H-Plane Tee Junction
H-Plane Tee Junction
An H-plane tee junction is a three-port microwave waveguide junction used to divide or combine microwave power within a waveguide system. It is formed by connecting an auxiliary waveguide arm to a main rectangular waveguide in such a way that the auxiliary arm lies in the H-plane of the dominant mode propagating in the main waveguide. The H-plane is the plane containing the direction of propagation and the magnetic-field vector of the dominant mode. This physical arrangement determines the electromagnetic coupling between the three ports and gives the H-plane tee its characteristic power-division and phase relationships. Because microwave signals are represented by traveling electromagnetic waves rather than by lumped circuit voltages and currents, the physical geometry and electromagnetic field distribution at the junction play an important role in determining how energy is transferred between the ports.
The H-plane tee is widely used in microwave engineering as a power-dividing and power-combining junction. When the H-arm is excited, the incident microwave power is coupled into the two collinear arms. Under ideal symmetric conditions, the two collinear ports receive equal-magnitude waves that are in phase. Conversely, signals applied appropriately to the two collinear ports can combine at the H-arm. This in-phase behavior is the characteristic feature of H-plane tee operation and distinguishes it from the E-plane tee, where excitation of the E-arm produces equal-magnitude waves at the two collinear ports with a phase difference of \(180^\circ\).
Definition and Basic Concept of an H-Plane Tee
An H-plane tee is defined as a three-port waveguide junction in which the auxiliary or side arm is connected to the main rectangular waveguide so that the junction lies in the H-plane of the dominant waveguide mode. The main waveguide contains two collinear ports, while the auxiliary waveguide forms the third port, commonly referred to as the H-arm. For a symmetric ideal junction, excitation of the H-arm produces equal-magnitude waves at the two collinear ports, and these waves are in phase.
The basic operation of an H-plane tee can be understood from the electromagnetic field distribution at the junction. When a microwave signal enters through the H-arm, the field configuration produces symmetrical coupling toward the two collinear arms. Under ideal symmetric conditions, the outgoing waves at the two collinear ports have equal magnitude and equal phase. Conversely, when signals are applied to the two collinear ports with the appropriate phase relationship, their fields can combine at the junction and produce a resultant wave at the H-arm. In particular, equal-amplitude, in-phase excitation of the two collinear ports corresponds to the sum-type operation of the H-plane tee.
The behavior of an H-plane tee is conveniently described using scattering parameters. At microwave frequencies, conventional measurements based directly on lumped-circuit voltage and current become less convenient because the waveguide is a distributed electromagnetic structure. Instead, the incident and outgoing traveling waves at each port are represented by normalized wave quantities. If the incident waves at the three ports are \(a_1\), \(a_2\), and \(a_3\), and the corresponding outgoing waves are \(b_1\), \(b_2\), and \(b_3\), the complete behavior of the three-port junction can be represented by a \(3\times3\) scattering matrix. The symmetry of the physical junction, together with reciprocity, matching, and lossless operation under ideal conditions, imposes specific relationships among the S-parameters. These relationships are used later to derive the ideal H-plane tee S-matrix.
Physical Construction of an H-Plane Tee
The physical construction of an H-plane tee begins with a main rectangular waveguide through which the microwave signal propagates. An additional waveguide section, known as the auxiliary or side arm, is connected to the main waveguide to form a three-port junction. The two ends of the original main waveguide form the two collinear ports, while the additional waveguide section forms the third port, known as the H-arm. The three waveguide arms meet at a common junction region where electromagnetic energy can be transferred from one port to the others.
The defining feature of the construction is the orientation of the auxiliary arm with respect to the electromagnetic field configuration of the dominant mode in the main rectangular waveguide. The auxiliary arm is positioned so that the junction lies in the H-plane of the dominant mode. This orientation determines the symmetry of the electromagnetic coupling between the H-arm and the two collinear arms. Under ideal symmetric conditions, excitation of the H-arm therefore produces equal-magnitude, in-phase waves at the two collinear ports.
In a conventional rectangular waveguide operating in its dominant TE10 mode, the electric and magnetic fields have specific spatial distributions within the waveguide. The H-plane is defined by the propagation direction and the magnetic-field direction of the mode. The side arm of an H-plane tee is arranged within this plane, which gives the junction its characteristic electromagnetic behavior. In practical microwave systems, the exact dimensions and geometry of the junction can be adjusted to obtain the required impedance matching, coupling, bandwidth, and power-division characteristics.
The three ports of the junction are physically distinct but electromagnetically interconnected through the common waveguide junction. Port 1 and Port 2 are typically located along the main waveguide and form the two collinear ports, while Port 3 is commonly assigned to the H-arm. This numbering is only a convention and is not a physical requirement. Different diagrams, textbooks, or examination problems may assign the H-arm to Port 1, Port 2, or Port 3. Regardless of the numbering convention, the physical construction remains the same: two collinear waveguide arms are connected through a common junction to a third auxiliary arm located in the H-plane.
Why It Is Called an H-Plane Tee
The name H-plane tee comes from the orientation of the junction with respect to the magnetic field of the dominant mode in the main rectangular waveguide. In a rectangular waveguide, the electromagnetic fields occupy specific spatial distributions determined by the propagation mode. For the dominant TE10 mode, the H-plane is defined as the plane containing the magnetic-field vector and the direction of wave propagation. The auxiliary arm of an H-plane tee is connected to the main waveguide so that the junction is formed in this H-plane. This electromagnetic orientation gives the junction its characteristic H-plane behavior.
The orientation of the side arm is the important feature that distinguishes an H-plane tee from an E-plane tee. In an H-plane tee, the auxiliary arm is connected in the H-plane of the dominant mode, whereas in an E-plane tee, the auxiliary arm is connected in the corresponding E-plane, which contains the electric-field vector and the direction of propagation. Although both structures are three-port waveguide junctions, their different electromagnetic geometries produce different relationships between the waves at the two collinear ports. For an ideal symmetric H-plane tee, excitation of the H-arm produces equal-magnitude waves at the two collinear ports that are in phase. This equal-phase relationship is a fundamental characteristic of H-plane tee operation.
The term H-plane therefore describes the electromagnetic orientation of the junction rather than simply its external physical shape. The letter H refers to the magnetic-field plane, just as the letter E refers to the electric-field plane in an E-plane tee. Understanding this naming convention is important because the plane in which the auxiliary arm is introduced determines the symmetry and phase relationship of the electromagnetic fields coupled between the three ports.
Other Names: Shunt Tee and Current Tee
The H-plane tee is also commonly known as a shunt tee because its auxiliary arm is connected to the main waveguide as a branch, producing a configuration that is analogous to a shunt connection in conventional transmission-line circuits. The two collinear arms form the main transmission path, while the H-arm provides an additional branch through which microwave energy can enter or leave the junction. The shunt terminology is therefore a circuit analogy used to describe the topology of the junction; the actual waveguide structure remains a distributed electromagnetic network rather than a lumped shunt circuit.
Another commonly used name for the H-plane tee is current tee. This terminology comes from the conventional equivalent-circuit interpretation of the junction, in which the H-plane tee is associated with a shunt or current-type junction. The term does not mean that the microwave junction operates according to ordinary low-frequency current division. Instead, its actual operation is governed by the electromagnetic fields and traveling waves within the waveguide. When the H-arm is excited under ideal symmetric conditions, the electromagnetic field couples equally into the two collinear arms, producing equal-magnitude, in-phase waves at those ports.
The different names therefore describe the same physical microwave component from different viewpoints. H-plane tee emphasizes the electromagnetic orientation of the junction, shunt tee emphasizes its branch-like connection to the main waveguide, and current tee refers to its conventional equivalent-circuit interpretation. These names should not be interpreted as different types of junctions. They refer to the same three-port waveguide structure and its characteristic H-plane mode of operation.
Basic Electromagnetic-Field Interpretation
The operation of an H-plane tee is fundamentally determined by the electromagnetic field distribution at the waveguide junction. When a microwave signal propagates through a rectangular waveguide, electric and magnetic fields are established according to the particular propagation mode. In the dominant TE10 mode, the electric and magnetic fields have specific spatial distributions and maintain the required relationship with the direction of propagation. When an auxiliary waveguide arm is connected in the H-plane, the resulting three-port geometry produces a symmetric electromagnetic coupling between the H-arm and the two collinear arms.
Consider a microwave signal incident at the H-arm, taken here as Port 3. Because an ideal H-plane tee is symmetric with respect to the two collinear arms, the incident wave at the H-arm couples equally into Ports 1 and 2. If the outgoing waves at the collinear ports are represented by \(b_1\) and \(b_2\), and the incident wave at the H-arm is represented by \(a_3\), the scattering relationships are
$ b_1=S_{13}a_3 $
and
$ b_2=S_{23}a_3 $
For an ideal symmetric H-plane tee, the two coupling coefficients are equal as complex quantities. Therefore,
$ \boxed{S_{13}=S_{23}} $
Because these are equal complex quantities, the equality implies both equal magnitude and equal phase. Thus,
$ \boxed{|S_{13}|=|S_{23}|} $
and
$ \boxed{\angle S_{13}=\angle S_{23}} $
This equal-phase relationship means that excitation of the H-arm produces equal-magnitude, in-phase waves at the two collinear ports. This is the fundamental electromagnetic characteristic of an ideal H-plane tee and is responsible for its sum-type behavior when the junction is excited from the collinear ports.
