sensors Article Discussion and Analysis of Dumbbell Defect-Ground-Structure (DB-DGS) Resonators for Sensing Applications from a Circuit Theory Perspective Lijuan Su , Paris Vélez , Jonathan Muñoz-Enano and Ferran Martín * CIMITEC, Departament d’Enginyeria Electrònica, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain; Lijuan.Su@uab.cat (L.S.); Paris.Velez@uab.cat (P.V.); Jonatan.Munoz@uab.cat (J.M.-E.) * Correspondence: Ferran.Martin@uab.cat Abstract: Microstrip transmission lines loaded with dumbbell defect-ground-structure (DB-DGS) resonators transversally oriented have been exhaustively used in microwave circuits and sensors. Typically, these structures have been modelled by means of a parallel LC resonant tank series connected to the host line. However, the inductance and capacitance of such model do not have a physical meaning, since this model is inferred by transformation of a more realistic model, where the DB-DGS resonator, described by means of a resonant tank with inductance and capacitance related to the geometry of the DB-DGS, is magnetically coupled to the host line. From parameter extraction, the circuit parameters of both models are obtained by considering the DB-DGS covered with semi-infinite materials with different dielectric constant. The extracted parameters are coherent and reveal that   the general assumption of considering the simple LC resonant tank series-connected to the line to describe the DB-DGS-loaded line is reasonable with some caution. The implications on the sensitivity, Citation: Su, L.; Vélez, P.; Muñoz-Enano, J.; Martín, F. when the structure is devoted to operating as a permittivity sensor, are discussed. Discussion and Analysis of Dumbbell Defect-Ground-Structure (DB-DGS) Keywords: slot resonator; dumbbell defect-ground structure (DB-DGS); microwave sensor; parame- Resonators for Sensing Applications ter extraction; microstrip technology from a Circuit Theory Perspective. Sensors 2021, 21, 8334. https:// doi.org/10.3390/s21248334 1. Introduction Academic Editor: Qammer The dumbbell defect-ground-structure (DB-DGS) resonator [1,2] is a slot resonator Hussain Abbasi exhaustively used in microwave engineering for the implementation of circuits (mainly fil- ters [3–9]) and sensors [10–13]. DB-DGS resonators are typically combined with microstrip Received: 5 November 2021 transmission lines. The typical topology of a DB-DGS-loaded microstrip line is depicted Accepted: 9 December 2021 in Figure 1, where the resonant element is transversally etched in the ground plane. This Published: 13 December 2021 structure exhibits a notched response with a transmission zero at the fundamental reso- nance frequency of the DB-DGS resonator, and the structure has been typically modelled Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in by means of a parallel resonant tank series-connected to the host transmission line [1,2] published maps and institutional affil- (see Figure 1b). iations. The notched response of DB-DGS-loaded microstrip lines can be exploited for the implementation of notch and stopband filters [3–6], as well as for the design of common- mode filters in differential lines [7,9]. However, slot resonators, including the DB-DGS resonator, are also of interest for the implementation of planar microwave sensors devoted to permittivity measurements, and to the characterization of materials (including solids and Copyright: © 2021 by the authors. liquids) [10–13]. The reason is that the resonance frequency of these slot resonators is very Licensee MDPI, Basel, Switzerland. sensitive to the dielectric properties of the material surrounding the resonator (in contact or This article is an open access article distributed under the terms and in close proximity to it). Thus, the canonical output variable in these permittivity sensors conditions of the Creative Commons is the resonance frequency of the resonant element (and eventually the notch magnitude) Attribution (CC BY) license (https:// very sensitive to the permittivity of the so-called material under test (MUT). creativecommons.org/licenses/by/ 4.0/). Sensors 2021, 21, 8334. https://doi.org/10.3390/s21248334 https://www.mdpi.com/journal/sensors Sensors 2021, 21, x FOR PEER REVIEW 2 of 12 Sensors 2021, 21, 8334 2 of 12 la la Z0, βla Lc’ Z0, βla P1 P2 P1 P2 Cc’ (a) (b) Figure 1. Typical Figure topology 1. Typical (a) and (a) topology most commonly and accepted accepted most commonly circuit model (b) model circuit of a microstrip (b) of a trans- microstrip mission line loadedline transmission with a transversally loaded oriented DB-DGS with a transversally oriented resonator, etched inetched DB-DGS resonator, the ground in theplane. ground Lc′plane. and CLc′0 are related 0 to the inductance and capacitance, respectively, of the DB-DGS resonator, c and Cc are related to the inductance and capacitance, respectively, of the DB-DGS resonator, whereas the transmission line (in black color) is described by the pair of transmission line sections whereas the transmission line (in black color) is described by the pair of transmission line sections with characteristic impedance Z0, length la, and phase constant β. The ground plane is depicted in with characteristic impedance Z , length l , and phase constant β. The ground plane is depicted in grey color. Losses are neglected in this0 model. a grey color. Losses are neglected in this model. For both circuit and sensor design, accurate circuit models of the structures under For both circuit and sensor design, accurate circuit models of the structures under consideration consideration (able(able to linkto the linkgeometry the geometry parameters to thetolumped parameters the lumped elements of theofmodel) elements the model) are very convenient. As mentioned, the most accepted circuit are very convenient. As mentioned, the most accepted circuit model of a