Branching Synthesis of Couplers and Hybrids

Branching Synthesis of Couplers and Hybrids: Rat-Race Hybrid S-Matrix Derivation

Branching Synthesis of Couplers and Hybrids is a technique used to realize microwave couplers and hybrid networks by connecting suitable transmission-line branches between the required ports. The basic idea is to first specify the desired admittance matrix of the microwave network and then synthesize that matrix using quarter-wavelength and three-quarter-wavelength TEM transmission-line sections. This method is particularly useful for realizing hybrid couplers because a quarter-wavelength transmission-line element produces a purely imaginary off-diagonal admittance while maintaining zero self-admittance at the design frequency. By selecting the characteristic admittance and electrical length of each branch appropriately, the required coupling and phase relationships between different ports can be obtained. A major application of this synthesis technique is the realization of the rat-race hybrid coupler, which can be synthesized from the S-matrix of a matched Magic Tee.

Basic Principle of Branching Synthesis

Consider a two-port TEM transmission-line section having characteristic admittance \(Y_{0A}\), connected between two ports. For a transmission line of characteristic admittance \(Y_0\) and length \(z\), the admittance matrix can be written as

For a quarter-wavelength section, \(z=\lambda/4\), the electrical length is \(\beta z=\pi/2\). Therefore, \(\cot(\pi/2)=0\) and \(\csc(\pi/2)=1\). The admittance matrix becomes

When this matrix is normalized with respect to the system characteristic admittance \(Y_0\), the normalized characteristic admittance is \(Y_{0A}/Y_0\). Hence, a quarter-wavelength TEM branch contributes zero diagonal admittance and a positive imaginary value to the corresponding off-diagonal elements. This property makes the quarter-wavelength branch particularly convenient for synthesizing a desired microwave admittance matrix.

Similarly, when the transmission-line length is \(3\lambda/4\), the electrical length becomes \(3\pi/2\). The resulting admittance matrix is

Thus, a three-quarter-wavelength TEM branch produces the same magnitude of off-diagonal admittance as a quarter-wavelength branch but with the opposite sign. This gives the synthesis process an important degree of freedom. A \( \lambda/4 \) branch is used when a positive imaginary coefficient is required, while a \(3\lambda/4\) branch is used when a negative imaginary coefficient is required. Therefore, any network whose admittance matrix has zero diagonal elements and purely imaginary off-diagonal elements can be synthesized by appropriately branching \( \lambda/4 \) and \(3\lambda/4\) TEM transmission-line elements.

Rat-Race Hybrid Synthesis from a Magic Tee

The rat-race hybrid can be synthesized by starting with the scattering matrix of a matched Magic Tee and then transforming it into a form whose corresponding admittance matrix can be directly realized using TEM transmission-line branches. Consider a matched Magic Tee having collinear ports 1 and 2, H-arm port 3, and E-arm port 4. For the chosen port numbering and phase convention, its scattering matrix is

This matrix represents the fundamental properties of the matched Magic Tee. The diagonal elements are zero, showing that all four ports are matched. Ports 1 and 2 are isolated from each other because \(S_{12}=S_{21}=0\). Similarly, ports 3 and 4 are isolated because \(S_{34}=S_{43}=0\). An excitation at the H-arm divides equally between the two collinear ports with the same phase, whereas an excitation at the E-arm divides equally between the two collinear ports with opposite phase. These equal-amplitude and phase relationships are the properties that must be reproduced by the synthesized hybrid.

Reference Plane Shift and Modified S-Matrix

To obtain an admittance matrix suitable for branching synthesis, the reference planes of the four ports are moved away from the Magic Tee junction by a distance corresponding to a quarter wavelength. This reference-plane shift does not destroy the matching or isolation properties of the network, but it changes the phase of the scattering parameters. If the reference plane of port \(i\) is shifted by an electrical distance \(q_i\), an S-parameter transforms according to

For a quarter-wavelength shift, the electrical distance is \(q=\pi/4\). Therefore, for a scattering parameter connecting two shifted ports, the total phase shift is

Consequently, the corresponding nonzero scattering parameters acquire a factor of \(e^{-j\pi/2}=-j\). With the selected reference planes, the modified scattering matrix can therefore be written as

The exact overall sign of the matrix can depend on the direction chosen for the reference-plane displacement and on the phase convention used for the ports. The important feature for the synthesis is that the modified matrix has purely imaginary nonzero elements and retains the required symmetry and isolation properties.