Now consider excitation from the two collinear ports. If the incident waves at Ports 1 and 2 are represented by \(a_1\) and \(a_2\), and the outgoing wave from the H-arm is represented by \(b_3\), the general scattering relationship is
$ b_3=S_{31}a_1+S_{32}a_2 $
For a reciprocal H-plane tee, the S-matrix is symmetric, so the relevant S-parameters satisfy
$ S_{31}=S_{13} $
and
$ S_{32}=S_{23} $
The H-plane symmetry condition gives \(S_{13}=S_{23}\). Therefore, the two coefficients contributing to the H-arm output are equal, and the previous equation becomes
$ \boxed{b_3=S_{13}a_1+S_{13}a_2} $
or, equivalently,
$ \boxed{b_3=S_{13}(a_1+a_2)} $
This equation provides the mathematical basis for the sum-type operation of the H-plane tee. When equal-amplitude, in-phase waves are applied to the two collinear ports, their contributions to the H-arm output have the same phase and therefore add constructively. If \(a_1=a_2\), the H-arm output becomes
$ b_3=2S_{13}a_1 $
On the other hand, if the two collinear ports are excited with equal amplitudes and opposite phases, such that \(a_2=-a_1\), the contributions cancel in the ideal symmetric junction:
$ b_3=S_{13}(a_1-a_1)=0 $
This cancellation follows directly from the equal coupling coefficients \(S_{31}=S_{32}\). It also demonstrates the complementary nature of the H-plane tee: the H-arm responds to the sum component of the two collinear-port excitations, while the equal-amplitude difference component does not couple to the H-arm under ideal symmetric conditions.
The electromagnetic behavior can therefore be summarized by the symmetry of the junction. Excitation of the H-arm produces equal-magnitude, in-phase waves at the two collinear ports. Conversely, excitation of the two collinear ports produces an H-arm output proportional to the sum of their incident waves. These field relationships provide the physical foundation for the S-parameter relationships that are subsequently derived using the properties of reciprocity, matching, and losslessness of the ideal H-plane tee.
Three Ports of the H-Plane Tee
An ideal H-plane tee is a three-port microwave waveguide junction consisting of two collinear ports and one auxiliary side port. The two ports located along the main waveguide are known as the collinear ports, while the auxiliary port is known as the H-arm. In the conventional port-numbering arrangement used throughout this analysis, the two collinear ports are designated as Port 1 and Port 2, while the H-arm is designated as Port 3. The port numbers are only a labeling convention, so they can be changed in different circuit diagrams or examination problems without changing the physical structure or electromagnetic operation of the junction.
Port 1 and Port 2 are the two collinear arms of the main waveguide. Their waveguide axes lie along the same straight line, which is why they are referred to as collinear ports. A microwave signal entering either collinear port propagates toward the junction, where its electromagnetic field interacts with the fields associated with the other two arms. For an ideal symmetric H-plane tee, the two collinear ports are electromagnetically equivalent with respect to the H-arm. This symmetry produces the characteristic equal-magnitude and equal-phase coupling between the H-arm and the two collinear ports.
Port 3 is the auxiliary port and is conventionally identified as the H-arm. The H-arm is connected to the main waveguide so that the junction is formed in the H-plane of the dominant mode. It provides an additional path through which microwave power can enter or leave the junction. When the H-arm is excited, the incident power is coupled into the two collinear ports. Under ideal symmetric conditions, the resulting waves at Ports 1 and 2 have equal magnitude and are in phase. Conversely, when the two collinear ports are excited with an appropriate phase relationship, their contributions can combine at the H-arm. In particular, the in-phase component of the two collinear-port excitations couples to the H-arm, while the equal-amplitude out-of-phase component cancels there under ideal symmetric conditions.
The three ports can therefore be represented by the incident traveling-wave quantities \(a_1\), \(a_2\), and \(a_3\), and the corresponding outgoing traveling-wave quantities \(b_1\), \(b_2\), and \(b_3\). The relationship between these incident and outgoing waves is described by the three-port scattering matrix:
$\begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix}=\begin{bmatrix} S_{11} & S_{12} & S_{13} \\ S_{21} & S_{22} & S_{23} \\ S_{31} & S_{32} & S_{33} \end{bmatrix}\begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix}$
Here, the first subscript of \(S_{ij}\) identifies the port at which the outgoing wave is observed, while the second subscript identifies the port at which the incident wave is applied. For example, \(S_{13}\) represents the ratio of the outgoing wave at Port 1 to the incident wave at Port 3 when the other ports are terminated in their reference impedances. Thus,
$S_{13}=\left.\frac{b_1}{a_3}\right|_{a_1=a_2=0}$
Similarly, \(S_{23}\) represents the transmission from Port 3 to Port 2, while \(S_{33}\) represents the reflection coefficient at Port 3 when Ports 1 and 2 are terminated in their reference impedances. This notation provides a systematic way to describe the reflection, transmission, and coupling behavior of the H-plane tee under different excitation conditions.
Operating Cases of an H Plane Tee
When Port 3 of an H Plane Tee is terminated with a matched load, its operation can be described by considering the excitation applied at the ports. The H Plane Tee has an important property that the signal entering the H arm is divided equally between the two collinear arms with the same phase. Similarly, signals applied equally and in phase at the two collinear ports combine constructively at the H arm. The following two cases explain the 3 dB power splitting and in phase power combining operation of the H Plane Tee.
Case 1: Input at Port 3 and No Inputs at Ports 1 and 2
Consider an input signal applied at Port 3, which is the H arm, while there are no input signals at Ports 1 and 2. Therefore, the input conditions are
\[ a_3\neq0,\qquad a_1=a_2=0 \]
For an ideal H Plane Tee, the signal entering through the H arm is divided equally between the two collinear arms. The output signals at Ports 1 and 2 have equal magnitude and are in phase. Therefore,
\[ b_1=\frac{1}{\sqrt{2}}a_3 \]
\[ b_2=\frac{1}{\sqrt{2}}a_3 \]
Since Port 3 is matched, there is no reflected wave at that port, giving
\[ b_3=0 \]
The power delivered to each collinear port is proportional to the square of the magnitude of its output wave. Therefore,
\[ |b_1|^2=\frac{1}{2}|a_3|^2 \]
and
\[ |b_2|^2=\frac{1}{2}|a_3|^2 \]
Thus, each output port receives half of the input power. In decibels, half of the input power corresponds to
\[ 10\log_{10}\left(\frac{1}{2}\right)=-3.01\text{ dB} \]
Hence, the power coming out of either Port 1 or Port 2 is approximately 3 dB down with respect to the input power at Port 3. Therefore, the H Plane Tee is called a 3 dB splitter.
Case 2: Equal Inputs at Ports 1 and 2 and No Input at Port 3
Now consider equal input signals applied simultaneously to Ports 1 and 2, while there is no input at Port 3. 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 H arm add constructively. The output waves at Ports 1 and 2 are
\[ b_1=\frac{1}{2}a-\frac{1}{2}a=0 \]
\[ b_2=-\frac{1}{2}a+\frac{1}{2}a=0 \]
Thus, there is no resulting output at Ports 1 and 2 because the reflected contributions cancel under the matched and symmetric condition.
At Port 3, however, the two input signals combine constructively. Therefore,
\[ b_3=\frac{1}{\sqrt{2}}a+\frac{1}{\sqrt{2}}a \]
or
\[ b_3=\sqrt{2}a \]
The two input signals arriving at Port 3 have the same phase. Hence, the signals are added in phase, producing a larger resultant signal at the H arm.
Therefore, when equal and in phase signals are applied to Ports 1 and 2, they combine at Port 3. This demonstrates the in phase power combining property of the H Plane Tee.
3 dB Splitting and In Phase Combining in H Plane Tee
The two operating cases show the fundamental behavior of an H Plane Tee. When Port 3 is excited, the input power is equally divided between Ports 1 and 2, with both output signals having the same phase. Each port receives half of the input power, corresponding to a 3 dB power division. Conversely, when equal and in phase signals are applied to Ports 1 and 2, the signals combine constructively at Port 3.
- Input at Port 3: Power is equally divided between Ports 1 and 2.
- Output power at each collinear port: Half of the input power, or approximately 3 dB below the input power.
- Phase relationship: Outputs at Ports 1 and 2 are in phase.
- Equal inputs at Ports 1 and 2: The signals combine constructively at Port 3.
