Design of a band-pass filter using stepped impedance resonators with floating conductors

Progress In Electromagnetics Research, Vol. 105, 31–48, 2010 DESIGN OF A BAND-PASS FILTER USING STEPPED IMPEDANCE RESONATORS WITH FLOATING CONDUCTORS M. D. C. Vel´azquez-Ahumada Department of Electronics and Electromagnetism University of Seville Av. Reina Mercedes s/n, Seville 41012, Spain J. Martel Department of Applied Physics 2 University of Seville Av. Reina Mercedes s/n, Seville 41012, Spain F. Medina Department of Electronics and Electromagnetism University of Seville Av. Reina Mercedes s/n, Seville 41012, Spain F. Mesa Department of Applied Physics 1 University of Seville Av. Reina Mercedes s/n, Seville 41012, Spain Abstract—A new type of miniaturized stepped impedance resonator (SIR) for bandpass filter applications is proposed in this paper. The new resonator incorporates a ground plane window with a floating conductor in the backside of the substrate. The ground plane window increase the characteristic impedance of the lines used to implement the inductive region of the quasi-lumped resonator, thus allowing some size reduction. Moreover, the presence of a floating conducting patch printed below the capacitive region of the resonator pushes up the first spurious band of the filter. A meaningful improvement of its out-of-band rejection level is then achieved. The coupling between adjacent resonators is also enhanced thus leading to wider achievable Corresponding author: F. Medina ([email protected]). 32 Vel´azquez-Ahumada et al. bandwidths. Some filter designs using the new resonator and other standard resonators are included for comparison purposes. 1. INTRODUCTION Miniaturization of printed planar microwave filters has been a popular research topic in recent years owing to the growing expansion of wireless and mobile communication systems operating at the lower end of the microwave spectrum. Recent examples of this kind of research can be found in [1–7]. Conventional distributed resonators are too large to be used at low frequencies when miniaturization is a key issue. Thus, these resonators have been progressively replaced by smaller structures. Half-wavelength resonators originally used to implement parallel-coupled line filters [8] were substituted by hairpin resonators such as those reported in [9]. A more compact version of the hairpin resonator is the square open-loop resonator [10], but this resonator still works as a distributed device that yields too large layouts at low frequencies. An additional drawback of distributed resonators is the degradation of the out-of-band rejection level because of the existence of undesired spurious harmonics. This problem can be overcome by using compact quasi-lumped resonators at the price of increasing losses. For instance, the folded hairpin resonator in [11] is much smaller than the conventional hal temairpin or openloop resonators owing to increased capacitance associated to two closely coupled sections. A similar effect can be attained using the miniaturized open-loop resonators proposed in [12], later modified by other authors [13] and now called folded stepped impedance resonators (folded SIR) [14]. This resonator can be viewed as a high characteristic impedance transmission line loaded with a low impedance coupled line pair. For relatively high values of the capacitance of the wide coupled lines section and for the lowest resonance frequency, the structure behaves as an LC quasi-lumped resonator. In such case, the inductance, L, is provided by the short section of high characteristic impedance microstrip line and the capacitance, C, is associated with the electrically short pair of coupled lines. The quasi-lumped nature of the resonator automatically alleviates the problem associated with the presence of the first spurious transmission band [15]. More sophisticated versions of SIR’s have been applied to filter design looking for compactness and wide upper stopband behavior [16]. Recently, a number of compact printed resonators have profited from the patterning of the backside metallization of the substrate [17– 20] in order to enhance their performance. These are known as defected Progress In Electromagnetics Research, Vol. 105, 2010 33 (a) (b) A A’ l4 l3 l2 l1 w0 wf lf s s0 s Figure 1. Folded stepped impedance resonator with ground plane window and floating conductor. (a) Top view; (b) bottom view. To clarify the window position in the bottom view, the contour of the upper side resonator has been drawn using dashed lines. ground structures (DGS). Following that research line, in this paper we propose the new resonator shown in Fig. 1. This resonator, a folded SIR with floating conductor, can be considered a modified version of the resonators proposed in [12, 14]. The ground plane area below the resonator is partially removed, in such a way that a metallic floating rectangular patch remains inside the ground window and below the coupled lines section. The presence of the window increases the characteristic impedance of the single strip section [21– 23] thus providing some size reduction. But what is more important in the context of this paper, the floating conductor provides a way to improve the out-of-band rejection level of filters based on this structure. This is a very important issue to avoid intermodulation distortion [24]. Roughly speaking, the first resonance frequency associated with the desired pass band is close to the one of the conventional folded SIR, while the second resonance frequency meaningfully pushed up. This fact allows a better rejection level above the first band. As additional benefits, the ground plane window opens a new coupling mechanism improving the coupling factor between neighbor resonators [25] thus making possible wider bandwidths. In brief, the new resonator allows us to built filters slightly smaller than the ones based on conventional folded SIRs with larger achievable bandwidths and better rejection level above the first pass band. 34 Vel´azquez-Ahumada et al. coupled lines single line Zseff s ,ε Zeeff e ,ε Zoeff o ,ε AA’ Figure 2. Transmission line circuit model of any folded SIR. The model is valid for conventional folded SIR and approximately accounts for the behavior of the folded SIR with floating conductor. 