Chapter 9

Waveguide Filters
9.0 INTRODUCTION
Besides being the dominant technology until some decades ago, waveguide technol-
ogy is still widely employed in the frequency range from 1 to 100 GHz because of
its specific advantages in terms of low loss and high power handling. This chapter is,
therefore, devoted to the design of the most common filters based on such technology.
Waveguide filters are typically band-pass filters due to the intrinsic high-pass behavior
of a waveguide. Thus, this chapter will almost exclusively consider this type of filter.
An exception is made for the low-pass filters discussed in Section 9.6.
Although various waveguide geometries can be adopted, such as rectangular, circular,
ridged, etc., the focus will be on rectangular waveguide (RW) filters, for it is the
most commonly employed technology. Nevertheless, with suitable modifications,
the concepts presented for the RW can be applied also to other geometries. Special
cases are represented by dual-mode cavity filters that are generally realized in circular
waveguide (CW) technology, described in Section 9.5, and the evanescent-mode ridge
waveguide filters described in Section 9.6.2.
Waveguide filters can be realized in a virtually unlimited variety of configurations,
basically by inserting various types of discontinuities (posts, irises, etc.) along a
uniform waveguide, or by connecting in various manners certain waveguide sections
or resonators. In the former case, the discontinuities are used to realize reactive
loads, either capacitive or inductive, either in series or in shunt configuration. In the
latter case, discontinuities serve as coupling elements between the waveguide lengths
or resonators. In both cases, the waveguide filter can be seen as the connection of

528 Electronic Filter Simulation & Design
distributed elements (waveguide lengths) and lumped elements realized in the form
of suitable discontinuities.1
The characterization of the discontinuities is therefore an essential step in the
design of waveguide filters. Equivalent circuit models have been developed in the past
for a variety of discontinuities (see for instance [1]–[3]). Such models are extremely
useful as they allow for a quick and in many cases accurate design. Nevertheless,
the availability of very accurate full-wave CAD tools makes it possible to achieve
extremely accuratedesignssothat theexperimental tuningofthe filter,stilla verycom-
mon practice, can be avoided in most cases. Without making obsolete the equivalent
circuit models, however, since they still offer a very good insight into the behavior
of the filters and an excellent starting point for the subsequent optimization, modern
CAD techniques represent nevertheless extremely powerful design tools which the
designer must be familiar with.
Commercial CAD tools are therefore systematically employed throughout this
chapter in order to:
a. provide accurate characterizations for the discontinuities
b. provide accurate predictions of the filter performance
c. optimize the filter designed using approximate techniques, thus providing a sort
of numerical tuning of the filter.
This chapter is organized as follows. After this introductory section, the electro-
magnetic wave propagation in waveguides is shortly summarized in Section 9.1,
while Section 9.2 is devoted to the realization of reactive elements employing the
most common waveguide discontinuities.
The classic band-pass filter structure, where shunt-inductive discontinuities are used
to load several lengths of waveguide, is presented in Section 9.3: the design procedure
and a design example for both narrow and wide-band filter are described.
Section 9.4 is devoted to the cross-coupled cavity filters, which are suitable to realize
elliptic and generalized Chebysheff filtering functions. In particular, the E-plane and
H-plane folded structures are described. To obtain size and mass reductions, pass-
band filters employing dual-mode cavities can be adopted. They are discussed in
Section 9.5.
Section 9.6 is devoted to waveguide low-pass filters, that are actually band-pass filters
with very wide stop-bands. Two specific examples are considered, the corrugated
waveguide filters and the evanescent mode ridge waveguide filters.
Numerous examples of filter designs are discussed and described throughout the
chapter; the CAD files employed are quoted in Section 9.7.
1It should be noted that this is essentially true only as long as the guided wavelength is substantially
larger than the length of the discontinuity, so that the latter can be seen as a lumped element—at higher
frequencies, the performance of the waveguide filter will degrade.

Waveguide Filters 529
9.1 PROPAGATION IN WAVEGUIDES
In this section, a brief account is provided of the main features of the electromagnetic
(EM) propagation in waveguides in order to highlight their peculiarities with respect
to the more conventional two-conductor transmission lines (such as the coaxial line).
The reader is supposed to be familiar with the fundamentals of EM propagation and
specifically with the propagation of plane waves.
A waveguide is a hollow metal tube where the electromagnetic field can propagate.
In contrast with the coaxial line, consisting of an inner and an outer conductor, a
waveguide is made of only one conductor. As a consequence, while a coaxial line can
be used from DC to high frequencies, the EM field in a waveguide can propagate
only above a cutoff frequency that is dependent upon the geometry of the waveguide
cross-section.
Three types of waveguides are employed in practice: rectangular, ridged, and circular,
the first one being by far the most common one. Its geometry is sketched in Figure 9.1:
a is the broader side and b is the narrower side.
To illustrate the concept of cutoff frequency, consider a plane wave propagating at
the frequency:
f = fc =
c0
2a
(9.1)
corresponding to a wavelength and a phase constant given, respectively, by:
λc = 2a β = βc =
2πfc
c0
= π/a (9.2)
where c0 = 1/
√
µ0ε0
∼
= 3 · 108
m/s is the phase velocity in free space.
It can be easily seen that, with reference to Figure 9.1, when such a plane wave
propagates along the x-axis, it bounces back and forth between the side walls of the
rectangular waveguide, creating a standing wave along the x-axis. The resulting EM
consists, in practice, of the superposition of two plane waves propagating in opposite
directions of the x-axis, with:
β = βx = ±
π
a
(9.3)
The E-field is directed along the y-axis, while the H-field is directed along the x-axis.
Figure 9.1
The rectangular
waveguide

When the frequency is increased above fc, the phase constant β = 2πf/c increases
as well, but the x-component cannot change because of the boundary conditions at
the side walls. As a consequence, a z-component arises such that:
β = β2
x + β2
z =
π
a
2
+ β2
z =
2πf
c
(9.4)
The phase constant along the waveguide axis z is, therefore:
βz = β2 −
π
a
2
= β 1 −
fc
f
2
(9.5)
In practice, the EM field consists of the superposition of two plane waves that propa-
gate in the xz plane at angles:
θ = ± sin−1 βz
β
= ± sin−1