- Application: The H Plane Tee can therefore operate as a 3 dB power splitter and an in phase power combiner.
Operation When the H-Arm Is Excited
Consider a microwave signal applied only to the H-arm, which is taken as Port 3. Ports 1 and 2 are terminated in their characteristic impedances, so there are no incident waves returning toward the junction from these ports. Under this condition, the only incident wave is \(a_3\). The incident wave at the H-arm interacts with the electromagnetic fields at the junction and couples power into the two collinear arms.
With only Port 3 excited, the outgoing waves at Ports 1 and 2 are described by the scattering-parameter equations
\[ b_1=S_{13}a_3 \]
and
\[ b_2=S_{23}a_3 \]
The H-plane tee is symmetric with respect to its two collinear ports. Therefore, when the H-arm is excited, the two collinear ports receive equal power. This gives the magnitude relationship
\[ \boxed{|S_{13}|=|S_{23}|} \]
For an ideal symmetric H-plane tee, the waves coupled from the H-arm to the two collinear ports are in phase. Thus, with a consistent choice of reference planes and phase reference, the corresponding S-parameters satisfy
\[ \boxed{S_{13}=S_{23}} \]
This relationship is the important phase characteristic of the H-plane tee. It means that the two waves have equal magnitude and equal phase. If \(S_{13}=A\), then \(S_{23}=A\), and the outgoing waves become
\[ b_1=Aa_3 \]
and
\[ b_2=Aa_3 \]
Hence, under the ideal symmetric condition,
\[ \boxed{b_1=b_2} \]
The equality of the two outgoing waves explains the sum-type behavior of the H-plane tee. The two collinear ports receive waves having the same amplitude and phase when the H-arm is excited. Consequently, when equal-amplitude and in-phase waves are applied simultaneously to the two collinear ports, their contributions toward the H-arm add constructively.
For an ideal lossless H-plane tee with the H-arm matched and the power divided equally between the two collinear ports, conservation of power requires the incident power at Port 3 to be divided equally between Ports 1 and 2. Therefore,
\[ |S_{13}|^2=|S_{23}|^2=\frac{1}{2} \]
and hence
\[ \boxed{|S_{13}|=|S_{23}|=\frac{1}{\sqrt{2}}} \]
With the conventional phase reference used for the ideal H-plane tee, the two coupling coefficients can therefore be written as
\[ \boxed{S_{13}=S_{23}=\frac{1}{\sqrt{2}}} \]
Because an ideal reciprocal junction satisfies \(S_{ij}=S_{ji}\), the corresponding reverse-direction coefficients are also equal:
\[ \boxed{S_{31}=S_{32}=\frac{1}{\sqrt{2}}} \]
Thus, the H-arm excitation produces two equal-power, in-phase waves at the collinear ports. This is the principal phase characteristic that distinguishes the H-plane tee from the E-plane tee. In an ideal E-plane tee, excitation of the side arm produces equal-magnitude waves at the two collinear ports with a \(180^\circ\) phase difference, whereas the H-plane tee produces equal-magnitude waves with the same phase.
It is important to distinguish the physical phase relationship from an isolated plus or minus sign in an S-matrix. The numerical signs of individual S-parameters can depend on the selected reference-plane and wave-direction conventions. The physically significant property is that the two responses from the H-arm have equal magnitude and equal phase under the chosen consistent reference convention.
Operation When the Collinear Ports Are Excited
The operation of the H-plane tee can also be understood by exciting the two collinear ports. Let \(a_1\) and \(a_2\) represent the incident waves entering the junction through Ports 1 and 2, respectively. The resulting outgoing wave at the H-arm, represented by \(b_3\), is determined by the contributions from both collinear ports.
The general S-parameter relationship for the H-arm is
$b_3=S_{31}a_1+S_{32}a_2$
For a reciprocal waveguide junction, the scattering matrix is symmetric, so
$S_{31}=S_{13}$
and
$S_{32}=S_{23}$
For the ideal H-plane tee, the two coupling coefficients associated with excitation of the H-arm are equal in magnitude and phase. With the standard reference-plane convention used for the ideal H-plane tee,
$S_{13}=S_{23}$
Therefore, by reciprocity,
$S_{31}=S_{32}$
and the output at the H-arm becomes
$b_3=S_{13}a_1+S_{13}a_2$
or
$\boxed{b_3=S_{13}(a_1+a_2)}$
This equation demonstrates the sum behavior of the H-plane tee. The contributions from Port 1 and Port 2 have the same coupling coefficient and therefore add constructively when the two incident waves have the same phase. If equal-amplitude in-phase signals are applied to the two collinear ports, then
$a_1=a_2$
and consequently
$b_3=S_{13}(a_1+a_1)$
which gives
$\boxed{b_3=2S_{13}a_1}$
For the ideal lossless and matched H-plane tee, the H-arm coupling coefficient has magnitude
$|S_{13}|=\frac{1}{\sqrt{2}}$
Therefore, for equal-amplitude in-phase excitation of the two collinear ports, the magnitude of the H-arm output is
$|b_3|=\sqrt{2}|a_1|$
This does not represent power gain because both collinear ports are supplying power to the junction. The total output power remains consistent with the lossless condition.
Thus, the contributions from the two collinear ports reinforce each other at the H-arm when the incident signals have equal amplitude and equal phase. This is the fundamental reason the H-plane tee is commonly described as a sum junction.
The opposite situation occurs when equal-amplitude signals are applied to Ports 1 and 2 with a \(180^\circ\) phase difference. In that case,
$a_1=-a_2$
and the output at the H-arm becomes
$b_3=S_{13}(a_1-a_1)=0$
Therefore, equal-amplitude but opposite-phase signals applied to the two collinear ports produce no output at the H-arm under ideal symmetric conditions. This complementary behavior means that the H-arm responds to the sum component of the two collinear-port excitations, while the difference component produces no H-arm output under ideal symmetric conditions.
The two operating conditions can therefore be understood together. When the H-arm is excited, the microwave power is divided equally between the two collinear ports, with the resulting waves having equal magnitude and equal phase. When the two collinear ports are excited, signals having equal amplitude and equal phase combine constructively at the H-arm, whereas equal-amplitude signals with opposite phase cancel at the H-arm. These relationships provide the physical basis for deriving the H-plane tee S-parameter matrix using reciprocity, symmetry, matching, and the lossless condition.
Power Division and Phase Relationship
The most important operating characteristic of an H-plane tee is the way in which microwave power is divided between its two collinear ports when the H-arm is excited. Consider an ideal symmetric H-plane tee in which Port 3 is the H-arm and Ports 1 and 2 are the collinear ports. When an incident wave is applied to Port 3, the electromagnetic energy entering the junction is coupled equally into Ports 1 and 2 because of the symmetry of the junction.
If \(a_3\) represents the incident wave at the H-arm, the outgoing waves at the two collinear ports are
$b_1=S_{13}a_3$
and
$b_2=S_{23}a_3$
For an ideal symmetric H-plane tee, the two waves have equal magnitude and the same phase. Therefore,
$\boxed{S_{13}=S_{23}}$
The equality of these two S-parameters means that the H-arm couples the incident energy equally into the two collinear arms. In terms of magnitude and phase, this relationship can be expressed as
$\boxed{|S_{13}|=|S_{23}|}$
and
$\boxed{\angle S_{13}=\angle S_{23}}$
Thus, the waves emerging from Ports 1 and 2 are in phase. This is the defining phase characteristic of an H-plane tee. It distinguishes the H-plane tee from an E-plane tee, in which excitation of the side arm produces equal-magnitude waves at the collinear ports with a \(180^\circ\) phase difference.
For an ideal lossless H-plane tee with the H-arm properly matched, \(S_{33}=0\), so no power is reflected back toward the H-arm when Port 3 is excited. Conservation of power therefore requires the total power delivered to Ports 1 and 2 to equal the incident power at Port 3. Hence,
$|S_{13}|^2+|S_{23}|^2=1$
Since the two magnitudes are equal,
$|S_{13}|^2=|S_{23}|^2$
and hence
$2|S_{13}|^2=1$
which gives
$\boxed{|S_{13}|=|S_{23}|=\frac{1}{\sqrt{2}}}$
Consequently, the ideal power division is
$\boxed{|S_{13}|^2=|S_{23}|^2=\frac{1}{2}}$
Therefore, one-half of the incident power at the H-arm is directed toward Port 1 and the other half is directed toward Port 2. With the standard reference-plane convention used for the ideal H-plane tee, the corresponding coupling coefficients are positive:
$\boxed{S_{13}=S_{23}=S_{31}=S_{32}=\frac{1}{\sqrt{2}}}$
The positive signs of these H-arm coupling coefficients indicate the equal-phase relationship between the H-arm and the two collinear ports under this convention. The negative signs in the ideal H-plane tee S-matrix occur in the reflection and transmission coefficients associated with the collinear ports, not in the H-arm coupling coefficients.
