диафрагмированные волноводные фильтры / e36f6e7b-f7b2-45d3-859e-56861867ca77

Abstract— A systematic design method for high-order dualband bandpass frequency selective surfaces (FSSs) with a low profile is derived from classical filter theory and presented here. To complement the design procedure, a multilayer double-slot resonator unit cell topology is proposed for realizing dual-band operations. For simplicity, the resonators are made to work for only a single polarization. To design the FSS, first, a classical dual-band bandpass filter circuit is designed by performing successive frequency transformations on a lowpass prototype. The filter is then transformed into a form resembling the equivalent circuit of the proposed multilayer FSS structure. Finally, the transformed filter is mapped to a set of FSS geometrical parameters. The method presents very few inherent limitations to realizing a diverse range of filter responses. The resulting designs lend themselves to fabrication since very few layers of metallization are required. Two FSSs with third-order passbands at 4 and 7 GHz but different passband characteristics are designed and verified numerically. One of the designs is fabricated and experimentally verified. The overall thickness of the designs is 0.08λl where λl is the free-space wavelength at 4 GHz. The unit cell size is approximately λl /8.

Index Terms— Filter theory, frequency selective surfaces (FSS), frequency transformation, multiband.

I. INTRODUCTION

FREQUENCY selective surfaces (FSSs) are 2-D periodic structures engineered to selectively transmit or reject electromagnetic waves of certain frequencies [1], often constructed by etching periodic metallization patterns on to dielectric substrates. The exact frequency response of an FSS depends on the design of the FSS unit cell as well as the angle of incidence and polarization of the impinging wave. Due to their filtering properties, FSSs are desirable in a wide range of applications including radomes [2], dichroic reflectors [3], radar cross section manipulation [4] as well as control of signal propagation inside complex wireless systems [5]. FSSs with multiple

Manuscript received February 5, 2018; revised July 6, 2018; accepted August 3, 2018. Date of publication August 21, 2018; date of current version October 29, 2018. (Corresponding author: Gengyu Xu.)

G. Xu is with The Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S 3H7, Canada (e-mail: xugengyu@gmail.com).

G. V. Eleftheriades and S. V. Hum are with the Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S 3H7, Canada (e-mail: gelefth@waves.utoronto.ca; sean.hum@utoronto.ca).

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/TAP.2018.2866503

independently controllable bands of operation are especially sought after in modern day communication infrastructures due to the multitude of emerging communication standards. In addition, it is desirable to have low profile FSSs with highly selective passbands or stopbands.

There is a large body of work on realizing high-order filtering responses using low-profile FSSs. For example, using some approximations and transformations of the equivalent circuit model (ECM) of their multilayer FSS structure, Behdad and Al-Joumayly [6], [7] demonstrated that the thickness of the dielectric substrate between adjacent metallization layers can be leveraged as a tuning tool to place the poles of the FSS at desired locations, and thus realize different types of highly selective filter responses. However, the substrate thickness cannot be changed conveniently in practical scenarios since only a few standard thicknesses are readily available.

FSSs with multiple operating bands have also been realized in several ways. Gao et al. [8] have constructed a dual-passband FSS based on transmission line resonators whose second and first harmonics can be individually tuned by loading the dielectric support with capacitive gratings. For practical reasons, the authors recommend a f2/ f1 ratio of around 1.6–2.7, where f2 and f1 are the center frequencies of the upper and lower bands, respectively.

Another approach for creating high-order dual-passband FSSs has been demonstrated by Al-Joumayly and Behdad [9]. To achieve two second-order passbands, a fourth-order singleband FSS with four layers of metallization has been designed. The four poles of this single-band filter are separated into two groups of two and a zero was inserted between them, creating two independent passbands with high out-of-band insertion loss. This approach is excellent for designs that require two closely spaced passbands.

Salehi and Behdad [10] proposed a way of realizing dualband operation using hybrid resonators with two different modes of operations. At each band, one mode of operation is dominant over the other. This technique is effective for applications demanding two passbands far apart from each other.

An FSS with three bands of operation was reported by Zhou et al. [11]. The triband response was realized by combining several easily tunable resonators into a large unit cell. The design is very intuitive but requires a large periodicity and exhibits relatively low out-of-band rejection.

