I. INTRODUCTION

Evanescent-mode waveguide (WG) bandpass filters have advantages over conventional half-wave resonator filters due to their compactness, a large range of achievable bandwidths, and a broad higher frequency stopband similar to low-pass filters. These filters may be realized with the E- plane screws [1–3] but their application requires a time-consuming filter tuning and is problematical at the millimeter-wave range. Application of nontouching E-plane fins of a small thickness [4, 5] and metal-finned or ridged WG short sections [6–12] led to “tuning-less” technology and provided the possibility of low-cost fabrication. At present, there are many papers in which various methods for the exact determination of the transverse electrical (TE) and transverse magnetic (TM) modes in the ridged WGs have been developed and have been compared. In addition to [6–12], we can cite, for example, [13–18].

Correspondence to: Anatoly Kirilenko; e-mail: kirilenko@ ire.kharkov.ua.

These methods, together with the well-known mode-matching technique, allow calculating the full-wave S-matrices of the ridged WG discontinuities and, as a result, to build an exact filter model based on the generalized S-matrix technique.

Several problems arise in the design of evanescent-mode filters with the ridged WG short sections. One of them appears if the K-inverter scheme is used as the first step in the filter design procedure. As a rule, an initially synthesized filter has a passband much wider than the specified one. Sometimes, this step was simply excluded from the design procedure or, as in [5], was replaced by the previously calculated look-up table for the scattering parameters of single fins, because the authors of [5] could not find a suitable relationship between the equivalent circuit elements of a short fin and its dimensions. A partial solution of the problem was found in [8] where quarter-wave resonators were used. To overcome this problem, in [10, 11] the relative filter bandwidth Wλ was replaced by Wλ/F 2 < F < 4 at the stage of the K-inverter prototype synthesis. The physical

reasons of the bandwidth widening of the initially synthesized filters were clarified in [12].

Usually, a below-cutoff filter housing is chosen to be one of the standard rectangular WGs, the cross section of which is smaller than that of the main WG [4–9]. To reach a wider highfrequency stopband, a smaller filter housing width and a narrow ridge gap have to be chosen. The impact of the smaller housing can display itself as an increase in ohmic loss and a decrease in breakdown microwave field intensity, and an appearance of additional difficulties in the fabrication of filters for millimeter-wave applications. In addition, filters with high skirt selectivity have, as a rule, long below-cutoff sections (ridge notches). At some specified rejection level, the lengths of the notches turn out to be so large that the overall filter length becomes comparable with filters based on the half-wave ridge sections.

Another problem occurs with the realization of the first (last) K-inverters. It arises mostly in the design of broadband filters loaded with rectangular WGs. The problem is that sometimes the desired value of those K-inverters cannot be reached even with a zero-length notch located between the first resonator and the input rectangular WG. Usage of a ridged WG section with a wider ridge gap in comparison with the other resonators can solve the problem of the first (last) K-inverter for some filter specifications [9]. A wider set of specifications can be realized with additional ridged WG transformers having a housing cross section larger than that of the filter resonator sections [8]. However, these transformers increase the overall filter length and complicate the filter fabrication.

In the present paper, a multistep initial filter synthesis is developed to solve the problems listed above. An unconventional choice of a ridged WG housing with an enlarged cross section and an insertion of inductive strips to ridge notches are the ways that are used to decrease ohmic loss, to extend the high-frequency stopband, and to reduce overall filter length. To avoid the problem of the first K-inverter in broadband filters, configurations with additional transformers are also considered. Rectangular WG sections close to a quarter-wave in length are used for these purposes.

The filter structures discussed here are based on the constant-gap ridged WG sections and are shown in Figure 1 in their longitudinal section (half only) and front views. The filters in Figures 1b and c have additional inductive strips

Figure 1. Filter configurations based on the doubleridge waveguide sections. (a) Filters formed by the simple-notch elements. (b) Filters formed by the notch- strip-notch elements. (c) Filters with additional transformers of a rectangular waveguide section type.

with the same thickness as the ridges. The strips are symmetrically placed between the neighboring resonators. The filters with transformers shown in Figure 1c are also analyzed for the strip-free structures identical to those shown in Figure 1a. All the filters are symmetrical relative to three coordinate planes and are built in a rectangular WG excited by the TE10 mode. From this, it follows that these structures may be considered as having a magnetic wall in the x = a/2 plane and an electrical wall in the y = b/2 plane. The latter allows us to consider a double-ridge structure as a single-ridge structure with the vertical dimensions reduced by half (similar topologies of single-ridge filters are also considered in the present paper).