Because the two waves produced at the collinear ports are also in phase, the H-plane tee exhibits sum-junction behavior. When signals are applied to the two collinear ports with equal amplitude and equal phase, their contributions at the H-arm reinforce one another.
The output at the H-arm can be written as
$b_3=S_{31}a_1+S_{32}a_2$
For a reciprocal junction,
$S_{31}=S_{13}$
and
$S_{32}=S_{23}$
Since \(S_{13}=S_{23}\) for the H-plane tee, the output becomes
$\boxed{b_3=S_{13}(a_1+a_2)}$
This equation clearly demonstrates the sum operation at the H-arm. If \(a_1=a_2\), the two contributions add constructively at the H-arm. If \(a_1=-a_2\), the two contributions cancel and the ideal output at the H-arm becomes zero.
Thus, the phase relationship is not merely a mathematical property of the S-matrix; it directly determines how microwave signals combine and divide at the junction. The H-plane tee therefore provides equal-power, in-phase division when the H-arm is excited and sum-type combining when equal-amplitude, in-phase signals are applied to the two collinear ports.
H-Plane Tee S-Parameter Matrix
The complete electromagnetic behavior of an H-plane tee can be represented by a \(3\times3\) scattering matrix. Consider the conventional port numbering arrangement in which Ports 1 and 2 are the two collinear ports and Port 3 is the H-arm. 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 S-parameter relationship is
$ \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} $
Each S-parameter \(S_{ij}\) represents the response observed at Port \(i\) due to an incident wave applied at Port \(j\), with the other ports terminated in their reference impedances. Therefore, the first subscript identifies the output port and the second subscript identifies the input port.
For example, \(S_{13}\) represents the wave emerging from Port 1 when Port 3 is excited, whereas \(S_{31}\) represents the wave emerging from Port 3 when Port 1 is excited. This distinction between the two indices is essential when deriving and identifying the S-matrix of the H-plane tee.
Reciprocity Condition
An ordinary passive waveguide H-plane tee is a reciprocal junction. For a reciprocal network, the scattering matrix is symmetric, so
$ \boxed{S_{ij}=S_{ji}} $
Therefore, the off-diagonal S-parameters satisfy
$ S_{12}=S_{21} $
$ S_{13}=S_{31} $
and
$ S_{23}=S_{32} $
Reciprocity therefore allows the corresponding transmission coefficients in opposite directions to be treated as equal under the same reference-plane convention.
Symmetry of the H-Arm Coupling
For an ideal symmetric H-plane tee, excitation of the H-arm produces equal-magnitude waves at the two collinear ports. The two waves are also in phase. With Port 3 designated as the H-arm, this gives
$ \boxed{S_{13}=S_{23}} $
The equality indicates that the two collinear ports receive equal responses when the H-arm is excited. In magnitude and phase form, the same physical condition can be expressed as
$ \boxed{|S_{13}|=|S_{23}|} $
and
$ \boxed{\angle S_{13}=\angle S_{23}} $
Using reciprocity, the corresponding coefficients in the third row also satisfy
$ \boxed{S_{31}=S_{32}} $
The equality of these coefficients represents the characteristic equal-magnitude, in-phase coupling between the H-arm and the two collinear ports. The exact numerical sign used in an S-matrix depends on the selected reference-plane and wave-direction conventions, but the physical phase relationship is the important property.
Matched H-Arm Condition
If the H-arm is assumed to be perfectly matched to the reference impedance, there is no reflected wave at Port 3 when Port 3 is excited alone. Therefore, its reflection coefficient is zero:
$ \boxed{S_{33}=0} $
Combining reciprocity, the symmetry of the ideal H-plane tee, and the matched H-arm condition gives the reduced S-matrix form
$ \boxed{ [S]= \begin{bmatrix} S_{11} & S_{12} & S_{13}\\ S_{12} & S_{22} & S_{13}\\ S_{13} & S_{13} & 0 \end{bmatrix} } $
At this stage, the remaining coefficients have not yet been assigned their numerical values. They are determined by applying the lossless condition together with the symmetry and idealized properties of the junction.
Lossless Condition
For an ideal lossless microwave junction, the total incident power must equal the total outgoing power. In S-parameter form, this condition is expressed by the unitarity of the scattering matrix:
$ \boxed{[S][S]^\dagger=[I]} $
Here, \([S]^\dagger\) denotes the conjugate transpose of the S-matrix and \([I]\) is the identity matrix. The lossless condition provides the equations required to determine the unknown coefficients in the idealized H-plane tee matrix.
For the standard ideal symmetric H-plane tee, the resulting normalized S-matrix is
$ \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} } $
This matrix represents the standard ideal reference-plane convention for an H-plane tee with the H-arm assigned to Port 3. The two \(1/\sqrt{2}\) terms in the third column show that excitation of the H-arm produces equal-amplitude responses at Ports 1 and 2. Their equal signs indicate an in-phase relationship under this standard phase convention.
By reciprocity, the same equal coupling appears in the third row. The zero at \(S_{33}\) represents the assumed matched H-arm. The four \(1/2\) terms form the collinear-port submatrix and describe the reflection and coupling relationships between Ports 1 and 2 in this idealized model.
Interpretation of the Ideal S-Matrix
The ideal H-plane tee matrix can therefore be written in terms of its characteristic elements as
$ \boxed{ S_{11}=S_{12}=S_{21}=S_{22}=\frac{1}{2} } $
and
$ \boxed{ S_{13}=S_{23}=S_{31}=S_{32}=\frac{1}{\sqrt{2}} } $
with
$ \boxed{S_{33}=0} $
These numerical relationships apply to the standard ideal symmetric H-plane tee model under the stated reference-plane convention. They should not be interpreted as universal numerical values for every practical H-plane tee, because real junctions can have losses, imperfect matching, asymmetry, and additional phase shifts caused by reference-plane choices.
The most important features to recognize in an examination are therefore the zero corresponding to the matched H-arm, the equal \(1/\sqrt{2}\) coupling coefficients between the H-arm and the two collinear ports, and the equal-sign coupling pattern that represents the in-phase relationship of the two collinear-port waves when the H-arm is excited.
For the conventional arrangement with Port 3 as the H-arm, the characteristic matrix is consequently
$ \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} } $
If a different port numbering convention is used, the physical H-plane tee does not change. Only the corresponding rows and columns of the S-matrix are rearranged. Therefore, the same physical relationships can appear in different matrix positions when the H-arm is assigned to Port 1, Port 2, or Port 3.
Reciprocity Property
An ideal passive waveguide H-plane tee is a reciprocal network. Reciprocity means that the transmission coefficient between two ports is equal to the corresponding transmission coefficient when the direction of excitation is reversed, provided the same reference-plane and port conditions are used. For a reciprocal network, the scattering matrix is symmetric and therefore satisfies
$\boxed{S_{ij}=S_{ji}}$
Applying this property to the three-port H-plane tee gives
$S_{12}=S_{21}$
$S_{13}=S_{31}$
$S_{23}=S_{32}$
Therefore, the general reciprocal S-matrix can be written as
$\boxed{[S]=\begin{bmatrix}S_{11}&S_{12}&S_{13}\\S_{12}&S_{22}&S_{23}\\S_{13}&S_{23}&S_{33}\end{bmatrix}}$
This property reduces the number of independent S-parameters that must be determined. Instead of treating all nine matrix elements as unrelated quantities, the three pairs \(S_{12}\) and \(S_{21}\), \(S_{13}\) and \(S_{31}\), and \(S_{23}\) and \(S_{32}\) are known to be equal.
H-Arm Phase Relationship
The defining electromagnetic characteristic of the H-plane tee is the phase relationship between the waves appearing at the two collinear ports when the H-arm is excited. With Port 3 designated as the H-arm, excitation at Port 3 produces equal-magnitude waves at Ports 1 and 2 that are in phase under the standard symmetric reference-plane convention. Therefore, the corresponding S-parameters satisfy
$\boxed{S_{13}=S_{23}}$
More generally, the physical condition is that the two coupling coefficients have equal magnitude and equal phase:
$\boxed{|S_{13}|=|S_{23}|}$
and
$\boxed{\angle S_{13}=\angle S_{23}}$
The equal-sign form \(S_{13}=S_{23}\) is therefore a convenient representation of the in-phase relationship when the reference planes and phase convention are chosen consistently. This distinction is important because the numerical signs of individual S-parameters can depend on the chosen reference-plane convention, whereas the physical phase relationship between the waves is the fundamental property.
Using reciprocity,
$S_{13}=S_{31}$
and
$S_{23}=S_{32}$
Therefore, the equality of the H-arm coupling coefficients also gives
$\boxed{S_{31}=S_{32}}$
Thus, when Port 3 is the H-arm, the equal coupling coefficients in the third row and third column form the characteristic S-parameter signature of the ideal H-plane tee.
