High-Performance Low-Pass Filter Using Stepped Impedance Resonator and Defected Ground Structure

A microstrip low-pass filter (LPF) using reformative stepped impedance resonator (SIR) and defected ground structure (DGS) is proposed in this paper. The proposed filter not only possesses the advantage of high frequency selectivity of SIR hairpin LPF with internal coupling, but also possesses the large stop-band (SB) bandwidth by adjusting the number and area of DGS units. The LPF proposed in this paper possesses the properties of miniaturization, wide SB, high selectivity, and low pass-band ripple (PBR) simultaneously. The characteristic parameters of the proposed LPF is that: the pass-band (PB) is 0~2 GHz, the PBR is 0.5 dB, the SB range is from 2.4 GHz to 9 GHz when the attenuation is under 20 dB, and the maximal attenuation could reach 45 dB in the SB. The size of this proposed LPF is 0.13λ × 0.09λ; λ is the corresponding wavelength of the upper PB edge frequency of 2 GHz.


Introduction
In recent years, the miniaturized microstrip low-pass filter (LPF) with high frequency selectivity and wide stop-band (SB) is widely used in a RF/microwave circuit such as satellite and mobile communication systems.However, based on step impedance resonator (SIR), the traditional microstrip LPFs implemented by Richard transform and Kuroda rule can only constitute the LPFs of the type of Butterworth and Chebyshev responses [1].The responses of these two filters with single-sections exhibit poor frequency selectivity.Therefore, in order to obtain a steep selecting edge, the number of filter sections must be increased [2,3].At the same time, the insertion loss in the pass-band (PB) and the physical size of the filter will be increased.On the basis of a previous study [4], Reference [5] proposed a PCB LPF using a semi-lumped parallel resonance circuit which exhibits an elliptic function response.In Reference [6], the coupling between two LP hairpin filters was performed to obtain a bandpass response.These kinds of filters achieve high frequency selectivity, but the SB width is not wide enough, and the attenuation out of the band is insufficient.
In order to realize a microstrip LPF with the characteristics of miniaturization, high selectivity and wide SB suppression at the same time, the research hotspot of the defected ground structure (DGS) of nearly two decades is combined with the SIR hairpin filter in this paper.The DGS was proposed to design LPF by Korean scholars D. Ahn et al. in 2001 [7].From then on, more and more research findings have used DGS to design a high performance microstrip filter.Nechel et al. [8] introduced metamodels in filter design by use of a DGS.The optimal time of the design process decreases drastically, however, the simple slot DGS analyzed in the reference is not suitable for filter design for high frequency selectivity.In Reference [9], asymmetrical Pi-shaped DGSs with Koch fractal curve were used to design compact LPF, however, the multiple asymmetrical fractal structures undoubtedly increase the filter's size.Zeng et al. [10] used compact DGSs with mutual capacity and inductive coupling to broaden the bandwidth and the rejection ratio in SB range; the shortcoming of this tight coupling design is that it increases the PB insertion loss.A compact LPF was designed by combining several fractal DGSs with a conventional multiple stub filter in Reference [11].However, the size of the filter designed in the reference appears insufficiently compact due to the usage of conventional multiple stub LPF in the top layer.
In this paper, we use the SIR hairpin resonator with internal coupling to make the filter more compact and use the properties of the DGS to widen the SB by optimization design.This paper is organized as follows.In Section 2, the electrical characteristics of one-, two-and three-DGS are investigated.A compact LPF with high frequency selectivity is proposed based on SIR hairpin with internal coupling in Section 3. A miniaturization, high selectivity and wide SB suppression microstrip LPF are simulated and experimentally validated in Section 4. Finally, Section 5 gives some concluding remarks.

