Wideband Printed Ridge Gap Rat-Race Coupler for Differential Feeding Antenna

In this paper, a wideband 3 dB hybrid 180° rat-race coupler is introduced in the printed ridge gap waveguide technology. It has simultaneous wide matching and isolation bandwidth with low output amplitude imbalance. It operates in the millimeter wave band from 25.8 to 34.2 GHz (27.96%) with 15 dB return loss and isolation, and ±0.5 dB output amplitude imbalance. The proposed design employing an open stub at the middle of the $3 \lambda /4$ branch line and quarter wavelength lines at all the ports of the coupler. The objective of the added open stub is to separate the output ports amplitudes around the -3 dB level by certain values depending on the required amplitude imbalance. The analytical derivation for the role of the added open stub is presented along with a parametric study on its effect on amplitude imbalance, matching, and isolation. This results in having two intersection points for the output ports instead of one of the conventional coupler and hence the amplitude imbalance bandwidth increases. The objective of the added quarter wavelength lines is to improve the matching and isolation bandwidths. First, the conventional rat-race coupler is presented and a bandwidth of 14.25% at 30 GHz is achieved. After that the rat-race with the added quarter wavelength lines is presented to illustrate the objective of the added quarter wavelength lines and a bandwidth of 19.44% is achieved. Finally, the rat-race with the quarter wavelength lines and the added stub is presented and a prototype is fabricated and measured. The s-parameters measurements are in a good agreement with the simulated ones.


I. INTRODUCTION
The Millimeter wave (mmw) band has gained a great attention in the research society. It opens up a broadband spectrum for a wide range of applications such as the next generation (5G) communication, automotive radar, remote sensing and security imaging [1]- [4]. Therefore, there is a need for new mmw devices to meet such huge spectrum. It is well-known that the conventional microstrip components is lossy at this frequency band. Thus, other waveguide structures become mandatory. One of the promising prototyping technologies is the ridge gap waveguide (RGW), where the propagating mode is the QTEM mode that has a small signal distortion. Another advantage of the RGW structure is that the wave propagates in air. Therefore, dielectric losses are eliminated. Moreover, the structure is closed which eliminates the radiation losses. There are two types of the ridge gap waveg-The associate editor coordinating the review of this manuscript and approving it for publication was Antonino Orsino . uide based on the fabrication technology. The first is the metal RGW introduced by Zaman et al. [5], Kildal [6], and Kildal et al. [7] where the computer numerical control (CNC) machining is used in the fabrication. The second is the printed RGW (PRGW) [8] where the printed circuit board (PCB) technology is used. RGW and PRGW technologies have been used in the design and implementation of several components such as couplers and antennas at different frequencies including 15 GHz [9], [10], 30 GHz [11]- [15], and 60 GHz [16]. In this paper, PRGW is used to implement a wideband, low cost, and light-weight rat-race coupler compared to the metallic one. More importantly, it is easily integrated with other devices and chips on PCB boards.
Rat-race coupler has important role in many microwave circuits. It provides equal power division with in-phase and out of phase characteristics, depending on the feeding port. It is used in mixers and differential feeding network. It has been implemented in different technologies such as microstrip line, substrate integrated waveguide (SIW), and ridge gap waveguide (RGW). Several techniques to improve the bandwidth and reduce the size are used. In [17], a full optimized rat-race coupler with quarter wavelength transformers at all ports is presented where a maximum bandwidth of 50.6% is achieved. However the impedance ratio is high (1:8.75) which leads to fabrication challenges. A technique to decrease the size and increase the bandwidth is implemented by using a 180 • phase inverter with only one quarter wavelength section instead of the 3λ/4 branch, using short lines reduces the frequency dependency of the rat-race coupler. This technique was first introduced theoretically by S. March in 1968 [18], where a shorted coupled line coupler section with λ/4 length is used instead of the 3λ/4 section. In [19], two quarter wavelength coupled line coupler are added to the structure in [18] to have different impedances at two ports of the rat-race coupler with wide bandwidth. The same technique is implemented in the coplanar waveguide [20], [21], finite-groundplane coplanar waveguide (FGCPW) [22], and inter digit coplanar waveguide [23]. This technique needs uniplanar waveguide structure that facilitates making the phase inverter either by using bonding wires or conductive tape without increasing the size of the structure. In [24], this technique is combined with adding quarter wavelength transformers at all ports to further increase the bandwidth. Also, replacing the 3λ/4 by a λ/4 left handed transmission line gives the same effect [25]- [27]. Moreover, lumped elements are used for size reduction by replacing the transmission lines [28]- [30] or loading the transmission lines and reduce its electrical length [31]. Also, slow wave waveguides have been used to reduce the size as in [32], [33]. Finally, cascading of rat-race couplers is used to have a wideband as in [34]. These techniques are valid for low operating frequencies as there is enough area to build left handed transmission line, slow wave structures, or to use lumped elements. However, at high frequencies, the devices become very small and there is no enough area to build artificial transmission lines or to use lumped elements with the current PCB technology.
For high frequency devices, new guiding structure technologies such as substrate integrated waveguide (SIW) and ridge gap waveguide are considered good candidates. The SIW has been used to build rat-race coupler [35]. However, as the propagating mode is TE 10 and the width of the waveguide is large, the λ/4 section can not be realized and 5λ/4 is used instead. This increases the frequency dependency of the structure and the bandwidth is only 10%. Then it has been optimized in [36] to reach 30%. Several techniques has been used to reduce the over all size of the SIW ratrace such as using half mode SIW [37], folded SIW [38] or ridge SIW with slow wave [39]. All of these works suffer from the dielectric losses in SIW technology. Considering the ridge gap waveguide, to the authors' best knowledge, only one article has been found [40] in metal RGW technology and using conventional rat-race with only 12.1% bandwidth around 16.5 GHz.
In this paper, a wideband rat-race coupler is designed at the millimeter wave band using the PRGW technology. The  addition of open stub and quarter wavelength transformers has greatly improved the impedance, isolation, and amplitude imbalance bandwidths. First, the general equations of the proposed coupler are presented along with a parametric study to show the effect of the added stub on the amplitude imbalance, and matching and isolation bandwidths. Second, the conventional rat-race coupler is designed in the PRGW to show its performance in this new technology. Third, the rat-race with the added quarter wavelengths is presented as a first step for a wide bandwidth (increasing the matching and isolation bandwidths). Also, multiple quarter wavelength transforms can be used to have different input and output impedances as in [19]. Finally, rat-race with the quarter wavelengths and the open stub at the middle of the 3λ/4 branch line is presented. The operating bandwidth is determined by having the return loss and the isolation better than 15 dB, and a coupling imbalance less than 1 dB (peak-to-peak).