2. CHARACTERIZATION OF SINGLE RESONATORS 2.1. Transmission Line Circuit Model Fig. 2 shows a transmission line circuit model that reasonably accounts for the electromagnetic behavior of folded SIR’s. For the folded SIR with floating conductor studied in this paper this is just an approximate model because the coupled section involves three conductors plus ground. Three, instead of two, quasi-TEM modes are then involved. However, it has been shown in the past that the even/odd mode theory reasonably applies to this case [26, 27]. The characteristic impedance and the effective dielectric constant of the single transmission line section, providing the inductive contribution, are Zsand εs eff respectively. The capacitive part of the resonator can be viewed as a coupled pair of microstrip lines whose electrical parameters are the even (e) and odd (o) mode characteristic impedances, Zeand Zo, and the modal effective permittivities, εe eff and εo eff. Note that this is the same circuit proposed in [11] for a folded hairpin resonator. The symmetry of the structure with respect to the AA0plane in Fig.1 allows us to consider separate even and odd mode excitations [11, 28]. Two Progress In Electromagnetics Research, Vol. 105, 2010 35 independent resonance conditions are then obtained: tan θs 2tan θo=Rzo (odd resonances) (1) tan θe=−Rze tan θs 2(even resonances) (2) where Rzo =Zo/Zs,RzeConf =Ze/Zs,θsis the electrical length of the single line section, and θe,o are the electrical lengths of the even (odd) modes of the coupled lines section. The first resonance frequency, f0, corresponds to the first odd mode resonance. This is the fundamental frequency determining the operation band of any filter based on these resonators. The first spurious resonance frequency, f1, corresponds to the first even mode resonance. 2.2. Minimum Size Conditions The degree of compactness of folded SIR’s can be described following the method in [28], where the physical length of the resonator is compared with the length of an open-loop resonator having the same fundamental frequency. This analysis yields the following optimal values for the electrical lengths of the single and coupled line sections Figure 3. Normalized electrical length at the first spurious resonance frequency (f1) of SIR’s versus θ1 susing the ratio Rze as a parameter. 36 Vel´azquez-Ahumada et al. (Rzo <1 is assumed): θ0 s= 2 tan−1sRzo(RzoPs−Po) RzoPo−Ps (3) θ0 o= tan−1sRzo(RzoPo−Ps) RzoPs−Po (4) where Ps=qεref eff /εs eff and Po=qεref eff /εo eff,εref eff being the effective permittivity of the transmission line used to implement the open-loop resonator (note that if Ps=1 we recover the results in [29] and if Ps=Po=1 we are in the case treated in [30]). In brief, we conclude that, for miniaturization purposes, the SIR must be designed so as to get values of Rzo,Psand Poas small as possible whereas θ0 sand θ0 o fulfill the minimum conditions in (3) and (4). 2.3. Pushing up the First Spurious Resonance It is our goal to maximize the ratio f1/f0. In other words, the electrical length of the SIR at f1should be as large as possible when compared with the electrical length of our reference half-wave resonator at f0 (i.e., π). Let us define the normalized electrical length at the first spurious resonance of a SIR as θN1 = (θ1 s+ 2θ1 e)/π. The electrical lengths θ1 sand θ1 eare, respectively, the lengths of the single line section and the even mode electrical length of the coupled section of the SIR at f1. In Fig. 3, we plot the values of θN1 as a function of θ1 susing the ratio Rze as a parameter. The case Rze = 1 corresponds to the half-wave open-loop resonator and, obviously, θN1 = 2 in such case. If Rze <1θN1 reaches a maximum when tan(θ1 s/2) = 1/√Rze (in that case θ1 e=θ1 s/2 + π/2) and a minimum when tan(θ1 s/2) = −1/√Rze (in that case θ1 e=θ1 s/2−π/2). On the contrary, if Rze >1, θN1 reaches a minimum when tan(θ1 s/2) = 1/√Rze (in that case θ1 e=θ1 s/2+π/2) and a maximum when tan(θ1 s/2) = −1/√Rzo (in that case θ1 e=θ1 s/2−π/2). This discussion is not completely equivalent to the similar analysis carried out in [28] for folded hairpin resonators because in [28] only the range π < θ1 s<2π(or θ1 s/2> θ1 e) was considered. Thus the authors of [28] conclude that, to increase the first spurious frequency, it is necessary that Rze >1. However, the reverse condition, i.e., Rze <1 might also lead to increase the first spurious frequency in the range 0< θ1 s< π (or θ1 s/2< θ1 e). An important reference concerning this discussion, but focused on the unfolded version of the conventional SIR, can be found in [31]. Progress In Electromagnetics Research, Vol. 105, 2010 37 l3(mm) 0.51.0 1.52.0 2.53.0 3.54.0 4.55.0 5.56.0 f1/f0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 lf=5 mm lf=4 mm lf=3 mm Conventional Figure 4. Variation of f1/f0as a function of the length of the coupled lines