In this paper, the trade-offs between out-of-band performance, filter length, power-handling capability, and insertion loss of both symmetrical and asymmetrical evanescent mode ridge rectangular waveguide filters are investigated. As a result, clear design methodologies for optimizing such performances are proposed. The developed methodologies are then applied to design several evanescent mode filters, and a complete performance analysis of the symmetrical and asymmetrical structures is performed. From the performance analysis results, the designer can choose the more appropriate filter topology and design strategy to satisfy the prescribed specifications.
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 Design requirements for passive waveguide filters in modern telecommunication systems for both space and terrestrial applications are becoming more and more restrictive. Low insertion loss, compact size, high skirt selectivity, wide spurious-free stopband, and suitable power-handling capability are usually required [Cameron et al., 2007]. These increasing demands have stimulated the advent and refinement of different types of filters during the last decades [Matthaei et al., 1980; Uher et al., 1993; Hunter, 2001]. The designer must choose the more suitable filter type to fulfill the set of specifications. In addition, the designer should exploit the structure free design parameters in order to obtain the best performance trade-off to satisfy the specs.
 From the different waveguide filter types, evanescent mode ridge filters, proposed originally by Craven  and Craven and Mok  and refined by Snyder , exhibit many attractive features. These filters provide an excellent out-of-band performance with an inherent wide stopband and sharp selectivity. In addition, they are very compact in comparison with other waveguide filter types [Uher et al., 1993]. Their foremost weaknesses come from losses and power considerations due to their small size. In spite of their moderate figures in insertion loss and power handling compared to other waveguide filter topologies, evanescent mode ridge waveguide filters are a suitable choice for moderate and wideband filters when compact size and excellent out-of-band response are involved. For instance, they can be used as preselector filters in input multiplexers, particularly in satellite applications where size is a major concern.
 A high research effort has been devoted to improve and take profit of the evanescent mode ridge filter excellent properties. They have been used to obtain all-pole bandpass and quasi low-pass responses of almost any width [Kirilenko et al., 1999], and recently novel configurations able to implement transmission zeros have been presented [Ruiz-Cruz et al., 2005, 2006]. Moreover, a wide range of topological variations have been proposed to improve a particular filter feature, mainly the stopband extent and/or the filter length [see Saad et al., 1986a, 1986b; Capurso et al., 2001; Kirilenko et al., 2002]. But in most cases, the improvement is obtained by sacrificing losses and power handling, and usually require harder and more expensive manufacture procedures.
 However, to the authors knowledge, there is a lack of studies on the balance between insertion loss, power handling, stopband performance, and filter length in evanescent mode filters. A first preliminary study focused on symmetrical ridge waveguide filters has been presented by Soto et al. . The present paper represents an important extension of this previous work. First, a more detailed parametric analysis has been performed, thus obtaining clearer and more defined design strategies. Second, both symmetrical and asymmetrical evanescent mode filters are considered. Third, a detailed performance analysis is performed, and the symmetrical and asymmetrical topologies are compared to determine the more appropriate configuration to satisfy a certain set of design specifications. In addition, a new and fast synthesis procedure of ridge filters is proposed. This procedure can help the designer in the application of the design strategies developed in this paper, since it can be used to make an initial adjustment of the filter physical dimensions and evaluate their impact in some key filter performances.
2. Parametric Analysis
 This paper is focused on conventional rectangular waveguide evanescent mode filters with rectangular cross-section metal inserts as capacitive elements (see Figure 1). The centered symmetrical configuration shown in Figure 1a, as well as the centered asymmetrical configuration with ridges only in the upper wall (see Figure 1b), will be considered and compared in order to establish the more appropriate one to refine each filter performance. For the sake of simplicity in the parametric analysis of the structure, all the rectangular shaped ridges are considered to be centered and to have the same width w and height h. In addition, the housing width ah and height bh are kept constant throughout all the structure.
 Since the evanescent mode filters under consideration can be decomposed into waveguide sections and planar steps, an accurate and efficient modal analysis technique based on an integral equation impedance representation and the Boundary Integral–Resonant Mode Expansion (BI-RME) method has been developed [Conciauro et al., 2000]. The design of the structure is performed by the systematic decomposition procedure proposed by Guglielmi . To compute the insertion loss and the electric field levels of the designed filters, the Ansoft HFSS commercial simulator has been used.