DB-DGS-loadedmodel of a DB-DGS-loaded microstrip line isline microstrip theisone thedepicted one depicted in Figure 1b, where in Figure the DB-DGS 1b, where the DB-DGS resonator is modelled resonator is modelled by means of the parallel resonant tank, series connected by means of the parallel resonant tank, series connected to the line. It hasto the line. It has been argued beenthatargued this model that this accounts modelfor the return accounts for current the return path in the ground current path in plane, the groundeitherplane, across either the nar- across row slot of the DB-DGS resonator (providing the capacitive effect), the narrow slot of the DB-DGS resonator (providing the capacitive effect), or surrounding or surrounding the apertures (inductive the apertures effect) [1,2,9]. (inductive effect)Thus, according [1,2,9]. to this physical Thus, according to this interpretation, the par- the physical interpretation, allel parallel combination combination of the inductance and the capacitance of the DB-DGS, as of the inductance and the capacitance of the DB-DGS, as well as the well as connection the connection of this parallel LC tank in series with the line, is clear. However,that of this parallel LC tank in series with the line, is clear. However, the fact the fact the resonator is etched inisthe that the resonator ground etched in theplane generates ground planesome doubtssome generates concerning doubtsthe validity the concerning of thevalidity above-mentioned physical interpretation of the above-mentioned of the modelofparameters. physical interpretation Indeed, in mi- the model parameters. Indeed, crostrip-to-slot-line transitions, in microstrip-to-slot-line it has been transitions, reported it has that the that been reported excitation of the slot the excitation lineslot of the is line due to magnetic coupling [14,15]. In the structure of Figure 1, rather is due to magnetic coupling [14,15]. In the structure of Figure 1, rather than a microstrip-to-than a microstrip-to- slot-line transition, slot-line a microstrip transition, line excites a microstrip a slotaresonator line excites with with slot resonator apertures in theinextremes, apertures the extremes, i.e., ai.e., DB-DGS a DB-DGSresonator. However, resonator. However, it is apparent that that it is apparent the driving the driving mechanism mechanism should be be should the same. Indeed, the same. in previous Indeed, paperspapers in previous [16,17],[16,17], a different model of a different the DB-DGS-loaded model of the DB-DGS-loaded mi- crostrip line, which microstrip line,accounts for the magnetic which accounts couplingcoupling for the magnetic betweenbetween the host line and the the host lineres- and the onantresonant element,element, has been hasreported. Actually, been reported. the simple Actually, modelmodel the simple of Figure 1 can1be of Figure canderived be derived fromfromsuch such model by element model by elementtransformation. transformation. However, However,the interpretation the interpretation of theofelements the elements of theofparallel the parallel resonant resonant tanktank of Figure of Figure 1, those 1, as as those describing describing thethe capacitanceand capacitance andinduct- inductance ance of of the theDB-DGS DB-DGSresonator, resonator,does doesnot notseem seemtotobebecorrect corrector,or,atat least, least,accurate. accurate. InIn this paper, this paper, allall these these aspects aspects areare discussed discussed in in detail, detail, andandthethe implications implications onon thethefunctionality functionality of the of theDB-DGS-loaded DB-DGS-loadedline lineasaspermittivity permittivitysensor sensorare areanalyzed. analyzed. 2. Model 2. Model and Analysis and Analysis Any Any planar planar microwave microwave resonator, resonator, including including both both metallic metallic and slot-based and slot-based resonators, resonators, can becan be modelled modelled by means by means of a LCofresonant a LC resonant tankvicinity tank in the in the ofvicinity of the fundamental the fundamental reso- resonance. Additionally, if the considered resonator is electrically nance. Additionally, if the considered resonator is electrically small, then the small, then the LC LC resonant tank resonant provides tank provides a good a good description of description the resonantofelement the resonant element broad in a relatively in a relatively band thatbroad band that extends significantly above the fundamental resonance. The extends significantly above the fundamental resonance. The DB-DGS resonator is the DB-DGS resonator is the complementary, or dual, counterpart of the so-called step-impedance-resonator complementary, or dual, counterpart of the so-called step-impedance-resonator (SIR) (SIR) [18,19]. Thus, likewise the SIR, the DB-DGS resonator can be considered an electrically [18,19]. Thus, likewise the SIR, the DB-DGS resonator can be considered an electrically small resonator, by virtue of the high contrast between the width of the narrow (central) small resonator, by virtue of the high contrast between the width of the narrow (central) slot and the apertures at the extremes. Whereas in the SIR the inductance is mainly due slot and the apertures at the extremes. Whereas in the SIR the inductance is mainly due to to the narrow strip and the capacitance to the metallic patches at the extremes [18,19], it Sensors 2021, 21, x FOR PEER REVIEW 3 of 12 Sensors 2021, 21, 8334 3 of 12 the narrow strip and the capacitance to the metallic patches at the extremes [18,19], it fol- lows from duality considerations [20–22] that the narrow slot of the DB-DGS resonator determines its capacitance, whilst the inductance is mainly determined by the size of the follows from duality considerations [20–22] that the narrow slot of the DB-DGS resonator apertures. However, in the model depicted in Figure 1b, the reactive element values Lc′ determines its capacitance, whilst the inductance is mainly determined by the size of the and Cc′ are not the inductance and capacitance, respectively, of the DB-DGS resonator. 