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3216 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 61, NO. 9, SEPTEMBER 2013

CAD Procedure for High-Performance Composite

Corrugated Filters

Fabrizio De Paolis, Member, IEEE, Rousslan Goulouev, Member, IEEE, Jingliang Zheng, and

Ming Yu, Fellow, IEEE

Abstract—The design of waveguide low-pass ﬁltersisaccom-

plished with a new method, where focus is on the upper stopband

performance rather than passband or roll-off requirements. Using

an efﬁcient multimode variational formulation, composite ﬁlters

are generated by direct optimization from an arbitrary number of

partial corrugated subelements, each showing mutual -mode

passband and different nonintersecting passbands corresponding

to higher order modes. The ﬂexibility of this method leads to

optimal ﬁlter solutions, having the designer full control over key

dimensional features, i.e., minimum gap and cavity length. It

is therefore possible to design low-pass ﬁlters exhibiting broad

stopbands free of spurious propagation, without penalties of

higher losses, lower power handling capability, larger size, and/or

increased manufacturing complexity. Simulated and measured

results demonstrate signiﬁcant advantages over ﬁlters designed

with conventional methods.

Index Terms—Computer-aided design (CAD), corrugated ﬁl-

ters, microwave ﬁlters, optimization, periodic structures.

I. INTRODUCTION

LOW-PASS ﬁlters ﬁnd extensive use in microwave and

millimeter wave systems for rejecting unwanted spectrum

over broad frequency ranges. One area of particular interest is

that of satellite communication systems, where far out-of-band

rejectionisrequiredinadditiontothenarrowbandﬁltering pro-

vided by the multiplexing network. Broad stopband, low inser-

tion loss, high-power handling capability, minimum size, and

simple manufacturing are the key requirements in similar appli-

cations. Unfortunately, these features are in conﬂict when de-

signing conventional ﬁlters, and a tradeoff has to be made. The

ultimate consequence is the increasing difﬁculty to meet tech-

nical requirements of modern applications.

Three categories of waveguide low-pass ﬁlters have been

studied by microwave engineers: stepped-impedance (corru-

gated), wafﬂe-iron, and ridged ﬁlters.

A. Limitations of Conventional Corrugated Filters

Corrugated waveguide ﬁlters were ﬁrst introduced by Cohn

[1] in the late 1940s. Later design methods using modern cir-

cuit theory and synthesis techniques were developed by Levy [2]

Manuscript received February 11, 2013; revised July 06, 2013; accepted July

16, 2013. Date of publication August 15, 2013; date of current version August

30, 2013.

F. De Paolis was with COM DEV Ltd., Cambridge, ON Canada N1R 3P2.

He is now with the European Space Agency, ESA-ESTEC, 2201AZ, Noordwijk,

The Netherlands (e-mail: Fabrizio.De.Paolis@esa.int).

R. Goulouev, J. Zheng, and M. Yu are with COM DEV Ltd., Cambridge, ON

Canada N1R 3P2 (e-mail: ming.yu@comdev.ca).

Color versions of one or more of the ﬁgures in this paper are available online

at http://ieeexplore.ieee.org.

Digital Object Identiﬁer 10.1109/TMTT.2013.2275451

and generalized for a wide class of tapered corrugated waveguide

ﬁlters. The inconvenience of spurious passbands corresponding

to high-order modes was reported in [2] and [3] and further in-

vestigated in [4]. The spurious passbands cannot be eliminated,

although their location can be predicted and eventually moved

up or down the frequency axis by changing the width of the cor-

rugated waveguide. The spurious bands are due to the E-plane

symmetry of structure, which is the baseline of this design. Since

most of the energy propagates in the dominant mode, usually only

narrow spurious components or “spikes” occur. While Hauth in-

troduced an efﬁcient analysis method [5] and described modi-

ﬁcations to the corrugated ﬁlter in order to use it as a bandpass

ﬁlter [6], Levy’s work was later generalized to the inhomoge-

neous case [7]. Although Levy’s latest approach improves the

ﬁlter rejection performance, two separate points remain open: 1)

high-order spurious spikes are quite difﬁcult to control (see [7,

p. 124]) and are not acceptable in many applications and 2) stan-

dard closed-form formulas are used, requiring major tradeoffs

among the stopband, insertion loss, and power handling (see [6,

p. 526]) and limiting the design ﬂexibility. More recent studies

proposing elaborate two-dimensional (2-D) proﬁles have been

published [8], [9], showing higher order mode suppression, but

the methods do not show low insertion loss, compact size, and

hardware simplicity as in this paper.

