диафрагмированные волноводные фильтры / 3b20849a-adae-48af-bedc-d82af4964eef

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 27 No. 6, 2008

pp. 1236-1248

q Emerald Group Publishing Limited 0332-1649

DOI 10.1108/03321640810905729

1. Introduction

Electromagnetic engineering applications require the design of evermore complex systems and devices, encompassing inhomogeneous, possibly exotic, materials and very complex geometrical structures. These kinds of problems can be very accurately handled numerically via, for example, the finite element (FE) method, which is among the most versatile numerical methods available (Silvester and Ferrari, 1996; Pelosi et al., 1998; Martini et al., 2003).

Even if modern computer can easily handle the FE solution of problems of somewhat large dimensions with respect to the working wavelength, in a synthesis procedure, or in a tolerance analysis, the number of subsequent, different, computations required can be so large that the whole process can easily become unaffordable.

In particular, as an example of relevant importance within passive devices, microwave and millimeter-wave waveguide filters may resort to dielectric parts (see, for example Shigesawa et al., 1989; Bui et al., 1984) to achieve the desired performances. These dielectric materials are naturally affected by their own tolerances in their permittivity and permeability values. Understanding how these tolerances reflect on the overall filter behavior is of great relevance to asses the device reliability

is obtained, matrix [D] accounting for the coupling between ME unknowns b and FE unknowns e. The non-null known term accounts for the incident mode feeding the device. This cannot be solved since it contains more unknowns than equations. To solve it the continuity of the electric field tangential to the ports must be explicitly imposed obtaining:

where [B] is again a coupling matrix while [A], for orthonormal modes, is the unit matrix.

3. Domain decomposition

If now we consider the problem domain V decomposed into N non-overlapping sub-domains Vi, as in Figure 2, some considerations are possible. First of all unknowns can be grouped according to the sub-domains to which they belong, then, coupling between sub-domains is naturally reduced to a coupling between the linear systems which can be written, separately, on each sub-domain.

Hence, first of all, vector e in equation (4) can be rearranged so as to have unknowns ordered on a sub-domain-by-sub-domain basis as eT ¼ ½eT1 ; eT2 ; . . . ; eTN &, being the generic ei relative to sub-domain Vi. In a first step let us assume that ei comprehends all the unknowns of Vi, hence also the ones on its boundary ›Vi. This implies that the same boundary unknown may appear twice or more, once for every sub-domain to which it belongs. To overcome the issue of duplicate entries for the unknowns each vector ei can be further decomposed into a vector eTi ¼ ½xTi ; yTi &, where xi contains the inner unknowns (unknowns that are in Vi but not on ›Vi) and yi contains the border unknowns (those on ›Vi). All the various yi can be easily condensed into a single y containing all unknowns on all boundaries, canceling out any possible redundancy.

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port 1

port3 Ω3

Ω1

Ω4 Ω2

pec

2 port

Figure 2.

Generic multiport device containing an inhomogeneous dielectric whose domain V is subdivided into four non-overlapping sub-domains V1, V2, V3

System (4) can hence be recast in the form:

equations (6) and (7) for all the other sub-domains to obtain a partial Schur complement. Then, within the iterative cycle, the complete Schur complement is computed, the system solved and the GSM computed.

4. Numerical results

The method discussed in the previous sections has been used to perform some analyses of the statistical behavior of microwave filters affected by tolerances on the value of the permittivity of their dielectric parts.

First the nominal response of the filter has been computed by assuming that the dielectrics exhibit their proper, nominal, value for the permittivity; then a series of simulations have been carried out, in the framework of a Monte-Carlo approach (Robert and Casella, 1999; MacKay, 1998) performed over a series of simulations in which the material parameters assume a possible value within their tolerance ranges. From the set of scattering parameters obtained a mean mS21 value of the S21 parameter, as well as its standard deviation sS21 , is computed.

Taking into account that, in a Gaussian distribution, 95 percent of samples are included in the range m ^ 2s, a suitable representation of the possible values for S21 can be achieved by plotting curve corresponding to mS21 as well as the two curves

mS21 þ 2sS21 and mS21 2 2sS21 . These will be presented in the following graphs. It is anyway worth noticing that the resulting statistical distribution for S21 is not

Gaussian, especially in the band-pass zone, where for physical reasons S21 can never be greater than 1 (or 0 dB) and the standard deviation, which is linked to the distance between the samples and their mean value, can anyway assume relatively large values. This implies that, mathematically and without any infringement of physical properties of passive devices, mS21 þ 2sS21 can be greater than one, hence letting the pertinent curve to go beyond the boundary of the graphs.

Since, the Monte-Carlo technique heavily relies on a large amount of different evaluations of the filter response any possible acceleration of its computation is highly appreciated. The efficiency of the proposed DD approach is here established and reported.

The first geometry analyzed is a broad band E-plane bandpass filter consisting of metal strips fabricated on a dielectric substrate inserted along the E-plane of a WR6 rectangular waveguide (Figure 3). Such a structure has been firstly presented in Bui et al. (1984). Please note that, since the dielectric metallized slab is inserted along the E-plane the device is actually uniform in that plane and can hence be analyzed in the H-plane.

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Note: Domain decomposition is done in two domains, one comprising only air, the other comprising the dielectric slab

Figure 3.

H-plane section of a millimeter-wave E-plane broadband filter

Figure 4.

Nominal response of the filter in Figure 3 when the slab permittivity is

1r ¼ 2.2

Figure 5.

Nominal response of the filter in Figure 3 when the slab permittivity is

1r ¼ 3.02

Frequency [GHz]

Note: The dotted box highlights the area which will be zoomed in the tolerance analysis results

Frequency [GHz]

Note: The dotted box highlights the area which will be zoomed in the tolerance analysis results

The mean mS21 curve and the mS21 ^ 2sS21 curves are calculated and the results are shown in Figures 6-7.

It is apparent how the response of this kind of filter is relatively stable with respect to permittivity tolerances in band, the ripple being unspoiled, while the bandwidth variation might pose an issue.

An estimate of the time needed for the FE analysis and the DD analysis is shown in Figure 8. It can be appreciated the higher computational efficiency of the method proposed in this paper. In this figure, the computing time attained on an Intel Celeron 1.4 GHz 512 MB RAM are reported. These times are not relevant in their absolute values, which depend on the machine available, but rather in their relative value,

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Frequency [GHz]

Notes: Solid line: nominal filter results; dashed line, mean value of the statistical results; dotted lines, bounding values mS21±2sS21

Figure 6.

Statistical behavior of the amplitude of the S21 parameter,

1r ¼ 2.2 ^ 0.04 case

Notes: Solid line: nominal filter results; dashed line, mean value of the statistical results; dotted lines, bounding values mS21±2sS21

Figure 7.

Statistical behavior of the amplitude of the S21 parameter,

1r ¼ 3.02 ^ 0.05 case

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Figure 8.

Computing time curves for the FE only tolerance analysis (dashed line) and the FE þ DD one (solid line) as a function of the number of iterations

Number of iterations N

Figure 9.

Band pass filter consisting of two parallel waveguides loaded with dielectric resonators (dimensions in millimeters)