0 apertures. However, in the model depicted in Figure 1b, the reactive element values Lc Nevertheless, and Cc 0 areasnot it will be shown later, the inductance Lc′ and Cc′ are and capacitance, related to the respectively, inductance of the DB-DGS Lresonator. c and capacitance Cc, respectively, of the DB-DGS resonator. Nevertheless, as it will be shown later, Lc 0 and Cc 0 are related to the inductance Lc and At the fundamental capacitance resonance Cc , respectively, frequency, of the a DB-DGS resonator exhibits a magnetic DB-DGS resonator. wall at itsAtsymmetry plane (the the fundamental axis of the resonance microstrip frequency, a line DB-DGSin Figure 1a). Consequently, resonator exhibits a magneticthe fundamental resonance of the DB-DGS resonator can be driven by wall at its symmetry plane (the axis of the microstrip line in Figure 1a). Consequently, means of a time-vary- ing the magnetic field with fundamental a non-negligible resonance of the DB-DGScomponent in the resonator canplane of thebyparticle, be driven means in of athetime- direction orthogonal to the magnetic wall. It is obvious that the magnetic varying magnetic field with a non-negligible component in the plane of the particle, in the field generated by the line satisfies direction orthogonalsuchto requirement. the magnetic Therefore, wall. It isthe line present obvious that theon top of the magnetic fieldDB-DGS generated can bymagnetically excite such the line satisfies the resonant requirement.element. Thus, athe Therefore, more linerealistic presentcircuit on topmodel of theof the DB-DGS DB-DGS-loaded microstrip transmission line is the one depicted can magnetically excite the resonant element. Thus, a more realistic circuit model of in Figure 2a [16,17]. In the thisDB-DGS-loaded model, Lc and Cc microstrip describe thetransmission DB-DGS resonator, as indicated before. Particularly, line is the one depicted in Figure 2a [16,17]. In the capacitance this model,Cc isLcdetermined and Cc describe by thethelength DB-DGSand resonator, width of the as narrow indicated (central) before.slot, whereas the Particularly, the capacitance size of the apertures Cc is determined by the length and width of the narrow (central) slot,Lwhereas is the main parameter determining the inductance value c. In orderthetosize describe of thethe magnetic apertures is coupling the mainbetween parameter thedetermining microstrip line theand the resonator, inductance valuewe Lc . In should ordermodel a portion to describe the of the linecoupling magnetic (with lengthbetweenl, where l < la) by means the microstrip line andof its theinductance resonator, we L and capacitance should model aC.portion By thisofmeans, the linewe can length (with define l,a where mutuall < inductance la ) by means M, of describing the L its inductance magnetic coupling between and capacitance C. By this themeans, line and wethecanDB-DGS define aresonator. In coherence mutual inductance M,with Figure the describing 1, the length of magnetic the transmission coupling between the line sections line and theinDB-DGS Figure 2resonator. is simply In lb =coherence la − l/2. with Figure 1, the length of the transmission line sections in Figure 2 is simply lb = la − l/2. (a) (b) Figure 2. Circuit Figure modelmodel 2. Circuit of the of DB-DGS-loaded microstrip the DB-DGS-loaded line with microstrip thewith line resonator magnetically the resonator cou- magnetically pledcoupled to the line (a) and to the transformed line (a) model (b). and transformed model (b). TheThe series branch series in the branch in circuit of Figure the circuit 2a can of Figure be transformed 2a can be transformed to a to parallel resonator a parallel resonator series-connected to an series-connected toinductor (Figure an inductor 2b).2b). (Figure TheThe elements of the elements series of the branch series in the branch in circuit the circuit of Figure 2b are of Figure related 2b are to the related elements to the of the elements original of the circuit original of Figure circuit 2a through of Figure 2a through the the following equations following [17,22] equations [17,22] L0 = L − L0 c (1) 𝐿 =𝐿−𝐿 (1) Lc 0 = Cc M2 ω02 (2) 𝐿 ′ = 𝐶 𝑀 ωLc (2) Cc 0 = 2 2 (3) M ω0 𝐿 𝐶 ′= (3) where ω02 = 1/Lc Cc = 1/Lc 0 Cc 0 (ω0 being 𝑀 ω the angular frequency at the fundamental resonance of the DB-DGS resonator). If we compare the circuit of Figure 2b with the one where 𝜔 = 1/𝐿 𝐶 = 1/𝐿 ′𝐶 ′ (𝜔 being the angular frequency at 0the fundamental reso- in Figure 1b, the difference is the presence of the inductance L and the capacitance C in nance of the DB-DGS resonator). If we compare the circuit of Figure 2b with the one in the circuit of Figure 2b, not present in the circuit of Figure 1b. However, as it will be later Figure 1b, the difference is the presence of the inductance L′ and the capacitance C in the demonstrated, L0 and C describe the transmission line section of length l. Such line section, circuit of Figure 2b, not present in the circuit of Figure 1b. However, as it will be later plus the lines of length lb in the circuit of Figure 2b, give the total line length la of the circuit demonstrated, L′ and C describe the transmission line section of length l. Such line section, of Figure 1b. According to the preceding paragraph, the circuits of Figures 1b and 2b or Figure 2a, are effectively equivalent, and, therefore, it is licit to consider valid the circuit of Figure 1b. However, in the circuit of Figure 1b, the reactive elements Lc 0 and Cc 0 do not correspond Sensors 2021, 21, 8334 4 of 12 to the inductance and capacitance of the DB-DGS resonator, as anticipated before. Never- theless, the reactive elements of the resonators