 1 −
fc
f
2

 = ± sin−1 λ
λc
(9.6)
with respect to the x-axis. As the frequency increases, the propagation constant and
the wavelength tend to those of the free space and the angle θ approaches 90◦
.
At frequencies below fc, the z-component βz of the propagation constant becomes
imaginary, implying that the electromagnetic field cannot propagate, but decays
exponentially.
9.1.1 TE and TM Modes
The EM field just described represents the simplest and most basic distribution of a
family of different configurations that can propagate in the rectangular waveguide.
Such configurations are called modes of the waveguide. It can be proved that there is
an infinite number of modes, each one characterized by a cutoff frequency and by its
specific field distribution.
Modes are usually classified as TEmn (transverse electric) or TMmn (transverse mag-
netic), depending on whether the axial component of the electric field or of the
magnetic field is zero, respectively. The pair of integer numbers m, n are related to
the field distribution within the waveguide cross-section.
9.1.2 Phase Constant
Although it has been derived in a special case, the formula (9.5) holds for any mode
of a waveguide, provided that the relevant cutoff frequency fcmn is specified. As an
example, Figure 9.2 shows the typical dispersion diagram of a rectangular waveguide,
where the phase constants of the various modes are plotted against the normalized
frequency. It is seen that as the frequency increases, the phase constants approach that

0 1 2 3 4 5 6 7 8
0
1
2
3
4
TEM
T
T T
β
z
f/fc, TE10
TE10
E01 E20
M11
Figure 9.2
Dispersion
diagram of a
rectangular
waveguide
of the TEM mode, which is linear with the frequency. A similar behavior is observed
for waveguides of any other shape.
9.1.3 Dominant Mode
The mode with the lowest cutoff frequency is called the dominant (or fundamental)
mode of a waveguide. The waveguide is normally used in the frequency range where
only the dominant mode can propagate, thus, above its cutoff frequency and below
the cutoff of the ﬁrst higher-order mode.
9.1.4 Guided Wavelength
From Equation 9.5, recalling the relation between the phase constant and the wave-
length, one obtains for the guided wavelength:
λg =
2π
βz
=
λ0
1 − fc
f
2
=
λ
1 − λ0
λc
2
(9.7)
with λ0 being the free-space wavelength.
9.1.5 Phase and Group Velocities
The phase velocity is deﬁned as the velocity of the propagation of the wave fronts
along the waveguide axis. It is given by:
vph =
ω
βz
=
c0
1 − fc
f
2
(9.8)

It can be observed that the phase velocity is greater than the velocity of light and
becomes infinite at the cutoff frequency. The group velocity is the velocity with
which a narrow-band signal propagates along the waveguide, and is given by:
vgr =
∂βz
∂ω
−1
= c0 1 −
fc
f
2
(9.9)
The group velocity is smaller than the velocity of light and becomes zero at cutoff.
9.1.6 Wave Impedance and Characteristic Impedance
The electric and magnetic fields in the cross-section of a waveguide are orthogonal to
one another. Their amplitudes are related by the wave impedance, which is given by:
ηTE =
η0
1 − fc
f
2
; ηTM = η0 1 −
fc
f
2
(9.10)
where η0 =
√
µ0/ε0 is the free-space impedance.
The characteristic impedance Z0 cannot be defined in a unique way, since neither the
voltage nor the current can, in general, be defined in an unambiguous way. There are,
therefore, various conventional definitions of waveguide impedance, all dependent
on the cross-sectional geometry, as shown in the case of the rectangular waveguide,
discussed next.
9.1.7 Rectangular Waveguide
The dominant mode of the rectangular waveguide is the TE10 mode. It possesses
three non-zero field components, namely Ey, Hx, Hz, whose distribution is shown in
Figure 9.3.
For evident reasons, the yz plane is called the E-plane of the waveguide, while the
xz-plane is called the H-plane of the waveguide. The Ey and Hx components have
Figure 9.3
Field distribution
of the dominant-
mode TE10 of the
rectangular
waveguide

maxima at the center of the cross-section and become zero at the side walls, while Hz
is zero at the center and maximum at the metal walls.
As was already seen, the cut-off frequency is:
fc10 =
c
2a
(9.11)
Three different definitions of the characteristic impedance of transmission lines are
in use. For a waveguide, the “power-voltage” definition is usually adopted:
Z0 = Z pv =
V · V∗
2P
= 2 · ηTE
b
a
(9.12)
where:
V =
b
0
E · dl (9.13)
is the line integral of the electric field along the y-axis at the center x = a/2 of the
cross-section and:
P =
1
2
a
0
b
0
E × H∗
· dS (9.14)
is the power flow along the waveguide axis.
9.1.8 Ridge Waveguide
The bandwidth of operation of a rectangular waveguide is limited on the lower end
by the cutoff frequency of the dominant TE10 mode and, on the upper end, by the
cutoff frequency of the second higher mode, which is usually the TE20 mode, whose
cutoff frequency is twice that of the TE10.
The insertion of one or two metal ridges at the center of the broad side where the
E-field is maximum (see Figure 9.4) has the effect of lowering the cutoff frequency
of the dominant mode, while the second cutoff frequency remains almost unaffected.
Figure 9.4
Single and
double-ridge
waveguides

r
θ
2a
Z
E
H
Figure 9.5
The circular
waveguide and
the field lines of
the dominant
TE11 mode; the
configuration
rotated by 90◦
(degenerate
mode) is also
possible
As a consequence, the usable bandwidth of the ridge waveguide is widened with
respect to the standard rectangular waveguide. The electric field is mostly confined in
the center of the cross-section, with a distribution similar to the TEM mode between
parallel plates: the closer the ridges, the wider the bandwidth. The price to be paid is
the increased conductor loss on the metal walls, thus, the field attenuation.
Since no closed-form expression is available for the cutoff frequency of the dominant
mode of the ridge waveguide, one has to resort to numerical computation or to graphs
(see [22], [23] in the “References” at the end of this chapter).
9.1.9 Circular Waveguide
The circular waveguide is employed in some specific applications where the circular
symmetry is exploited, such as in a rotary joint, or in specific components, such
as phase shifter or dual-mode filters, that exploit the presence of degenerate modes
(see the following paragraphs). The geometry is shown in Figure 9.5.
Because of its circular symmetry, any mode of the circular waveguide can exist in two
orthogonal configurations, the field lines being simply rotated by 90◦
, one to another.
Each pair of such modes constitutes a pair of “degenerate” modes: They have identical
propagation characteristics (cutoff frequency, phase constant, etc.).2
The dominant mode of the circular waveguide is the TE11 mode. Its cutoff frequency
is given by:
fc =
1
2π
1.841c0
a
(9.15)
The field lines of the transverse components of the E- and H-fields are shown in
Figure 9.5b. As already specified, a degenerate TE11 mode can also be supported,
with the field lines rotated by 90◦
. By short-circuiting a section of circular waveguide,
2An exception is represented by those modes characterized by a first index 0—for example, the TE0n
modes. Such modes are independent of the coordinate θ and are, therefore, non-degenerate.

