диафрагмированные волноводные фильтры / b0603d30-ec11-4335-9f54-7d8b96fc6793

Abstract—A design methodology is presented which applies band-pass filter principles to form wideband small-aperture evanescent-mode waveguide antenna designs. This approach permits matching of the complex antenna aperture admittance of an evanescent-mode open-ended waveguide to a real impedance generator, and thereby to an SMA coaxial feeding probe. A reflection coefficient of less than 10 dB was obtained over a measured bandwidth of 15%, from 2.30–2.68 GHz, in 2.72 GHz cutoff waveguide. Capacitive posts and evanescent-mode waveguide sections are used to form the resonators/coupling elements of the antenna’s filtering sections. Measured realised gain variation from 3.5–4.7 dBi is observed across the bandwidth. The antenna’s maximum aperture dimension is at the upper frequency in the bandwidth and so it is suitable for use in a wide angle scanning phased array.

Index Terms—Electrically small aperture, evanescent waveguide, filtering antenna, phased arrays.

I. INTRODUCTION

P HASED arrays are a well-established technique to produce steerable directive beams [1]–[3]. Waveguide or cavity type radiators are particularly suitable for applications that call for medium to high power handling as well as highly isolated array elements (e.g. for wide angle scanning). Despite these favourable characteristics, waveguide and cavity radiators for practical phased array systems suffer from limited bandwidth and/or large geometrical dimensions. On one hand, apertures of propagating waveguides involve dimensions of the order of greater than a half wavelength, which in a phased array arrangement leads to unwanted grating lobes; this becomes increasingly significant as the operating frequency shifts higher with respect to the waveguide cutoff, hence posing additional constraints to the available bandwidth. On the other hand, the impedance of unmodified waveguide apertures exhibits a strong reactive impedance [4], hence matching is not trivial, particularly over a broad bandwidth. Although flaring the waveguide can improve matching, this increases the footprint

Manuscript received October 19, 2011; revised February 08, 2012; accepted September 18, 2012. Date of publication September 21, 2012; date of current version December 28, 2012. This work was supported by the UK Engineering and Physical Science Research Council under Grant EP/E01707X/1 The work of P. Ludlow was supported by Powerwave Technologies and by the Department of Learning (DEL) for Northern Ireland.

The authors are with The Institute of Electronics, Communications and Information Technology, Queen’s University Belfast, Queen’s Island, Belfast BT3 9DT, Northern Ireland, U.K. (e-mail: v.fusco@ecit.qub.ac.uk).

Digital Object Identifier 10.1109/TAP.2012.2220314

and hence is not suitable for phased arrays. Cavity radiators similarly suffer from narrow bandwidth stemming from their resonant operation.

Evanescent-mode open-ended waveguide radiators are formed from an open-ended waveguide that is operated below its cutoff frequency and are therefore more compact electrically than above-cutoff open-ended waveguide antennas. Although this feature can alleviate the limitations associated with the emergence of grating lobes in a phased array, the electromagnetic fields in an evanescent-mode waveguide are reactive and decay exponentially; hence the excitation of the open-ended aperture over a substantial bandwidth remains a significant challenge.

Open-ended waveguide antennas with aperture dimensions may be dielectric-filled to enable propagation [5]–[8].

The designs of [5], [6] use waveguide with dielectric

filling and an evanescent-mode air gap at a distance from the aperture to match from the waveguide’s characteristic impedance to free space. In [5] an 8% bandwidth for reflection

The design technique of [7] uses partially dielectric loaded rectangular waveguides—whereby the centre of the waveguide is air-filled and allows propagation of a pseudo-TEM wave—to achieve antenna designs that have aperture dimensions equal to and at the operating frequency,

for 10 dB reflection coefficient bandwidths of 0.65% and 1.2%, respectively.

In [9], a rectangular evanescent-mode waveguide aperture is matched to free space through loading of the waveguide with double ring resonators printed on to an E-plane oriented dielectric substrate that is centred in the waveguide. The design exploits the principle that backward wave propagation in an evanescent-mode waveguide is possible if the waveguide is loaded in such a way that the effective permeability in the transversal direction of the waveguide is negative. A bandwidth

In [8], [10], [11] loading of the waveguide aperture, with protruding dielectric [8] or variable width strips [10], [11], is used along with more complex feeding arrangements to enable wideband waveguide matching to free space. Reference [8] proposes exciting a dielectric-filled waveguide using an L-shaped feed,

whereby the dielectric (which has ) protrudes in a con- ical-shape from the aperture. A match over a 27% bandwidth is achieved, for aperture width/height across the band-

width. In [10], [11], square evanescent-mode waveguide phased array antennas are presented that are impedance matched to free space through using near-fed coaxial probes that resonate variable width strips placed across the aperture, with a dielectric sheet air-spaced from the aperture to permit wide-angle scan-