The Electrical Properties of DGS
The DGS etched on the grounding metal plate for the microstrip circuit can be square-headed, semi-circular-headed dumbbell-shaped, and snake-shaped patterns.In this paper, the square-headed dumbbell-shaped DGS is applied.The 3-D model of one-, two-and three-DGS structures are shown in Figure 1a-c.
Electronics 2019, 8, x FOR PEER REVIEW 2 of 12 introduced metamodels in filter design by use of a DGS.The optimal time of the design process decreases drastically, however, the simple slot DGS analyzed in the reference is not suitable for filter design for high frequency selectivity.In Reference [9], asymmetrical Pi-shaped DGSs with Koch fractal curve were used to design compact LPF, however, the multiple asymmetrical fractal structures undoubtedly increase the filter's size.Zeng et al. [10] used compact DGSs with mutual capacity and inductive coupling to broaden the bandwidth and the rejection ratio in SB range; the shortcoming of this tight coupling design is that it increases the PB insertion loss.A compact LPF was designed by combining several fractal DGSs with a conventional multiple stub filter in Reference [11].However, the size of the filter designed in the reference appears insufficiently compact due to the usage of conventional multiple stub LPF in the top layer.
In this paper, we use the SIR hairpin resonator with internal coupling to make the filter more compact and use the properties of the DGS to widen the SB by optimization design.This paper is organized as follows.In Section 2, the electrical characteristics of one-, two-and three-DGS are investigated.A compact LPF with high frequency selectivity is proposed based on SIR hairpin with internal coupling in Section 3. A miniaturization, high selectivity and wide SB suppression microstrip LPF are simulated and experimentally validated in Section 4. Finally, Section 5 gives some concluding remarks.

The Electrical Properties of DGS
The DGS etched on the grounding metal plate for the microstrip circuit can be square-headed, semi-circular-headed dumbbell-shaped, and snake-shaped patterns.In this paper, the square-headed dumbbell-shaped DGS is applied.The 3-D model of one-, two-and three-DGS structures are shown in Figure 1a-c      The transmission zero of the one-DGS structure shifts to the lower frequency band with the increasing length of B as shown in Figure 2a.The frequencies of the transmission zero are 6.28 GHz, 5.62 GHz, 4.96 GHz, and 4.69 GHz, respectively.Further study shows that the frequencies of transmission zeros decrease with the increasing areas of DGS.Actually, the electrical property of the model of one DGS under the microstrip is equivalent to that of a parallel L-C resonator circuit.The growing area of the DGS corresponds to increasing the equivalent inductance of the parallel resonance circuit; the transmission zero therefore moves to the low frequency with increasing area of the DGS.
Similarly, the simulation frequency response characters of the models consisted of two-and three-DGS units which are presented in Figure 1b,c and Figure 2b,c, respectively.The distance between adjacent DGSs for two-and three-DGS models are D /2 = 3 mm.The sizes of the other parts are set up identical to the one-DGS case.The transmission zero of the one-DGS structure shifts to the lower frequency band with the increasing length of B as shown in Figure 2a.The frequencies of the transmission zero are 6.28 GHz, 5.62 GHz, 4.96 GHz, and 4.69 GHz, respectively.Further study shows that the frequencies of transmission zeros decrease with the increasing areas of DGS.Actually, the electrical property of the model of one DGS under the microstrip is equivalent to that of a parallel L-C resonator circuit.The growing area of the DGS corresponds to increasing the equivalent inductance of the parallel resonance circuit; the transmission zero therefore moves to the low frequency with increasing area of the DGS.
Similarly, the simulation frequency response characters of the models consisted of twoand three-DGS units which are presented in Figure 1b,c and Figure 2b,c, respectively.The distance between adjacent DGSs for two-and three-DGS models are D/2 = 3 mm.The sizes of the other parts are set up identical to the one-DGS case.
The transmission zeros of two-and three-DGS structures also move to the low frequency band with the increasing length of B as shown in Figure 2b,c.By comparing the Figure 2a-c, the frequency positions of transmission zeros decrease with the increasing number of DGS, and the width of the transition band (TB) between PB and SB greatly reduced in the three-DGS case.The interpretation of this phenomenon can be illustrated in Figure 3.The transmission zeros of two-and three-DGS structures also move to the low frequency band with the increasing length of B as shown in Figure 2b,c.By comparing the Figure 2a-c, the frequency positions of transmission zeros decrease with the increasing number of DGS, and the width of the transition band (TB) between PB and SB greatly reduced in the three-DGS case.The interpretation of this phenomenon can be illustrated in Figure 3.
The equivalent circuits of the three-DGS model in Figure 2c.
The equivalent circuit model of the three-DGS model can be modeled by three parallel L-C resonators, as shown in the black dashed box in Figure 3.The mutual coupling effects between each resonator are ignored because of a greater distance between the adjacent DGSs.The relations of the electric parameters for this circuit can be expressed as Via simple mathematical transformation, Equation (1) converts into Equation (2) ( ) ( ) The single equivalent parallel L-C resonator in the red dotted line in Figure 3 reflects the relations of the electric parameters in Equation (2).Because the area of the DGS mainly affects the equivalent inductance of the parallel resonator, the three lengthening side lengths of B triple the parallel inductance.From DGS simulation in Figure 2, it is known that the structure of DGS has the characteristics of low-pass filtering.Its low-pass characteristics can be adjusted by changing the defected area and shape etched on the grounding plate.This characteristic is applied to broaden the SB width of the SIR hairpin LPF in Section 3.

