Hybrid Ring (Rat-Race Junction)
Hybrid Ring or Rat-Race Junction: Construction, Operation and S-Matrix Analysis
The Hybrid Ring, also called a Rat-Race Junction or 180° Hybrid, is a four-port passive microwave device used for equal power division, power combining, sum and difference operations, and isolation between selected ports. Its operation is based on the controlled phase relationship between signals travelling around an annular transmission-line structure through paths having different electrical lengths. At the design frequency, the two signal paths can combine constructively at one port and destructively at another port, producing equal-amplitude outputs at selected ports while maintaining isolation at the remaining port. The ideal Hybrid Ring is a reciprocal, matched, and lossless four-port network, and these properties can be used systematically to derive its complete \(4\times4\) scattering matrix.
The name Rat-Race Junction comes from its ring-shaped construction, in which four ports are connected to an annular transmission line at appropriate locations. The total mean circumference of the ring is normally \(3\lambda/2\) at the design frequency. The four ports are not separated by four equal \(\lambda/4\) sections around the complete ring. Instead, three successive sections are typically \(\lambda/4\), while the remaining section is \(3\lambda/4\), giving a total electrical length of \(3\lambda/2\). This unequal path arrangement is essential because the difference between the two possible paths between appropriate ports is \(\lambda/2\), producing the required \(180^\circ\) phase difference for cancellation.
Construction and Operating Principle of the Hybrid Ring
Choose a suitable planar microwave device to split power into equal halves with 180 degree phase shift
A Hybrid Ring can be understood as a four-port extension of the power-splitting concept of a conventional Tee junction. Instead of using a simple three-port junction, the transmission line is formed into an annular ring and four ports are coupled to it at carefully selected positions. The ring contains transmission-line sections with electrical lengths chosen so that signals travelling in clockwise and anticlockwise directions reach the output ports with either the same phase or a \(180^\circ\) phase difference. When the two signals arrive with the same phase, they add constructively and produce a maximum output. When they arrive with opposite phase, they cancel. Therefore, the Hybrid Ring does not rely only on geometrical power division; its most important characteristic is the controlled phase relationship between the two signal paths.
The ideal Hybrid Ring is designed for a particular operating frequency because the electrical lengths of the ring sections are frequency dependent. At the design frequency, the required \(\lambda/4\), \(3\lambda/4\), and \(\lambda/2\) phase relationships are obtained accurately. If the frequency changes, the electrical lengths expressed in wavelengths also change, so the phase cancellation is no longer perfect. Consequently, an ideal Hybrid Ring provides exact isolation only at its design frequency, whereas a practical Hybrid Ring exhibits finite isolation over a limited operating bandwidth. Manufacturing tolerances, conductor loss, dielectric loss, junction discontinuities, and impedance mismatch also cause small leakage terms in a practical device.
Port Arrangement and Reference S-Matrix Convention
Explain the working mechanism of a Hybrid Ring (Rat-Race Coupler) using S-matrix parameters
Since the numerical positions of the S-parameters depend on the way the four ports are numbered, a specific port convention must be established before deriving the scattering matrix. For the analysis below, the ports are numbered so that an excitation at Port 1 produces equal-amplitude outputs at Ports 2 and 4, while Port 3 is isolated. With this convention, Port 1 and Port 3 form one isolated pair, while Port 2 and Port 4 form the other isolated pair. A different textbook may number the physical ports differently, in which case the same physical Hybrid Ring will have a different-looking S-matrix even though its electrical behavior is unchanged.
The port numbering is therefore not something that should be memorized independently of the physical arrangement. If two ports are renumbered, both the corresponding row and the corresponding column of the S-matrix must be interchanged. Changing only a row or only a column would destroy reciprocity and would no longer represent the same physical network.
General 4×4 S-Matrix of the Hybrid Ring
The derivation begins with the completely general four-port scattering matrix. At this stage, no coefficient is assumed to be zero and no numerical value is assigned to any transmission coefficient. Since the Hybrid Ring is a reciprocal network, the symmetric form can be written directly as
\[ [S]= \begin{bmatrix} S_{11}&S_{12}&S_{13}&S_{14}\\ S_{12}&S_{22}&S_{23}&S_{24}\\ S_{13}&S_{23}&S_{33}&S_{34}\\ S_{14}&S_{24}&S_{34}&S_{44} \end{bmatrix} \]
The use of \(S_{ij}=S_{ji}\) represents the reciprocity of the passive Hybrid Ring. The diagonal terms represent reflection coefficients at the four ports, while the off-diagonal terms represent transmission between different ports. The purpose of the following steps is to determine which of these coefficients must be zero, which coefficients must have equal magnitude, and which coefficients must have opposite or equal phase.
Applying the Matching Condition
An ideal Hybrid Ring is assumed to be perfectly matched at all four ports. Therefore, no signal is reflected back into the same port when that port is excited under the ideal matched condition. Mathematically, this means that all four diagonal elements of the scattering matrix are zero:
\[ S_{11}=S_{22}=S_{33}=S_{44}=0 \]
Substituting the matching condition into the general matrix gives
\[ [S]= \begin{bmatrix} 0&S_{12}&S_{13}&S_{14}\\ S_{12}&0&S_{23}&S_{24}\\ S_{13}&S_{23}&0&S_{34}\\ S_{14}&S_{24}&S_{34}&0 \end{bmatrix} \]
The matrix still contains twelve potentially nonzero off-diagonal terms, but reciprocity has already reduced these to six independent transmission coefficients. The physical operation of the Hybrid Ring now provides the additional conditions required to identify the isolated port pairs.
Applying the Isolation Conditions
According to the selected port convention, Port 1 and Port 3 are isolated. Therefore, a signal applied at Port 1 does not appear at Port 3, and a signal applied at Port 3 does not appear at Port 1. Hence,
\[ S_{13}=S_{31}=0 \]
The second isolated pair is Port 2 and Port 4. Therefore,
\[ S_{24}=S_{42}=0 \]
Applying these conditions gives the reduced matrix
\[ [S]= \begin{bmatrix} 0&S_{12}&0&S_{14}\\ S_{12}&0&S_{23}&0\\ 0&S_{23}&0&S_{34}\\ S_{14}&0&S_{34}&0 \end{bmatrix} \]
This reduced matrix is the most useful starting point for determining the numerical S-parameters. It shows that every port has two possible transmission paths and one isolated port. The remaining coefficients are not yet assigned numerical values because their magnitudes and relative phases must still be established.
Applying the Equal Power Division Property
The Hybrid Ring is an equal-power four-port hybrid. When a single port is excited, the available power is equally divided between the two non-isolated output ports. Therefore, the magnitudes of the corresponding transmission coefficients must be equal. For excitation at Port 1, the two active paths are represented by \(S_{12}\) and \(S_{14}\), so
\[ |S_{12}|=|S_{14}| \]
Similarly, for excitation at Port 3,
\[ |S_{23}|=|S_{34}| \]
Because the network is lossless and the input power is divided equally between two output paths, each active transmission coefficient has power magnitude \(1/2\). Therefore,
\[ |S_{12}|^2=|S_{14}|^2=\frac{1}{2} \]
and
\[ |S_{23}|^2=|S_{34}|^2=\frac{1}{2} \]
Consequently,
\[ |S_{12}|=|S_{14}|=|S_{23}|=|S_{34}|=\frac{1}{\sqrt{2}} \]
The \(1/\sqrt{2}\) coefficient is therefore a direct indication of equal power division. Since power is proportional to the squared magnitude of an S-parameter, a coefficient of \(1/\sqrt{2}\) corresponds to one-half of the available power.
