This is the accepted version of the article: Selga Ruiz, Jordi; Coromina, Jan; Vélez Rasero, Paris; [et al.]. «Miniaturised and harmonic-suppressed rat-race couplers based on slow-wave transmission lines». IET Microwaves, Antennas & Propagation, Vol. 13, issue 9 (July 2019), p. 1293-1299. DOI 10.1049/iet-map.2018.5379 This version is avaible at https://ddd.uab.cat/record/221394 under the terms of the license Miniaturized and Harmonic-Suppressed Rat-Race Couplers Based on Slow-Wave Transmission Lines Jordi Selga1, Jan Coromina1*, Paris Vélez1, Armando Fernández-Prieto2, Jordi Bonache1, Ferran Martín1 1 CIMITEC, Departament d’Enginyeria Electrònica, Universitat de Barcelona, 08193 Bellaterra, Spain 2 Departamento de Electrónica y Electromagnetismo, Universidad de Sevilla, 41012 Sevilla, Spain * jan.coromina@uab.cat Abstract: In this paper, a compact rat-race hybrid coupler with harmonic suppression based on slow-wave transmission lines (SW-TL) is presented. Such artificial lines are implemented by periodic loading a host microstrip line with series meandered inductors and shunt patch capacitors. The presence of both loading elements has a twofold effect, i.e., phase velocity reduction (due to the enhancement of the effective inductance and capacitance of the periodic line), and the generation of a controllable stop band in the frequency response (due to the Bragg effect, inherent to periodicity). It is shown that by designing the unit cell of the periodic line with an electrical length of 45°, at least the first five harmonic bands of the rat-race coupler are efficiently suppressed, keeping the band of interest unaltered. Moreover, 82% size reduction, as compared to the ordinary coupler, is achieved in the reported SW-TL-based prototype. or both capacitive and inductive loading, whereas the papers 1. Introduction [25],[27] present branch-line couplers with size reduction Size reduction and spurious/harmonic suppression based on T-shaped structures. However, fewer efforts have have been (and are still) two challenges in the design of been dedicated to achieve that objective in rat-race couplers distributed microwave components. Miniaturization is of [9],[29],[40]-[48]. In [24],[26],[29],[45], miniaturization in special interest in device topologies involving multiple the considered couplers is achieved by means of open-stub transmission line sections, such as high-order filters or loaded lines. In [43],[47], the authors use optimization couplers, among others. Spurious and harmonic bands are techniques (based on space mapping) for the implementation inherent to distributed components, and may be the cause of of harmonic suppressed rat-race couplers based on semi- interferences or undesired signals that may degrade system lumped elements. In the works [44],[48], the authors use functionality. Therefore, the suppression of such bands is a fractal and T-shaped structures, respectively. Finally, in [40]- due in certain applications, and achieving that purpose [42],[46], the capacitive or capacitive/inductive loaded lines without the penalty of increasing circuit size is of the highest are used to reduce the size of the proposed couplers. In this interest. paper, based on the periodic LC-loaded SW-TLs first Slow-wave transmission lines (SW-TL) based on proposed in [49], we report a miniature rat-race hybrid periodic structures are good candidates to simultaneously coupler able to suppress at least up to the fifth harmonic band. reduce circuit size and reject spurious/harmonic bands [1]- As compared to previous works devoted to size reduction and [3]. In such artificial lines, the phase velocity is smaller than harmonic suppression of rat-race couplers, in this paper we the one of their ordinary counterparts implemented on the report a design methodology based on the Floquet analysis of same substrate. Therefore, signal wavelengths are reduced, periodic structures, and we justify the convenience to use unit with direct impact on circuit compactness. Moreover, due to cells with a specific electrical length (to be discussed later) in the Bragg effect, related to periodicity, periodic SW-TLs order to efficiently suppress the harmonic bands leaving the exhibit stop bands at controllable frequencies, useful for band of interest unaltered. spurious/harmonic suppression. Most