Matched H-Arm Condition
If the H-arm is perfectly matched to the reference impedance, an incident wave entering Port 3 produces no reflected wave back toward the source. The reflection coefficient at Port 3 is therefore zero. In S-parameter notation, this condition is expressed as
$\boxed{S_{33}=0}$
The condition \(S_{33}=0\) means that Port 3 is matched under the reference-plane and termination conditions used to define the S-parameters. In a practical waveguide tee, the discontinuity at the junction can produce reflections, so suitable mechanical design or tuning may be required to obtain a desired match. The ideal theoretical model assumes that the H-arm is perfectly matched.
It is important not to assume that \(S_{11}\) and \(S_{22}\) must also be zero merely because the H-arm is matched. The condition \(S_{33}=0\) specifically describes the reflection coefficient at the H-arm. The reflection coefficients at the two collinear ports are determined separately from the symmetry, reciprocity, and lossless conditions of the complete junction.
Reduced S-Matrix
We can now combine the three important properties obtained so far. Reciprocity gives \(S_{ij}=S_{ji}\), the H-plane phase relationship gives \(S_{13}=S_{23}\) under the standard symmetric reference-plane convention, and the matched H-arm condition gives \(S_{33}=0\).
Starting from the reciprocal matrix
$[S]=\begin{bmatrix}S_{11}&S_{12}&S_{13}\\S_{12}&S_{22}&S_{23}\\S_{13}&S_{23}&S_{33}\end{bmatrix}$
we substitute
$S_{23}=S_{13}$
and
$S_{33}=0$
to obtain
$\boxed{[S]=\begin{bmatrix}S_{11}&S_{12}&S_{13}\\S_{12}&S_{22}&S_{13}\\S_{13}&S_{13}&0\end{bmatrix}}$
This reduced matrix contains four apparently unknown quantities: \(S_{11}\), \(S_{12}\), \(S_{22}\), and \(S_{13}\). The remaining coefficients are already related through reciprocity and the H-arm phase relationship. The next step is to use the symmetry of the two collinear ports together with the lossless condition to obtain additional equations and determine these unknown quantities.
Lossless and Unitary Property
An ideal waveguide H-plane tee is assumed to be lossless. This means that no microwave power is dissipated inside the junction. The total incident power must therefore equal the total outgoing power. In S-parameter theory, a lossless network has a unitary scattering matrix. The mathematical condition is
$\boxed{[S][S]^\dagger=[I]}$
where \([S]^\dagger\) denotes the conjugate transpose of the S-matrix and \([I]\) is the identity matrix. For the reduced H-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 diagonal elements of this product provide power-conservation equations, while the off-diagonal elements provide orthogonality equations. These equations determine the magnitudes and relationships among the remaining S-parameters.
Step-by-Step Derivation of the Unitary Equations
The first diagonal element is obtained by multiplying the first row of \([S]\) by the first column of \([S]^\dagger\):
$\begin{bmatrix}S_{11}&S_{12}&S_{13}\end{bmatrix}\begin{bmatrix}S_{11}^{*}\\S_{12}^{*}\\S_{13}^{*}\end{bmatrix}=1$
Therefore,
$S_{11}S_{11}^{*}+S_{12}S_{12}^{*}+S_{13}S_{13}^{*}=1$
Using \(SS^{*}=|S|^2\), we obtain
$\boxed{|S_{11}|^2+|S_{12}|^2+|S_{13}|^2=1}$
The second diagonal element is obtained from the second row and second column:
$\begin{bmatrix}S_{12}&S_{22}&S_{13}\end{bmatrix}\begin{bmatrix}S_{12}^{*}\\S_{22}^{*}\\S_{13}^{*}\end{bmatrix}=1$
which gives
$\boxed{|S_{12}|^2+|S_{22}|^2+|S_{13}|^2=1}$
The third diagonal element is particularly useful because the third row contains two identical coupling coefficients:
$\begin{bmatrix}S_{13}&S_{13}&0\end{bmatrix}\begin{bmatrix}S_{13}^{*}\\S_{13}^{*}\\0\end{bmatrix}=1$
Therefore,
$S_{13}S_{13}^{*}+S_{13}S_{13}^{*}=1$
giving
$2|S_{13}|^2=1$
and hence
$\boxed{|S_{13}|=\frac{1}{\sqrt{2}}}$
This result confirms that, for the ideal matched and lossless junction, the incident power at the H-arm is divided equally between the two collinear ports.
Next, consider the off-diagonal element in the first row and third column. Because this is an off-diagonal element of the identity matrix, its value must be zero:
$\begin{bmatrix}S_{11}&S_{12}&S_{13}\end{bmatrix}\begin{bmatrix}S_{13}^{*}\\S_{13}^{*}\\0\end{bmatrix}=0$
Therefore,
$S_{11}S_{13}^{*}+S_{12}S_{13}^{*}=0$
Factoring \(S_{13}^{*}\),
$S_{13}^{*}(S_{11}+S_{12})=0$
Since \(|S_{13}|=1/\sqrt{2}\), \(S_{13}\) is nonzero. Therefore,
$\boxed{S_{11}+S_{12}=0}$
which gives
$\boxed{S_{11}=-S_{12}}$
Similarly, the second row and third column give
$S_{12}S_{13}^{*}+S_{22}S_{13}^{*}=0$
and therefore
$\boxed{S_{22}=-S_{12}}$
Combining the two relationships gives
$\boxed{S_{11}=S_{22}=-S_{12}}$
The remaining diagonal equation can now be used to determine the magnitude of the collinear-port coefficients. From
$|S_{11}|^2+|S_{12}|^2+|S_{13}|^2=1$
and
$|S_{13}|^2=\frac{1}{2}$
we obtain
$|S_{11}|^2+|S_{12}|^2=\frac{1}{2}$
Since \(S_{11}=-S_{12}\), their magnitudes are equal:
$|S_{11}|=|S_{12}|$
Therefore,
$2|S_{12}|^2=\frac{1}{2}$
which gives
$|S_{12}|^2=\frac{1}{4}$
and hence
$\boxed{|S_{12}|=\frac{1}{2}}$
Because \(S_{11}=-S_{12}\) and \(S_{22}=-S_{12}\), the corresponding magnitudes are also
$\boxed{|S_{11}|=|S_{22}|=\frac{1}{2}}$
Solve the Equations to Obtain the Unknown S-Parameters
The unitary equations have now established the magnitudes of the unknown coefficients and the relationships among them. We have obtained
$\boxed{|S_{13}|=\frac{1}{\sqrt{2}}}$
and
$\boxed{|S_{11}|=|S_{12}|=|S_{22}|=\frac{1}{2}}$
The phase relationships are determined by
$S_{11}=-S_{12}$
and
$S_{22}=-S_{12}$
so that
$\boxed{S_{11}=S_{22}=-S_{12}}$
At this stage, the unitary equations determine the relative sign between the collinear-port coefficients, but they do not by themselves determine which coefficient is positive and which is negative. A consistent choice of reference planes and wave-phase convention is required to assign the final signs.
For the commonly used textbook convention, the H-arm coupling coefficients are chosen as positive:
$S_{13}=S_{23}=\frac{1}{\sqrt{2}}$
With one standard phase convention, the remaining coefficients are then written as
$S_{11}=S_{22}=\frac{1}{2}$
and
$S_{12}=-\frac{1}{2}$
By reciprocity,
$S_{21}=S_{12}=-\frac{1}{2}$
and
$S_{31}=S_{13}=\frac{1}{\sqrt{2}}$
$S_{32}=S_{23}=\frac{1}{\sqrt{2}}$
Together with the matched H-arm condition \(S_{33}=0\), all nine S-parameters are determined for this particular reference-plane convention.
An equally valid phase convention can reverse the signs of the collinear-port coefficients while preserving the physical behavior and the unitary condition. Therefore, the physically important result is the relative relationship
$S_{11}=S_{22}=-S_{12}$
rather than memorizing the isolated signs without considering the reference convention.
Complete Ideal H-Plane Tee S-Matrix
Using the commonly adopted reference-plane convention described above, the complete ideal reciprocal and lossless H-plane tee S-matrix is
$\boxed{[S]=\begin{bmatrix}\frac{1}{2}&-\frac{1}{2}&\frac{1}{\sqrt{2}}\\-\frac{1}{2}&\frac{1}{2}&\frac{1}{\sqrt{2}}\\\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0\end{bmatrix}}$
This matrix contains several important physical features. The diagonal element \(S_{33}=0\) represents the matched H-arm. The equal positive values \(S_{13}=S_{23}=1/\sqrt{2}\) show that excitation of the H-arm produces equal-magnitude, in-phase waves at the two collinear ports. The values \(S_{11}=S_{22}=1/2\) represent the reflection coefficients at the two collinear ports under this reference-plane convention, while \(S_{12}=S_{21}=-1/2\) represents the coupling between the two collinear ports.