Applying the Lossless or Unitary Condition
The exact phase relationships and the consistency of the magnitudes can be established using the lossless condition. For an ideal lossless microwave network, the scattering matrix must be unitary:
\[ [S][S]^\dagger=[I] \]
Applying this condition to the reduced Hybrid Ring matrix gives normalization equations from the diagonal elements of the product. For example, the first row must have unit magnitude, giving
\[ |S_{12}|^2+|S_{14}|^2=1 \]
Similarly, the second row gives
\[ |S_{12}|^2+|S_{23}|^2=1 \]
The third row gives
\[ |S_{23}|^2+|S_{34}|^2=1 \]
and the fourth row gives
\[ |S_{14}|^2+|S_{34}|^2=1 \]
These equations confirm that the active transmission coefficients must have magnitude \(1/\sqrt{2}\) when the two output paths have equal power. The off-diagonal equations obtained from the same unitary condition ensure that the corresponding rows are orthogonal. These orthogonality relationships are consistent with the destructive-interference conditions responsible for port isolation.
Final Ideal 4×4 Hybrid Ring S-Matrix
Using the matching condition, reciprocity, isolation, equal-power division, phase relationships, and lossless condition, the ideal Hybrid Ring can be represented for the selected port convention by
\[ \boxed{ [S]= \frac{1}{\sqrt{2}} \begin{bmatrix} 0&1&0&-1\\ 1&0&1&0\\ 0&1&0&1\\ -1&0&1&0 \end{bmatrix}} \]
An equivalent form may contain a common phase factor such as \(-j\), depending on the choice of reference planes and the phase convention used for the ports. Such a common phase factor does not change the power division or isolation characteristics. What is physically important is the relative phase between the transmission paths, particularly the opposite signs that produce the required \(180^\circ\) phase relationship.
Pattern of the Hybrid Ring S-Matrix
The final matrix has a useful pattern that can be used in examinations instead of memorizing every individual S-parameter position. The ideal four-port Hybrid Ring contains eight zero elements and eight nonzero elements. Four of the eight zero elements are the diagonal terms caused by perfect matching:
\[ S_{11}=S_{22}=S_{33}=S_{44}=0 \]
The remaining four zero elements occur as two reciprocal isolation pairs:
\[ S_{13}=S_{31}=0 \]
\[ S_{24}=S_{42}=0 \]
All eight nonzero terms have magnitude \(1/\sqrt{2}\). For the particular phase convention used above, six nonzero terms are positive \(1/\sqrt{2}\) and two are negative \(1/\sqrt{2}\). The locations of the negative terms are not universal because they depend on port numbering and reference-plane phase conventions. Therefore, the safest method is to identify the physical isolation pairs and phase relationships first rather than memorizing the exact position of every positive and negative coefficient.
Unlike some interconnected Magic Tee networks, the ideal four-port Hybrid Ring does not contain \(1/2\) transmission coefficients. Its active coefficients are \(1/\sqrt{2}\) because the device performs a single equal power division between two output paths. A \(1/2\) coefficient can appear in other microwave networks when a signal experiences two successive equal power divisions, but that should not be confused with the basic ideal four-port Rat-Race Junction.
Case 1: Input Applied at Port 1
When Port 1 is used as the input port, the relevant transmission coefficients are the elements associated with Port 1 in the scattering matrix. From the final matrix,
\[ S_{11}=0,\qquad S_{21}=\frac{1}{\sqrt2},\qquad S_{31}=0,\qquad S_{41}=-\frac{1}{\sqrt2} \]
Thus, Port 1 is perfectly matched and Port 3 is isolated. Equal magnitudes occur at Ports 2 and 4 because
\[ |S_{21}|=|S_{41}|=\frac{1}{\sqrt2} \]
The corresponding powers are therefore equal:
\[ |S_{21}|^2=|S_{41}|^2=\frac12 \]
The negative sign associated with \(S_{41}\) indicates that the signals appearing at Ports 2 and 4 have a \(180^\circ\) relative phase. Port 3 receives no output because the clockwise and anticlockwise contributions arriving there have a path difference of \(\lambda/2\), producing destructive interference. This is the fundamental isolation mechanism of the Hybrid Ring.
Case 2: Input Applied at Port 3
When Port 3 is excited, the relevant coefficients are
\[ S_{13}=0,\qquad S_{23}=\frac{1}{\sqrt2},\qquad S_{33}=0,\qquad S_{43}=\frac{1}{\sqrt2} \]
Therefore, Port 1 is isolated, while Ports 2 and 4 receive equal-amplitude outputs. In this case, both transmission coefficients have the same sign:
\[ S_{23}=S_{43}=\frac{1}{\sqrt2} \]
Hence the two output signals at Ports 2 and 4 have the same phase under the selected reference-plane convention. This is the complementary operating condition to the Port 1 excitation case. The same physical Hybrid Ring can therefore provide either an in-phase or opposite-phase relationship depending on which of the two appropriate input ports is excited.
Case 3: Simultaneous Signals at Ports 1 and 3
The sum and difference behavior becomes particularly clear when two signals are applied simultaneously at Ports 1 and 3. It is not necessary to introduce separate incident-wave variables to understand this operation. The relevant S-parameters can be examined directly. At Port 2, the two relevant transmission coefficients are
\[ S_{21}=\frac{1}{\sqrt2} \]
and
\[ S_{23}=\frac{1}{\sqrt2} \]
Since these coefficients have the same magnitude and phase, the contributions associated with Ports 1 and 3 combine constructively at Port 2. Consequently, the Port 2 response is proportional to the sum of the two input signals.
At Port 4, the corresponding coefficients are
\[ S_{41}=-\frac{1}{\sqrt2} \]
and
\[ S_{43}=\frac{1}{\sqrt2}. \]
These coefficients have opposite signs. Therefore, the contributions at Port 4 combine according to the difference of the two input signals. The Hybrid Ring consequently provides a sum output at one port and a difference output at another port. If the two input signals are equal in amplitude and phase, the difference component cancels and the sum output becomes maximum. If the two input signals are equal in amplitude but opposite in phase, the sum component cancels while the difference output becomes dominant.
Case 4: Unequal Signals at Ports 1 and 3
If unequal signals are applied simultaneously at Ports 1 and 3, neither the sum output nor the difference output necessarily becomes zero. The output at the sum port is determined by the combination represented by \(S_{21}\) and \(S_{23}\), while the output at the difference port is determined by \(S_{41}\) and \(S_{43}\). Therefore, the Hybrid Ring can separate two input signals into their sum and difference components even when their amplitudes are unequal.
In terms of the S-parameters, the sum-path relationship is governed by
\[ S_{21}+S_{23} \]
while the difference-path relationship is governed by
\[ S_{41}+S_{43}. \]
Since
\[ S_{41}=-S_{21} \]
and
\[ S_{43}=S_{23}, \]
the two output paths naturally represent complementary sum and difference operations. This property is one of the main reasons the Hybrid Ring is classified as a \(180^\circ\) hybrid.
Case 5: Input Applied at Port 2
Because the Hybrid Ring is reciprocal, the response for Port 2 excitation can be obtained directly from the second column of the S-matrix. The relevant coefficients are
\[ S_{12}=\frac{1}{\sqrt2}, \qquad S_{22}=0, \qquad S_{32}=\frac{1}{\sqrt2}, \qquad S_{42}=0. \]
Therefore, excitation at Port 2 produces equal-amplitude outputs at Ports 1 and 3, while Port 4 is isolated. The corresponding power division is
\[ |S_{12}|^2=|S_{32}|^2=\frac12. \]
This case demonstrates why the exact positions of the zeros should not simply be memorized. When the input port changes, the corresponding isolated port also changes according to the physical symmetry of the Hybrid Ring.