in both models (Figure 2a,b) are intimately related. By introducing ω02 = 1/Lc Cc in Equations (2) and (3) one obtains: M2 Lc 0 = (4) Lc L2c Cc 0 = Cc (5) M2 providing the relationship between the resonator elements in both models. Note, however, that there is a fundamental difference between the models of Figure 2a,b. Whereas the series branch in the model of Figure 2a depends on four reactive parameters (L, M, Lc and Cc ), the series branch in the circuit model of Figure 2b, equivalent to Figure 1b, contains only three reactive elements (L0 , Lc 0 and Cc 0 ). Thus, from the information of the reactive parameters of the circuit of Figure 2b, easily obtained from parameter extraction [22–24], it is not possible to obtain M, Lc and Cc . By contrast, L can be easily isolated from Equation (1). In order to independently obtain M, Lc and Cc from Lc 0 and Cc 0 , at least one of the three reactive parameters M, Lc or Cc must be determined from an independent method. Let us discuss this aspect in the next section, where the implications of the transformation Equations (1)–(3) or Equations (4) and (5), on the functionality of the DB-DGS-loaded line as permittivity sensor are discussed. 3. Parametric Analysis and Implications for Permittivity Sensing Let us consider as case study a microstrip transmission line loaded with a DB-DGS resonator with the dimensions depicted in the caption of Figure 3, where the specific topology is depicted. The width of the line (W = 1.1 mm) provides a 50-Ω characteristic impedance with the considered substrate parameters (dielectric constant εr = 10.2 and thickness h = 1.27 mm). The response of this structure inferred from electromagnetic simulation is depicted in Figure 3b. From this response, we have extracted the parameters of the circuit of Figure 2b, i.e., L0 = 5.12 nH, Lc 0 = 4.10 nH, Cc 0 = 3.63 pF and C = 2.21 pF (the parameter extraction method reported in [22–24] has been used). Actually, such parameters correspond to the DB-DGS coupled to a line section of length l = 12 mm (note that the obtained values of L0 and C are compatible with a 50-Ω characteristic impedance line of the considered length, i.e., 12 mm). With such parameters, the circuit response is found to be roughly undistinguishable from the electromagnetic response in the considered frequency range. However, as mentioned, it is not possible to infer M, Lc and Cc by means of Equations (1)–(3), or Equations (4) and (5). In the circuit of Figure 2a, only the capacitance C, invariable as compared to the one of Figure 2b, and L (= L0 + Lc 0 = 9.21 nH), can be univocally determined. In order to be able to obtain the circuit parameters of the original model of Figure 2a, we have considered an auxiliary structure, i.e., a microstrip line loaded with an identical DB-DGS resonator, but folded and rotated 90◦ , as depicted in Figure 4 (this structure can also be useful for permittivity sensing). In this structure, the magnetic coupling cancels because the symmetry plane of the particle, a magnetic wall, is orthogonally oriented with regard to the line axis. However, there is an electric dipole in the vertical direction, since the resonator is folded [17]. Therefore, with such orientation and folding, the DB-DGS resonator is excited exclusively by the electric field generated by the line, with a significant component in the vertical direction. The circuit model for this structure is thus the one shown in Figure 5. The parameters of this model can be easily extracted (the method reported in [23] has been applied), and the results are as follows: L” = 9.27 nH, C” = 2.07 pF, Cc ” = 4.23 pF and Lc ” = 3.68 nH (the electromagnetically simulated and circuit response are depicted in Figure 4b). As it can be appreciated, L” ≈ L and C” ≈ C, i.e., the parameters describing the line section of length l do not appreciable change by rotating and folding the DB-DGS resonator. In this paper, we do assume that the capacitance Cc of the unfolded Sensors 2021, 21, 8334 5 of 12 DB-DGS does not change as compared to the one of the folded particles, Cc ”. This latter hypothesis is reasonable, since the capacitance of the resonator mainly depends on the length and width of the narrow (central) slot and it is not expected that such capacitance varies significantly by rotating and folding the DB-DGS. By considering that Cc ” = Cc , the remaining parameters of the circuit of Figure 2a, i.e., M and Lc , can be perfectly determined from inversion of Equations (4) and (5), or Equations (1)–(3), and the results are M = 3.80 nH and Lc = 3.52 nH. Let us clarify that in the circuit of Figure 5, the capacitance Cc ” can be univocally determined from parameter extraction. However, from the extracted parameters of the circuit of Figure 2b, it is not possible through inversion of Equations (4) and (5) or Equations (1)–(3), to univocally determining the elements of the series branch in the circuit of Figure 2a, particularly M, Lc and Cc . It is neither possible to infer these elements Sensors 2021, 21, x FOR PEER REVIEWfrom direct parameter extraction, since, given a response, for example the one of Figure 3b, 5 of 12 corresponding to the structure of Figure 3a, there are many combinations of the triad M, Lc and Cc providing such response, as inspection of Equations (1)–(3) and Equations (4) and (5) reveals. (a) (b) Sensors 2021, 21, x FOR PEER REVIEW Figure 6 of 12 Figure3.3.Layout Layout(a) (a)and andfrequency frequency response (b) (b) of ofthe theconsidered consideredDB-DGS. DB-DGS. DB-DGS DB-DGS dimensions dimensions are: are: length length of of thethe narrow narrow slot slot 2l2l==24 24mm, mm,width width of of the the slot slot 0.2 0.2 mm, mm,side sidelength lengthofofthe thesquare squareapertures apertures 1212mm. The electromagnetic simulations are lossless. mm. The electromagnetic simulations are lossless. In order to be able