Corrugated ﬁlterspreviouslyreportedarebasedonwave-

guide structures susceptible to propagation of high-order modes

within the design stopband. Modes of order higher than the

dominant mode are always present in any practical microwave

systems due to discontinuities associated with both manufac-

turing inaccuracies (e.g., asymmetries and misalignments) and

external mode exciters (e.g., transformers, bends, and adapters).

Sincetheﬁlter out-of-band mode composition is, in principle,

unknown, the worst case excitation must be taken into account

if rejection speciﬁcations are to be met without spurious effects.

Design methods employed in the conventional ﬁlter types

are based on distributed or lumped element circuits. The target

of those design methods is the synthesis of an equi-ripple

ﬁlter response by passband and roll-off requirements. As a

result, the in-band frequency response may well reproduce

ideal polynomials such as Chebyshev or Zolotarev. In practice,

thoseﬁlters may not meet the rejection requirements as well.

In the out-of-band frequency range, the original model used

for in-band synthesis is no longer valid. Therefore, conven-

tional design methods ignore information about critical points

corresponding to high-order modes, which form passbands or

stopbands.However,theonlyeffectivemethodtodesigna

spurious-free ﬁlter must be based on the proper distribution of

those critical points over the required out-of-band frequency

range, in order to guarantee full coverage of reﬂection zeros by

transmission zeros.

0018-9480 © 2013 IEEE

DE PAOLIS et al.: CAD PROCEDURE FOR HIGH-PERFORMANCE COMPOSITE CORRUGATED FILTERS 3217

Fig. 1. Symmetric E-plane cavity. (a) The 2-D view. (b) Equivalent network.

B. Limitations of Wafﬂe-Iron and Ridge Filters

Waf ﬂe-iron ﬁlters [10], [11] are susceptible to generation

of spurious spikes in the passband, roll-off and stopband [11],

[12]. These spikes are caused by excitation of waveguide

modes not considered by established design methods. Limited

power handling is another major disadvantage. Wafﬂe-iron

ﬁlters are therefore rarely employed in applications requiring

both spurious-free and high power performance.

Evanescent-mode ridge waveguide ﬁlters [13], [14] require

quite small gap sizes in order to provide sufﬁcient out-of-band

rejection in most applications. Therefore, the quality factor and

peak power-handling capability of the ridged ﬁlter are signiﬁ-

cantly lower than that of nonevanescent type of ﬁlters. Tuning

could also be required to compensate for the high sensitivity to

manufacturing tolerance. Therefore, ridged ﬁlters are not suit-

able for those applications requiring very low losses or high

power-handling capability.

C. Alternatives to Conventional Synthesis Techniques for

Harmonic Filter Design

In addition to rejection performance, conventional synthesis

techniques limit the low-pass ﬁlter designer to control other im-

portant features, such as insertion loss, power handling, size, and

manufacturability, which are not directly associated with ﬁlter

prototypes. On the other hand, recent developments in computer

technologies have made possible direct optimization of whole

ﬁlter structures for speciﬁc requirements. As preliminarily shown

in [15] and online,1optimization methods would allow one to by-

pass the constraints of synthesis techniques, offering distinct ad-

vantages in terms of (multimode) rejection performance and de-

sign ﬂexibility. An optimal practical low-pass ﬁlter could then

look quite different from any classical prototype.

II. THEORY

A. Cavity Model and Transmission Zeros

The concept of direct-coupled cavities is the fundamental

model describing reﬂection and transmission properties of a cor-

rugated ﬁlter as a nonuniform waveguide in multimode condi-

tion of propagation. In order to illustrate the basic properties

of a single cavity element as the building block of high-perfor-

mance low-pass ﬁlters, a simple equivalent network is consid-

ered ﬁrst. Fig. 1(a) shows the symmetric corrugation composed

by a T-junction loaded by a shorted waveguide stub, whereas

the equivalent network following a model from Marcuvitz [16]

1[Online]. Available: http://www.goulouev.com/structures/newﬁl.htm

Fig. 2. Transmission response of a single cavity element.