two resonant modes can thus be supported at the same resonant frequency, just by
rotating the field lines 90◦
to each other. This property is exploited in the discussion
on dual-mode filters in Section 9.5.
9.2 REACTIVE ELEMENTS IN WAVEGUIDE
Reactive elements required in the design of waveguide filters are realized in the
form of reactive discontinuities embedded in the waveguide structure. Although a
virtually unlimited variety of discontinuities can be employed, in most cases, simple
waveguide obstacles, such as irises and posts, both inductive and capacitive, are used
in practical applications.
The design of such discontinuities is made on the basis of simplified closed-form
expressions, while in the general case, one has to resort to full wave simulations,
accounting for the reactive effects associated with the discontinuity.
In this section, we present some useful expressions for the design of the most common
reactive waveguide discontinuities. The reader is referred to the vast literature on this
subject—for example, [3], for further and more complete information.
9.2.1 Shunt-Inductive Obstacles
Shunt inductors are typically realized in the form of metallic diaphragms or cylindri-
cal postsparallelto thetransverseelectricfield,and thus,intheE-planeofarectangular
waveguide. The circular waveguide shunt inductors can be realized as an annular
window in a metallic plate.
Table9.1shows thegeometriesofshunt-inductiveobstaclesin rectangularandcircular
waveguides, along with the corresponding model for the associated inductance. As
already mentioned, the lumped models of Table 9.1 are based on the assumption that
the longitudinal dimension of the obstacle is negligible compared with the guided
wavelength; thus, the diaphragm must be thin. The same holds for the post.
9.2.2 Shunt-Capacitive Obstacles
Shunt capacitances can be realized in the form of metallic obstacles perpendicular
to the transverse E-field—such as H-plane diaphragms in rectangular waveguide or
as annular obstacles placed in the cross-section of a circular waveguide. Such discon-
tinuities are provided in Table 9.2.
Regarding the H-plane post, the expression for the normalized shunt capacitance
is rather complex and can be found in [3] in the “References” at the end of this
chapter.

Table
9.1
E-Plane
Metallic
Obstacles
Equivalent
Normalized
Shunt
Circuit
Discontinuity
Geometry
Susceptance
Symmetrical
inductive
iris
in
rectangular
waveguide
d
a
b
B
=
2π
βa
cot
2
π
d
2a
1
+
aγ
3
−3π
4π
sin
2
πd
a
where,
β
=
ω
2
εµ
−
π
a
2
,
γ
3
=
3π
a
2
−
ω
2
εµ
(9.16)
Unsymmetrical
inductive
iris
in
rectangular
waveguide
d
t
a
b
B
=
2π
βa
cot
2
π
d
2a
1
+
csc
2
π
d
2a
(9.17)
B
Inductive
post
in
rectangular
waveguide
2t
a/2
a/2
b
B
=
4π
βa
ln
a
πt
−
1
+
2
a
πt
2
∞
n
=
3,5,...
π
aγ
n
−
1
n
sin
2
nπt
a
−1
where
γ
n
=
nπ
a
2
−
ω
2
εµ
(9.18)
Annular
window
in
circular
waveguide
2R
2r
d
B
=
λ
g
r
2R
π
d
2
0.162
J
2
1
3.83r
R
where
λ
g
=
λ
1−
λ
2.64
R
2
(9.19)

Table
9.2
H-Plane
Metallic
Obstacles
Equivalent
Circuit
Discontinuity
Geometry
Normalized
Shunt
Susceptance
Symmetrical
capacitive
iris
a
b
d
B
=
2βb
π
ln
csc
πd
2b
+
2π
bγ
2
−
1
cos
4
πd
2b
where
γ
2
=
2π
b
2
−
β
2
(9.20)
Unsymmetrical
capacitive
iris
d
b
a
B
=
4βb
π
ln
csc
πd
2b
+
π
bγ
1
−
1
cos
4
πd
2b
where
γ
1
=
π
b
2
−
β
2
(9.21)
B
Capacitive
post
2t
b/2
b/2
a
Annular
obstacle
in
circular
waveguide
2
R
2r
d
B
=
r
λ
0
−
π
d
R
2
1
0.269
J
2
1
2.405
r
R
where
λ
g
=
λ
1−
λ
2.61
R
2
(9.22)

9.3 SHUNT-INDUCTIVE LOADED FILTER
This is probably the most common type of band-pass waveguide filter. It consists of
a waveguide section loaded with shunt inductive discontinuities, typically, in RW,
E-plane irises, or posts. Observe that by adopting a different point of view, waveguide
lengths between consecutive discontinuities can also be seen as waveguide cavities,
with each cavity coupled to the next and previous ones in such a way that there is
a unique path for the electromagnetic wave traveling from the input to the output of
the filter. Such filters are, therefore, also classified as direct coupled filters [2]. They
can be represented by the equivalent circuit shown in Figure 9.6, consisting of N
transmission line sections loaded with N + 1 shunt inductances.
9.3.1 Design Procedure
The design of pass-band direct-coupled filters consists of four main steps: a) the
synthesis of the low-pass prototype; b) the synthesis of the band-pass filter; c) design
of the waveguide filter structure; and d) optimization. Such steps are described in the
following paragraphs.
Synthesis of the Low-Pass Prototype
This step consists of identifying the low-pass prototype,3
shown in Figure 9.7—that
is, its order N and the component values gn (n = 0, 1, 2, . . . , N + 1). To this end,
the low-pass to band-pass transform
ω
ω1
=
2
wλ
λg0 − λg
λg0
(9.23)
is adopted,
where:
•
ω
ω1
is the normalized radian frequency of the low-pass prototype.
• wλ =
λg1 − λg2
λgo
is the fractional bandwidth. (9.24)
• λg1 and λg2 are the guided wavelengths at the band edges.
• λg0 = (λg1 + λg2)/2.
Z0
θ ≈ π
B1 B2
BN+1 Z0
θ ≈ π
Z0 Z0
Figure 9.6
Equivalent circuit
of a shunt
inductive loaded
waveguide filter
3The low-pass prototype was described in Section 2.4.The network in Figure 9.7 coincides with the one
in Figure 2.23b.

