диафрагмированные волноводные фильтры / 0f3a5702-9a20-41ab-9046-ed352fa9947d

Abstract—A new class of three-dimensional (3-D) bandpass frequency-selective structures (FSSs) with multiple transmission zeros is presented to realize wide out-of-band rejection. The proposed FSSs are based on a two-dimensional (2-D) array of shielded microstrip lines with shorting via to ground, where two different resonators in the substrate are constructed based on the excited substrate mode. Furthermore, metallic plates of rectangular shape and “T-type” are inserted in the air region of shielded microstrip lines, which can introduce additional resonators provided by the air mode. Using this arrangement, a passband with two transmission poles can be obtained. Moreover, multiple transmission zeros outside the passband are produced for improving the out-of-band rejection. The operating principles of these FSSs are explained with the aid of equivalent circuit models. Two examples are designed, fabricated, and measured to verify the proposed structures and circuit models. Measured results demonstrate that the FSSs exhibit high out-of-band rejection and stable filtering response under a large variation of the incidence angle.

Index Terms—Frequency-selective structure (FSS), microstrip line, multiple transmission zeros.

I. INTRODUCTION

F REQUENCY-SELECTIVE surfaces are spatial filters for electromagnetic waves, which have been widely investigated over the past few decades for many applications, such as antenna subreflectors, radomes, and polarizers [1], [2]. These spatial filters can be designed to exhibit bandpass or bandstop response [2], while the bandpass frequency-selective surface is more widely used. A traditional bandpass frequency-selec- tive surface, which consists of a two-dimensional (2-D) periodic array of slots/apertures etched out of a conducting plate, exhibits poor filtering characteristics, such as low selectivity and unstable angular response. Although cascading a number of these 2-D surfaces with dielectric spacers can improve the filtering performance [2]–[5], these designs are difficult to obtain wide out-of-band rejection with stable response under a large

variation of the incidence angle.

Realization of a bandpass filtering response with wide out-of-band rejection has been extensively studied for planar microstrip filters [6]–[8]. Usually, wide out-of-band rejection

Manuscript received April 22, 2013; revised August 13, 2013; accepted August 16, 2013. Date of publication September 16, 2013; date of current version October 02, 2013.

The authors are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 (e-mail: ezxshen@ntu. edu.sg).

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMTT.2013.2279776

is obtained by introducing transmission zeros at desired finite frequencies, which can be realized by using stubs [6], coupled lines [7], or electromagnetic bandgap (EBG) structures [8]. Unlike microstrip filters, only a few investigations for fre- quency-selective surfaces have been reported by introducing transmission zeros at desired frequencies. A three-layer fre- quency-selective surface where two patch arrays are coupled through a 2-D periodic array of slots was proposed in [9] and one transmission zero was produced in the lower rejection band. Three-dimensional (3-D) frequency-selective surfaces based on substrate integrated waveguide (SIW) structures were reported in [10]–[12], where transmission zeros were obtained by employing couplings between different resonant modes in these SIW cavities. A third-order frequency-selective surface with a tunable transmission zero was recently presented in [13], where the transmission zero was created by introducing a hybrid resonator in one of the layers. In [14], a 3-D frequency-selective surface based on a circular tube element was described. Most of the above existing surfaces can introduce a transmission zero close to the passband, realizing high selectivity near the passband. However, few designs can realize high selectivity together with wide out-of-band rejection characteristics.

Recently, a concept of 3-D frequency-selective structure (FSS) was reported in [15], where multiple transmission zeros/poles can be produced by a 3-D FSS with multimode resonators. Based on this concept, a number of 3-D FSSs based on a 2-D periodic array of shielded microstrip line resonators were proposed in [15]–[19]. Because two quasi-TEM modes (air and substrate modes) are excited in the shielded microstrip line, transmission poles and zeros at desired frequencies can be realized by using resonances and couplings of multiple propagation modes. Unfortunately, the structures described in [15]–[19] exhibit only a bandstop filtering response together with low-pass characteristics in lower frequencies. Although combining an array of shielded microstrip lines with an array of rectangular waveguides can exhibit quasi-elliptic bandpass filtering response [20], this structure still has difficulty realizing wide out-of-band rejection.

