A Design of Microstrip Lowpass Filter with Wide Rejection Band and Sharp Transition Band Utilizing Semi-Circle Resonators

In this paper, a microstrip lowpass filter with –3 dB cut-off frequency of 2.1 GHz consisting of three cascaded resonators with different semi-circle patches and four suppressors employing radial stubs has been proposed. To indicate the role of each employed microstrip transmission line in the structure of its resonance cell, the equations of the transfer function and transition zero of the resonator have been calculated, separately. The designed filter has been constructed and tested, and a good agreement between the results of simulation and measurement has been achieved. In the whole rejection region, a return loss better than +0.28 dB and a 19.656 GHz stopband bandwidth with high rejection level of 32 dB have been obtained. Moreover, a flat insertion loss close to zero in the passband and sharp cutoff slope (203.57 dB/GHz) can verify the desired frequency response. The proposed filter has a high figure of merit equal to 24241.69.


Introduction
Microstrip lowpass filters (LPFs) are one of the key blocks that are employed widely to suppress undesired frequencies.To meet this purpose, various LPFs have been designed.For example, in [1], loaded radial-shape patches and meandered main transmission line have been used and a microstrip lowpass filter with quasi-elliptic response has been presented, but, this filter has a gradual transition band and low level of suppression.In order to widen the stopband bandwidth of LPFs, several hairpin resonators have been designed [2][3][4][5][6].By adopting this cell in [2], the stopband region has been extended.However, the circuit occupies a large area and the transition band is not sharp enough.A technique to design a LPF is using radial shaped stubs in the structure of hairpin resonator, which has been carried out in [3], but this method does not have a steep cut-off.In [4], to design a LPF another form of stepped impedance hairpin units have been employed.However, this filter does not have a wide stopband.In [5], another technique to present a LPF on the basis of an application of shunt open-stubs coupled-line in the circuit of hairpin cell has been reported.However, this filter does not have a sharp transition band.In [6], defected ground structure (DGS) is utilized to design a LPF with sharp rejection, although this circuit cannot be used on the metal surface.To present a LPF with new features, rat-race directional couplers have been utilized to function as bandstop transversal filtering sections (TFSs), in [7].However, employing this method leads to a large surface without achieving an acceptable expanded stopband.In [8], a quasi--slot resonator and open stubs is adopted and a LPF has been proposed, but its stopband and transition band performances are not satisfactory.A microstrip LPF using triangular and radial patch resonators has been proposed [9], but this filter suffers from gradual cut-off.To obtain a LPF featuring wide stopband, a cascaded microstrip coupledline hairpin unit and semi-circle stepped-impedance shunt stubs have been utilized, in [10].In [11], by using LC resonant structures and transformed radial stubs a LPF has been designed.However, the proposed filters in [10] and [11] suffer from gradual roll-off rate and relatively large circuit size.Two other techniques to design LPFs with steep transition band are employing asymmetric high-low impedance patches and LC tank resonators, which have been reported in [12] and [13], respectively.In [14], a lowpass filter employing polygon patch resonant cells, T-shaped resonators and two different suppressing cells has been designed, although, this filter does not have a steep skirt performance.One more method to present a LPF with acceptable frequency response is cascading resonators with polygon patches [15], but it occupies a large area.In [16], employing modified hairpin resonator with long straight slots leads to designing a lowpass filter with sharp skirt performance and wide rejection band, but its return loss performance is not good enough.Some other methods to present symmetric LPFs with sharp transition slope are utilizing flabelliform resonators and triangular patches [17] and [18], although these circuits suffer from a narrow stop-band bandwidth and large circuit size, respectively.In this letter, a microstrip lowpass filter (LPF) operating at -3-dB cut-off frequency of 2.1 GHz has been designed.In the structure of the proposed filter two main resonators with semi-circle patches to create a sharp transition band and four suppressing units to reject the spurious frequencies in the frequency response have been employed.