The Imaginary Smith Chart was used in [12], [13] to design a narrowband match to the aperture admittance of an evanes- cent-mode open-ended waveguide. In [12] this resulted in a narrowband antenna design with a simulated bandwidth of 3.2% for reflection coefficient 10 dB and a maximum realised gain

of 3.71 dBi. Here a capacitive post and a length of evanescentmode waveguide were used to achieve the desired matching, with the structure in size across the bandwidth. In

[13] a reconfigurable evanescent-mode open-ended waveguide antenna was presented that permitted tuning of reflection coefficient at the coaxial input port across an operating frequency range of 19.4%, with a maximum realised gain of 5 dBi and the maximum dimension of the antenna . This is achieved

through the use of a printed iris with a shunt varactor diode connected across the aperture of the waveguide and a length of evanescent-mode waveguide.

In this paper a new design technique is outlined that enables matching to the aperture admittance of an evanescent-mode open-ended waveguide antenna, and thereby to free space. The proposed technique is based on well-established evanes- cent-mode waveguide filter theory [14] and is therefore efficient and may be readily applied to higher order designs. The iterative design technique is very straightforward and allows the design parameters to be established without extensive electromagnetic simulation. Moreover, the proposed scheme dispenses the need for dielectric blocks, as used in [5]–[8], which can be costly and/or add mass, or other protruding elements, as used in [10], [11]. Furthermore, the feeding arrangement used is much simpler than that of [8], [10], [11] as a standard SMA coaxial feed inserted through the broad wall of the waveguide is used to excite the antenna. The structure of the paper is as follows. In Section II, a detailed design procedure is derived for a wideband antenna design in evanescent-mode waveguide that uses band-pass filter techniques with resonators

to achieve the wideband match. In Section III, a match at 2.4 GHz to a rectangular waveguide aperture with is

designed using this procedure, using the circuit-based simulator, ADS [15], and electromagnetic simulator, CST [16]. The measured input matching and farfield results are then presented in Section IV and compared with simulated results.

II. DESIGN PROCEDURE FOR A WIDEBAND SMALL-APERTURE

EVANESCENT-MODE WAVEGUIDE ANTENNA

According to the proposed method the antenna is designed such that the radiating antenna aperture forms the last resonator and the load resistance/conductance of the filter circuit. If a load to be matched has a reactive part, perfect power transmission to

Fig. 1. (a) (i) T and (ii) equivalent lumped circuit models for an evanescentmode waveguide section ( and ) (b) Evanescent-mode waveguide filter model [14].

the load is possible only at discrete frequencies, and not over a band of frequencies [17]. Furthermore, it will usually be found that the overall transmission can be improved if at least a small amount of power is reflected at all frequencies [17]. It is convenient to characterize the complex load to be matched by its decrement, which is defined in [17] as:

For TE evanescent modes, T or circuits may be used as an equivalent lumped circuit model of a section of waveguide, as shown in Fig. 1(a). The approximation to lumped circuits becomes closer as the frequency of operation moves further below cutoff, with exact correspondence established sufficiently far below the cutoff frequency [18]. A correction factor, , is derived in [14] to account for the deviation in the frequency dependencies of the evanescent-mode waveguide elements and the equivalent lumped elements; while the resonator slope parameters of the lumped resonator equivalents have a pole at infinite frequency, the evanescent-mode waveguide elements will have a pole at the cutoff frequency of the waveguide and therefore, especially for frequencies closer to cutoff, the resonator slope will be greater than for the lumped equivalent.

(2)

By manipulation of the equivalent circuit model such that negative shunt elements of equal magnitude to the series element are introduced [14], an admittance inverter can be formed, as shown in Fig. 1(b). Such an inverter is only possible if the shunt inductive elements in Fig. 1(b) add up algebraically to the original value in Fig. 1(a)(ii). Using the resonance condition, the required shunt capacitance values can be found from:

The value of each resonator therefore depends on the length of the adjacent evanescent-mode waveguide sections and how

far the operating frequency is below cutoff (which in turn determines the values of and ). The value of each J-inverter is determined by:

(4)

The J-inverter values are therefore also determined by the length of evanescent-mode waveguide used and how far the operating frequency is below cutoff.

According to the theory presented in [17], the value of each inverter depends on the value of the slope parameters of the adjacent resonators that are coupled by the inverter, with the first and last J-inverters enabling matching to source and load resistances of a specified value at each side of the filter. In a distributed coupled resonator filter, for instance, the resonator slope parameters are often chosen to be of a convenient value, e.g. by using transmission line of in length as resonators—the value of J-inverters required may therefore be easily determined. However it is not possible in this case to arbitrarily select resonator parameters and then design J-inverters based on these values since both J-inverter and resonator slope parameters depend on the length of each evanescent-mode waveguide section. The value of each J-inverter and resonator therefore have to be co-designed—an iterative design process is described in [18] for narrow to moderate bandwidths.