This paper proposes a new type of 3-D bandpass FSS with multiple transmission zeros for achieving high selectivity and wide stopband characteristics. It is modified from the structure of shielded microstrip lines with shorting via briefly reported in [18] for bandstop response. Two design examples are presented to explain the operating principle and to verify existence of multiple transmission zeros. In the first design with two transmission zeros, a rectangular metallic plate is inserted in the air path of the shielded microstrip-line structure. Due to this inserted plate, two other resonators in the air path are introduced

Fig. 1. Generalized equivalent circuit model of a 3-D FSS with multiple resonators (modes).

to produce two transmission zeros at higher frequencies, while resonators in the substrate path remain unchanged and provide the passband with two transmission poles. In the second design with three transmission zeros, a “T-type” metallic piece is inserted in the air path of every unit cell, which can provide three transmission zeros, thus leading to an enhanced out-of-band rejection at both sides of the passband.

This paper is organized as follows. Section II describes a generalized equivalent circuit model for a 3-D FSS with multiple transmission zeros. Section III introduces a dual-band bandstop FSS, on which our proposed FSSs are based. Its operating principle is explained in detail based on an equivalent circuit model. Section IV proposes a 3-D bandpass FSS with two transmission zeros. A prototype is designed, fabricated, and tested to verify our concept and to demonstrate the filtering performance of the 3-D FSS. In Section V, an improved 3-D bandpass FSS is presented to realize three transmission zeros. A design example is also given and measured results are in good agreement with simulated ones. Finally, concluding remarks are given in Section VI.

II. OPERATING PRINCIPLE

Fig. 1 shows a generalized equivalent circuit model of a 3-D FSS unit cell containing a number of resonators (or propagation

can provide additional transmission zeros at desired frequencies, which can either improve the selectivity of a bandpass FSS or increase the operating bandwidth of a bandstop FSS. Based on this equivalent circuit model, a 3-D FSS can easily produce a desired number of transmission zeros/poles at finite frequencies by controlling resonances of these resonators, leading to a quasi-elliptic or elliptic filtering response [15]–[20]. It should be noted that a traditional 2-D frequency-selective surface, including only one resonator or one mode, is actually the special case of this equivalent circuit model for 3-D FSSs, with only one transmission pole/zero and, hence, suffers from poor

useful to briefly review the operating principle of the 3-D FSS in [15]. When the electric (-) field of an incident plane wave is

perpendicular to the strip lines (TE polarization), two propagation paths [15], the air and substrate paths inside the shielded microstrip line, are formed, linking the input and output ports. Each path can be seen as a resonator ( 1, 2), where air and substrate quasi-TEM modes are excited, respectively. At lower frequencies, most signals go from one port to the other through the air path, thus leading to a low-pass response. At higher frequencies, signals can pass through both paths. Because the guided wavelength of the substrate mode is smaller than that of the air mode, the substrate-mode resonator will resonate first. When the phase difference between signals along the substrate and air paths is 180 degrees at the output port, transmission zeros may then be produced.

In this paper, the dual-band bandstop FSS to be described in Section III includes two substrate-mode resonators and , and an air-mode resonator linking the input and output ports. These two transmission zeros are mainly determined by and . provides a transmission pole at a very high frequency. The bandpass FSS with two transmission zeros introduced in Section IV is constructed by blocking the air path of the dual-band bandstop FSS with an inserted metallic plate.

Consequently, is replaced with and , while

and are unchanged. Transmission poles in the passband are provided by and , and transmission zeros at the upper stopband are produced by and . The bandpass FSS with three transmission zeros presented in Section V also contains three resonators , , and , which are similar to the dual-band bandstop FSS. However, the inserted “T-type” metallic piece produces strong coupling between and

as well as between and , thus providing transmission zeros at both sides of the passband.

III.DUAL-BAND BANDSTOP FSS

A.Description of the Structure

Fig. 2 shows the perspective view of a 3-D dual-band bandstop FSS, which was briefly described in [18]. and denote the zenith and azimuthal angles of the incident plane wave, respectively. In this paper, we present the equivalent circuit model and design equations for this dual-band bandstop FSS to gain insight into the operating principle of our proposed bandpass FSSs with wide out-of-band rejection. It should be mentioned that the dual-band bandstop FSS is a modified version of the simple array of shielded microstrip lines in [15], which exhibits only single-band filtering response. The difference between them is that a shorting via is introduced at the center of the microstrip line connecting the microstrip line and ground, as shown in Fig. 2. Periods along the - and -axes are denoted by and , respectively. Thickness of the FSS along the -axis is denoted by . The diameter of via hole is represented by .