section (l3). The dashed line corresponds to a conventional folded SIR. Solid lines correspond to folded SIR with floating conductors having different values of lfand s= 0.1 mm. In conclusion, the double goal of optimum miniaturization and maximum ratio f1/f0can be reached provided the conditions discussed in this section and (3) and (4) are simultaneously fulfilled. But this is not a trivial task because usually the involved electrical parameters are not independent. For a given substrate, effective dielectric constants and characteristic impedances are strongly interleaved. However, additional flexibility would be obtained if the values of Rzo and Rze could be separately controlled. In such case, the designer could independently tune, within a certain range of values, the positions of the fundamental and the first spurious resonance frequencies. This is the advantage provided by the structure proposed in this paper. In the conventional folded SIR the electric field in both the odd and even modes is strongly confined inside the dielectric substrate and Ze tends to be similar to Zo. Therefore the miniaturization leads to a lack of control of the first spurious frequency because the value of Zecan not be tuned independently of Zo. This is not the case for the modified folded SIR in Fig. 1. For the first (odd) resonance, the floating conducting patch behaves, roughly speaking, as a grounded conductor. The parameters of the coupled lines Zoand εo eff are, approximately, the same as those the conventional folded SIR. Therefore, we would not expect an important modification of the fundamental resonance frequency with respect to that of the conventional folded SIR. However, for the first even resonance, the behavior of the floating conductor is far away of being a virtual ground plane. In such case, the electric field 38 Vel´azquez-Ahumada et al. is not confined in the substrate and, even if Zois chosen to be very low, a wide range of values of Zeis achievable tuning the dimensions of the floating conductor. Thus, the designer has a higher level of control on the separation of the two first resonance frequencies. This feature of the the structure in Fig. 1 is exploited in this paper. In Fig. 4, we plot the ratio f1/f0as a function of l3for several values of the parameter lf(SIR with floating conductor) and for the conventional folded SIR. The new structure allows us to obtain values of this ratio larger than the ratio provided by conventional folded SIR. 2.4. Some Practical Examples In order to check the theoretical considerations above, we have numerically studied the behavior of several electrical parameters of conventional folded SIR’s and the modified SIR in this paper. The resonators are supposed to be printed on a commercial substrate of relative dielectric permittivity εr= 10.2 and thickness h= 0.254 mm. This substrate has been used later to obtain experimental results. The printed circuit is enclosed in a metallic box whose dimensions have been taken into account in the calculations. The physical size of the inductive region has been chosen to be the same for all the resonators. Therefore, the dimensions w0= 0.37 mm, l1= 3.075 mm, l2= 1.68 mm are the same for all of them. We also keep constant the value l4= 3.025 mm. Using the fast solver described in [32] we can compute the required electrical parameters. For the conventional SIR we have Zs= 39.72 Ω, εs eff = 7.06, Zo= 7.69 Ω, εo eff = 8.09, Ze= 8.93 Ω and εe eff = 9.33. Note that, as predicted, Zeand Zo are low, as required for miniaturization, but close to each other, what is not suitable to get high values of f1/f0. For the folded SIR with floating conductor Zs= 61.85 Ω, εs eff = 4.97. For this structure, the values of Zo,εo eff,Zeand εe eff depend on the dimensions of the floating conductor, sand lfin Fig. 1. Our computations show that odd mode parameters of the coupled section slightly depend on sbut are invariant with respect to lf. For values of sranging from 0.1 mm to 1.0 mm, Zo varies between 7.7 and 9.2 Ω. These values are similar to the value of Zo for the conventional folded SIR. Although εs eff is smaller for the newly proposed resonator, the relatively high value of Zscompensates this effect and makes f0slightly smaller in the case of the SIR with floating conductor. However, it is the effect on f1what is more important. The values of Zefor our resonator quickly increases with the value of lfand is not difficult to obtain values of this parameter 5 or 6 times higher than the value of 8.93 Ω corresponding to the conventional folded SIR. Progress In Electromagnetics Research, Vol. 105, 2010 39 d(mm) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 d d Coupling factor,K Figure 5. Coupling factor versus the distance between adjacent resonators for the different resonator geometries used to design filter A. Black solid line: folded SIR with floating conductor; grey solid line: conventional folded SIR; black dashed line: open-loop resonator. 3. EXAMPLES OF BANDPASS FILTER DESIGNS Several filters with similar specifications have been designed and fabricated using conventional open-loop, folded SIR resonators, and the newly proposed folded SIR with floating conductor. 3.1. Example 1: Three Poles Filter Let us consider the following specs for filter A: order N= 3, central frequency f0= 900 MHz, ripple = 0.1 dB and fractional bandwidth = 4%. The same substrate considered in the previous section has been used. The filters were placed inside a metallic enclosure whose dimensions were 40 ×20 mm2. 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