2.3 Output cavity

This cavity is the second resonator of the module. A cavitybacked slot antenna is incorporated into this cavity, by following the procedure described in [5, 10]. This leads to an antenna consisting of a radiating element conformed by two meandered slots of length k/2 each one, located at the lower metallization of the cavity (Fig. 1). With the meandering of the slots, the radiating element is inductively loaded and, thus, its resonance frequency is lowered. This allows fitting the slots into the cavity dimensions. The common origin point of the meandered slots is located at the point of maximum current density. The two current components necessary for the accomplishment of a slot antenna are established in this way: the circulating component, responsible for the resonant condition and the perpendicular component, responsible for the radiation.

In this manner, an integrated filter-antenna module is obtained, in which the filtering function is possible thanks to the appropriate coupling of two resonant cavities, one of which is perturbed by the introduction of two meandered slots. These slots, at the time, are the elements that enable the radiation by the module. Therefore, the filtering and radiation functions are performed in a simultaneous way.

3 Design of the filter-antenna module

Since the output cavity of the filter-antenna module is perturbed by the inductive load associated with the meandered slots etched on the lower metallization of it, the resonance frequency of this cavity is different from the resonance frequency of the input cavity. Hence, the filterantenna module can be modeled as an asynchronouslytuned coupled-resonator circuit [11, 12]. A block diagram for this model is presented in Fig. 2.

In Fig. 2, the boxes labeled Bn(x) represent the resonators and the labeled Jn,n?1 are the admittance inverters which model the different kinds of couplings present in the structure:

•J01: Coupling between the generator (internal impedance of 50 X) and the first resonator. This is realized by

means of the access line.

Fig. 2 Model of a coupled-resonator circuit

•J12: Coupling between the two resonators. This is realized by means of the inter-cavity slots.

•J23: Coupling between the second resonator and the external media. This is realized by means of the radiating elements (meandered slots).

The radiation of the meandered slots is modeled by means of the radiation resistance (Rrad).

From this model, the filter-antenna module can be designed as a coupled-resonator circuit. Then, the design procedure consists of three stages:

1.Theoretical determination of the design parameters: Input external quality factor (QE-in), coupling coefficient (k12) and output external quality factor (QE-out) [11–13]. It can be performed for different response characteristics: Chebyshev, Butterworth, linear phase or elliptic. In this case, the design of two filter-antenna modules, one at 2 GHz and the other at 5 GHz, is done for a Chebyshev response.

2.Characterization of the design parameters as a function of the physical dimensions that control them: As a result of this stage, the characterization curves of

Design parameters versus Physical dimensions are obtained.

3.Determination of the required value for the physical dimensions: This is carried out by crossing the required theoretical value of the design parameter with the corresponding characterization curve obtained in the previous stage.

3.1 Theoretical determination of the design parameters

The theoretical determination of the design parameters was carried out by following the directives given in [11–13]. The synthesis of the filter response was performed according to [13]. From the synthesized response, the external quality factors (QE-in, QE-out) and the coupling coefficient (k12) were extracted, according to [11, 12].

3.1.1 Extraction of the theoretical frequency response

In [13], a general procedure to synthesize a filtering response with Chebyshev characteristic is presented. This general procedure allows the synthesis of asymmetrical responses, which correspond to asynchronously-tuned coupled-resonator circuits, as the filter-antenna module to be designed.

The input parameters for the synthesis procedure are listed below:

•Number of coupled resonators (N)

•Central frequency (x0)

•-3 dB fractional bandwidth (FBW)

•Minimum return loss in bandpass (RL)

•Transmission zeros (zeros of the S21 parameter)

As defined, for the design procedure presented in this paper, the number of coupled resonators is always 2 (N = 2). The central frequency, fractional bandwidth and minimum return loss are varied according to the requirements. Given that the frequency response of the filterantenna module does not involve a transmission parameter (S21 parameter), because the module is a 1-port network, a pair of fictitious transmission zeros must be specified as input to the synthesis method, in order to obtain the two required reflection zeros (zeros of the S11 parameter).

The synthesis of the filtering response is based on the expression of the S11 and S21 parameters as ratios of second order polynomial functions of the variable frequency (x), as indicated in (1) and (2).

where e is a constant that normalizes the S21 parameter to the constant ripple value of the bandpass region and is given by (3).

The polynomials F2(x), E2(x) and P2(x) are all second order polynomials. The polynomial P2(x) is readily obtained from the specified transmission zeros. The polynomials F2(x) and E2(x) can be found as indicated in [13]. With these three polynomials, the frequency response of the circuit is completely specified, given that the S11 and S21 parameters can be determined from (1) to (3), as a function of x.

3.1.2 Computation of the coupling matrix (M)

The coupling matrix (M) is a 2 9 2 matrix containing the coupling coefficients of the circuit [13]. For the design of a second order circuit, the only coupling coefficient of interest is k12, which corresponds to the elements M(1,2) = M(2,1). The coupling matrix can be computed from the admittance parameters of the two-port network formed by the circuit to be designed (specifically from the y21 and y22 admittance parameters). From the scattering parameters (S11 and S21) found by means of (1) to (3), the admittance parameters (y21 and y22) can be obtained by using a linear transformation. These admittance parameters can be expanded in partial fractions, as shown in (4).

where the rij,n are the residuals of the partial fractions expansion, and the ki are the poles of both admittance parameters.

From the partial fractions expansion, the elements (tij) of an auxiliary matrix T, with rows of orthogonal vectors, are found by using (5)

r21;k

t1k ¼ p ð5aÞ

r22;k

for k = 1, 2.

In order to compute the coupling matrix corresponding to a prototype coupled-resonator circuit, the matrix T must be orthonormal, which means that its rows, apart from being orthogonal, must be also normal, that is, they must have magnitude equals to one. In order to accomplish this, a pair of normalization constants must be calculated, as shown in (6).

X2

k¼1

Now, the elements (tij0) of a new orthonormal matrix T0 can be computed as indicated in (7).

for k = 1, 2.

Finally, the coupling matrix (M0) of a prototype coupledresonator circuit is computed by using (8).

The normalization constants determined in (6) can be seen as the turns ratio of two transformers connected at the terminations of the coupled-resonator circuit of Fig. 2 in order to get normalized termination resistances (R10 and R20) [11]. It is illustrated in Fig. 3.