In the filtering antennas presented in [19], [20] an n-pole filter is firstly designed, using propagating waveguide sections that are coupled by capacitive or inductive irises; a radiating slot antenna element is then built on or inside the nth resonator and the distance to the nth coupling element adjusted as required. Note that the slot antenna elements used in [19], [20] have a bandwidth equal to or greater than that of the filter section, with the main drive of the work in each case being compact integration of filter and antenna. In the evanescent-mode waveguide antenna outlined here band-pass filter techniques are instead applied to enable wideband antenna design in evanescent-mode waveguide.

The difficulty in achieving this is that the coupling element and resonator values in evanescent-mode waveguide filters are interdependent as outlined above and an iterative procedure is therefore necessary, since changing the nth resonator will affect the nth coupling element and thereby the (n-1)th resonator etc. The use of an iterative technique in the design hence enables wideband antenna design in evanescent-mode waveguide, which has smaller transverse dimensions than unmodified propagating waveguide and may therefore be used in a wide-angle scanning phased array.

Modification of the theory outlined in [14] and [18] had to be made to accommodate matching to the complex shunt load represented by the aperture of an evanescent-mode open-ended waveguide antenna. A band-pass filter circuit in evanescentmode waveguide that matches to a shunt resonant antenna load is as shown in Fig. 2 for a filter with order and will be used in the remainder of this section to illustrate the proposed synthesis procedure.

The Q factor of the load has to meet the following criterion [17]:

(5)

Fig. 2. Evanescent-mode waveguide filter with that uses J-inverter coupled resonators matched to a shunt resonant antenna load.

component of the adjacent evanescent-mode waveguide section,

By combining (5) and (6) it is possible to derive an expression for that allows the quality factor for the load to meet the criterion given in (5):

(7)

However there is a further criterion that has to be met with regard to —the value of the J-inverter, , is defined by , as shown in (4); however also has to meet the criterion for a J-inverter used to couple two resonators in a band-pass filter circuit:

(8)

By manipulating (8) it is possible to derive a second expression for ; note that the expression for uses the value of derived above:

(9)

The load decrement, , was defined in (1) as being the reciprocal of the product ; similarly, a source decrement, , may also be defined, which for an filter equals:

(10)

On the source side of the filter the quality factor for the source has to satisfy the following criterion to give reflection coefficient 10 dB across the bandwidth:

(11)

It is now possible to derive an expression for that allows the quality factor of the source to meet the criterion stated in (11):

(12)

However there is a further criterion that has to be met with regard to —by manipulating (8) it is possible to derive an expression for ; note that the expression for uses the value of derived above:

(13)

To ensure that matching is as desired for the filter design it is essential that and , in order to ensure that the criteria stated in (5) and (11) are satisfied. The terms that may be varied in (7), (9), (12) and (13) are , , and . The remaining terms in each expression can then be fixed using: (i) the centre frequency, (ii) the parameters of the antenna load, , , (iii) the values for the value of defined by the load (iv) the parameters of the evanescent-mode waveguide at the centre frequency i.e. characteristic reactance, , correction factor for the resonators, , and propagation constant, , which is defined according to the waveguide width, , for the definition of characteristic impedance used here, i.e. wave impedance.

As a starting point for the iteration, values of and are input to (9) and (13), with the values of and adjusted until and . This allows lengths of evanes- cent-mode waveguide and to be derived that will satisfy

the matching requirements for particular values of and . However, a further step in the iteration is now required since the values of and are also dependent on the centre inverter,

. An expression for may be derived using (8) that will allow to be of the value required by the filter design. Note that the values for and defined at the beginning of the iteration are used in the expression for .

(14)

The iterated value of , , may therefore be written as follows, with the value of given by and the value of given by (18):

The design process for a wideband small-aperture evanes- cent-mode open-ended waveguide antenna that uses a band-pass filter matching network of order may therefore be summarized as follows:

i.Define waveguide height/width, , and the centre frequency, , at which the antenna design is to operate.

ii.Calculate the characteristic reactance, , correction factor for the resonators, , and propagation constant, for these waveguide dimensions at the centre frequency.

iii.For the given waveguide dimensions evaluate the aperture admittance characteristics around this frequency and define an equivalent circuit.

iv.Define the reflection coefficient that is required over the band and from [17] find the value of the load decrement, , required and thereby the values for the filter design.

