E-Plane Tee Junction
An E-plane tee junction is a three-port waveguide junction used to divide or combine microwave power in a controlled manner. It is formed by connecting an auxiliary rectangular waveguide to the main rectangular waveguide in such a way that the axis of the auxiliary arm is parallel to the electric field of the dominant mode in the main waveguide. The main waveguide contains two collinear ports, identified as Port 1 and Port 2, while the auxiliary arm forms Port 3, which is called the E-arm. Because the side arm is positioned in the E-plane of the main waveguide, the resulting three-port junction is known as an E-plane tee. It is also commonly called a series tee or voltage tee because of its characteristic electric-field and voltage relationships.
The physical construction of an E-plane tee can be understood by considering a rectangular opening made along the broad wall of the main waveguide. An additional waveguide section is connected through this opening so that three separate waveguide ports are formed. Ports 1 and 2 lie along the original direction of propagation and are therefore called the collinear ports. Port 3 is positioned as the side arm, and its axis is parallel to the electric-field direction of the dominant mode in the main waveguide. The location and orientation of this side arm are important because the electromagnetic field distribution determines how efficiently energy can be transferred between the E-arm and the two collinear arms.
The name E-plane tee comes directly from the orientation of the side arm with respect to the electric field. In the dominant mode of a rectangular waveguide, the electric field has a definite transverse direction across the guide. When the axis of the auxiliary arm is made parallel to this electric-field direction, the junction lies in the E-plane of the main waveguide. This arrangement produces a distinctive phase relationship between the two collinear ports. The E-plane tee therefore behaves differently from an H-plane tee, where the side arm is associated with the magnetic-field plane. The E-plane tee is particularly important in microwave systems because its field configuration allows it to perform differential power division and combination.
Ports of an E-Plane Tee
An ideal E-plane tee has three ports. Port 1 and Port 2 are the collinear ports because they lie along the same main waveguide axis. These ports provide the normal input and output path through the main guide. Port 3 is the E-arm, which is the auxiliary waveguide connected to the main guide in the E-plane. The E-arm is therefore the port through which a signal can be introduced into the junction and divided between the two collinear arms, or through which signals arriving at the collinear ports can combine and produce an output.
The three-port nature of the junction means that an electromagnetic wave can be incident at any one of the three ports, while waves can emerge from all three ports depending on the excitation and matching conditions. The behavior of these waves is described conveniently using scattering parameters. In microwave engineering, S-parameters are particularly useful for waveguide junctions because voltage and current measurements become difficult at microwave frequencies, whereas incident and reflected traveling waves can be measured and related directly to the power flow at each port.
Power Division and Phase Relationship
The most important characteristic of an E-plane tee is the phase relationship between the waves appearing at Ports 1 and 2 when the E-arm is excited. Suppose a microwave signal is applied only to Port 3. Because of the symmetry of the junction, the incident power from the E-arm is divided between the two collinear arms. For an ideal symmetric junction, the magnitudes of the waves appearing at Ports 1 and 2 are equal. However, unlike an ordinary in-phase power divider, the two waves have opposite phase. Their phase difference is 180°.
Thus, when Port 3 is excited, the waves emerging from Ports 1 and 2 satisfy the relationship
$ \boxed{ S_{13}=-S_{23} } $
The negative sign does not mean that the magnitude of the transmitted signal is negative. It represents a phase reversal of \(180^\circ\). Therefore, if the magnitude of \(S_{13}\) is equal to some value \(A\), then \(S_{23}\) has the same magnitude but opposite phase:
$ S_{13}=A $
$ S_{23}=-A $
Consequently,
$ |S_{13}|=|S_{23}| $
while
$ \angle S_{23} = \angle S_{13}+180^\circ $
This equal-magnitude and opposite-phase behavior is the defining electrical characteristic of the E-plane tee. It is also the reason the E-plane tee is often described as a difference junction. When signals are applied to Ports 1 and 2, the resulting field at the E-arm depends on their difference rather than simply their sum.
Operation When the E-Arm Is Excited
Consider a signal entering the junction through Port 3 while Ports 1 and 2 are available for transmission. The electromagnetic field entering through the E-arm interacts with the field distribution in the main waveguide. Because the junction is symmetric with respect to the two collinear arms, equal power is directed toward Ports 1 and 2 in an ideal structure. However, the orientation of the electric field causes the fields at the two collinear ports to have opposite polarity. Therefore, the output at one collinear port is the phase-reversed version of the output at the other collinear port.
If the incident wave at Port 3 is represented by \(a_3\), then the outgoing waves at Ports 1 and 2 can be represented as
$ b_1=S_{13}a_3 $
and
$ b_2=S_{23}a_3 $
Using the E-plane phase relationship,
$ S_{23}=-S_{13} $
we obtain
$ b_2=-S_{13}a_3 $
Thus, the two output waves have equal magnitude and opposite phase when the E-arm is excited under ideal symmetric conditions.