Recently, Zhao et al. [12] demonstrated a novel FSS topology which takes advantage of interlayer electromagnetic coupling to fit multiple resonators into a single unit cell. Many resonators are joined together in series using an intricate network of vias and traces to create the desired multiband response.

Most techniques reported in the literature employ ad hoc approaches to achieve high-order multiband behavior, leaving the task of realizing specific passband characteristics to full-wave (FW) numerical optimization. This leads to a limited design space, unoptimized frequency responses and/or time-consuming fine-tuning processes. A generalized and systematic methodology that can efficiently realize different response types (e.g., maximally flat or equiripple) is highly desirable.

In this paper, we propose a semianalytic technique for synthesizing thin dual-band bandpass FSSs derived from classical filter theory. Passband bandwidth and spacing are restricted only by manufacturing tolerances, but not the technique itself. In addition, many types of filter responses are achievable based on classical filter synthesis whereby frequency scaling and transformations are performed on a lowpass prototype whose parameters are obtained from appropriate filter tables. Accompanying the design procedure, a double complementary dogbone slot (DCDS) unit cell topology is proposed for realizing dual-band operations.

To validate the proposed technique, two third-order FSSs with different response types are designed and verified numerically. Both designs have very low profiles, which serve to preserve their filtering performance even when a wave is incident at large oblique angles or has a nonplanar wavefront [13]. One of the designs is fabricated and experimentally verified.

II. CIRCUIT-BASED DESIGN

In order to take full advantage of the tools available from classical filter theory, we begin our design process in the circuit domain. We start by designing a lowpass filter (LPF) prototype (Fig. 1) based on the required frequency response characteristics (i.e., maximally flat, equiripple, and so on). The values of {gi } can be obtained from appropriate filter tables [14]. The input impedance of this prototype can be expressed as a continued fraction ZLPF which depends on the frequency ω, its 3-dB cutoff frequency ωc , {gi }, and characteristic impedance Zo.

To convert the LPF prototype into the desired dual-bandpass filter (DBPF), we can perform successive frequency transformations [15], whereby the LPF is first transformed into a bandpass filter (BPF) by applying (1a) [16]. The BPF is

Fig. 2. Graphical depiction of the successive transformations.

Fig. 3. Typical dual-band bandpass filter.

subsequently transformed into a DBPF using (1b) [17]

Here, ω denotes the frequency of the BPF and denotes the frequency of the DBPF. ω− and ω+ are the lower and upper cutoff frequencies of the BPF, respectively. 1 and 2 denote the center frequencies of the lower passband and upper passband of the DBPF, respectively.

The successive transformations are depicted in Fig. 2, in which −1 , +1 , −2 , and +2 denote the lower and upper cutoff frequencies of the first and the second passbands, respectively.

Substituting (1b) into (1a) and replacing all instances of ω in ZLPF with the resulting expression yields ZDBPF, which when deconstructed, can be recognized as the input impedance of the DBPF circuit depicted in Fig. 3. From this expression, the required values of {Li{1,2}} and {Ci{1,2}} (i = 1, 2, . . . , N ) for achieving the desired band characteristics can be obtained.

Note that (1) leaves three critical frequencies which are free to choose in the DBPF response: 1, 2, and any one of { −1 , +1 , −2 , +2 }. This implies the bandwidths of the two passbands are related to each other and we may only specify one of them. As an illustrative example, we will

We use the factor F here to keep the equations concise. Substituting (2) into (1a) before performing the aforementioned frequency transformations ensures that the circuit

parameters of the DBPF (i.e., Fig. 3) are functions of design variables 1, 2, and −1 only.

Although Fig. 3 represents an intuitive implementation of a classical DBPF filter, it is hard to directly realize this type of circuit in FSS form due to the potentially large series elements in the even-numbered stages. Alternative implementations consisting of only shunt elements can be realized using admittance inverters. However, in order to reduce the overall profile of the FSS, we must avoid using bulky implementations such as λ/4 transformers which correspond to λ/4 thick dielectric slabs in an FSS. One possible solution using negative inductors is shown in Fig. 4. The inductive μ -networks each behaves approximately as a J inverter with

where we judiciously choose o to be equal to the geometric mean of 1 and 2. Since the admittance inversion is exact only at o, the two passbands of the DBPF will experience slight distortion mainly in the form of bandwidth alteration and pole displacement. The closer a band is to o, the less distortion it will experience. Fortunately, both forms of distortion can be fixed at a later stage via circuit-based optimization (see Section V).