II. METHODS OF SOLVING

THE BOUNDARY VALUE PROBLEMS FOR KEY FILTER ELEMENTS

The filter geometry can be divided into some key WG elements (double-side rectangular-to- rectangular and rectangular-to-ridged WG junctions, WG bifurcation) with calculated S-matrices, which enable us to use the generalized S-matrix

356 Kirilenko, Rud, and Tkachenko

technique to obtain the characteristics of separate components and the filter as a whole. The mode-matching technique is used to solve the full-wave problems of rectangular-to-rectangular and rectangular-to-ridged WG junctions. For the last junction, the problem of a ridged WG mode basis has its own importance and is considered in Section III.

It should be noted that the WG bifurcation should be used as a key element in various “electromagnetic images.” On the one hand, the key problem of a rectangular WG bifurcation has to be solved in the full TE–TM mode basis to obtain the full-wave S-matrices of inductive strips placed between two ridged WG sections being characterized with their own TE–TM mode basis. The 3D problems for the structures uniform along the y-axis may be solved with two quasiscalar 2D problems regarding the longitudinal-section TE y and TM y modes, which have the electrical and magnetic fields located in the plane perpendicular to the y-axis. On the other hand, the transverse resonance method applied in Section III for the search of a ridged WG mode basis requires the knowledge of S-matrices of a parallel-plate WG bifurcation in the bases of modes having only one magnetic or electrical field component polarized along the bifurcating semiplate edge. Solutions of the corresponding Neumann and Dirichlet problems have much in common with the solutions for the above-mentioned TE y and TM y modes.

These scalar problems were considered in detail in [19]. The solutions developed there were based on the moment method. At first, the corresponding scattering problems were reduced to the first kind Fredholm integral equation for the electric field distribution function E x at the WG junction aperture. The E x function was expanded into a series in terms of the firstor second-kind Chebyshev polynomials with a necessary weight. In the TE y or TM y mode problem, respectively, this is

Nbif

ETE x = 1 − x/a1 2 −1/2 DnT2n−1 x/a1

n=1

Nbif

ETM x = 1 − x/a1 2 1/2 DnU2n−1 x/a1

n=1

where Dn are unknown coefficients and a1 is the width of a narrow WG. By using T2n−1 x/a1 or U2n−1 x/a1 as the basis and testing functions, we succeeded in reducing the corresponding problem to a system of linear algebraic equations. The

matrix elements of the latter take the form of a sum of two slow-converging series involving the Bessel functions and propagation constants of the modes taken into account in the wide and narrow WGs. Using the Kummer technique enabled us to essentially speed up the convergence of these series and to ultimately obtain a fast-converging algorithm for calculating the WG bifurcation S- matrix. The needed asymptotic expressions were found with the aid of the Hankel expansion for the integer-index Bessel functions of large arguments and the expansion of mode propagation constants as series in inverse powers of the mode number, m 1. As a result, we developed an algorithm that exceeds the known ones both in efficiency and in accuracy. For a small-thickness (t/a ≤ 0 2) bifurcating plate, the 1%-accuracy

S-matrix elements may be obtained at Nbif = 10 and Mbif ≈ 8 − 10 Nbif where Mbif is the number of modes taken into account in the wide and

narrow WGs.

III. TRANSVERSE-RESONANCE METHOD IN THE PROBLEM OF A RIDGED WG MODE BASIS

There are two types of ridged WGs that may be used in the evanescent-mode bandpass filters design: double-ridge (see Fig. 2a) and singleridge (Fig. 2b) WGs having a housing ar × br, ridge thickness t, and gap w. For our purposes, both guided structures can be considered as the magnetic-wall symmetrical ones regarding the plane x = a/2. In addition to that, the first structures have an electrical-wall symmetry plane at

Figure 2. Cross section of ridged waveguides, (a) double-ridge, (b) single-ridge.

y = br/2 that is why they can be analyzed as equivalent single-ridge WGs with brsng = brdbl /2,

w sng = w dbl /2. The latter allows us to choose a single-ridge configuration as a basic one for developing a numerical algorithm for the calculation of both singleand double-ridge WG mode bases.