Case 6: Unequal or Simultaneous Inputs Involving Port 2
When signals are simultaneously applied at Ports 1 and 2, the response at the other ports is determined by the corresponding S-parameter pairs. At Port 3, the relevant terms are \(S_{31}\) and \(S_{32}\). Since
\[ S_{31}=0 \]
while
\[ S_{32}=\frac1{\sqrt2}, \]
the Port 3 response is determined only by the signal associated with Port 2. At Port 4, the relevant coefficients are \(S_{41}\) and \(S_{42}\), where
\[ S_{41}=-\frac1{\sqrt2}, \qquad S_{42}=0. \]
Therefore, Port 4 responds only to the Port 1 excitation in this particular simultaneous-input arrangement. This illustrates an important feature of the Hybrid Ring: because of its isolation structure, certain input combinations remain separated from particular output ports regardless of the amplitude of the other input.
Hybrid Ring S-Matrix Variations with Different Port Numbering
Different textbooks, diagrams, and examination questions may use different port numbering arrangements for the same physical Hybrid Ring. Consequently, the S-matrix may appear different even though the device has exactly the same physical characteristics. The correct approach is therefore not to memorize that a particular element such as \(S_{13}\) must always be zero. Instead, first identify the physical isolated-port pairs and then determine the corresponding zero elements.
For the reference convention used here, the isolated pairs are
\[ S_{13}=S_{31}=0 \]
and
\[ S_{24}=S_{42}=0. \]
If the physical ports are renumbered, these zero positions move with the corresponding ports. At the same time, the \(1/\sqrt2\) transmission coefficients move to the appropriate new positions. The physical properties remain unchanged: every port remains matched in the ideal model, the two isolated pairs remain isolated, the active transmission paths remain equal in magnitude, and the required relative phase relationships remain present.
Pattern for Identifying a Hybrid Ring from Its S-Matrix
A Hybrid Ring can be recognized from its S-matrix by checking a sequence of characteristic patterns. First, an ideal matched four-port Hybrid Ring has zero diagonal elements:
\[ S_{11}=S_{22}=S_{33}=S_{44}=0. \]
Second, the matrix is symmetric because the network is reciprocal:
\[ S_{ij}=S_{ji}. \]
Third, two pairs of ports are isolated, giving four additional zero elements outside the diagonal. Fourth, the remaining eight transmission coefficients have magnitude \(1/\sqrt2\), corresponding to equal power division. Finally, the signs or phases of these nonzero coefficients must produce the required constructive and destructive interference. These characteristics together are much more reliable for identifying a Hybrid Ring than memorizing one particular matrix arrangement.
For the reference matrix, the pattern can be summarized as follows: there are four diagonal zeros, four off-diagonal isolation zeros, and eight active transmission coefficients, each having magnitude \(1/\sqrt2\). There are no \(1/2\) coefficients in the ideal four-port Hybrid Ring. The exact number and location of positive and negative entries can change with the chosen port numbering and reference-plane phase convention, so the physical phase relationship should always be used as the final test.
Effect of Changing Port Numbering
When two port labels are exchanged, both the corresponding row and column of the S-matrix must be exchanged. For example, if Ports 1 and 3 are renumbered, the first and third rows must be interchanged and the first and third columns must also be interchanged. This operation produces the S-matrix for the new labeling without changing the physical device. Interchanging only one row or only one column would produce an incorrect matrix because the relationship between input and output ports would no longer be represented consistently.
The same principle applies to any permutation of the four ports. Therefore, when an examination question presents a Hybrid Ring with unfamiliar port numbering, the student should first redraw or inspect the physical arrangement, identify the two isolated port pairs, identify the two equal-power output paths, and then map those relationships into the S-matrix. This method avoids memorization and works for different diagrams and textbook conventions.
Important Difference Between \(1/\sqrt2\) and \(1/2\) Terms
The coefficient \(1/\sqrt2\) in the Hybrid Ring S-matrix represents an amplitude transmission coefficient corresponding to one-half of the power. Since the power ratio is proportional to the square of the magnitude of the S-parameter,
\[ \left|\frac1{\sqrt2}\right|^2=\frac12. \]
Therefore, when a signal is equally divided between two output ports, each output receives half of the available power. A coefficient of \(1/2\), on the other hand, would correspond to a power ratio of \(1/4\) if it were an S-parameter amplitude coefficient:
\[ \left|\frac12\right|^2=\frac14. \]
Hence, \(1/2\) should not be interpreted as the standard equal-power coefficient of the basic four-port Hybrid Ring. The appearance of \(1/2\) in a different microwave network generally indicates that the signal has undergone additional power division or that the network has a different structure.
Ideal Hybrid Ring Versus Practical Hybrid Ring
The ideal S-matrix contains exact zeros because the mathematical model assumes perfect matching, perfect symmetry, lossless transmission lines, and exact phase relationships. A practical Hybrid Ring cannot satisfy all of these assumptions perfectly. Small manufacturing errors can change the electrical lengths of the ring sections, junction discontinuities can introduce impedance mismatch, and conductor and dielectric losses can reduce the available output power. As a result, the theoretically zero isolation coefficients become small but nonzero values, and the theoretically equal \(1/\sqrt2\) transmission coefficients may differ slightly in magnitude.
The phase cancellation is also frequency dependent. At the design frequency, the relevant paths have the required electrical-length difference and produce maximum cancellation at the isolated port. Away from that frequency, the phase difference deviates from the ideal value, so some leakage power appears at the nominally isolated port. Thus, practical Hybrid Rings are normally specified over a finite bandwidth rather than being treated as perfectly ideal over all frequencies.
Key Properties of the Hybrid Ring
The important properties of an ideal Hybrid Ring can therefore be summarized through its S-matrix. It is a four-port network with all ports ideally matched, it is reciprocal, it is lossless, and it provides equal power division between the appropriate output ports. Two port pairs are isolated, while the remaining transmission paths have magnitude \(1/\sqrt2\). The relative signs of the transmission coefficients establish the required \(180^\circ\) phase relationships. Because of these characteristics, the Hybrid Ring can operate as both a power divider and a power combiner and can also perform sum and difference operations.
The most important examination rule is to identify the physical pattern rather than memorize the numerical position of individual S-parameters. Look for four zero diagonal terms, two reciprocal isolated port pairs, eight active coefficients of magnitude \(1/\sqrt2\), a symmetric matrix caused by reciprocity, and phase relationships that satisfy the constructive and destructive interference conditions. If the port numbering changes, the corresponding rows and columns change together, but the physical properties of the Hybrid Ring remain the same.
Hybrid Ring S-Matrix Derivation
The scattering matrix of an ideal Hybrid Ring can be derived systematically from the physical properties of the four-port network. Rather than assuming the final numerical matrix directly, the derivation begins with a completely general \(4\times4\) S-matrix in which every element is represented by its corresponding scattering parameter \(S_{ij}\). The physical properties of the Rat-Race Junction are then applied one at a time. Matching conditions identify the diagonal elements, reciprocity relates corresponding forward and reverse transmission coefficients, while the sum and difference characteristics determine which transmission coefficients are equal and which have opposite signs. Finally, the lossless condition is applied through the unitary relationship \([S][S]^\dagger=[I]\), which provides the equations required to determine the magnitudes of the remaining unknown coefficients.