Investigation of Proposed SIR Hairpin LPF
To realize the function of filtering, a filter requires one or more units of resonant structure.The resonant elements of SIR structure for filter designing is an effective way to realize compact filters.Lung-Hwa Hsieh et al. proposed a LPF using a SIR hairpin unit [2].On this basis, combined with M. Makimoto's monograph [12], the following LPF with high attenuation edges are proposed.The cell structure is shown in Figure 4.
The equivalent circuit of the resonator shown in Figure 4 can be regarded as a parallel connection of a single transmission line and two parallel lines with internal coupling.The characteristic impedance and electric length of the single transmission line are S Z and T θ , respectively.The characteristic impedance (the electric length) of the even and odd modes of the parallel lines with The equivalent circuit model of the three-DGS model can be modeled by three parallel L-C resonators, as shown in the black dashed box in Figure 3.The mutual coupling effects between each resonator are ignored because of a greater distance between the adjacent DGSs.The relations of the electric parameters for this circuit can be expressed as Via simple mathematical transformation, Equation (1) converts into Equation ( 2) The single equivalent parallel L-C resonator in the red dotted line in Figure 3 reflects the relations of the electric parameters in Equation (2).Because the area of the DGS mainly affects the equivalent inductance of the parallel resonator, the three lengthening side lengths of B triple the parallel inductance.From DGS simulation in Figure 2, it is known that the structure of DGS has the characteristics of low-pass filtering.Its low-pass characteristics can be adjusted by changing the defected area and shape etched on the grounding plate.This characteristic is applied to broaden the SB width of the SIR hairpin LPF in Section 3.

