Received October 14, 2021, accepted November 1, 2021, date of publication November 8, 2021, date of current version November 12, 2021. Digital Object Identifier 10.1109/ACCESS.2021.3125775 On the Modeling of Microstrip Lines Loaded With Dumbbell Defect-Ground-Structure (DB-DGS) and Folded DB-DGS Resonators LIJUAN SU 1 , (Member, IEEE), JONATHAN MUÑOZ-ENANO 1 , (Member, IEEE), PARIS VÉLEZ 1 , (Senior Member, IEEE), JESÚS MARTEL 2 , (Senior Member, IEEE), FRANCISCO MEDINA 2 , (Fellow, IEEE), AND FERRAN MARTÍN 1 , (Fellow, IEEE) 1 GEMMA/CIMITEC, Departament d’Enginyeria Electrònica, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain 2 Grupo de Microondas, Universidad de Sevilla, 41012 Sevilla, Spain Corresponding author: Ferran Martín (ferran.martin@uab.es) This work was supported in part by the Ministerio de Economía y Competitividad (MINECO)-Spain under Project PID2019-103904RB- I00, in part by the Generalitat de Catalunya under Project 2017SGR-1159, in part by the Institució Catalana de Recerca i Estudis Avançats (who awarded Ferran Martín), and in part by the Fondo Europeo de Desarrollo Regional (FEDER) funds. The work of Lijuan Su and Paris Vélez was supported by the Juan de la Cierva Program under Project IJCI-2017-31339 and Project IJC2019-040786-I. The work of Jonathan Muñoz-Enano was supported by the Secreteraria d’Universitats i Recerca (Gen. Cat.) and European Social Fund for the Formació d’Investigadors (FI) grant. ABSTRACT This paper presents a lumped-element equivalent circuit model of microstrip lines loaded with dumbbell defect-ground-structure (DB-DGS) resonators, etched in the ground plane. The model is valid for ordinary (i.e., unfolded) DB-DGSs, as well as for folded DB-DGSs with an arbitrary aperture angle and relative orientation between the line and the resonator. It is shown that in folded, or partially folded, DB-DGS resonators both magnetic and electric coupling between the line and the resonator should be considered, except for a particular DB-DGS orientation, namely, the one where the symmetry plane of the particle (a magnetic wall) is orthogonal to the line axis. In this case, the particle is exclusively excited by the electric field generated by the line. It is also shown that the circuit model of a microstrip line loaded with an unfolded DB-DGS resonator transversally oriented to the line can be derived from the general model by considering the effects of opening the particle, i.e., a reduction of the electric coupling. In the extreme situation where the DB-DGS is completely opened (unfolded), the electric coupling vanishes, and the particle is exclusively driven by the magnetic field generated by the line. This effect is taken into account in the model by considering that the capacitance between the line and the inner metallic region of the folded, or partially folded, DB-DGS depends on the aperture angle of the particle, and it is null when the particle is unfolded. The models are validated by parameter extraction and comparison of the circuit responses with the responses inferred by electromagnetic simulation and measurement. INDEX TERMS Dumbbell DGS (DB-DGS), defect ground structures (DGS), folded DB-DGS, mixed coupling, microstrip technology, parameter extraction. I. INTRODUCTION S-shaped CSRR [8], the magnetic LC resonator [9], [10] Defect-ground-structure (DGS) resonators are slot (or aper- or the dumbbell defect-ground-structure (DB-DGS) res- ture) resonators etched in the ground plane of planar trans- onator [1], [11], among others. These DGS resonant elements mission lines, typically, microstrip lines [1]–[3]. There are have been used in a wide variety of applications. Examples many types of DGS resonators, including the slot resonator include the implementation of microwave circuits and filters (linear or circular) [4], the complementary split ring resonator with reduced size and superior performance (exploiting the (CSRR) [5], the complementary spiral resonator [6], [7] the small electrical size and specific electromagnetic properties of the considered DGS resonators) [11]–[23], as well as The associate editor coordinating the review of this manuscript and planar antennas [24]–[27]. DGS resonators are also very approving it for publication was Jenny Mahoney. interesting for the design of highly sensitive microwave This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 150878 VOLUME 9, 2021 L. Su et al.: On Modeling of Microstrip Lines Loaded With DB-DGS and Folded DB-DGS Resonators sensors, mainly, although not exclusively, devoted to permit- certain applications involving inter-resonator’s coupling (for tivity measurements and characterization of material prop- example, filters based on coupled folded SIRs [41] and cou- erties [28]–[36]. Resonant sensors exhibit high sensitivities pled folded DB-DGSs [21], [22] have been proposed). Also, to variations in the complex permittivity of the material depending on the length of the narrow slot of the DB-DGS surrounding it. Moreover, DGS-based sensors present a sig- particle, its shape factor might be too extreme, unless it is nificant advantage as compared to sensors implemented by folded. means of metallic sensing elements, i.e., isolation of the DGS resonators, including DB-DGS and folded DB-DGS material under test (MUT) from the host line [37]. Specifi- resonators, are preferentially combined with microstrip lines, cally, in microstrip sensors based on DGS sensing resonators, by etching them in the ground plane, beneath the line. By this the MUT should be placed in the backside of the substrate, means, the resonant particle is under the influence of the elec- in contact with the DGS. Therefore, the MUT is backside tromagnetic field generated by the line, a necessary condition isolated from the line, due to the presence of the ground plane, to excite the particle and obtain the desired functionality. only altered by the etched DGS resonator. It should also be However, for the design of microwave circuits or sensors indicated that such backside isolation of the MUT is espe- based on microstrip