. \]

The resulting waves are

\[ b_1=\frac{A}{\sqrt{2}}, \qquad b_2=-\frac{A}{\sqrt{2}}, \qquad b_3=0, \qquad b_4=0. \]

Therefore, the E-arm divides the incident power equally between the two collinear ports, but the two output waves have opposite phase.

\[ \boxed{S_{14}=\frac{1}{\sqrt{2}},\qquad S_{24}=-\frac{1}{\sqrt{2}}} \]

The E-arm is matched and the H-arm is isolated:

\[ \boxed{S_{44}=0,\qquad S_{34}=0} \]

Thus, E-arm excitation produces equal-amplitude, opposite-phase outputs at the two collinear ports.

Port 1 Excited

When Port 1 is excited, let \(a_1=A\) and all other incident waves be zero:

\[ a_1=A,\qquad a_2=a_3=a_4=0. \]

The output waves are

\[ b_1=0, \qquad b_2=0, \qquad b_3=\frac{A}{\sqrt{2}}, \qquad b_4=\frac{A}{\sqrt{2}}. \]

Therefore, Port 1 is matched, Port 2 is isolated, and the incident signal divides equally between the H-arm and E-arm.

\[ \boxed{S_{11}=0,\qquad S_{21}=0} \]

The two side arms receive equal-magnitude waves:

\[ \boxed{S_{31}=S_{41}=\frac{1}{\sqrt{2}}} \]

Thus, excitation of one collinear port produces equal power division between the H-arm and E-arm, while the other collinear port remains isolated.

Port 2 Excited

When Port 2 is excited, let \(a_2=A\) and all other incident waves be zero:

\[ a_2=A,\qquad a_1=a_3=a_4=0. \]

The resulting waves are

\[ b_1=0, \qquad b_2=0, \qquad b_3=\frac{A}{\sqrt{2}}, \qquad b_4=-\frac{A}{\sqrt{2}}. \]

Therefore, Port 2 is matched and Port 1 is isolated. The incident signal is divided equally between the H-arm and E-arm, but the two side-arm outputs have opposite phase.

\[ \boxed{S_{22}=0,\qquad S_{12}=0} \]

and

\[ \boxed{S_{32}=\frac{1}{\sqrt{2}},\qquad S_{42}=-\frac{1}{\sqrt{2}}} \]

Ports 1 and 2 Excited Simultaneously

The simultaneous excitation of Ports 1 and 2 demonstrates the sum and difference operation most clearly. With \(a_3=a_4=0\), the relevant output equations are

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

and

\[ \boxed{ b_4=\frac{a_1-a_2}{\sqrt{2}} } \]

For equal in-phase inputs, \(a_1=a_2=A\),

\[ b_3=\sqrt{2}A,\qquad b_4=0. \]

Hence, the H-arm receives the combined signal while the E-arm is isolated.

For equal opposite-phase inputs, \(a_1=A\) and \(a_2=-A\),

\[ b_3=0,\qquad b_4=\sqrt{2}A. \]

Hence, the E-arm receives the difference signal while the H-arm is isolated.

Excitation Pattern of the Ideal Magic Tee

This table provides a useful way to understand the Magic Tee without memorizing individual S-parameters. Each single-port excitation produces two nonzero output coefficients of magnitude \(1/\sqrt{2}\), while the directly isolated or matched ports produce zero coefficients.

Recognizing the Magic Tee S-Matrix by Its Numerical Pattern

An ideal Magic Tee can also be recognized directly from the numerical pattern of its S-matrix. This is particularly useful in examination problems where the port names are not explicitly stated and the student must identify the physical arrangement from the matrix.

For the standard ideal Magic Tee matrix used in this discussion, the entries are only

\[ \boxed{ 0,\qquad +\frac{1}{\sqrt{2}},\qquad -\frac{1}{\sqrt{2}} } \]

There are eight zero elements and four nonzero elements of magnitude \(1/\sqrt{2}\) with positive or negative signs in the complete \(4\times4\) matrix.