is derived in Fig. 1(b). The -network admittance parameters

are calculated as

(1)

where the admittance is zero for

(2)

The reﬂection and transmission coefﬁcients of the single

cavity can be expressed in elementary functions as follows:

(3)

in which

(4)

Simple inspection of (3) shows that when is zero, the

transmission coefﬁcient is also zero. Each extracted cavity

then forms transmission zeros (Fig. 2) located between fre-

quency bands of moderate transmission. Therefore the cavity

element can be used as a low-pass unit or high-pass unit. In

the vicinity of a given transmission zero , we consider a

bandwidth where the transmission coefﬁcient magnitude

shall be smaller than the corresponding speciﬁed value .

There is also a plurality of frequency points where the reﬂection

coefﬁcient is approximately equal to zero. This happens at

the cut-off of each mode and at the frequency points where

is near inﬁnity.

B. Composite Corrugated Filter

Composite corrugated ﬁlters are generated via direct connec-

tion of quasi-periodic E-plane tapered waveguides (partial

ﬁlters), internally matched without intermediate transforming

networks (Fig. 3). The composite structure is formed by

cavities in series, with coupling gap ,

length and spacing . The cavity length is much shorter than

3218 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 61, NO. 9, SEPTEMBER 2013

Fig. 3. Composite corrugated ﬁlter consisting of partial ﬁlters.(a)Sideview.(b)Topview.

. The cavity height is gradually tapered from at

the ends to at the center of each partial ﬁlter.

Although the structure of Fig. 3 physically resembles Levy’s

inhomogeneous ﬁlter, there is a substantial difference from the

electrical viewpoint. The ﬁlter considered is based on short ex-

tracted cavities, realizing transmission zeros in the stopband

(only ﬁrst root for each cavity is used). The order

is not associated with the number of in-band reﬂection zeros

and cannot be unambiguously associated with performance re-

quired. Therefore, is a secondary parameter and can be ini-

tially set to a value of 20 2 (for third-harmonic rejection). This

approach differs from classic corrugated ﬁlter theory where an

-pole mode response is synthesized using a capacitive

iris or stepped-impedance transmission line model [2].

Assuming a gap relatively smaller than the waveguide

widths , we can limit our consideration to only modes

in the analysis of the stopband. An image frequency for the

mode of the th partial is predicted by

(5)

which is a generalized version of the formula given in [2] and

where is a frequency point of the dominant mode and is

the waveguide width in inches of the th ﬁlter.

The multimode transmission responses can be represented as

schematic graphs for the ﬁlters to generate mode charts. Se-

lecting partial ﬁlters with different widths, the passbands corre-

sponding to spurious modes can be positioned relatively

to each other in such a way that no mutual frequency points

exist. In this case any of the modes carrying spurious fre-

quency spectrum in one of the partial ﬁlters will be rejected by

the other partial ﬁlter. This condition is called “strong,” since

it provides rejection of spurious frequency spectrum carried by

all modes except the dominant one. When the position of spu-

rious passbands corresponding to both partial ﬁlters intersects

each other and the intersecting modes have different symmetry

indexes, the condition is called “weak.” A weak condition also

guarantees that no spurious mode can propagate through the

overall ﬁlter provided that ideal symmetry is maintained. Both

conditions of spectrum superposition can be adopted for design

using (5). However, it shall be mentioned that the weak con-

dition, in practice, provides less rejection due to conversion be-

tween symmetric and asymmetric modes when asymmetries are

introduced by manufacturing inaccuracies.

The criterion used to select the dimension (expressed in

inches) is based on imposing the cross-polarized wave-

guide mode to not propagate through the ﬁlter, i.e.,

(6)

where is the dominant mode cut-off frequency of the lowest

partial ﬁlter, is the highest frequency point of the spec-

iﬁed stopband and 0.5 is an empirical coefﬁcient to account for

the cavity loading effect.

The lengths deﬁne the cavity’s Q-factor at the transmission

zeros and are optimally selected using the empirical formula

(7)

where increasing the value enlarges the transmission zeros

bandwidth , at the price of stronger spurious effects of higher

order resonances.