periodic SW-TLs are either based on capacitive 2. SW-TLs with LC loading and design equations [4]-[15] or inductive [16]-[19] loading, and few of them are The circuit schematic and topology of the considered implemented by combining both reactive elements [20]-[23] LC-loaded SW-TLs are depicted in Fig. 1, where k, l and Z0 (other SW-TLs loaded with distributed elements or are the phase constant, length and characteristic impedance, combinations of distributed/semi-lumped components have respectively, of the host line, and the loading reactive been also reported [24]-[30]). The advantage of simultaneous elements are designated by Lls and Cls. The dispersion relation inductive and capacitive (LC) loading is major design (neglecting losses) in the regions of propagation and the flexibility and the possibility to achieve further levels of characteristic (Bloch) impedance of such lines are given by miniaturization, as compared to SW-TLs based on a single the following equations [2],[50] reactive element. In [23], a compact and harmonic suppressed branch-line coupler based on LC-loaded SW-TLs was cos  l   A (1) reported. There are many papers in the recent literature devoted to the miniaturization and harmonic suppression of  jB B' ZB   (2) branch line couplers [9],[12],[24]-[39]. Among them, the sin  l  sin  l  devices presented in [31],[33],[35],[38] use either capacitive 1 with series connected to the loading inductances Lls/2. For such  L C Z  circuit model, the cutoff frequency is found to be [2],[15] A  cos( kl )   ls  ls 0   sin( kl )   2Z 0 2  fC  1 (5) L C  ( L  Lls )(C  C ls )  ls ls  2 cos 2 (kl / 2) (3) 2 whereas the electrical length of the unit cell can be  L2  2 Lls Cls 2 Z 0  expressed as B '  Lls cos( kl )   Z 0  ls   sin( kl )   4Z0 2   l   2f 0 ( L  Lls )( C  C ls ) (6) L C  2 3 N  C ls  Z 02 sin 2 ( kl / 2)  ls ls cos 2 ( kl / 2 ) (4) 4 Therefore, the ratio between fC and f0 is In (1),  is the phase constant of the loaded line, and A fC 2N 2 and B appearing in (1) and (2), respectively, are the first row   (7) f0  l elements of the transmission ABCD matrix of the unit cell. The design specifications in the SW-TL are the The shorter transmission line sections of the rat-race electrical length of the unit cell, l, and the characteristic coupler have a phase of  = /2. Therefore, the considered impedance, ZB, at the design frequency, f0. The electrical values of the unit cell electrical length must satisfyl = /2N. length of the unit cell is dictated by the number of considered In order to suppress the first harmonic of the coupler, at 3f0, cells, N, and by the required phase, , of the whole SW-TL the following condition must be satisfied: fC/f0 < 3. This under consideration, that is, l = /N. To determine N, or l, means that either l = /2 (N = 1), or l = /4 (N = 2). it should be taken into account that the onset of the first stop However, l = /4 is preferred for two main reasons: (i) fC is band of the periodic structure, fC, is intimately related to l further away from f0 (consequently preserving the coupler (which can only take discrete values as long as N is an integer response in the region of interest), and (ii) the stop band is and  is a design specification). Obtaining the analytical extended up to higher frequencies (thus enhancing the dependence of fC on l from (3) is not possible. However, a harmonic suppression capability of the coupler). Thus, we rough approximation can be inferred from the lumped can conclude that the coupler will consist of three quarter element equivalent circuit model of the unit cell, with the wavelength ( = /2) SW-TL sections (each one with N = 2) transmission line sections replaced with a shunt capacitor, C, and one SW-TL section with  = 3/2 (and N = 6). in parallel with Cls, and with a pair of series inductors, L/2, Besides (1) and (2), an additional design equation is the slow wave ratio P1 l/2 l/2 P2 v pL  /  kl , Lls/2 Zo, k Zo, k Lls/2 swr    (8) Cls v po  / k l (a) which provides the level of miniaturization (in length) of the SW-TL. In (8), vpl and vpo are the phase velocities of the loaded and unloaded lines. 3. Rat-race coupler design The design process of the rat-race coupler consists of the following steps: 1) In the first step, the topology of the unit cell, i.e., the one of Fig. 1 (a), described to a good approximation in the region of interest by the circuit model, also shown in Fig. 1 (b), is introduced, and the operating frequency, f0, is set to a certain value (dictated by specifications). 