Hybrid-Pair Ratio Adjustable Power Splitters for Add-on RF Shimming and Array-Compressed Parallel Transmission

Abstract

Purpose:

A ratio adjustable power splitter (RAPS) circuit was recently proposed for add-on RF shimming and array-compressed parallel transmission. Here we propose a new RAPS circuit design based on off-the-shelf components for improved performance and manufacturability.

Theory and Methods:

The original RAPS used a pair of home-built Wilkinson splitter and hybrid coupler connected by a pair of connectorized coaxial cables. Here we propose a new hybrid-pair RAPS (or HP-RAPS) circuit that replaces the home-built circuits with two commercially available hybrid couplers and replaces connectorized cables with interchangeable microstrip lines. We derive the relation between the desired splitting ratio and the required phase shifts for HP-RAPS and investigate how to generate arbitrary splitting ratios using paired meandering and straight lines. Several HP-RAPSs with different splitting ratios were fabricated and tested on the workbench and MRI experiments.

Results:

The splitting ratio of an HP-RAPS circuit has a tan or cot dependence on the meandering line’s additional length compared to the straight line. The fabricated HP-RAPSs exhibit accurate splitting ratios as expected (<4% deviations) and generate transmit fields that well agree with predicted fields. They also demonstrated a low insertion loss of 0.33 dB, high output isolation of −26 dB, and acceptable impedance matching of −16 dB.

Conclusion:

A novel HP-RAPS circuit was developed and implemented. It is easy-to-fabricate/reproduce with minimal expertise. It also preserves the features of the original RAPS circuit (ratio-adjustable, small footprint, etc.) with lower insertion loss.

1. Introduction

Transmit field $\left(B_1^{+}\right)$ inhomogeneity is a well-known challenge in ultrahigh field MRI owing to wave behavior and destructive interference in human tissue [1, 2, 3]. To address this, RF shimming was proposed and widely used to manipulate $B_1^{+}$ [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. The desired amplitude/phase applied to individual coils in RF shimming can be realized by independent RF power amplifiers, or using add-on RF shimming devices, such as power splitting circuits, phase shifters, and/or power combining circuits [7, 9, 15, 16, 17]. The latter design typically provides lower RF shimming ability compared to independent power amplifiers, but with the benefit of much simpler structure, lower cost and smaller footprint. In particular, the combination of add-on RF shimming devices and multiple independent RF power amplifiers can provide additional degrees of freedom to manipulate the transmit field beyond the scanner’s hardware limitation via array-compressed parallel transmission (acpTx) and related techniques [18, 19].

Compared to phase shifts that can be easily adjusted by cable lengths, manipulating transmitted amplitudes without wasting power is much more challenging. Although Wilkinson [21] and T-shaped splitters can be designed with arbitrary ratios, their splitting ratio is determined by the trace width and thereby is fixed in manufacturing. To address this problem, in Yan et. al [22] we proposed a ratio-adjustable power splitter (RAPS) circuit which improved on a previous combination of equal splitters with adjustable attenuators [23]. The results demonstrated that RAPS circuits are a high-power, non-magnetic, high output isolation, compact, and inexpensive design that meets the desirable features for the add-on shimming and acpTx.

However, there are some weaknesses of the previous RAPS circuit that we have discovered in practice. First, building a high-performance RAPS at high frequencies (298 MHz) requires the builder to be experienced in RF circuits owing to parasitic capacitance/inductance in the circuits and nonideality of real components. Second, previous work used connectorized coaxial cables to realize the phase shifts, which induces non-negligible loss and becomes laborious when many RAPS circuits are needed. Third, in practice there is unavoidable power dissipation in the RF resistor (100 Ω) in the Wilkinson splitter of the RAPS circuit due to coil mismatching and residual coil decoupling, raising potential safety concerns.

To solve these problems, we propose a new design based on off-the-shelf components and interchangeable microstrip phase shifters. Specifically, the Wilkinson splitter is replaced with a 3-dB hybrid coupler for two reasons. First, compared to power splitters, non-magnetic hybrid couplers are more widely available and affordable from manufacturers, and they can be obtained with the surface-mount or drop-in socket packages that occupy minimum space and suit the RAPS application. Second, by using a pair of hybrids, the terminated RF resistor (50 Ω) can be easily moved away from the circuit area by extended RF cables, without crossing the circuit area. The connectorized cables were further replaced with interchangeable microstrip lines as the latter are easy-to-produce in printed circuit boards (PCBs). The new RAPS design is referred to herein as a hybrid-pair RAPS circuit, or HP-RAPS for short.