Fig. 3 provides the simulated -parameter results of the dual-band bandstop FSS using a commercial full-wave simulator CST Microwave Studio (CST-MWS). It is observed that two transmission zeros are obtained at and . Compared with the narrowband design in [15], an extra transmission zero is introduced at . This is attributed to the shorting via hole between the microstrip line to ground, which introduces an extra short-circuited resonator, in addition to the existing

Fig. 2. Structure of the 3-D dual-band bandstop FSS. (a) Perspective view.

(b) Top view of a unit cell in the plane. (c) Side view of a unit cell in the plane.

Fig. 3. Comparison of -parameter results of CST-MWS with those obtained from the equivalent circuit model in Fig. 5(a) for the bandstop FSS (TE incidence, ).

open-circuited microstrip line resonator. The detailed operating principle can be explained by an equivalent circuit model in Section III-B.

B. Equivalent Circuit Model Analysis

As discussed in [21], the free-space region representing the incoming plane wave can be equivalent to a parallel-plate waveguide when the polarization of the incoming wave is perpendicular to the strips. Based on this, a waveguide structure provided in Fig. 4 can be used to model the dual-band bandstop FSS shown in Fig. 2. The air-to-microstrip line discontinuities between air and the FSS are denoted by dashed lines in Fig. 4. According to the descriptions in [22], the structure shown in Fig. 4 can be regarded as an E-plane bifurcation in a parallel-plate

waveguide, whose equivalent circuit model is established in Fig. 5(a). In this equivalent circuit model, represents the discontinuity between the parallel-plate waveguide and the air path of the FSS. denotes the discontinuity between the waveguide and the substrate region. The transmission line and represent the air and substrate propagation paths, respectively. is the inductance of the shorting via hole. As pointed out in [23], the equivalent circuit model can be divided into two series subnetworks I and II, as shown in Fig. 5(a). The transfer matrices of these two subnetworks can be expressed, respectively, as

(1a)

(1b)

where , and and are the electrical lengths of the air and substrate paths. By connecting these two subnetworks in series, the impedance parameters of the equivalent circuit model can be written as follows:

(2a)

(2b)

After the impedance matrix of the equivalent circuit model shown in Fig. 5(a) is obtained, we can then calculate the scattering parameters [23] using the following equations:

(3a)

(3b)

where is the characteristic impedance of the parallel-plate waveguide.

On the other hand, the circuit parameters shown in Fig. 5(a) can be estimated from physical dimensions of the structure. and can be approximately evaluated as

(5)

Fig. 4. Unit cell of the dual-band bandstop FSS with a parallel-plate waveguide.

Fig. 5. Equivalent circuit model and -field distributions for the dual-band bandstop FSS at transmission-zero frequencies and . (a) Equivalent circuit model. (b) -field distribution at . (c) -field distribution at .

The characteristic impedances ( and ) and electrical lengths ( and ) of the air and substrate paths can be obtained by using the Eigen-mode Solver of CST-MWS [21]. It is noted that the periodic boundary of the structure shown in Fig. 4 can be replaced by perfect magnetic conductor (PMC) boundary under normal incidence. Moreover, the propagation constants ( and ) of these two modes can also be calculated, as discussed in [21], and the electrical lengths of the air and substrate paths can then be estimated as and , respectively.

By substituting (4)–(6) into (1)–(3), we can obtain the -parameter results of the equivalent circuit model when physical dimensions of the FSS are given. Fig. 3 compares the simulated frequency responses for the proposed FSS from CST-MWS and those obtained from the equivalent circuit model using Advanced Design System (ADS) 2012, where a good agreement can be observed. The physical dimensions of the dual-band bandstop FSS and its corresponding circuit parameters are listed in Table I. It is obvious that, when

than 10 GHz, there is a discrepancy between the -parameter results from equivalent circuit model and EM simulation. This is mainly because the values of capacitors and are

assumed to be frequency-independent in the equivalent circuit model, while the air-to-microstrip line discontinuities vary at high frequencies.

