E-H Plane Tee (Magic Tee)
What Is an E-H Plane Tee or Magic Tee?
An E-H Plane Tee, commonly known as a Magic Tee or Hybrid Tee, is a four-port microwave waveguide junction formed by combining the electromagnetic properties of an E-plane tee and an H-plane tee. It is an important passive microwave component used for power division, power combining, signal addition and subtraction, impedance measurement, and other microwave network applications.
The Magic Tee consists of two collinear ports and two side-arm ports. The two collinear ports belong to the main waveguide, while the two side arms are arranged in the electric-field and magnetic-field planes of the waveguide. Because the E-arm and H-arm have different phase relationships with the collinear ports, the junction can perform both sum and difference operations.
The ideal Magic Tee is therefore more than a simple power divider. Its four-port structure allows microwave signals to be combined or divided with a controlled phase relationship. This makes the E-H plane tee particularly useful in microwave engineering wherever signals must be separated into sum and difference components.
Why Is It Called a Magic Tee?
The name Magic Tee comes from the unusual isolation and power-combining properties of the junction. When signals are applied to particular ports with the appropriate magnitude and phase, the signals can combine constructively at one side arm while cancelling at the other side arm. Similarly, excitation of either side arm produces specific phase relationships at the two collinear ports.
For example, when equal signals are applied to the two collinear ports with the same phase, their combined effect appears at the H-arm while the E-arm is ideally isolated. When equal signals are applied with opposite phase, the combined effect appears at the E-arm while the H-arm is ideally isolated.
This ability to produce sum and difference signals through a single waveguide junction gives the device its characteristic name. The term "magic" therefore refers to its special microwave signal-routing behavior rather than to any unusual physical mechanism.
Four-Port Configuration of a Magic Tee
A Magic Tee is a four-port waveguide junction. Unlike an ordinary E-plane tee or H-plane tee, which has three ports, the E-H plane tee contains two collinear ports and two side arms. These four ports provide separate paths for the input and output microwave signals.
For the conventional port numbering used in the S-parameter analysis, the ports are designated as follows:
- Port 1: First collinear port
- Port 2: Second collinear port
- Port 3: H-arm or Sum port
- Port 4: E-arm or Difference port
The incident waves entering the four ports can be represented by \(a_1\), \(a_2\), \(a_3\), and \(a_4\), while the corresponding outgoing waves are represented by \(b_1\), \(b_2\), \(b_3\), and \(b_4\). The complete microwave behavior of the four-port junction can therefore be described by a \(4\times4\) scattering matrix.
The general S-parameter relationship is
$\boxed{\begin{bmatrix}b_1\\b_2\\b_3\\b_4\end{bmatrix}=[S]\begin{bmatrix}a_1\\a_2\\a_3\\a_4\end{bmatrix}}$
where \([S]\) is the four-port scattering matrix of the Magic Tee.
Collinear Ports of the Magic Tee
Ports 1 and 2 are called the collinear ports because they lie along the same main waveguide axis. These ports are located at opposite ends of the principal waveguide section and provide the two primary signal paths through the junction.
The collinear ports play an important role in the sum and difference operation of the Magic Tee. When signals are applied simultaneously to Ports 1 and 2, their relative phase determines which side arm receives the resulting signal.
If equal signals are applied to the two collinear ports with the same phase, their fields combine in a manner that produces an output at the H-arm. If the two signals have equal magnitude but opposite phase, the corresponding field distribution produces an output at the E-arm instead.
Thus, Ports 1 and 2 can be regarded as the two input or output ports through which the sum and difference operations are formed.
H-Arm or Sum Port
Port 3 is conventionally designated as the H-arm or Sum port. It is connected to the junction in the H-plane of the main waveguide. The H-arm has the characteristic property that excitation of this port produces equal-magnitude and in-phase waves at the two collinear ports.