 In order to design the evanescent mode filters, we have tuned the ridge waveguide section lengths ti and the below-cutoff waveguide section lengths li, since a modification to these parameters does not require to recompute the waveguide modes and hence the filter design procedure is sped up. As a result, the housing dimensions ah, bh and the ridge height h and width w are free design parameters that must be previously fixed. They should be used to obtain the trade-off between insertion loss, power handling, length, and out-of-band performance required by the filter design specifications.
 Following the procedure described by Soto et al.  a parametric analysis has been carried out to find out the effect of the free design parameters in the filter performances. A 2-pole Chebychev filter centered at 7 GHz with 280 MHz bandwidth and 15 dB return loss ended in WR137 access ports was taken as reference filter. As a result, several tables and graphs relating the free design parameters values and the filter performances have been obtained. The collected data will be analyzed in detail in sections 2.1–2.4.
 In addition, the modal spectrum of the ridge waveguides have also been characterized. As shown in Figure 2, a ridge depth h increase reduces the fundamental mode cutoff frequency whereas the cutoff frequency of the other represented modes hardly change, so that the monomode operating region increases. Concerning the ridge width w, a parabolic variation is observed in the fundamental mode cutoff wavelength, with a maximum value at wopt which is around 0.35ah to 0.45ah. To obtain wopt for an asymmetrical ridge waveguide, the following expression can be used:
with x = h/bh, y = bh/ah. In the symmetrical case, x = 2h/bh and y = bh/2ah must be taken. The validity of this approximation comprises the range 0.25 < y < 2, which covers all the practical applications. The coefficients Ai and Bi are collected in Table 1.
 The cutoff frequencies of the higher-order modes collected in Figure 2 are also very important for the filter performances, particularly the out-of-band response.
 The dimensions of the filters designed in section 4 reveal that Figure 2b is the typical modal chart to be used in the ridge choice for symmetrical evanescent mode filters. On the contrary, Figure 2a is very appropriate for the asymmetric filter ridge choice, since the asymmetric ridge modal spectrum can be extracted from the spectrum of a symmetric double-ridge waveguide with double height and the same w, h and ah.
2.1. Insertion Loss
 Assuming vacuum for the dielectric material, we have only considered the filter losses due to finite conductivity conductors. Nonetheless, for waveguide filters with good conductors the losses are very low and their computation is not a simple task (for instance, the response ripple can hide small insertion loss variations required to perform the parametric analysis). To overcome the difficulties, a metallic conductor with moderate conductivity has been considered (steel stainless, σ = 1.1 · 106 S/m), and the losses have been measured at the second reflection zero of the ideal response. In addition, to compensate for the deviation caused by ∣S11∣ ≠ 0 the following modified insertion loss figure has been considered:
which represents the ratio of the total input power to the power not dissipated inside the filter. Finally, to obtain a more realistic value, this figure is scaled to copper conductors by correcting the material conductivity factor.
Figure 3 compiles the resulting graphs for the modified insertion loss (scaled to copper conductors).
 Concerning the ridge width, the parametric analysis reveals that the losses increase is negligible for w < 0,5wopt and a dramatic increase is observed when w > 0,65wopt for both symmetrical and asymmetrical topologies. Hence, to reduce insertion loss it is highly recommended to keep w < 0,65wopt.
 Losses also tend to raise with the ridge depth h, and their change rate increases with the housing height. As a result, the effect of the ridge depth h in losses is more important for asymmetrical filters. To reduce insertion loss, mainly for asymmetrical filters, a small value of h should be chosen.
 The free design parameter with the greatest influence in the filter insertion loss is the housing width ah. As expected, a wider housing provides lower losses in both symmetrical and asymmetrical configurations. As a result, to optimize losses it is of paramount importance to take the wider housing that allows to satisfy the remaining filter specifications. The housing height bh, on the other hand, is not relevant provided that the housing is high enough (especially in symmetrical filters). Anyway, in order to reduce the filter losses a higher housing height normally proves to be more appropriate.
2.2. Power-Handling Capability
 In microwave passive devices, assuming a good thermal design, the power handling capability is usually restricted by multipaction and ionization breakdown [Yu, 2007]. The multipaction breakdown limits the maximum voltage between two close conductors plates whereas the ionization breakdown is related to the maximum electric field in the device. At extremely low pressure conditions (for instance, in satellite applications), multipaction is the limiting factor.