Fig. 5(b) and (c) illustrates the -field lines (in the plane)

of the proposed bandstop FSS at these two transmission-zero frequencies and . It is seen that, at frequency , the

-field vectors in the substrate path have the same magnitude and phase at both sides of via hole. The substrate path can be equivalent to two short-circuited resonators , which con-

sist of half of the microstrip line and the via hole. At frequency , the -field vectors at both sides of via hole along the mi-

crostrip line still have the same magnitude but are out of phase. In this case, the substrate path can be represented as an open-cir- cuited resonator , which comprises only the microstrip line. At both frequencies, signals that pass through the air path have reverse phase shifts with those from the substrate path at the output port. Since the air path can only resonate at a higher frequency than that of the substrate path, it is concluded that two transmission-zero frequencies are mainly determined by the substrate-mode resonators and . This means that these two transmission-zero frequencies should be close to the resonant frequencies of and , respectively, while the air path has limited influence on the locations of these two transmis- sion-zero frequencies. In addition, as observed in Fig. 3, a trans- mission-pole around 12 GHz is obtained from the simulated result of the equivalent circuit model. This transmission pole can be attributed to a resonance of the air path, which is seen as an open-circuited resonator . At this frequency, signals propagating through the air path and substrate path have the same phase shift at the output port. Moreover, this transmission-pole frequency is realized at around 13.5 GHz from EM simulation, where a 1.5-GHz frequency difference can be observed. This difference is mainly due to the fact that the equivalent circuit

parameter representing the air-to-microstrip line discontinuity is assumed to be frequency-independent in the equivalent circuit model.

Fig. 6 shows the transmission-zero frequencies for different diameters of the via hole and different widths of strip lines obtained from equivalent circuit model and EM simulation. Again, a good agreement can be achieved, which verifies the validity of the equivalent circuit model for the dual-band bandstop FSS. Furthermore, only affects the resonant frequency and a smaller leads to a lower . This is because inductance is mainly determined by diameter , which is only included in resonator . The width of microstrip line can affect and

because its effect is included in both resonators. Finally, it may be mentioned that the periods and have similar effects (smaller values leading to wider bandwidth) on the overall frequency response to the single-band structure in [15] because they are structurally the same except the shorting via hole.

IV. BANDPASS FSS WITH TWO TRANSMISSION ZEROS

Based on the understanding gained from the dual-band bandstop FSS, it is easy to develop a new FSS with bandpass response by blocking one of these two propagation paths in the structure. In the sections, we will show how to modify the dualband bandstop FSS to realize a new type of bandpass FSS with wide stopband response.

Fig. 7. Structure of the 3-D bandpass FSS with two transmission zeros.

(a) Perspective view. (b) Top view of a unit cell in the plane. (c) Side view of a unit cell in the plane.

A. Description of the Structure

Fig. 7 shows the topology of the 3-D bandpass FSS with two transmission zeros and a unit cell. It is seen that, in every unit cell, the microstrip line and the shorting via hole is unchanged, compared with the structure in Fig. 2. Moreover, a thin metallic plate is inserted in the air path. The distances between the metallic plate and two ends of the microstrip line are and

. In this structure, the substrate propagation path is the same to the FSS in Fig. 2. However, the air propagation path is blocked by the inserted metallic plate because the inserted metallic plate connects the microstrip line and the top PEC, as shown in Fig. 7. Signals coupled to the air path will be reflected by the inserted metallic plate. Therefore, the air quasi-TEM mode cannot propagate from the input port to output port though it is still excited. Fig. 8 presents the simulated -parameter results of the bandpass FSS shown in Fig. 7, where a bandpass filtering response with two transmission poles at and and a high-frequency stopband with two transmission zeros ( and ) can be observed. Its operating principle will be explained in Section IV-B.

B. Equivalent Circuit Model Analysis

Compared to the dual-band bandstop FSS, only a metallic plate is inserted into the air path in this bandpass FSS. The equivalent circuit model illustrated in Fig. 9 can be employed to explain the operating principle of the bandpass FSS, which is modified from Fig. 5(a). Since a metallic plate can be seen as a perfect conductor, a short-circuited line is used to represent the metallic plate. The air path is then divided into two reflecting paths, which are represented with two short-circuited transmission lines and . On the other hand, the equivalent circuit of the substrate path for the bandpass FSS is the

Fig. 8. Comparison of -parameters of CST-MWS with those obtained from the equivalent circuit model in Fig. 9 for the bandpass FSS (all parameters are the same as those in Table I: 5.3 mm, 4 mm, 13.87 GHz, 18.37 GHz, TE incidence, ).

