диафрагмированные волноводные фильтры / efeaf691-be11-44f8-a9ac-2fba652cfa73

Spectral approach to the synthesis ofnietal grating angular and frequency ifiters

Sergei L. Senkevich

Institute ofRadiophysics and Electronics ofUkrainian Academy ofSciences. 12 Proscura a, Kharkov, 310085, Ukraine

ABSTRACT

A non4raditional method of synthesis of metal grating angular and frequency bandpass filters based on the analysis of spectrum of complex eigen frequencies of resonators forming a ifiter is described. Two types of filters, the resonators of which are formed by two infinite metallic strip periodic gratings in H- or E-plane, are considered. It was shown that filters of the first type has spurious passbands, they are more difficult in manufacturing and suggestion method has an advantage over traditional approach in high frequency range.

INTRODUCTION

A number of works (see, for example, [1]) are devoted to examination ofangular bandpass filters and frequency selective surfaces capable ofcarrying out a suppression of side lobes of directional pattern. As a rule, these filters consist ofa number

of dielectric layers, separated by air interlayers. Excessive weight, cost and high sensitivity to changing the polarization characteristics, low level of suppression out of pass band etc. should be related to disadvantages of them. Changing

dielectric layers to metal gratings is one of possible ways to clear the troubles [2J.

Frequency bandpass filters (FBPF) and angular bandpass filters (ABPF) consist of a number of diffraction gratings having equal period 1 (see Fig. 1,a) and different width of slots d/1, separated by dielectric layers of permitivity , having depth h/1. Vector E(H) of incident field is parallel to 01', that allows to reduce the problem to scalar one.

Fig. 1. Geometry ofthe filters (a) and basis element (b)

Conventional approach to synthesis of such devices can be used Let us consider the range of incident angles p and frequencies i: = i/A (A is wavelength in the free space) such that only zeroth harmonic of diffracted field propagates f 3].

Coining the variable z = cosqfor the synthesis of FBPF at fixed angle q' or for synthesis ABPF at fixed frequency Kone can use the synthesis procedure, developed in [4] for designing the waveguide ifiters. Periodical grating is considered here as impedance invertor (K-invertor) and the space between them is considered as half-wave resonator. However, as it will be shown such an approach does not provide required synthesis accuracy in the value region xcosço >0.85. Approaches realized on the numerical optimization of structure geometiy [5] are not optimal too, so far as they do not allow to built them in intricate simulating complexes because of poor time characteristics and because of the absence ensured results of

synthesis.

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So, development of fast and fairly accurate procedures of frequency selective device synthesis is still the "hottest" problem. In present work an approach, which is based on spectral theory of open resonators, is realized and StUdied It allows to use direct technique of synthesis of ifiter geometry without using low frequency prototypes, and it provides higher accuracy of design in comparison with the techniques by the help of which resonator boundaries (K-invertors) are synthesized, since the resonators are calculated here accounting higher modes.

This work is devoted to development of synthesis procedure and examination of frequency bandpass filters and angular bandpass filters, realized on the base of strip gratings. The cases of E- and H-polarization of incident field vector with respect of generator of grating are considered. Algorithms of synthesis and analysis of electromagnetic characteristics of

FBPF and ABPF have been developed on the base of generalized S-martix techniques, in which multimode scattering matrices ofkey blocks are developed on the base of regorous relations.

1. DIFFRACTION PROBLEM

Cross-section of base structure by the plane ZOX, that is perpendicular to generators of semi-infinite plates, is shown in

Fig. l,b. Vector E (E-polarization) or H (H-polarization) of exciting field is parallel to 01'. Using the periodicity condition along OX axis let us consider the field on the interval x (0, 1). Exciting the structure from the region 1 by a number of

Floque waves of the type of

qx

p =0, 1 ,2, ... (H-polanzation)

we obtain generalized S-matrix of an element. We coined here following designations:

are chosen as basis and test functions. As the result of this the solution reduces to infinite system of linear algebraical equations (SLAE). For E-polarization we obtain

p=O,±l,±2,...,(r=1);p=1,2,3,...,(r=2);

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=bdA - + (N)co sbUA

(N)oo—=u

001 3!dS/ ion ccz

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Using generalized S-matrix techniques reduces the solution of spectral problem for i-th resonator (see Fig. 2,b) to the solution of SLAE ofthe type of (I —S22ES22E) =0. One can show that solution ofthis problem can be approximated by the solutions offinite dispersion equations

Here S22 is matrix of reflection coefficients ofthe Floque waves from the grating, E is diagonal matrix with the elements of the type of exp(4trihI/ 1), describing attenuation or phase accumulation of spatial harmonics along the resonator length h.

For E-polarization oscillation frequencies with odd number of half-waves (symmetrical modes) along resonator length are solution of the equation det(I + SEE) () and oscillation frequencies with even number of half-waves (antisymmetrical modes) are the solution of the equation det(I —S22E) = 0 . For H-polarization the situation changes to reversal.