 Focusing on multipactor breakdown [Hatch and Williams, 1958], the multipactor susceptibility curves [Woode and Petit, 1989] show that for practical microwave components the maximum voltage is directly proportional to the gap between both plates. Therefore, to derive the power-handling capability it is very useful to compute the following equivalent electric field for a 1W input excitation:
where g is the ridge gap and s is the integration path, both defined in Figure 4. The subscript max denotes that this equivalent electric field is obtained using the integration path where the above integral is maximum, which is found at the center of a ridge of the structure.
 The maximum equivalent electric field (3) obtained for a 1 W input excitation in combination with the curves in the work of Woode and Petit  allow to obtain the filter power-handling capability related to multipactor breakdown.
 At higher pressures the limiting factor is the ionization breakdown, which is originated in a zone with an intense electric field. The characterization of this breakdown is more complex since close to the sharp corners of the ridges the electric field is singular. Depending on several parameters, the ionization breakdown will be originated in the distributed uniform field in the gap between ridges or in the localized field singularity in the ridge sharp edges as discussed by Olsson et al. . Therefore, the equivalent electric field Emax could not be appropriate to obtain the maximum input power at the filter ports when the ionization breakdown is originated at the ridge corners. Anyway, even in that case, this figure can be used to compare the power-handling capabilities of different evanescent mode filters, since a higher value of Emax implies a greater field in the ridge corners and therefore a poorer power-handling capability.
 The equivalent electric field (3) in terms of the design free parameters for symmetrical and asymmetrical evanescent mode filters is depicted in Figure 5.
 The results of our parametric analysis reveal that the maximum equivalent electric field depends on the gap between ridges g exponentially. Hence, the filter power-handling capability is maximized by increasing the housing height bh and by reducing the ridge depth h for a bh set. A wider housing should be preferred since it allows less deep ridges, and the ridge gap can be increased. Moreover, Emax can be slightly reduced by means of a wider ridge. Nevertheless, if the ridge width is chosen to be greater than wopt, the resonator cutoff frequency starts to increase and the resonator must be enlarged to compensate this effect, reducing the stopband (or increasing the filter length). To avoid the stopband and/or length degradation, the ridge depth must be increased to reduce the resonator lengths, thus diminishing the gap and the filter power-handling performance. Therefore, the optimum choice of the ridge width is w ≃ wopt in order to increase the ridge gap (or even a slightly wider ridge, that can provide a very small improvement since it can take profit of the slow cutoff frequency variation around wopt shown in Figure 2).
 Finally, asymmetrical filters can provide higher ridge gaps and therefore withstands higher input power levels.
2.3. Filter Length
 For the sake of space, the filter length information has been summarized in Figures 3 and 5 with marks. For usual filters, the inverter (i.e., housing section between ridges) lengths are the main part of the filter length. Although a narrower housing increases the resonator lengths, it drastically reduces the inverter lengths, so the housing width must be reduced to diminish the filter length. The remaining free design parameters are less important than the housing width to the filter total length. Anyway, a higher housing height should be chosen to obtain a shorter structure. Regarding the ridge dimensions, to reduce the filter length it is convenient to increase the ridge width w (up to wopt since for w > wopt the filter length reduction is insignificant) and then increase their depth h.
2.4. Spurious-Free Stopband Response
 Several factors can produce a first spurious response before the upper frequency of the requested stopband, fusb. The first factor arises from the propagation and resonance of the TE10 mode in the housing rectangular waveguide. The avoidance of this resonance restricts the maximum housing width ah to:
where x is a term that takes into account that the housing sections resonate at the frequency where their lengths become approximately λg/2, instead of at the TE10 cutoff frequency.
 The second resonance of the ridge sections is another effect that can end the filter stopband. This TE102 resonance comes from the periodic response of the resonators, and can be controlled by a suitable choice of the ridge dimensions w and h to keep the ridge electrical length under half guide wavelength in the filter stopband.
