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

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all the dielectric resonators whose relative permittivity is affected by tolerances. In this way the variations are enclosed on the V2 sub-domain only.

Figure 10 shows the nominal response of the filter. Figure 11 shows the results of the tolerance analysis. In a first-run set, 22 samples for the dielectric constant randomly generated in the range of tolerance 3.78 ^ 0.05. These random values were considered equal for all the four dielectric blocks.

Since, the assumption that all the blocks of dielectric exactly exhibit the same permittivity is somewhat too restrictive, a second set of analyses has been then performed, now considering a larger number of samples (100). The dielectric constant is again randomly generated in the range of tolerance 3.78 ^ 0.05 but, this time, each of

FE-DD based permittivity tolerance

1245

	0
	– 10
	– 20
[dB]	– 30
Parameter	– 40
	– 50
	– 60
21
S
	– 70
	– 80
	– 90	9.0	9.5	10.0	10.5	11.0	11.5	12.0	12.5
	8.5

Frequency [GHz]

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

Figure 10.

Nominal behavior of the filter in Figure 9

	0
[dB]	– 5
Parameter	– 10
21
S	– 15
				Nominal value
				mS	21	±2sS
				mS
					21		21
	– 20	9.2	9.3	9.4		9.5		9.6	9.7	9.8	9.9
	9.1

Frequency [GHz]

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

Figure 11.

Statistical behavior of the amplitude of the S21 parameter, in the case in which all the dielectric resonators are assumed to have exactly the same permittivity value within the tolerance range (1r ¼ 3.78 ^ 0.05)

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COMPEL	the four dielectric blocks do exhibit a different effective value of the permittivity within
27,6	the allowable range. In this way, a more accurate estimation of the dispersion of S21
	around its mean value has been obtained. The results are shown in Figure 12, and in
	this case the filter indeed shows a better behavior since the totally random values for
	the permittivities have uncorrelated effects which do tend to cancel out each other,
	leading to overall smaller deviations. To better appreciate this slightly better behavior,
1246	Figure 12 shows also, as thin dotted lines, the two curves relative to mS21 ^ 2sS21 of the

previous case, shown in Figure 11. The reduction of the variance in the statistic is not big but noticeable.

In both cases this geometry, as compared to the previous, presents not only a negative effect on bandwidth due to tolerances, but also a noticeable worsening of the in-band ripple, which makes the filter less reliable and more critical.

An estimate of the time needed for a FE analysis and a DD analysis is shown in Figure 13, again performed on the machine described earlier. It can be appreciate the high-computational efficiency of the method proposed in this paper. In particular, for the latter set of simulation, featuring 100 analyses, the FE-DD technique is 2.6 times faster. The edge of the FE-DD technique is here smaller and this is due to the fact that the portion of the overall domain V belonging to the V2 is larger.

5. Conclusions

In this paper, an original, fast, method based on DD theory for microwave waveguide filter permittivity tolerance analysis has been presented.

Two different types of microwave filters have been investigated and speed up results shown. From the analyses reported, a good speed up in a tolerance analysis loop can be appreciated. Some considerations on the different effects of the permittivity tolerances on the two different filter geometry have also been presented.

Geometrical tolerance analyses, which require re-meshing, will be matter of future work.

Figure 12.

Statistical behavior of the amplitude of the S21 parameter, in the case in which all the dielectric resonators are assumed to have each a different permittivity value within the tolerance range

(1r ¼ 3.78 ^ 0.05)

	0
[dB]	– 5
Parameter	– 10
21
S	– 15
				Nominal value
				mS	21	±2sS
				mS
					21		21
	– 20	9.2	9.3	9.4		9.5		9.6	9.7	9.8	9.9
	9.1

Frequency [GHz]

Notes: Solid line: nominal filter results; dashed line, mean value of the statistical results; dotted lines, bounding values mS21±2sS21. Thinner dotted line shows the bounding mS21±2sS21 lines of Fig. 11

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	350
			FE
	300		FE+DD
time [s]	250
	200
Computing	150
	100
	50
	0	10	20	30	40	50	60	70	80	90	100
	1
				Number of iterations N

FE-DD based permittivity tolerance

1247

Figure 13.

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

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About the authors

Giacomo Guarnieri was born in Florence, Italy on June 5, 1978. He graduated in Electronic Engineering, cum laude, at the University of Florence in 2003, where he also gained the PhD in 2007. His research interests are numerical techniques for electromagnetics with a particular attention to time domain methods and advanced finite elements techniques.

Giuseppe Pelosi is a Full Professor of Electromagnetic Fields at the University of Florence and was a Visiting Scientist at McGill University, Montreal, Quebec, Canada from 1993 to 1995 and Professor at the University of Nice-Sophie Antipolis, France in 2001. His research spans numerical and asymptotic techniques for electromagnetic engineering, applied to antennas, circuits, microwave and millimeter-wave devices and scattering problems. He is also very active in the divulgation of electromagnetic engineering and telecommunications history. He is co-author of over 300 scientific publications on international referred journals and to national/international conferences and has been a Guest Editor of several special issues of international journals. He is also co-author of three books. He is a Fellow of the IEEE, since 2000.

Lorenzo Rossi was born in Arezzo, Italy on December 14, 1979. He graduated in Electronic Engineering, cum laude, at the University of Florence in 2005. He is currently pursuing the PhD degree in RF, microwave and electromagnetics at the Department of Electronics and Telecommunication of the University of Florence. His research interests are numerical techniques for electromagnetics, with a particular attention to finite element method application in solving electromagnetic problems.

Stefano Selleri was born in Viareggio, Italy, on December 9, 1968. He obtained his degree, cum laude, in Electronic Engineering and the PhD in Computer Science and Telecommunications from the University of Florence in 1992 and 1997, respectively. In 1992, he was a Visiting Scholar at the University of Michigan, Ann Arbor, MI; in 1994 at the McGill University, Montreal, Canada; in 1997 at the Laboratoire d’Electronique of the University of Nice – Sophia Antipolis. From February to July 1998 he was a Research Engineer at the Centre National d’Etudes Telecommunications France Telecom and a Visiting Professor at the Polytechnic University of Madrid in 2007. He is currently an Assistant Professor at the University of Florence, where he conducts research on numerical modeling of microwave, devices and circuits with particular attention to numerical optimization. Stefano Selleri is the corresponding author and can be contacted at: stefano.selleri@unifi.it

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This article has been cited by:

1.Giacomo Guarnieri, Giuseppe Pelosi, Lorenzo Rossi, Stefano Selleri. 2010. A Domain Decomposition Technique for Efficient Iterative Solution of Nonlinear Electromagnetic Problems. IEEE Transactions on Antennas and Propagation 58:12, 4090-4095. [CrossRef]

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