Figure 9.7
Low-pass
prototype
It should be reminded that, in contrast to TEM transmission lines, waveguides are
dispersive so that λg is not proportional with 1/f .
The band-pass filter specifications are then converted into those of the low-pass
prototype, which is then synthesized using the procedures described in Chapter 2—
that is, its order N and the parameters (g0, g1, . . . , gN+1) are computed (see the design
example).
Synthesis of the Band-Pass Circuit of Figure 9.6
The Nth
-order low-pass prototype of Figure 9.7 is transformed into the ladder
band-pass filter prototype of Figure 9.8a using the low-pass to band-pass frequency
transform (9.25), repeated here for the reader’s convenience:
ω = Fband-pass(ω) =
ω0
ω2 − ω1
ω
ω0
−
ω0
ω
(9.25)
C1
R0
L2
CN
RN+1
L1
C2 L4
C4
L3 C3
L4
RA
L0
RB
C0 L0
C0
K01 K12 KN N+1
K01 K12
π π
Z0 Z0
(a)
(b)
(c)
φ1
2
Z0
π π
φ1
2
φ2
2
φ2
2
φN+1
2
φN+1
2
B1 B2 BN+1 Z0
(d)
Z0 Z0
KN N+1
Figure 9.8
Conversion of the
band-pass
prototype (a) into a
filter made of
K-inverters and
series LC
resonators (b),
K-inverters and
half-wave line
resonators (c),
and a shunt-
inductance loaded
filter (d)

where:
• ω0 =
√
ω1ω2 is the center angular frequency of the band-pass ﬁlter.
• ω = ω2 − ω1 is the pass-band width.
• δω = ω2−ω1
ω0
is the fractional bandwidth.
Observe that all resonators, both parallel and series, have the same resonant fre-
quency ω0.
The band-pass prototype of Figure 9.8a cannot be realized in the form of a wave-
guide ﬁlter. Using the properties of impedance inverters, however, it can be converted
into a network composed of K-inverters4
and series LC resonators, as shown in
Figure 9.8b.
An ideal K-inverter is characterized by the following impedance matrix:
[Z] =
0 jK
jK 0
(9.26)
or, equivalently, by the chain matrix:
[T] =
0 jK
j / K 0
(9.27)
The basic property that isused hereis theequivalence illustrated inFigure 9.9. Assume
that Yp represents a parallel LC resonator:
Yp = jωC +
1
jωL
(9.28)
Using Equation 9.26, it can be easily proved that Figure 9.9 is equivalent to
Figure 9.9, with Zs representing a series LC resonator:
Zs = K2
Yp = jω(K2
C) +
1
jω(L/K2)
(9.29)
The value of K2
can be chosen arbitrarily; this degree of freedom can be used to
select appropriate values for the impedance level of Zs.
Zs = K2
Yp
Zs = K 2Yp
Figure 9.9
Conversion of a
shunt admittance
Yp into a series
impedance Zs
using a pair of K
inverters
4Impedance inverters were discussed in Section 2.5.2.2. Figure 9.9 coincides with Figure 2.32b, apart
from a 180◦ phase shift not considered in the latter one.

The filter in Figure 9.8b consists of identical series resonators C0 L0 and is loaded
with RA and RB at theinputand outputports, respectively;the values ofthe K-inverters
are as follows:
K01 =
RA L0ω0δω
g0g1ω1
, K j, j+1 j=1,...,N−1
=
δω
ω1
L2
0ω2
0
gj gj+1
, KN,N+1 =
RB L0ω0δω
gN gN+1ω1
(9.30)
where:
• ω0 =
√
ω1ω2 is the center angular frequency of the band-pass filter.
• ω1 = 1 is the normalized cutoff frequency of the low-pass prototype filter.
• δω = ω2−ω1
ω0
is the fractional bandwidth.
The details of the full conversion of Figure 9.8a into Figure 9.8b are omitted here,
the general procedure being about the same as that of Section 2.4.2.2.
The network in Figure 9.8b, however, is not yet suitable to waveguide realization,
since it still involves lumped elements (the series resonators). A second conver-
sion is then applied to replace the series L0C0 resonators with transmission lines
of impedance Z0 and length λ/2 at center frequency. Such conversion is shown in
Figure 9.10 and is approximately true only when the impedance level R, which loads
the two circuits, is low. Such equivalence is based on the calculation of the reac-
tive slope parameter of the series lumped-element resonator and the reactive slope
parameter of a half-wave short-circuited transmission line [1] of impedance Z0.
The reactive slope parameter is defined as:
x =
ω0
2
dX(ω)
dω
ω=ω0
(9.31)
where X(ω) is the frequency-dependent reactance of a series resonator. Since
X = ωL − 1
ωC
is the reactance of a lumped-element series resonator LC, then:
x =
ω0
2
dX(ω)
dω
ω=ω0
= ω0L (9.32)
is the reactive slope parameter for a lumped-element series resonator LC.
C
L
Z0
2
2
=
λg
λ
π
R R R R
L C
when
R low
Z0
θ = π
0
0
Figure 9.10
Equivalence
between a series
LC resonator and a
half-wavelength
transmission line
when low
impedance loads
the circuits