The most important feature for identifying an H-plane tee is therefore the equal coupling relationship
$\boxed{S_{13}=S_{23}}$
when Port 3 is the H-arm. This indicates equal magnitude and equal phase at the two collinear ports under the standard symmetric reference-plane convention. The corresponding ideal matrix satisfies reciprocity and the lossless unitary condition and provides the mathematical basis for analyzing the power division, signal combining, and phase behavior of the H-plane tee.
Physical Meaning of Every Matrix Element
The scattering matrix of an H-plane tee provides a complete mathematical description of how incident microwave signals are reflected, transmitted, and coupled between its three ports. For a three-port network, the relationship between the incident waves \(a_1,a_2,a_3\) and the outgoing waves \(b_1,b_2,b_3\) is written as
$\begin{bmatrix} b_1\\ b_2\\ b_3 \end{bmatrix}=\begin{bmatrix} S_{11} & S_{12} & S_{13}\\ S_{21} & S_{22} & S_{23}\\ S_{31} & S_{32} & S_{33} \end{bmatrix}\begin{bmatrix} a_1\\ a_2\\ a_3 \end{bmatrix}$
The first subscript of \(S_{ij}\) identifies the port at which the outgoing wave is observed, while the second subscript identifies the port at which the incident wave is applied. More precisely, \(S_{ij}\) is defined as
$S_{ij}=\left.\frac{b_i}{a_j}\right|_{a_k=0,\;k\neq j}$
Thus, \(S_{ij}\) 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 so that their incident waves are zero.
The diagonal elements \(S_{11}\), \(S_{22}\), and \(S_{33}\) are reflection coefficients at Ports 1, 2, and 3, respectively. For example, \(S_{11}\) represents the reflection at Port 1 when Port 1 is excited and the other ports are properly terminated. Similarly, \(S_{22}\) represents the reflection at Port 2, while \(S_{33}\) represents the reflection at Port 3. If a particular port is perfectly matched under the specified reference conditions, its corresponding diagonal reflection coefficient is zero.
The off-diagonal elements represent transmission or coupling between different ports. Thus, \(S_{12}\) describes the signal appearing at Port 1 when Port 2 is excited, whereas \(S_{21}\) describes the signal appearing at Port 2 when Port 1 is excited. Likewise, \(S_{13}\) and \(S_{23}\) describe the coupling from Port 3 to Ports 1 and 2, respectively.
For an H-plane tee in which Port 3 is the H-arm, the most important transmission coefficients are \(S_{13}\) and \(S_{23}\). When Port 3 is excited, the signal divides between the two collinear ports with equal magnitude and equal phase. Therefore, the defining relationship is
$\boxed{S_{13}=S_{23}}$
This equality means that the two waves have the same magnitude and the same phase under the ideal symmetric condition. Unlike the E-plane tee, there is no negative sign between these two coefficients in the conventional reference-plane arrangement used here.
Reciprocity provides the corresponding reverse-direction relationships:
$S_{31}=S_{13}$
$S_{32}=S_{23}$
Consequently, when Port 3 is the H-arm, the coupling between the H-arm and the two collinear ports is characterized by equal-magnitude and equal-phase S-parameters in both directions.
The element \(S_{12}\) represents transmission directly between the two collinear ports. It describes the wave emerging at Port 1 when Port 2 is excited. By reciprocity, this coefficient is equal to \(S_{21}\):
$\boxed{S_{12}=S_{21}}$
Thus, every element of the H-plane tee matrix has a physical interpretation in terms of reflection or transmission between the three ports. Understanding the meaning of each element is particularly useful when interpreting an unfamiliar S-matrix in an examination because the physical behavior can be identified from the mathematical pattern rather than relying only on memorization.
All S-Matrix Variations When the H-Arm Is Port 1, Port 2, or Port 3
The physical structure of an H-plane tee does not change when the port numbering is changed. Only the positions of the corresponding S-parameters within the scattering matrix change. Since the first index of \(S_{ij}\) identifies the output port and the second index identifies the input port, changing the port labels requires the corresponding rows and columns of the S-matrix to be rearranged together.
For an ideal reciprocal, lossless, and matched H-plane tee, the two collinear ports receive equal-magnitude waves that are in phase when the H-arm is excited. Therefore, if Port \(k\) is the H-arm and Ports \(i\) and \(j\) are the two collinear ports, the characteristic relationship is
$\boxed{S_{ik}=S_{jk}}$
The ideal H-plane tee S-matrix also contains opposite-sign coupling terms between the two collinear ports in the conventional reference-plane convention used here. These signs are required by the lossless and unitary conditions of the ideal three-port network. Therefore, the signs must be retained when the S-matrix is rearranged for different port-numbering arrangements.
H-Arm as Port 1
If Port 1 is designated as the H-arm, Ports 2 and 3 become the two collinear ports. Starting from the standard arrangement and interchanging the corresponding rows and columns gives the ideal S-matrix
$\boxed{[S]=\begin{bmatrix} 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\[4pt] \frac{1}{\sqrt{2}} & \frac{1}{2} & -\frac{1}{2} \\[4pt] \frac{1}{\sqrt{2}} & -\frac{1}{2} & \frac{1}{2} \end{bmatrix}}$
The first column represents excitation of the H-arm. The waves appearing at Ports 2 and 3 therefore have equal magnitude and equal phase:
$\boxed{S_{21}=S_{31}=\frac{1}{\sqrt{2}}}$
Because Port 1 is the matched H-arm, its reflection coefficient is zero:
$\boxed{S_{11}=0}$
By reciprocity, the corresponding elements in the first row are also equal:
$\boxed{S_{12}=S_{13}=\frac{1}{\sqrt{2}}}$
The coupling between the two collinear ports contains the opposite sign required by the lossless three-port condition:
$\boxed{S_{23}=S_{32}=-\frac{1}{2}}$
Thus, the complete pattern for Port 1 as the H-arm is a zero at \(S_{11}\), equal positive \(1/\sqrt{2}\) coupling between the H-arm and each collinear port, and a pair of opposite-sign \(1/2\) terms between the collinear ports.
H-Arm as Port 2
If Port 2 is designated as the H-arm, Ports 1 and 3 become the two collinear ports. Rearranging the rows and columns of the ideal H-plane tee matrix gives
$\boxed{[S]=\begin{bmatrix} \frac{1}{2} & \frac{1}{\sqrt{2}} & -\frac{1}{2} \\[4pt] \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\[4pt] -\frac{1}{2} & \frac{1}{\sqrt{2}} & \frac{1}{2} \end{bmatrix}}$
The second column represents excitation of the H-arm. Therefore, the waves appearing at Ports 1 and 3 have equal magnitude and equal phase:
$\boxed{S_{12}=S_{32}=\frac{1}{\sqrt{2}}}$
Since Port 2 is the matched H-arm,
$\boxed{S_{22}=0}$
Reciprocity gives the corresponding relationships in the second row:
$\boxed{S_{21}=S_{23}=\frac{1}{\sqrt{2}}}$
The two collinear ports are coupled through the negative terms
$\boxed{S_{13}=S_{31}=-\frac{1}{2}}$
Again, the negative sign belongs to the coupling between the two collinear ports in the chosen reference-plane convention and should be retained when identifying the corresponding matrix.
H-Arm as Port 3
The most commonly used numbering arrangement places the H-arm at Port 3, while Ports 1 and 2 form the two collinear arms. In this arrangement, the ideal S-matrix is
$\boxed{[S]=\begin{bmatrix} \frac{1}{2} & -\frac{1}{2} & \frac{1}{\sqrt{2}} \\[4pt] -\frac{1}{2} & \frac{1}{2} & \frac{1}{\sqrt{2}} \\[4pt] \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \end{bmatrix}}$
The third column corresponds to excitation of the H-arm. Therefore, the waves appearing at Ports 1 and 2 are equal in magnitude and equal in phase:
$\boxed{S_{13}=S_{23}=\frac{1}{\sqrt{2}}}$
The H-arm is matched, so the corresponding diagonal element is
$\boxed{S_{33}=0}$
By reciprocity, the third row also contains equal coupling coefficients:
$\boxed{S_{31}=S_{32}=\frac{1}{\sqrt{2}}}$
The two collinear ports are coupled with opposite signs:
$\boxed{S_{12}=S_{21}=-\frac{1}{2}}$
This sign relationship is an important part of the ideal H-plane tee S-matrix under the reference-plane convention used here. Together with the \(1/\sqrt{2}\) H-arm coupling coefficients and the zero \(S_{33}\) term, it ensures that the ideal matrix satisfies the required lossless and unitary conditions.