Qi = — Re ; /(2 liii ) is considered further as quality factor ofeigen modes.

In numerical examination the behavior of eigen frequencies has been studies on the first "physical" sheet K [9] in the fourth quadrant of the region where 0 < Rei. < t . In diffraction problem this region of changing Rek corresponds to the case when only zeroth Floque wave propagates in the regions 1,3 (see Fig. 2,b). Solution of dispersion equations has been carried out by the Neutonn method. Real eigen frequencies of closed ( d=O ) resonator have been chosen as initial approximation.

c_,31 4

(4)

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antisymmetrical modes, N and N are the numbers of them in shaded band from Fig. 3. It is easily to see that these expressions are generalization ofthe results obtained in [1 lJ. Unknown real constants are obtained from known solutions of

diffraction problem in limit points. So, we have S (0) = 1, S(O) 0 at normal excitation (q 0) of the periodical

resonator by E-polarized plane wave and S (0) O, (0) 1 for the case of Hpolarization

Numerical experiment has been carried out in order to examine the properties of representations .Normal incidence (q' 0 ) ofE-polarized wave on the resonator of the geometry h/i = 0.5 , d'l 0.503 , e = 1.0, e2 = 2. 07 has been considered

Real parts of eigen frequencies oftwo symmetrical (,c 0.542 —iO.0248, O.907—iO.0114) and one antisymmetrical (k1 O.977—iO.O764) modes were thrown into single mode range ( 0 <Reic < I here). Frequency dependencies of transition coefficient in power SI2, obtained on the base of algorithm (1}, and the result of reconstruction of insertion loss characteristic by three poles, obtained on the base of (4), are shown in Fig. 4 by solid and dashed lines, respectively.

Locations ofthe poles are marked in lower part ofthe Fi& 4 by starlets. It is seen that approximation curve describes closely the behavior of insertion loss characteristic in nonresonant region and it is coincident with rigorous solution near the resonances with graphical accuracy.

Usually resonators with such complex spectrum are not used in applied problems and it is desirable to choose the geometry of resonators such as only one mode will exist in operating frequency region. In our case such a situation takes place ussually if s closes to and the mode corresponding to the shortest length of resonator, i e. to eigen frequency icr, is chosen as operating one. In this case insertion loss characteristic has only one passing resonance that is described clearly by the expression (4), in which only one pole is present. The representation inthe vicinity of resonance is more accurate, the oscillation has more quality. In this case using (4) we can obtain in the vicinity of transition resonance for real frequencies:

confirms reasonability of using coined determination for quality. In the case of E-polarization, for example, for the synthesis procedure in frequency region these expressions are transformed into

For the synthesis in angular region 'p = Req +ihnp they have following form

Similar expressions are obtained also for the case of Hpolarization. Variable z is used further.

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3. DIRECT SYNTHESIS METHOD.

The presence of analytical expressions (5) for scattering matrix elements of the resonator in the operating range of frequencies allows to propose direct scheme of synthesis of the filters without using low frequency prototypes. Let us choose

reference planes of resonator (Fig. 5,a) such as &gS=O and connect to them circuit sections of electrical length (E)0 /2

(Fig. 5,b) and let us coin new variable x =(Re; — Re ;) / Re; where Re r( is value of the parameter at central frequency (angle) of the filter. Then one can write

where (E is the length of connected section at z =Rez0. For the filters with quarter-waves coupling e0 equals to ir/2. Let us consider the resonator with the load having the reflection coefficient Tat the frequency Rez0. Accounting that S <<1 in the vicinity of passing resonance one can obtain

b)

Fig.5. Synthesized elements of the filter

As usually [41 , in ifiter synthesis on the first step the input task is proposed as four characteristic frequency and three level of loss for the approximating polynomial ( Chebyshev, Bauerworth), that determinates pass bands and stop bands, steepness of the filter front, ripple level of frequency characteristic in the pass band. Input task is marked in figures by

triangles (see, for example, Fig. 6) The number of ifiter resonators N is determinated by this task. Using (6), (7) and accounting geometrical symmetry of the ifiter and resonators one can obtain reflection coefficient for any number of resonators. So, for example, for three resonators design we have

JSJ=2cos771cos(if1+(f2)—co5772=2x(Q2—2Q1)+8Q1(Q1+Q2+r/2)2x3

where Q1 =Q3 are unknown qualities of extreme resonators, Q2 is unknown quality of central resonator. Comparing this expression with approximating polynomial and equation coefficients at equal degrees x we obtain simple system of

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equations for determination of resonator qualities, that is frequently solved analytically. So, for example, for the

Batterworth filter, using the same assumptions as for obtaining (7), we have

L = (1_J$i2 )' 1+IJ2 JkI2<<1 1 + (p3x3)2 , /43 =1SJo5 /X.5

Q=2Q1 , 8Q1(Q1÷Q2÷,r/2)2=p3

where index 0.5 points to the fact that the value is deterininated by the level of reflected (passed) power of 0.5.