Operation When the Collinear Ports Are Excited
The same phase relationship can be viewed in the reverse direction. If signals are applied to Ports 1 and 2, their fields interact at the junction and produce a resulting field in the E-arm. Because the E-plane tee responds to the difference between the fields arriving from the two collinear arms, the output at Port 3 depends on the relative amplitude and phase of the two input signals. This makes the E-plane tee fundamentally different from a junction in which two equal in-phase signals simply reinforce one another at the third port.
For incident waves \(a_1\) and \(a_2\) at Ports 1 and 2, respectively, the output wave at Port 3 is expressed generally as
$ b_3=S_{31}a_1+S_{32}a_2 $
For the E-plane tee, reciprocity and the E-plane phase relationship give
$ S_{31}=S_{13} $
and
$ S_{32}=S_{23}=-S_{13} $
Therefore,
$ \boxed{ b_3=S_{13}a_1-S_{13}a_2 } $
or equivalently,
$ \boxed{ b_3=S_{13}(a_1-a_2) } $
This equation clearly shows the difference behavior of the E-plane tee. If equal in-phase waves are applied to the two collinear ports, then \(a_1=a_2\), and their contributions to Port 3 cancel:
$ a_1=a_2 $
therefore,
$ b_3=S_{13}(a_1-a_1)=0 $
On the other hand, if the two collinear ports receive equal-amplitude waves that are \(180^\circ\) out of phase, then \(a_1=-a_2\). In this case, the contributions reinforce at the E-arm and a strong output can be produced at Port 3. The E-plane tee therefore provides a convenient means of combining or separating microwave signals according to their relative phase.
Important Properties of an Ideal E-Plane Tee
The E-plane tee is a three-port waveguide junction in which the side arm is connected in the plane of the electric field of the main waveguide. Because the side arm lies in the E-plane, the junction exhibits a unique phase-reversal property that distinguishes it from the H-plane tee. When signals are divided or combined through the E-arm, the resulting waves at the collinear ports have equal magnitude but opposite phase.
1. Excitation of the E-Arm
When a microwave signal is incident at the E-arm (Port 3), the power divides equally between the two collinear ports (Ports 1 and 2). However, the two output signals are 180° out of phase with each other.
Therefore,
$ S_{13}=-S_{23} $
and by reciprocity,
$ S_{31}=-S_{32} $
For an ideal symmetric E-plane tee, the magnitudes of these coefficients are equal:
$ |S_{13}|=|S_{23}|=\frac{1}{\sqrt{2}} $
Thus, excitation of the E-arm produces equal-amplitude but opposite-phase waves at the two collinear ports.
2. Equal In-Phase Signals Applied at the Collinear Ports
When two signals of equal magnitude and the same phase are applied simultaneously to Ports 1 and 2, the contributions arriving at the E-arm cancel each other because of the opposite-sign coupling associated with the E-plane tee.
Therefore,
$ b_3=0 $
The E-arm does not respond to the sum of the collinear-port signals.
3. Equal Opposite-Phase Signals Applied at the Collinear Ports
When equal signals with opposite phase are applied to Ports 1 and 2, the contributions reaching the E-arm add constructively.
As a result, a strong output appears at the E-arm.
This behavior makes the E-arm a difference port because it responds to the difference between the signals applied at the two collinear ports.
4. Difference-Port Operation
The E-arm is commonly called the difference port because it produces an output when the signals applied to the two collinear ports differ in phase or amplitude.
For equal-amplitude inputs,
$ a_1=a $
$ a_2=-a $
the output at the E-arm becomes maximum because the two contributions add instead of cancel.
5. Reciprocity Property
The ideal E-plane tee is a reciprocal microwave junction. Therefore, its scattering matrix is symmetric and satisfies
$ S_{ij}=S_{ji} $
Consequently,
$ S_{13}=S_{31} $
and
$ S_{23}=S_{32} $
This means that the transmission characteristics are identical in both directions between any pair of ports.
6. Symmetry Property
Because the two collinear arms are physically identical, the E-plane tee is symmetrical about its center plane.
Therefore, the coupling from the E-arm to the two collinear ports has equal magnitude:
$ |S_{13}|=|S_{23}| $
The only difference is the phase reversal represented by the negative sign.
7. Matched E-Arm Condition
For an ideal E-plane tee, the E-arm is assumed to be perfectly matched. Therefore, no power is reflected back into the E-arm when it is excited.
Hence,
$ S_{33}=0 $
This matched-port condition is frequently used during S-matrix derivations.
8. Power Division Characteristic
When the E-arm is excited, the incident power is divided equally between the two collinear ports.
