A Compact Multiband BPF Using Step-impedance Resonators with Interdigital Capacitors

A compact multiband band-pass filter design for applications of GSM, Wi-MAX and WLAN systems is presented. The design is based on the resonant characteristics of step-impedance and interdigital capacitor resonators with overlap cross coupling structure. The fabricated filter has been operated at the fundamental, first and second harmonic resonant frequencies of 1.8 GHz, 3.7 GHz, and 5.2 GHz, respectively. The experimental results of the fabricated filter agree very well with the simulation expectations using IE3D package. The proposed filter has good performances, while the resonator size can be reduced from λ/2 to λ/8, resulting in the most compact multiband band-pass filter compared with the others using transmission line resonators .


Introduction
Nowadays, wireless communications are necessary in daily life.Filter is the device that can remove or reject needless signals.It is an important component in RF frontend for both transmitter and receiver.There are four types of filters including high-pass filter, low-pass filter, bandpass filter and band-stop filter.Band-pass filter (BPF) is one popular filter because it can operate from lower cutoff frequency to higher cutoff frequency.The band-pass filters can be developed from different materials and structures.Microstrip is a proper design because of its characteristics; simple design, lighter weight, lower cost, compact size and higher operating frequency range.Most of wireless systems can be operated in two or more frequency bands.More advanced techniques were applied to design band-pass filters such as step-impedance resonators (SIR) used to control harmonic frequencies as desired in dual band BPFs with hairpin structure [1], comb structure [2], and parallelcouple structure [3].The triple-band BPFs were developed using tri-section step-impedance resonator (TSSIR) with inter-coupled hairpin structure to reduce size and obtain high out of band suppression [4], open stub-embedded TSSIRs arranged in anti-parallel coupled structure to generate an extra zero between harmonics [5], folded hairpin structure and new coupling scheme to improve performance and reduce size [6], [7], cross-coupled arrangement for sharpen passband skirts [8], and pseudo interdigital coupling structure to reduce size and improve stopband rejection [9].All above techniques, the resonators were still large due to they were designed with electrical length of λ/2.The slow-wave open-loop resonators [10][11][12] were also proposed for harmonics suppression and size reduction smaller than half a wavelength.In addition, the SIRs with capacitive load at the end were proposed for dual band [13] and triple band [14] responses, resulting in further size reduction from λ/2 to λ/4.Whereas, the technique using interdigital capacitor seems to have higher capacitance than capacitive load, therefore, novel microstrip interdigital bandstop filters [15], dual-mode BPF based on interdigital [16] were presented, resulting in further size reduction.
In order to improve the filter performances, the cross coupling technique was employed with open-loop resonators [17] and hairpin resonators [18].The synthesis of cross-coupled resonators was done by [19].This technique can improve the insertion loss and out-of-band responses due to transmission zero appearance.Also, the technique of overlap cross coupling was proposed for more improvement of insertion loss value at higher frequency to be lower than 3 dB [13].
In this paper, we have developed from step-impedance resonator that can control harmonic frequency as desired for triple-band BPF and taken interdigital capacitor that can reduce size of the resonators.The proposed technique can reduce the resonator size from ⁄2 to ⁄8, which it is the most compact size compared with other transmission line resonators in literatures.In addition, the techniques of multi-section feeding for multiband matching [20], [21] and overlap cross coupling for insertion loss reduction will be applied.The proposed filter can response for multiband operation of GSM, Wi-MAX, and WLAN systems.The details of filter design will be shown in the next section.Then, the proposed filter will be implemented and meas-ured as detailed in Sec. 3. Finally, conclusion will be in the last section.

Step-impedance Resonator with Interdigital Capacitor
The proposed multiband BPF has been designed on the GML 1000 substrate with relative permittivity ε r of 3.2, loss tangent of 0.004 and thickness of 0.762 mm.Structure of the proposed filter is shown in Fig. 1, consisting of four resonators connected by using overlap cross coupling technique.Each resonator has been designed using step-impedance resonator (SIR) with interdigital capacitors at both ends as illustrated in Fig. 2(a).The desired fundamental, first, and second harmonic resonances appear at 1.8 GHz, 3.7 GHz, and 5.2 GHz, respectively; the simulated result by IE3D software is shown in Fig. 3.
The interdigital capacitor structure connected with SIR can reduce size of the filter which it can control the   second and third resonances.It is found that this technique can reduce the electrical length of resonators from λ g /2 to λ g /8.In Fig. 2(a), the characteristic impedances of SIR in the transmission line are Z 1 and Z 2 , Z 3 is the characteristic impedance within interdigital capacitor, C i is the interdigital capacitor , β is the propagation constant, θ a is the electrical length of transmission line, θ b is the electrical length within interdigital capacitor, and l is the length of transmission line.The parameter θ a is 2(θ 1 + θ 2 ).Assuming θ 1 = θ 2 = θ, the characteristic impedance (Z a ) can be determined from (1), while the impedance relation ratio (K) is characteristic impedance ratio between Z 1 and Z 2 .1) tan tan Figure 2(b) shows the equivalent circuit of this resonator, which can be determined from ABCD matrix [22].The parameters Z a and C i are related to three resonant frequencies that can be determined as Then, the designed resonator parameters can be obtained as follows:

Cross-coupled Structure
The structure of cross-coupled multiband BPF consists of four step-impedance interdigital capacitor resonators.The cross-coupled technique is used to design the proposed filter which can improve matching, resulting in lower insertion loss.The determined coupling coefficient between a pair of resonators can be found by using where f l and f h are lower and higher peak frequencies of the simulated response, respectively.To calculate both values, the coupled distance between a pair of resonators must be tuned, resulting in the double peak response [22], [23].
From the coupling effect of our proposed filter, these values can be investigated by simulation using full wave method of moment (MOM) software package from IE3D.
The coupling distances D 1,2 , D 1,4 , D 2,3 and D 3,4 and coupling coefficients can be simulated for each pair of resonators.Graphs obtained from simulation can be classified into three types as electric coupling, magnetic coupling, and mix coupling as depicted in Figs. 4 (a  -24.693 dB at the frequencies of 1.8 GHz, 3.7 GHz, and 5.2 GHz, respectively.The insertion loss at the second harmonic is about -3 dB, then overlap coupling technique has been utilized for improvement.

Overlap Cross Coupling
After connecting the resonators with 50 Ω feeds as shown in Fig. 5, it has been found that the result has not matched at the second and third harmonic frequencies.The multi-section feeds have been then proposed to connect multiband resonators [21].The feeds consist of tapered and multi-step sections, whose characteristic impedance corresponds to those operations L and C. The designed multisection feeding line parameters can be obtained as follows: The group delay, time domain performance, of the proposed filter can be analyzed.Figure 7 shows the simulated group delay of the proposed filter.It is shown that the  simulated group delays are constant over three desired passbands, which are about -60 ns at the fundamental frequency band and about -25 ns at both harmonic frequency bands.The negative values could be caused by the slow wave effect.However, this can guarantee that fundamental and harmonic frequency bands can be controlled as desired and the proposed filter can work very well for all bands.
Then, the overlap coupling technique [10] can be used by shifting up and down the resonators 3 and 4 along the vertical axis with a distance of h as shown in Fig. 8.The distance h is varied between 0.5 to 2.0 mm with constant coupling coefficients (D 12 , D 23 , D 34 , and D 14 ).The results of each resonant frequency are illustrated in Fig. 9.It can be clearly seen that at odd frequencies (1.8 GHz, and 5.2 GHz), the effects by varying parameter h are displayed in Fig. 9(a) and Fig. 9(c) whereas even frequency is not affected obviously when varying parameter h as shown in Fig. 9(b).In Fig. 9(a), it is found that the return loss (S 11 ) is approximate -10 dB and the insertion loss (S 21 ) is about -3 dB; however, at h = 0.5 mm, the return loss (S 11 ) is less than -20 dB.In Fig. 9(b), the insertion loss (S 21 ) and the return loss (S 11 ) are slightly changed by varying parameter h.Finally, Fig. 9(c) shows the return loss (S 11 ) less than -10 dB and insertion loss (S 21 ) nearby -3 dB; nevertheless, at h = 0.5 mm, the return loss (S 11 ) is less than -20 dB.It can be summarized that overlap parameter h = 0.5 mm is the best choice for insertion loss improvement of the proposed filter as shown in Fig. 9.The insertion loss and return loss caused by overlap coupling are demonstrated in Tab. 1.

Implementation and Results
After designing the multiband BPF by techniques as mentioned, this section presents and discusses property of coupling between resonators connected by cross coupling.There are appropriated parameters of coupling between resonators as D 12 = D 34 = 0.56 mm, D 23 = 0.76 mm and D 14 = 0.67 mm.The illustration of overlap coupling between resonators 3 and 4 along vertical axis has been done by simulation of electric and magnetic fields at all frequency ranges using IE3D program as results shown in Figs.11-13 where increasing coupling coefficient results to insertion    loss (S 21 ) lower than 3 dB.There are three types of cross couplings as electric, magnetic and mix couplings.Normally, there are transmission fields from resonator 1 to resonator 2, resonator 2 to resonator 3, and resonator 3 to resonator 4.There is no transmission field between resonator 1 to resonator 3 and resonator 2 to resonator 4.