The Procedure of Designing
This section introduces the utilized resonance cells, the procedure of analyzing their transfer functions and also explains the design process, separately.The schematic of the primary resonance cell utilizing semi-circle patch and its equivalent LC circuit have been depicted in Fig. 1.As can be observed from Fig. 1b, L 1P is the inductance caused by the main transmission line.The employed semi-circle patch and the high-impedance transmission line are modeled by L 2P and C 2P , respectively.Moreover, C 0P and C 1P represent the capacitance between the microstrip structure and the ground.Clearly, the cut-off frequency, the slope of the transition band and also the stopband bandwidth can be affected by the location and attenuation level of transition zeros.Furthermore, the impact of each employed microstrip transmission line of the resonator on the frequency response can be justified by calculating the transfer function of the equivalent LC circuit.Thus, to achieve transition zeros of the primary resonator, the transfer function based on the equivalent LC circuit will be extracted.
The equations of transfer function and transition zero of the primary resonator shown in Fig. 1b are given by ( 1) and ( 2), respectively, as follows: Note that R in the transfer function stands for the resistance of matching.As can be observed in (7), the location of transmission zero can be controlled by altering the values of capacitance and inductance determined by C 2P and L 2P .To illustrate the influences of changing C 2P and L 2P on the scattering parameters, their corresponding microstrip realizations can be considered instead.Thus, several full-wave simulations versus the widths and lengths of the transmission lines of the primary semi-circle resonator have been carried out.The behavior of the above-mentioned resonance cell against changing the values of A, B and C has been illustrated in Fig. 2. As it can be seen, by enhancing these variables with the shown steps, the transition zero will shift to lower frequencies creating a sharper transition band, as it was anticipated by (7).Moreover, these increases can affect the operating frequency of the resonator, as it is observed from Figs.According to the simulation results shown in Fig. 1c, the designed resonator has a -3 dB cut-off frequency of 1.987 GHz.In the whole passband region, the insertion loss is almost equal to zero.Furthermore, the presented resonator has a transmission zero at 3.323 GHz with corresponding attenuation level of +48.059 dB leading to having  an acceptable transition band.The schematic of the main resonator and its equivalent lumped circuit have been illustrated in Fig. 3. Similar to the primary resonator, to clarify how the frequency response of the resonator can be controlled, the equation of transition zero can be obtained according to Fig. 3b as follows: According to (9), the transition zero can be controlled by changing the values of L 4M and C 5M .To explain the impacts of these lumped elements on the frequency response, their corresponding microstrip realizations are utilized, instead.
In order to illustrate how changing the dimensions of the mentioned corresponding microstrip lines can influence the frequency response, several full-wave simulations versus D, E and F have been plotted in Figs.4a, b and c, respectively.As it can be seen from Fig. 4, by enhancing the values of all these three variables with the determined steps, the transition zero will shift to lower frequencies creating a sharper skirt performance.Moreover, the increases in D and F lead to decreasing the operating frequency, but changing the value of E does not have any significant effect on the cut-off frequency.The values of lumped elements of the shown LC circuit in Fig. 3b are [19]: L 3M = 5.612 nH, L 4M = 1.554 nH, C 3M = 0.0013 pF, C 4M = 0.439 pF and C 5M = 1.33 pF.Note that, the values of inductors and capacitances have been obtained based on RO4003 substrate with a thickness of 0.508 mm and the permittivity of 3.38.The EM simulation and the frequency response of the lumped circuit of the main resonator have been depicted in Fig. 3c, which are in good agreement.As can be observed, the return loss and insertion loss in the passband region are almost +21.331 and +0.069 dB, respectively.Moreover, the first main resonance cell makes a transition zero at 3.467 GHz with an attenuation level of +55.717 dB.Thanks to this transmission zero a suppressing band covering an acceptable frequency range from 2.257 to 8.637 GHz with corresponding attenuation level of 15 dB has been obtained.Obviously, the main resonator suffers from gradual skirt performance and also a poor stopband characteristic.
The schematic of the second main resonance cell employing another form of semi-circle patch and its equivalent LC circuit are depicted in Fig. 5a and b.To control the frequency response characteristics of this resonator, the impacts of variables on the frequency response should be determined like the first main resonator.Thus, the equations of transfer function and transmission zeros of the resonator shown in Fig. 5a based on its LC circuit have been extracted as follows: Likewise, the influences of changing the values of the valuables determined in ( 10)-( 21) affecting the transfer function and the transmission zeros can be studied on the basis of their corresponding microstrip realizations.As it is clear, the values of L 1 , L 2 , C 1 and C g can be utilized to control the location of transition zeros of the second main resonator, effectively.Therefore, to study their impacts on the frequency response of the depicted resonator in Fig. 5a, the dimensions of their corresponding microstrip realizations have been changed.
Several full-wave simulations versus the most effective dimensions, i.e.G, H and K have been plotted in Fig. 6.As it can be seen, by enhancing the above-mentioned values with the shown steps, the both transition zeros will move to lower frequencies leading to creating a sharper transition band.Note that, with the exception of the value determined by K, by increasing the dimensions of the semi-circle patches, the operating frequency of the second resonator will shift to lower frequencies.As it can be seen from the simulation results illustrated in Fig. 5c, the operating frequency of the presented resonator with semi-circle patches is located on 2.08 GHz.
The stopband of the designed resonator rejects spurious frequencies from 2.32 GHz to 7.682 GHz with corresponding attenuation level of 15 dB.The insertion loss is around +0.092 dB from DC to 1.78 GHz which proves  a desired response in the pass band.Furthermore, the proposed resonator makes two transmission zeros at 3.223 GHz and 3.613 GHz with corresponding attenuation levels of +67.591 dB and +67.417 dB, respectively, leading to having an acceptable stopband characteristic.However, to widen the stopband region, suppressing cells are required.
Utilizing these suppressing units makes some transmission nulls, thus transmission peaks will be suppressed and consequently the stopband region will be expanded.One of the most influential suppression cells to extend the stopband bandwidth is high-low impedance resonator with radial patches.Figure 7a and b illustrates the schematic and the frequency response of these suppressing cells.As it can be seen from the frequency response, the designed suppressing cell creates three transition zeros at 11.75 GHz, 15.61 GHz and 16.44 GHz with corresponding attenuation levels of +53.769, +51.758 and +57.922 dB.Owing to these transition zeros, the stopband of this cell can suppress spurious frequencies from 10 to 23.78 GHz with a desired level of suppression.
Finally, by combining the first, second and the designed suppressing resonators, a microstrip LPF with -3dB cut-off frequency of 2.1 GHz has been designed.Furthermore, the presented LPF has a transmission zero around the cut-off frequency, which leads to an acceptable skirt performance.In addition, a suppressing factor equal to 32.14 dB over the illustrated frequency range has been achieved.