lines loaded with DB-DGSs or with cially important in microfluidic sensors [28], [34], [36], since folded DB-DGSs, accurate circuit models are needed. Like the fluidic channel plus the necessary mechanical accessories, the SIR resonator, the DB-DGS resonator (ordinary or folded) present also in the backside of the substrate, do not perturb the can be considered to be electrically small, by virtue of the electromagnetic field generated by the line. high contrast between the width of the narrow (central) slot and the width of the apertures at the extremes. Therefore, a microstrip line loaded with a DB-DGS or a folded DB-DGS resonator can be described by means of a lumped element circuit model consisting of reactive elements and, eventu- ally, with electric and/or magnetic couplings (obviously, the effects of losses can be accounted for by introducing resis- tances in the model, although this aspect is not considered in this paper). The model of a microstrip line loaded with an ordi- nary (unfolded) DB-DGS resonator transversally oriented has been reported in the literature, and it consists of a parallel resonant tank series-connected to the line [1], [11], [42]. However, by folding the resonator, the reported model is completely different (even the topology, as shown in Fig. 1), since there is a capacitive effect between the line and the inner metallic region of the resonator, which must be included in the model. In this paper, a single model able to describe these two FIGURE 1. Typical topology of a SIR (a), DB-DGS (b), folded SIR (c) and extreme cases (folded and unfolded DB-DGS), as well as the folded DB-DGS (d) resonator. The metallic regions are depicted in grey. general case of a line loaded with a partially folded DB-DGS is proposed. This is achieved by introducing a capacitance in According to the preceding paragraph, it is clear that DGS the model that depends on the folding level of the resonant resonators are of high interest in microwave engineering. element, being null for the ordinary (unfolded) resonator Among them, the so-called dumbbell DGS (DB-DGS) res- [Fig. 1(b)], maximum for the completely folded DB-DGS onator [1], [11], [13], [19], the complementary (or dual) resonator [Fig. 1(d)], and taking intermediate values for res- counterpart of the step impedance resonator (SIR) [38], [39] onators partially folded. The transition from the circuit model (see Fig. 1), has been exhaustively used in the design of accounting for the response of the folded version to the circuits, filters, and sensors. For example, DB-DGS res- circuit model corresponding to the regular unfolded one is not onators have been applied to the suppression of common- obvious, and constitutes the main contribution of the present mode noise in microstrip differential lines [40]. DB-DGS paper. Additionally, the model is able to describe the structure resonators are also of interest for the implementation of when the folded DB-DGS resonator is rotated 90◦ . stopband filters, by virtue of the notched response of a The paper is organized as follows. In section II, the pro- DB-DGS-loaded microstrip line [13], [20]. Finally, let us posed model for a DB-DGS-loaded microstrip line is pre- mention that the achievable sensitivities in DB-DGS-based sented, and it is particularized to specific (canonical) cases, frequency-variation sensors are superior to those that can be i.e., unfolded DB-DGS resonator, and folded DB-DGS res- obtained in other DGS-based sensors, as recently reported onator with transverse and longitudinal orientation. Model in [35]. A variant of the DB-DGS resonator is the folded validation, by comparing the circuit response with extracted DB-DGS resonator, the dual counterpart of the folded SIR parameters and the response inferred from electromagnetic (see Fig. 1). These folded particles may be of interest in simulation and measurement is presented in section III. VOLUME 9, 2021 150879 L. Su et al.: On Modeling of Microstrip Lines Loaded With DB-DGS and Folded DB-DGS Resonators Section IV discusses some interesting aspects related to the direction orthogonal to the plane of the particle at the funda- elements of the model. Finally, the main conclusions are mental resonance, and a magnetic dipole moment orthogonal highlighted in section V. to the symmetry plane, a magnetic wall. Therefore, the folded DB-DGS resonator can be excited either by a vertical time- varying electric field, or by a time-varying magnetic field applied in the normal direction to its plane of symmetry. Therefore, with the orientation shown in Fig. 3, the particle can only be excited by means of the electric field generated by the line, with a significant component in the vertical direction, but not with the magnetic field (note that the microstrip line supports a quasi-TEM mode with a negligible component of the magnetic field orthogonal to the symmetry plane of the DB-DGS resonator). However, by rotating the particle 90◦ FIGURE 2. Typical topology (a) and circuit model (b) of a microstrip line loaded with a transversally oriented DB-DGS resonator. The microstrip (Fig. 4), the symmetry plane of the resonator is perfectly central line is in black color, whereas the metallization of the ground aligned with the symmetry plane of the line, also a magnetic plane is depicted in grey. wall, and the particle is excited by both the magnetic and the electric field generated by the line (mixed coupling). This mixed coupling has been also reported in reference to CSRR-loaded microstrip lines [43] (indeed, the coupling mechanisms in CSRR and folded DB-DGS resonators are similar). According to the previous words, the circuit models of the microstrip line loaded with the folded DB-DGS resonators for both orientations are those depicted in Figs. 3(b) and 4(b). In the circuit of Fig. 3(b), L and C are the inductance and FIGURE 3. Typical topology (a) and circuit model (b) of a microstrip line capacitance, respectively, of the line section on top of the loaded with a folded DB-DGS resonator oriented with the symmetry plane folded