The spacing between cavities is set to the minimum value

technologically feasible. Small values of widen the overall

stopband, simultaneously reducing the ﬁlter loss and length. In

addition, a minimum value of maximizes the ratio, ex-

ploiting fringing-ﬁeld effects to further increase the peak power-

handling capability in vacuum [17].

For each partial ﬁlter, the minimum cavity depth de-

pends on the highest speciﬁed frequency point of stopband

, where rejection is required. The maximum value for

—related to the ﬁrst root of when is a

variable—can be approximately evaluated as the stub length at

resonance frequency, including an empirical loading factor

(8)

The ideal composite corrugated ﬁlter presents optimal distri-

bution of transmission zeros formed by each cavity so that any

frequency point of the design stopband is within the rejection

bandwidth of at least one cavity. At the same time, the ta-

pering function has to be adequately smooth in order to mini-

mize the change of the scattering properties along the structure.

DE PAOLIS et al.: CAD PROCEDURE FOR HIGH-PERFORMANCE COMPOSITE CORRUGATED FILTERS 3219

A good compromise is found choosing the sine distribution as

tapering function as

(9)

where .

The parameter is linked to the ﬁlter roll-off and deﬁnes the

transmission zeros distribution from the roll-off, i.e., the trans-

mission zeros cluster moves up in frequency when increasing

.Thesmaller -values (between 1.0 and 1.5) lead to sharp but

narrowband rejection, which is recommended for rejecting up

to the second harmonic. The larger -values (between 1.5 and

3.0) lead to a smooth but wideband rejection and are better for

third-harmonic rejection. Imposing the value, the parameter

is determined as a root of equation as

(10)

where, for a given partial ﬁlter,

is the dominant mode transmission,

is the vector of cavity heights, and is the characteristic

frequency where the ﬁlter dominant mode transmission equals

.

The nonlinear equation system of (9) and (10) can be easily

solved with numerical methods.

The use of extracted cavities with sinusoidal distribution

could result in nonideal passband return loss and near-band re-

jection, because reﬂection zeros can appear around the roll-off

range. However, for low-pass ﬁlters, this is irrelevant when

compliance to speciﬁcations is considered: passband return

loss can be easily optimized (if required) as the initial structure

is already internally matched; near-band performance is not a

low-pass design driver, as its function is to provide far-band

and harmonic rejection.

III. OPTIMIZATION TOOLS AND DESIGN PROCEDURE

A. Mode Chart

The mode chart is the analytical tool from (5), used to perform

preliminary spectrum superposition and determine the optimum

waveguide widths and roll-off frequencies of each th ﬁlter.

B. Fast Multimode Simulator

The fast multimode simulator is a code that loads the

schematic ﬁle with the ﬁlter design dimensions and outputs

frequency response data.

The code model is based on a variational formulation

that takes both fundamental and higher order modes into ac-

count. Fig. 4(a) depicts a cavity of length formed by two

step-junction discontinuities, in the general case of asymmetric

rectangular-to-rectangular waveguide discontinuities with

different accessible modes. A rigorous formulation in terms of

Y- and B-matrices is presented in [18], which can be used for

computing of the multimodal -parameters of generic uniaxial

waveguide discontinuities. Although with such an approach the

Fig. 4. Single waveguide cavity. (a) The 3-D view. (b) Equivalent network.

cavity element of Fig. 4(a) can be characterized by a cascade

of -matrices of two step-junctions, using an independent

model for the double discontinuity gives the advantage of a

more efﬁcient computational algorithm. The expressions of the

Y-matrix members are expressed as

(11)

where the -values are the aperture integrals on both accessible

ports and as

and ,,and are the wave admittances, propagation con-

stant and transverse electric ﬁelds for the th mode in the larger

waveguide. and are the transverse electric ﬁelds for the

th and th mode of both accessible waveguides. The analytical

formulation is given in the Appendix. The case of a single-step

discontinuity can also be solved using a similar formulation.

The rigorous model introduced above is used for fast and ac-

curate modeling of a generally non-uniform corrugated wave-

guide using sequential cascading of multi-modal -matrices.

The process of simulation takes about 2 s on a standard PC for

a typical ﬁlter structure with 20 corrugations, using one acces-

sible mode and approximately 200 localized modes swept over

600 discrete frequency points.