 The higher-order modes are the last factor that must be considered to satisfy the stopband specification. These modes can provide an unwanted bridge to transfer energy along the structure ridge and housing sections. Using modal charts similar to Figure 2 we can choose the ridge dimensions w and h to avoid the propagation of higher-order modes in the ridge sections before the stopband end. An extremely important case for evanescent mode waveguide filters, and usually the limiting factor, is the TE01-like mode or, in LSE/LSM terminology, the LSE01-like mode. The generation of higher LSE/LSM modes are favored by the waveguide wall parallelism, and the ridge perpendicularity to top and bottom of the waveguide. These modes, when propagate, travel directly from input to output and contribute to the termination of the filter stopband. The first of these modes is the TE01/LSE01-like mode, that starts to propagate at nearly the same frequency in the housing and the ridge resonators. Although from symmetry considerations this mode should not be excited, manufactured structures never hold a perfect symmetry and alignment, and as it will be shown in section 4, in practice the condition
must limit the housing height to avoid TE01/LSE01 related spikes.
3. Design Methodology
3.1. Design Strategies
 From the parametric analysis it can be concluded that for optimizing insertion loss, power handling, and filter length, it is convenient to choose the highest housing height. Hence, according to (5), for any design bh should be fixed to bh,max.
 To reduce the filter losses and increase its power-handling capability the widest housing must be chosen. For both strategies the housing width can be obtained by starting from (4) with x = 0.15 and then increasing ah while the length and the out-of-band specification is fulfilled. In case that the designed filter does not satisfy these specifications, a narrower housing should be taken.
 In addition, to optimize the power-handling capability, the maximum ridge gap must be obtained by choosing the lower ridge depth h. This can be accomplished by setting w ≃ wopt and then reducing h whereas the first inverter section can be manufactured, the filter total length condition is fulfilled, and the ridge section length increase does not introduce the TE102 resonance in the stopband. On the other hand, to optimize the filter insertion loss it is also recommended to take the lower ridge depth h but with the width set to w ≃ 0.65wopt as the losses increase dramatically for w > 0.65wopt. If a compromise between insertion loss and power handling must be sought, an intermediate value for w could be taken.
 Conversely, the design procedure to optimize filter length and out-of-band response require a manufactured filter with the narrowest housing. The following algorithm can be used to obtain the housing width and the remaining design free parameters:
 1. For the housing dimensions set, take w = wopt.
 2. Increase the ridge depth h up to obtain the smaller ridge length that can be successfully manufactured.
 3. Whereas the first inverter section can be manufactured, reduce ah and go to 1.
 This algorithm provides the shorter filter and maximizes the stopband simultaneously. The differences between filter length or stopband optimization comes from the housing height bh. Out-of-band optimization is more restrictive in terms of bh, since as the upper stopband frequency fusb is increased, the housing height must be reduced together with the housing width to avoid spurious responses from TE01-like modes (see (5)).
 In summary, two pair of quite similar strategies have been presented. The first pair is for optimizing insertion loss and power-handling capability, whilst the other pair is for filter length and out-of-band response. Both pair of strategies take the highest housing height allowed by (5) to satisfy the out-of-band specification, and a very similar value for the ridge width w. However, they have the opposite interest regarding the housing width ah and the ridge height h. Sometimes a suitable trade-off between these dimensions must be found to fulfill the filter specs.
 To conclude this section, it is worth to remark that several techniques have been proposed in the literature to implement narrower housings, such as the stepped-wall approach [Snyder, 1983] or the ridge transformer [Nanan et al., 1991]. At expense of insertion losses and power handling, these techniques can improve the filter spurious-free band and, for high-order filters, they can also provide shorter structures. Although these topologies are out of the scope of this paper, most of the procedures and conclusions presented herein can be easily extended to these geometries.
3.2. Approximate Synthesis Procedure
 The design strategies described in section 3.1, although provide a systematic design procedure considering different key filter performances, give general guidelines to the designer that require some effort and time to be put into practice. In fact, only the housing height is clearly defined by equation (5), whereas the value of the ridge width, although known in terms of wopt, cannot be obtained until a good approximation for the housing width is available.
 In this section, we are going to outline a new and fast synthesis procedure that provides an approximation for the filter physical dimensions. From these dimensions, the filter performance in terms of length and spurious-free stopband will be evaluated. By using this technique, and guided by the design strategies proposed in this paper, the practitioner can obtain in reduced time a very good estimation for the filter housing width (the main parameter related to the length and the spurious-free stopband). From the knowledge of the housing dimensions, and with a suitable value for the ridge height, the ridge width can be accurately computed by using (1). Therefore, a good starting point can be obtained to apply the design strategies proposed in this work.