The Results of Simulation and Measurement
The photograph of the proposed LPF has been depicted in Fig. 9a.The presented LPF has been designed, constructed and measured.The fabricated LPF has been constructed on a substrate with the thickness of 0.508 mm, the permittivity of 3.38 and the loss tangent of 0.0021.The results of simulation and measurement of the scattering parameters have been illustrated in Fig. 9b.As it can be observed, the -3 dB operating frequency of the filter has been placed on 2.1 GHz.In the entire passband region the insertion loss is close to +0.1 dB from DC to 1.855 GHz, which indicates a flat response.As it can be seen, close to the cut-off frequency two transmission zeros (TZs) at 2.382 and 2.897 GHz are created which lead to a steep transition band.Owing to these TZs, a sharp roll-off rate equal to (203.57dB/GHz) has been obtained, which proves a desired skirt performance.
The stopband region covers a wide range, suppressing spurious frequencies from 2.294 up to 21.95 GHz with corresponding rejection level of 32 dB.Table 1 illustrates the performance comparison of the other published works and the proposed LPF.In the shown table the roll-off rate ξ is defined as follows: In this equation α max is the 40dB attenuation point, α min is the 3dB attenuation point, f S is the 40dB stop-band frequency, and f C is the -3dB cutoff frequency.
The relative stop-band bandwidth (RSB) is defined as shown below: The suppression factor (SF) is defined as shown in (24).For example, when the stop-band suppression is 32 dB, the corresponding SF will be 3.2.stop-band supression 10 dB SF  .
The normalized circuit size (NCS) is given by: where λ g is the guided wavelength at -3dB cut-off frequency.
The architecture factor (AF) denotes the circuit complexity.When we have a two-dimensional design, we define AF = 1.Similarly, for three-dimensional designs, we have AF = 2.
At the end, the figure-of-merit (FOM) is defined as follows: .