Since each port receives half of the total power,
$ |S_{13}|^2=|S_{23}|^2=\frac{1}{2} $
which gives
$ |S_{13}|=|S_{23}|=\frac{1}{\sqrt{2}} $
The two outputs therefore have equal magnitude but opposite phase.
Ideal E-Plane Tee Properties
The defining characteristics of an ideal E-plane tee can be summarized as follows:
- Port 3 is the E-arm or Difference Port.
- Excitation of the E-arm produces equal-magnitude outputs at Ports 1 and 2.
- The two output waves are 180° out of phase.
- The E-arm responds to the difference between signals applied at the collinear ports.
- The coupling coefficients satisfy \(S_{13}=-S_{23}\).
- The E-arm is ideally
Operating Cases of an E Plane Tee
The operation of an E Plane Tee can be understood by considering different input conditions at its three ports. The scattering behavior changes depending on which port is excited and whether the input signals are applied individually or simultaneously. For an ideal E Plane Tee, the signals appearing at the two collinear ports have equal magnitude, while their phase relationship depends on the excitation condition. The following three cases describe the important operating conditions of an E Plane Tee.
Case 1: Input at Port 3 and No Input at Ports 1 and 2
Consider an input signal applied at Port 3, which is the E arm, while there are no input signals at the two collinear ports. Therefore, the input conditions are
\[ a_3 \neq 0,\qquad a_1=a_2=0 \]
For an ideal E Plane Tee, the signal entering through the E arm divides equally between the two collinear arms. However, the two output signals have equal magnitude and are 180° out of phase. Thus, the output waves at Ports 1 and 2 are
\[ b_1=\frac{1}{\sqrt{2}}a_3 \]
\[ b_2=-\frac{1}{\sqrt{2}}a_3 \]
There is no output at Port 3 itself for an ideal matched E Plane Tee, so
\[ b_3=0 \]
The negative sign in the expression for \(b_2\) indicates the 180° phase difference between the waves appearing at Ports 1 and 2. Therefore, when the E arm is excited, the input power is equally divided between the two collinear arms with opposite phase.
Case 2: Equal Inputs at Ports 1 and 2 and No Input at Port 3
Now consider equal input signals applied simultaneously to the two collinear ports, while there is no input at the E arm. The input conditions are
\[ a_3=0,\qquad a_1=a_2=a \]
Since equal signals are applied to Ports 1 and 2 with the same phase, their contributions at the E arm cancel each other. The output waves at the collinear ports are therefore
\[ b_1=\frac{1}{2}a+\frac{1}{2}a=a \]
\[ b_2=\frac{1}{2}a+\frac{1}{2}a=a \]
For the E arm, the contributions from the two collinear ports have opposite phase. Hence, they cancel at Port 3, giving
\[ b_3=0 \]
Thus, equal in phase signals applied to the two collinear ports do not produce an output at the E arm. This demonstrates the cancellation property of the E Plane Tee.
Case 3: Input at Port 1 and No Inputs at Ports 2 and 3
Finally, consider an input signal applied only at Port 1, while Ports 2 and 3 have no incident waves. The input conditions are
\[ a_1\neq0,\qquad a_2=a_3=0 \]
The input signal at Port 1 is divided between the other two ports according to the scattering characteristics of the E Plane Tee. The resulting output waves are
\[ b_1=\frac{1}{2}a_1 \]
\[ b_2=\frac{1}{2}a_1 \]
\[ b_3=\frac{1}{\sqrt{2}}a_1 \]
The signal appearing at Port 3 represents the coupling between the collinear arm and the E arm. The E Plane Tee therefore provides controlled power division and phase relationships between its ports, making it useful for signal combining, power division, and microwave network applications.
Phase Relationship in an E Plane Tee
The three operating cases demonstrate the fundamental phase property of an E Plane Tee. When the E arm is excited, the signals appearing at the two collinear arms have equal magnitude but opposite phase. This 180° phase difference is the defining characteristic of the E Plane Tee and is responsible for cancellation when equal in phase signals are applied to the collinear arms.
Therefore, the E Plane Tee can be used wherever microwave signals must be divided or combined according to their phase relationship. In particular, it can distinguish between signals that are in phase and signals that are out of phase, which is useful in microwave bridges, balanced circuits, mixers, and other microwave systems.