In Fig. 4, the negative shunt inductors −Lt, + can be

i i 1

absorbed into the positive inductors which they are in parallel

with (L or L + ). Since the values of Lt, + are arbitrary,

i2 (i 1)2 i i 1

we pick them to be equal to μoμr hi,i+1 , the equivalent inductances of transmission line sections with lengths hi,i+1 .

This allows us to replace the positive series inductors Lt, +

i i 1

with transmission lines if hi,i+1 are small compared to the wavelength and if we compensate for the capacitances of the

lines appropriately as shown in Fig. 5, in which C t, + =

i i 1

i,i+1 hi,i+1 and Zi,i+1 = μo /i,i+1 . Note that hi,i+1 can be equal to the thickness of any available substrate. For simplicity, we will henceforth assume that all transmission line

sections have the same length h, relative permittivity r , and characteristic impedance Z1.

Absorbing the negative capacitors into the shunt capacitors (Ci2) of each stage, we arrive at the circuit shown in Fig. 6. As an example, the circuit parameters for a third order filter

The circuit in Fig. 6 has an equivalent form which is illustrated in Fig. 7. The two circuits are related by the

Fig. 7 can be interpreted as the equivalent circuit seen by an electromagnetic plane wave impinging upon an N -layer FSS if we set Zo as 377 , the wave impedance of free space. Each layer of this FSS is modeled by a double LC resonator and is separated from its neighbors by dielectric substrates with thickness h and relative permittivity (Zo/Z1)2. To physically implement this hypothetical FSS, we first derive the required values of {Cia }, {Cib }, {Lia }, and {Cib } using (4), (5), and (7). This provides useful insight into the required behavior of each layer of the FSS such as the required pole-zero locations of the resonators. In Section III we propose a unit cell topology suitable for practical realization of the FSS. In sections IV and V, we describe how to leverage the aforementioned insights in our design process.

Fig. 8. DCDS unit cell.

Fig. 9. ECM for a single layer of DCDS surface.

III. PROPOSED UNIT CELL

For simplicity, we will assume that the input wave is x -polarized. This means we only need to design the response of the unit cell for that polarization and can ignore the response when the cell is rotated about the z-axis.

To realize the desired dual-band bandpass response, we propose a very intuitive double complementary dogbone slot (DCDS) unit cell design as depicted in Fig. 8, which shows one period of DCDS metallization sandwiched between dielectric slabs of thickness hu and hl . Such a structure can be modeled by the equivalent circuit shown in Fig. 9, assuming the incident wave is x -polarized. The double LC resonators model the equivalent wave impedance of the metallic layer. As illustrated in Fig. 10, the capacitances Ca and Cb originate

Fig. 10. Working principle of the DCDS unitcell.

from the two slots perpendicular to the E field while the inductances La and Lb originate from the strips parallel to

the E field. To increase the capacitances, one may reduce the gap sizes (gca , gcb ) or increase their lengths (wca , wcb ). To increase the inductances, one may increase the length of the strips (lla , llb ) or reduce their widths (wla , wlb ). The short transmission line sections of lengths hu and hl are used to model the total dielectric supports above and below the metal surface, respectively. The characteristic impedance of the transmission lines is Z1 = Zo/√ r where Zo is the wave

At any of the two resonant frequencies, the corresponding resonator behaves as an open circuit and the input impedance seen by the impinging wave becomes that of Zo transformed by a very short transmission line section; as a consequence there are two peaks in the transmission spectrum of the single-layer DCDS FSS in the neighborhoods of its two resonance frequencies. By Foster’s reaction theorem, between these two poles, there must be a transmission zero.

The selectivity of an isolated shunt LC resonator can be characterized by the factor

It is a dimensionless quantity related to the quality factor of the resonator. A higher ξi leads to a more selective (sharper) transmission peak near ωi . Specifying ωi and ξi is sufficient to uniquely determine Li and Ci . When two shunt LC resonators are placed in series as in Fig. 9, increasing the ξi of one of the resonators has the effect of shifting the transmission zero toward the resonance frequency of that resonator. By tuning ξa and ξb to a specific ratio, it is possible to place the zero anywhere in between ωa and ωb .