resonator, with inductance Lc (mainly related to the orthogonal to the line axis. apertures) and capacitance Cc (mostly determined by the cen- tral slot). The capacitor C accounts for the electric coupling between the line and the resonant element, the single coupling mechanism for the DB-DGS orientation of Fig. 3. In the circuit of Fig. 4(b), electric coupling is preserved. However, magnetic coupling between the line and the resonator should arise, and such coupling is accounted for by means of the mutual inductances, designated as M /2. By rotating the folded DB-DGS 90◦ , it is expected that the reactive elements of the circuit of Fig. 3(b) do not change appreciably, except Lc FIGURE 4. Typical topology (a) and circuit model (b) of a microstrip line (obviously, provided the DB-DGS dimensions and substrate loaded with a folded DB-DGS resonator oriented with the symmetry plane aligned with the line axis. are identical in both cases). The reason is that the current path between the inner and the outer region of the folded DB- DGS might be influenced by the presence of the line strip II. CIRCUIT MODEL OF A MICROSTRIP LINE LOADED on top of the particle. Consequently, the parameters L, C WITH A DB-DGS RESONATOR WITH ARBITRARY and Cc extracted from the circuit of Fig. 3(b) are assumed to ORIENTATION AND FOLDING LEVEL be valid in the circuit of Fig. 4(b). The mutual inductance, The canonical topologies and circuit models of DB-DGS- M , and the new Lc value of the rotated resonator can be loaded microstrip lines are depicted in Figs. 2, 3 and 4. inferred by curve fitting the electromagnetically simulated (or In Fig. 2(a), the DB-DGS is unfolded and transversally measured) response of the structure of Fig. 4(a) (an aspect to oriented with regard to the line axis. The circuit model, be discussed later). as reported in the literature [1], [11], [42], consists of a paral- Let us now consider the DB-DGS orientation of Fig. 4, but lel resonant tank, series connected to the line. By contrast, with the resonator partially folded, see Fig. 5, as given by the in the structures of Figs. 3 and 4, the DB-DGS resonator angle of aperture, θ (note that for the extreme cases of θ = is folded. However, in Fig. 3, the symmetry plane of the 0◦ and θ = 90◦ , the resulting DB-DGS topology is the one of resonant particle is transversally oriented to the line, whereas the folded and ordinary –unfolded– resonator, respectively). it coincides with the line axis in Fig. 4 (the difference is that As far as 0 ≤ θ < 90◦ , the particle exhibits an electric dipole one particle is rotated 90◦ with regard to the other). The folded in the vertical direction and therefore it can be electrically DB-DGS resonator exhibits an electric dipole moment in the excited by the line. Nevertheless, the electric coupling should 150880 VOLUME 9, 2021 L. Su et al.: On Modeling of Microstrip Lines Loaded With DB-DGS and Folded DB-DGS Resonators FIGURE 5. Topology of a microstrip line loaded with a folded DB-DGS resonator with an arbitrary angle of aperture (a), zoom view of the aperture arm (b), and equivalent circuit model valid for any arbitrary angle of aperture (c). decrease as the angle of aperture increases, and such coupling FIGURE 6. Circuit model of Fig. 5(c) for θ = 90◦ (a) transformed model must vanish for θ = 90◦ . This dependence of the electric with unification of the inductance and mutual coupling (b), and coupling with the angle can be taken into account by means of transformed circuit (c). an angle-dependent coupling capacitance, C(θ). Reasonably, such capacitance can be considered to be the capacitance of the line section of length d corresponding to the projection of In the limit where θ = 0◦ , the circuit of Fig. 5(c) transforms to the extension of the particle in the line axis [see Fig. 5(b)]. the model of Fig. 4(b). By contrast, for an angle of aperture If C is the capacitance when θ = 0◦ , it follows that the of θ = 90◦ , corresponding to the topology of the unfolded angle-dependent coupling capacitance is simply DB-DGS resonator, the model of Fig. 5(c) should be the one C(θ ) = Cpul · d = C · cos(θ) (1) of Fig. 2(b). It is apparent that for θ = 90◦ , the coupling capacitance is null (i.e., C(θ = 90◦ ) = 0), whereas the capac- provided d = l·cosθ and C = Cpul · l, where Cpul is the itances of the extremes are simply C/2 (the line capacitance). per-unit length capacitance of the line. The result is the π-model depicted in Fig. 6(a), where the res- On the other hand, if the model of the structure must onant tank (i.e., the DB-DGS resonator) is excited exclusively describe the resonant element plus the line section of length l, by the mutual inductance M . This model can be transformed the decrease in the capacitance C caused by the angle of to the one of Fig. 6(b), and, finally, to the one of Fig. 6(c), aperture should be compensated by a pair of capacitances, where the equivalent inductance L’ and the capacitances C/2, corresponding to the line portions of length (l− d)/2 outside can be considered to describe the line section in the model of the area of influence of the DB-DGS, see Fig. 5. Obvi- Fig. 2(b) corresponding to the length l. Therefore, the lumped ously, such capacitances are simply [C − C(θ )]/2, and are element circuit model of Fig. 6(c) is equivalent to the one of directly connected to ground in the circuit model. Note that Fig. 2(b). The element values of the series branch in the circuit with the considered DB-DGS orientation of Fig. 5, the mag- of Fig. 6(c) can be derived by first obtaining the impedance netic coupling between the line and the DB-DGS resonator of the series branch, i.e. [44], is preserved, regardless of the angle of aperture. However,   modelling the variation of such coupling, i.e., the change  M 2  in the mutual inductance, with the angle of aperture does Zs = jω · L + (2) ω 2 Lc ( ω02 − 1)   not seem to be an easy task with the considered geometry. Therefore, such mutual inductance can be considered to be Such impedance can be expressed as angle dependent and a fitting parameter. Note that