Theory

Figure 1a shows a diagram of the HP-RAPS circuit. It comprises a pair of hybrid couplers connected by a pair of phase shifters. A simple explanation of the HP-RAPS is as follows. First, the left hybrid coupler divides the input RF signal (V_in) evenly into two signals ($V_1=-j/\sqrt{2}{V}_{\text{in}}$,$V_2=-1/\sqrt{2}{V}_{\text{in }}$). Then, the phase shifters add additional phase delays (φ1 and φ2) to V₁ and V₂. Finally, the right hybrid coupler in Figure 1a sums the two signals as $V_{\text{o}1}=-1/\sqrt{2}\left(j{V}_3+V_4\right)=V_{\text{in}}/\sqrt{2}\left(\text{exp}\left(-j{\phi }_2\right)-\text{exp}\left(-j{\phi }_1\right)\right)$ and $V_{\text{o}2}=-1/\sqrt{2}\left(V_3+j{V}_4\right)=j{V}_{\text{in }}/\sqrt{2}\left(\text{exp}\left(-j{\phi }_2\right)+\text{exp}\left(-j{\phi }_1\right)\right)$. Therefore, the splitting ratio between the two output ports only depends on the phase difference between the phase shifters (|V_o1|/|V_o2| = tan (|φ₁ − φ₂|/2)) or |V_o2|/|V_o1| = cot (|φ₁ − φ₂|/2). A comprehensive derivation of the amplitude relationships between the circuits’ input and output ports is provided in the Appendix.

Figure 1.

(a): Circuit diagram of the Hybrid-Pair Ratio Adjustable Power Splitter (HP-RAPS). (b-c): Top view and side view of the HP-RAPS model. The straight and meandering lines were 2.54 cm apart to match the size of hybrid coupler. The length of the straight and meandered lines were both 2.68 cm. The meander’s width (a and b) was swept to investigate its relation with the phase difference and thus the splitting ratio, while the trace width of meandering conductor (w) was varied for impedance optimization. The gap between adjacent meanders (g) in the meandering line was kept to a fixed value of 0.9 mm. The width of the straight line was kept to a fixed value of 1.18 mm based on the standard equations. (d): Photograph of a fabricated HP-RAPS.

2. Methods

2.1. Simulation

To realize a phase difference between the two branches, one microstrip line was kept straight and the other was meandered. For a straight microstrip line, the trace width for a desired impedance and phase shift can be calculated from literature or textbooks [24]. However, the optimal trace width and phase shift of the meandered microstrip line deviates from the standard equations due to the parasitic capacitance/inductance between the adjacent meanders [25]. To find the optimal trace width for 50-Ω characteristic impedance and to find a relationship between the meander’s size and its phase shift, we performed a set of full-wave electromagnetic (EM) simulations using HFSS (Ansys, Canonsburg, PA).

Figure 1b shows the dimensions of the simulated microstrip lines. The dielectric constant and thickness of the substrate were set to 10.2 and 1.27 mm, respectively, which correspond to the parameters of Rogers 3010 material. Based on an impedance calculation for standard microstrip line [24], the trace width of the straight microstrip line was set to 1.18 mm to achieve 50-Ω characteristic impedance. The straight microstrip line had a fixed phase delay of 25.3 degrees. The trace width (w in Figure 1b) of the meandering line was varied from 0.72 mm to 1.12 mm in steps of 0.01 mm to find the width that produces 50-Ω characteristic impedance. After the optimal trace width was determined, the phase shifts of meandering lines with different widths (a and b in Figure 1b) were calculated in full-wave EM simulations.

2.2. Hardware Fabrication

Figures 1b–d show the design model and a photograph of an HP-RAPS circuit with a desired split ratio of 1:1. The circuit comprises a motherboard with two off-the-shelf hybrid couplers (model IPP2026, Innovative Power Products, Holbrook, NY), a daughterboard (i.e., phase shifter board) with one straight and one meandering microstrip line, and a 3D-printed cover that fastens the two boards together. Model IPP2026 was chosen as the hybrid coupler herein as it exhibits a superior performance at the frequencies near 300 MHz, with insertion loss < 0.1 dB and amplitude/phase imbalance < 0.05 dB/0.5 degrees. The motherboard was made of 1.27-mm-thick Rogers 4003C board while the daughterboard was made of a 1.27-mm-thick Rogers 3010 board. Hybrid couplers were provided with the drop-in package with a size of 3.43 cm × 1.27 cm. The motherboard also serves as a ground plane for the whole circuit. Daughterboards can be interchanged to realize arbitrary power split ratios.