2) From the value of swr (a design parameter determining the length reduction of the unit cell), the characteristic impedance ZB (determined by the specific application, and equal to ZB = 70.71 Ω in a rat-race hybrid coupler), and the required electrical length of the unit cell (b) (l = 45º in our case), we calculate the element parameters of the schematic of Fig. 1(a), that is, Lls, Cls, Z0 and kl. Fig. 1. Circuit schematic (a) and topology (b) of the SW- Concerning kl, it is directly determined from equation (8), TL (unit cell) based on simultaneous inductive and provided swr is set to a certain value and l is an input capacitive loading. The (roughly) triangular shape of the parameter (l = 45º in our case). For the determination of Z0, capacitor patches is considered in order to accommodate Cls and Lls, there is some freedom as long as these three such patches in the inner region of the coupler (see Fig. 3). parameters are given by two equations (i.e., equations 1 and 2). The constrains are related to the fact that extreme values of Cls and Lls must be avoided since then these elements must 2 be implemented in planar form, as quasi-lumped elements. 180 l (degree) Therefore, these elements are determined from an iterative LC-loaded TL eq.circuit process, where Z0 is first set to a value equal to ZB, and Lls and 135 LC TL EM Sim. Lare univocally obtained from (1) and (2). Then, the layout 90 of these elements is inferred from Keysight Momentum, where the dimensions are optimized in order to obtain the 45  l (f0)= 45º required reactances at the design frequency. Once these layouts are obtained, it is determined if the lateral dimensions 80 of both the meander and the patch are comparable or are very  () different. In the second case, Z0 is varied in order to 60 ZB(f0) = 70.71 compensate the lateral size difference between the quasi- 40 lumped reactive elements, taking into account that an increase 20 of Z0 increases Cls and decreases Lls, and vice versa. 3) In the third step, once Z0, Lls and Cls have been 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 determined, the individual layouts of the meander, patch and Frequency (GHz) line are assembled, and it is verified if the design goals are satisfied (i.e., if ZB and l inferred from electromagnetic Fig. 2. Dependence of l and ZB with frequency. The simulation satisfy the requirements), and the complete layout considered substrate is the Rogers RO4003C with dielectric of the unit cell is optimized if necessary. Note that the length constant r = 3.55, thickness h = 1.524 mm and loss tangent reduction factor cannot exactly coincide with the one given tan = 0.0022. Unit cell dimensions are indicated in the by the selected value of swr, since the meander and the patch caption of Fig. 3. have finite dimensions. 4) In the next step, the complete circuit layout is assembled, and the topology of the patches is somehow modified, if necessary, to accommodate half of the patches in the inner region of the coupler (note that each capacitance Cls is implemented by means of two parallel patches in order to avoid an excessive patch size). 5) Finally, the electromagnetic simulation of the whole structure is carried out, and it is optimized again if necessary. The operating frequency of the designed coupler has been set to f0 = 0.825 GHz (central frequency of the E-UTRA Band 20 for LTE). For a rat-race coupler with equal power division between the output ports (hybrid coupler), the (a) characteristic impedance of the constitutive lines should be ZB = 70.71 Ω. By considering a slow wave ratio of swr = 0.25, and taking into account that the electrical length of the unit cell should be set to l = /4 (for the reasons explained before), it follows (using 8) that kl = /16. Following the previous design procedure, the impedance of the host line and the reactive elements have been found to be Z0 = 80 Ω, Lls = 8.251 nH and Cls = 1.471 pF. From these values, we have generated the layout of the unit cell [Fig. 1(b)], by implementing the loading inductances and capacitances by means of meander inductors and (roughly) triangular patch capacitor, respectively. The