Comparison of the H-Plane Tee S-Matrix Variations
| H-Arm | Collinear Ports | H-Arm Phase Relationship | Matched H-Arm Condition | Ideal S-Matrix |
|---|---|---|---|---|
| Port 1 | Ports 2 and 3 | \(S_{21}=S_{31}\) | \(S_{11}=0\) | \(\displaystyle \begin{bmatrix} 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\[4pt] \frac{1}{\sqrt{2}} & -\frac{1}{2} & \frac{1}{2} \\[4pt] \frac{1}{\sqrt{2}} & \frac{1}{2} & -\frac{1}{2} \end{bmatrix}\) |
| Port 2 | Ports 1 and 3 | \(S_{12}=S_{32}\) | \(S_{22}=0\) | \(\displaystyle \begin{bmatrix} -\frac{1}{2} & \frac{1}{\sqrt{2}} & \frac{1}{2} \\[4pt] \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\[4pt] \frac{1}{2} & \frac{1}{\sqrt{2}} & -\frac{1}{2} \end{bmatrix}\) |
| Port 3 | Ports 1 and 2 | \(S_{13}=S_{23}\) | \(S_{33}=0\) | \(\displaystyle \begin{bmatrix} -\frac{1}{2} & \frac{1}{2} & \frac{1}{\sqrt{2}} \\[4pt] \frac{1}{2} & -\frac{1}{2} & \frac{1}{\sqrt{2}} \\[4pt] \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \end{bmatrix}\) |
General Pattern of the Three S-Matrix Variations
Although the position of the H-arm changes with the port numbering, all three matrices have the same physical characteristics. The diagonal element corresponding to the matched H-arm is zero. The two S-parameters connecting the H-arm to the two collinear ports are equal in magnitude and phase, with magnitude \(1/\sqrt{2}\). The two collinear-port reflection coefficients are \(-1/2\), while the transmission coefficients between the two collinear ports are \(+1/2\) under the conventional reference-phase choice used here.
Therefore, for any port \(k\) designated as the H-arm and Ports \(i\) and \(j\) designated as the two collinear ports, the characteristic conditions are
$\boxed{S_{kk}=0}$
and
$\boxed{S_{ik}=S_{jk}=\frac{1}{\sqrt{2}}}$
Using reciprocity, the corresponding reverse-direction coefficients are
$\boxed{S_{ki}=S_{kj}=\frac{1}{\sqrt{2}}}$
For the conventional phase reference used in the ideal H-plane tee matrix, the two collinear-port reflection coefficients are
$\boxed{S_{ii}=S_{jj}=-\frac{1}{2}}$
while the transmission coefficients between the two collinear ports are
$\boxed{S_{ij}=S_{ji}=\frac{1}{2}}$
Thus, the ideal H-plane tee contains one zero element, two collinear-port reflection coefficients of \(-1/2\), two collinear-port transmission coefficients of \(+1/2\), and four H-arm coupling coefficients of \(1/\sqrt{2}\).
Identifying the H-Arm from an Unknown S-Matrix
When an examination problem gives an S-matrix without explicitly stating which port represents the H-arm, the port numbering should not be assumed in advance. The H-arm can be identified from the mathematical pattern of the matrix. For an ideal matched H-plane tee, the diagonal element corresponding to the H-arm is zero. Therefore, the first step is to inspect the three diagonal elements \(S_{11}\), \(S_{22}\), and \(S_{33}\) and locate the zero element.
If the zero occurs at \(S_{kk}\), then Port \(k\) is the H-arm under the ideal matched condition. After locating the zero diagonal element, examine the corresponding row or column. The two S-parameters connecting the H-arm to the two collinear ports must have equal magnitude and equal phase.
Mathematically, if Port \(k\) is the H-arm and Ports \(i\) and \(j\) are the collinear ports, the defining relationships are
$\boxed{S_{kk}=0}$
and
$\boxed{S_{ik}=S_{jk}}$
For a reciprocal network, the corresponding reverse-direction relationship is
$\boxed{S_{ki}=S_{kj}}$
For the ideal normalized model, the equal H-arm coupling coefficients have magnitude \(1/\sqrt{2}\).
For example, if the zero diagonal element is \(S_{33}\), then Port 3 is the H-arm. The corresponding third column must contain equal coupling coefficients:
$\boxed{S_{13}=S_{23}=\frac{1}{\sqrt{2}}}$
Similarly, reciprocity gives
$\boxed{S_{31}=S_{32}=\frac{1}{\sqrt{2}}}$
If instead \(S_{11}=0\), Port 1 is the H-arm and the corresponding relationship is
$\boxed{S_{21}=S_{31}=\frac{1}{\sqrt{2}}}$
Likewise, if \(S_{22}=0\), Port 2 is the H-arm and
$\boxed{S_{12}=S_{32}=\frac{1}{\sqrt{2}}}$
This method is more reliable than memorizing the position of the H-arm in one particular S-matrix because the port numbering can be changed without changing the physical behavior of the junction.
Sign and Magnitude Pattern
The sign and magnitude of the S-parameters provide an important method for recognizing an ideal H-plane tee. When the H-arm is excited, the two collinear ports receive equal-magnitude waves that are in phase. Therefore, the two S-parameters describing transmission from the H-arm to the collinear ports have equal magnitude and the same phase.
For the common arrangement in which Port 3 is the H-arm and Ports 1 and 2 are the collinear ports, the characteristic H-arm coupling coefficients are
$\boxed{S_{13}=S_{23}=\frac{1}{\sqrt{2}}}$
Because the ideal H-plane tee is reciprocal, the corresponding reverse-direction coefficients are also equal:
$\boxed{S_{31}=S_{32}=\frac{1}{\sqrt{2}}}$
Thus, the four S-parameters associated with coupling between the H-arm and the two collinear ports have equal magnitude and equal phase:
$\boxed{S_{13}=S_{23}=S_{31}=S_{32}=\frac{1}{\sqrt{2}}}$
The important point is that the two waves produced at the collinear ports by excitation of the H-arm are equal in magnitude and in phase. This is the defining phase relationship of the H-plane tee.
The complete ideal H-plane tee matrix also contains a specific sign relationship between the reflection and transmission coefficients of the two collinear ports. With Port 3 designated as the H-arm, the conventional ideal matrix is
$\boxed{[S]=\begin{bmatrix}-\frac{1}{2} & \frac{1}{2} & \frac{1}{\sqrt{2}}\\[4pt]\frac{1}{2} & -\frac{1}{2} & \frac{1}{\sqrt{2}}\\[4pt]\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0\end{bmatrix}}$
The diagonal reflection coefficients of the two collinear ports are therefore
$\boxed{S_{11}=S_{22}=-\frac{1}{2}}$
while the transmission coefficients between the two collinear ports are
$\boxed{S_{12}=S_{21}=\frac{1}{2}}$
The H-arm is perfectly matched, so its reflection coefficient is
$\boxed{S_{33}=0}$
Therefore, the ideal H-plane tee has one zero element, two negative collinear-port reflection coefficients equal to \(-1/2\), two positive collinear-port transmission coefficients equal to \(1/2\), and four equal H-arm coupling coefficients equal to \(1/\sqrt{2}\).
The sign and magnitude pattern can therefore be summarized as
$\boxed{\begin{bmatrix}-\frac{1}{2} & \frac{1}{2} & \frac{1}{\sqrt{2}}\\[4pt]\frac{1}{2} & -\frac{1}{2} & \frac{1}{\sqrt{2}}\\[4pt]\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0\end{bmatrix}}$
For examination purposes, the most important distinction is that the H-arm produces equal-magnitude, equal-phase waves at the two collinear ports. The negative signs occur in the collinear-port reflection coefficients in this conventional reference-phase choice and should not be confused with the H-arm phase relationship.
Physical Interpretation of the Sign Pattern
The sign pattern of the ideal H-plane tee follows from the lossless and unitary nature of the three-port junction. The H-arm coupling coefficients are positive and equal, which represents the equal-phase response at the two collinear ports when the H-arm is excited. The negative signs of \(S_{11}\) and \(S_{22}\), together with the positive value of \(S_{12}=S_{21}\), ensure that the different rows and columns of the scattering matrix remain orthogonal as required by the unitary condition.
For example, using the third row and the first row of the ideal matrix, the orthogonality condition gives
$\left(-\frac{1}{2}\right)\left(\frac{1}{\sqrt{2}}\right)+\left(\frac{1}{2}\right)\left(\frac{1}{\sqrt{2}}\right)+\left(\frac{1}{\sqrt{2}}\right)(0)=0$
Similarly, the second and third rows satisfy the same orthogonality condition. Thus, the sign arrangement is not arbitrary; it is required for the ideal lossless three-port S-matrix to satisfy
$\boxed{[S][S]^\dagger=[I]}$
The complete matrix therefore simultaneously expresses reciprocity, equal H-arm power division, equal phase at the collinear ports, matching of the H-arm, and conservation of microwave power.
Port Interchange and Row/Column Interchange
Changing the port numbering of a three-port network does not change the physical H-plane tee. It only changes which physical port is assigned to Port 1, Port 2, or Port 3. Since every S-parameter has two indices, changing a port number requires the corresponding row and column of the S-matrix to be rearranged.
The first index of \(S_{ij}\) identifies the output port and therefore determines the row of the S-matrix. The second index identifies the incident port and therefore determines the column. Consequently, whenever two ports are interchanged, both their corresponding rows and columns must be interchanged.