Investigation of Proposed SIR Hairpin LPF
To realize the function of filtering, a filter requires one or more units of resonant structure.The resonant elements of SIR structure for filter designing is an effective way to realize compact filters.Lung-Hwa Hsieh et al. proposed a LPF using a SIR hairpin unit [2].On this basis, combined with M. Makimoto's monograph [12], the following LPF with high attenuation edges are proposed.The cell structure is shown in Figure 4.
The equivalent circuit of the resonator shown in Figure 4 can be regarded as a parallel connection of a single transmission line and two parallel lines with internal coupling.The characteristic impedance and electric length of the single transmission line are Z S and θ T , respectively.The characteristic impedance (the electric length) of the even and odd modes of the parallel lines with internal coupling are Z pe , Z po (θ pe , θ po ), respectively.The A matrix of the single transmission line and the parallel coupling line are expressed as A I and A I I , respectively.
Electronics 2019, 8, x FOR PEER REVIEW 5 of 12 internal coupling are pe Z , po Z ( pe θ , po θ ), respectively.The A matrix of the single transmission line and the parallel coupling line are expressed as I A and II A , respectively.
The matrix T A is defined as the total A matrix of the above two parallel circuits. where The matrix A T is defined as the total A matrix of the above two parallel circuits. where where D t = A t due to D 1 = A 1 and D 2 = A 2 .A load impedance Z L is connected to the terminal in the circuit.The input admittance Y I can be calculated based on Equation (5): The resonance condition is met under Y I = 0 and Z L = ∞, therefore, substituting the calculated value in Equation (5), the resonance condition is obtained as follows: Based on the resonance condition of Equation ( 8), the characteristics of the above resonator can be analyzed using the microwave simulator method [12].Figure 5 shows the geometry of the resonator connected with two tapped microstrip lines and its equivalent circuit diagram.
Electronics 2019, 8, x FOR PEER REVIEW 6 of 12 The resonance condition is met under I Y = 0 and L Z = ∞ , therefore, substituting the calculated value in Equation ( 5), the resonance condition is obtained as follows: Based on the resonance condition of Equation ( 8), the characteristics of the above resonator can be analyzed using the microwave simulator method [12].Figure 5 shows the geometry of the resonator connected with two tapped microstrip lines and its equivalent circuit diagram.The nonideal inductance L M in Figure 5b is the equivalent inductance of L 3 , L 2 , L 3 and two 45 • bends in Figure 5a.L B is the equivalent inductance of L 4 .The non-ideal capacitances C S , C P are equivalent to the capacitances generated by two open coupling lines, in which C S is generated by the coupling effect of two lines; C P is generated by the coupling between the terminal microstrip line and the ground plate; and Z 0 is the characteristic impedance of two feeding lines.The equivalent circuit in Figure 5b shows that the proposed structure has the property of an elliptic function-like LPF.The corresponding circuit model in Figure 5a is simulated by ADS software.The simulation is to design a LPF with a cut-off frequency of 2 GHz under the condition that the relative dielectric constant of the microstrip substrate is 10.8 and the thickness of the substrate is 0.5 mm.The simulated schematic in ADS software is shown in Figure 6a.To improve the frequency selectivity, the structural dimensions of the hairpin in Figure 5a are obtained through software optimizing measures, which are listed in Table 1.The transmission characteristics of the filters tuned by software are shown in Figure 6b.As shown in Figure 6b, the proposed single SIR filter possesses the characteristic of high selectivity and produces two attenuation poles within a finite frequency band.This is the transmission characteristic of the three-branch elliptic function LPF.The disadvantage of this filter is that the SB width is narrow.Although the SB width can be properly enlarged by changing the size of each part of the SIR, it sacrifices the original characteristics of high frequency selectivity.Next, the SB width will be extended without changing the above structure sizes.That is to say, without changing the high edge frequency selectivity of the filter, the SB bandwidth is extended.The method is to introduce DGS structure.

Model Simulation and Sample Measurement
The proposed filter model is shown in Figure 7 using the design method of combining DGS with an SIR hairpin filter.The optimized dimensions of the SIR hairpin filter on the top layer are identical to those in Figure 5a.The sizes of the dielectric substrate and the three DGS patterns are the same with those in Figure 1c

MSub Goal OptimGoal2
RangeMax [1]=6GHz RangeMin [1]=3.5GHz RangeVar As shown in Figure 6b, the proposed single SIR filter possesses the characteristic of high selectivity and produces two attenuation poles within a finite frequency band.This is the transmission characteristic of the three-branch elliptic function LPF.The disadvantage of this filter is that the SB width is narrow.Although the SB width can be properly enlarged by changing the size of each part of the SIR, it sacrifices the original characteristics of high frequency selectivity.Next, the SB width will be extended without changing the above structure sizes.That is to say, without changing the high edge frequency selectivity of the filter, the SB bandwidth is extended.The method is to introduce DGS structure.