Introduce the following magnetic Hertzian vector (the time-dependence exp−iωt is assumed) to characterize the TEn mode field in a ridged WG,

where ζn = k2 − χ2n 1/2 is the propagation constant of the TEn mode, χn is the TEn mode cut-

off wave number to be determined, Nnh is the normalization constant. In eg. (1), ϕnh x y is a piecewise-given scalar function corresponding to the cross-section division in Figure 2b. In doing so, a ridged WG configuration may be considered as a bifurcated parallel-plate WG with shortened arms. Keeping this conception and magnetic wall symmetry in mind, we can write the following rep-

resentation for ϕnh x y in the regions 0 and 1:

ϕnh x y

M0

Here, Mj is the number of terms (waves) taken into account in the field Fourier-expansion in

may be considered as complex amplitudes of for-

ward and backward traveling waves, respectively;

ωnmj = χ2n − βmj 2 1/2. With the aid of eqs. (1) and (2), one can find the expressions for the TEn

mode field components following to the known relationships. The TMn mode field components may be obtained in the same way by using the electrical Hertzian vector.

To determine the cutoff wave numbers χn for the TEn or TMn modes and the sets of coefficients

Anmj and Bnmj j = 0 1 for the field expansion

(2), one can use the mode-matching technique and one can reduce the eigenvalue problem to a homogeneous matrix equation. However, a more suitable technique for the solution of such a problem is the transverse resonance technique based on the known generalized S−matrix of the parallel-plate WG bifurcation with a magnetic wall in the plane of symmetry. According to the S-matrix transverse resonance technique, we can write the pole-free homogeneous coupled matrix equation in the vec-

tors of unknowns B 0 = Bnm0 Mm=0 0 and A 1 =Anm1 Mm=1 0 in eq. (2) as

I S 00 E 0 2 B 0 S 01 E 1 2 A 1 = 0

(3)

S 10 E 0 2 B 0 + I S 11 E 1 2 A 1 = 0

While obtaining eq. (3), the matrix relationships,

A 0 = ±E 0 B 0

B 1 = ±E 1 A 1

(4)

were used. They follow from the boundary conditions for the Hzi [upper signs in eqs. (3)–(4)] or the Ezi (lower signs) field components on

diag exp iωnmj dj Mm=j 0 1 j = 0 1, where d0 = w and d1 = br − w. Putting to zero the determinant

of the homogeneous matrix equation (3),

gives a dispersion equation for determining the desired set of cutoff wave numbers χn of the TEn or TMn modes for the given parameters of a ridged WG. Note that, due to complex-valued S-matrix elements, the function D χ is also a complex-valued one even if χ is real valued.

The results obtained by the developed algorithm were compared with the ones presented in [13, 15, 17, 18]. The agreement with the known results was very good. The CPU time needed for calculation of cutoff wave numbers and mode-function coefficients of 24 TE and 12 TM magnetic-wall-symmetry modes in a singleridge WG is 3–4 min of CELERON 500 PC, if

Nbif = 10, Mbif = 100 in the WG bifurcation key problems and M0 = M1 = 25 in eq. (2) and in the

similar expansion for TM modes.

358 Kirilenko, Rud, and Tkachenko

IV. FILTER DESIGN PROCEDURE

The developed design procedure contains the following stages:

(1)Searching for the ridged WG geometry.

(2)Initial filter synthesis with an iterative correction of a K-inverter prototype.

(3)Searching for the initial transformer geometry (for filters with transformers only).

(4)Interpolation of filter element S-matrices.

(5)Filter response optimization.

We use the following set of parameters as a filter specification: fa and fd (low and high side rejection frequencies), fb and fc (low and high edge frequencies of the passband), La and Ld (attenuation at fa and fd), Lr (a level of insertion loss ripples in the passband).