II. THEORY
In this section, the general geometry of the rat race coupler with quarter wave transformers and a stub at the middle of the 3λ/4 branch line is analyzed using odd and even mode analysis. The geometry of the proposed rat race is shown in Fig. 1 and the even and odd equivalent circuits are shown in Fig. 2. The s-parameters of the rat race is calculated from the VOLUME 8, 2020 even and odd analysis as follows [42].
where S 11e ,S 21e , S 11o , and S 21e are calculated by applying the even and odd symmetry respectively on the rat race. The even symmetry leaves open stubs while the odd symmetry leaves short stubs. The ABCD matrices of different sections of the rat race for even symmetry are the total ABCD matrix for the even mode is given by (9) For the odd mode analysis, the second and forth ABCD matrix are The total ABCD matrices for the odd case is ABCD odd total = ABCD 1e ABCD 2o ABCD 3e ABCD 4o ABCD 1e (12) From these general equations for the proposed rat race, the conventional rat race is obtained by making Z q = Z 0 , 2Z 0 , and L stub 0. The rat race with quarter wave transformers is obtained by making Z q = Z 0 / 4 √ 2, Z 1 = Z 0 and L stub 0.
In order to illustrate the effect of the added stub on the amplitude imbalance, we take Z stub = 1 √ 2 Z 0 ,and Z q = Z 0 . Therefore the ABCD 2e can be written as For small value of L stub and at the center frequency, where θ = βL stub , and using the trigonometric Identity that After that we substitute in the ABCD matrix and get the s-parameters as follows By taking the absolute values, Equations (21) to (24) are obtained.
These equations illustrate that the amplitude of S 21 is greater than 1 √ 2 and increase with proportion to the added stub length. Also, the amplitude of S 41 becomes less than 1 √ 2 and decrease with proportion to the stub length. From the known behavior of the S 21 and S 41 (makes shape and shape around the center frequency, respectively), they will intersect at two different points around the center frequency and hence increase the amplitude imbalance bandwidth. Moreover, it is observed that the change of the matching and isolation is very small and proportional to θ 2 .
In order to prove the concept that adding a stub and quarter wave transformers improves the bandwidth for the general case, a parametric study on the effect of the stub is carried out using the ABCD matrix circuit model with Figure 3 shows the effect of the stub length on the amplitude imbalance of the output ports, while its effect on the matching and isolation levels is depicted in Fig. 4. Unlike the conventional rat race, which has one intersection point between the output ports, the addition of the stub provides a wide amplitude imbalance bandwidth. This is achieved by having the outputs intersect at two points instead of one. On the other hand, as the stub length increases, the separation between the intersection points increases resulting in deterioration of the output amplitude imbalance, the matching,  and the isolation. Therefore, optimization is needed to have a wide bandwidth with an acceptable amplitude imbalance, matching, and isolation. In this work, return loss and isolation are better than 15 dB with output ports imbalance less than 1 dB.

III. COAXIAL TO RIDGE GAP FEEDING DESIGN
The proposed device is fed by a coaxial connector. Therefore, a transition from coaxial to PRGW is designed.The unit cell of PRGW of [41] is used, where the material is Roger RT6002 (ε r = 2.94 and tan δ = 0.0012) and the resulted band gap is from 22.307 to 43.095 GHz. In this work, a simplified version of the previous work [41] is introduced where the matching pins have been removed to simplify the fabrication procedure, as a moderate bandwidth (from 25 to 35 GHz) is needed. The geometry of the back-to-back configuration is shown on Fig. 5 and the associated s-parameters are shown in Fig. 6. Dimensions of the transition are in Table 1. The return loss is better than 15 dB from 25.4 GHz till 36 GHz and the associated insertion loss is higher than 0.4 dB.