−
=
−
=
=
B
Z
K
K
Z
Y
B
B
2
tan−1
,
0
0
0
φ
Figure 9.11
The impedance
inverter can be
obtained by a
series inductor and
two transmission
lines of negative
length (φ negative)
The reactive slope parameter for a half-wave, short-circuited, non-dispersive
transmission line of impedance Z0 is [1]:
x =
π
2
Z0 (9.33)
By replacing ω0L with πZ0/2, Equation 9.30 reduces to:
K01
Z0
=
π
2
wλ
g0g1ω1
,
K j, j+1
Z0 j=1,...,N−1
=
πwλ
2ω1
1
√
gj gj+1
,
KN,N+1
Z0
=
π
2
wλ
gN gN+1ω1
(9.34)
where we have put, for convenience, RA = RB = Z0, and the waveguide fractional
bandwidth wλ has been used instead of δω.
A practical realization of a K-inverterconsists of a shunt inductancecascaded between
two transmission line sections of negative length, as shown in Figure 9.11.
B̂ =
B
Y0
=
Z0
K
−
K
Z0
, φ = − tan−1 2
B̂
The chain matrix of the two-port network in Figure 9.11 can be computed as the
product of its constituents5
and can easily found to be:


cos φ
2
jZ0 sin φ
2
jY0 sin φ
2
cos φ
2


1 0
jB 1
cos φ
2
jZ0 sin φ
2
jY0 sin φ
2
cos φ
2
=


cos φ
2 − B̂
2 sin (φ) jZ0 sin (φ) − B̂ sin2 φ
2
jY0 sin (φ) + B̂ cos2 φ
2
cos φ
2
− B̂
2
sin (φ)


(9.35)
where we have put B̂ = B/Y0 and Y0 = 1/Z0.
5Because of the property (1.94).

For Equation 9.35 to be the chain matrix of a K-inverter (Equation 9.27), we must
impose
t11 = t22 = cos
φ
2
−
B̂
2
sin(φ) = 0 (9.36)
It, therefore, results in:
φ = tan−1 2
B̂
(9.37)
Using Equation 9.37, after some algebraic manipulations, we obtain:
t12 = jZ0


 1 +
B̂
2
2
−
B̂
2


; t21 = jY0


 1 +
B̂
2
2
+
B̂
2


 (9.38)
By comparing it to Equation 9.27, we conclude that the network in Figure 9.5 be-
haves as a K-inverter with
K = Z0


 1 +
B̂
2
2
−
B̂
2


 (9.39)
Solving Equation 9.39 for B, we obtain:
B̂ =
Z0
K
−
K
Z0
(9.40)
The negative-length transmission lines in Figure 9.11 cannot, of course, be realized by
themselves; in practice, when cascaded with line sections of the same characteristic
impedance and positive lengths, their realization consists of a mere corresponding
shortening of the latter transmission lines. Observe that K is frequency-dependent.
Replacing the K-inverters in the filter shown in Figure 9.8c with their realization in
Figure 9.11, we finally obtain the circuit shown in Figure 9.8d, which has the same
structure as Figure 9.6. Observe that the angles φi are negative, so the line lengths
between the inductors are shorter than λ/2.
Design of the Waveguide Filter
In practice, to design the waveguide filter in Figure 9.6, one has simply to select
one of the possible options seen in Section 9.2.1 (Table 9.1) for realizing the shunt
inductances. The dimensions of the discontinuities are determined by inverting the
formulas expressing their inductances, while the waveguide lengths correspond to the
electrical lengths of the equivalent circuit in Figure 9.6. In the case of the rectangular
waveguides, for example, the inductor can be realized as an inductive post or iris, the
latter being the most common choice, as discussed in Section 9.2.1.

The width of each iris is determined by inversion of the formula (9.16) in Table 9.1.
Observe that such formula is valid in the limited case of zero thickness of the metallic
iris and provides a good approximation in most practical cases—the narrower the
width of the iris, the better. In the case of a thick iris, one has to resort to more accurate
formulas or to computer optimization based on full-wave analysis to accurately deter-
mine the iris’ thickness. A similar procedure is followed in the case of other inductive
discontinuities, such as posts or E-plane longitudinal diaphragms.
Finally, the length of the i-th waveguide section between the consecutive irises i
and i + 1 is easily computed by the corresponding electrical length θi at the center
frequency ω0
li =
λg0
2π
θi =
λg0
2π
π +
φi
2
+
φi+1
2
(9.41)
where λg0 is the guided wavelength at center frequency. Note that since φi and φi+1
are negative, the waveguide length is somewhat shorter than λg0/2.
Optimization
At the end of the design procedure, the filter can be fabricated and measured. Never-
theless, because of the various approximations involved, the experimental response
may significantly differ from the ideal one. As a consequence, either the filter is to
be experimentally adjusted using suitable tuning elements or the design of the filter
is optimized using numerical tools based on full-wave models. A specific design
example is illustrated in the next section.
9.3.2 Design Example
We consider the design of a fifth-order iris-coupled band-pass filter in rectangular
waveguide (sketched in Figure 9.12), with a Chebysheff response with the following
specifications:
• Rectangular waveguide WR90, a = 22.86 mm, b = 10.16 mm
• Pass-band ripple RP = 0.01 dB (pass-band return loss ∼
= 26.5 dB )
B1 B2 BN+1
B3
θ
1
=
π
+
φ
1
2
φ
2
2
+
Z0 Z0
Z0
Z0
a
Z
X
φ
2
2
φ
3
2
+
θ
2
=
π
+
Figure 9.12
Cross-section
(xz plane) of the
filter discussed
in Section 9.4.2,
realized in RW
technology using
inductive irises