A. Searching for the Ridged WG Geometry

The following considerations have to be taken into account when choosing the dimensions ar and br (see Fig. 2) for the ridged WG housing that forms ridge notches. At first, the notches must be below cutoff at least at frequencies f < fb. On the other hand, it should be kept in mind that the smaller ar, the wider the filter stopband because the first spurious high-frequency passband (for strip-free filters especially) may be generated by a half-wave resonance in the above-cutoff notches. In regard to the housing height, in many cases the value of br = b is best suited for the filters based on the double-ridge WG sections. For the filters based on the single-ridge WG sections, such a housing is acceptable only if the passband is narrow 1–2% . Sometimes one has to choose the values of ar a/2 and br b/2 for these filters.

As for the values of the ridge thickness t and the gap w, they should be chosen according to the possibilities of manufacturing process, permissible voltage breakdown level, ohmic loss, etc. These parameters must provide the cutoff frequency of

the ridged WG dominant mode fcut < fb. For many filter specifications, it is enough to choose

the parameter w in such a way that provides the

value of fcut to coincide with the one for the outer a × b rectangular WGs.

B. Initial Filter Synthesis with an Iterative Correction of

a K-Inverter Prototype

A proper design procedure starts with the calculation of K-inverter values of a low-pass Chebyshevtype prototype. This stage enables one to obtain the initial data needed at the stages of filter synthesis and optimization. To provide the design of narrow and wide band filters with an arbitrary number of resonator sections, Rhodes’ scheme described in [20] for the initial determination of K-inverter values is used. Simple notches, as shown in Figure 1a, or notch-strip-notch elements, as shown in Figure 1b, play the role of K-inverters in the filters under consideration.

Based on the single-mode resonance condition,

arg S11jj + arg S11j+1 j+1 + 2ζ1rj = 2qπ

q = 0 1 2 (6)

where ζ1 is the propagation constant of the dominant mode in the jth section, one can find the required length rj of that section at the filter central frequency f0. The value of q may be different depending on the character of section loads. Usually, they choose a value of rj that is the closest to the half-wave one, for which π/ζ1 = λg0/2. In this case, the obtained filter cavities will have Q-factors that are closer to the prototype ones with the λg0/2-sections. The peculiarity of the evanescent-mode filters design is caused by the capacitive character of sectionended loads. In the majority of cases, the phases of notch reflection coefficients are small and negative in sign. As a consequence (but not always), the single-mode resonance equation (6) contains the solution for q = 0 if the value of rj λg0/2. Therefore, apart from the geometry with the sections slightly different from λg0/2, it is possible to use a geometry with very short sections. However, conventional procedures of the circuit-theory synthesis are oriented on the half-wave resonators only. A direct usage of shortened sections generated by eq. (6) with q = 0 will lead to drastically low-Q filter resonators and consequently will lead to the degradation of the filter skirt selectivity.

Taking into account that the initial design is carried out only as an initial step before the numerical optimization based on the full-wave exact model, it turns out that it is possible to limit oneself by a crude procedure of prototype correction. The main goal of this procedure is to make the Q-factors of shortened resonators close

to those of half-wave resonators. It is clear that this goal may be achieved if the reduction of the resonator volume is compensated by a reduction of the resonator couplings (by the reduction of K-inverters in our case). The needed reduction coefficient k is found by the procedure described in [12].

The reduction coefficient k = Ksh/K may be calculated for the central filter section only and then this coefficient is supplied to other K-inverters. The exceptions are the first and the last K-inverters that are multiplied by √k. As a rule, the geometry produced at this first step of the K-inverter tuning yields a response with an essentially corrected skirt selectivity, however it remains to be improved before running the optimization procedure. Here, an iterative procedure of a more exact choice of the reduction coefficient k is employed as the second step of the K-inverter correction. It consists in the search for a k-value, which satisfies the following conditions:

(1) insertion loss at the stopband edge frequencies must be better than the specified ones, (2) insertion loss at least at one of the passband edge frequencies must be lower than 3 − 5 Lr.