Model Simulation and Sample Measurement
The proposed filter model is shown in Figure 7 using the design method of combining DGS with an SIR hairpin filter.The optimized dimensions of the SIR hairpin filter on the top layer are identical to those in Figure 5a.The sizes of the dielectric substrate and the three DGS patterns are the same with those in Figure 1c except that the length of the middle GDS B 2 is slightly longer than those of the two side GDS B 1 .The optimized lengths of B 1 and B 2 are different in order to widen the valid SB width.The electrical parameters of the dielectric substrate and the size of each part of the filter are listed in Table 1.
Electronics 2019, 8, x FOR PEER REVIEW 8 of 12 SB width.The electrical parameters of the dielectric substrate and the size of each part of the filter are listed in Table 1.
(a) (b) The proposed filter is fabricated and shown in Figure 8.The Keysight N5224A PNA network analyzer is used to measure the S-parameters of the proposed LPFs. Figure 9 presents the simulated and measured results.The red lines are the S-parameters for a single SIR hairpin filter.These results obtained by full-wave electromagnetic simulation coincide with the results achieved by the circuit simulation using ADS software in Figure 6.
From the simulation results in Figure 9, it can be seen that the transmission coefficient of 21,dB S is less than 0.38 dB from 0 Hz to 2 GHz, and the SB width is from 2.42 GHz to 8.2 GHz (see the blue curve) with a rejection of greater than 20 dB.Fortunately, the testing results show a broader SB width from 2.42 GHz to 9 GHz (see the black curve) with the same frequency selectivity and the attenuation in SB.The upper SB value even exceeds 10 GHz if the maximum attenuation degree is determined as 14.45 dB.The simulation and testing results are in good agreement.The proposed filter is fabricated and shown in Figure 8.The Keysight N5224A PNA network analyzer is used to measure the S-parameters of the proposed LPFs. Figure 9 presents the simulated and measured results.The red lines are the S-parameters for a single SIR hairpin filter.These results obtained by full-wave electromagnetic simulation coincide with the results achieved by the circuit simulation using ADS software in Figure 6.
From the simulation results in Figure 9, it can be seen that the transmission coefficient of S 21,dB is less than 0.38 dB from 0 Hz to 2 GHz, and the SB width is from 2.42 GHz to 8.2 GHz (see the blue curve) with a rejection of greater than 20 dB.Fortunately, the testing results show a broader SB width from 2.42 GHz to 9 GHz (see the black curve) with the same frequency selectivity and the attenuation in SB.The upper SB value even exceeds 10 GHz if the maximum attenuation degree is determined as 14.45 dB.The simulation and testing results are in good agreement.
Electronics 2019, 8, x FOR PEER REVIEW 8 of 12 SB width.The electrical parameters of the dielectric substrate and the size of each part of the filter are listed in Table 1.
(a) (b) The proposed filter is fabricated and shown in Figure 8.The Keysight N5224A PNA network analyzer is used to measure the S-parameters of the proposed LPFs. Figure 9 presents the simulated and measured results.The red lines are the S-parameters for a single SIR hairpin filter.These results obtained by full-wave electromagnetic simulation coincide with the results achieved by the circuit simulation using ADS software in Figure 6.
From the simulation results in Figure 9, it can be seen that the transmission coefficient of 21,dB S is less than 0.38 dB from 0 Hz to 2 GHz, and the SB width is from 2.42 GHz to 8.2 GHz (see the blue curve) with a rejection of greater than 20 dB.Fortunately, the testing results show a broader SB width from 2.42 GHz to 9 GHz (see the black curve) with the same frequency selectivity and the attenuation in SB.The upper SB value even exceeds 10 GHz if the maximum attenuation degree is determined as 14.45 dB.The simulation and testing results are in good agreement. where is the load reflection coefficient, and in Figure 10b, it can be found that the first resonant frequency ,1 R f = 1 GHz.Therefore, the SIR hairpin filter is a resonant element.
The quality-factor (Q) of a filter is the parameter to present the ratio of reactive power to active power.In the bandpass filter, the higher Q values mean the narrower bandwidth of the filter [13].Based on IN Z , the quality-factor (Q) value is given by Q = ( ) f .This is because the very accurate frequency points for resonance failed to take using the discrete digital computing method.
Figure 10.The input impedance of the SIR hairpin resonant filter.For the hairpin resonant filter, the simulated S-parameters shown in Figure 9 by the red curves can be used to obtain the input impedance Z I N [1]: where is the load reflection coefficient, and Z 0 and Z L are the characteristic impedance of the transfer line and the load impedance, respectively.The Z 0 and Z L in our manuscript are also set to 50 Ω.Therefore, the Z I N = 50(1 + S 11 )/(1 − S 11 ) and its values are shown in Figure 10.From the Z I N in Figure 10b, it can be found that the first resonant frequency f R,1 = 1 GHz.Therefore, the SIR hairpin filter is a resonant element.The quality-factor (Q) of a filter is the parameter to present the ratio of reactive power to active power.In the bandpass filter, the higher Q values mean the narrower bandwidth of the filter [13].Based on Z I N , the quality-factor (Q) value is given by Q = Im(Z I N )/Re(Z I N ).The Q values reach 1120, 1214 and 0.86 at the two attenuation poles f A,1 = 2.75 GHz, f A,2 = 3.14 GHz and the cut-off frequency f C = 2 GHz, respectively.The Q values theoretically would be 0 at the resonant frequency points.The Q values at resonant frequency points f R,1 , • • • , f R,6 are very small values rather than zero, e.g., Q f R,1 = 0.0078 at f R,1 .This is because the very accurate frequency points for resonance failed to take using the discrete digital computing method. where is the load reflection coefficient, and in Figure 10b, it can be found that the first resonant frequency ,1 R f = 1 GHz.Therefore, the SIR hairpin filter is a resonant element.
The quality-factor (Q) of a filter is the parameter to present the ratio of reactive power to active power.In the bandpass filter, the higher Q values mean the narrower bandwidth of the filter [13].Based on IN Z , the quality-factor (Q) value is given by Q = ( ) f .This is because the very accurate frequency points for resonance failed to take using the discrete digital computing method.
Figure 10.The input impedance of the SIR hairpin resonant filter.According to the S-parameters calculated in Figure 9, we present the insertion loss (IL) in decibel form based on the equation of IL dB = 10lg|S 21 | 2 = 2S 21,dB [1,14,15].Therefore, the values of IL dB are twice the values of S 21,dB over the range of the investigated frequency.
The group delay as the derivative of the phase varies as an important parameter in microwave filter design.The smaller the group delay variation, the better the flat property of the designed filter.The simulation and measured values of group delay are shown in Figure 11.From this figure, the group delay varies from 0.23 to 0.78 ns in the pass band of 0~2 GHz.That is to say, the maximum variation of group delay is about 0.55 ns, representing a good flat property of the designed filter.
According to the S-parameters calculated in Figure 9 The group delay as the derivative of the phase varies as an important parameter in microwave filter design.The smaller the group delay variation, the better the flat property of the designed filter.The simulation and measured values of group delay are shown in Figure 11.From this figure, the group delay varies from 0.23 to 0.78 ns in the pass band of 0~2 GHz.That is to say, the maximum variation of group delay is about 0.55 ns, representing a good flat property of the designed filter.Some LPFs proposed in the references are compared with this paper in Table 2.The proposed LPF in this paper has smaller pass-band ripple (PBR) by comparing with the LPF in Reference [10].The size of our designed LPF is less than those in Reference [9,11] with similar SB and TB width.The attenuation in stop-band (ASB) of LPF in Reference [2] is better than that of our design, however, the size of the designed filter in the reference is larger than our proposed filter in this paper.Therefore, the LPF proposed in this paper possesses the properties of miniaturization, wide SB, high selectivity, and low PBR simultaneously.