1. General Four-Port S-Matrix
Since the Hybrid Ring is a four-port microwave network, its most general scattering matrix contains sixteen S-parameters. Before applying any physical property, the matrix is written completely in terms of \(S_{ij}\). For a reciprocal network, the matrix will eventually become symmetric, but this condition should be introduced as a property rather than assumed at the beginning of the derivation. Therefore, the general form can be written as
\[ [S] = \begin{bmatrix} S_{11} & S_{12} & S_{13} & S_{14}\\ S_{21} & S_{22} & S_{23} & S_{24}\\ S_{31} & S_{32} & S_{33} & S_{34}\\ S_{41} & S_{42} & S_{43} & S_{44} \end{bmatrix} \]
Here, \(S_{ij}\) represents the ratio of the wave emerging from Port \(i\) to the wave incident at Port \(j\), with all other ports terminated in matched loads. At this stage, no element is assumed to be zero and no two elements are assumed to be equal. The matrix therefore represents every possible reflection and transmission path among the four ports. The objective of the derivation is to use the known characteristics of the Hybrid Ring to eliminate physically impossible paths and determine the numerical values of the remaining transmission coefficients.
2. Applying the Matching Condition
An ideal Hybrid Ring is assumed to be perfectly matched at all four ports. A matched port produces no reflected wave when it is excited while all other ports are terminated in their characteristic impedances. In terms of the scattering matrix, the reflection coefficient at Port \(i\) is represented by the diagonal element \(S_{ii}\). Therefore, perfect matching requires every diagonal element of the four-port S-matrix to be zero.
Hence,
\[ S_{11}=S_{22}=S_{33}=S_{44}=0 \]
Substituting the matching condition into the general matrix gives
\[ [S] = \begin{bmatrix} 0 & S_{12} & S_{13} & S_{14}\\ S_{21} & 0 & S_{23} & S_{24}\\ S_{31} & S_{32} & 0 & S_{34}\\ S_{41} & S_{42} & S_{43} & 0 \end{bmatrix} \]
The zero diagonal elements are an important identifying feature of the ideal Hybrid Ring. However, they do not by themselves determine the complete S-matrix because the twelve off-diagonal transmission coefficients are still unknown.
3. Applying Reciprocity
The Hybrid Ring is constructed from passive reciprocal transmission-line sections and junctions. In the absence of nonreciprocal materials or active components, the network satisfies the reciprocity condition. For a reciprocal microwave network, transmission from Port \(j\) to Port \(i\) is equal to transmission from Port \(i\) to Port \(j\). Therefore, the scattering parameters satisfy
\[ S_{ij}=S_{ji} \]
The individual reciprocity relationships are
\[ S_{12}=S_{21} \]
\[ S_{13}=S_{31} \]
\[ S_{14}=S_{41} \]
\[ S_{23}=S_{32} \]
\[ S_{24}=S_{42} \]
\[ S_{34}=S_{43} \]
Therefore, the matrix becomes symmetric and can be written as
\[ [S] = \begin{bmatrix} 0 & S_{12} & S_{13} & S_{14}\\ S_{12} & 0 & S_{23} & S_{24}\\ S_{13} & S_{23} & 0 & S_{34}\\ S_{14} & S_{24} & S_{34} & 0 \end{bmatrix} \]
Reciprocity has now reduced the number of independent unknown coefficients, but the physical phase relationships of the Hybrid Ring must still be applied to determine which of these coefficients are equal, opposite in sign, or zero.
4. Applying the Isolation Condition
Consider excitation at Port 1. According to the operating principle of the Rat-Race Junction, the input power at Port 1 is divided equally between Ports 2 and 4, while Port 3 is isolated. Therefore, no signal appears at Port 3 when Port 1 alone is excited. This means that the transmission coefficient between Port 1 and Port 3 must be zero.
Hence,
\[ S_{13}=0 \]
By reciprocity,
\[ S_{31}=0 \]
Now consider excitation at Port 3. Port 1 is isolated under this excitation condition. The same isolation relationship therefore follows naturally from the reciprocal operation of the network. Thus, the zero \(S_{13}\) and \(S_{31}\) elements represent the ideal isolation between the sum and difference ports.
Substituting this condition gives
\[ [S] = \begin{bmatrix} 0 & S_{12} & 0 & S_{14}\\ S_{12} & 0 & S_{23} & S_{24}\\ 0 & S_{23} & 0 & S_{34}\\ S_{14} & S_{24} & S_{34} & 0 \end{bmatrix} \]
5. Equal-Amplitude and Equal-Phase Condition
When Port 1 is excited, the signal divides equally between Ports 2 and 4 and the two output signals have the same phase. Therefore, the transmission coefficients from Port 1 to Ports 2 and 4 must have equal magnitude and equal phase. Under the selected reference-plane convention, this relationship can be written as
\[ S_{12}=S_{14} \]
By reciprocity, the corresponding reverse transmission coefficients also satisfy
\[ S_{21}=S_{41} \]
Similarly, when Port 3 is excited, equal-amplitude signals appear at Ports 2 and 4 but with a \(180^\circ\) phase difference. Therefore, the corresponding coefficients must have equal magnitude and opposite sign:
\[ S_{23}=-S_{34} \]
By reciprocity, this also gives the corresponding reverse-direction relationship.
\[ S_{32}=-S_{43} \]
These relationships are the mathematical representation of the two fundamental operating modes of the Hybrid Ring. The equal signs represent the in-phase sum behavior, whereas the opposite signs represent the \(180^\circ\) phase reversal associated with the difference behavior.
6. Reduced S-Matrix Before Numerical Evaluation
Applying the matching, reciprocity, isolation, and phase relationships progressively reduces the original matrix to a form containing only the coupling coefficients that are physically permitted by the Hybrid Ring structure. Let the common magnitude of the equal-phase coupling be represented directly by the S-parameter \(S_{12}\), while the difference-mode coupling is represented by \(S_{23}\). The matrix can therefore be expressed as
\[ [S] = \begin{bmatrix} 0 & S_{12} & 0 & S_{12}\\ S_{12} & 0 & S_{23} & S_{24}\\ 0 & S_{23} & 0 & -S_{23}\\ S_{12} & S_{24} & -S_{23} & 0 \end{bmatrix} \]
The symmetry of the physical structure further requires equal coupling from the two input/output ports to the appropriate sum and difference ports. Thus, the coupling relationships can be reduced consistently so that the final matrix contains the equal-amplitude sum-mode and difference-mode coefficients. At this stage, however, their numerical magnitudes have not yet been determined. These values must come from the lossless condition.
7. Applying the Lossless Condition
An ideal Hybrid Ring is a passive lossless microwave network. Therefore, the total power entering the network must equal the total power leaving the network. In S-parameter form, a lossless network satisfies the unitary condition
\[ [S][S]^\dagger=[I] \]
where \([S]^\dagger\) is the Hermitian transpose of the scattering matrix and \([I]\) is the identity matrix. This condition means that every row and every column of the scattering matrix must have unit power norm and that different rows or columns must be orthogonal to one another.