If Port 3 is excited, the output waves at Ports 1 and 2 have the same magnitude and the same phase under ideal symmetric conditions. In normalized S-parameter form, the characteristic relationship is
$\boxed{S_{13}=S_{23}}$
The H-arm is called the Sum port because equal in-phase signals applied to the two collinear ports combine at this port. In an ideal Magic Tee, the E-arm is isolated during this operation.
Therefore, the H-arm is associated with the sum operation of the junction and with equal-phase coupling to the two collinear ports.
E-Arm or Difference Port
Port 4 is conventionally designated as the E-arm or Difference port. It is connected to the junction in the E-plane of the main waveguide. The E-arm has a phase relationship opposite to that of the H-arm.
When Port 4 is excited, equal-magnitude waves appear at Ports 1 and 2, but the two waves have opposite phase. Consequently, the corresponding S-parameters have equal magnitude but opposite signs under the conventional reference-phase choice:
$\boxed{S_{14}=-S_{24}}$
The E-arm is called the Difference port because equal-magnitude signals applied to the two collinear ports with a \(180^\circ\) phase difference combine at the E-arm. At the same time, the H-arm is ideally isolated from this difference excitation.
The E-arm therefore provides the difference operation, while the H-arm provides the sum operation.
Physical Combination of E-Plane and H-Plane Tees
The physical structure of a Magic Tee can be understood by considering it as a combination of an E-plane tee and an H-plane tee. The two collinear arms form the main waveguide path, while one side arm is arranged in the E-plane and the other side arm is arranged in the H-plane.
The side arm connected through the E-plane is called the E-arm, while the side arm connected through the H-plane is called the H-arm. Because these two arms interact with the electromagnetic field in different ways, they produce different phase relationships at the collinear ports.
The E-plane section contributes the opposite-phase behavior associated with the Difference port, whereas the H-plane section contributes the in-phase behavior associated with the Sum port. Combining these two properties in one four-port junction produces the characteristic hybrid behavior of the Magic Tee.
This physical arrangement also explains the isolation between the two side arms. Under ideal conditions, excitation of the H-arm does not produce an output at the E-arm, and excitation of the E-arm does not produce an output at the H-arm.
Thus, the Magic Tee combines two complementary three-port tee behaviors into a single four-port microwave junction.
Basic Operating Principle of a Magic Tee
The fundamental operating principle of a Magic Tee is based on the superposition of electromagnetic fields and the different phase relationships associated with the E-arm and H-arm. The relative magnitude and phase of signals applied to the ports determine how the resulting electromagnetic fields combine at the remaining ports.
When equal-amplitude signals are applied to Ports 1 and 2 with the same phase, the signals combine at the H-arm. The E-arm ideally receives no signal because the corresponding field contributions cancel there. This is the sum operation.
When equal-amplitude signals are applied to Ports 1 and 2 with a \(180^\circ\) phase difference, the signals combine at the E-arm. The H-arm ideally receives no signal because the corresponding field contributions cancel there. This is the difference operation.
The two fundamental operations can therefore be summarized as
$\boxed{\text{In-phase inputs}\Rightarrow\text{Sum port (H-arm)}}$
and
$\boxed{\text{Opposite-phase inputs}\Rightarrow\text{Difference port (E-arm)}}$
The reverse operation is also important. Exciting the H-arm produces equal-magnitude, in-phase waves at the two collinear ports, while exciting the E-arm produces equal-magnitude, opposite-phase waves at the two collinear ports.
Under ideal conditions, the two side arms are isolated from each other. Therefore, an excitation applied to the H-arm does not appear at the E-arm, and an excitation applied to the E-arm does not appear at the H-arm. This isolation is one of the most important properties that distinguishes the Magic Tee from an ordinary three-port tee.
The ideal Magic Tee can therefore function as both a power divider and a power combiner. Depending on which port is excited and on the phase relationship between the signals, the junction can divide microwave power equally or combine signals into either the sum or difference port.