to obtain the circuit parameters of the original model of Figure 2a, we have considered an auxiliary structure, i.e., a microstrip line loaded with an identical DB-DGS resonator, but folded and rotated 90°, as depicted in Figure 4 (this structure can also be useful for permittivity sensing). In this structure, the magnetic coupling cancels because the symmetry plane of the particle, a magnetic wall, is orthogonally oriented with regard to the line axis. However, there is an electric dipole in the vertical direction, since the resonator is folded [17]. Therefore, with such orientation and folding, the DB-DGS resonator is excited exclusively by the electric field generated by the line, with a significant component in the vertical direction. The circuit model for this structure is thus the one shown in Figure 5. The parameters of this model can be easily extracted (the method re- ported in [23] has been applied), and the results are as follows: L″ = 9.27 nH, C″ = 2.07 pF, Cc″ = 4.23 pF and Lc″ = 3.68 nH (the electromagnetically simulated and circuit response are depicted in Figure (a)4b). As it can be appreciated, L″ ≈ L and C″ (b) ≈ C, i.e., the parameters describing Figure the line Figure 4.4. Layout Layout (a)section (a) and of length andlossless lossless l do not frequency frequency appreciable response response (b) change (b)ofofthe the folded foldedbyandrotating and 90° and DB-DGS. 90◦rotated rotated folding DB-DGS. the DB-DGS Dimensions resonator. are identical In to this those paper, of Dimensions are identical to those of Figure 3. we Figure do 3. assume that the capacitance C c of the unfolded DB-DGS does not change as compared to the one of the folded particles, Cc″. This latter hypothesis Note is reasonable, that the inductance since ofthethe capacitance rotated and of folded the resonator resonator mainly variesdepends slightly, on the as com- length and width pared to the one of the narrow (central) slot and it is not expected that such unfolded and transversally oriented DB-DGS, i.e., Lc ” 6= Lc . This is capacitance varies significantly attributed to the fact bythat rotating and folding the current the DB-DGS. path from the innerBy considering to the that Cregion outer metallic c″ = Ccof , the the remaining parameters folded particle is veryofdifferent the circuit of Figure from 2a, i.e., M the inductive and Lpath current c, canin bethe perfectly unfolded determined resonator. Moreover, from inversionthe ofinfluence Equations of the (4) line andstrip onEquations (5), or the inductance of and (1)–(3), the DB-DGS the resultsis more are Msevere = 3.80if nHtheand particle is folded Lc = 3.52 nH. Let andusoriented as shown clarify that in thein Figure circuit of4a. Figure 5, the capacitance Cc″ can be univocally determined from parameter extraction. However, from the extracted pa- rameters of the circuit of Figure 2b, it is not possible through inversion of Equations (4) Figure 5. Circuit model of the microstrip line loaded with the folded and rotated (90°) DB-DGS and (5) or Equations (1)–(3), to univocally determining the elements of the series branch resonator. in the circuit of Figure 2a, particularly M, Lc and Cc. It is neither possible to infer these (a) (b) Sensors 2021, 21, 8334 6 of 12 Figure 4. Layout (a) and lossless frequency response (b) of the folded and 90° rotated DB-DGS. Dimensions are identical to those of Figure 3. Circuit model model of of the the microstrip microstrip line line loaded loaded with with the the folded folded and and rotated rotated (90 ◦ DB-DGS Figure 5. Figure 5. Circuit (90°)) DB-DGS resonator. resonator. To gain To gain insight insight on onthe thebehavior behaviorofofthe thereactive reactive elements elements when when dimensions dimensions areare modified, modi- we have considered variations in the length of the narrow fied, we have considered variations in the length of the narrow slot of the DB-DGS slot of the DB-DGS (keeping (keep- the the ing other dimensions other dimensions unaltered). unaltered). This Thisshould shouldmainly mainly(although (although not not exclusively) affect exclusively) affect the capacitance of the resonant particle, and it is reasonable to consider the capacitance of the resonant particle, and it is reasonable to consider the effects of var- the effects of varying such dimension. The reason is that in the functionally of the structure under ying such dimension. The reason is that in the functionally of the structure under consid- consideration as permittivity sensor, the material under test (MUT), placed on top of the eration as permittivity sensor, the material under test (MUT), placed on top of the DB- DB-DGS, modifies the capacitance of the particle. Nevertheless, it should be mentioned that DGS, modifies the capacitance of the particle. Nevertheless, it should be mentioned that it is also expected that the inductance of the particle, L , as well as the mutual inductance, it is also expected that the inductance of the particle, Lcc, as well as the mutual inductance, M, experience a certain variation by varying the length of the narrow slot of the DB-DGS M, experience a certain variation by varying the length of the narrow slot of the DB-DGS resonators. By contrast, such inductive elements are not altered by the presence of a resonators. By contrast, such inductive elements are not altered by the presence of a die- dielectric material (MUT) on top of the DB-DGS. lectric material (MUT) on top of the DB-DGS. Figure 6 depicts the electromagnetically simulated responses that are obtained by Figure 6 depicts the electromagnetically simulated responses that are obtained by varying the slot length in a quantity ∆l as compared to the nominal value. As expected, varying the slot length in a quantity Δl as compared to the nominal value. As expected, the resonance frequency decreases with the slot length. The extracted