Consider the row associated with Port 1. The two nonzero coupling coefficients correspond to the equal-amplitude outputs at Ports 2 and 4. The diagonal element and the isolated-port coefficient are both zero. Therefore, the normalization condition for this row becomes
\[ |S_{12}|^2+|S_{14}|^2=1 \]
Since the two output paths have equal magnitude,
\[ |S_{12}|=|S_{14}| \]
therefore,
\[ 2|S_{12}|^2=1 \]
giving
\[ |S_{12}|=\frac{1}{\sqrt{2}} \]
Consequently,
\[ |S_{14}|=\frac{1}{\sqrt{2}} \]
The same argument applies to the difference-mode excitation. Since Port 3 divides the input power equally between Ports 2 and 4, the two corresponding coupling coefficients must also each have magnitude \(1/\sqrt{2}\). Therefore,
\[ |S_{23}|=|S_{34}|=\frac{1}{\sqrt{2}} \]
8. Final Ideal Hybrid Ring S-Matrix
Combining the magnitude conditions with the previously established phase relationships gives the normalized ideal four-port scattering matrix. Under the selected port numbering and reference-plane convention, the final matrix is
\[ [S] = \frac{1}{\sqrt{2}} \begin{bmatrix} 0 & 1 & 0 & 1\\ 1 & 0 & 1 & 0\\ 0 & 1 & 0 & -1\\ 1 & 0 & -1 & 0 \end{bmatrix} \]
This matrix contains the complete ideal behavior of the Rat-Race Junction under the selected port-numbering convention. The zero diagonal elements represent perfect matching, the symmetric arrangement confirms reciprocity, the \(S_{13}=S_{31}=0\) terms represent isolation between Ports 1 and 3, and the \(1/\sqrt{2}\) coefficients represent equal power division between the coupled ports. The positive and negative signs distinguish the sum and difference modes of the Hybrid Ring.
It is important to understand that the exact position of the positive and negative coefficients depends on the chosen port numbering and reference-plane convention. The physical behavior should therefore be identified from the relationships between the S-parameters rather than by memorizing one matrix arrangement. In particular, the two ports receiving the sum-mode signal have equal magnitude and equal phase, while the two ports associated with the difference mode have equal magnitude and opposite phase. If the port numbering is changed, both the corresponding rows and columns of the S-matrix must be changed together.
Prove that the ports of a Hybrid Ring are matched and mutually isolated when properly terminated.
For a 4-port hybrid ring, the scattering matrix can be written as
\[ [S]=\frac{1}{\sqrt{2}} \begin{bmatrix} 0 & -j & 0 & -j\\ -j & 0 & -j & 0\\ 0 & -j & 0 & -j\\ -j & 0 & -j & 0 \end{bmatrix} \]
However, this simplified form does not represent the phase relationships of the standard rat-race correctly. A more useful idealized representation for proving matching and isolation is to consider the even and odd excitation behavior.
Ports are matched
A port is matched when there is no reflected wave at that port. Mathematically,
\[ S_{11}=S_{22}=S_{33}=S_{44}=0 \]
Therefore, the reflection coefficient at every port is zero:
\[ \Gamma_1=\Gamma_2=\Gamma_3=\Gamma_4=0 \]
Hence,
\[ \boxed{\text{All ports of the Hybrid Ring are matched.}} \]
This means that when power is applied to any port, no power is reflected back to the source.
Ports are mutually isolated
Two ports are mutually isolated when there is no power transferred directly between them. In terms of the scattering matrix,
\[ S_{ij}=0 \]
for the isolated port pairs.
For example, if port 1 is excited, the ideal Hybrid Ring produces outputs at the appropriate ports with equal magnitude but with the required phase relationship. At the isolated port, the two waves arriving through the two paths of the ring are equal in magnitude and opposite in phase.
Thus,
\[ V_{\text{isolated}}=V_1+V_2 \]
where
\[ V_1=-V_2 \]
Therefore,
\[ V_{\text{isolated}}=0 \]
and hence
\[ \boxed{S_{ij}=0} \]
for the corresponding isolated ports.
The isolation occurs because the Hybrid Ring has two possible paths between particular ports. The ring circumference is designed so that the waves traveling along these paths have the required phase difference.
At the isolated port, the two waves arrive with equal amplitudes and a \(180^\circ\) phase difference:
\[ A+A(-1)=0 \]
Therefore, they cancel completely.
At the output ports, the waves instead combine constructively, giving the desired power division. Thus, for an ideal properly terminated Hybrid Ring,
\[ \boxed{S_{11}=S_{22}=S_{33}=S_{44}=0} \]
which proves that all ports are matched, while the appropriate off diagonal scattering coefficients are zero,
\[ \boxed{S_{ij}=0} \]
which proves that the corresponding ports are mutually isolated.
Therefore, a properly terminated Hybrid Ring is a matched and isolated multiport network.
Operation of Hybrid Ring as a 3 dB In Phase and Out of Phase Power Splitter
3 dB Power Splitting Operation of a Hybrid Ring
A Hybrid Ring operates as a 3 dB power splitter because the input power is divided equally between two output ports. Depending on the port through which the signal is applied, the two output signals can be either in phase or out of phase. The phase relationship is determined by the electrical lengths of the two paths through the ring.
3 dB In Phase Power Splitter
Consider a signal applied to the sum port of the Hybrid Ring. The input signal travels around the ring in two different directions and reaches the two output ports. The ring dimensions are selected such that the signals arriving at the two output ports have the same phase.
If the input signal is represented by \(V_i\), the two output voltages have equal magnitude and equal phase. Therefore,
\[ V_2=V_3 \]
For an ideal 3 dB splitter, the power is divided equally between the two output ports. Hence,
\[ P_2=P_3=\frac{P_i}{2} \]
The corresponding voltage or wave amplitude at each output is therefore reduced by a factor of \(\sqrt{2}\):
\[ b_2=\frac{1}{\sqrt{2}}a_1 \]
\[ b_3=\frac{1}{\sqrt{2}}a_1 \]
The two output waves have the same phase, so
\[ \angle b_2=\angle b_3 \]
Thus, the Hybrid Ring acts as a 3 dB in phase power splitter.
3 dB Out of Phase Power Splitter
Now consider excitation at the difference port of the Hybrid Ring. The input signal again divides into two waves that travel around the ring in opposite directions. Because of the different electrical path lengths, the waves arrive at the two output ports with a phase difference of \(180^\circ\).
Therefore, the two output waves have equal magnitude but opposite phase:
\[ V_2=-V_3 \]
As before, the input power is divided equally between the two output ports:
\[ P_2=P_3=\frac{P_i}{2} \]
The corresponding wave amplitudes can be represented as
\[ b_2=\frac{1}{\sqrt{2}}a_1 \]
and
\[ b_3=-\frac{1}{\sqrt{2}}a_1 \]
Therefore,
\[ \angle b_2-\angle b_3=180^\circ \]
Hence, the Hybrid Ring acts as a 3 dB out of phase power splitter.
Why the Power Division Is 3 dB
The term 3 dB comes from the equal division of power between two output ports. If the input power is \(P_i\), each output receives half of the input power:
\[ P_o=\frac{P_i}{2} \]
The power division in decibels is therefore
\[ 10\log_{10}\left(\frac{P_o}{P_i}\right) = 10\log_{10}\left(\frac{1}{2}\right) \]
which gives
\[ 10\log_{10}\left(\frac{1}{2}\right)=-3.01\text{ dB} \]
Thus, each output port receives approximately 3 dB less power than the input.
In Phase and Out of Phase Operation
The two operating modes of the Hybrid Ring can therefore be summarized as follows.
- In phase operation: the two output signals have equal magnitude and \(0^\circ\) phase difference.
- Out of phase operation: the two output signals have equal magnitude and \(180^\circ\) phase difference.
- Power division: in both cases, the input power is divided equally between the two output ports.
- Power at each output: \(P_i/2\), corresponding to approximately \(3\text{ dB}\) below the input power.
A Hybrid Ring can operate as both a 3 dB in phase power splitter and a 3 dB out of phase power splitter. When the sum port is excited, the two output signals have equal magnitude and are in phase. When the difference port is excited, the two output signals have equal magnitude but differ in phase by \(180^\circ\). In both cases, the input power is divided equally between the two output ports, giving a \(3\text{ dB}\) power division.