In addition to the 1:1 ratio, we also fabricated microstrip line boards for desired output ratios of 1:2.5, 1:5 and 1:7.5. Note that all ratios in this work are amplitude rather than power ratios. The PCBs were fabricated using a Protomat S103 milling machine (LPKF Laser & Electronics AG, Garbsen, Germany). The motherboard and daughterboards were pressed together tightly by the 3D-printed cover and four brass screws. In this way, the strip conductors of the daughterboard were connected to the hybrid couplers’ pins and the ground of the daughterboard was connected to the motherboard.

2.3. Bench test and $B_1^{+}$ mapping experiment

The four constructed daughterboards (ratios of 1:1, 1:2.5, 1:5 and 1:7.5) and entire HP-RAPS circuits were tested on the workbench using a four-port vector network analyzer (Keysight 5071C). Additionally, HP-RAPS circuits with different ratios were further validated using $B_1^{+}$ mapping experiments on a Nova birdcage coil (Nova Medical, Wilmington, MA, USA) and a home-built ICE-decoupled 8-coil transmit array [26]. Note that the 1:7.5 ratio was not validated in the $B_1^{+}$ mapping experiment as the transmit power of low-amplitude output port is too low and thereby its $B_1^{+}$ map is too noisy. Two kinds of $B_1^{+}$ mapping validations were performed: single-coil validation and two-coil validation. All MRI experiments were performed on a Philips Achieva 7T scanner (Philips Healthcare, Best, Netherlands).

For the single-coil experiment, each output of the HP-RAPS was alternately connected to the coil, with the unused output terminated with 50 Ω through a high-power 30-dB attenuator. Two axial $B_1^{+}$ maps (a $B_1^{+}$ map for each output) for each ratio were acquired in a 15 cm gel phantom. We also acquired an additional scan with no HP-RAPS circuit in-line as a baseline. The unused port of the birdcage coil and the unused 7 coils of the 8-coil Tx array were terminated with 50 ohms and not driven. This single-coil experiment validated the splitting ratios by comparing the $B_1^{+}$ magnitudes of two outputs.

To further validate the complex weights of the HP-RAPS circuits, two-coil validations were performed where the two outputs of HP-RAPS were connected to quadrature ports of the birdcage coil or to two neighbouring coils of the Tx array. One axial $B_1^{+}$ map with two ports or two coils simultaneously driven was acquired for each ratio. Individual $B_1^{+}$ maps (each port of birdcage coil or each used coil of the Tx array) were acquired to predict the combined $B_1^{+}$ maps which were calculated as: $B_{1,\text{pred}}^{+}=B_{1,\text{c}1}^{+}\times w_{\text{in},\text{o}1}+B_{1,\text{c}2}^{+}\times w_{\text{in},\text{o}2}$, where $B_{1,\text{c}1}^{+}$ and $B_{1,\text{c}2}^{+}$ are individual $B_1^{+}$ maps of port/coil 1 and 2, and w_in,o1 and w_in,o2 are the complex weights between the input and outputs of the HP-RAPS circuits. This two-coil experiment validates the expected complex weights (magnitude and phase) by comparing the measured two-coil $B_1^{+}$ maps with the predicted maps. $B_1^{+}$ maps were acquired using the DREAM method [27] with the same parameters and normalized to the same input power for all scans (FOV = 220 × 220 mm³, voxel size = 2 × 2 × 3 mm³). In all experiments, a Nova 32-channel receive-only coil was used for signal reception.

3. Results

3.1. Simulation Results

Figure 2a shows the impedance match (S₁₁) at one end of the meandering line, with the other end terminated by 50 Ω. The figure shows that the optimal trace width of the meandering line is approximately 0.9 mm, which yields an excellent match of <−45 dB (solid lines in Figure 2). The optimal trace width remains the same as the desired splitting ratio changes, i.e., the meander’s width (a and b in Figure 1b) changes. This indicates there is no need to re-optimize the trace width when designing daughterboards for different splitting ratios.