parameters of the the resonance frequency decreases with the slot length. The extracted parameters of the circuit of Figure 2b are shown in Table 1. The circuit responses inferred from the extracted circuit of Figure parameters 2b are for each shown case in Table are also 1. Theincircuit depicted Figureresponses inferred from 6 for comparison the extracted purposes, and the parameters for each case are also depicted in Figure 6 for comparison agreement is good in all the cases. By applying the previous procedure, the parameters purposes, and theof agreement the model of is Figure good in2aallhave the cases. By applying been obtained, and the the previous results areprocedure, also depicted the in parameters Table 1. Theof the model ofofFigure capacitance 2a have been the resonator, obtained, Cc , and and the results the transformed are also capacitance, Cc 0depicted in Table , both increase with1. The capacitance the slot length, asofexpected, the resonator, Cc, and but it can the that be seen transformed Lc , Lc 0 and capacitance, M experience Cc′,an both increase appreciable with the slot length, as expected, but it can be seen that L c, Lc′ and M experience variation, as well. The important aspect of this analysis is that Cc and Cc increase with 0 an appre- the ciable variation, as well. The important aspect of this analysis is that slot length. However, the capacitance that has a physical interpretation, i.e., the one of the C c and Cc′ increase with DB-DGS the slot length.(attributed resonator However,to the thecapacitance narrow slot), thatishas a physical Cc , rather thaninterpretation, Cc 0 . i.e., the one of the DB-DGS resonator (attributed to the narrow slot), is Cc, rather than Cc′. Table 1. Circuit parameters of the circuits of Figure 2 for different lengths of the narrow slot of the DB-DGS resonator. ∆l (mm) L0 /L (nH) Lc 0 /Lc (nH) Cc 0 /Cc (pF) M (nH) −2 4.96/8.73 3.77/3.15 3.28/3.93 3.45 −1 5.06/9.02 3.96/3.34 3.44/4.08 3.64 0 5.12/9.21 4.10/3.52 3.63/4.23 3.80 1 5.26/9.54 4.27/3.67 3.77/4.39 3.96 2 5.18/9.54 4.36/3.87 4.01/4.52 4.11 3 5.29/9.80 4.51/4.02 4.17/4.69 4.26 21, 21, x FOR PEER REVIEW 7 of 12 Sensors 2021, 21, 8334 7 of 12 (a) (b) Figure 6. Responses (circuit Figure and electromagnetic) 6. Responses (circuit andofelectromagnetic) the DB-DGS-loaded of themicrostrip line for microstrip DB-DGS-loaded various line for various lengths of the narrow lengths of the narrow slot of the DB-DGS. (a) Magnitude of the transmission magni- slot of the DB-DGS. (a) Magnitude of the transmission coefficient; (b) coefficient; (b) magnitude tude of the reflection coefficient. of the Minus reflection coefficient. Δl means sign inMinus signreducing the reducing in ∆l means length of the the length narrowofslot. the narrow slot. Table 1. Circuit parameters of the Let us circuits next of Figure consider 2 for different the effects lengths of varying theof the narrow slot capacitance of the of the DB-DGS resonator, DB-DGS resonatorbut. rather than by tailoring particle dimensions (narrow slot length), by covering the DB- DGS by means of a dielectric MUT with semi-infinite length in the vertical direction, and Δl (mm) L′/L (nH) Lc′/Lc (nH) Cc′/Cc (pF) M (nH) with lateral dimensions extending far beyond the boundaries of the DB-DGS (semi-infinite −2 4.96/8.73 3.77/3.15 3.28/3.93 3.45 approximation). This approximation, which means that the electric field generated by −1 5.06/9.02 3.96/3.34 3.44/4.08 3.64 the DB-DGS does not reach the limits of the MUT, is not a sensor requirement. However, 0 it 5.12/9.21 eases the analysis, to4.10/3.52 be carried out later, and 3.63/4.23 does not represent 3.80 a loss of generality. 1 5.26/9.54 4.27/3.67 The responses of the DB-DGS-loaded line with nominal dimensions3.77/4.39 3.96(i.e., with narrow slot 2 5.18/9.54 4.36/3.87 4.01/4.52 length 2l = 24 mm) for MUTs with various dielectric constant are depicted 4.11 in Figure 7a. By 3 5.29/9.80the DB-DGS with covering 4.51/4.02 a dielectric material, 4.17/4.69 Cc should change, 4.26but not Lc and M, as indicated before. Consequently, according to Equations (4) and (5), Cc 0 should vary with Let us next consider the effects the dielectric of varying constant of the MUT, but not Lc 0of the capacitance . Ifthe weDB-DGS infer theresonator, circuit responses for the but rather than by tailoring different particle constant dielectric dimensions (narrow of the slot length), MUT using by covering the different valuesthe Cc 0 that are required of DB- DGS by means oftoaadjust dielectric the MUT with semi-infinite resonance frequency to length in the vertical the simulated direction,the value (keeping andother parameters invariable), with lateral dimensions the agreement extending far beyond with thethe simulations boundaries is very of the The values of Cc 0 and Cc good.(semi-infi- DB-DGS for the nite approximation). different This dielectricwhich approximation, constants means arethat shown in Tablefield the electric 2. Let us call Cby generated c,air the value of Cc the DB-DGS doescorresponding not reach the limitsto the of DB-DGS the MUT, surrounded by airrequirement. is not a sensor (with dielectric constant εMUT = 1). For However, a semi-infinite it eases the analysis, to be carriedMUT withand out later, arbitrary does not dielectric represent constant, a loss ofthe capacitance generality. The of the resonator should vary according to [25,26] responses of the DB-DGS-loaded line with nominal dimensions (i.e., with narrow slot length 2l = 24 mm) for MUTs with various dielectric constant are depicted in Figure 7a. By ε r,eq + ε MUT covering the DB-DGS with a dielectric material, C Ccc should = Cc,airchange, but not Lc and M, as (6) ε +1 indicated before. Consequently, according to Equations (4) andr,eq (5), Cc′ should vary with the dielectric constant where of ε r,eqthe MUT, is the but notequivalent so-called