More specifically, the standard matrix contains

The important correction here is that the ideal Magic Tee S-matrix does not contain \(1/2\) or \(-1/2\) terms. Those values belong to the ideal three-port H-plane tee and E-plane tee matrices discussed earlier. The ideal four-port Magic Tee instead contains \(0\), \(+1/\sqrt{2}\), and \(-1/\sqrt{2}\) terms.

The Structural Pattern Behind the Matrix

The numerical pattern is more useful than memorizing the exact positions of the elements. In the standard matrix, the two collinear ports have zero reflection coefficients and are isolated from each other:

\[ \boxed{ S_{11}=S_{22}=S_{12}=S_{21}=0 } \]

The two side arms are also matched and isolated from each other:

\[ \boxed{ S_{33}=S_{44}=S_{34}=S_{43}=0 } \]

The remaining eight elements describe coupling between the collinear ports and the two side arms. Six have positive sign and two have negative sign in the particular reference convention used here.

The H-arm coupling has the same sign for both collinear ports:

\[ \boxed{ S_{13}=S_{23}=S_{31}=S_{32} =\frac{1}{\sqrt{2}} } \]

The E-arm coupling has opposite signs for the two collinear ports:

\[ \boxed{ S_{14}=S_{41} =\frac{1}{\sqrt{2}}, \qquad S_{24}=S_{42} =-\frac{1}{\sqrt{2}} } \]

Therefore, the most important visual pattern is that the H-arm column and row contain equal-sign coupling to the two collinear ports, whereas the E-arm column and row contain opposite-sign coupling.

Identifying the Port Arrangement Without Memorizing the Matrix

When an examination gives an unknown ideal Magic Tee S-matrix, the port arrangement can be identified from the pattern rather than from memorization. First locate the two ports that form an isolated pair. Their mutual transmission coefficients are zero. These are the two collinear ports.

Next identify the remaining two ports. These are the H-arm and E-arm. Both are matched, so their diagonal elements are zero, and they are isolated from each other:

\[ \boxed{ S_{kk}=S_{\ell\ell}=S_{k\ell}=S_{\ell k}=0 } \]

The distinction between the H-arm and E-arm is then obtained from the signs of their coupling coefficients with the two collinear ports. The H-arm has equal-phase coupling, whereas the E-arm has opposite-phase coupling.

Thus, the identification process is

\[ \boxed{ \text{Find isolated pair} \;\longrightarrow\; \text{identify collinear ports} \;\longrightarrow\; \text{inspect signs} \;\longrightarrow\; \text{identify H-arm and E-arm} } \]

This approach is much safer than memorizing that “Port 3 is always H” or “Port 4 is always E,” because port numbering can be changed without changing the physical junction.

Magic Tee S-Matrix Variations for Different Port Numbering

Changing the port numbering does not change the physical Magic Tee. It only changes the positions of the S-parameters in the matrix. Because each S-parameter has an output-port index and an input-port index, changing a port number requires the corresponding row and column to be rearranged together.

There are multiple equivalent numbering arrangements. The following table shows the distinct arrangements obtained by assigning different ports as the H-arm and E-arm while keeping the two remaining ports as the collinear pair. The signs shown follow the same reference-plane convention used throughout this article.

The matrices generated by port permutation must always be checked using the physical conditions rather than by relying only on their numerical appearance. In particular, the H-arm must couple equally and in phase to the two collinear ports, the E-arm must couple equally with opposite phase, the H-arm and E-arm must be isolated, and the two collinear ports must also be isolated.

Important: the first table above is the standard and most commonly used numbering convention. For examination purposes, it is safer to identify the physical roles from the zero pattern and the relative signs rather than attempting to memorize every possible permutation of the \(4\times4\) matrix.