Scattering Parameters of the E-Plane Tee
Because an E-plane tee is a three-port microwave junction, its complete behavior can be described by a \(3\times3\) scattering matrix. Each element of this matrix describes the response at one port resulting from an incident wave applied at another port while all remaining ports are terminated in their reference impedances. The incident waves are represented by \(a_1\), \(a_2\), and \(a_3\), while the corresponding outgoing waves are represented by \(b_1\), \(b_2\), and \(b_3\). The general scattering relationship for the three-port junction is
$ \boxed{ \begin{bmatrix} b_1\\ b_2\\ b_3 \end{bmatrix} = \begin{bmatrix} S_{11} & S_{12} & S_{13}\\ S_{21} & S_{22} & S_{23}\\ S_{31} & S_{32} & S_{33} \end{bmatrix} \begin{bmatrix} a_1\\ a_2\\ a_3 \end{bmatrix} } $
The matrix contains nine scattering coefficients because there are three possible incident ports and three possible output ports. For example, \(S_{11}\) represents the reflection coefficient at Port 1 when Port 1 is excited and the other ports are matched. Similarly, \(S_{13}\) represents the wave emerging from Port 1 when Port 3 is excited. The third index identifies the excited port, while the first index identifies the port at which the response is observed.
General S-Matrix and Reciprocity
For a reciprocal waveguide junction, the scattering matrix is symmetric. Reciprocity means that the transmission behavior between two ports is the same when the direction of propagation is reversed, provided the same reference conditions are maintained. Mathematically, reciprocity gives
$ \boxed{ S_{ij}=S_{ji} } $
Therefore, the transmission coefficients satisfy
$ S_{12}=S_{21} $
$ S_{13}=S_{31} $
and
$ S_{23}=S_{32} $
These relationships reduce the number of independent coefficients that must be determined. For the E-plane tee, reciprocity is combined with the special phase relationship between the two collinear ports to obtain the particular form of its scattering matrix.
Opposite Phase Relationship in the S-Matrix
When Port 3 is excited, equal-magnitude waves appear at Ports 1 and 2 but with a \(180^\circ\) phase difference. Therefore, the corresponding scattering coefficients satisfy
$ \boxed{ S_{23}=-S_{13} } $
Using reciprocity,
$ S_{32}=S_{23} $
and
$ S_{31}=S_{13} $
so the same phase relationship appears in the reverse direction:
$ \boxed{ S_{32}=-S_{31} } $
This relationship is fundamental because it mathematically represents the \(180^\circ\) phase difference between the two collinear arms. The two coefficients have equal magnitude but opposite sign.
Matched E-Arm Condition
If the E-plane tee is properly matched at Port 3, an incident wave entering the E-arm does not return toward the source. In S-parameter terminology, the reflection coefficient looking into Port 3 is zero. Therefore,
$ \boxed{ S_{33}=0 } $
A practical E-plane tee may require tuning elements such as screws, posts, or suitable waveguide matching structures to obtain the desired matching condition. Under the ideal matched condition, the absence of reflection at Port 3 is represented directly by \(S_{33}=0\).
It is important, however, to distinguish this condition from a statement that all three diagonal elements must necessarily be zero. The condition \(S_{33}=0\) specifically represents matching at the E-arm. The values of \(S_{11}\) and \(S_{22}\) depend on the complete three-port network and the matching conditions imposed at the collinear ports. In an ideal lossless three-port junction, a completely matched condition at all three ports is not generally possible because a reciprocal lossless three-port network cannot simultaneously satisfy all-port matching and isolation requirements. Therefore, \(S_{11}=S_{22}=S_{33}=0\) should not be imposed indiscriminately on the ideal E-plane tee derivation.
Reduced S-Matrix of the E-Plane Tee
Starting with the general three-port matrix and applying reciprocity together with the E-plane phase relationship and the matched E-arm condition gives
$ [S]= \begin{bmatrix} S_{11} & S_{12} & S_{13}\\ S_{12} & S_{22} & -S_{13}\\ S_{13} & -S_{13} & 0 \end{bmatrix} $
This is the important reduced form of the E-plane tee scattering matrix. Instead of nine completely independent quantities, only four quantities remain to be determined:
$ S_{11},\qquad S_{12},\qquad S_{22},\qquad S_{13} $
The remaining coefficients are obtained from these four through reciprocity and the \(180^\circ\) phase relationship. The next step is therefore to use the lossless property of the ideal junction to determine their magnitudes and relationships.