this applies also to the inductance Lc of the particle, since the current path  L 0c  varies with θ. Zs = jω · L − L 0c + (3) 1 − L 0c C 0c ω2 Figure 5(c) depicts the lumped element equivalent cir- cuit model corresponding to a microstrip line loaded with a and it is apparent that this is the impedance of the series DB-DGS partially folded and oriented as shown in Fig. 5(a). branch in the circuit of Fig. 6(c), with element values VOLUME 9, 2021 150881 L. Su et al.: On Modeling of Microstrip Lines Loaded With DB-DGS and Folded DB-DGS Resonators given by L 0 = L − Lc0 (4a) Lc0 = Cc M 2 ω02 (4b) Lc Cc0 = (4c) M 2 ω02 and ω02 = 1/Lc Cc = 1/Lc0 Cc0 (ω0 is the angular frequency of the DB-DGS resonator). In the previous analysis, it has been considered that the magnetic coupling between the apertures of the DB-DGS resonator can be neglected. This is reasonable as far as the distance between the apertures is significant, the case consid- ered in this paper even for the folded DB-DGS (with θ = 0◦ ). Nevertheless, if the distance between the apertures is small and, consequently, inter-aperture coupling cannot be neglected, the effect is a modification of the equivalent induc- tance of the DB-DGS. The specific value is simply L + M ’/2, where M ’ is the mutual coupling between the apertures. The analysis has also neglected the effect of losses, since the main interest of the paper is to propose and validate a unified circuit model (the one of Fig. 5) able to predict the behavior of the DB-DGS-loaded microstrip line and the coupling mechanisms for any arbitrary angle of aperture, rather than to obtain an accurate circuit response. Moreover, in the specific structures used for validation, to be discussed next, losses are small, since low-loss microwave substrates (i.e., with low loss tangents) are considered, and ohmic losses, FIGURE 7. (a) Electromagnetic, circuit and measured responses of the structure of Fig. 3; (b) photograph of the fabricated structure (ground related to the metallic layers, are also small. plane side). III. MODEL VALIDATION For model validation, the different DB-DGS and folded The extracted parameters, inferred by means of the method DB-DGS based microstrip lines (all exhibiting notched of [45], are: L = 9.27 nH, C = 2.07 pF, Lc = 3.69 nH, responses) are simulated by means of an electromagnetic and Cc = 4.23 pF. The circuit response, also included in solver (the Keysight Momentum commercial simulator). From the figure, is in good agreement with the electromagnetic such electromagnetic simulations, the circuit parameters are simulation, which indicates that the model provides an accu- extracted and used to obtain the circuit responses, which are rate description of the structure. Such structure has been compared to the full-wave electromagnetic simulations. The fabricated by means of the LPKF-H100 drilling machine, considered substrate is the Rogers RO3010 with dielectric available in the laboratory of the research group [Fig. 7(b)]. constant εr = 10.2 and thickness h = 1.27 mm. In such The measured response, depicted in Fig. 7(a), is also in good substrate the width of a 50- line is found to be W = agreement with the circuit and electromagnetic responses 1.1 mm, the considered value. Concerning the dimensions The small discrepancies are mainly due to the effects of of the DB-DGS, the apertures are square shaped and exhibit losses, and to fabrication related tolerances (radiation losses a side length of l/2, whereas the total length of the narrow are small, as inferred from the simulated loss factor, not slot is 2l, with l = 12 mm. The width of the slot is 0.2 mm. shown, calculated by excluding ohmic an dielectric losses). For the folded DB-DGS with θ = 0◦ , the distance between It should be clarified that for parameter extraction, the effects the apertures is also l/2, and such distance progressively of the 50- access lines, with length lb , have been omitted, increases as the angle of aperture increases (see Fig. 3(a) for but then, in the simulations, such lines have been included. details). In the measurements, the port extension function of the vector The first structure under consideration is the microstrip network analyzer (VNA) has been used to correct the phase line loaded with the folded DB-DGS oriented as depicted shift introduced by the connectors. in Fig. 3(a), since this structure exhibits exclusively electric Next, the electromagnetic response of the structure that coupling and the circuit model is, thus, relatively simple, see results from the structure of Fig. 3 by rotating the folded Fig. 3(b). Moreover, parameter extraction with such circuit DB-DGS 90◦ has been inferred [Fig. 4(a)]. The hypothesis model has been reported in the literature [45]. The electro- that such rotation in the orientation of the resonator does magnetic response of this structure is depicted in Fig. 7(a). not modify the circuit parameters of the line (including the 150882 VOLUME 9, 2021 L. Su et al.: On Modeling of Microstrip Lines Loaded With DB-DGS and Folded DB-DGS Resonators and M (θ = 90◦ ) = 4.0 nH. Since the electric coupling in this configuration vanishes (the DB-DGS is excited exclusively by means of the magnetic field generated by the line), the cir- cuit is symmetric with regard to the bisecting plane between the input and output ports, in agreement with the topology of the structure, also symmetric with regard to that plane. Thus, in this case, it follows that matching is identical from both ports (i.e., S11 = S22 ). The measured response, also included in Fig. 9(a), is in reasonably good agreement with the simulated and circuit responses (the origin of the small discrepancies has been indicated in reference to the previous results). FIGURE 8. (a) Electromagnetic, circuit and measured responses of the structure of Fig. 4; (b) photograph of the fabricated structure (ground plane side). S11 and S22 magnitude for both EM simulation and circuit simulation are nearly undistinguishable. coupling capacitance) and the capacitance of the resonator, has been made, as anticipated