C. Transformer Synthesizer

The transformer synthesizer is a code based on conventional

matching techniques, which loads simulation ﬁle with initial

speciﬁcation and generates a partial ﬁlter matched around the

central frequency of passband via step transformer. Such tech-

niques are widely known and therefore not repeated here. The

program outputs schematic ﬁle with design dimensions. Time of

the synthesis process takes less than one second on a standard

PC, when applied to typical ﬁlters with 20 corrugations.

3220 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 61, NO. 9, SEPTEMBER 2013

D. Optimization

For design optimization, an error function to be minimized is

constructed with the cavity spacing and transformer lengths

as optimization variables as

(12)

where and are the sample frequencies within the passband

and the stopband, and are the reﬂection and transmission

targets and is the vector of the ﬁlter lengths to be optimized.

Only waveguide lengths are used to tune the passband response,

thus the procedure is very fast since only a step characterization

is required all over the optimization process. Time of the opti-

mization process normally takes less than 10 min on a standard

PC when applied to typical ﬁlters with 20 corrugations. This

step can be often omitted, particularly in all cases where stan-

dard return loss level (around 20 dB) are required.

E. Summary of the CAD Procedure

The design procedure comprises the following steps.

Step 1) Filter widths and roll-off frequencies are selected

using the mode chart generated from (5).

Step 2) With the aid of the multimode simulator, initial pa-

rameters of each partial ﬁlter are determined using

the design formulas (6)–(10).

Step 3) The partial ﬁlters are ﬁrst matched individually using

the transformer synthesizer and then directly con-

nected into a composite ﬁlter.

Step 4) With the aid of the multimode simulator, check that

speciﬁed stopband is clear from spurious

spikes.

Step 5) If required, ﬁne-tune the passband return loss and

verify performance.

The procedure requires running many simulations of different

waveguide modes for several design iterations. Any modal anal-

ysis technique can be in principle employed, being the proce-

dure completely general. However, it is found that the approach

becomes particularly fast using the multimode variational for-

mulation of Section III, where most of complex matrix manip-

ulations are replaced by 2-D sums. The overall design process

is then a matter of few minutes, including optimization.

The fundamental difference of this method with respect to

synthesis emerges now in its full extent. Here, the multimode

stopband performance is determined before the passband im-

plementation, imposing the largest and the smallest values,

then tapering the corrugation heights. As near-band transmis-

sion zeros rely on selected -values, the ﬁlter steepness is con-

trolled well independently of the order or gap size .The

advantages resulting from this approach are the broad rejec-

tion free of spurious bands and the optimal dimensional

features. Previous methods proceed somewhat backwards: an

equi-ripple passband is synthesized as a ﬁrst step, losing in-

formation about spurious modes and leading to small-gap and

lengthy structures associated with ﬁlter prototypes.

Similarly to [7], bandwidth limitation is the disadvantage of

using composite ﬁlters. The maximum bandwidth is limited by

the mode cut-off frequency of the waveguide of smallest

width and the lowest roll-off frequency (actual limitation can

Fig. 5. Mode chart for the ﬁrst four modes: , , ,and

.

be easily estimated with the mode-chart). Using this approach

fractional bandwidths up to 20% are typically achieved. These

encompass the majority of applications.

IV. DESIGN EXAMPLE:-BAND OUTPUT FILTER

A. Speciﬁcation Review and Mode-Chart—Step 1)

In the frame of a recent -band space ﬂight program, a

low-pass ﬁlter was required after high-power ampliﬁcationina

remote location of the spacecraft. The ﬁlter must provide broad

rejection free of spurious bands, according to the fol-

lowing speciﬁcations:

Passband 11.2 to 12.5 GH z;

Insertion loss over passband 0.12 dB;

Return loss over passband 26 dB;

Rejection:

from 17.3 to 18.1 GHz 80 dB ( -band Rx);

from 22.4 to 25.0 GHz 60 dB (second harmonic);

from 33.6 to 37.5 GHz 60 dB (third harmonic);

Power handling

(multipactor)

6205 W;

Length 6 in;

Waveguide interface WR75.

Note: good performance over a broader frequency range

would be desirable as a design target. Speciﬁcally, the ﬁlter

should feature a passband between 10.7 and 12.7 GHz with

return losses greater than 26 dB. Moreover, a clean rejection

level should be maintained for frequencies between 17.3 and

40.0 GHz. Although not formal requirements, these targets

would lead to a more generic design suitable to a variety of

-band high-power applications.