Unitary Property of a Lossless E-Plane Tee
An ideal waveguide junction is assumed to be lossless, meaning that no microwave power is dissipated as heat or lost through radiation. Consequently, the total power entering the junction must equal the total power leaving the junction. For a lossless scattering network, this condition is expressed mathematically by the unitary property of the S-matrix:
$ \boxed{ [S][S]^\dagger=[I] } $
Here, \([S]^\dagger\) denotes the conjugate transpose of the scattering matrix and \([I]\) is the identity matrix. The conjugate transpose is obtained by taking the complex conjugate of every element and then transposing the matrix. For the reduced E-plane tee matrix,
$ [S]= \begin{bmatrix} S_{11} & S_{12} & S_{13}\\ S_{12} & S_{22} & -S_{13}\\ S_{13} & -S_{13} & 0 \end{bmatrix} $
the conjugate transpose is
$ [S]^\dagger= \begin{bmatrix} S_{11}^{*} & S_{12}^{*} & S_{13}^{*}\\ S_{12}^{*} & S_{22}^{*} & -S_{13}^{*}\\ S_{13}^{*} & -S_{13}^{*} & 0 \end{bmatrix} $
Therefore, the unitary condition becomes
$ \begin{bmatrix} S_{11} & S_{12} & S_{13}\\ S_{12} & S_{22} & -S_{13}\\ S_{13} & -S_{13} & 0 \end{bmatrix} \begin{bmatrix} S_{11}^{*} & S_{12}^{*} & S_{13}^{*}\\ S_{12}^{*} & S_{22}^{*} & -S_{13}^{*}\\ S_{13}^{*} & -S_{13}^{*} & 0 \end{bmatrix} = \begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1 \end{bmatrix} $
The multiplication must be performed carefully because each element of the resulting matrix comes from the product of one row of \([S]\) and one column of \([S]^\dagger\). The three diagonal elements provide power-conservation equations, while the off-diagonal elements provide orthogonality conditions. The first diagonal equation is obtained from row 1 and column 1, the second from row 2 and column 2, and the third from row 3 and column 3. These equations are the basis for determining the unknown scattering coefficients.
Mathermatically
Derivation of the Unitary Equations
The reduced scattering matrix of the ideal E-plane tee is
$ [S]= \begin{bmatrix} S_{11} & S_{12} & S_{13}\\ S_{12} & S_{22} & -S_{13}\\ S_{13} & -S_{13} & 0 \end{bmatrix} $
For a lossless junction, the scattering matrix satisfies
$ [S][S]^\dagger=[I] $
The right-hand side is the identity matrix:
$ [I]= \begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1 \end{bmatrix} $
This means that every diagonal element of the product must be equal to \(1\), while every off-diagonal element must be equal to \(0\). This point is important when deriving the equations because the diagonal and off-diagonal terms have different right-hand sides. The diagonal terms represent normalization and power conservation, whereas the off-diagonal terms represent orthogonality between different rows or columns of the unitary matrix.
Derivation of Equation 6
To obtain the first equation, take the first row of \([S]\):
$ \begin{bmatrix} S_{11}&S_{12}&S_{13} \end{bmatrix} $
and multiply it by the first column of \([S]^\dagger\):
$ \begin{bmatrix} S_{11}^{*}\\ S_{12}^{*}\\ S_{13}^{*} \end{bmatrix} $
The dot product is
$ S_{11}S_{11}^{*} + S_{12}S_{12}^{*} + S_{13}S_{13}^{*} $
The corresponding element of the identity matrix is \(1\). Therefore,
$ S_{11}S_{11}^{*} + S_{12}S_{12}^{*} + S_{13}S_{13}^{*} =1 $
Using
$ SS^{*}=|S|^{2} $
we obtain
$ \boxed{ |S_{11}|^{2} + |S_{12}|^{2} + |S_{13}|^{2} =1 } \tag{6} $
This equation must contain \(S_{11}\), \(S_{12}\), and \(S_{13}\) because these are the three elements present in the first row of the scattering matrix. Therefore, the expression \( |S_{11}|^2+|S_{11}|^2+|S_{11}|^2=1 \) is not the correct result. Equation 6 states that when Port 1 is excited, the normalized incident power is distributed among all possible outgoing waves represented by the three scattering coefficients in the first row.
Derivation of Equation 7
The second equation is obtained from the second diagonal element of the product. Take the second row of \([S]\):
$ \begin{bmatrix} S_{12}&S_{22}&-S_{13} \end{bmatrix} $
and multiply it by the second column of \([S]^\dagger\):
$ \begin{bmatrix} S_{12}^{*}\\ S_{22}^{*}\\ -S_{13}^{*} \end{bmatrix} $
The dot product is
$ S_{12}S_{12}^{*} + S_{22}S_{22}^{*} + (-S_{13})(-S_{13}^{*}) $
The two negative signs in the third term cancel:
$ (-S_{13})(-S_{13}^{*}) = S_{13}S_{13}^{*} $
Therefore, the complete expression becomes
$ S_{12}S_{12}^{*} + S_{22}S_{22}^{*} + S_{13}S_{13}^{*} =1 $
Writing the result in magnitude form gives
$ \boxed{ |S_{12}|^{2} + |S_{22}|^{2} + |S_{13}|^{2} =1 } \tag{7} $
Equation 7 is the power-conservation condition associated with excitation at Port 2. Notice carefully that the first term is \(S_{12}\), not \(S_{11}\), because the first element of the second row is \(S_{12}\). Similarly, the second term is \(S_{22}\), and the third term is \(S_{13}\) because the third element of the second row is \(-S_{13}\). The negative sign disappears when the coefficient is multiplied by its complex conjugate.