topologies of the meander and patch, as well as the whole topology of the unit cell, have been optimized by means of the Keysight Momentum commercial software, as indicated before. The dependence of the electrical length of the unit cell (b) and characteristic impedance with frequency is depicted in Fig. 2, where it can be seen that the required values at f0 are Fig. 3. Layout (a) and photograph (b) of the designed SW- obtained to a very good approximation. It should be TL coupler. The dimensions, in reference to Fig. 1, are: mentioned that the unconventional shape of the patches is W = 1.40 mm, lh = 6.18 mm, lp = 5.52 mm, Wp1 = 4.34 mm, justified by the need to accommodate such patches in the Wp2 = 1.70 mm, Ss = 0.60 mm, hs = 4.865 mm, ls = 1 mm, inner region of the rat-race topology, shown in Fig. 3(a). The Ws = 0.20 mm, Wcell = 13.70 mm, l = 8.18 mm. The access photograph of the fabricated device is depicted in Fig. 3(b). lines have a width of 3.36 mm, corresponding to 50  The size of the coupler is only 18% the size of the ordinary characteristic impedance. The area of the indicated counterpart. (dashed-line) hexagon is 1544 mm2 (whereas the area of the conventional coupler is 8590 mm2). 3 4. Results and comparison to other compact 0 couplers S31 0 S-Parameters (dB) -5 The response of the coupler (including the lossy -10 electromagnetic simulation and measurement) is depicted in -10 S-Parameters (dB) S41 Fig. 4. The measurements have been carried out by means of -20 -15 0 1 2 the 4-port Agilent PNA N5221A network analyzer, and calibration has been done using an electronic calibration -30 (Ecal Module N7554A). The proposed coupler exhibits a high harmonic rejection efficiency, with more than 17 dB -40 suppression up to at least 10 GHz (which means that at least S31 EM Simulation the first five harmonics are strongly suppressed). The phase S31 Measurement -50 S41 EM Simulation balance for the  and  ports in the vicinity of f0, depicted in Fig. 5, also validates the coupler functionality in the region of S41 Measurement interest. -60 0 1 2 3 4 5 6 7 8 9 10 The simulated response of the ordinary coupler, designed at the same frequency and considering identical Frequency (GHz) substrate, is compared to the simulated response of the (c) designed SW-TL-based coupler in Fig. 6. It can be 0 0 appreciated that the response of the SW-TL-based coupler is S32 EM Simulation S32 Measurement roughly indistinguishable from the one of the ordinary -5 -10 S42 EM Simulation coupler in the region of interest (i.e., around f0), which means S42 Measurement -10 S-Parameters (dB) that the functionality of the coupler in such region is -20 -150 compatible with the elimination of harmonic bands and with S32 1 S42 2 a substantial size reduction. -30 0 0 -40 -10 -15 -50 S-Parameters (dB) -20 -30 -60 -45 0 1 2 3 4 5 6 7 8 9 10 -30 S11 S22 -60 0 1 2 Frequency (GHz) (d) -40 S11 EM Simulation S11 Measurement Fig. 4. Response of the designed and fabricated SW-TL- -50 S22 EM Simulation based rat-race coupler. (a) Matching from ports 1 (-port) S22 Measurement and 2 (-port); (b) isolation between ports 1 and 2; (c) -60 power division from port 1; (d) power division from port 0 1 2 3 4 5 6 7 8 9 10 (2). Zoom views are shown in the insets. Frequency (GHz) (a) 90 0 Phase Balance |S41)-S31)| (deg) 0 EM Simulation -15 80 Measurement -10 -30 70 S-Parameters (dB) -20 -45 60 -60 0 S21 EM Simulation 1 2 50 -30 S21 Measurement 40 -40 30 20 -50 10 -60 0 0 1 2 3 4 5 6 7 8 9 10 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Frequency (GHz) Frequency (GHz) (b) (a) 4 270 0 S31 Phase Balance |S42)-S32)| (deg) 225 -10 S-Parameters (dB) S41 180 -20 135 -30 90 -40 S31 SW-TL 45 EM Simulation -50 S31 Conventional Measurement S41 SW-TL 0 -60 S41 Conventional 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0 1 2 3 4 5 6 7 8 9 10 Frequency (GHz) (b) Frequency (GHz) (b) Fig. 5. Phase balance for the  (a) and  (b) ports. 