Table 9.3
Parameters of the Filter in Section 9.3.2
k gk i, j Ki j /Z0 B̂j φj
0, 6 1 01, 56 0.2700 3.4337 −0.5273
1, 5 0.7563 12, 45 0.0555 17.962 −0.1108
2, 4 1.3049 23, 34 0.0384 26.003 −0.0768
3 1.5773 − − − −
• Pass-band edges f1 = 9.9 GHz, f2 = 10.1 GHz (thus, f0 = 10 GHz)
• Minimum stop-band attenuation L = −30dB at fa = 10.35 GHz.
The parameters gk of the Chebysheff low-pass prototype are quoted in Table 9.3,
column 2. The values Kij of the inverters are calculated from Equation 9.34 and are
quoted in column 4. The normalized susceptances B̂ j and the electrical lengths φj
realizing the impedance inverters are calculated by Equations 9.37 and 9.40, and are
shown in columns 5 and 6.
Finally, the dimensions of the waveguide filter are obtained by converting the
susceptances B̂ j into the apertures dj of the irises using Equation 9.16, listed in
Table 9.1, and by calculating the cavity lengths lj using Equation 9.41. The results
are listed in Table 9.4.
Figure 9.13 shows the performance of the filter with a full-wave simulation using a
mode-matching tool.
It has to be stressed that the filter has been designed based on the assumption of zero-
thickness diaphragms; therefore, in practical cases, it is to be expected that the actual
performance of the filter may differ considerably from the predicted one should the
metal thickness be non-negligible with respect to the wavelength. A further cause of
discrepancies between predicted and actual performances is due to the rounding of
the metal edges and of the inner corners as a result of the milling fabrication process.
To avoid lengthy and costly experimental tuning procedures, a further optimization
Table 9.4
Physical Dimensions of the Waveguide
Filter with Zero-Thickness Irises
Physical Dimension Value, mm
Irises 1, 6 (d1,d6) 8.676
2, 5 (d2,d5) 4.320
3, 4 (d3,d4) 3.640
Cavity length 1, 5 (l1,l5) 17.80
2, 4 (l2,l4) 19.30
3 (l3) 19.40

9.5 9.7 9.9 10.1 10.3 10.5
−80
−60
−40
−20
0
20
log
10
(|s
21
|)
Frequency, GHz
−60
−40
−20
0
20
←s21
s11
→
20
log
10
(|s
11
|)
Figure 9.13
Full-wave
simulation
(mode-matching)
of the synthesized
filter
step of the filter is necessary in such cases, using full-wave models to account for the
finite thickness of the diaphragms.
To better illustrate this point, we have assumed that the waveguide filter is to be
manufactured using a milling process that results in a curvature radius r = 1.5 mm
of the rounded edges and a diaphragm thickness of t = 1.5 mm.
First, afast mode-matching tool has been used to optimize the filter structure consider-
ing ideal inner edges. Then, with a commercial tool CST Microwave Studio (CST),
the curvature radius of the rounded edges has been involved in the optimization.
The new filter dimensions are listed in Table 9.5. Significant differences from the
original values of Table 9.4 can be observed.
The optimized filter has finally been fabricated in two halves joined along the H-plane,
each half being milled in the E-plane, as shown in Figure 9.15. Figure 9.14 shows the
comparison between the measurements and the theoretical simulations using CST.
The discrepancies between theory and experiment do not degrade the performance of
the actual filter (the specifications are met) and are due to fabrication tolerances.
Table 9.5
Physical Dimension of the Optimized Filter
Irises 1, 6 (d01,d56) 11.137
2, 5 (d12,d45) 6.8815
3, 4 (d23,d34) 6.0513
2, 4 (l2,l4) 18.64
3 (l3) 18.85
Irises Thickness 1.5
Radius of Blended Edge 1.5

9.5 9.7 9.9 10.1 10.3 10.5
−80
−60
−40
−20
0
simulated
measured
s11→
←s21
20
log
10
(|s
21
|)
Frequency, GHz
−60
−40
−20
0
20
20
log
10
(|s
11
|)
Figure 9.14
Full-wave
simulation (CST) of
the optimized filter
9.3.3 Design Procedure for Wide-Band Filter
The design procedure described in the previous section for the shunt-inductive loaded
waveguide filter in Figure 9.6 can be adopted for band-pass filters with moderate
bandwidths, no more than 20 percent for TEM lines and even lower for dispersive
structures, such as waveguides [16].
For wider bandwidths, the design procedure can be developed based on the quarter-
wave transformer or on the distributed low-pass prototype in Figure 9.16. Due to its
distributed nature, the latter can indeed also be used as a band-pass filter when θ = π
and, thus, is also called the half-wave prototype. In Figure 9.17, an example of an
eleventh-order prototype response is plotted versus the electrical length θ.
Such a prototype can be synthesized directly from the impedance values of a quarter-
wave transformer [2]. The half-wave prototype approach has the advantage of
Figure 9.15
Fifth-order
direct-coupled
cavity filter realized
in aluminum

Figure 9.16
Distributed
low-pass or (for
θ = π) half-wave
prototype network
generating impedance levels, with subsequent sections alternating between above
and below unity. In the quarter-wave transformer approach, on the contrary, the
impedance level increases monotonically from input to output, yielding very different
impedance levels.
The impedance values of half-wave filters have been calculated by Levy for
Chebysheff approximation in [15] for different filter orders, bandwidths, and VSWR
ripples. In such tables, the fractional bandwidth w is actually the fractional bandwidth
of the corresponding quarter-wave transformer, defined as:
wtransf =
4θ0
π
(9.42)
which, looking at Figure 9.17, is two times that of the filter fractional bandwidth:
wfilter =
2θ0
π
=
wtransf
2
(9.43)
Such convention has to be taken into account when the tables in [15] are used.
The design procedure can be summarized as follows:
a) Given the filter specifications:
• VSWR pass-band ripple
• Waveguide fractional bandwidth wλ of the filter, as defined in Equation 9.24
• Number of sections
0
−120
−90
−60
−30
0
2π − θ0
π − θ0 π + θ0
θ0
20
log
10
(|s
21
|)
θ, Rad
−50
−40
−30
−20
−10
0
10
| |
|
|
|
| |
3π/2 2π
π
π/2
←s21
s11→
20
log
10
(|s
11
|)
Figure 9.17
Response of the
distributed
low-pass filter
prototype
(eleventh-order
example): the
response is plotted
with respect to the
electrical length θ
of its transmission
lines