Detailed S-Matrix Properties

The modified S-matrix is symmetric, so the network satisfies the reciprocity condition

It is also purely imaginary, which means that taking its complex conjugate gives

Because the Magic Tee is assumed to be lossless, its scattering matrix must also satisfy the unitary condition

Since the matrix is symmetric, \( [S]^T=[S] \), and because it is purely imaginary, \( [S]^\dagger=-[S] \). Therefore, the unitary condition becomes

This important result allows the S-matrix to be converted into an admittance matrix in a particularly simple form. For a network normalized to the reference characteristic admittance \(Y_0\), the relationship between the scattering matrix and admittance matrix is

Using the matrix properties above, the expression can be simplified. Since matrix multiplication is associative, the expression may be rearranged as

Multiplying the numerator and denominator in the matrix sense by the appropriate factors gives

Using the lossless property \(S^2=-U\), the expression reduces to a matrix directly proportional to \(S\). With the reference-plane and sign convention used in the synthesis, the required normalized admittance matrix is represented by

For the selected phase convention, the synthesized admittance matrix therefore has the form

The sign may appear reversed if the opposite reference-plane convention is used. The physical synthesis is unchanged because the required positive and negative imaginary admittance coefficients are realized by selecting \( \lambda/4 \) or \(3\lambda/4\) branches accordingly.

Identifying the Required Branches

The most important step in the branching synthesis is to compare the required admittance matrix with the admittance matrix of individual TEM branches. The diagonal elements of the required matrix are zero, so there are no self-admittance terms that need to be synthesized separately. The off-diagonal elements determine which pairs of ports must be connected and whether the connection requires a quarter-wavelength or three-quarter-wavelength transmission-line section.

For this rat-race hybrid, the required direct connections are between ports 1 and 3, ports 1 and 4, ports 2 and 3, and ports 2 and 4. There are no direct coupling elements between ports 1 and 2 or between ports 3 and 4 because the corresponding admittance matrix elements are zero. Thus, the synthesis naturally produces four transmission-line branches connecting the two groups of ports.

Branch Between Ports 1 and 3

The admittance matrix requires a positive imaginary off-diagonal coefficient between ports 1 and 3. Therefore, a quarter-wavelength TEM transmission-line element is inserted between these two ports. Its normalized characteristic admittance is selected as

Thus, the branch between ports 1 and 3 has an electrical length of \( \lambda/4 \) and normalized characteristic admittance \(1/\sqrt{2}\). This branch contributes the required positive imaginary mutual admittance between ports 1 and 3.

Branch Between Ports 1 and 4

The same type of matrix coefficient is required between ports 1 and 4. Therefore, another quarter-wavelength TEM element is inserted between ports 1 and 4. Its normalized characteristic admittance is also

This branch produces the required mutual admittance between ports 1 and 4. Because the branch has the same characteristic admittance and the same electrical length as the branch between ports 1 and 3, the two paths contribute equal magnitude coupling from port 1 to ports 3 and 4.

Branch Between Ports 2 and 3

The admittance matrix also requires a positive imaginary coefficient between ports 2 and 3. Consequently, a quarter-wavelength TEM element is inserted between ports 2 and 3. Its normalized characteristic admittance is again

This branch therefore has the same electrical length and characteristic admittance as the branches between ports 1 and 3 and ports 1 and 4. The equal branch values are responsible for maintaining the required equal-amplitude power division of the hybrid.

Branch Between Ports 2 and 4

The required admittance coefficient between ports 2 and 4 has the opposite sign from the coefficients of the other three required connections. A quarter-wavelength branch would produce the wrong sign. Therefore, a three-quarter-wavelength TEM element is inserted between ports 2 and 4. Its normalized characteristic admittance remains

The use of \(3\lambda/4\) instead of \( \lambda/4 \) reverses the sign of the imaginary mutual admittance while preserving its magnitude. This is the key reason for the apparently different branch length in the rat-race structure. The three-quarter-wavelength branch supplies the negative coefficient required by the admittance matrix and establishes the necessary phase relationship between the two output paths.