Lc′. If we dielectric infer the circuit constant responses for the defined in [26], of the substrate, different dielectric constant which takesofinto the account MUT using the the different finite values thickness of the of Csubstrate. c′ that are required Namely, if the substrate Sensors 2021, 21, 8334 8 of 12 satisfies the semi-infinite approximation, the capacitance Cc is given by (6) with ε r,eq replaced with ε r , the substrate dielectric constant [25,26]. However, if the field lines generated in the slot of the DB-DGS resonator reach the opposite substrate interface, then the capacitance should be calculated according to (6). Equation (6) assumes that the capacitance of the resonator Cc is given by the parallel combination of the capacitances of the upper (MUT) and lower (substrate) subspaces of the slot, a hypothesis that is valid provided there is a quasi-magnetic wall in the plane of the DB-DGS resonator. In [26], it was demonstrated that for any reasonable combination of substrate thickness, substrate dielectric constant, and slot width, it can be considered that the plane of the particle is a magnetic wall to a very good approximation, thereby pointing out the validity of (6). The physical interpretation of ε r,eq is clear, namely, it is the value of the dielectric constant that a hypothetical semi-infinite substrate should exhibit in order to provide a contribution to the capacitance of the particle identical to the one of the actual (finite) substrate. The method for the calculation of ε r,eq is simple. Let us call f 0,air and f 0,MUT the resonance frequencies of the DB-DGS resonator when it is unloaded and loaded with a certain (semi-infinite) MUT, respectively. The ratio of these frequencies is s f 0,air ε r,eq + ε MUT = (7) f 0,MUT ε r,eq + 1 Provided the inductance of the DB-DGS resonator does not vary with the presence of the MUT. Since these frequencies can be easily obtained from electromagnetic simulation, it follows that ε r,eq can be inferred from (7). The calculated value is ε r,eq = 6.30. Obviously, ε r,eq does not depend on the MUT. It only depends on the characteristics of the substrate and slot width. Thus, the value of ε r,eq should be the same, regardless of the considered value of the dielectric constant of the MUT, ε MUT . This aspect has been corroborated by considered different ε MUT values, providing roughly identical values of ε r,eq (not shown). Indeed, the value of ε r,eq given above is an average value, but the standard deviation is very small. The capacitance values given by (6) are also included in Table 2 and are very close to the values of Cc inferred by inversion of (4–5) using the values of Cc 0 adjusted to fit the resonance values of the different MUTs. The circuit responses with the values of Cc calculated by means of (6) are compared to the simulated responses in Figure 7b, and the agreement is also very good. From (4–5), it follows that Cc 0 should exhibit a dependence with εMUT given by L2c ε r,eq + ε MUT Cc 0 = 2 Cc,air (8) M ε r,eq + 1 Thus, if we call Cc 0 air = Cc,air L2c /M2 , it is clear that the capacitance of the model of Figure 2b has a dependence on εMUT identical to the one of the capacitance of the circuit of Figure 2a, the actual capacitance of the DB-DGS resonator. Moreover, the resonance frequency is identical in both models. Therefore, it is obvious that the sensitivity does not depend on whether the circuit model of Figure 2b or the more realistic model of Figure 2a is considered. Sensors 2021, 21, 8334 9 of 12 Table 2. Circuit parameters for different values of the dielectric constant of the MUT. εMUT Cc 0 (pF) Cc (pF) Cc (pF) [From (6)] 1 3.63 4.23 4.22 2 4.14 4.81 4.81 3 4.63 5.39 5.39 4 5.13 5.97 5.97 5 5.65 6.57 6.55 6 6.13 7.13 7.13 7 6.59 7.67 7.71 8 7.12 8.29 8.29 9 7.58 8.82 8.87 10 8.11 9.44 9.45 11 8.61 10.02 10.02 To gain more insight on this latter aspect, let us calculate the sensitivity, namely, the variation experienced by the output variable, the resonance frequency, with the input variable, the dielectric constant of the MUT, i.e., d f0 S = (9) dε MUT If the model of Figure 2b is considered, the previous expression can be decomposed in two terms as follows d f 0 dCc 0 S = (10) dCc 0 dε MUT where the first and second terms are given by d f0 1 f0 0 = −2· 0 (11) dCc Cc and dCc 0 Cc 0 air = (12) dε MUT εr + 1 respectively. Introducing (11) and (12) in (10), and using (8), the sensitivity is found to be 1 f0 S = − · (13) 2 ε r,eq + ε MUT Alternatively, if the model of Figure 2a is the considered one, the sensitivity is also given by (10), with Cc 0 replaced with Cc . Equations (11) and (12) are also valid with Cc 0 replaced with Cc , and Cc 0 air replaced with Cc,air . Note, however, that the resulting sensi- tivity is also given by (13), since the resonance frequency is identical in both models (see Equations (1)–(3). Therefore, according to this analysis, it can be unequivocally concluded that despite the fact that the capacitance Cc 0 does not correspond to the capacitance of the DB-DGS resonator, using the model of Figure 2b for design purposes, as usually done with an eye towards simplification, is perfectly licit. The sensitivity given by (13) is compared with the sensitivity inferred from the simu- lated data (resonances) of Figure 7, and the agreement is very good, as shown in Figure 8. This points out the validity of the previous sensitivity analysis. Note, however, that in- ferring the sensitivity from experimental results is difficult due to the lack of data points. Namely, the number of potential samples with different dielectric constant is limited. Nev- ertheless, we have experimentally validated the reported circuit models by covering the sensor structure of Figure 3 with various semi-infinite materials, particularly uncladded substrates available in our laboratory. Figure 9 depicts the