Figure 2.

Impedance matching (a) and insertion loss (b) of the microstrip lines with different trace widths. The solid line represents the meandering line, and the dotted line represents the straight line.

Figure 2b plots insertion losses of the straight (dotted line) and the meandering lines (solid line). As the ratio comes closer to 1:1, the total trace length of the meandering line increases and thus the insertion loss increases. However, the meandering line still has a low insertion loss of 0.16 dB. As shown in the Supporting Information Figure S1 and Eq. S4, the splitting ratio (R) and the meandering line’s additional total trace length compared to the straight line (Δl) have a simple linear relationship, and there is no need to perform an EM simulation and optimization for every splitting ratio, which simplifies the design and fabrication of daughterboards.

3.2. Bench Test Results

Four daughterboards with different meandering line widths (a and b in Figure 1b) were fabricated to achieve target ratios of 1:1, 1:2.5, 1:5, and 1:7.5. The trace widths of the straight and meandering lines were chosen to be 1.18 mm and 0.9 mm based on the optimization results in Figure 2. The meandering line widths (a and b) were chosen based on Supporting Information Eq. S4. Table 1 lists measured splitting ratios, the impedance matches of the microstrip lines and the insertion losses of the entire fabricated circuits, with comparison to simulated results. For both the simulation and bench tests, the overall power loss of the HP-RAPS circuit was calculated for each ratio using S-parameters as: $\text{1- }{S}_{13}^2-S_{14}^2$, where port 1 was the input, and ports 3 and 4 were the outputs.

Table 1.

Comparison between the simulations from ideal parameters and fabricated boards. $S_{11}^{\text{strt }}$ and $S_{11}^{\text{med }}$ are reflection coefficients at one end of the straight and meandering lines, respectively, with the other end terminated with 50 Ω. IL_HP-RAPS is the insertion loss of the entire HP-RAPS circuit.

Ratio (simulated or target)	1	2.5	5	7.5
Ratio (measured)	1.01	2.25	4.74	7.33
$S_{11}^{\text{strt }}$ (simulated, dB)	−42.40	−42.42	−42.40	−45.45
$S_{11}^{\text{med }}$ (simulated, dB)	−46.06	−45.00	−48.37	−49.32
$S_{11}^{\text{strt }}$ (measured, dB)	−27.18	−27.54	−27.20	−27.20
$S_{11}^{\text{med }}$ (measured, dB)	−18.03	−16.96	−18.38	−20.07
ILHP-RAPS (simulated, dB)	−0.0971	−0.0798	−0.0591	−0.0243
ILHP-RAPS (measured, dB)	−0.3534	−0.3592	−0.3298	−0.3229

For the straight lines, the measured impedance match was approximately constant at −27 dB (i.e., 0.2% return loss). For the meandering lines, the measured impedance match varied slightly with different ratios, over a range of −17 to −20 dB (i.e., 1–2% return loss). As shown in Table 1, the measured ratios were close to the expected/simulated ratios, with an average amplitude splitting ratio error of 4.0% and a worst-case amplitude splitting ratio error of 5.2%. The good agreement between the measured ratios and expected/simulated ratios validates the accuracy of Supporting Information Eq. S4 in predicting the splitting ratio with a simple expression.

Figure 3a plots measured transmission coefficients between the input and the two outputs as a function of frequency (S₁₃ and S₁₄). Note that the ratio between S₁₄ and S₁₃ corresponds to the splitting ratio. Figures 3b and 3c show the impedance matching of all ports (S₁₁, S₃₃ and S₄₄) and isolation between the output ports (S₃₄), respectively. The plots show that the two output ports (port 3 and port 4) were well-isolated, with S₃₄ <−25 dB for different ratios. The impedance matching varied from −15.3 to −17.5 dB for the different ratios (i.e., 2–3% return loss).

Figure 3.

Bench tests of the boards with nominal amplitude splitting ratios of 1:1 (a, e, and i), 1:2.5 (b, f, and j), 1:5 (c, g, and k), and 1:7.5 (d, h, and l). The corresponding power splitting ratios expressed in dB are: 0 dB, 7.96 dB, 13.98 dB and 17.50 dB. The power splitting (dB) is shown in a)-d), port matching conditions in e)-h), and isolation between the output ports in i)-l).