H-Arm Excitation
When a microwave signal is applied to the H-arm or Sum port, the signal is coupled equally to the two collinear ports. Since the H-arm is associated with the magnetic-field plane of the main waveguide, the electromagnetic field distribution produces equal-magnitude waves with the same phase at Ports 1 and 2.
Using Port 3 as the H-arm, an incident wave \(a_3\) produces outgoing waves \(b_1\) and \(b_2\) at the two collinear ports. The corresponding S-parameter relationships are
$\boxed{b_1=S_{13}a_3}$
and
$\boxed{b_2=S_{23}a_3}$
For an ideal symmetric Magic Tee, the two coupling coefficients are equal:
$\boxed{S_{13}=S_{23}}$
Therefore, an excitation at the H-arm produces two equal-magnitude and in-phase signals at the collinear ports. At the same time, the E-arm is ideally isolated from the H-arm.
E-Arm Excitation
When the E-arm or Difference port is excited, the signal is also divided between the two collinear ports. However, the phase relationship is different from that of the H-arm. The two output waves have equal magnitude but are \(180^\circ\) out of phase.
With Port 4 designated as the E-arm, an incident wave \(a_4\) produces outgoing waves at Ports 1 and 2 according to
$\boxed{b_1=S_{14}a_4}$
and
$\boxed{b_2=S_{24}a_4}$
For an ideal Magic Tee, the two coefficients have equal magnitude and opposite signs:
$\boxed{S_{14}=-S_{24}}$
Thus, excitation of the E-arm produces equal-magnitude but opposite-phase waves at the two collinear ports. The H-arm is ideally isolated from this excitation.
Equal-Magnitude and In-Phase Output from the H-Arm
The defining output characteristic of the H-arm is its equal-magnitude and equal-phase coupling to the two collinear ports. When Port 3 is excited, the power is divided equally between Ports 1 and 2 under ideal matched conditions.
The corresponding S-parameters satisfy
$\boxed{S_{13}=S_{23}}$
For an ideal lossless Magic Tee, each collinear port receives one-half of the incident power from the H-arm. Since microwave power is proportional to the squared magnitude of the normalized wave amplitude,
$|S_{13}|^2=|S_{23}|^2=\frac{1}{2}$
Therefore,
$\boxed{|S_{13}|=|S_{23}|=\frac{1}{\sqrt{2}}}$
With the conventional reference-phase choice, the coefficients have the same sign, so the two output waves are in phase. This equal-phase behavior is why Port 3 is called the Sum port.
Equal-Magnitude and Opposite-Phase Output from the E-Arm
The E-arm has the complementary phase behavior. When Port 4 is excited, the signal divides equally between the two collinear ports, but the resulting waves have opposite phase.
The defining relationship is
$\boxed{S_{14}=-S_{24}}$
For an ideal lossless Magic Tee, the magnitudes of the two coupling coefficients are equal:
$|S_{14}|=|S_{24}|=\frac{1}{\sqrt{2}}$
Therefore, the two collinear-port outputs contain equal power, but their phases differ by \(180^\circ\). This opposite-phase relationship is the characteristic property of the Difference port.
The distinction between the two side arms can therefore be summarized as
$\boxed{\text{H-arm: equal magnitude, same phase}}$
$\boxed{\text{E-arm: equal magnitude, opposite phase}}$
Isolation Between the E-Arm and H-Arm
One of the most important properties of an ideal Magic Tee is the isolation between the E-arm and H-arm. When a signal is applied to one side arm, no signal ideally appears at the other side arm.
With Port 3 as the H-arm and Port 4 as the E-arm, the isolation condition is expressed as
$\boxed{S_{34}=0}$
This means that an incident wave entering Port 4 does not produce an outgoing wave at Port 3. By reciprocity,
$\boxed{S_{43}=0}$
Therefore, the complete isolation relationship is
$\boxed{S_{34}=S_{43}=0}$
Physically, this isolation occurs because the field distribution produced by excitation of one side arm has the appropriate symmetry to cancel at the other side arm. The E-arm and H-arm therefore provide separate difference and sum signal paths.