For example, if Port 1 and Port 2 are interchanged, Row 1 must be exchanged with Row 2 and Column 1 must be exchanged with Column 2. Therefore, the general rule is
$\boxed{\text{Port interchange}\Rightarrow\text{corresponding row interchange and corresponding column interchange}}$
This rule is important when different forms of the H-plane tee S-matrix appear in an examination. The numerical positions of the S-parameters may change when the port numbering changes, but the physical relationships of the junction remain unchanged.
For an ideal matched H-plane tee, the diagonal element associated with the H-arm is zero. Therefore, if Port 1 is the H-arm, the zero appears at \(S_{11}\). If Port 2 is the H-arm, the zero appears at \(S_{22}\). If Port 3 is the H-arm, the zero appears at \(S_{33}\).
The equal H-arm coupling coefficients also move with the H-arm. For example, when Port 3 is the H-arm,
$\boxed{S_{13}=S_{23}=S_{31}=S_{32}=\frac{1}{\sqrt{2}}}$
When Port 1 is the H-arm, the corresponding relationship becomes
$\boxed{S_{21}=S_{31}=S_{12}=S_{13}=\frac{1}{\sqrt{2}}}$
When Port 2 is the H-arm, the corresponding relationship becomes
$\boxed{S_{12}=S_{32}=S_{21}=S_{23}=\frac{1}{\sqrt{2}}}$
The remaining S-parameters associated with the two collinear ports are rearranged at the same time. Their signs must remain consistent with the selected reference-plane phase convention and the lossless unitary condition.
Common Examination Patterns
Questions on the H-plane tee commonly test the physical operation of the junction, the phase relationship between the collinear ports, reciprocity, the matched-port condition, power division, and the mathematical properties of the S-matrix. A common examination problem provides an S-matrix and asks the student to identify which port is the H-arm.
For an ideal matched H-plane tee, the fastest method is to inspect the diagonal elements. The diagonal element corresponding to the matched H-arm is zero. Therefore, if \(S_{11}=0\), Port 1 is the H-arm. If \(S_{22}=0\), Port 2 is the H-arm. If \(S_{33}=0\), Port 3 is the H-arm.
Once the H-arm has been identified, inspect the corresponding row and column. The two coupling coefficients between the H-arm and the collinear ports must have equal magnitude and equal phase. For example, if Port 3 is the H-arm,
$\boxed{S_{13}=S_{23}=S_{31}=S_{32}=\frac{1}{\sqrt{2}}}$
Another common examination question asks what happens when the H-arm is excited. For an ideal symmetric and lossless H-plane tee, the incident power at the H-arm is divided equally between the two collinear ports. Therefore,
$|S_{13}|^2=|S_{23}|^2=\frac{1}{2}$
and hence
$\boxed{|S_{13}|=|S_{23}|=\frac{1}{\sqrt{2}}}$
The two coefficients have the same phase, so the waves emerging from the two collinear ports are in phase. This equal-phase behavior is the defining feature of the H-plane tee when its H-arm is excited.
A frequently tested conceptual point is the difference between an H-plane tee and an E-plane tee. When the H-arm of an H-plane tee is excited, the two collinear ports receive equal-magnitude, in-phase waves. In an E-plane tee, the corresponding waves at the two collinear ports have equal magnitude but opposite phase. Thus, the relative phase of the two collinear-port outputs provides an important way to distinguish the two junctions.
Reciprocity is another common examination topic. For a reciprocal H-plane tee, the S-matrix is symmetric and satisfies
$\boxed{S_{ij}=S_{ji}}$
Thus, if \(S_{13}\) is known, \(S_{31}\) has the same value. Similarly, \(S_{23}=S_{32}\) and \(S_{12}=S_{21}\).
The lossless condition is also frequently tested. For an ideal lossless junction, the S-matrix is unitary and satisfies
$\boxed{[S][S]^\dagger=[I]}$
This condition guarantees conservation of microwave power and imposes orthogonality conditions between the rows and columns of the S-matrix.
Using the following consistent reference-plane phase convention for the Port 3 H-arm arrangement,
$\boxed{[S]=\begin{bmatrix}-\frac{1}{2} & \frac{1}{2} & \frac{1}{\sqrt{2}}\\[4pt]\frac{1}{2} & -\frac{1}{2} & \frac{1}{\sqrt{2}}\\[4pt]\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0\end{bmatrix}}$
the first and second rows are orthogonal because
$\left(-\frac{1}{2}\right)\left(\frac{1}{2}\right)+\left(\frac{1}{2}\right)\left(-\frac{1}{2}\right)+\left(\frac{1}{\sqrt{2}}\right)\left(\frac{1}{\sqrt{2}}\right)=0$
The magnitude of each row also satisfies the power-conservation condition. For example, the first row gives
$\left|-\frac{1}{2}\right|^2+\left|\frac{1}{2}\right|^2+\left|\frac{1}{\sqrt{2}}\right|^2=\frac{1}{4}+\frac{1}{4}+\frac{1}{2}=1$
Therefore, the matrix is consistent with the lossless condition.
A typical derivation problem may provide reciprocity, symmetry, matching, equal power division, and losslessness, and ask the student to derive the complete S-matrix. For the standard arrangement with Port 3 as the H-arm, one consistent ideal representation is
$\boxed{[S]=\begin{bmatrix}-\frac{1}{2} & \frac{1}{2} & \frac{1}{\sqrt{2}}\\[4pt]\frac{1}{2} & -\frac{1}{2} & \frac{1}{\sqrt{2}}\\[4pt]\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0\end{bmatrix}}$
Important Examination Tips
Tip 1: Do not memorize only the position of the H-arm. Port numbering can be changed. Instead, identify the zero diagonal element and then check whether the corresponding H-arm coupling coefficients are equal in magnitude and phase.
Tip 2: Remember the defining phase relationship. An H-plane tee produces equal-magnitude and equal-phase waves at the two collinear ports when the H-arm is excited.
$\boxed{S_{ik}=S_{jk}}$
Tip 3: For an ideal matched H-plane tee, the diagonal element corresponding to the H-arm is zero.
$\boxed{S_{kk}=0}$
Tip 4: For a reciprocal network, remember that the S-matrix is symmetric.
$\boxed{S_{ij}=S_{ji}}$
Tip 5: When port numbering changes, interchange both the corresponding row and column. Never interchange only one of them.
Tip 6: Do not confuse the H-plane tee with the E-plane tee. The H-plane tee has an equal-phase relationship between the two collinear-port outputs when the H-arm is excited, whereas the E-plane tee has an opposite-phase relationship.
Tip 7: Be careful when comparing H-plane tee S-matrices from different textbooks. The signs of individual S-parameters can depend on the reference-plane phase convention. The physically important H-plane characteristic is that the two H-arm coupling coefficients have equal magnitude and equal phase under the chosen convention, while the complete matrix must satisfy reciprocity and the lossless unitary condition.
Conceptual Examination Questions
- What is an H-plane tee junction?
- Why is the H-plane tee also called a shunt tee or current tee?
- Why is the side arm of an H-plane tee associated with the magnetic-field plane?
- Which ports are called the collinear ports in an H-plane tee?
- Which port is called the H-arm?
- What happens when the H-arm of an ideal H-plane tee is excited?
- What is the phase relationship between the waves appearing at the two collinear ports when the H-arm is excited?
- What is the relationship between \(S_{13}\) and \(S_{23}\) when Port 3 is the H-arm?
- What is the matched H-arm condition when Port 3 is the H-arm?
- What reciprocity property does the S-matrix of a reciprocal H-plane tee satisfy?
- Why is \(S_{ij}=S_{ji}\) valid for a reciprocal H-plane tee?
- What is the lossless condition for the S-matrix of an ideal H-plane tee?
- How can the H-arm be identified if the port numbering is not given?
- How does the S-matrix change when the H-arm is changed from Port 3 to Port 1?
- What happens to the rows and columns when two ports of the H-plane tee are interchanged?
- Why must both the row and column be interchanged when a port number is changed?
- How can an H-plane tee be distinguished from an E-plane tee by examining the relative phase of the coupling coefficients?
- What is the significance of the \(1/\sqrt{2}\) terms in the ideal H-plane tee S-matrix?
- Why are the two coupling coefficients associated with H-arm excitation equal?
- How can an unknown S-matrix be identified as belonging to an ideal H-plane tee?
For examination problems, the most useful pattern to remember is that the H-arm of an ideal matched H-plane tee corresponds to a zero diagonal element, while excitation of that port produces equal-magnitude and equal-phase waves at the two collinear ports. Therefore, the essential identification pattern is
$\boxed{S_{kk}=0,\qquad S_{ik}=S_{jk}}$
where \(k\) represents the H-arm and \(i,j\) represent the two collinear ports. For the ideal normalized case, the magnitude of the H-arm coupling coefficients is \(1/\sqrt{2}\).