With two independently controllable poles, a single-layer DCDS surface constitutes only a first-order dual-band filter. However, comparing Fig. 9 with Fig. 7, it is clear that the N th-order DBPF we wish to create may be implemented using N layers of periodic DCDS surfaces joined together by N − 1 dielectric substrates. We just need to carefully place the poles and zero of each layer, whose required locations can be found from (4), (5), and (7).

Note that a key difference between the proposed method and those previously reported such as [6] is that the thickness

accordingly. This leads to a much larger selection of achievable filters. In this regard, our methodology is comparable to that discussed in [9]. However, the process presented here is much more systematic, granting more precise control over the exact behavior of the passbands. In addition, there are no inherent limitations on the passband bandwidths or spacing. The only restriction arises from the minimum achievable feature size of the manufacturing process, which places a lower limit on the size of the capacitive gaps and inductive strips and hence bounds the maximum realizable capacitance and inductances.

Several simple extensions can be made to the DCDS unit cell to enhance its performance. For instance, it is often desirable to make the unit cell size of an FSS as small as possible so as to fit more unit cells within a limited physical area. However, it is difficult to achieve sufficiently high capacitances and/or inductances with a DCDS surface when the periodicity is small. One solution is to use a variation of the DCDS unit cell employing interdigitated capacitors (IDCs) and meandering inductors (Fig. 11). The length and number of fingers can be adjusted to tune the capacitance of the IDC while the inductance can be controlled by the length and number of turns in the meandering inductor.

As mentioned earlier, the proposed DCDS unit cell behaves as intended only for x-polarized plane waves; thus it is only suitable for applications in which the polarization of the input wave is known (e.g., as tools to enhance antenna performance). Comparable cell topologies that exhibit a bandpass response for both polarizations are available [21]. To increase the number of passbands, more slot resonators may be placed in series into the unit cell using more involved design procedures. However, the unit cell size might need to be increased in order to accommodate the additional resonators.

IV. PHYSICAL IMPLEMENTATION

Having identified an appropriate constituent unit cell for the dual-band FSS, we now wish to map the equivalent circuit parameters obtained from (4), (5), and (7) to a set of geometrical parameters for the physical structure. To do this, each stage of Fig. 7 is mapped individually to a single DCDS layer. Then, they are combined together to form the complete design. The mapping of each layer is accomplished via an iterative tuning process involving FW as well as circuitbased simulations in which the DCDS layer is tuned to exhibit the same pole-zero locations as its circuit counterpart. Each iteration of the FW simulations is performed with all dielectric layers but only one metallic layer (the one being mapped) present while its desired pole-zero locations are obtained by simulating the circuit design with all transmission line sections but only one double LC resonator. Note that it is imperative to include all dielectric layers in the FW simulations despite the exclusion of all but one metallic layer because the capacitance of a slot resonator is dependent on the total dielectric thickness

single-layer DCDS surface may be placed at desired locations by tuning the capacitive gaps and the inductive strips as described in Section III.

the zero location by varying the ratio ξa /ξb while holding ωa and ωb constant. One possible way of increasing ξi while maintaining ωi , for example, would be to decrease gci and increase wli .

Matching the location of the zero only ensures that the ratio between ξa and ξb is correct. In addition to that, we must match the transmission amplitude at another arbitrary point. This guarantees ξa and ξb are individually correct and is achieved by simultaneously increasing or decreasing ξa and ξb by the same factor, while holding ωa and ωb constant.

V. COUPLING COMPENSATION

The ECM shown in Fig. 7 does not capture interlayer electromagnetic coupling interactions, which can be significant in designs with low profiles. Some important manifestations of the coupling interaction are the alteration of the bandwidth of the FSS from its designed values as well as the displacement of poles and zeros in addition to the contribution of the approximations described in Section II. Fortunately, it is possible to efficiently compensate for these effects via circuit-based optimization, rather than brute force FW optimization, using a process similar to one previously proposed. Incidentally, this process also alleviates the distortion caused by the circuit approximations described in Section II.