It should be noted that the predictable reduction coefficient k = 2 was used in [8] for one of the special filter configurations with the quarterwave ridged WG sections.

C. Searching for the Initial Transformer Geometry

The values of the terminal K-inverters are conventionally more than the ones for internal elements, and the wider passband, the larger these K-inverters. In the case of a transformer-free filter (see Fig. 1a and b), a step junction between the main WG and the section of the belowcutoff housing WG plays the role of the terminal K-inverters. Tuning of such a K-inverter is realized by changing the terminal notches length l0 = ln+1, however, for some specifications, the K-inverter turns out to be greater than it may be even at l0 = ln+1 = 0. In this case, the filter configuration with matching transformers (see Fig. 1c) has to be employed.

These transformers are not aimed at reaching the total matching of the ridged WG with the main one but at providing a possibility to tune end filter elements (K-inverters) by varying the length of terminal notches. Here, the necessary geometryatr btr ltr of the rectangular WG transformer similar to the quarter-wave one is searched for

0 ≤ ltr ≤ λg/2 by the optimization procedure with usage of the goal function providing the maximum matching of the input rectangular WG joined directly with the ridged one. Usually, the transformer geometry with atr = a, btr = br turns out to be the best one. After determining the transformer geometry, the design is the same as for transformer-free filters.

D. Multimode Tuning of Filter Resonators

The initially synthesized filter geometry is rather an estimation one because it is obtained with the procedure based on the single-mode approximation. In fact, the role of the higher modes cannot be ignored both for the half-wave filter resonators and for the shortened ridged WG resonators especially. The fringing fields interaction leads to shifting and narrowing the filter passband regarding to the specified one. Therefore, each iterative step of the K-inverter prototype correction described in the previous subsection has to be performed with taking into account the higher mode interaction. The multimode tuning stage consists in tuning each filter resonator to the required central frequency f = f0 by varying the resonator length rj. Filter resonator is bounded usually by various reflecting elements, therefore in the general case a nonzero minimum of its reflection coefficient can be observed at f = f0.

This stage is very fast as the search for the desired resonator length is performed at the fixed frequency and does not require a recalculation of the S-matrices of WG elements forming the resonator. After tuning all the resonators, the passband central frequency of an initially synthesized filter is really located at the required point f = f0. However, as a rule, the obtained passband remains narrower in comparison with the specified one due to the frequency dispersion of all the filter components.

E. Interpolation of Filter Element S-Matrices

The deviation of the response of the initially synthesized filter from the specified one requires using the optimization procedure to refine the filter geometry. Such a procedure is a

360 Kirilenko, Rud, and Tkachenko

time-consuming one if the full-wave filter model is used directly. RAM storing of the matrices of frequency-independent coupling integrals for all the WG junctions solves this problem only in part.

An essential reduction in the CPU time can be reached if one substitutes the full-wave models of filter elements for their interpolation models. The latter ones are obtained by a 2D interpolation of the exact S-matrices of filter elements on the given grid of the frequency and one of geometrical parameters. As the latter, the length of notch or transformer is considered. The set of exact S-matrix element values is calculated for the required number ng of interpolation points connected to a geometrical parameter at the

The procedure is realized on the basis of Newton’s formulas of a power p for a “forward” interpolation. Depending on the required calculation accuracy, one can choose ng = 3 − 10, p = 2 − 4. In doing so, the number of frequency points is

defined as nf = fend − fbeg /1f , where fbeg and fend are the beginning and the end of a frequency range where the filter elements have to be inter-

polated. Using the interpolation models reduces the CPU time consumption in five to seven times while the filter optimization is performed.

F. Filter Response Optimization

The optimization procedure is based on solving the minimax many-variable problems by the method of steepest descent. It should be noted that the characteristics of considered evanescentmode filters with shortened ridged WG sections are very sensitive to small variations in the lengths of sections. It suggests that the possible range of variation of the set of filter geometrical parameters at a filter optimization must be smaller for the filter section lengths and must be greater for the notch and transformer lengths. As a rule, a multistep optimization procedure has to be applied to achieve the desired result.