Isolation Between the Two Collinear Ports
An ideal Magic Tee also has an important isolation property between the two collinear ports. When Port 1 is excited, the signal does not directly appear at Port 2 under the ideal matched and symmetric conditions. Similarly, excitation of Port 2 does not directly produce an output at Port 1.
This condition is represented by
$\boxed{S_{12}=0}$
and, by reciprocity,
$\boxed{S_{21}=0}$
Hence,
$\boxed{S_{12}=S_{21}=0}$
The zero values do not mean that Ports 1 and 2 are physically disconnected. Instead, the geometry and field symmetry of the Magic Tee cause the direct transmission between these two ports to cancel ideally. The applied signal is instead coupled to the appropriate side-arm modes.
Sum and Difference Operation
The most useful functional property of a Magic Tee is its ability to perform sum and difference operations. The result depends on the relative phase of the signals applied to the two collinear ports.
When two equal-amplitude signals are applied to Ports 1 and 2 with the same phase, their contributions add at the H-arm. The corresponding contributions at the E-arm cancel. Therefore, the H-arm acts as the Sum port.
This can be represented conceptually as
$\boxed{\text{In-phase inputs at Ports 1 and 2}\Rightarrow\text{output at H-arm}}$
When two equal-amplitude signals are applied to Ports 1 and 2 with a \(180^\circ\) phase difference, their contributions cancel at the H-arm and add at the E-arm. Therefore, the E-arm acts as the Difference port.
$\boxed{\text{Opposite-phase inputs at Ports 1 and 2}\Rightarrow\text{output at E-arm}}$
The two operations can therefore be summarized by the field symmetry of the junction. The H-arm responds to the even or sum component of the two collinear-port signals, whereas the E-arm responds to the odd or difference component.
Power Division and Power Combining
A Magic Tee can operate as both a power divider and a power combiner. When either side arm is excited, the incident power is ideally divided equally between the two collinear ports.
For H-arm excitation, the two collinear ports receive equal power:
$\boxed{|S_{13}|^2=|S_{23}|^2=\frac{1}{2}}$
For E-arm excitation, the same equal-power division occurs:
$\boxed{|S_{14}|^2=|S_{24}|^2=\frac{1}{2}}$
The difference is not in the amount of power delivered to the two collinear ports, but in their phase relationship. H-arm excitation produces equal-phase outputs, whereas E-arm excitation produces opposite-phase outputs.
The reverse process is power combining. When suitable signals are applied simultaneously to the two collinear ports, the Magic Tee combines them at one of the side arms according to their relative phase. Equal in-phase signals combine at the H-arm, while equal opposite-phase signals combine at the E-arm.
Thus, the same junction can perform two complementary functions: it can divide power from a side arm into two collinear paths, or combine signals from the collinear paths into either the Sum port or the Difference port.
Physical Interpretation of \(S_{34}=S_{43}=0\)
The parameters \(S_{34}\) and \(S_{43}\) describe transmission between the two side arms. With Port 3 defined as the H-arm and Port 4 defined as the E-arm, \(S_{34}\) represents the wave emerging from the H-arm when the E-arm is excited.
Therefore,
$\boxed{S_{34}=0}$
means that an ideal excitation at the E-arm produces no output at the H-arm. Similarly,
$\boxed{S_{43}=0}$
means that an excitation at the H-arm produces no output at the E-arm.
Because the Magic Tee is reciprocal, these two isolation parameters are equal. Hence,
$\boxed{S_{34}=S_{43}=0}$
This is the mathematical expression of E-arm to H-arm isolation.
Physical Interpretation of \(S_{12}=S_{21}=0\)
The parameters \(S_{12}\) and \(S_{21}\) describe direct transmission between the two collinear ports. Specifically, \(S_{12}\) represents the output at Port 1 caused by excitation at Port 2, while \(S_{21}\) represents the output at Port 2 caused by excitation at Port 1.