Why Equations 6 and 7 Are Different
Equations 6 and 7 can be written side by side:
$ |S_{11}|^{2} + |S_{12}|^{2} + |S_{13}|^{2} =1 $
and
$ |S_{12}|^{2} + |S_{22}|^{2} + |S_{13}|^{2} =1 $
Both equations represent conservation of power, but they correspond to different incident ports. Equation 6 is associated with Port 1, whereas Equation 7 is associated with Port 2. Because Ports 1 and 2 are the two collinear arms of a symmetric E-plane tee, these equations will later provide a relationship between the reflection coefficients \(S_{11}\) and \(S_{22}\). It is important not to assume their equality before applying the symmetry and unitary conditions.
Derivation of Equation 8
The third diagonal element is obtained from the third row of \([S]\):
$ \begin{bmatrix} S_{13}&-S_{13}&0 \end{bmatrix} $
and the third column of \([S]^\dagger\):
$ \begin{bmatrix} S_{13}^{*}\\ -S_{13}^{*}\\ 0 \end{bmatrix} $
Taking their dot product gives
$ S_{13}S_{13}^{*} + (-S_{13})(-S_{13}^{*}) + (0)(0) =1 $
Again, the two negative signs cancel:
$ S_{13}S_{13}^{*} + S_{13}S_{13}^{*} =1 $
Therefore,
$ 2|S_{13}|^{2}=1 $
Dividing both sides by \(2\),
$ |S_{13}|^{2} = \frac{1}{2} $
Taking the square root,
$ \boxed{ |S_{13}|=\frac{1}{\sqrt{2}} } \tag{8} $
Because \(S_{23}=-S_{13}\), it follows immediately that
$ |S_{23}|=|S_{13}|=\frac{1}{\sqrt{2}} $
The corresponding powers are
$ |S_{13}|^{2} = |S_{23}|^{2} = \frac{1}{2} $
Thus, when the E-arm is excited under the ideal matched and lossless conditions, one-half of the incident power is directed toward Port 1 and one-half toward Port 2. The two output waves have equal power but opposite phase.
Derivation of Equation 9
Equation 9 must be derived from an off-diagonal element of the unitary matrix product. Unlike the diagonal elements, an off-diagonal element of the identity matrix is zero. Therefore, the correct right-hand side is \(0\), not \(1\).
Consider the element in the first row and third column of \([S][S]^\dagger\). Take the first row of \([S]\):
$ \begin{bmatrix} S_{11}&S_{12}&S_{13} \end{bmatrix} $
and multiply it by the third column of \([S]^\dagger\):
$ \begin{bmatrix} S_{13}^{*}\\ -S_{13}^{*}\\ 0 \end{bmatrix} $
The multiplication gives
$ S_{11}S_{13}^{*} + S_{12}(-S_{13}^{*}) + S_{13}(0) =0 $
Therefore,
$ S_{11}S_{13}^{*} - S_{12}S_{13}^{*} =0 $
Factoring \(S_{13}^{*}\),
$ S_{13}^{*} \left( S_{11}-S_{12} \right) =0 $
From Equation 8, \(S_{13}\) is non-zero because
$ |S_{13}|=\frac{1}{\sqrt{2}} $
Therefore, \(S_{13}^{*}\neq0\), and the only way the product can be zero is
$ \boxed{ S_{11}=S_{12} } \tag{9} $
This is the correct relationship obtained from the off-diagonal unitary condition. The expression
$ S_{13}S_{11}^{*} - S_{13}S_{12}^{*} =1 $
does not satisfy the unitary matrix equation because the corresponding off-diagonal element of the identity matrix is zero. The correct derivation therefore leads to an equality between the appropriate scattering coefficients rather than an equation equal to \(1\).
E-Plane Tee Junction: S-Parameter Identification and Port Interchange
How to Identify the E-Arm from the S-Matrix
The E-arm can be identified from the S-matrix even when the port numbering is changed. The E-arm is not identified simply by assuming that it is Port 3. Instead, examine the S-parameters associated with each port and identify the port whose excitation produces equal-magnitude waves at the other two ports with a phase difference of \(180^\circ\).
Quick Identification Tip: For the ideal E-plane tee S-matrix, regardless of how the ports are numbered, the matrix contains one zero, four terms equal to \(1/2\), and four terms with magnitude \(1/\sqrt{2}\). The E-arm can be identified from the two \(1/\sqrt{2}\) terms that have opposite signs. Therefore, if you see a pair of \(+1/\sqrt{2}\) and \(-1/\sqrt{2}\) in the same column, that column corresponds to excitation of the E-arm.