How 180 phase shift?
How Is the 180° Phase Shift Produced?
The 180° phase shift in a Hybrid Ring is produced by the difference in electrical path lengths traveled by the two waves around the ring. When a signal enters the Hybrid Ring, it divides into two waves that travel in opposite directions around the ring.
For one output port, the two waves travel paths of equal electrical length. Therefore, they arrive at the output port with the same phase and combine constructively.
For the other output port, the two waves travel paths whose lengths differ by
\[ \Delta l=\frac{\lambda}{2} \]
The phase difference produced by a path difference is
\[ \Delta\phi=\frac{2\pi}{\lambda}\Delta l \]
Substituting \(\Delta l=\lambda/2\),
\[ \Delta\phi= \frac{2\pi}{\lambda} \left(\frac{\lambda}{2}\right) \]
Therefore,
\[ \Delta\phi=\pi=180^\circ \]
Thus, the two waves arrive at the output port with equal magnitude but opposite phase. If the two waves are represented as
\[ V_1=A\angle0^\circ \]
and
\[ V_2=A\angle180^\circ=-A \]
their resultant voltage is
\[ V_{\text{out}}=V_1+V_2 \]
\[ V_{\text{out}}=A+(-A)=0 \]
Hence, the waves cancel each other at the isolated port due to destructive interference.
In simple terms, the Hybrid Ring produces the required phase relationship through its electrical path lengths:
\[ \boxed{\Delta l=0\quad\Rightarrow\quad\Delta\phi=0^\circ} \]
This produces in phase signals.
\[ \boxed{\Delta l=\frac{\lambda}{2}\quad\Rightarrow\quad\Delta\phi=180^\circ} \]
This produces out of phase signals.
Therefore, the \(180^\circ\) phase difference required for out of phase operation is a direct result of the half wavelength path difference between the two waves traveling through the Hybrid Ring.
Power Interpretation of the \(1/\sqrt{2}\) Coefficients
The coefficient \(1/\sqrt{2}\) has a direct power interpretation. If one port of an ideal Hybrid Ring is excited with normalized incident power, the amplitude of each of the two equal output waves is \(1/\sqrt{2}\) times the incident amplitude. Since microwave power is proportional to the square of the wave amplitude, the power delivered to each output is
\[ \left|\frac{1}{\sqrt{2}}\right|^2 = \frac{1}{2} \]
Therefore, each output receives half of the available input power. The total output power is
\[ \frac{1}{2}+\frac{1}{2}=1 \]
which confirms power conservation for the ideal lossless network. The negative sign in a coefficient such as \(-1/\sqrt{2}\) does not mean negative power. It represents a \(180^\circ\) phase reversal because power depends on the magnitude squared of the coefficient. Thus, \(+1/\sqrt{2}\) and \(-1/\sqrt{2}\) have the same power magnitude but different phase.
Important Pattern for Identifying the Hybrid Ring S-Matrix
The Hybrid Ring S-matrix should not be memorized only according to the physical position of a particular port because different textbooks and diagrams may use different port-numbering conventions. A more reliable method is to identify the physical relationships represented by the matrix. For the ideal four-port Rat-Race Junction, the diagonal elements are zero because all ports are matched. One pair of ports is isolated, represented by zero transmission coefficients between those ports. The remaining nonzero transmission coefficients have equal magnitude, and their signs indicate whether the corresponding signals are in phase or \(180^\circ\) out of phase.
The main identification pattern is therefore based on the following conditions:
- Matched ports: \(S_{11}=S_{22}=S_{33}=S_{44}=0\).
- Reciprocity: \(S_{ij}=S_{ji}\).
- Equal power division: the magnitude of each nonzero coupling coefficient is \(1/\sqrt{2}\).
- Sum behavior: two equal-magnitude coupling coefficients have the same phase.
- Difference behavior: two equal-magnitude coupling coefficients have opposite phase.
- Isolation: the appropriate sum and difference ports have zero transmission between them.
- Losslessness: \([S][S]^\dagger=[I]\).
These relationships remain useful even when the ports are renumbered. When the port numbering changes, the correct procedure is to interchange the corresponding row and column together. Interchanging only a row or only a column changes the physical meaning of the S-parameters and can produce an incorrect matrix. Therefore, the safest approach is to first identify the sum port, difference port, and the two coupled ports from the physical diagram, and then construct the corresponding S-matrix using matching, reciprocity, phase, isolation, and lossless conditions.
Hybrid Ring S-Matrix Variations for Different Port Excitations
The final S-matrix of a Hybrid Ring should not be understood only as a single fixed arrangement of positive and negative terms. The physical behavior remains the same when the ports are renumbered, but the positions of the S-parameters change because the corresponding rows and columns of the scattering matrix are also changed. Therefore, a useful way to analyze examination problems is to understand the response for excitation at each port and then identify the sum, difference, and isolated ports from the S-matrix. This approach avoids memorizing several separate matrices and makes it easier to recognize a Hybrid Ring even when a different port-numbering convention is used.
1. Excitation at Port 1
When Port 1 is excited, the Hybrid Ring operates as an equal power divider. The incident signal travels around the ring in both directions and reaches Ports 2 and 4 through paths that produce constructive interference. Therefore, Ports 2 and 4 receive equal-amplitude signals having the same phase. Port 3, on the other hand, receives two contributions that arrive with a relative phase difference causing cancellation. Hence, Port 3 is isolated from Port 1 under the ideal operating condition.
From the ideal S-matrix,
\[ S_{21}=\frac{1}{\sqrt{2}} \]
and
\[ S_{41}=\frac{1}{\sqrt{2}} \]
while
\[ S_{31}=0 \]
Thus, the input power at Port 1 is divided equally between Ports 2 and 4. Since each amplitude coefficient has magnitude \(1/\sqrt{2}\), the corresponding power coefficient is
\[ \left|\frac{1}{\sqrt{2}}\right|^2=\frac{1}{2} \]
Therefore, each of Ports 2 and 4 receives one-half of the input power, while Port 3 receives ideally zero power.
2. Excitation at Port 3
Port 3 provides the complementary operating mode of the Hybrid Ring. When Port 3 is excited, the signal again divides equally between Ports 2 and 4, but the two output signals now have opposite phase. The signal reaching Port 1 through the two paths cancels ideally, so Port 1 becomes isolated. Therefore, Port 3 acts as the difference port, while Port 1 acts as the corresponding sum port.
The relevant S-parameters are
\[ S_{23}=\frac{1}{\sqrt{2}} \]
and
\[ S_{43}=-\frac{1}{\sqrt{2}} \]
while
\[ S_{13}=0 \]
The magnitudes of \(S_{23}\) and \(S_{43}\) are identical, so the output powers at Ports 2 and 4 are equal. However, the negative sign in \(S_{43}\) indicates a \(180^\circ\) phase difference between the two outputs. The negative sign therefore represents phase reversal and does not indicate negative power.
3. Two-Input Operation at Ports 2 and 4
The most important combining operation of the Hybrid Ring occurs when two signals are applied simultaneously at Ports 2 and 4. The network then produces a sum component at Port 1 and a difference component at Port 3. This is the reverse operation of the single-port power-divider behavior described above. Instead of dividing one signal into two outputs, the Hybrid Ring receives two signals and separates their common and differential components.