before. However, the induc- tance of the resonator, Lc , is considered to be an adjustable parameter, as far as it is expected to change by rotating the DB-DGS resonator, as justified in the preceding section. Thus, the inductance Lc and the mutual inductance M have been inferred by curve fitting, and the resulting values have been found to be: Lc = 2.80 nH and M = 2.86 nH. Figure 8(a) FIGURE 9. (a) Electromagnetic, circuit and measured responses of the depicts the circuit, electromagnetic and measured responses, structure of Fig. 2; (b) photograph of the fabricated structure (ground where the agreement is, again, very reasonable (the slight plane side). frequency shift in the measured resonant frequency is due to fabrication related tolerances). Note that, due to the lack of Finally, we have considered a partially folded DB-DGS symmetry with regard to the bisecting plane between the input resonator with folding level given by θ = 45◦ . According to and the output ports, the S-parameter matrix is not symmetric, the model, see Fig. 5, the line inductance, L, is preserved, i.e., S11 6 = S22 . as compared to the previous cases, whereas the line capaci- The next step has been to consider the structure of Fig. 2, tance corresponding to the length l is divided in three parts, all with completely unfolded (θ = 90◦ ) and transversally ori- dependent on the folding angle θ. Nevertheless, it is assumed ented DB-DGS resonator. The same hypothesis assumed that the contribution of the three capacitances, C, does not before is adopted here, i.e., L, C, and Cc are not altered vary in comparison to the previous cases. According to (1), by unfolding the resonator. Thus, again, Lc and M are fit- the capacitance portion coupled to the resonant element is ting parameters. The values providing the best agreement C(θ = 45◦ ) = 1.46 pF, whereas the capacitances of the between the electromagnetic simulation and circuit response, extremes are [C −C(θ = 45◦ )]/2 = 0.30 pF. It is also assumed see Fig. 9(a), have been found to be Lc (θ = 90◦ ) = 3.51 nH that the capacitance of the resonator Cc does not vary. Thus, VOLUME 9, 2021 150883 L. Su et al.: On Modeling of Microstrip Lines Loaded With DB-DGS and Folded DB-DGS Resonators the fitting parameters are, again, Lc and M , and the val- Table 1 (the simulations are not shown). Note that C(θ ), rather ues that have provided the best fitting have been found than fitted, it has been calculated by means of expression (1), to be Lc (θ = 45◦ ) = 2.85 nH and M (θ = 45◦ ) = 3.36 nH. taking into account that C(θ = 0◦ ) = 2.07 pF. The table Figure 10(a) depicts the simulated, circuit, and measured includes also the coupling coefficient, k, between the line responses for θ = 45◦ . Naturally, in this case, S11 6 = S22 , inductance and the√inductance of the DB-DGS resonator, as expected on account of the lack of symmetry due to partial defined as k = M / LLc . folding. In the circuit model, the mixed coupling destroys the symmetry, similar to the case of Figs. 4 and 8. TABLE 1. Dependence of model parameters with the angle θ. According to the results of Table 1, the inductance of the resonant element, Lc , does not appreciably change for angles of aperture below 45◦ , and it increases with θ when θ > 45◦ . By contrast, M experiences a progressive growth with θ for angles between 0◦ < θ < 90◦ . However, the coupling coefficient, k, does not significantly vary with θ. It slightly increases with the angle of aperture of the DB-DGS, but it tends to saturate as θ approaches 90◦ . This behavior is attributed to the fact that, for angles of aperture close to 90◦ , the magnetic field generated by the line impacts only the narrow slot region of the DGS in the vicinity of the line strip. Consequently, the coupling coefficient should not sig- nificantly vary with θ for apertures close to 90◦ . By contrast, for small aperture angles, the regions of the folded DB-DGS under the influence of the magnetic field generated by the line are not restricted to the narrow slot region beneath the line. The square-shaped apertures may interact with the line, but such interaction is expected to decrease with θ. It is believed that the increase of Lc with the aperture angle is related to FIGURE 10. (a) Electromagnetic, circuit and measured responses of the structure of Fig. 5 and θ = 45◦ ; (b) photograph of the fabricated structure the progressively decreasing influence of the strip line on (ground plane side). this parameter, as such angle increases (i.e., by increasing the angle, the influence of the line is less severe). This increase According to the results presented in Fig. 10(a), the agree- in Lc is thought to be the reason of an increase in M , provided ment between the circuit, electromagnetic and measured k, the coupling coefficient, is roughly constant. Nevertheless, responses is also reasonable. These results validate the pro- an exact calculation of the coupling coefficient and an accu- posed model of Fig. 5, where the capacitance of the line rate explanation of the involved phenomenology is complex, section l is divided in three different portions, which depend since the geometry of the considered folded DB-DGS is not on the angle of aperture of the DB-DGS resonator. Neverthe- simple. less, the total capacitance C neither depends on the angle of Let us analyze in more detail the line loaded with aperture of the resonator, nor on the angle of orientation of the unfolded and transversally oriented DB-DGS resonator. the DB-DGS resonator. Using expressions (4), the parameters of the transformed model of Fig. 6(c) are found to be L’ = 4.72 nH, Lc0 = 4.55 nH IV. DISCUSSION and Cc0 = 3.26 pF. Obviously, the capacitances C/2 of the To gain more insight on the dependence of C(θ ), Lc (θ), and shunt branches of the model of Fig. 6 are not affected by M (θ) with the folding (aperture) angle, we have simulated the transformation. Let us now directly infer the parameters (but not fabricated) additional DB-DGS-loaded microstrip of the circuit model of Fig. 6(c), excluding C, by parameter lines with