3.3. $B_1^{+}$ mapping Tests

Figures 4a and b show $B_1^{+}$ maps measured across output ratios in single-coil experiments. The calculated ratios from measured $B_1^{+}$ maps were 1:1.05, 1:2.22, and 1:4.91 using one port of the Nova birdcage coil, and were 1:1.01, 1:2.17, and 1:4.81 using one coil of the 8-channel loop array. These $B_1^{+}$ ratios were calculated by dividing average $B_1^{+}$ over the high-signal area. In both cases, the measured $B_1^{+}$ ratios and bench-test ratios matched well, with errors less than 5%. Figures 4c and d show the predicted and measured $B_1^{+}$ maps and their difference of two-coil experiments. As mentioned above, predicted $B_1^{+}$ maps were calculated based on the individual $B_1^{+}$ map of each port/coil and the complex weights between input and two outputs of HP-RAPS circuits, and the measured $B_1^{+}$ maps were measured with the HP-RAPS circuits inserted between the RF amplifier and the two ports/coils. Most of the $B_1^{+}$ errors were in areas of low MR signal in the DREAM images, where image division in the $B_1^{+}$ calculation amplified noise in the maps. For both the birdcage coil and the ICE-decoupled Tx array, the predicted and measured $B_1^{+}$ maps closely match each other across different splitting ratios. These $B_1^{+}$ mapping results further show that the HP-RAPS circuits function well, and are also consistent with the theory, simulation, and bench test results. Note that the ICE-decoupled 8-channel Tx array has an average isolation of −14.8 dB between neighbouring coils. Therefore, the $B_1^{+}$ mapping results in Figures 4b and d indicate HP-RAPS circuit works well even for Tx arrays with medium decoupling performance.

Figure 4.

$B_1^{+}$ mapping results with different HP-RAPS circuits (nominal splitting ratios of 1, 2.5, and 5). (a): $B_1^{+}$ mapping results of the single-coil experiment using one port of the Nova birdcage coil. (b): $B_1^{+}$ mapping results of the single-coil experiment using one coil of the 8-channel ICE-decoupled loop array [26]. (c): $B_1^{+}$ mapping results of the two-coil experiment using two quadrature ports of the Nova birdcage coil. (d): $B_1^{+}$ mapping results of the two-coil experiment using two coils of the 8-channel ICE-decoupled loop array. In single-coil experiments, one coil was alternately connected to one output of the HP-RAPS circuit, with the other output terminated with 50 Ω. In the two-coil experiments, two coils were connected to the two outputs of HP-RAPS circuits.

4. Discussion

4.1. Reproducibility and Reliability

Reproducibility of HP-RAPS circuits is highly desired in massive splitter networks as it can significantly ease circuit fabrication and improve the stability and consistency of different devices. The consistency of different HP-RAPS circuits is mainly determined by the hybrid couplers. We tested up to 25 hybrid couplers (with the same model) on the bench and found they exhibit high consistency, with negligible amplitude/phase variations. In this work, these PCB boards were fabricated with an in-house PCB milling/drilling machine and the copper was not covered by any coating. This might lead to oxidization of copper and thus reliability issues over the long term. The oxidization can be avoided by using proper surface finish technologies which are commonly available from commercial PCB board manufacturers.

4.2. Arbitrary Splitting Ratio

Although Wilkinson and T-shaped power splitters can theoretically achieve arbitrary splitting ratios, they are primarily limited to lower splitting ratios (lower than 1:4 power splitting ratio) due to the realizable high-impedance transmission lines in practice [28, 29]. By using HP-RAPS design, however, we can construct a circuit with an extremely large amplitude splitting ratio difference of 1:7.5, with a negligible error of 2.3%. Note that the 1:7.5 amplitude splitting ratio means a power splitting ratio up to 1:56.25 which is almost impossible to realize with typical Wilkinson or T-shaped splitters.

4.3. Wide Bandwidth for Other Static Fields

The off-the-shelf hybrid couplers used in this work have a wide bandwidth, from 250 MHz to 500 MHz. Therefore, this design can be directly transferred to other ultrahigh fields such as 9.4 T and 10.5 T where the $B_1^{+}$ inhomogeneity is even more severe. This also can be seen from Figure 3 that the impedance matching and port isolation was maintained over a large bandwidth. The only thing that needs to be taken care of is that, for a given size of the microstrip lines, the phase shift changes at different RF frequencies.