If the signals applied at Ports 2 and 4 are represented by \(V_2\) and \(V_4\), the output at Port 1 is proportional to their sum:
\[ V_1=\frac{1}{\sqrt{2}}\left(V_2+V_4\right) \]
while the output at Port 3 is proportional to their difference:
\[ V_3=\frac{1}{\sqrt{2}}\left(V_2-V_4\right) \]
These relationships show directly why the Hybrid Ring is useful as a sum-and-difference network. Port 1 responds to the component that is common to both input signals, whereas Port 3 responds to the difference between them. The sign convention depends on the selected port numbering and reference-plane convention, but the physical distinction between the sum and difference outputs remains unchanged.
3.1 Equal In-Phase Inputs
Suppose equal signals are applied at Ports 2 and 4 with the same phase. Let the two signals have equal amplitude and equal phase. In that case, the two signals add constructively at Port 1. At Port 3, the contributions have opposite signs in the S-matrix and therefore cancel.
For equal inputs,
\[ V_2=V_4 \]
and consequently,
\[ V_3 = \frac{1}{\sqrt{2}} \left(V_2-V_4\right) = 0 \]
At the sum port,
\[ V_1 = \frac{1}{\sqrt{2}} \left(V_2+V_4\right) \]
Therefore, equal in-phase signals applied at Ports 2 and 4 combine at Port 1, while Port 3 is ideally isolated. This is the fundamental sum operation of the Hybrid Ring.
3.2 Equal Opposite-Phase Inputs
If equal-amplitude signals are applied at Ports 2 and 4 but have a \(180^\circ\) phase difference, the situation is reversed. The signals cancel at the sum port and combine at the difference port. Mathematically, the second input can be represented by the negative of the first input:
\[ V_4=-V_2 \]
Therefore, the sum output becomes
\[ V_1 = \frac{1}{\sqrt{2}} \left(V_2+V_4\right) = 0 \]
At Port 3,
\[ V_3 = \frac{1}{\sqrt{2}} \left(V_2-V_4\right) \]
and the difference component therefore appears at Port 3. Thus, equal but opposite-phase inputs are directed to the difference port, while the sum port becomes ideally isolated.
3.3 Unequal Inputs at Ports 2 and 4
When the two signals applied at Ports 2 and 4 are unequal, neither the sum nor the difference is generally zero. Consequently, both Port 1 and Port 3 can produce output signals. The sum port receives the combined contribution of the two input signals, whereas the difference port receives the residual difference between them. This operating condition is particularly useful when the Hybrid Ring is used in balanced microwave circuits where the two input signals do not have identical amplitudes.
For unequal input signals,
\[ V_2\neq V_4 \]
the outputs are
\[ V_1 = \frac{1}{\sqrt{2}} \left(V_2+V_4\right) \]
and
\[ V_3 = \frac{1}{\sqrt{2}} \left(V_2-V_4\right) \]
Therefore, both outputs contain useful information about the two input signals. Port 1 represents their sum, while Port 3 represents their difference. If the amplitudes become equal and the phase becomes identical, the difference output approaches zero. If the inputs become equal in magnitude but opposite in phase, the sum output approaches zero.
4. Excitation at Port 2
Excitation at Port 2 provides another way to understand the reciprocal nature of the Hybrid Ring. According to the S-matrix, a signal entering Port 2 is coupled equally to Ports 1 and 3. The corresponding coefficients have equal magnitude, but their signs are different according to the selected reference-plane convention. Port 4 is isolated from Port 2 under the ideal condition.
From the ideal matrix,
\[ S_{12}=\frac{1}{\sqrt{2}} \]
and
\[ S_{32}=\frac{1}{\sqrt{2}} \]
while
\[ S_{42}=0 \]
Thus, the input at Port 2 is divided equally between Ports 1 and 3, with the appropriate phase relationship determined by the selected S-matrix convention. Port 4 receives no ideal output. Because the network is reciprocal, this response is exactly related to the behavior observed when the corresponding output ports are used as inputs.
5. Excitation at Port 4
Port 4 provides the complementary response to Port 2. When Port 4 is excited, equal-amplitude signals are produced at Ports 1 and 3, while Port 2 becomes isolated. The phase relationship is opposite to the response produced by excitation at Port 2, as required by the signs in the S-matrix.
The corresponding coefficients are
\[ S_{14}=\frac{1}{\sqrt{2}} \]
and
\[ S_{34}=-\frac{1}{\sqrt{2}} \]
while
\[ S_{24}=0 \]
Therefore, excitation at Port 4 produces equal power division between Ports 1 and 3, with a phase relationship determined by the positive and negative signs of the corresponding S-parameters. Port 2 is ideally isolated from Port 4.
6. Complete Port-Excitation Pattern
The behavior of all four ports can be summarized by observing which port is isolated and which two ports receive equal-amplitude signals. This is more useful than memorizing individual equations because the same physical pattern remains valid when the numbering convention is changed.
For the port numbering used in this article, the single-port excitation behavior can be represented as follows:
- Port 1 excited: Ports 2 and 4 receive equal-amplitude in-phase signals, while Port 3 is isolated.
- Port 3 excited: Ports 2 and 4 receive equal-amplitude opposite-phase signals, while Port 1 is isolated.
- Port 2 excited: Ports 1 and 3 receive equal-amplitude signals with the phase relationship determined by the matrix signs, while Port 4 is isolated.
- Port 4 excited: Ports 1 and 3 receive equal-amplitude signals with the complementary phase relationship, while Port 2 is isolated.
This pattern shows that the four ports can be divided into two functional pairs. Ports 1 and 3 act as the sum and difference ports, while Ports 2 and 4 form the pair through which the two signals can be combined or divided. When the device is excited from either of the sum or difference ports, the signal is divided between the other pair. When the device is excited from the two paired ports simultaneously, the network performs sum and difference operations at the remaining two ports.
7. How to Identify a Hybrid Ring When Port Numbering Changes
Different textbooks, diagrams, and examination questions may assign different numbers to the four ports of a Hybrid Ring. As a result, the positive and negative \(1/\sqrt{2}\) terms may appear in different positions in the S-matrix. The correct approach is therefore not to memorize the position of every coefficient. Instead, identify the physical relationship between the ports and then construct or interpret the matrix according to that relationship.
The first feature to identify is the zero diagonal elements. For an ideal matched Hybrid Ring,
\[ S_{11}=S_{22}=S_{33}=S_{44}=0 \]
Next, identify the isolated pair. If two ports are isolated from each other, the corresponding off-diagonal coefficients are zero. For the numbering used here,
\[ S_{13}=S_{31}=0 \]
and
\[ S_{24}=S_{42}=0 \]
The remaining nonzero coefficients have magnitude \(1/\sqrt{2}\). Their signs indicate the phase relationship. Two coefficients with the same sign represent equal phase under the selected reference-plane convention, whereas coefficients with opposite signs represent a \(180^\circ\) phase difference.
8. Pattern of Zero and \(1/\sqrt{2}\) Terms
The ideal Hybrid Ring has a particularly recognizable S-matrix pattern. There are four zero diagonal elements because every port is matched. In addition, there are four off-diagonal zero elements corresponding to the two isolated port pairs, with each physical isolation relationship appearing twice because of reciprocity. The remaining eight off-diagonal elements have magnitude \(1/\sqrt{2}\). Thus, the matrix contains twelve zero entries in total and eight nonzero entries, each having magnitude \(1/\sqrt{2}\).
The nonzero coefficients can be either positive or negative depending on the chosen reference-plane and port-numbering convention. The important physical information is not simply the number of positive or negative signs but their relative arrangement. The signs must produce the required equal-phase sum behavior and \(180^\circ\) difference behavior while also satisfying reciprocity and the lossless condition.