Fig. 5 shows the mode chart, where both strong and weak

conditions are used for spurious suppression ( 0.805 in,

0.595 in).

DE PAOLIS et al.: CAD PROCEDURE FOR HIGH-PERFORMANCE COMPOSITE CORRUGATED FILTERS 3221

Fig. 6. Multimode simulator frequency response of the ﬁrst four modes: (a) ,(b) ,(c) ,(d) , and (e) .

B. Multimode Design and Simulation—Steps 2)–4)

Using (6) and (7) at the highest speciﬁed frequency point

37.5 GHz and imposing ,theﬁrst set of design

dimensions was determined ( 0.08 in, ,

0.09 in). The minimum cavity height was calculated from (8)

0.08 in . Setting the order of the two partial ﬁlters

, then selecting and

solving the two resulting nonlinear equation systems (for both

cases: ,0.22 in), the other elements of the

height vector are calculated. The two partial ﬁlters are matched

to WR75 via two-step transformers at one end and then con-

nected together without any transformer at the other end.

It can be noted how (6)–(10) provide very good initial values,

so that the fast multimode simulator is only used for minor

adjustments of while monitoring the spurious perfor-

mance (Fig. 6).

C. Fine Tuning and Performance Veriﬁcation—Step 5)

The passband performance of Fig. 6(a) shows good matching

without any full-wave global optimization. In order to meet

the stringent return loss requirement, minor optimization of

the cavity spacing and transformer lengths was necessary in

this example. Final lengths are within 0.010 in of the starting

values.

The ﬁnal RF performance, including insertion loss and peak

voltages, was veriﬁed via commercial numerical tools (CST Mi-

crowave Studio). Fig. 7 depicts a 3-D view of the ﬁnal structure

half-shell ready for manufacturing.

The ﬁlter response in Fig. 6(a) may look nonoptimal in terms

of polynomial prototypes: there is no correspondence between

number of poles and number of cavities and one reﬂection ap-

pears in the roll-off range around 15 GHz. However, the ﬁlter

structure looks really optimal from a more practical perspective:

Fig. 7. 3-D mechanical model of the fabricated ﬁlter (one half-shell shown).

the large gap size shows promise of high power handling, low

loss, and low sensitivity to mechanical tolerance; the short cav-

ities and simple matching networks results in a compact size

(length is less than 6 in); the ﬁlter body is straightforward from

the manufacturing viewpoint and can be realized using conven-

tional clamshell techniques.

D. Experimental Results

The ﬁlter was built from two aluminum body halves using

industry standard machining process, silver-plated and fastened

by mounting screws.

Figs. 8 and 9 show the low-power -parameters measured

results, without any tuning. The in-band performance (Fig. 8)

shows a return loss greater than 28 dB and an insertion loss

around 0.10 dB. The measurement of Fig. 9 was carried out over

the out-of-band frequency range from 13.0 to 40.0 GHz divided

into three subbands, i.e., 13.0 to 20.0 GHz, 20.0 to 30.0 GHz,

and 30.0 to 40.0 GHz. The ﬁlter WR75 waveguide interface is

3222 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 61, NO. 9, SEPTEMBER 2013

Fig. 8. Measured in-band performance. (a) Return loss. (b) Insertion loss.

Fig.9. Measuredout-of-bandperformanceoverseveralfrequencyranges.(a)Rejectionfrom13.0to20.0GHz.(b)Rejectionfrom20.0to30.0GHz.(c)Rejection

from 30.0 to 40.0 GHz.

connected to the dominant mode waveguide corresponding to

each subband through waveguide tapers, ensuring the measured

signal is converted into the mode. In order to achieve

worst case excitation of both symmetric and asymmetric modes

within the speciﬁed stopband, special mode exciters (H-plane

bends) are placed at the ﬁlter input and output ports [4]. The re-

sulting multimode standing wave between tapers is reduced by

attenuating sections installed before the exciters and calibrated

out. As can be seen in Fig. 9, the out-of-band range is free of

spurious bands beyond the third harmonic. Only minor

narrowband spikes are observed. Such spikes are mainly due to

the mode, in agreement with simulation of Fig. 6(b). It is

important to remark that a “clean” out-of-band rejection—simi-

larly to the plots of Fig. 6(a)—is obtained when the ﬁlter is used

in a symmetric waveguide system, where high-order modes are

considerably lower [4].