response for such materials (in- dicated), and the agreement between experimental, circuit and electromagnetic simulation is good. Small discrepancies are due to fabrication related tolerances, and to the fact that Sensors 2021, 21, x FOR PEER REVIEW 10 of 12 Sensors Sensors2021, 2021,21, 21,x xFOR FORPEER PEERREVIEW REVIEW 1010ofof1212 Nevertheless, we have experimentally validated the reported circuit models by covering Sensors 2021, 21, 8334 Nevertheless, Nevertheless, the we sensor structure wehave have experimentally experimentally of Figure validated validated 3 with various the thereported reported semi-infinite circuit models modelsby bycovering circuitparticularly materials, covering unclad- 10 of 12 the ded sensor thesubstrates structure sensor structure of Figure of Figure available 3 with various 3 with various in our laboratory. semi-infinite Figuresemi-infinite 9 depicts thematerials, materials, particularly responseparticularly unclad- unclad- for such materials ded dedsubstrates (indicated), andavailable substrates available ininour the agreement ourlaboratory. laboratory. between Figure Figure99depicts experimental,depicts the theresponse circuit response and for forsuch suchmaterials electromagnetic materials simu- (indicated), (indicated), and and the theagreement agreement between between experimental, experimental, circuit circuit and and electromagnetic electromagnetic lation is good. Small discrepancies are due to fabrication related tolerances, and to the simu- simu- fact the lation samples lation is good. is good.are actually Small Small stacked discrepancies discrepancies slabs, are are with due due to tocertain air fabrication fabricationgap effect related related between tolerances, tolerances, that the samples are actually stacked slabs, with certain air gap effect between them (there them and and (there to tothe the are fact fact not that that thick the the enough samples samples aresubstrates actually are actually so as stacked to stacked consider slabs, slabs,withthem with semi-infinite). certain certain are not thick enough substrates so as to consider them semi-infinite). air gap air gap effect between effect between them them (there (there are arenot notthick thickenough enoughsubstrates substratessosoasastotoconsider considerthem themsemi-infinite). semi-infinite). Responses(circuit Figure7.7.Responses Figure (circuitand andelectromagnetic) electromagnetic)ofofthe theDB-DGS-loaded DB-DGS-loadedmicrostrip microstripline linefor forvarious various Figure Figure7.7.Responses dielectric dielectric constants Responses constants of(circuit ofthethe (circuitand MUT. electromagnetic) and(a) MUT. (a) Circuit electromagnetic) Circuit ofofthe responses responses the DB-DGS-loaded obtained with DB-DGS-loaded obtained with Cmicrostrip inferred cmicrostrip Cc inferred line by by curve for forvarious curve line fitting; various fitting; (b) dielectric dielectric (b) constants circuit circuit ofof constants responses responses the theMUT. MUT. obtained obtained with C(a) c(a) with Circuit CCircuitresponses(6).obtained responses c calculated calculated fromfrom obtained (6). with withCC c inferred c inferredby bycurve curvefitting; fitting;(b) (b) circuit circuitresponses responsesobtained obtainedwith withCCc calculated c calculated from from(6). (6). Figure 8. Sensitivity inferred from theory and from electromagnetic simulation. Figure Figure8.8. Figure Sensitivityinferred 8.Sensitivity Sensitivity inferredfrom inferred from theory fromtheory and theoryand from andfrom electromagnetic fromelectromagnetic simulation. electromagneticsimulation. simulation. Figure 9. Experimental validation. The dielectric constant of the considered samples is 10.2 (Rogers RO3010), 4.4 (FR4), 3.55 (Rogers RO4003C) and 1 (air), with 4 pieces of the slabs stacked together (around 4 mm) for each type of samples. Sensors 2021, 21, 8334 11 of 12 4. Conclusions In conclusion, in this paper we have pointed out that the generalized assumption that the circuit model of a microstrip line loaded with a dumbbell defect-ground-structure (DB-DGS) resonator transversally etched in the ground plane is a simple parallel resonant tank series-connected to the line is correct, but with some considerations. Specifically, the reactive elements of such simple model do not actually correspond to the inductance and capacitance of the DB-DGS resonator. Such reactive elements are the result of a circuit transformation from a more realistic model, where the DB-DGS resonator is magnetically coupled to the host microstrip line. However, the analysis carried out in this paper reveals that, from a practical viewpoint, the simplified model can be used not only for circuit design purposes, but also for the prediction of the sensitivity, a relevant performance parameter in applications of the considered DB-DGS-based structure to dielectric constant sensing. Author Contributions: Conceptualization, L.S. and F.M.; methodology, L.S.; analysis, L.S., P.V., J.M.-E. and F.M.; validation, L.S., P.V., J.M.-E. and F.M.; writing manuscript, L.S. and F.M.; supervision, F.M.; project administration, F.M.; funding acquisition, L.S. and F.M. All authors have read and agreed to the published version of the manuscript. Funding: This work was supported by MINECO-Spain (project PID2019-103904RB-I00), by Gener- alitat de Catalunya (project 2017SGR-1159), by Institució Catalana de Recerca i Estudis Avançats (who awarded Ferran Martín), and by FEDER funds. J. Muñoz-Enano acknowledges Secreteraria d’Universitats i Recerca (Gen. Cat.) and European Social Fund for the FI grant. L. Su acknowledges the Juan de la Cierva Program for supporting her through Project IJC2019-040786-I. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results. References 1. Kim, C.S.; Park, J.S.; Ahn, D.; Lim, J.B. 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