Range of Frequency
At fundamental frequency (1.8 GHz) as seen in Fig. 11, it is found that the gap between resonators 1 and 4 on the left side of Fig. 11(a) has electric coupling.Also, there is E-field between resonators 1 and 4 more than Efield between resonators 2 and 3, which the structure has magnetic coupling.While the H-field between resonators 2 and 3 is higher than the H-field between resonators 1 and 4 as shown on the right side of Fig. 11(a).After shifting up the resonators 3 and 4 along vertical axis as shown in Fig. 11(b), the E-field and H-field are similar to Fig. 11(a).There are cross fields between resonators 1 and 3, and between resonators 2 and 4, which are called mix coupling.Because there are E-field and H-field concomitant, when increasing overlap coupling distance, mix coupling has been increased.This will result in improvement of insertion loss value.However, at fundamental frequency 1.8 GHz, there is less cross coupling, the insertion loss improvement could not be noticed as shown in Fig. 8(a).At the second harmonic frequency (3.7 GHz), the E-field and H-field are displayed in Figs.12(a) and (b), respectively.The cross coupling fields between resonators 1 and 3, and between resonators 2 and 4 when overlap at the second harmonic frequency is higher than the fundamental frequency.As a result, the insertion loss at the second harmonic frequency is improved comparing with the fundamental frequency as shown in Fig. 9(b).At the third harmonic frequency (5.2 GHz), both E-field and H-field are shown in Figs.13(a) and (b), respectively.It has been found that there is strong effect of cross coupling when overlap compared with the fundamental and second harmonic frequencies, resulting in more improvement in insertion loss as shown in Fig. 9(c).It can be concluded that the cross coupling fields caused by overlap resonators have more effect at higher frequency and less effect at low frequency.
After all, we have fabricated the proposed multi-band BPF with the size of 4 × 2.3 cm 2 as shown in Fig. 14

Conclusions
A compact multi-band BPF using overlap coupling of step-impedance with interdigital capacitor resonators has been proposed to operate at 1.8 GHz, 3.7 GHz, and 5.2 GHz, respectively.This proposed BPF has the most compact size comparing with the recent BPF whose size can be reduced from λ /2 to λ /8.The results show good agreement between simulation and measurement with return loss more than 10 dB and insertion loss less than 3.0 dB.The filter also demonstrates good performances and it can support GSM, Wi-MAX, and WLAN systems.However, the filter bands can be adjusted to any frequency corresponding to application bands by changing the resonator lengths and interdigital capacitive values.
W 1 = 1.15 mm, W 2 = 2.39 mm, W 3 = 0.578 mm, W 4 = 0.975 mm, L 1 = 13.885mm, L 2 = 7.03 mm, L 3 = 8.192 mm, and L 4 = 2.357 mm.The resonator size is approximately ⁄8.It has been found that the proposed filter has a smaller size compared with the recent published filer designs that has the resonator size of about ⁄4 [14].However, most of the designed filters have the resonator sizes of ⁄2 at the fundamental frequency [1].

Fig. 6 .
Fig. 6.The BPF with multi-section line: (a) the details of designed feeding line and (b) the simulated result.

Fig. 15 .
Fig. 15.Comparison results of simulated and measured responses of S 11 and S 21 .
. The simulation results show the resonant frequencies at 1.8 GHz, 3.7 GHz, and 5.2 GHz as designed.The values of return losses (S 11 ) at the resonances are -20.86 dB, -13.86 dB, and -24.57dB, respectively.The values of insertion losses (S 21 ) are -2.64 dB, -2.40 dB, and -2.74 dB, and bandwidth are 110 MHz, 200 MHz, and 220 MHz, respectively.The proposed multi-band BPF was measured by Agilent 8791ES network analyzer, resulting in the same resonant frequencies as simulated.From the measurement results, the values of measured return losses (S 11 ) are -17.6 dB, -19.7 dB, and -15.9 dB, and bandwidth are 121 MHz, 224 MHz, and 233 MHz, respectively.The values of insertion losses (S 21 ) at the resonances are -2.83dB, -2.82 dB, and -2.96 dB, respectively.The comparison results of simulated and measured responses of return loss and insertion loss are shown in Fig. 15.It is clearly seen that the simulated and measured resonant results are at the same frequencies.Moreover, values of both measurement insertion losses and return losses are almost similar to simulation.

Type of Loss 1.8 GHz 3.7 GHz 5.2 GHz Insertion loss (S 21 ) dB
Comparison results of simulated responses of cross coupling and overlap coupling.