 The synthesis starts from an initial guess of the design free parameters ah, bh, w and h, and the waveguide port dimensions ain, bin. As a result, it returns the length of the filter ridge and housing sections.
 To represent the step between the input port and the housing, the model described by Craven and Mok  has been considered. The evanescent mode housing sections of length li are modeled by the traditional circuit composed of an admittance inverter Ji loaded with a shunt admittance jBi at each side. The values of these elements are:
where η is the intrinsic impedance of the medium, and fc,h = c/2ah denotes the TE10 cutoff frequency in the housing waveguide.
 On the other hand, Figure 6 shows the equivalent circuit of an step between the housing and the ridge waveguide. The ith ridge section of length ti is replaced by the cascade connection of this equivalent circuit followed by a transmission line section of length ti and again the equivalent circuit that models the step between the ridge and the housing waveguides. The turn ratio ni and the capacitance Ci of the equivalent circuit are easily obtained by matching the input impedance of an EM simulation of the step between both waveguides.
 For a filter with ridge sections of different lengths, a frequency mapping valid for all the resonators from the filter prototype to an standard synthetizable low-pass prototype cannot be defined. Therefore, an approximation must be performed. Following the traditional synthesis techniques [see, e.g., Craven and Mok, 1971], the ridge resonators are replaced by a lumped capacitance that represents the ridge main effect. Using this approximation, the filter order and the housing section lengths are computed. In a further step in the synthesis procedure, the resonators will be replaced by a more accurate model to obtain a better estimation of the ridge section lengths. Anyway, the capacitive approximation causes a reduction in the realized bandwidth of the filter that increases with the filter bandwidth. A better representation would be obtained by introducing a transmission line in the ridge representation, but then it will not be possible to define a unique frequency mapping valid for all the filter resonators.
 Once the ridge resonators are replaced with a capacitance, the frequency mapping can be defined. For the majority of narrow and moderate bandwidth filters, the admittance inverters are nearly constant with frequency. In such cases, the following low-pass to passband transformation is established to link the equivalent low-pass prototype with the resulting prototype of the evanescent mode filter:
where ωc,h = 2πfc,h. The center radian frequency ω0 and the bandwidth parameter Δ can be computed from the filter cutoff radian frequencies ω1 and ω2 after applying:
 In case of the filter relative bandwidth is greater than 10%, or the housing is well below cutoff (fc,h > 3f0), it is more accurate to extract a frequency dispersion factor for the inverters, as suggested by Snyder . Including this frequency dispersion factor, the frequency transformation will be:
 The parameter Δ includes the relative bandwidth of the filter as well as the reactance slope of the resonators in both cases.
 According to the filter requirements, the appropriate frequency transformation is used to translate the attenuation specifications to the equivalent normalized low-pass prototype. The filter order N and normalized low-pass parameters gi are then computed in the usual way [Matthaei et al., 1980].
 Next, turning back the transformation to the evanescent mode prototype, the following equations that involve the lengths li are obtained:
where R′ is obtained from the equivalent circuit of the step between the access port and the housing [see Craven and Mok, 1971], and γ0,h = .
 To compute the lengths li from (12)–(14), an iterative procedure must be implemented. Initially, the lengths are assumed to be so large that coth (γ0,hli) ≃ 1. Next, (12)–(14) are used iteratively to refine the housing section lengths li. Convergence is achieved in very few iterations.
 In addition to the capacitive effect considered to deduce (8)–(14), the ridge sections have a distributed behavior. Therefore, to compute the ridge section length, more accurate results are obtained considering a distributed ridge section instead of an equivalent shunt capacitance (for instance, the one given by Chen ). Therefore, the equivalent circuit shown in Figure 7a is considered for the ith resonator. The shunt admittances j1,i and j2,i are usually equal to the admittances of the housing sections connected at the input and output of the ridge section, respectively (see (7)). However, in the first and last resonators, the shunt admittances jX′J12 and jX′JN+12 must be also included. These shunt admittances come from the equivalent circuit of the input and output steps as described by Craven and Mok .
 To obtain the resonator lengths, the shunt admittances are grouped and replaced by a section of short-circuited ridge transmission line. As a result, the equivalent circuit in Figure 7b is obtained, with stub lengths
where λg,r and Y0,r denote the fundamental mode wavelength and characteristic admittance of the ridge waveguide. The resonator physical length ti is deduced forcing an equivalent resonator length of λg,r/2 at the filter center frequency f0.