different angles of aperture of the folded resonator. extraction from the electromagnetically simulated response The parameters, inferred by curve fitting, are summarized in (see Fig. 9(a)). The results are, L’ = 5.46 nH, Lc0 = 4.37 nH 150884 VOLUME 9, 2021 L. Su et al.: On Modeling of Microstrip Lines Loaded With DB-DGS and Folded DB-DGS Resonators and Cc0 = 3.40 pF i.e., in reasonably good agreement with the an unfolded and transversally oriented DB-DGS resonator, parameters of the series branch obtained by means of expres- the lumped elements are restricted to the description of the sions (4). Figure 11 compares the circuit responses inferred resonant element. Note that in the model of Fig. 2(b), the line by considering both sets of parameters. The figure includes sections account for the whole line, whereas in the model also the EM simulation of the structure (also depicted in of Fig. 6(c), the line sections describe only the access lines Fig. 9(a)). The agreement between the three responses is by excluding the central part of length l. Thus, according to good. these words, the values of L 0 and C in the circuit of Fig. 6(c) should be coherent with the inductance and capacitance of the 50- line corresponding to the considered line length l. Indeed, such coherence is verified as far as the characteristic impedance given by these reactive parameters, i.e., L 0 /C, p coincides to a very good approximation with the nominal characteristic impedance of the line (50 ). Particularly, by considering L 0 inferred from p the transformations (4), i.e., L’ = 4.72 nH, one obtains L 0 /C = 47.8 , whereas with L’ = 5.46 nH (extracted value), the characteristic impedance is found to be L 0 /C = 51.4 . In both cases, p C = 2.07 pF. On the other √ hand, the electrical length of the line section l is βl = ω L 0 C. At the resonance frequency of the DB-DGS resonator, f0 = 1.305 GHz, the electrical length is found to be βl = 0.81 rad for L’ = 4.72 nH, and βl = 0.87 rad for L’ = 5.46 nH. These phases are in reasonable agreement with the nominal electrical length of the line section of length l, inferred from the transmission line calculator LineCalc (included in Keysight ADS) using FIGURE 11. Comparison between the circuit response of the structure of the parameters of the considered substrate (0.87 rad). Con- Fig. 9(a), given by the model of Fig. 6(c), obtained by the two sets of sequently, it can be concluded that L 0 and C in the model reactive parameters indicated in the text, i.e., from the transformation equations (4) and from direct parameter extraction. The of Fig. 6(c) describe the line section of length l, whereas electromagnetically simulated response is also included. the access lines of length lb are accounted for by the corre- sponding transmission line sections. In the model of Fig. 2(b), the access lines of length la describe the complete line strip. Nevertheless, it is clear according to this discussion that the models of Figs. 2(b) and 6(c) are equivalent. Another interesting aspect to discuss is the discrepancy in the phase response of S11 (between circuit/electromagnetic simulation and measurement) for the structure of Fig. 7 (the difference in magnitude has been attributed to the effects of losses, and due to the fact that losses preclude from obtaining a ‘‘true’’ reflection zero). The difference in the phase of S11 is more subtle, and further insight can be inferred from the representation of S11 in the Smith Chart, as shown in Fig. 12. At the reflection zero frequency, when losses are negligi- ble, there is a cross through the center of the Smith Chart (with, theoretically, total transmission), and therefore the phase changes suddenly (jumps) from roughly −90◦ to +90◦ , corresponding approximately to a vertical cross through the FIGURE 12. Detail of S11 corresponding to the structure of Fig. 7, center of the Smith Chart. However, in measurement, due to represented in the Smith Chart. the effect of losses, the trace of S11 crosses the horizontal axis of the Smit Chart (where the impedance is purely resistive) There is a difference between the model of Fig. 2(b) and the slightly to the left of the center, and therefore the phase model of Fig. 6(c), namely, the presence of L 0 and C in the jump is exactly from −180◦ to +180◦ . So, this higher (360◦ ) latter. Because the general lumped element model of Figure 5 jump in measurement is explained by this fact. Therefore, this accounts for the DB-DGS plus the line section of length l, the apparent high discrepancy in S11 between measurement and model of Fig. 6 should include such line section, and, indeed, simulated values is in reality an artifact (as it can be appreci- L 0 and C model such line section. By contrast, in the model ated in the Smith Chart, the difference between measurements of Fig. 2, the usual model of a microstrip line loaded with and simulations is not very significant). VOLUME 9, 2021 150885 L. Su et al.: On Modeling of Microstrip Lines Loaded With DB-DGS and Folded DB-DGS Resonators To end this section let us mention that if a gap capacitor is researchers working in circuit modelling, planar filters, and etched in the strip line, aligned with the narrow slot of the planar microwave sensors. transversally oriented folded, or partially folded, DB-DGS resonator, then it is expected that the notched response REFERENCES switches to a bandpass response. However, the corresponding [1] C. S. Kim, J. S. Park, D. Ahn, and J. B. Lim, ‘‘A novel 1-D periodic defected modelling is out of the scope of this paper. Let us also empha- ground structure for planar circuits,’’ IEEE Microw. 