Fast Pattern Recognition for Examination Problems

For an ideal Magic Tee, the fastest recognition method is to look for the characteristic zero and \(1/\sqrt{2}\) pattern. The matrix has eight zeros, six positive \(1/\sqrt{2}\) terms, and two negative \(1/\sqrt{2}\) terms under the reference convention used here. There are no \(1/2\) or \(-1/2\) terms in this ideal four-port Magic Tee matrix.

The two isolated pairs immediately reveal the structure. One isolated pair consists of the two collinear ports, while the other isolated pair consists of the H-arm and E-arm. The H-arm is then recognized by equal-sign coupling to the collinear ports, while the E-arm is recognized by opposite-sign coupling.

Therefore, the essential pattern to remember is

\[ \boxed{ \begin{array}{c} \text{Collinear ports: isolated from each other}\\[4pt] \text{H-arm and E-arm: isolated from each other}\\[4pt] \text{H-arm: equal-phase coupling}\\[4pt] \text{E-arm: opposite-phase coupling}\\[4pt] \text{All nonzero coupling magnitudes: }1/\sqrt{2} \end{array} } \]

Once this pattern is recognized, a change in port numbering does not require memorizing a completely new physical rule. The rows and columns simply move together, while the same physical relationships remain unchanged.

# Identifying an Unknown Magic Tee S-Matrix

Identifying an Unknown Magic Tee S-Matrix

When an S-matrix is given in an examination without explicitly stating which ports correspond to the E-arm, H-arm, and collinear arms, the port numbering should not be assumed in advance. An ideal Magic Tee can be identified directly from the mathematical pattern of its S-parameters. The most useful approach is to check the matrix systematically using its four-port structure, isolation relationships, coupling signs, matching conditions, reciprocity, and lossless condition.

Four-Port Structure

A Magic Tee is a four-port microwave junction consisting of two collinear ports, one H-arm or sum port, and one E-arm or difference port. The four ports are represented by the incident waves \(a_1,a_2,a_3,a_4\) and outgoing waves \(b_1,b_2,b_3,b_4\). The scattering relationship is

$ \begin{bmatrix} b_1\\ b_2\\ b_3\\ b_4 \end{bmatrix} = [S] \begin{bmatrix} a_1\\ a_2\\ a_3\\ a_4 \end{bmatrix} $

For the commonly used numbering convention, Ports 1 and 2 are the collinear ports, Port 3 is the H-arm or sum port, and Port 4 is the E-arm or difference port. The ideal Magic Tee S-matrix is

$ \boxed{ [S]= \begin{bmatrix} 0 & 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\\[4pt] 0 & 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}\\[4pt] \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 & 0\\[4pt] \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0 & 0 \end{bmatrix} } $

This normalized form is important when identifying an unknown matrix because the four coupling coefficients associated with the E-arm and H-arm have magnitude \(1/\sqrt{2}\). The collinear-port coupling coefficients \(S_{12}\) and \(S_{21}\) are zero because the two collinear ports are isolated in the ideal Magic Tee.

First Check: Locate the Isolated Port Pairs

The first useful step is to look for zero off-diagonal elements. In an ideal Magic Tee, the two collinear ports are isolated from each other, and the E-arm and H-arm are also isolated from each other.

Therefore, the characteristic isolation relationships are

$ \boxed{S_{12}=S_{21}=0} $

and

$ \boxed{S_{34}=S_{43}=0} $

These two pairs of zeros are a strong indication that the matrix represents a Magic Tee. The first pair represents isolation between the two collinear ports, while the second pair represents isolation between the H-arm and E-arm.

Thus, when an unknown four-port S-matrix is given, a useful first question is whether it contains two isolated port pairs having this characteristic structure. If these isolation relationships are absent, the matrix does not represent the ideal Magic Tee under the stated port convention.

Identify the H-Arm from Equal Positive Coupling

The H-arm is identified by its equal-phase coupling to the two collinear ports. When the H-arm is excited, equal-magnitude waves appear at the two collinear ports with the same phase.