RSB SF
Based on the above definitions, it is easily understood that, the calculated roll-off rate ξ, RSB, SF, NCS and AF of the proposed filter lead to a figure-of-merit equal to 24241.69.
As it can be observed, the skirt performance of the designed filter is sharper than the other published articles, which is because of the first and second resonators.Utilizing the depicted suppressing unit shown in Fig. 7a, leads to obtaining an acceptable relative stopband bandwidth RSB, which is 1.78.The closest RSB to this have been cited in [9], [11] and [14] (1.65, 1.72 and 1.666, respectively).The suppressors suppress spurious frequencies with a suppressing factor of 32 dB, which is better than the other works.Clearly, the proposed filters in [11] and [17] have the closest suppressing factor to this work.As it can be observed, the presented filter, in spite of occupying 262.44 mm 2 at the operating frequency of 2.1 GHz, has brought about a figure of merit equal to 24241.69, which is much more than the other published articles.

Conclusion
A low pass filter using semi-circle patch resonant cells and four suppressing cells with high-low impedance patches has been proposed and fabricated.According to the equivalent LC circuit of each resonator, the locations of transition zeros of the mentioned resonance cells have been calculated.The frequency response verifies a good in-band and out-band performances based on the measurement results.The proposed lowpass filter demonstrates a sharp skirt performance, a wide rejection band, low insertion loss and low return loss in the passband region and also occupies a small area (16.2 × 16.2 mm 2 ).According to the measurement results, the figure of merit of the proposed filter is equal to 24241.69.

Fig. 1 .
Fig. 1.(a) The configuration of the primary resonator with semi-circle patch.(b) Its equivalent LC circuit.(c) The EM simulation result and the frequency response of LC circuit of the primary structure.
2b and c.The values of inductors and capacitors of the shown LC circuit in Fig. 1b are [19]: L 1P = 5.112 nH, L 2P = 1.554 nH, C 0P = 10.5 fF, C 1P = 27.9 fF, C 2P = 1.48 pF.The result of EM simulation and the frequency response of LC circuit of the introduced resonator have been depicted in Fig. 1c.

Fig. 2 .
Fig. 2. The behavior of the proposed primary resonator against changing: (a) the value of A, (b) the value of B, and (c) the value of C.

Fig. 3 .
Fig. 3. (a) The configuration of the first main resonator.(b) Its equivalent lumped circuit.(c) The EM simulation result and the frequency response of LC circuit of the main structure.

Fig. 4 .
Fig. 4. The behavior of the main resonator against changing (a) the value of D, (b) the value of E and (c) the value of F.

Fig. 5 .
Fig. 5. (a) The configuration of the second main resonator with different semi-circle patches.(b) Its equivalent LC circuit.(c) The EM simulation result of the resonator and the frequency response of its LC circuit.

Fig. 6 .Fig. 7 .
Fig. 6.The behavior of the second main resonator against changing (a) the value of G, (b) the value of H and (c) the value of K.

Fig. 8 .
Fig. 8.The configuration of the proposed lowpass filter.

This Work 203.57 1.61 3.2 0.208×0.208 1 24241.69 Tab. 1.
The comparison of the abilities of the proposed LPF and previous works.