The impedance values Zi of the distributed prototype in Figure 9.16 are obtained
from the tables in (see [15] in the “References” at the end of this chapter).
b) The prototypein Figure9.16 is convertedinto thatin Figure9.8c, whereK-inverters
have been introduced so as to set all sections to the same impedance Z0.
The i-th impedance inverter is then calculated in such a way as to keep the same
VSWRi as in the original network:
Ki
Z0
=
1
√
VSWRi
(9.44)
where VSWRi = Zi+1
Zi
is the voltage standing-wave ratio at the i-th impedance step
of the prototype in Figure 9.16.
c) As shown in Section 9.3.1, each impedance inverter K can be realized as a
shunt-inductive load of normalized susceptance B̂, with two negative-transmission
line φ/2 at its sides. These values are calculated again from the normalized K value
calculated by Equation 9.42, using Equations 9.40 and 9.37, respectively.
d) The inductive susceptances are realized in waveguide technology. This can be
done with symmetrical zero-thickness irises using Equation 9.16 of Table 9.1,
while the distances between irises are obtained from Equation 9.41. Of course, any
other inductive obstacles, such as those shown in Table 9.1 for RW, can be used
as well.
It should be noted that once the impedance steps of the half-wave prototype are
replaced by shunt inductances and the transmission lines are dispersive, such as with
the waveguide, the response of the filter will be slightly different from the prototype.
In particular, the response will be asymmetrical, with a much lower attenuation in the
upper stop band. Because of the second harmonic response—which is nearer to the
first one as the filter bandwidth becomes wider—the upper transition band tends to
vanish with increasing bandwidths. This is the reason why, for very large waveguide
bandwidths, such filters are called pseudo-high-pass filters [1].
9.3.4 Design Example
We consider the design of a ninth-order, iris-coupled band-pass filter in rectangular
waveguide (sketched in Figure 9.12), with the following specifications:
• Rectangular waveguide WR90, a = 22.86 mm, b = 10.16 mm
• Pass-band VSWR ripple= 1.2(RL = 20 dB)
• Pass-band edges f1 = 8.5 GHz, f2 = 11.5 GHz
From Equation 9.24, the waveguide fractional bandwidth is 54 percent.
Using the preceding specifications, the following impedance values for a half-wave
prototype with 110 percent (wtransf = 2wfilter) of fractional bandwidth are read
from [15]:

Table 9.6
Design Parameter Values
i VSWRi Ki /Z0 B̂i di , mm li , mm
1, 9 1.4160 0.8404 0.34959 16.972 12.698
2, 8 2.0292 0.7020 0.72252 14.829 13.804
3, 7 2.7916 0.5985 1.0723 13.495 14.582
4, 6 3.3517 0.5462 1.2845 12.857 14.963
5 3.6115 0.5262 1.3742 12.616 15.071
Z0 = 1
Z1 = 1.416
Z2 = 0.6978
Z3 = 1.948
Z4 = 0.5812
Z5 = 2.099
Using the procedure just described, we obtain the parameters quoted in Table 9.6,
where di are the widths of the irises and li are the lengths of the waveguide sections.
The full-wave CST simulation of the filter thus synthesized shows good results after
the design procedure (Figure 9.18), with an excellent return loss in the whole specified
band. As can be noted, however, the upper stop-band behavior is rather poor because
7 8 9 10 11 12 13
−100
−80
−60
−40
−20
0
20
log
10
(|s
21
|)
Frequency, GHz
−40
−30
−20
−10
0
10
←s21
s11→
20
log
10
(|s
11
|)
Figure 9.18
Full-wave
simulation of a
filter designed
under the
assumption of
irises with zero
thickness after the
design procedure

7 8 9 10 11 12 13
−100
−80
−60
−40
−20
0
20
log
10
(|s
21
|)
Frequency, GHz
−40
−30
−20
−10
0
10
←s21
s11→
20
log
10
(|s
11
|)
Figure 9.19
Simulated
full-wave response
of the filter after
optimization to
account for the
finite thickness
(0.8 mm) of the
irises
of the second harmonic response of the filter, which is centered at the frequency fh
for which λgh = λg0/2.
In the present case, λg0 = 39.7, so that fh = 16.27 GHz.
An improved filter behavior in the upper transition band can be obtained by increasing
the order of the filter.
To account for the finite thickness of the irises, a subsequent full-wave optimization
has been applied. The optimized response for an iris thickness of 0.8 mm is shown in
Figure 9.19; the corresponding filter dimensions are quoted in Table 9.7.
Table 9.7
Optimized Filter Dimensions
Irises 1, 10 (d1,d9) 16.450
2, 9 (d2,d9) 13.975
3, 8 (d3,d8) 13.053
4, 7 (d4,d7) 12.745
5, 6 (d5,d6) 12.637
2, 8 (l2,l8) 13.590
3, 7 (l3,l7) 14.244
4, 6 (l4,l6) 14.474
5 (l5) 14.542
Irises Thickness 0.8

9.4 CROSS-COUPLED CAVITY FILTERS
The filters considered in the previous section can be viewed as the cascade of wave-
guide cavities, coupled one to the next through inductive diaphragms or posts. There-
fore, this class of filters is also called “direct-coupled cavity filters,” since coupling
occurs only in a sequential manner from input to output. The resulting out-of-band
attenuation is monotonically increasing to infinity as the frequency departs from the
pass-band. Direct-coupled cavity filters, thus,typically exhibita Chebysheff response.
In cross-coupled cavity filters, the introduction of additional couplings between
nonconsecutive cavities allows one to generate transmission zeros at finite frequen-
cies so as to improve the selectivity of the filter. The degree of freedom provided by
the cross couplings may also be employed to improve the phase characteristics of the
filter by generating transmission zeros on the real axis of the complex frequency plane
s = σ + jω.
The transfer function of a two-port network can be written as:
T(s) = s21(s) =
P(s)
E(s)
(9.45)
where s21 is the transmission coefficient. In order to ensure the physical realizability
of the network, the following conditions must be satisfied:
• P(s) and E(s) are polynomials whose coefficients are real and positive.
• E(s) has zeros with negative real part.
• The degree of P(s) is not higher than E(s).
• The amplitude of T (s) may not exceed unity.
It can be demonstrated that couplings between nonconsecutive cavities produce
finite transmission zeros—that is, finite zeros of P(s). In particular, Kurzrok [13] for
coaxial cavity filters, and then Easter and Powell [14] for waveguide filters, demon-
strated that attenuation poles at finite frequencies can be produced by an additional
coupling between the first and last resonators of the direct-coupled cavity filter. In
the case of Chebysheff approximations, all such zeros are located at infinity—that is,
P(s) does not possess finite zeros.
In the general case, in cross-coupled filters, the zeros of P(s) can be located both on
the real and imaginary axes of the complex s plane, their location depending of the
filter topology and on the couplings between resonators.
Transmission zeros on the imaginary axis (s = jω) affect the amplitude response,
producing an attenuation pole at the corresponding frequency, which substantially
improves the filter selectivity with respect to the Chebysheff response. Transmission
zeros on the real axis (s = σ) can be used to improve the phase linearity and thus the
group delay of the filter.