 According to this procedure, the lengths of the housing and ridge sections are obtained. Therefore, the filter total length can be calculated. In addition, it is possible to estimate the stopband extent of the filter since the three factors described in section 2.4 can be computed. The resonance of the TE10 mode in the housing sections appears when its length turns to be λg,h/2, namely:
 Another limiting factor is the second resonance of the resonators. This resonance can be easily evaluated looking for the next resonance of the equivalent circuit in Figure 7b. Finally, the third limiting factor is the propagation of the TE01/LSE01 in the housing due to ridge misalignments in the manufacture procedure. The cutoff frequency of this mode is f = c/2bh, that must be kept above the filter stopband.
 The method provides a very good estimation of the stopband limit, and tends to overestimate the total filter length providing a reduction in the realized filter bandwidth. Anyway, the limiting factor of the housing width is usually the stopband specification, and hence a good estimation of the housing width can be achieved. The precision of the proposed method depends on the accuracy of the pure capacitive model for the ridge sections used to define the frequency mappings (8)–(11). This representation is valid for narrowband filters but degrades when the filter bandwidth increases. Therefore, worse results (longer filters with narrower bandwidths and poorer filter starting dimensions) are obtained for wideband filters. This effect is more remarkable in the case of asymmetrical ridge resonators, where the gap is higher and the pure capacitive model is less accurate.
 Finally, although no method has been provided to compute the filter maximum power-handling capability, the approximate synthesis technique, in combination with the design strategies, can be used to maximize the ridge gap and therefore the power-handling capability.
4.1. Performance Analysis of Filter Topologies
 To test the design strategies described in section 3.1 and determine the optimal performances of the evanescent mode ridge waveguide filters, four different specifications have been considered. A symmetrical and an asymmetrical filter have been designed to fulfill each specification, so that both filter topologies can be compared in terms of insertion losses, power handling, length, and stopband. For all the filter specifications, a 0.02 dB ripple Chebychev bandpass response with 300 MHz bandwidth centered at 10 GHz has been considered. The filter access ports were standard WR-90 waveguides.
 The first pair of filters, denoted onward as filters A, have been designed to reduce losses with a stopband from 10.4 to 17 GHz providing a rejection greater than 40 dB. Filters B have been optimized to improve power-handling capability with the same out-of-band specifications of filters A. The symmetrical and asymmetrical filters C have been designed to reduce length while keeping the spurious responses above 20 GHz. Finally, in filters D, the stopband extent has been optimized.
 Following the design strategies described in this paper, and using the full-wave modal analysis tool and the segmentation design technique referenced in section 2, the four pairs of filters have been designed. Five-order evanescent mode filters have been required in all cases to satisfy the stopband specifications. Table 2 compares the dimensions and performances for each designed structure. The manufacture constraints consisted on a minimum length of 0.25 mm for the first inverter and ridge sections longer than 0.9 mm.
Table 2. Symmetrical and Asymmetrical Filter Dimensions and Performances for the Four Specifications Considered
l1 = l6 (mm)
t1 = t5 (mm)
l2 = l5 (mm)
t2 = t4 (mm)
l3 = l4 (mm)
 As it was inferred in section 2.1, the wider housing satisfying the out-of-band requirements must be sought to optimize insertion loss. However, the ridge gap and hence the power-handling capability can be improved by choosing a slightly narrower housing. This housing gives a small margin to fulfill the stopband specs since the spurious spikes go up in frequency. The margin can be invested in a ridge section length increase due to a higher ridge gap choice, resulting in slightly better power-handling figures (compare filters A and B in Table 2).
 For optimizing the out-of-band response and filter length, the narrower housing should be chosen. Comparing filters D with filters C in Table 2, to increase the stopband above 20 GHz a lower housing height must be taken to keep the TE01 mode below cutoff in the stopband (see (5)). This lower housing increases the first discontinuity in the structure and reduces the housing section length l1. Hence, when the spurious-free band is optimized, a wider housing is required to be able to implement the first inverter and a longer filter is obtained.