The purpose of the first step in the filter optimization is to symmetrize the filter response without its degradation within both stopbands. The following goal function allows us to take into

account the features of the first optimization step:

Here, x is the vector of objective filter geometrical parameters; La Lr Ld are the specified values; L fa and L fd are the actual filter insertion loss calculated at the frequencies fa and fd; Nb is the number of frequency points in the filter passband; L fi is the insertion loss value calculated at the ith frequency point. Two last sums contain the terms with the insertion loss values L fq calculated at the qth frequency point positioned between fa and fb or between fc and fd. The val-

ues of Lql and Lqr are calculated at the corresponding points according to a linear law by which the filter response is approximated within the corresponding subranges. Usually, Nr l = 5 is quite enough for obtaining the required skirt selectivity. The last sums in eq. (7) are introduced to avoid the displacement of response pulses out of the filter passband and the degradation of the filter skirt selectivity due to these phenomena. Without taking into account the above-mentioned sums in eq. (7), elimination of the latter consequences increases the filter optimization time drastically or is impossible at all.

As objective parameters, the lengths of notches lj and ridged WG sections rj are considered (as well as the transformer length ltr if necessary). For the filters shown in Figure 1b, all strips are of the same length sj = s specified previously, and notches adjacent to the jth strip have the equal lengths lj. The number of filter sections n determines the number of objective parameters as well as the weight coefficient W = 10n. Due to the symmetry of filter configuration, only the half of its parameters has to be included into the vector x.

The further optimization steps provide the finetuning of the filter geometry and are aimed mainly at improving the passband characteristics. These steps are performed with W = 1 and Nr l = 2 in eq. (7) and are repeated until the filter characteristics satisfy the specification or become most

close to it. If one of objective parameters comes to a limiting value of the interpolation range, the re-interpolation of a corresponding filter element is done before entering the next optimization step.

All the final characteristics (insertion loss, return loss, voltage standing-wave ratio, and group delay) of the optimized filter are calculated with the exact filter model.

V. NUMERICAL AND

EXPERIMENTAL RESULTS

The filters shown in Figure 1b and c have not been considered in the literature, therefore they are paid special attention. They have a set of advantages over strip-free structures that are caused by the properties of notch-strip-notch elements bounding the filter resonators. However, the question of how strips influence the possibility of obtaining the shortened ridged sections with the desired response is not so obvious and requires additional study.

A. Influence of Strips on Resonator Properties

To clarify the role of strips in forming the geometry of filter sections with the specified properties, ridged WG resonators bounded by notch-strip- notch elements with a varying strip length were analyzed. The curves in Figure 3 show the influence of the strip length s on the notch length l and resonator length r at the specified resonant frequency f0 = 10 GHz and the quality factor Q ≈ 100. The resonators as well as input–output ports are formed by the double-ridge WGs with ar × br = 10 16 × 10 16 mm, t = 1 mm. Two cases of the ridge gaps w = 0 5 and w = 1 2 mm are

Figure 3. Dimensions of the double-ridge waveguide resonators vs. the strip length.

considered. For comparison, the parameters of simple-notch resonators with the same f0 and Q are: l = 6 81, r = 0 525 mm for w = 0 5 mm and l = 8 2, r = 1 31 mm for w = 1 2 mm.

As one can see in Figure 3, increasing the strip length s leads to the reduction of notches’ length l and to an appreciable increase in the resonator length r, especially in the case of w = 1 2 mm. However, even at s = 0 6 mm the overall length l4 of these resonator structures (l4 = 4l + 2s + r = 10 8 mm) is less than the length of simple-notch structures (l4 = 2l + r = 17 71 mm). The fact that strips can essentially reduce the lengths of belowcutoff notches is obvious due to the fact that strips are strong scattering obstacles themselves. Inductive character of strips causes a reduction of the negative-value phase terms arg S11 in the resonant condition (6). The larger strip length, the larger the value of (−arg S11 and, as a result, the larger the resonator length r.