Conclusions
Firstly, the square-headed dumbbell-shaped DGS is introduced.The transmission characteristics of one-, two-and three-DGS units are analyzed and compared by the 3D numerical simulation software HFSS.The results show that the structure of DGS has the property of low-pass filtering, and the SB attenuation extremum frequency decreases with the increase of the area and number of DGS units.
Secondly, a compact LPF is designed by using SIR hairpin resonator with internal coupling, which is equivalent to three-branch elliptic function LPF.The optimal sizes of the SIR LPF are obtained by the engineering electromagnetic software ADS of Agilent company.From the result of Some LPFs proposed in the references are compared with this paper in Table 2.The proposed LPF in this paper has smaller pass-band ripple (PBR) by comparing with the LPF in Reference [10].The size of our designed LPF is less than those in Reference [9,11] with similar SB and TB width.The attenuation in stop-band (ASB) of LPF in Reference [2] is better than that of our design, however, the size of the designed filter in the reference is larger than our proposed filter in this paper.Therefore, the LPF proposed in this paper possesses the properties of miniaturization, wide SB, high selectivity, and low PBR simultaneously.

Conclusions
Firstly, the square-headed dumbbell-shaped DGS is introduced.The transmission characteristics of one-, two-and three-DGS units are analyzed and compared by the 3D numerical simulation software HFSS.The results show that the structure of DGS has the property of low-pass filtering, and the SB attenuation extremum frequency decreases with the increase of the area and number of DGS units.
Secondly, a compact LPF is designed by using SIR hairpin resonator with internal coupling, which is equivalent to three-branch elliptic function LPF.The optimal sizes of the SIR LPF are obtained by the engineering electromagnetic software ADS of Agilent company.From the result of the simulation,

Figure 2 .
Figure 2. Frequency responses of return loss and insertion loss of the model of (a) one-DGS, (b) two-DGS and (c) three-DGS.