For the standard numbering in which Port 3 is the H-arm, the relevant coefficients are

$ \boxed{ S_{13}=S_{23}=\frac{1}{\sqrt{2}} } $

By reciprocity, the reverse-direction coefficients are also equal:

$ \boxed{ S_{31}=S_{32}=\frac{1}{\sqrt{2}} } $

Therefore, the H-arm is associated with two equal coupling coefficients having the same phase. In the common real-valued reference convention, these coefficients appear as two positive \(1/\sqrt{2}\) terms.

This provides a powerful examination shortcut. If one port has equal positive coupling to two other ports, that port can be identified as the H-arm, provided the remaining isolation and reciprocity conditions are also satisfied.

Identify the E-Arm from Opposite-Sign Coupling

The E-arm has a different phase relationship from the H-arm. When the E-arm is excited, equal-magnitude waves appear at the two collinear ports but with opposite phase.

For the standard numbering in which Port 4 is the E-arm,

$ \boxed{ S_{14}=-S_{24} } $

For the ideal normalized matrix, this becomes

$ \boxed{ S_{14}=\frac{1}{\sqrt{2}}, \qquad S_{24}=-\frac{1}{\sqrt{2}} } $

Reciprocity gives the corresponding reverse-direction relationship:

$ \boxed{ S_{41}=\frac{1}{\sqrt{2}}, \qquad S_{42}=-\frac{1}{\sqrt{2}} } $

Therefore, the E-arm can be recognized by the equal magnitudes and opposite signs of its coupling coefficients to the two collinear ports. The opposite signs represent the \(180^\circ\) phase difference between the two collinear-port outputs.

Check the Diagonal Elements

For an ideal, perfectly matched Magic Tee, all four ports are matched. Therefore, every diagonal S-parameter is zero:

$ \boxed{ S_{11}=S_{22}=S_{33}=S_{44}=0 } $

The diagonal elements represent the reflection coefficients at their respective ports. Thus, zero diagonal elements indicate that no incident power is reflected back into the corresponding port under the reference termination conditions.

Finding four zero diagonal elements is therefore another important indication that the proposed matrix represents an ideal matched Magic Tee. However, zero diagonal elements alone are not sufficient for identification because other matched four-port networks can also have zero reflection coefficients. The coupling and isolation pattern must also be checked.

Check Reciprocity

An ideal passive Magic Tee is reciprocal. Therefore, its S-matrix is symmetric and must satisfy

$ \boxed{ S_{ij}=S_{ji} } $

For the standard matrix, this means

$ S_{12}=S_{21},\qquad S_{13}=S_{31},\qquad S_{14}=S_{41} $

and

$ S_{23}=S_{32},\qquad S_{24}=S_{42},\qquad S_{34}=S_{43} $

If an unknown matrix does not satisfy this symmetry, it cannot represent the ideal reciprocal Magic Tee under the same reference-plane convention.

Check the Unitary Condition

Because an ideal Magic Tee is lossless, its S-matrix must conserve total incident power. Mathematically, the matrix must satisfy the unitary condition

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

where \([S]^\dagger\) is the conjugate transpose of the scattering matrix and \([I]\) is the identity matrix.

For the standard matrix, consider the first row:

$ \left|S_{13}\right|^2+ \left|S_{14}\right|^2 = \frac{1}{2}+\frac{1}{2}=1 $

Thus, the total normalized power associated with the first row is unity. The same condition holds for every row.

The orthogonality of different rows provides an additional check. For example, the first and second rows give

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

Therefore, the rows are orthogonal, as required for a lossless network.

A Fast Examination Identification Method

For an examination problem, the unknown Magic Tee S-matrix can be identified efficiently by following a fixed sequence. First, confirm that the matrix is a four-port matrix. Next, look for the two characteristic isolated port pairs. For the standard numbering, these are \(S_{12}=S_{21}=0\) and \(S_{34}=S_{43}=0\).

Then inspect the coupling coefficients. The H-arm must couple equally and with the same phase to the two collinear ports, while the E-arm must couple equally in magnitude but with opposite phase. Finally, verify that the diagonal elements are zero for the ideal matched case, the matrix is symmetric for reciprocity, and the matrix satisfies the unitary condition for a lossless junction.