4.4. Data Availability

Complete specifications required to build HP-RAPS circuits with different ratios are freely available at https://github.com/XinqiangYan/HP-RAPS, including Solidworks drawings, Gerber files for PCB fabrication, and a bill of materials with part numbers and vendor information. More discussion about the impedance matching, insertion loss, cost and power capability can be found in Supporting Information.

5. Conclusion

A Hybrid Pair-Ratio Adjustable Power Splitter (HP-RAPS) circuit was theoretically analyzed, designed, simulated, fabricated, and validated in bench tests and MRI experiments. In addition to benefits inherited from the original RAPS circuit (high-power handling, compact size, and ratio-adjustable), the HP-RAPS circuit uses off-the-shelf components and an interchangeable microstrip line phase shifter that possesses additional characteristics that are useful in practice, including ease of fabrication and reproduction, a lower insertion loss, and the ability to move the RF resistor away from the circuit.

6. Appendix

HP-RAPS Circuit Scattering Parameters and Input-output Relations

The scattering (S)-matrices of a 3-dB hybrid coupler (S_hc) and a pair of phase shifters (S_ps) can be represented as:

$$\left[\begin{array}{c}{V}_{\text{in}}^{-}\\ V_{\text{iso}}^{-}\\ V_1^{-}\\ V_2^{-}\end{array}\right]={\mathit{\text{S}}}_{\text{hc}}\left[\begin{array}{c}{V}_{\text{in}}^{+}\\ V_{\text{iso}}^{+}\\ V_1^{+}\\ V_2^{+}\end{array}\right],\left[\begin{array}{c}{V}_1^{-}\\ V_2^{-}\\ V_3^{-}\\ V_4^{-}\end{array}\right]={\mathit{\text{S}}}_{\text{ps}}\left[\begin{array}{c}{V}_1^{+}\\ V_2^{+}\\ V_3^{+}\\ V_4^{+}\end{array}\right]\text{ and }\left[\begin{array}{c}{V}_3^{-}\\ V_4^{-}\\ V_{\text{o}1}^{-}\\ V_{\text{o}2}^{-}\end{array}\right]={\mathit{\text{S}}}_{\text{hc}}\left[\begin{array}{c}{V}_3^{+}\\ V_4^{+}\\ V_{\text{o}1}^{+}\\ V_{\text{o}2}^{+}\end{array}\right]$$

where

$${\mathit{\text{S}}}_{\text{hc}}=-\frac{1}{\sqrt{2}}\cdot \left[\begin{array}{cccc}0& 0& j& 1\\ 0& 0& 1& j\\ j& 1& 0& 0\\ 1& j& 0& 0\end{array}\right]\text{ and }{\mathit{\text{S}}}_{\text{ps}}=\left[\begin{array}{cccc}0& 0& e^{-j{\phi }_1}& 0\\ 0& 0& 0& e^{-j{\phi }_2}\\ e^{-j{\phi }_1}& 0& 0& 0\\ 0& e^{-j{\phi }_2}& 0& 0\end{array}\right]$$

Then transmission matrices can be obtained from the scattering matrices by the following rules [24]:

$$\left[\begin{array}{cc}\left[T_{\text{I},\text{I}}\right]& \left[T_{\text{I},\text{II}}\right]\\ \left[T_{\text{II},\text{I}}\right]& \left[T_{\text{II},\text{II}}\right]\end{array}\right]=\left[\begin{array}{cc}\left[S_{\text{I},\text{II}}\right]-\left[S_{\text{I},\text{I}}\right]{\left[S_{\text{II},\text{I}}\right]}^{-1}\left[S_{\text{II},\text{II}}\right]& \left[S_{\text{I},\text{I}}\right]{\left[S_{\text{II},\text{I}}\right]}^{-1}\\ -{\left[S_{\text{II},\text{I}}\right]}^{-1}\left[S_{\text{II},\text{II}}\right]& {\left[S_{\text{II},\text{I}}\right]}^{-1}\end{array}\right]$$ (1)

where for a four port network with scattering matrix [S_mn](m,n = 1, 2, 3, 4), [S_M,N] = [S_mn] (M,N = I, II), m = 2M − 1 or 2M, n = 2N − 1 or 2N. The transmission matrices of hybrid coupler (T_hc) and pair of phase shifter (T_ps) are calculated to be:

$${\mathit{\text{T}}}_{\text{hc}}=\frac{1}{\sqrt{2}}\cdot \left[\begin{array}{cccc}j& 1& 0& 0\\ 1& j& 0& 0\\ 0& 0& -j& 1\\ 0& 0& j& -1\end{array}\right]\text{ and }{\mathit{\text{T}}}_{\text{ps}}=\left[\begin{array}{cccc}{e}^{-j{\phi }_1}& 0& 0& 0\\ 0& e^{-j{\phi }_2}& 0& 0\\ 0& 0& e^{j{\phi }_1}& 0\\ 0& 0& 0& e^{j{\phi }_2}\end{array}\right]$$

The transmission matrix of the HP-RAPS (T_RAPS) is calculated by multiplying three transmission matrices in series: T_RAPS = T_hcT_psT_hc:

$${\mathit{\text{T}}}_{\text{RAPS }}=\frac{1}{2}\cdot \left[\begin{array}{cccc}-e^{-j{\phi }_1}+e^{-j{\phi }_2}& j{e}^{-j{\phi }_1}+j{e}^{-j{\phi }_2}& 0& 0\\ j{e}^{-j{\phi }_1}+j{e}^{-j{\phi }_2}& e^{-j{\phi }_1}-e^{-j{\phi }_2}& 0& 0\\ 0& 0& -e^{j{\phi }_1}+e^{j{\phi }_2}& -j{e}^{j{\phi }_1}-j{e}^{j{\phi }_2}\\ 0& 0& -j{e}^{j{\phi }_1}-j{e}^{j{\phi }_2}& e^{j{\phi }_1}-e^{j{\phi }_2}\end{array}\right]$$ (2)

The scattering matrix can be calculated from the transmission matrix as:

$$\left[\begin{array}{cc}\left[S_{\text{I},\text{I}}\right]& \left[S_{\text{I},\text{II}}\right]\\ \left[S_{\text{II},\text{I}}\right]& \left[S_{\text{II},\text{II}}\right]\end{array}\right]=\left[\begin{array}{cc}\left[T_{\text{I},\text{II}}\right]{\left[T_{\text{II},\text{II}}\right]}^{-1}& \left[T_{\text{I},\text{I}}\right]-\left[T_{\text{I},\text{II}}\right]{\left[T_{\text{II},\text{II}}\right]}^{-1}\left[T_{\text{II},\text{I}}\right]\\ {\left[T_{\text{II},\text{II}}\right]}^{-1}& -{\left[T_{\text{II},\text{II}}\right]}^{-1}\left[T_{\text{II},\text{I}}\right]\end{array}\right]$$ (3)

Thereby, the calculated HP-RAPS circuit’s scattering matrix is:

$${\mathit{\text{S}}}_{\text{RAPS }}=-j{e}^{j\left({\phi }_1+{\phi }_2\right)/2}\left[\begin{array}{cccc}0& 0& \text{sin}\left(\frac{{\phi }_1-{\phi }_2}{2}\right)& \text{cos}\left(\frac{{\phi }_1-{\phi }_2}{2}\right)\\ 0& 0& \text{cos}\left(\frac{{\phi }_1-{\phi }_2}{2}\right)& -\text{sin}\left(\frac{{\phi }_1-{\phi }_2}{2}\right)\\ \text{sin}\left(\frac{{\phi }_1-{\phi }_2}{2}\right)& \text{cos}\left(\frac{{\phi }_1-{\phi }_2}{2}\right)& 0& 0\\ \text{cos}\left(\frac{{\phi }_1-{\phi }_2}{2}\right)& -\text{sin}\left(\frac{{\phi }_1-{\phi }_2}{2}\right)& 0& 0\end{array}\right]$$ (4)

The circuit’s output ratio is

$$S_{\text{in},\text{o}1}/S_{\text{in},\text{o}2}=S_{1,3}/S_{1,4}=\text{tan }\left(|{\phi }_1-{\phi }_2|/2\right)$$ (5)

or

$$S_{\text{in},\text{o}2}/S_{\text{in},\text{o}1}=S_{1,4}/S_{1,3}=\text{cot }\left(|{\phi }_1-{\phi }_2|/2\right)$$ (6)