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(IACC), Ghaziabad, India, Feb. 2013, pp. 1573–1576. LIJUAN SU (Member, IEEE) was born in [26] F. A. L. B. Konkyana and B. A. Sudhakar, ‘‘A review on microstrip Qianjiang, Hubei, China, in 1983. She received antennas with defected ground structure techniques for ultra-wideband the B.S. degree in communication engineering and applications,’’ in Proc. Int. Conf. Commun. Signal Process. (ICCSP), the M.S. degree in circuits and systems from the Chennai, India, Apr. 2019, pp. 930–934. Wuhan University of Technology, Wuhan, China, [27] D. Gopi, A. R. Vadaboyina, and J. R. K. K. Dabbakuti, ‘‘DGS based in 2005 and 2013, respectively, and the Ph.D. monopole circular-shaped patch antenna for UWB applications,’’ Social Netw. Appl. Sci., vol. 3, no. 2, p. 198, Feb. 2021. degree in electronic engineering from the Univer- [28] A. Ebrahimi, W. Withayachumnankul, S. Al-Sarawi, and D. Abbott, sitat Autonoma de Barcelona, Barcelona, Spain, ‘‘High-sensitivity metamaterial-inspired sensor for microfluidic dielec- in 2017. From November 2017 to December 2019, tric characterization,’’ IEEE Sensors J., vol. 14, no. 5, pp. 1345–1351, she worked as a Postdoctoral Researcher with the May 2014. Flexible Electronics Research Center, Huazhong University of Science and [29] A. Ebrahimi, W. Withayachumnankul, S. F. Al-Sarawi, and D. Abbott, Technology, Wuhan. She is currently a Postdoctoral Researcher under the ‘‘Dual-mode behavior of the complementary electric-LC resonators loaded support of the Juan de la Cierva Program with the CIMITEC, Universitat on transmission line: Analysis and applications,’’ J. Appl. Phys., vol. 116, Autonoma de Barcelona. Her current research interests include development no. 8, Aug. 2014, Art. no. 083705. of novel microwave sensors with improved performance for biosensors, [30] L. Su, J. Mata-Contreras, P. Vélez, and F. Martín, ‘‘Splitter/combiner dielectric characterization of solids and liquids, defect detection, and indus- microstrip sections loaded with pairs of complementary split ring res- trial processes. onators (CSRRs): Modeling and optimization for differential sensing applications,’’ IEEE Trans. Microw. Theory Techn., vol. 64, no. 12, pp. 4362–4370, Dec. 2016. JONATHAN MUÑOZ-ENANO (Member, IEEE) [31] L. Su, J. Mata-Contreras, P. Vélez, A. Fernández-Prieto, and F. Martín, was born in Mollet del Vallès, Barcelona, Spain, ‘‘Analytical method to estimate the complex permittivity of oil Samples,’’ in 1994. He received the bachelor’s degree in elec- Sensors, vol. 18, no. 4, p. 984, 2018. tronic telecommunications engineering and the [32] P. Velez, K. Grenier, J. Mata-Contreras, D. Dubuc, and F. Martin, ‘‘Highly- master’s degree in telecommunications engineer- sensitive microwave sensors based on open complementary split ring ing from the Universitat Autonoma de Barcelona resonators (OCSRRs) for dielectric characterization and solute concen- tration measurement in liquids,’’ IEEE Access, vol. 6, pp. 48324–48338, (UAB), in 2016 and 2018, respectively, where he 2018. is currently pursuing the Ph.D. degree, which is [33] J. Yeo and J.-I. Lee, ‘‘High-sensitivity microwave sensor based on an focused on the development of microwave sensors interdigital-capacitor-shaped defected ground structure for permittivity based on metamaterials concepts for the dielectric characterization,’’ Sensors, vol. 19, no. 3, p. 498, Jan. 2019. characterization of materials and biosensors. [34] P. Vélez, J. Muñoz-Enano, M. Gil, J. Mata-Contreras, and F. Martín, ‘‘Dif- ferential microfluidic sensors based on dumbbell-shaped defect ground structures in microstrip technology: Analysis, optimization, and applica- PARIS VÉLEZ (Senior Member, IEEE) was born tions,’’ Sensors, vol. 19, no. 14, p. 3189, Jul. 2019. in Barcelona, Spain, in 1982. He received the [35] J. Munoz-Enano, P. Velez, C. Herrojo, M. Gil, and F. Martin, ‘‘On the degree in telecommunications engineering, spe- sensitivity of microwave sensors based on slot resonators and frequency cializing in electronics, and the Electronics Engi- variation,’’ in Proc. Int. Conf. Electromagn. Adv. Appl. (ICEAA), Granada, neering degree and the Ph.D. degree in electrical Spain, Sep. 2019, pp. 0112–0115. engineering from the Universitat Autònoma de [36] J. Muñoz-Enano, P. Vélez, M. Gil, E. Jose-Cunilleras, A. Bassols, and Barcelona, Barcelona, in 2008, 2010, and 2014, F. Martín, ‘‘Characterization of electrolyte content in urine samples respectively. His Ph.D. thesis concerned common through a differential microfluidic sensor based on dumbbell-shaped mode suppression differential microwave circuits defected ground structures,’’ Int. J. Microw. Wireless Technol., vol. 12, based on metamaterial concepts and semi-lumped no. 9, pp. 817–824, Nov. 2020. resonators. During his Ph.D., he was awarded with a pre-doctoral teaching [37] J. Muñoz-Enano, P. Vélez, M. Gil, and F. Martín, ‘‘Planar microwave resonant sensors: A review and recent developments,’’ Appl. Sci., vol. 10, and research fellowship by the Spanish Government, from 2011 to 2014. no. 7, p. 2615, 2020. From 2015 to 2017, he was involved in the subjects related to metamaterials [38] M. Makimoto and S. Yamashita, ‘‘Compact bandpass filters using stepped sensors for fluidics detection and characterization at LAAS-CNRS through impedance resonators,’’ Proc. IEEE, vol. 67, no. 1, pp. 16–19, Jan. 1979. a TECNIOSpring fellowship, co-founded by the Marie Curie Program. His [39] J. Naqui, M. Durán-Sindreu, J. Bonache, and F. Martín, ‘‘Implementation current research interests include the miniaturization of passive circuits of shunt connected series resonators through stepped-impedance shunt RF/microwave and sensors-based metamaterials through Juan de la Cierva stubs: Analysis and limitations,’’ IET Microw., Antennas Propag., vol. 5, Fellowship. He is a Reviewer of the IEEE TRANSACTIONS ON MICROWAVE pp. 1336–1342, Aug. 2011. THEORY AND TECHNIQUES and for other journals. VOLUME 9, 2021 150887 L. Su et al.: On Modeling of Microstrip Lines Loaded With DB-DGS and Folded DB-DGS Resonators JESÚS MARTEL (Senior Member, IEEE) received FERRAN MARTÍN (Fellow, IEEE) was born in the Licenciado and Ph.D. degrees in physics Barakaldo, Vizcaya, Spain, in 1965. He received from the University of Seville, Seville, Spain, in the B.S. degree in physics from the Universitat 1989 and 1996, respectively. Since 1990, he has Autònoma de Barcelona (UAB), in 1988, and the performed research with the Microwave Group, Ph.D. degree, in 1992. University