For the standard numbering, the complete identification pattern is therefore

$ \boxed{ \begin{aligned} &S_{11}=S_{22}=S_{33}=S_{44}=0,\\ &S_{12}=S_{21}=0,\\ &S_{34}=S_{43}=0,\\ &S_{13}=S_{23}=S_{31}=S_{32}=\frac{1}{\sqrt{2}},\\ &S_{14}=S_{41}=\frac{1}{\sqrt{2}},\\ &S_{24}=S_{42}=-\frac{1}{\sqrt{2}} \end{aligned} } $

The key distinction is therefore simple: the H-arm produces equal-magnitude, equal-phase coupling to the two collinear ports, whereas the E-arm produces equal-magnitude, opposite-phase coupling. The isolated H-arm and E-arm, together with the isolated collinear ports, complete the characteristic Magic Tee pattern.

Examination Shortcut

A useful way to recognize an ideal Magic Tee without calculating every element is to remember its structural pattern rather than memorizing only one matrix. There are four zero diagonal elements, two additional isolated port pairs, four coupling terms of magnitude \(1/\sqrt{2}\) associated with the H-arm and E-arm, and opposite signs for the two E-arm coupling paths.

Once the equal-phase pair is found, the associated port is the H-arm. Once the equal-magnitude opposite-phase pair is found, the associated port is the E-arm. The remaining two ports are the collinear ports. This pattern remains physically valid even when the port numbering is changed; only the positions of the corresponding rows and columns change.

Possible question

2082 Bhadra (BEI) 

• Design a model of a three-port network and define its characteristic parameters. Prepare the S-Matrix of a rectangular Magic Tee having shorted both E-arms and H-arms and explain how it behaves.

2081 Bhadra (BEI) 

• Identify and explain, along with a neat diagram, the properties of the passive microwave device described by the following S-Matrix:

  \[ S= \begin{bmatrix} S_{11} & S_{12} & S_{13} & S_{14} \\ S_{12} & S_{22} & S_{13} & -S_{14} \\ S_{13} & S_{13} & 0 & 0 \\ S_{14} & -S_{14} & 0 & 0 \end{bmatrix} \]

2080 Bhadra (BEI)

• Draw a neat diagram of a Magic Tee and derive its S-parameters.

2078 Chaitra (BEX)

• Identify and explain the properties of a microwave passive device having the following S-Matrix with necessary diagram:

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

2078 Chaitra (BEX) 

• Write a short note on the combinational effect of two Magic Tees connected in H-plane.

2077 Chaitra (BEX) 

• Explain the properties of two Magic Tees if one connects their E-arms and derive its S-matrix.

2075 Bhadra (BEX) 

• Which of the passive microwave device is explained by this S-matrix? Judge the condition and explain its characteristics:

  \[ [S] = \begin{bmatrix} S_{11} & 0 & S_{13} & S_{14} \\ 0 & S_{22} & S_{13} & S_{14} \\ S_{13} & -S_{13} & 0 & 0 \\ S_{14} & S_{14} & 0 & 0 \end{bmatrix} \]

2074 Bhadra (BEX) 

• Identify and explain the properties of a microwave passive device having the following S-Matrix:

  \[ \begin{bmatrix} S_{11} & S_{12} & S_{13} & S_{14} \\ S_{21} & S_{22} & S_{13} & -S_{14} \\ S_{13} & S_{13} & 0 & 0 \\ S_{14} & -S_{14} & 0 & 0 \end{bmatrix} \]

2073 Magh (BEX) 

• Choose a suitable passive microwave device to split power into half and explain its properties.

2072 Ashwin (BEX) 

• What are waveguide junctions? Describe the operational principles of Magic Tee based on S-parameters.

2072 Magh (BEX) 

• Describe Magic Tee based on S-parameters. Differentiate between dominant and degenerate modes.