0 To demonstrate the validity of the circuit schematic of -10 S-Parameters (dB) Fig. 1(a) in the region of interest, the lossless electromagnetic simulation of the designed coupler of Fig. 3(a) (inferred from -20 Keysight Momentum) is compared to the circuit simulation S32 (obtained from the schematic simulator of Keysight ADS) in -30 S42 Fig. 7. Note that for a proper comparison, losses have been excluded in the electromagnetic simulation (in coherence -40 with the schematic model, where losses are not present). In spite of the relative complexity of the considered device (with -50 12 unit cells), the agreement between the circuit and S32 SW-TL S42 SW-TL electromagnetic simulation is good. Consequently, the circuit -60 S32 Conventional S42 Conventional model and the design procedure of the coupler, detailed in the previous section, are validated. It should be mentioned that 0 1 2 3 4 5 6 7 8 9 10 the validity of the model is restricted to the frequency region Frequency (GHz) where the meanders and patches are adequately described by (c) an inductance and a capacitance, respectively. For that reason, the comparison is limited to 2.5 GHz in Fig. 7. Fig. 6. Comparison of the responses of the ordinary and SW-TL-based couplers. Only the matching from port 1 (a), and power division from ports 1 (b) and 2 (c) are included. 0 0 -10 S-Parameters (dB) -10 -20 S-Parameters (dB) -20 -30 -30 -40 -40 S11 Sch. Simulation -50 S11 SW-TL S11 EM Simulation S11 Conventional -50 S22 Sch. Simulation -60 S22 EM Simulation 0 1 2 3 4 5 6 7 8 9 10 -60 0.5 1.0 1.5 2.0 2.5 Frequency (GHz) Frequency (GHz) (a) (a) 5 0 but their harmonic suppression capability is very limited or null. The coupler presented in [43] exhibits a very good -10 combination of size reduction and harmonic suppression, with only 15% the size of the ordinary counterpart, and S-Parameters (dB) suppression up to the 6th harmonic. In the coupler presented -20 in this work, the size reduction is comparable (18%) to the one in [43], and up to the 5th harmonic is suppressed. However, it is remarkable that in the present work the overall -30 harmonic suppression level of the coupler is better than the S21 Sch. Simulation one found in [43]. Therefore, the proposed SW-TL-based -40 S21 EM Simulation coupler, implemented by artificial lines with simultaneous inductive and capacitive loading, is found to be competitive in terms of combination of size and harmonic suppression. -50 Moreover, it has been clearly pointed out that the electrical 0.5 1.0 1.5 2.0 2.5 length of the unit cell of the considered periodic lines should Frequency (GHz) not be arbitrary. Particularly, it has been found that unit cells (b) with 45º electrical length are required to maintain the coupler 0 response in the region of interest unaltered and maximize the stopband for harmonic suppression. Concerning the useful operating range of the coupler, this is given by the bandwidth. Also in Table 1, we have S-Parameters (dB) included the bandwidth for S11, defined (in this work) as the -10 frequency band where matching is better than 15 dB, the bandwidth for power splitting, where the response deviates no more than  0.25 dB from the value at f0, and the bandwidth for phase balance (in this case with a tolerance of -20 S31 Sch. Simulation  5º). The previous bandwidths are compared to those of the S31 EM Simulation other couplers, and we can conclude that the proposed coupler S41 Sch. Simulation is also competitive in this regard. S41 EM Simulation To end this section, let us justify the reason for using -30 periodic structures, rather than a filtering non-periodic 0.5 1.0 1.5 2.0 2.5 structure, or a structure with some weighting function, for the Frequency (GHz) purpose of spurious suppression and size reduction. (c) Periodic structures provide stop bands due to the well known Bragg effect. However, as compared to filters (where 0 periodicity is sacrificed), periodic structures have intrinsic limitations such as moderate stopband bandwidth and the presence of ripple in the pass bands. Several works have been S-Parameters (dB) published (e.g. [51]), where, by sacrificing periodicity -10 (through tapering and by continuously varying the unit cell dimensions), ripple is minimized, and the stopband bandwidth is enhanced. In the present work, the intention is to achieve S32 Sch. Simulation -20 harmonic suppression and simultaneously reduce the size of S32 EM Simulation the coupler by means of the slow wave effect related to the S42 Sch. Simulation presence of inductances and capacitances, which effectively S42 EM Simulation reduce the phase velocity of the constitutive artificial lines. -30 Since we do not need, strictly speaking, a stop band