Complete Branching Arrangement of the Rat-Race Hybrid

The four required branches can therefore be summarized directly from the synthesized admittance matrix. Between ports 1 and 3, a \( \lambda/4 \) branch of normalized characteristic admittance \(1/\sqrt{2}\) is used. Between ports 1 and 4, another \( \lambda/4 \) branch with normalized characteristic admittance \(1/\sqrt{2}\) is used. Between ports 2 and 3, a third \( \lambda/4 \) branch with normalized characteristic admittance \(1/\sqrt{2}\) is used. Finally, between ports 2 and 4, a \(3\lambda/4\) branch with normalized characteristic admittance \(1/\sqrt{2}\) is used. There are no direct branches between ports 1 and 2 or between ports 3 and 4 because the corresponding off-diagonal admittance elements are zero.

• Between ports 1 and 3: \( \lambda/4 \), normalized characteristic admittance \(1/\sqrt{2}\).
• Between ports 1 and 4: \( \lambda/4 \), normalized characteristic admittance \(1/\sqrt{2}\).
• Between ports 2 and 3: \( \lambda/4 \), normalized characteristic admittance \(1/\sqrt{2}\).
• Between ports 2 and 4: \(3\lambda/4\), normalized characteristic admittance \(1/\sqrt{2}\).

The physical structure formed by these four branches is the rat-race hybrid coupler. The three \( \lambda/4 \) branches and one \(3\lambda/4\) branch provide the required electrical path lengths and phase relationships. The longer \(3\lambda/4\) path is especially important because it introduces the required additional phase relative to the quarter-wavelength paths. As a result, the network can reproduce the sum and difference characteristics associated with the original Magic Tee while using TEM transmission-line elements.

Why the Three-Quarter-Wavelength Branch Is Required

The use of a \(3\lambda/4\) branch is not simply a choice of a physically longer transmission line. Its purpose is to obtain the opposite sign of the mutual admittance. A \( \lambda/4 \) branch has an admittance matrix containing \(+jY_{0A}\) in its off-diagonal positions, whereas a \(3\lambda/4\) branch contains \(-jY_{0A}\). Therefore, changing the branch from \( \lambda/4 \) to \(3\lambda/4\) changes the sign without changing the magnitude of the coupling coefficient.

This sign reversal is essential for the rat-race hybrid because one of the four required coupling coefficients has the opposite phase relationship from the other three. If all four branches were made \( \lambda/4 \) long, the resulting admittance matrix would not have the required sign pattern and the network would not reproduce the desired Magic Tee behavior. The \(3\lambda/4\) branch therefore provides the phase reversal required for proper hybrid operation.

Physical Meaning of the Synthesized S-Matrix

The synthesized network retains the main characteristics of the Magic Tee. The diagonal elements of the S-matrix are zero, indicating that the ports are matched. The zero elements between ports 1 and 2 indicate isolation between the two collinear ports, while the zero elements between ports 3 and 4 indicate isolation between the two remaining ports. The nonzero elements have equal magnitude \(1/\sqrt{2}\), indicating equal power division. Since the power associated with an S-parameter is proportional to the square of its magnitude, each nonzero coupling path has a power ratio of

Thus, when a single port is excited, the available power is divided equally between the two appropriate output ports. The signs or \(j\)-phase factors determine whether the corresponding output waves are in phase or opposite in phase. This phase relationship is what allows the rat-race hybrid to perform sum and difference operations.

Final Result

The branching synthesis method begins by selecting the required S-matrix of the desired hybrid, shifting the reference planes to obtain a purely imaginary symmetric S-matrix, and converting that matrix into the corresponding admittance matrix. Because the resulting admittance matrix has zero diagonal elements and purely imaginary off-diagonal elements, it can be physically realized using \( \lambda/4 \) and \(3\lambda/4\) TEM transmission-line sections. For the rat-race hybrid obtained from the Magic Tee, four branches are required: three quarter-wavelength branches and one three-quarter-wavelength branch. All four branches have normalized characteristic admittance \(1/\sqrt{2}\), while their electrical lengths are selected according to the sign of the corresponding admittance coefficient.

The final branch arrangement is therefore the key result of the synthesis: ports 1 and 3 are connected by a \( \lambda/4 \) branch, ports 1 and 4 are connected by a \( \lambda/4 \) branch, ports 2 and 3 are connected by a \( \lambda/4 \) branch, and ports 2 and 4 are connected by a \(3\lambda/4\) branch. This arrangement forms the rat-race hybrid and realizes the required matched, reciprocal, isolated, equal-power and phase-controlled behavior of the original Magic Tee network.