We first propose a complete model for the FSS which explicitly accounts for coupling effects. The nature of coupling for the multilayer DCDS FSS is found to be mostly inductive, which can be captured by adding mutual inductances to every pair of inductors in the circuit. Capacitive coupling is not required for modeling the behavior of the FSS to sufficient accuracy. As an illustrative example, the complete ECM for a three-layer surface is shown in Fig. 12. In general, the coupling

interactions can be captured by a 2N × 2N mutual inductance matrix in which the entries relating inductors to themselves as well as the entries relating two inductors of the same stage are zero.

As outlined previously [22], we first design the FSS without regards for coupling. This yields an initial structure whose frequency response deviates from the desired one. We then extract the inductive coupling coefficients of this structure by inserting its incomplete ECM (Fig. 7) parameters into the complete ECM (Fig. 12) and finding the set of {M j,k } that results in the best match with its FW-simulated response. Then, we set {Lia , Lib , Cia , Cib } as optimization variables while fixing {M j,k } and try to tune the parameters until they are able to produce the desired frequency response by appropriately compensating for coupling. Comparing the optimal set {Lia , Lib , Cia , Cib } (which provides the best match with the target response) to the original unoptimized parameters provides intuitive directions on how the geometries of each DCDS layer should be tuned. Fortunately, the equivalent inductances and capacitances of each slot can be tuned by moderate amounts without drastically affecting interlayer coupling. This means that each layer of the FSS can be tuned individually in absence of others before they are combined again to form the complete optimized design. Since each tuning iteration requires FW simulation with only one metallic layer rather than all N layers, it is very computationally efficient. For the most accurate realization of the desired frequency response, multiple iterations of the proposed procedure can be performed.

VI. NUMERICAL RESULTS

Following the proposed methodology, we design two FSSs with third-order passbands at 4 and 7 GHz. Each FSS is designed based on a different lowpass prototype. Design A is to have as little ripple as possible in the passbands, thus it is based on a Butterworth prototype. Design B is to have larger bandwidths and sharper transition regions than design A, while permitting larger passband ripples; it is based on a Chebychev prototype. The lower band fractional bandwidths are specified to be 15% for design A and 20% for design B. To reiterate, these characteristics are arbitrarily chosen and specified using (2), (4), and (5). They will not meet exactly by the initial

Fig. 13. Geometrical parameters of design A (dimensions in millimeters).

Fig. 14. FW and ECM simulated response of design A at normal incidence.

design. However, through the iterative compensation method described in Section V, we are able to match them very closely.

Both designs have an interlayer dielectric thickness of 3.15 mm (0.08 λ at 4 GHz) with r = 3. The thickness is set as such to account for bonding layers. The unit cell size of both designs is 9.5 mm (0.125 λ at 4 GHz).

A. Minimal Ripple FSS

The optimized equivalent circuit parameters for Design A which compensate for coupling effects are given in Table I(a), while the extracted coupling coefficients are given in Table I(b). The geometric parameters of the FSS are shown in Fig. 13.

Fig. 14 shows the transmission and reflection spectra of the design at normal incidence. The response exhibits highly selective passbands at the intended locations. The fractional

Fig. 16. Design A reflection spectrum (scanning H-plane).

bandwidths for the lower and the upper bands are approximately 14.7% and 8%, respectively. The results obtained from the equivalent circuit matches those of the FW simulation almost exactly. While the response is not completely free of ripples due to approximations applied during design (e.g., limited bandwidth of the J inverter and transmission line approximation for the substrate), the maximum magnitude of the ripples has been reduced to merely 0.25 dB via the compensation method described previously.

To numerically characterize the sharpness of the transition regions, we examine the four bandedges of the frequency response. For each edge, we define a f which is the frequency interval it takes for the transmission coefficient to rise from −20 to −3 dB or to drop from −3 to −20 dB. For design A, we have f equal to 0.34, 0.17, 0.27, and 0.34 GHz for the four edges.

Figs. 15–18 illustrate the response of the design at various oblique incidence angles. Evidently, the performance is stable when scanning in both the E-plane as well as the H-plane even at large angles. The two planes of incidence are illustrated in Fig. 19.

B. Equiripple FSS

Design B, while having the same thickness and unit cell size as design A, is to have a larger bandwidth and sharper cutoff at both bands. In order to achieve this, we allow larger ripples in the passbands. To that end, it is designed based on a Chebychev LPF prototype, and optimized to have 3-dB ripples in the lower band and 2-dB ripples in the upper band. While these ripples may be too large in practical applications, the intent here is to demonstrate precise control over the