filter, but 0.5 1.0 1.5 2.0 2.5 a structure integrated within the coupler able to suppress the Frequency (GHz) harmonic bands, the possible effects of ripple are not relevant (d) in this case. For this main reason, tapering (or weighting) is not necessary in our structure. On the other hand, harmonic Fig. 7. Comparison of the lossless coupler response suppression of (at least) the first five bands is achieved, which inferred from circuit and electromagnetic simulation in the is considered to be enough from a practical viewpoint. For all region of interest. (a) Matching from ports 1 (-port) and 2 these reasons, the consideration of a purely periodic (-port); (b) isolation between ports 1 and 2; (c) power reactively-loaded artificial line makes sense in this work. division from port 1; (d) power division from port (2). Moreover, the structure must behave as a transmission line with an effective phase velocity and characteristic The proposed SW-TL-based coupler is compared to impedance, and its design on the basis of Floquet analysis, as other compact couplers in Table 1 (the relative size is in it is done in the paper (mainly through equations 1, 2 and 8), comparison to the size of the ordinary coupler) and in Table 2. is simple and effective. All these important considerations are The couplers of refs. [41], [42] and [48] are extremely small, not so simple by weighting. 6 Table 1 Comparison with previously reported miniaturized Rat-Race couplers Frequency Matching Power splitting Phase balance Relative Harmonic Suppression Ref. Harmonic Suppression of operation bandwidth (%) bandwidth (%) bandwidth (%) Size (%) Level (dB) * (GHz) (S11 <  15 dB) Σ/Δ ( 0.25 dB) Σ/Δ ( 5º) [9] 32-46 NO --- 1.8 44.4 -/- -/- [29] 31 1st, 2nd, 3rd 30 / 40 / 18 0.5 58.7 23/23 17.6/17 [40] 15 1st 40 1 50 11/11 10.6/9.4 [41] 8 1st 40 1.44 24.3 12.5/12 -/10 [42] 3.9 NO --- 0.9 16.6 10/11 9/8.7 [43] 15 1st, 2nd , 3rd, 4th, 5th ,6th 12 / 38 / 32 / 28 /18 /10 1 42 20/21 -/15.5 [44] 27 NO --- 1 10 7/10 25/18 [45] 21.5 1st, 2nd 22 / 30 2.45 18.3 -/- 22/24.4 [46] 15 NO --- 2 15 -/- -/- [47] 17.8 1st, 2nd, 3rd, 4th 20 / 35 /45 / 30 1 24.3 20/20 10/13 [48] 5 NO --- 0.9 31.5 -/- 17/14 This 18 1st, 2nd , 3rd, 4th, 5th 17 / 36 / 36 / 43 / 35 0.825 46 20/23 17/15 work * The worst rejection level from output ports has been chosen. Table 2 Miniaturization methods used by the contributions applications, combining the coupler functionality with low- reviewed in Table 1 pass filtering is needed. Thus, space is saved if both functionalities are achieved with a single device, as Ref. Miniaturization Method demonstrated in this work. Moreover, the impact on miniaturization is further enhanced by means of the proposed [9] Artificial transmission lines slow-wave transmission lines. This combination of size and [29] Shunt open stubs harmonic suppression is competitive. Note that a figure of merit in devices with harmonic/spurious suppression is the [40] Capacitor loading number of suppressed spurious bands and the rejection level. With the reported coupler, potential interfering signals at the [41] Slow-wave effect output ports of the coupler (e.g., UWB signals) are avoided. Finally, the response of the coupler in the region of interest [42] Shunt-stub-based artificial transmission lines has been kept unaltered. [43] Slow-wave resonant structures 6. Acknowledgements [44] Low impedance and fractal-shaped resonant cells This work was supported by MINECO-Spain under [45] Periodic stepped-impedance Projects TEC2016-75650-R and TEC2017-84724-P, by Generalitat de Catalunya under project 2017SGR-1159, by [46] T-shaped PBG cells ICREA (who awarded Ferran Martín), by the Spanish Junta de Andalucía under project P12-TIC-1435, and by FEDER [47] Capacitive-inductive loading funds. [48] Asymmetrical T-structures 7. References This Slow-wave TLs with capacitive-inductive Loading work [1] K. Wu, “Slow wave structures,” in Encyclopedia of Electrical and Electronics Engineering, J. G. Webster, Ed. New York: Wiley, 1999, vol. 19, pp. 366–381. 5. Conclusions [2] F. Martín, Artificial Transmission Lines for RF and A compact rat-race coupler with harmonic Microwave Applications, John Wiley, Hoboken (NJ), 2015. suppression capability, based on slow-wave transmission [3] S. Seki, and H. 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