Fig. 9. Equivalent circuit model of the 3-D bandpass FSS with two transmission zeros.

same as that for the dual-band bandstop FSS. Consequently, the transfer matrix of the modified equivalent circuit model can be written as

(7)

Subsequently, we can calculate the scattering parameters of the equivalent circuit model as follows [23]:

(8a)

(8b)

Similarly, by substituting (4)–(6) into (7) and (8), we can compute the -parameter results of the proposed FSS. Fig. 8 shows the frequency responses for the bandpass FSS using CST-MWS and those obtained from the equivalent circuit model, where a good agreement can be observed. The electrical lengths and can be obtained by calculating the propagation constant of the air mode [21]. As discussed earlier, there is a discrepancy between these results in higher frequencies. This is due to the fact the lumped elements and are not very accurate in higher frequencies and higher-order modes are not included in the equivalent circuit model.

As seen from the equivalent circuit model shown in Fig. 9, two resonators and are constructed in the substrate

path and they are similar to the substrate-mode resonators in the dual-band bandstop structure in Section III. As discussed earlier, signals may only pass through the substrate path and are reflected by the air path because the air path is totally blocked by the inserted metallic plate. Therefore, a passband with two transmission poles can be formed around the resonant frequencies of resonators and , as shown in Fig. 9. Beyond the passband, the substrate path exhibits strong reflection due to the shorting via hole inside the substrate. Moreover, the air path is represented by two short-circuited transmission lines, as illustrated in Fig. 9. These two transmission lines will resonate at frequencies higher than the passband because the wavelength in the air path is relatively longer than that in substrate. Transmission zeros can then be achieved close to the resonant frequencies of these two short-circuited transmission lines represented by two resonators and , as shown in Fig. 9. The main difference between and is the electrical length, which leads to different resonant frequencies.

Since transmission poles of the bandpass FSS are determined by resonators and , the variation tendency of transmis- sion-pole frequencies should be similar to that of transmissionzero frequencies shown in Fig. 6. On the other hand, these two transmission zeros are mainly controlled by two air-mode resonators and respectively. Fig. 10 illustrates the resonant frequencies of transmission poles/zeros varied with the lengths of and (or the location of the metallic plate). It is observed that, when adjusting , the transmission-zero frequency

varies accordingly, while the other transmission-zero frequency remains unchanged. When tuning changes accordingly, while is very stable, as shown in Fig. 10(b). It is concluded that these two transmission zeros can be designed independently. Moreover, the transmission-pole frequencies and are not affected when varying and . This indicates that we can adjust the bandwidth of the stopband while keeping the passband unchanged, which provides more freedom for the FSS design. On the other hand, the ripple of the upper stopband is determined by the separation of these two transmission-zero frequencies. It is obvious that the stopband ripple increases (or decreases) when enlarging (or reducing) the separation of these two transmission zeros. In a special case of 4.5 mm, these two transmission zeros are merged into one, as shown in Fig. 10. This is because resonators and have the same resonant length when the metallic plate is placed in the middle of the microstrip line.

C. Design Example

In order to verify the design concept, a prototype of the bandpass FSS with a center frequency of 8.0 GHz is fabricated and measured. In this design, we use a Rogers 4230 substrate (

, 1.524 mm, ) to fabricate all the unit cells. The final dimensions of the FSS unit cell are as follows:

frequency of 8.0 GHz.

Fig. 11 presents simulated transmission and reflection coefficients of the designed FSS for both TE and TM polarizations

under various oblique incidence angles. It is observed that the desired filtering response of the proposed FSS is achieved for TE polarization only. This is because the electric field component of the incident plane wave is perpendicular to the strips for TE polarization, which can excite air and substrate quasi-TEM modes of the shielded microstrip line, as described in the previous subsections. Furthermore, because of its small cell size, the proposed FSS exhibits stable frequency response under oblique incidence of angles up to 60 . For TM polarization, air and substrate quasi-TEM modes cannot be excited because the electric field component of the incident plane wave is parallel to the strips. Signals are then reflected because both the air and substrate paths are shorted by the metallic plate and via hole.

Fig. 12(a) shows a photograph of the fabricated prototype, which is approximately 200 mm 180 mm in size and consists

of 40 50 unit cells. The metallic plate for providing transmission zeros is realized using aluminum due to its light weight. In the final realization, a long aluminum plate (200 mm in length) is used between each of two adjacent microstrip-line circuit boards because the length of the aluminum plate in every unit cell is equal to the period along the -axis, as shown in Fig. 12(b).