The pattern can be written as
\[ \boxed{ \text{One }0,\quad \text{four }\frac{1}{2},\quad \text{four }\pm\frac{1}{\sqrt{2}} } \]
The two terms \(+1/\sqrt{2}\) and \(-1/\sqrt{2}\) indicate that the two collinear-port waves have equal magnitude and are \(180^\circ\) out of phase when the E-arm is excited.
Examination Tip: If the S-matrix is rearranged because the port numbering is changed, do not memorize the position of the zero or the \(1/\sqrt{2}\) terms. Instead, look for the pair
\[ \boxed{ +\frac{1}{\sqrt{2}} \quad\text{and}\quad -\frac{1}{\sqrt{2}} } \]
The column containing this opposite-sign pair identifies the E-arm. The diagonal zero in that same position provides an additional confirmation for an ideal matched E-plane tee.
If Port \(k\) is the E-arm and the other two ports are \(i\) and \(j\), the characteristic relationship is
\[ \boxed{S_{ik}=-S_{jk}} \]
For a reciprocal E-plane tee, the corresponding row also satisfies
\[ \boxed{S_{ki}=-S_{kj}} \]
Therefore, the main pattern used to identify the E-arm is
\[ \boxed{\text{Equal magnitude + opposite sign}} \]
or
\[ \boxed{S_{ik}=-S_{jk}} \]
In an ideal matched E-plane tee, the diagonal S-parameter corresponding to the E-arm is also zero. Thus, when identifying the E-arm from an S-matrix, two conditions can be checked together:
\[ \boxed{S_{kk}=0} \]
and
\[ \boxed{S_{ik}=-S_{jk}} \]
Here, \(k\) represents the E-arm, while \(i\) and \(j\) represent the two collinear ports.
All S-Matrix Variations for Different E-Arm Port Numbering
Changing the port numbering does not change the physical operation of the E-plane tee. Only the positions of the rows and columns in the S-matrix change. The defining property remains the same: excitation of the E-arm produces two equal-magnitude waves at the collinear ports that are \(180^\circ\) out of phase.
1. Port 3 as the E-Arm
When Port 3 is the E-arm and Ports 1 and 2 are the collinear ports, the ideal E-plane tee has
\[ \boxed{ [S]= \begin{bmatrix} \frac{1}{2} & \frac{1}{2} & \frac{1}{\sqrt{2}}\\ \frac{1}{2} & \frac{1}{2} & -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0 \end{bmatrix} } \]
The third column corresponds to excitation at the E-arm. Therefore,
\[ S_{13}=\frac{1}{\sqrt{2}} \]
and
\[ S_{23}=-\frac{1}{\sqrt{2}} \]
Hence,
\[ \boxed{S_{13}=-S_{23}} \]
By reciprocity,
\[ \boxed{S_{31}=-S_{32}} \]
2. Port 1 as the E-Arm
If Port 1 is designated as the E-arm, Ports 2 and 3 become the collinear ports. Reordering the rows and columns of the standard E-plane tee matrix gives
\[ \boxed{ [S]= \begin{bmatrix} 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}} & \frac{1}{2} & \frac{1}{2}\\ -\frac{1}{\sqrt{2}} & \frac{1}{2} & \frac{1}{2} \end{bmatrix} } \]
The first column corresponds to excitation at the E-arm. Therefore,
\[ S_{21}=\frac{1}{\sqrt{2}} \]
and
\[ S_{31}=-\frac{1}{\sqrt{2}} \]
Thus,
\[ \boxed{S_{21}=-S_{31}} \]
By reciprocity,
\[ \boxed{S_{12}=-S_{13}} \]
3. Port 2 as the E-Arm
If Port 2 is designated as the E-arm, Ports 1 and 3 become the collinear ports. The corresponding S-matrix is
\[ \boxed{ [S]= \begin{bmatrix} \frac{1}{2} & \frac{1}{\sqrt{2}} & \frac{1}{2}\\ \frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}}\\ \frac{1}{2} & -\frac{1}{\sqrt{2}} & \frac{1}{2} \end{bmatrix} } \]
The second column corresponds to excitation at the E-arm. Therefore,
\[ S_{12}=\frac{1}{\sqrt{2}} \]
and
\[ S_{32}=-\frac{1}{\sqrt{2}} \]
Hence,
\[ \boxed{S_{12}=-S_{32}} \]
By reciprocity,
\[ \boxed{S_{21}=-S_{23}} \]
Comparison of the E-Plane Tee S-Matrix Variations
| E-Arm | Collinear Ports | Opposite-Phase Relationship | Ideal S-Matrix |
|---|---|---|---|
| Port 1 | Ports 2 and 3 | \(S_{21}=-S_{31}\) | \[ \begin{bmatrix} 0 & 1/\sqrt{2} & -1/\sqrt{2}\\ 1/\sqrt{2} & 1/2 & 1/2\\ -1/\sqrt{2} & 1/2 & 1/2 \end{bmatrix} \] |
| Port 2 | Ports 1 and 3 | \(S_{12}=-S_{32}\) | \[ \begin{bmatrix} 1/2 & 1/\sqrt{2} & 1/2\\ 1/\sqrt{2} & 0 & -1/\sqrt{2}\\ 1/2 & -1/\sqrt{2} & 1/2 \end{bmatrix} \] |
| Port 3 | Ports 1 and 2 | \(S_{13}=-S_{23}\) | \[ \begin{bmatrix} 1/2 & 1/2 & 1/\sqrt{2}\\ 1/2 & 1/2 & -1/\sqrt{2}\\ 1/\sqrt{2} & -1/\sqrt{2} & 0 \end{bmatrix} \] |
General Rule for Identifying the E-Arm
When an S-matrix is given in an examination without explicitly identifying the E-arm, first examine the diagonal elements. For an ideal matched E-plane tee, the diagonal element corresponding to the E-arm is zero. After locating the zero diagonal element, examine the corresponding column. The two off-diagonal elements in that column have equal magnitude and opposite signs.