For an ideal Magic Tee, these direct transmission coefficients are zero:
$\boxed{S_{12}=S_{21}=0}$
This means that an excitation applied to one collinear port does not produce a direct transmitted wave at the other collinear port under the ideal junction conditions. Instead, the signal is distributed between the E-arm and H-arm according to the symmetry and phase relationships of the junction.
The zero values of \(S_{12}\) and \(S_{21}\), together with the zero values of \(S_{34}\) and \(S_{43}\), reveal the two important isolation pairs of the ideal Magic Tee:
$\boxed{S_{12}=S_{21}=0}$
for the two collinear ports, and
$\boxed{S_{34}=S_{43}=0}$
for the E-arm and H-arm.
These relationships form the foundation for deriving the complete four-port S-parameter matrix of the ideal Magic Tee. The next step is to express these physical properties mathematically using the general \(4\times4\) scattering matrix and then apply reciprocity, matching, symmetry, and lossless conditions.
General 4 × 4 S-Matrix of an E-H Plane Tee
An E-H plane tee, commonly known as a Magic Tee, is a four-port microwave junction. Because the junction has four possible incident waves and four corresponding outgoing waves, its scattering behavior is described by a \(4\times4\) S-matrix. The S-matrix provides a complete mathematical relationship between the waves incident on the four ports and the waves leaving those ports.
For the conventional port numbering, 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 incident and outgoing waves at these four ports are represented by \(a_1,a_2,a_3,a_4\) and \(b_1,b_2,b_3,b_4\), respectively.
Incident Waves \(a_1,a_2,a_3,a_4\)
In S-parameter analysis, \(a_i\) represents the normalized incident wave entering Port \(i\). Therefore, for the four-port Magic Tee, the incident-wave vector is
\[ \mathbf{a}= \begin{bmatrix} a_1\\ a_2\\ a_3\\ a_4 \end{bmatrix} \]
The individual quantities have the following physical meanings:
- \(a_1\) is the incident wave entering Collinear Port 1.
- \(a_2\) is the incident wave entering Collinear Port 2.
- \(a_3\) is the incident wave entering the H-arm or Sum port.
- \(a_4\) is the incident wave entering the E-arm or Difference port.
An incident wave at one port can produce outgoing waves at one or more of the other ports, depending on the coupling and isolation properties of the junction.
Outgoing Waves \(b_1,b_2,b_3,b_4\)
Similarly, \(b_i\) represents the normalized outgoing wave leaving Port \(i\). The outgoing-wave vector is therefore
\[ \mathbf{b}= \begin{bmatrix} b_1\\ b_2\\ b_3\\ b_4 \end{bmatrix} \]
The individual quantities represent:
- \(b_1\) is the outgoing wave from Collinear Port 1.
- \(b_2\) is the outgoing wave from Collinear Port 2.
- \(b_3\) is the outgoing wave from the H-arm.
- \(b_4\) is the outgoing wave from the E-arm.
The scattering matrix connects the incident-wave vector to the outgoing-wave vector according to
\[ \boxed{\mathbf{b}=[S]\mathbf{a}} \]
General \(4\times4\) S-Matrix
Before applying the physical properties of the Magic Tee, the most general four-port scattering matrix can be written as
\[ \boxed{ [S]= \begin{bmatrix} S_{11} & S_{12} & S_{13} & S_{14}\\ S_{21} & S_{22} & S_{23} & S_{24}\\ S_{31} & S_{32} & S_{33} & S_{34}\\ S_{41} & S_{42} & S_{43} & S_{44} \end{bmatrix} } \]
The matrix contains sixteen S-parameters. Each parameter describes the response at one output port resulting from excitation at one input port, with the remaining ports terminated in their reference impedances.