 With regard to the comparison between symmetrical and asymmetrical topologies, for the same specifications the asymmetrical filters provide better insertion loss and power-handling capability. The only exception comes from filters B. In this case, the insertion loss are almost equal because the losses increase with the ridge width w starts from a lower w in the asymmetrical topologies (compare Figures 3a and 3c). Anyway, the results in Table 2 reveal that asymmetrical filters can be easily designed to provide simultaneously better losses and power-handling capability than any symmetrical counterpart. In addition, even though slightly shorter filters can be obtained from symmetrical configurations optimized to reduce length, for evanescent mode filters demanding good figures in insertion loss and power handling with moderate stopband the asymmetrical filters are substantially shorter. As a result, the use of the symmetrical topology is only recommended when an extreme optimization of the out-of-band performance or the filter length is requested in combination with poor requirements in losses and power handling.
4.2. Experimental Results
 An evanescent mode ridge waveguide filter in C band has been designed and manufactured to validate the design procedures proposed in this paper with real measurements. An all-pole 249 MHz bandwidth Chebychev response centered at 6.966 GHz has been chosen. The return loss has been set to 22 dB. The filter was manufactured from an aluminium block by milling techniques, and was ended with standard WR-137 ports. A silver-plated coating was applied to the conductor walls to reduce the ohmic losses.
 The filter was designed to optimize insertion loss with a rejection greater than 40 dB between 7.2 and 14 GHz. An asymmetrical configuration was chosen since it can fulfill the stopband demands and provides lower insertion loss than the symmetrical topology. A seventh-order filter was required to satisfy the stopband requirement at 7.2 GHz. Figures 8 and 9 depict the simulated and measured filter responses. The dimensions obtained from the structure dimensional control were employed to perform the simulations. The measured passband insertion loss was 0.35 dB, which translates into a quality factor of about 3600. In Figure 10 the pieces of the asymmetrical filter and the final assembled structure are shown. The filter total length (excluding access ports) was 102.13 mm. These results can be compared with the data from the symmetrical prototype reported by Soto et al.  which was designed to optimize the filter length under the same specifications. The resulting symmetrical filter was a more compact structure of 73.62 mm length, but the quality factor drops to nearly 2800. These results are consistent with the comparative study performed in section 4.1 and validate many conclusions of this study.
Figure 8 shows the excellent agreement between the measured and simulated bandpass responses of the asymmetrical prototype. The slight variations can be attributed to manufacture tolerances. On the other hand, Figure 9 reveals a good agreement between measured and simulated responses in the stopband frequency, but the measured response is plagued with very narrow spikes. For this filter the housing height constraint was not applied, since at that time we were not aware of constraint (5) and a priori seemed that bh did not affect to the out-of-band performance. Therefore, the design strategies lead us to take a higher housing to reduce insertion loss. Nonetheless, spurious appeared in the measured stopband response. The authors studied these spurious and proved that they came from asymmetries due to manufacture tolerances and assembly misalignments that excited unexpected modes. In fact, the first spike appears exactly at the TE01 cutoff frequency. From these results, we deduced the limitation on the housing height (5) that must be always enforced in order to avoid spikes in the stopband.
 In this paper, a complete performance analysis and comparison of symmetrical and asymmetrical configurations of evanescent mode ridge waveguide filters have been carried out. In addition, the compromises between insertion loss, power handling, out-of-band response, and length have been deeply studied. From this paper, the designer can choose the best topology to satisfy a prescribed set of specifications. Furthermore, he can also select the design strategy to be followed in order to achieve the best performance trade-off.
 This research stresses the importance of taking into account key performances such as insertion loss, power-handling capability, and filter length in the design of microwave filters, and that a careful choice of the filter physical topology and dimensions must be sought in order to obtain the best compromise to satisfy the filter specifications. Although this practical work has been focused on conventional evanescent mode ridge waveguide filters, most of the results and conclusions can be easily extended to other evanescent mode topologies, and the same procedure can be followed to investigate these compromises in other waveguide devices of interest to the microwave industry.
 The authors thank Thales Alenia Space for the manufacturing and measurement of the filter prototypes. The authors would also like to thank Christoph Ernst, ESTEC-ESA, Noordwijk, Netherlands, for a fruitful discussion about this work during the 36th European Microwave Conference held at Manchester, UK. The authors would also like to acknowledge the suggestions and contributions of the anonymous reviewers. This work was supported by Thales Alenia Space España, S.A., under research project AEO/UPV/01.3005, and by Ministerio de Educación y Ciencia, Spanish Government, under the coordinated Research Project TEC2007-67630-C03-01.