For the convention used in this article, the ideal matrix contains four positive \(1/\sqrt{2}\) terms and two negative \(1/\sqrt{2}\) terms when counted as unique symmetric positions. Because the matrix is symmetric, each off-diagonal term appears twice. Therefore, the complete matrix contains eight positive \(1/\sqrt{2}\) entries and four negative \(1/\sqrt{2}\) entries. The remaining four diagonal and four reciprocal-isolation pairs give the required zero pattern.
9. Port Renumbering Rule
When the port numbering of a Hybrid Ring is changed, the S-matrix must be transformed consistently. If two ports exchange their numbers, the corresponding rows and columns must both be exchanged. A row represents the output port index, while a column represents the input port index. Changing only one of them would change the physical meaning of the S-parameters and would not represent a simple renumbering of the same network.
For example, if Ports 1 and 3 are interchanged, the first and third rows must be interchanged and the first and third columns must also be interchanged. The resulting matrix may look different from the original matrix, but it describes exactly the same physical Hybrid Ring. Therefore, an examination question may present a matrix with a different arrangement of positive and negative coefficients and still represent the same Rat-Race Junction.
The safest identification method is therefore to check the physical properties rather than the exact visual arrangement of the matrix. Verify that the ports are matched, verify reciprocity, locate the isolated port pairs, identify the equal-amplitude coupling coefficients, and then examine their relative signs to determine the sum and difference relationships. Finally, the complete matrix should satisfy the lossless condition
\[ [S][S]^\dagger=[I] \]
This procedure allows the Hybrid Ring S-matrix to be identified correctly even when the port names or numbering are different from the standard arrangement.
Derivation of the Characteristic S-Matrix from the Rat-Race Geometry: The characteristic S-matrix of the rat-race hybrid can be obtained directly from the relative electrical lengths of the transmission-line paths around the ring. The total perimeter of the rat-race is \(3\lambda/2\), or \(1.5\lambda\). The ports are arranged so that, for the relevant port pairs, one signal path has an electrical length of \(\lambda/4\), while the other path has an electrical length of \(3\lambda/4\). Since the two paths differ by \(\lambda/2\), the waves traveling through them acquire a phase difference of \(180^\circ\).
\[ \Delta l=\frac{3\lambda}{4}-\frac{\lambda}{4} =\frac{\lambda}{2} \] \[ \Delta\phi = \frac{2\pi}{\lambda}\Delta l = \frac{2\pi}{\lambda}\frac{\lambda}{2} = \pi = 180^\circ \]To obtain the S-matrix, first consider an excitation at port 1. The incident wave divides into the two branches connected to the other ports. Because the two relevant paths have equal branch characteristics, the two output waves have equal magnitude. The total ring length and the port locations determine their relative phase. At ports 3 and 4, the two contributions arrive through paths having the required phase relationship, so the output amplitudes are obtained by adding the corresponding wave contributions.
For an ideal matched rat-race hybrid, the power supplied at a single input port is equally divided between the two output ports. Therefore, the magnitude of each nonzero transmission coefficient is
\[ |S_{ij}|=\frac{1}{\sqrt{2}} \]Since the network is matched, there is no reflected wave at the excited port. Hence, for excitation at port 1,
\[ S_{11}=0 \]The geometry also provides isolation between ports 1 and 2, so
\[ S_{12}=0 \]The waves reaching ports 3 and 4 have equal magnitude. With the selected port numbering and reference-plane convention, they have the same phase for excitation at port 1. Therefore,
\[ S_{31}=S_{41}=\frac{1}{\sqrt{2}} \]Thus, the first column of the S-matrix is
\[ \begin{bmatrix} S_{11}\\ S_{21}\\ S_{31}\\ S_{41} \end{bmatrix} = \frac{1}{\sqrt{2}} \begin{bmatrix} 0\\ 0\\ 1\\ 1 \end{bmatrix} \]Now consider excitation at port 2. Again, the incident power divides equally between the two available output paths, so the magnitudes of the two nonzero scattering coefficients are \(1/\sqrt{2}\). Port 2 is matched and is isolated from port 1, giving
\[ S_{22}=0,\qquad S_{21}=0 \]The important difference is the relative path length. One contribution reaches port 4 through the \(3\lambda/4\) path, while the corresponding path to port 3 has the shorter electrical length. The additional half-wavelength difference produces a \(180^\circ\) phase reversal. Therefore, the waves at ports 3 and 4 have opposite signs for excitation at port 2:
\[ S_{32}=\frac{1}{\sqrt{2}}, \qquad S_{42}=-\frac{1}{\sqrt{2}} \]The second column is therefore
\[ \begin{bmatrix} S_{12}\\ S_{22}\\ S_{32}\\ S_{42} \end{bmatrix} = \frac{1}{\sqrt{2}} \begin{bmatrix} 0\\ 0\\ 1\\ -1 \end{bmatrix} \]Because the rat-race hybrid is a reciprocal network, its scattering matrix satisfies
\[ S_{ij}=S_{ji} \]Therefore, the remaining elements are obtained from the corresponding elements already determined from excitations at ports 1 and 2. In particular,
\[ S_{13}=S_{31}=\frac{1}{\sqrt{2}} \] \[ S_{14}=S_{41}=\frac{1}{\sqrt{2}} \] \[ S_{23}=S_{32}=\frac{1}{\sqrt{2}} \] \[ S_{24}=S_{42}=-\frac{1}{\sqrt{2}} \]The diagonal elements remain zero because all four ports are matched, while the required isolated port pairs remain represented by zero elements. Combining these results gives the characteristic S-matrix of the rat-race hybrid:
\[ \boxed{ [S] = \frac{1}{\sqrt{2}} \begin{bmatrix} 0 & 0 & 1 & 1\\ 0 & 0 & 1 & -1\\ 1 & 1 & 0 & 0\\ 1 & -1 & 0 & 0 \end{bmatrix} } \]The matrix therefore follows directly from the physical geometry: the \(1.5\lambda\) total perimeter provides the required ring structure, while the \(\lambda/4\) and \(3\lambda/4\) relative paths create a \(180^\circ\) phase difference. Equal path coupling gives the magnitude \(1/\sqrt{2}\), the matched ports produce zero diagonal elements, isolation produces the zero coupling terms, and reciprocity completes the remaining S-parameters. The resulting matrix is the characteristic S-matrix of the ideal rat-race hybrid for the selected port and phase convention.
Examination-Oriented Method for Solving Hybrid Ring Variations
When a Hybrid Ring question appears with an unfamiliar port numbering, the most reliable method is to begin with the physical diagram rather than attempting to recall a memorized matrix. First identify the two ports that form the sum and difference pair. Then determine which remaining ports receive equal-amplitude signals when one of those ports is excited. The port that receives no signal is the isolated port. After identifying these relationships, write the general S-matrix using \(S_{ij}\) terms and apply matching and reciprocity. The phase relationship then determines whether the corresponding coefficients have equal signs or opposite signs. Finally, the lossless condition confirms the magnitude \(1/\sqrt{2}\) and verifies the completed matrix.
This approach is especially useful when an examination problem asks for the S-matrix of a Hybrid Ring with a different port arrangement. Instead of memorizing separate matrices for every possible numbering system, remember the physical pattern: matched ports have zero diagonal terms, isolated ports have zero transmission between them, equal power division produces coefficients of magnitude \(1/\sqrt{2}\), equal-phase outputs have the same sign, opposite-phase outputs have opposite signs, and a reciprocal lossless network must satisfy both symmetry and the unitary condition. These properties are sufficient to reconstruct the appropriate matrix for any consistent port-numbering convention.