At the first glance, the observed reduction of the structure length l4 is a positive factor, however, it plays a certain negative role as well. The matter is that the longer the resonator, the lower the frequency, at which this resonator becomes a half-wave one. This can lead to the appearance of parasitic spikes within the high-frequency stopband, i.e., to the reduction of its width. For example, the frequency of the half-wave resonance in the structures with w = 0 5 mm reduces from f = 28 9 to f = 26 2 GHz when the strip length changes from s = 0 to s = 0 5 mm. Therefore, it is better to choose a strip size as small as possible within the requirements on rigidity of the filter all-metal insert or the features of its fabrication. For comparison, the corresponding simple-notch resonator has the first parasitic spike at f = 22 4 GHz. In contrast to a notch- strip-notch structure, the latter is generated by the half-wave resonance in long above-cutoff notches. Similar properties take place for the resonator structures in the single-ridge WGs.

B. Comparative Analysis of Narrow Band Filters of Different Configurations

Consider the possibilities of narrow band filters based on the singleor double-ridge WG sections with or without additional inductive strips. As an example, four-pole filters in the WR-90 waveguide with 2% bandwidth centered at f0 = 9 GHz are chosen. All the designed filters provide the passband return loss not worse than

Figure 4. Frequency responses of the four-pole filters based on the single-ridge waveguide sections bounded by simple notches (curve 1) and notch-strip-notch elements (curve 2, s = 0 3 mm). WR−90 input–output waveguide, WR−42 waveguide as a filter housing, t =

l4 = 1 448, l1 = l3 = 3 787, l2 = 4 172, r1 = r4 = 2 777, r2 = r3 = 4 201, l4 = 41 24 .

26–28 dB. The insertion loss response within a broad high-frequency stopband was calculated

as L = 10 log Mm=1 Pm, where M is the number

of propagating modes in the output WG at the

given frequency, and Pm is the power of the mth propagating mode.

Figure 4 demonstrates the responses of the single-ridge four-pole filters. Both simple-notch (curve 1) and notch-strip-notch (curve 2) filters have a sharp spike in the vicinity of the cutoff frequency of the outer WG TE11 and TM11 modes. The observed spurious passbands for the simplenotch filter are caused by the half-wave resonance in the long notches that become above-cutoff ones at f ≈ 14 GHz. As for the notch-strip-notch filter, its stopband is limited by f = 24 GHz due to a similar resonance but in the ridge sections.

Figure 5 shows the responses of three doubleridge type filters. Curve 1 corresponds to the conventional simple-notch filter with the WR-42 housing, whereas the other curves are obtained for a housing whose height coincides with the one of the outer WGs. As can be seen, the possibilities of the simple-notch filter are similar to those shown in Figure 4. The filter with the enlarged housing and ridge gap (curve 2) has wider stopband and smaller longitudinal size that is far from obvious result. As for any simplenotch filter, the half-wave resonance within long above-cutoff notches limits the width of the stopband. The usage of notch-strip-notch elements further makes smaller the longitudinal filter size

Figure 5. Frequency responses of the four-pole filters based on the double-ridge waveguide sections bounded by simple notches (curves 1 and 2) and notch-strip- notch elements (curve 3, s = 0 3 mm). WR-90 input– output waveguide, WR−42 waveguide (filter 1) and ar × br = 10 668 × 10 16 mm waveguide (filters 2 and 3) as filter housings, t = 1 mm. Filter dimensions (in millimeters): 1 − w = 0 5, l0 = l4 = 0 912, l1 = l3 =

11 374, l2 = 12 238, r1 = r4 = 6 485, r2 = r3 = 8 559, (l4 = 66 90); 2 − w = 1 0, l0 = l4 = 3 502, l1 = l3 = 13 629, l2 = 14 926, r1 = r4 = 1 483, r2 = r3 = 1 618,l4 = 55 39 ; 3 − w = 1 0, l0 = l4 = 3 476, l1 = l3 = 4 122, l2 = 4 597, r1 = r4 = 1 830, r2 = r3 = 2 248,l4 = 41 69 .

(compare the values of overall filter lengths l4 presented in Figures 4 and 5 captions) and better the filter stopband characteristics. This is confirmed by the curve 2 in Figure 4 and the curve 3 in Figure 5.