v.Define values of and and find values of and

that satisfy (7), (9), (12) and (13) such that and .

vi.Input these values of and and into (14)–(16) and determine and .

vii.Repeat steps (v)–(vi) with different starting values of and , which may be higher or lower than the previous ones depending on the values of and calculated in step (vi), with the aim that both converge to unity, i.e. if is 1 then a higher starting value of

should be chosen and vice versa, similarly for . The convergence criteria for the synthesis procedure is therefore that , , , . Once the iteration process is complete, the values of lengths , and are then known (from (7), (9), (14) and (12), (13) respectively). The value of required shunt capacitances is given by:

(17)

(18)

(19)

(20)

The value of is determined by the iteration and so cannot be freely chosen; from design experience its value is generally one-quarter to half of the value of . From (7) and (12) it is possible to define maximum bounds for the bandwidth, , over which the load may be matched for the reflection coefficient desired, since . From (7):

come of the iteration. The variation in the denominator of (21)

significantly for frequencies closer to the cutoff frequency of an evanescent-mode waveguide, in particular for waveguides with a higher ratio [5]. It is thereby evident that the maximum

achievable bandwidth is higher for waveguides with a higher ratio or for centre frequencies closer to the waveguide cutoff

frequency, although obviously must be less than

. Transverse waveguide dimensions may therefore be chosen to give a wider bandwidth, depending on the physical space

onators used and the reflection coefficient desired across the band as this determines the value of for the design and thereby

the values used. With a greater number of resonators used, the desired reflection coefficient may be obtained with a lower value of [17]—this is equivalent to the Bode-Fano criterion

[21], which states that with a greater number of resonators used, the desired reflection coefficient may be obtained with a higher value of Q-bandwidth product. According to [17] there is an inverse relationship between and and therefore if a greater

number of resonators are used to obtain a match with a particular reflection coefficient across the bandwidth, a higher value of is required in the design. Hence, according to (21) with more resonators used a higher bandwidth may be obtained, however there are rapidly diminishing returns for each increase in [17].

III. DESIGN OF WIDEBAND SMALL-APERTURE

EVANESCENT-MODE WAVEGUIDE ANTENNA

To validate the synthesis procedure, an evanescent-mode waveguide antenna with a centre frequency of 2.4 GHz has

. Full-wave electromagnetic simulators (here CST Microwave Studio was used) may be used to determine the aperture admittance of an open-ended waveguide. The aperture admittance of a waveguide with and

varies as shown in Fig. 3 with frequency:

The next step in the design process is to define an equiva-

were found to give a good lumped parallel LC approximation to the aperture susceptance over the band from 2.1–2.7 GHz, as shown in Fig. 3. Although the nominal value of

Approximately 8–10 iterations are typically required to resolve

TABLE I

DIMENSIONS OF WIDEBAND SMALL-APERTURE EVANESCENT-MODE

WAVEGUIDE ANTENNA [LENGTHS IN MM, CAPACITANCES IN PF]

to two decimal places (whereby the lengths are in mm and the inductance values are in nH). This gives the lengths of evanes-

The final parameters of the evanescent-mode waveguide antenna are as shown in Table I.

Values of (and therefore ) and

are required to complete the iteration, meaning that a bandwidth of 20.4% should be obtainable with the design. The next step in the design process is to export the aperture admittance characteristics from CST, such that the antenna may be simulated in a circuit simulator (here Agilent ADS was used). Care should be taken when co-simulating waveguide elements in CST and ADS as the wave impedance definition of characteristic impedance is used in CST while the power-voltage defini-

coaxial probe be used to excite the antenna, so some distance between the coaxial probe and first obstacle is therefore required, as shown in Fig. 4.

In Fig. 5, the variation in simulated reflection coefficient with frequency at the input to the antenna is shown for the cases

, a bandwidth of 20.2% (2.23–2.73 GHz) is also obtained, with the addition of a short section of evanescent-mode waveguide between source and first obstacle not having much effect on the bandwidth. Simulation in ADS has shown that the antenna design works as intended, with matching over a 20.2% bandwidth being obtained.

The next step in the design process is to convert the design from a circuit-based implementation into a physical design—to achieve this it is necessary to use an electromagnetic simulator such as CST. The lengths of evanescent-mode waveguide used

Fig. 7. Fabricated wideband small-aperture evanescent-mode waveguide antenna (a) perspective view (b) front view.

TABLE II

DIMENSIONS OF WIDEBAND SMALL-APERTURE EVANESCENT-MODE

WAVEGUIDE ANTENNA [MM]

Fig. 8. Variation in measured/simulated reflection coefficient of the wideband small-aperture evanescent-mode waveguide antenna with frequency.

design. The filtering antenna presented in [20] is much larger