IV. CONVENTIONAL RAT-RACE COUPLER
In this section, the conventional rat-race coupler is presented. The geometry of the structure is shown in Fig. 7, where the ring line impedance is √ 2 Z 0 . The lengths of the short branches are λ g /4, and the long branch has 3λ g /4 length. The line width is 1.38 mm and the inner and outer radius of the ring coupler are R 1 = 2.475 mm, R 2 = 3.325 mm respectively. The s-parameters are shown in Fig. 8 where the bandwidth is from 28.09 to 32.4 GHz based on the previously mentioned criteria. It is clear that the bandwidth is narrow (14.25%) and also the phase difference between the differential out put ports changes rapidly, VOLUME 8, 2020   when moving away from the center frequency as shown in Fig. 13.

V. RAT-RACE COUPLER WITH QUARTER WAVE TRANSFORMER
In order to improve the bandwidth of the rat race, quarter wave transformers are used while maintaining a 50 input line impedance. The structure is shown in Fig. 9 where W q = 1.9 mm, L q = 2.6 mm, R 1 = 1.8 mm, and R 2 = 3.4 mm. The s-parameters are shown in Fig. 10 where   the bandwidth is from 27.17 to 33.02 GHz (19.44%). It is clear that there are enhancements in both the amplitude imbalance bandwidth and the phase response (Fig. 13).

VI. RAT-RACE COUPLER WITH QUARTER WAVE TRANSFORMER AND STUB
For further improvement in the output amplitude imbalance, a stub is used in the 3/4λ branch line as perturbation to have the S 21 and S 31 equal to each other at two intersection points and hence a wider amplitude imbalance bandwidth.  The geometry of the coupler with the stub is shown in Fig. 11 where W q = 1.7 mm, L q = 2.5 mm, R 1 = 1.85 mm, R 2 = 3.35 mm, W stub = 0.8 mm, and L stub = 2.4 mm. The resulted s-parameters are shown in Fig. 12. The bandwidth is from 25.84 to 34.24 GHz (27.96%) which is larger than the two previous configurations. Figure 13 shows a comparison between the differential output phase response in the previous three cases. The use of quarter wavelength transformers has increased the bandwidth as well as the stability of the output phase difference.

VII. MEASUREMENT AND DISCUSSION
The fabricated prototype of the rat-race with quarter wavelength transformers and a stub is shown in Fig. 14, where the ridge line layer, the upper copper layer, and the spacer are manufactured using the conventional printed circuit board technology (PCB). The metal base (Fig. 14 (d)) has been   drilled also with the PCB machining and used to hold the 2.4 coaxial connectors. The whole structure is assembled with plastic screws as shown in Fig. 14 (e). The (N52271A) PNA network analyzer has been used in measuring the s-parameters. The measured results along with the simulated ones are shown in Fig. 15 and Fig. 16 for the s-parameters amplitude and output phase difference, receptively. The behavior of the measured results matches very well with simulation results as there are two intersection points in the output ports which improve the output amplitude imbalance bandwidth, as can be seen from Fig. 17. The deviation of the measured results with the simulated ones may comes from the fabrication tolerance and the alignment mismatch. On Table 2, a brief comparison between the proposed coupler and other couplers is presented where the focused is on the SIW and RGW structures, as they are the promising technologies for the millimeter wave band. The proposed coupler has simultaneous both wide bandwidth and small amplitude imbalance compared to the referenced work. Moreover, it has low insertion loss as the wave propagates in an air gap region (no dielectric losses) and the structure is closed (no radiation losses). Most of the designed conventional rat-race coupler has a narrow bandwidth less than 13% [38]- [40] while the proposed one has 27.96%. In addition, SIW-based structures have higher losses compared to the proposed PRGW structure. This is due to the dielectric losses as the wave propagates in a dielectric medium while in the proposed structure, the wave propagates mainly in an air gap region. The SIW-based work in [36] and [37] are implemented in the microwave band with either a low return loss (12 dB) as in [37] or a large amplitude imbalance (±1.35 dB) as in [36].

VIII. CONCLUSION
A wideband rat-race coupler built in the PRGW technology with low amplitude imbalance and 15 dB matching and isolation levels is presented. A circuit model of the proposed rat race is presented to establish the bases for the proposed design. Then, the conventional rat-race is presented. After that, quarter wavelength transformers are used to improve the impedance matching bandwidth and the output phase difference stability. Finally, a stub is added to the 3λ 0 /4 branch to improve the amplitude imbalance between the output ports and to further increase the bandwidth. Moreover, a study of the effect of the added stub on both amplitude imbalance, and matching and isolation bandwidths is presented. A prototype of the last configuration is fabricated and measured. The proposed configuration has 27.96% bandwidth around 30 GHz with return loss and isolation better than 15 dB with less than 1 dB output coupling imbalance.