The ﬁlter was high-power tested for multipactor breakdown.

The setup consisted of an RF pulsed source (power level

6205 W,test frequency 11.717 GHz, duty cycle 2),

a thermal-vacuum chamber (control temperature 82 C,

pressure 510 torr) and a Cesium radioactive source for

electron seeding. Close-to-carrier noise, third harmonics, and

e-probe were used as detection methods [19]. No multipactor

events occurred up to the maximum power level required.

TAB L E I

-BAND FILTER PERFORMANCE SUMMARY

Simulated with CST Microwave Studio.

No breakdown detected.

Table I summarizes the predicted and measured ﬁlter perfor-

mance versus the technical requirements. It was possible not

DE PAOLIS et al.: CAD PROCEDURE FOR HIGH-PERFORMANCE COMPOSITE CORRUGATED FILTERS 3223

only to meet the stringent speciﬁcations, but also to provide

good rejection (between 55 to 80 dB approximately) over

a much broader range than required. Similar rejection levels

cannot be guaranteed by conventional corrugated ﬁlters and are

normally achievable only using wafﬂe-iron or ridged ﬁlters,

possibly with the approach recently reported in [8] and [9];

however, the price is reduced power handling, higher loss,

and/or increased manufacturing complexity. To the best of

the authors’ knowledge, using previously published design

methods, it is not possible to simultaneously fulﬁll similar sets

of requirements.

V. C ONCLUSION

Computer-aided design based on multimode variational tech-

niques takes minutes and leads to optimal solutions meeting

modern requirements for waveguide low-pass ﬁlters. It is shown

that composite corrugated structures can be utilized to form

a multimode cluster of transmission zeros through fast design

and optimization of the cavity heights. The designer then has

full control over key dimensional features (gap ,length ,and

spacing ) that can be optimally preselected at the beginning of

the design process. The following improvements are introduced

with respect to previous techniques.

1) Rejection of harmonic spurious bands: it is possible to ef-

fectively design low-pass ﬁlters with spurious rejection for

both fundamental and higher-order modes, irrespective of

the excitation conditions of the system.

2) Enhanced design ﬂexibility: it is possible to design low-

pass ﬁlters with large gap size and short lines, without jeop-

ardizing the harmonic spurious performance. This leads to

a number of distinct beneﬁts in parallel:

• low insertion loss;

• high-power handling capability;

• small length;

• simple industrialization (no tuning, low sensitivity

to mechanical tolerance, and standard manufacturing

process)

Many designs have been successfully built for low and high

power applications, ranging from the -band to -band. Sim-

ulation and measurement results of a -band example demon-

strated the feasibility of the method, in particular when rejection

speciﬁcations up to the third harmonic are to be met without spu-

rious. In many practical cases, out-of-band rejection, insertion

loss, power handling capability, size, and manufacturing com-

plexity no longer need to be traded off.

APPENDIX

With reference to Fig. 4(a), transverse electric and magnetic

ﬁelds of a smaller waveguide on junction areas or can be

represented in terms of Harrington’s voltage and current factors

[18] as follows:

(13)

(14)

where and correspond respectively to the sum

and difference of incident and reﬂected wave amplitudes of the

th mode at the accessible waveguides 1 and 2. is

the cross-section electric ﬁeld of the th mode in the smaller

waveguide normalized on or .

Electric and magnetic ﬁelds in the larger waveguide cavity

are expressed as superposition of standing waves as

(15)

(16)

where represents the modal amplitude of the th standing

wave shorted at and , respectively.

Projecting on eigenfunction basis the boundary condi-

tions for electric ﬁelds at the junction planes ( is the wall area

of the larger waveguide at the junction)

for

for (17)

for

for (18)

we obtain

(19)

Substituting

(20)

then

(21)

Projecting on eigenfunction basis the continuity con-

ditions of magnetic ﬁeld at apertures

for (22)

for (23)

and manipulating the resulting equations, we obtain

(24)

where

(25)

Substituting (21) into (24) and introducing the index for the

th mode in the smaller waveguide, we obtain the expressions

(11) of the Y-matrix.