Physical Meaning of Every \(S_{ij}\)
The first subscript of \(S_{ij}\) identifies the port where the outgoing wave is observed, while the second subscript identifies the port where the incident wave is applied. Thus, \(S_{ij}\) is defined as
\[ \boxed{ S_{ij}= \left. \frac{b_i}{a_j} \right|_{\text{all other ports matched}} } \]
The sixteen elements can therefore be interpreted systematically.
Reflection coefficients: The diagonal elements \(S_{11},S_{22},S_{33}\), and \(S_{44}\) describe reflection at Ports 1, 2, 3, and 4, respectively.
Transmission and coupling coefficients: The off-diagonal elements describe coupling between different ports. For example, \(S_{13}\) represents the wave emerging from Port 1 when Port 3 is excited, while \(S_{31}\) represents the wave emerging from Port 3 when Port 1 is excited.
For the Magic Tee, these coefficients have additional relationships because of its symmetry, reciprocity, and isolation properties.
Reciprocity Condition
An ideal passive Magic Tee is 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 S-matrix is symmetric and satisfies
\[ \boxed{S_{ij}=S_{ji}} \]
For the four-port junction, this gives
\[ S_{12}=S_{21},\qquad S_{13}=S_{31},\qquad S_{14}=S_{41} \]
\[ S_{23}=S_{32},\qquad S_{24}=S_{42},\qquad S_{34}=S_{43} \]
This symmetry reduces the number of independent S-parameters that must be determined.
H-Arm Phase Relationship
When the H-arm is excited, the signal is divided equally between the two collinear ports. The two output waves have equal magnitude and equal phase. With Port 3 designated as the H-arm, this condition is expressed as
\[ \boxed{S_{13}=S_{23}} \]
By reciprocity, the corresponding reverse-direction coefficients also satisfy
\[ \boxed{S_{31}=S_{32}} \]
Therefore, excitation of the H-arm produces an in-phase sum signal at the two collinear ports.
E-Arm Phase Relationship
When the E-arm is excited, the signal is divided equally between the two collinear ports, but the two output waves have opposite phase. Therefore, when Port 4 is the E-arm,
\[ \boxed{S_{24}=-S_{14}} \]
By reciprocity, the corresponding reverse-direction coefficients satisfy
\[ \boxed{S_{42}=-S_{41}} \]
Thus, the E-arm produces a difference signal at the two collinear ports. The two output waves have equal magnitude and a phase difference of \(180^\circ\).
Isolation Between the H-Arm and E-Arm
One of the most important properties of an ideal Magic Tee is the isolation between its two side arms. The H-arm and E-arm are isolated from each other. Therefore, an excitation applied to the H-arm does not produce an output at the E-arm, and an excitation applied to the E-arm does not produce an output at the H-arm.
For Port 3 as the H-arm and Port 4 as the E-arm, this condition is
\[ \boxed{S_{34}=S_{43}=0} \]
This isolation is a fundamental feature of the ideal Magic Tee and is responsible for its ability to separate sum and difference signals.
Isolation Between the Two Collinear Ports
For an ideal Magic Tee, the two collinear ports are also isolated from each other. Therefore, an excitation applied at Port 1 does not directly appear at Port 2, and an excitation applied at Port 2 does not directly appear at Port 1.
The corresponding S-parameters are
\[ \boxed{S_{12}=S_{21}=0} \]
This isolation occurs because the contributions associated with the E-arm and H-arm have opposite and equal phase relationships in the appropriate combinations, resulting in cancellation at the other collinear port.
Matched Side-Arm Conditions
In an ideal Magic Tee, the H-arm and E-arm are assumed to be perfectly matched to the junction. Consequently, no portion of the incident wave at either side arm is reflected back toward the source. Therefore, their reflection coefficients are zero:
\[ \boxed{S_{33}=0} \]
and
\[ \boxed{S_{44}=0} \]
These matched-port conditions are important when deriving the complete ideal S-matrix. Once the symmetry, reciprocity, phase relationships, isolation, and matching conditions are applied, the general sixteen-element matrix can be reduced to a much simpler form.