of Seville. He is currently a Profes- From 1994 to 2006, he was an Associate sor with the Department of Applied Physics II, Professor of electronics with the Departament University of Seville, where he was the Head of d’Enginyeria Electrònica, UAB. Since 2007, the Department, from 2010 to 2018. His current he has been a Full Professor of electronics. research interests include the numerical analysis of In recent years, he has been involved in different planar transmission lines, the modeling of planar microstrip discontinuities, research activities, including modeling and simulation of electron devices the design of passive microwave circuits, the techniques in microwave mea- for high frequency applications, millimeter wave and THz generation sys- surements, and the proprieties of artificial media. tems, and the application of electromagnetic bandgaps to microwave and millimeter wave circuits. He is very active in the field of metamaterials and their application to the miniaturization and optimization of microwave circuits and antennas. Other topics of interests include microwave sensors and RFID systems, with special emphasis on the development of high data FRANCISCO MEDINA (Fellow, IEEE) was born capacity chipless-RFID tags. He is currently the Head of the Microwave in Puerto Real, Cádiz, Spain, in 1960. He received Engineering, Metamaterials and Antennas Group (GEMMA Group), UAB, the Licenciado (5-year) degree (summa cum laude) and the Director of CIMITEC, a Research Center on Metamaterials sup- in physics (electronics) and the Ph.D. degree (cum ported by TECNIO (Generalitat de Catalunya). He has organized several laude) in physics (microwaves) from the Univer- international events related to metamaterials and related topics, including sity of Seville, Seville, Spain, in 1983 and 1987, Workshops at the IEEE International Microwave Symposium (2005 and respectively. 2007) and the European Microwave Conference (2009, 2015, and 2017), and Afterwards, he spent one year with the group the Fifth International Congress on Advanced Electromagnetic Materials in headed by Prof. Henri Baudrand at ENSEEIHT, Microwaves and Optics (Metamaterials 2011), where he acted as the Chair Toulouse, France, as a Postdoctoral Researcher of the Local Organizing Committee. He has authored or coauthored over thanks to another competitive fellowship granted by the Spanish and French 650 technical conference papers, letter, journal articles, and book chapters. governments. Then he joined as an Assistant Professor, the group headed He is the coauthor of the book on Metamaterials entitled Metamaterials with by Prof. Manuel Horno at the Department of Electronics and Electromag- Negative Parameters: Theory, Design and Microwave Applications (John netism, Physics Faculty, University of Seville, in 1987, where he became an Wiley & Sons Inc.), an author of the book Artificial Transmission Lines for Associate Professor, in 1990, and then a Full Professor, in 2009. Since 1998, RF and Microwave Applications (John Wiley & Sons Inc.), a co-editor of he has been the Head of the Microwave Group, University of Seville. During the book Balanced Microwave Filters (Wiley/IEEE Press), and a coauthor of the last 38 years, he has been working on several topics related to applied the book Time-Domain Signature Barcodes for Chipless-RFID and Sensing electromagnetism and microwave engineering, including the modeling of Applications (Springer). He has generated 21 Ph.D., has filed several patents planar transmission lines, antennas and circuits using optimized integral on metamaterials, and has headed several Development Contracts. He has equation methods, (bi)anisotropic structures, metamaterials, and periodic acted as a guest editor for six special issues on metamaterials and sensors electromagnetic structures. He has also contributed to the design of a variety in five international journals. He is a fellow of the IET. He is a member of of novel planar passive devices. This research has mainly been reported in the IEEE Microwave Theory and Techniques Society (IEEE MTT-S). He is a co-edited book Balanced Microwave Filters (John Wiley & Sons Inc. and also a member of the Technical Committees of the European Microwave IEEE-Press 2018), a dozen book chapters, more than 165 JCR journal articles Conference (EuMC) and International Congress on Advanced Electromag- and more than 340 conference contributions. netic Materials in Microwaves and Optics (Metamaterials). He serves or has Dr. Medina was a recipient of a Research Fellowship of the Spanish served as a member of the Editorial Board for IET Microwaves, Antennas Government during his Ph.D. training period. He was a recipient of a and Propagation, International Journal of RF and Microwave Computer- Salvador de Madariaga Fellowship during a four months visiting scholar stay Aided Engineering, and Sensors. Among his distinctions, he has received with Prof. Yang Hao at QMUL, London, U.K. A few of these conference the 2006 Duran Farell Prize for Technological Research, holds the Parc de or journal contributions have received ‘‘best paper awards,’’ including the Recerca UAB–Santander Technology Transfer Chair. He was a recipient of recently awarded 2020 IEEE Microwave Magazine Best Paper Award. He is three ICREA ACADEMIA Awards (calls 2008, 2013, and 2018). He is a also the Editor-in-Chief of International Journal of Microwave and Wireless Reviewer of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES Technologies. He has also been a reviewer for more than 75 different and the IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, among many journals, mainly in the fields of microwave and antenna engineering, optics, other journals. and applied physics. He has also co-organized the 2017 IEEE-MTT NEMO Symposium, Seville, Spain, and collaborates as a TPC member or a reviewer with a number of major conferences in the fields of microwave engineering, antenna engineering, and electromagnetic theory. Since 2017, he has been the President of the Spanish URSI Committee. 150888 VOLUME 9, 2021