3224 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 61, NO. 9, SEPTEMBER 2013

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Fabrizio De Paolis (S’04–M’12) received the Laurea

degree in electronic engineering from “La Sapienza”

University of Rome, Rome, Italy, in 2004.

In 2005, he joined COM DEV, Cambridge, ON,

Canada, as an Advanced Member of Technical Staff,

where he was involved with the electromagnetic

modeling and design of ﬁlters and multiplexers for

space applications. Since 2009, he has been with the

European Space Research and Technology Centre

(ESTEC), Noordwijk, The Netherlands, where he

is responsible for industrial and R&D activities in

the area of RF payload equipment and technologies. His current interests

encompass the areas of ﬁlters and multiplexers, nonlinear effects in passive

microwave devices, high-power ampliﬁcation, and satellite communication

systems.

Rousslan Goulouev (M’02) was born in Nebit-Dag, USSR, on February 9,

1961. He received the Engineer-Physicist Diploma from the Moscow Institute

of Physics and Technology, Moscow, Russia, specializing in RF and electronic

devices.

Currently, he is with COM DEV, Cambridge, ON, Canada, where he is a Se-

nior Member of Technical Staff designing microwave ﬁlters for space applica-

tions. His interests are microwave ﬁltering structures, design process automa-

tion, numerical methods and boundary problem solving.

Dr. Goulouev is a Professional Engineer of Ontario.

Jingliang Zheng received the B.S. degree from Bei-

jing University of Posts and Telecommunications,

Beijing, China, in 1982, and the M.S. and Ph.D.

degrees in electrical engineering from Tsinghua

University, Beijing, China, in 1984 and 1988,

respectively.

For one year, he was with the Beijing Design In-

stitute of Telecommunications, Beijing, China, where

he was involved with wireless communication. From

1989 to 1993, he was involved with electromagnetic

ﬁeld simulation with the Swiss Federal Institute of

Technology, Zurich, Switzerland. From 1994 to 1998, he was involved with an-

tenna and antenna array design with DSO National Laboratories, Singapore. He

was then an Engineer with GHz Technologies Inc., Montreal, QC, Canada, for

two years. In 2000, he joined the Research and Development Department, COM

DEV., Cambridge, ON, Canada, where he is currently a Principal Member of

Technical Staff, involved with the development of computer-aided design soft-

ware for design, simulation, and optimization of microwave circuits for space

applications.

Ming Yu (S’90–M’93–SM’01–F’09) received the

Ph.D. degree in electrical engineering from the

University of Victoria, Victoria, BC, Canada, in

1995.

In 1993, while working on his doctoral disser-

tation part-time, he joined COM DEV, Cambridge,

ON, Canada, as a Member of Technical Staff, where

he was involved in designing passive microwave/RF

hardware from 300 MHz to 60 GHz for both space-

and ground-based applications. He was also a

principal developer of a variety of COM DEV’s core

design and tuning software for microwave ﬁlters and multiplexers, including

computer-aided tuning software in 1994 and a fully automated robotic diplexer

tuning system in 1999. His varied experience also includes being the Manager

of Filter/Multiplexer Technology (Space Group) and Staff Scientist of Cor-

porate Research and Development (R&D). He is currently the Chief Scientist

and Director of R&D. He is also an Adjunct Professor with the University of

Waterloo, Waterloo, ON, Canada. He holds an NSERC Discovery Grant from

2004–2015 with the University of Waterloo. He has authored or coauthored

over 100 publications and numerous proprietary reports. He holds eight patents

with six more pending. At COM DEV, he is responsible for overseeing the

development of company R&D Roadmap and next-generation products and

technologies, including high-frequency and high-power engineering, elec-

tromagnetic-based computer-aided design and tuning for complex and large

problems, and novel miniaturization techniques for microwave networks.

Dr. Yu is a IEEE Distinguished Microwave Lecturer from 2010 to 2012. He

is MTT Filter committee Chair (MTT-8) since 2010 and also served as Chair of

TPC-11. He is an associate editor of the IEEE TRANSACTIONS ON MICROWAVE

THEORY AND TECHNIQUE. He was the recipient of the 1995 and 2006 COM DEV

Achievement Award for the development a computer-aided tuning algorithms

and systems for microwave ﬁlters and multiplexers.