Figure 2 .
Figure 2. Frequency responses of return loss and insertion loss of the model of (a) one-DGS, (b) two-DGS and (c) three-DGS.

Figure 3 .
Figure 3.The equivalent circuits of the three-DGS model in Figure 2c.

Figure 5 .Figure 5 .
Figure 5. (a) Layout of LPF constructed by SIR hairpin with internal coupling; (b) equivalent circuit diagram of microstrip LPF.

Figure 6 .
Figure 6.SIR hairpin LPF composed of single resonator with internal coupling.(a) The schematic diagram modeled in Agilent ADS software.(b) The simulation transmission characteristics.

Figure 6 .
Figure 6.SIR hairpin LPF composed of single resonator with internal coupling.(a) The schematic diagram modeled in Agilent ADS software.(b) The simulation transmission characteristics.

Figure 7 .
Figure 7. LPF model based on SIR and DGS.(a) 3-D model diagram and (b) DGS pattern diagram on the backside of the proposed filter.

Figure 8 .Figure 7 .
Figure 8.The fabricated microstrip LPF based on SIR and DGS: (a) top layer and (b) bottom layer.

Figure 7 .
Figure 7. LPF model based on SIR and DGS.(a) 3-D model diagram and (b) DGS pattern diagram on the backside of the proposed filter.

0 10 .
Z and L Z are the characteristic impedance of the transfer line and the load impedance, respectively.The 0 Z and L Z in our manuscript are also set to 50 Ω .Therefore, the IN Z From the IN Z Re INZ .The Q values reach 1120, 1214 and 0.86 at the two attenuation poles ,1 A f =2.75 GHz, ,2 A f =3.14 GHz and the cut-off frequency C f =2 GHz, respectively.The Q values theoretically would be 0 at the resonant frequency points.The Q values at resonant frequency points ,1 R f ,  , ,6 R f are very small values rather than zero, e.g.,

Figure 9 .
Figure 9.The simulation and measured results of S-parameters.

Figure 9 .
Figure 9.The simulation and measured results of S-parameters

0 10 .
Z and L Z are the characteristic impedance of the transfer line and the load impedance, respectively.The 0 Z and L Z in our manuscript are also set to 50 Ω .Therefore, the IN Z From the IN Z 14 GHz and the cut-off frequency C f =2 GHz, respectively.The Q values theoretically would be 0 at the resonant frequency points.The Q values at resonant frequency points,1 R f ,  , ,6 R fare very small values rather than zero, e.g.,

Figure 10 .
Figure 10.The input impedance of the SIR hairpin resonant filter.Figure 10.The input impedance of the SIR hairpin resonant filter.(a) Global view and (b) Partial view.

[ 1 ,
, we present the insertion loss (IL) in decibel form based on the equation of 14,15].Therefore, the values of dB IL are twice the values of 21,dB S over the range of the investigated frequency.

Figure 11 .
Figure 11.The simulation and measured values of group delay of the designed filter.

Figure 11 .
Figure 11.The simulation and measured values of group delay of the designed filter.

Table 1 .
. Figure2ashows of the whole model is about L × W × H .The detailed numerical values of the model are listed in Figure2ashows the simulated frequency responses of the return loss 11 S and the insertion loss 21 S of one-DGS case when the length B of DGS are 5.8, 6.8, 7.8, and 8.3 mm, respectively.

SIR hairpin filter Values (mm) DGS Values (mm)
except that the length of the middle GDS 2 B is slightly longer than those of the two side GDS 1 B .The optimized lengths of 1 B and 2 B are different in order to widen the valid

Table 2 .
Performance comparisons between the reported filters and the proposed filter in this paper (SB: stop-band, TB: transition band, PBR: pass-band ripple, ASB: attenuation in stop-band).

Table 2 .
Performance comparisons between the reported filters and the proposed filter in this paper (SB: stop-band, TB: transition band, PBR: pass-band ripple, ASB: attenuation in stop-band).