Meanwhile, the aluminum plate has both thickness and height of 2 mm.

The fabricated bandpass FSS is then measured by the free-space method using two horn antennas in a controlled

environment of an anechoic chamber. The measurement set-up has been described in [20]. Fig. 13 shows the measured and simulated -parameter results of the fabricated FSS under TE polarization and two incidence angles (0 and 40 ). It is seen that the fabricated FSS can successfully provide a passband with two transmission poles at 7.5 and 8.8 GHz and a stopband with two transmission zeros at 11.9 and 18.4 GHz at . The measured 3-dB bandwidth of the passband is 2.72 GHz, which indicates a relative bandwidth of 34%. The measured insertion loss at the center frequency of the passband is 0.8 dB under the normal incidence, which is certainly greater than the simulated one (0.2 dB). A bigger measured insertion loss may be due to the surface roughness of the fabricated aluminum plates and the connection loss between aluminum plates and copper strips printed on substrates. The measured bandwidth of stopband better than 15-dB rejection is more than 8 GHz, which indicates a very wide out-of-band rejection. It is observed that there is a 10-dB difference between the measured and simulated results for the rejection around 15 GHz. Such a discrepancy is mainly attributed to the small variation in the location of the metallic plates, which determines the locations of transmission zeros and controls the ripple level of the upper stopband. As seen in Fig. 13, the first measured transmission zero moves toward to a lower frequency compared with the simulated one,

Fig. 12. (a) Photograph of the fabricated 3-D bandpass FSS shown in Fig. 7.

(b) Assembly details.

Fig. 13. Simulated and measured results of the fabricated 3-D bandpass FSS shown in Fig. 12 under TE polarization and oblique incidence angles.

while the second measured transmission zero shifts to a higher frequency. The separation between these two transmission zeros increases in the measured results, thus resulting in a higher ripple in the upper stopband.

V. BANDPASS FSS WITH THREE TRANSMISSION ZEROS

Although the 3-D bandpass FSS shown in Fig. 7 exhibits wide stopband in the upper side of passband, the selectivity in the

Fig. 14. Structure of the 3-D bandpass FSS with three transmission zeros.

(a) Perspective view. (b) Top view of a unit cell in the plane. (c) Side view of a unit cell in the plane.

lower side of the passband is not high because no transmission zero is introduced except the one at zero frequency. In many applications, it is desired that a fast roll-off characteristic at both sides of the passband should be realized. For this purpose, another new 3-D bandpass FSS with three transmission zeros is investigated in this section.

A. Description of the Structure

Fig. 14 illustrates the 3-D bandpass FSS with three transmission zeros and its unit-cell topology. In this structure, the substrate path is also the same as the structures shown in Figs. 2 and 7. Unlike the rectangular metallic plates used in Fig. 7, “T-type” metallic pieces are employed in the air path of this FSS. In order to simplify the design and fabrication procedure, each “T-type” metallic piece is inserted at the center of the microstrip line, which also connects the center shorting via hole. Furthermore, the height of the “T-type” metallic piece is smaller than that of the air path , which means the air propagation path still exists between the input and output ports. Fig. 15 illustrates the simulated -parameter results of the bandpass FSS shown in Fig. 14, where a bandpass filtering response with two transmission poles at and can be observed. Furthermore, it is seen that one transmission zero at is produced below the passband, thus leading to a sharp roll-off for the lower side of the passband. Two transmission zeros at and are also produced at the upper side of the passband and a wideband out-of-band rejection can also be obtained. Although this structure was first reported in [25], its detailed operating principle,

Fig. 15. Comparison of -parameters of CST-MWS with those obtained from the equivalent circuit model in Fig. 16 for the bandpass FSS with three trans-

Fig. 16. Equivalent circuit model of the 3-D bandpass FSS with three transmission zeros.

equivalent circuit model, and measured results have not yet been presented.