To verify the proposed design procedure, the four-pole filter in the a × b = 23 × 10 mm rectangular WG has been fabricated. Specification for the filter design was the same as for the filters from Figure 5. The comparison of theoretical and measured data is shown in Figure 6. As one can see, the results are in a good agreement except of the passband location. The observed difference is caused mainly by the inaccuracy in the gap and resonator length fabrication. The wider gap and shorter resonator sections led to the expected small high-frequency shift of the response of the fabricated filter. In addition to ohmic loss, non-symmetry of the actual filter configuration is a reason for an increased level of ripples within the passband (measured Lr ≈ 0 6 dB instead of the specified one Lr = 0 01 dB).

C. Broadband Filters

The above filter configurations can provide the passbands up to 10%. To achieve wider passbands, additional transforming elements have to be used. The simplest of them are sections of rectangular

Figure 6. Response of the designed filter (solid curve) and that calculated with the measured dimensions of the fabricated filter (dashed curve). The crosses mark the measured data. Designed filter dimensions (in millime-

ters): ar br = 11 5 10, t = 2, w = 1 8, s = 0 3, l0 = l4 = 4 104, l1 = l3 = 3 595, l4 = 4 063, r1 = r4 = 3 072, r2 = r3 = 4 061. Actual filter dimensions: ar br = 11 47 10,

t = 1 97, w = 1 82, s = 0 28, l0 = 4 12, l1 = 3 625, l2 = 4 075, l3 = 3 62, l4 = 4 08, r1 = 3 05, r2 = 4 045, r3 = 4 04, r4 = 3 05.

WGs incut between outer WGs and filter housing (see Fig. 1c). Transformer initial dimensions are found according to the procedure described in Section IV, and further only the length ltr is estimated during the filter optimization.

Figure 7 demonstrates the possibilities of two broadband eight-pole filters with the passband 12–15 GHz. Both filters provide the return loss not worse than 25 dB over the specified passband. As expected, the notch-strip-notch filter (see curve 1) has an improved skirt selectivity and a wider high-frequency stopband in comparison with the strip-free filter (curve 2) due to shorter notches. The latter lead to the reduction of the filter size (compare the values of l4 presented in the Fig. 7 caption). Similar filters formed by the single-ridge sections have a narrower high-frequency stopband that is caused by a denser spectrum of modes in the regular WG sections of all the filter components.

VI. CONCLUSION

The full-wave models, algorithms, and tuning procedures presented here are a powerful tool for the exact design of evanescent-mode bandpass filters of various configurations built on the shortened ridged WG sections. It has been shown that using below-cutoff housings with a cross section enlarged in height (at least, for the filters with narrow and moderate passbands) leads

Figure 7. Frequency responses of the eight-pole transformer-loaded filters based on the double-ridge waveguide sections bounded by the notch-strip-notch elements (curve 1, s = 0 2 mm) and simple notches (curve 2). WR−75 input–output waveguide, WR−34 waveguide as a filter housing, t = 1, w = 0 25 mm.

1 594, r2 = r7 = 2 421, r3 = r6 = 2 331, r4 = r5 = 2 244,

5 355, l4 = 5 601, r1 = r8 = 1 051, r2 = r7 = 1 135, r3 = r6 = 0 904, r4 = r5 = 0 823, (l4 = 53 07).

to the configurations with an improved performance that is important for their application in the millimeter-wave range. Insertion of additional inductive strips to ridge notches enables one to reduce the filter longitudinal size and essentially to increase the width of the stopband. Simple transformers, formed by rectangular WG sections, may be used in the design of broadband filters (up to 20–25%). The design capabilities of the developed computer-aided design (CAD) tool are confirmed by a good agreement of the simulated filter characteristics with the measured data.

ACKNOWLEDGMENTS

The authors thank Dr. P. Pramanick for his interest in this work and helpful discussions, D. Kulik for taking an active part in the CAD software development, E. Sverdlenko for carrying out the measurements, and Professor A. Nosich for his help in preparing the manuscript.

REFERENCES

1.G. Graven and C.K. Mok, The design of evanescent-mode waveguide band-pass filters for a prescribed insertion loss characteristic, IEEE Trans Microwave Theory Tech 19 (1971), 295–308.