Therefore, the identification rule is
\[ \boxed{S_{kk}=0} \]
and
\[ \boxed{S_{ik}=-S_{jk}} \]
The three possible cases are
\[ \boxed{\text{Port 1 E-arm: }S_{11}=0,\quad S_{21}=-S_{31}} \]
\[ \boxed{\text{Port 2 E-arm: }S_{22}=0,\quad S_{12}=-S_{32}} \]
\[ \boxed{\text{Port 3 E-arm: }S_{33}=0,\quad S_{13}=-S_{23}} \]
The most important pattern to remember is that the E-arm produces equal-magnitude and opposite-phase waves at the two collinear ports.
\[ \boxed{\text{E-arm excitation}\Rightarrow \text{equal magnitude}+\text{\(180^\circ\) phase difference}} \]
Important Questions and Examination Tips
- Which microwave junction produces equal-magnitude but \(180^\circ\) out-of-phase waves at its two collinear ports when the E-arm is excited?
- In an E-plane tee, what are Port 1 and Port 2 called, and what is Port 3 called?
- Which type of waveguide tee has its side-arm axis parallel to the electric field of the main waveguide?
- Which waveguide tee is also known as a series tee or voltage tee?
- Why is the junction called an E-plane tee?
- What happens when a microwave signal is applied to the E-arm of an E-plane tee?
- What happens when signals are applied through the two collinear ports of an E-plane tee?
- What is the general \(3\times3\) S-matrix of a three-port E-plane tee?
- What is the relationship between \(S_{13}\) and \(S_{23}\) when Port 3 is the E-arm?
- What is the value of \(S_{33}\) when Port 3 is a perfectly matched E-arm?
- What symmetry property does the S-matrix of a reciprocal E-plane tee satisfy?
- Write the S-matrix of an ideal reciprocal and lossless E-plane tee when Port 3 is the E-arm.
- If Port 1 is the E-arm, what is the relationship between \(S_{21}\) and \(S_{31}\)?
- If Port 2 is the E-arm, what is the relationship between \(S_{12}\) and \(S_{32}\)?
- How can the E-arm be identified from an S-matrix when the port numbering is changed?
- What is the general S-parameter condition that identifies the E-arm?
- If Port 3 is excited, what is the phase relationship between the waves appearing at Ports 1 and 2?
- If equal in-phase signals are applied to Ports 1 and 2, what happens at the E-arm?
- If Ports 1 and 2 are excited with equal amplitudes but opposite phases, what happens at the E-arm?
- What happens to an S-matrix when two ports of a three-port network are interchanged?
- Which rows and columns must be interchanged when Port 1 and Port 2 are exchanged?
- Which rows and columns must be interchanged when Port 1 and Port 3 are exchanged?
- Which rows and columns must be interchanged when Port 2 and Port 3 are exchanged?
Tip: In an examination, do not identify the E-arm merely by looking for a particular port number. Port numbering can be changed. Instead, locate the zero diagonal element for an ideal matched E-plane tee and then check the corresponding column for two equal-magnitude S-parameters with opposite signs.
Tip: Remember the general pattern
\[ \boxed{S_{kk}=0,\qquad S_{ik}=-S_{jk}} \]
where \(k\) is the E-arm and \(i,j\) are the two collinear ports.
Tip: When ports are interchanged, both the corresponding rows and columns of the S-matrix must be interchanged because the first index of \(S_{ij}\) identifies the output port and the second index identifies the input port.