B. Equivalent Circuit Model Analysis

Fig. 16 shows the equivalent circuit model of the bandpass FSS with three transmission zeros. Since there is a narrow gap between the “T-type” metallic piece and the top PEC shown in Fig. 14(b) and (c), a lumped capacitor is introduced to represent this region. Based on this, can be approximately seen as a parallel plate capacitor and can be calculated as

(9)

where is the permittivity in free space. The transfer matrix of the air path can be written as follows:

(10)

Replacing (1a) with (10) and then substituting (10) and (1b) into (2) and (3), we can compute the -parameter results of the equivalent circuit model. Fig. 15 compares the simulated frequency responses for this bandpass FSS with three transmission zeros using CST-MWS and those obtained from its equivalent circuit model. A good agreement can be observed except for a

small frequency shift at high frequencies above 10 GHz. This discrepancy is mainly because the equivalent circuit parameters ( and ) of air-to-microstrip line discontinuities are

assumed to be frequency-independent in the equivalent circuit model.

Since the substrate path is unchanged, two transmission

respectively. Therefore, the impedance of the substrate path is inductive before the resonance of . This means the

substrate path can be equivalent to an inductor at frequencies lower than . Meanwhile, the air path is equivalent to a large

capacitor due to the introduction of a narrow gap. Therefore, a transmission zero at is produced when the sum of the

impedances of these two paths is zero. At a frequency higher

when the sum of these two impedances is zero. At very high

, the sum of the impedances of these two paths will be zero again, which is similar to the second transmission zero of the wideband bandstop design in [15]. Based on the above analyses, it is concluded that two transmission zeros close to the passband are produced by the introduced narrow gap between the metallic piece and the top ground (PEC). Fig. 17 illustrates

the first transmission zero moves to a lower frequency and the second transmission-zero frequency increases. This is because

the structure is the same as the dual-band bandstop structure shown in Fig. 2, and the structure is the bandpass structure shown in Fig. 7 when 0 mm.

C. Simulated and Measured Results

A prototype of this bandpass FSS is fabricated and measured. Rogers 4230 substrate (, 1.524 mm,

) is also used to fabricate all the unit cells. Fig. 18(a) shows photographs of the fabricated prototype, which is approximately 180 mm 160 mm in size and consists of 30 27 unit cells. In order to facilitate the fabrication process, all of the dimensions of this FSS are enlarged based on those in Fig. 15. After a quick optimization using CST-MWS, the final

eral unit cells. It is noted that the microstrip lines printed on the Rogers 4230 substrate are cut into a number of 2-D pieces. Each 2-D circuit piece contains 30 unit cells. These 2-D pieces are stacked together using spacers and rods. The height of every

spacer is 4 mm, which is equal to the height of the air region

. “T-type” metallic pieces are also realized using aluminum. In the assembling, each “T-type” metallic piece is inserted into the via hole. This can make sure that the metallic pieces are tightly secured at the center of the microstrip line.

It is noted that periods along - and -axes are and , respectively, where is the free-space wavelength at the center frequency of 8.2 GHz. This also indicates that the fil-

tering response of this FSS should be stable under a large variation of incidence angle because of its small cell size. Fig. 19 shows measured and simulated -parameter results of the fabricated FSS under TE polarization and two incidence angles (0 and 40 ). It is observed that the frequency performance of the proposed FSS is stable under different incidence angles. Furthermore, it is seen that two transmission poles (at 7.9 and 8.4 GHz) in the passband and three transmission zeros in the stopbands (at 6.1, 10, and 17.9 GHz) are obtained. Due to these three transmission zeros, this bandpass FSS can exhibit high selectivity at both sides of the passband and a wideband out-of- band rejection in higher frequencies. The measured bandwidth of high-frequency stopband better than 12-dB rejection is more than 8 GHz. The measured 3-dB bandwidth of the passband is about 1.4 GHz, which is a little narrower than the simulated one. The measured and simulated insertion losses at the center frequency of 8.2 GHz are 1.2 and 0.4 dB, respectively, under normal incidence. Again, a bigger measured insertion loss is mainly caused by the surface roughness of the fabricated alu-

Fig. 19. Simulated and measured results of the fabricated 3-D bandpass FSS with three transmission zeros under oblique incidence angles (TE polarization).

minum plates. Compared with simulated results, the locations of measured transmission zeros/poles are shifted. Meanwhile, the measured rejection ripples in the stopband are higher than those obtained by simulations. Such discrepancies between measured and simulated results are attributed to assembly tolerances and measurement errors.

D. Discussion

Although three transmission zeros allow the FSS to significantly improve the close-to-passband selectivity, the rejection