So, the synthesis procedure was reduced to searching the geometry of resonators with given quality on the central frequency ofthe ifiter Re ç=(Re zj Re r)V2 where Re r and Re r are left and right limits ofpass bani This problem can be solved by different techniques. In given case geometrical dimensions of resonators h and d1 (i 1N ) are searching from the system of nonlinear equations, obtained on the base of above mentioned dispersion equation (3) for single resonator [131

Re {det[I — A(Re r, ,Im ç ,h1 ,d, )}} =0

Im{det[I — A(Re ro Im J 4 )]} =0

A=(S22E)2, Imç—Rer0/(2Q1), i=1÷N.

Solution is obtained by Neutonn method. After the calculation of dimensions of the resonators quarter-wave coupling sections between them are restored on the base of regorous model taking into account the phases of reflection coefficients from the boundaries of resonators. At the final step two adjacent boundaries of neighboring resonators and quarter-wave coupling section (Fig. 5,b) at the central frequency of the filter is replaced by one diffraction grating with equivalent electromagnetic properties (for the zeroth harmonic). It allows to obtain the same filter topology as in the case of using K- invertors 141.

Above-cited procedure of direct synthesis of filters is not obligatory. As it is not difficult to note the expressions for resonator qualities were obtained at practically those assumptions which made possible using CircUitS of prototypes on concentrated parameters for the synthesis of electromagnetic structures [4].. Then values g, obtained from the analysis of

the properties oflow frequency prototype on concentrated elements can be used for determination of Q-factor values of the resonator [4J. Resonator qualities are obtained from the expression Q1 =2Q1 I g1 , where Qf is quality of the filter in the

whole and it equals to ratio ofthe central frequency ofthe filter to the width ofits pass band. However all the rest of steps earned out on the base of multimode models have be valid in order to save design accuracy.

Clearly that the region of input tasks for pass band , beingrealized, for such two approaches are close. Since proposed synthesis procedure seems applicable for design of the filters having pass band of not more 15%. Increasing the synthesis

accuracy at the broadening the pass band is possible using distributed prototypes [141 for the calculations of values Q.

4. NUMERICAL RESULTS

As the first example let us consider 2.5% pass band FBPF, input task for which is marked in Fig 6 by triangles.

Frequency characteristics of developed three resonator designs are shown by solid line for the case of E-polarization and by dashed line for H-polarization (Fig.6,a). The curves coincide with graphical accuracy in all angle range, then only the results for ço= 0 are presented here. Diffraction loss (ripple level) in passband are 0.2 dB. Lengths of resonators are

h1 =h3 =13.22 mm,h2 = 12.62 mm forH-polarizationandh1 =h3 = 11.34 mm, h2 =11.99 mm(gratingperiodis 10mm) for E-polarization. Strip widths of gratings are d1 =d4 = 0.489 mm, d2 = d3 = 0.026 mm for H-polarization and

d1 = d4 = 6.035 mm, d2 = d3 =2.88 mm for E-polarization. Characteristics of both designs completely meet initial task,

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used. Angle characteristics of this FBPF on frequency f=12.15 GGz are shown by solid line for the case of Epolarization on Fig.6,b, those on the frequencyf= 12.05 GGz are presented by the line with points.

I

Fig.6. Frequency (a) and angle characteristics of FBPF

Insertion loss characteristics for the ABPF are plotted on the Fig7 by solid line for the case of E-polarization and by dashed line for H-polarization. Task data are marked by triangle. Angular filter were designed with Chebyshev characteristics. Filters were synthesized on the frequency j=12. 15 GGz. AngIe characteristics are symmetrical with respect to ça=O, that is why they are shown in the region from 0 to 90° Synthesized designs consisted of three resonators. We notice that their characteristics are in good agreement with inpit task and insertion loss in the passband did not exeed 0.2 dB, but for H-polarization there is spurious passband in the vicinity of çô=80°.

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The example of synthesis of FBPF for xcosçO.93 is shown on Fig. 8. Dashed line corresponds the case of using conventional approach based on [4], solid line corresponds the synthesis procedure of present work. We notice that improper results have been obtained in the first case, whereas in the second case loss characteristic is completely meets the initial task.

CONCLUSION

In the report direct technique of synthesis of frequency bandpass and angular bandpass filters on the base of periodical structures has been considered Algorithms of analysis and synthesis are developed on the base of rigorous models.

Examples of synthesis of ABPF and FBPF for E- and H-pobrization have been citeci It was revealed that in the case of H-

polarization designs are less technological and while designing ABPF they have spurious passband at q8O0. Advantage of pro technique in comparison with conventional one while designing the devices in high frequency region of single

mode range has been shown. The synthesis procedure, being used, can be applicable for design of low frequency filters, rejection filters, different matching devices both in quasi-optical and in waveguide Circuits.

ACKNOWLEDGMENTS

This work was supported by the International Scientific Foundation under Grant K6U100.

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