The coupling between two nonconsecutive cavities produces a number of transmis-
sion zeros equal to the number of cavities that are skipped by the path between such
cavities.Let N betheorderofthefilterand,thus, thenumberofcavities.The maximum
number of transmission zeros is obtained by coupling the input (first cavity) with the
output (last cavity) and is, therefore, equal to N − 2. Additional cross couplings
between subsequent cavities provide additional degrees of freedom to the designer,
but do not increase the number of transmission zeros in the complex plane.
In principle, by producing a direct coupling between the input and output ports, two
additional zeros can be produced, thus equating the degree N of the filter.
9.4.1 Elliptic and Generalized Chebysheff
Filtering Functions
In contrast with Chebysheff filters, Cauer or elliptic-type filters possess transmission
zeros at finite frequencies suitably located in the stop-band so as to produce an
equiripple response in the stop-band and increase the selectivity of the filter, while at
the same time keeping an equiripple response in the pass-band.
In many practical applications, however, rather than an equiripple attenuation in the
stop-band, it is required to place the transmission zeros at prescribed frequencies so as
to have a higher flexibility in the design. Cameron [4] has thus used the generalized
Chebysheff function, where the location of the transmission zeros can be chosen
arbitrarily, still keeping the in-band equiripple response as in the conventional
Chebysheff filters:
|s21(ω)|2
=
1
1 + ε2C2
N (ω)
(9.46)
where:
• CN (ω) = cosh
N
n=1
cosh−1
(xn) (9.47)
• xn =
ω − 1
ωn
1 − ω
ωn
(9.48)
In the preceding formulas, jωn = sn is the location of the n-th zero on the imaginary
frequency axis, while ε is related to the filter’s return loss (RL) by:6
ε =
1
10
RL
10 − 1
(9.49)
By inspecting Equation 9.47, one can easily recognize the following properties:
• when |ω| = 1, then |xn| = 1, CN = 1, |s21|2
= 1
1+ε2
• when |ω| ≤ 1, then |xn| ≤ 1, CN ≤ 1
6Obtained by substituting Equation 2.4 into Equation 1.45.

C = 1
L = 1
2
L = 1
2
C = 1
L = 1
C = 1
L = 1
R R
M1,2 M2,3 M3,4 MN−1,N
M1,3 M2,4 M3,N
M1,4
M2,N
C = 1
L = 1
2
L = 1
2
L = 1
2
L = 1
2
M1,N
Figure 9.20
Cross-coupled filter prototype network
• when |ω| ≥ 1, then |xn| ≥ 1, CN ≥ 1
• when |ω| → ωn, then CN → ∞, |s21(ωn)|2
→ 0
Observe that when all zeros tend to infinity, the Nth-order generalized Chebysheff
function reduces to the conventional Nth-order Chebysheff polynomial:7
CN (ω) ωn→∞ = cosh[N cosh−1
(ω)]
9.4.2 Coupling Matrix Description for Narrow-Band
Cross-Coupled Filters
Figure 9.20 shows the equivalent circuit representation of a general cross-coupled
filter, where each cavity is coupled to all remaining ones. Such a circuit allows one
to implement generalized Chebysheff responses of both even and odd orders, both
symmetrical and asymmetrical, with given transmission zeros on the real and
imaginary axes.
All LC cells, when isolated—that is, when Mij = 0—are normalized, assuming that
they resonate at the same radian frequency ω0 = 1 rad/s and have unit impedance level
√
L/C. Therefore, L = C = 1, and the filter is thus fully described by the coupling
parameters Mi j (i = j) between the i-th and j-th cells and by the load resistance R at
both ends of the filter. Both Mij and R are normalized with respect to the fractional
bandwidth w, the respective denormalized values being:
Kij = Mijw, Rd = Rw (9.50)
Similarly, a shift from ω0 = 1 to ω0 = ω0 of the center frequency of the filter is
obtained by changing the value of all Ls and Cs using:
L = C =
1
ω0
(9.51)
7This equation coincides with Equation 1.43, although the latter uses the circular cosine instead of the
hyperbolic cosine.

In this manner, the unit impedance level is maintained. To change the impedance level
to Z0, one has to multiply L and R by Z0 and divide C by Z0.
9.4.2.1 Outline of the Coupling Matrix Synthesis Procedure
The synthesisofthe filterprototypein Figure9.20 consistsof determining thecoupling
matrix MandtheloadresistanceRforgiventransmission zeros.Thegeneralprocedure
has been developed by Atia and Williams [6] and Cameron [4, 5].
From the filter specifications:
• Order of the filter (even or odd)
• Prescribed transmission and group delay equalization zeroes,
• Symmetrical or unsymmetrical filter response
The synthesis procedure consists of three basic steps [4].:
1. Polynomial synthesis of the transfer function: From the filter specifications and
the zero locations, the generalized Chebysheff function is computed by means of
recursive techniques.
2. Synthesis of the coupling matrix: The admittance parameters Y of the network are
generated from the transfer function computed in step 1 using a partial fraction
expansion and an orthonormalized procedure. The M matrix is then evaluated from
the admittance matrix.
3. Reduction of the coupling matrix: The M matrix resulting from step 2 is usually a
full matrix, implying that each cavity is coupled to all others. This can hardly be
realized in practice. The third step, therefore, consists of reducing M to a form,
which, depending on the filter topology, contains a number of zero elements and
can so be realized in practice. This is done using similarity transforms (plane
rotations).
A fundamental topology, whose transform procedure from the full matrix is de-
scribed in [4], is the folded topology that applies to both even and odd order filters, as
sketched in Figure 9.21 The circles represent the resonators, the full lines the direct
couplings and the broken line the cross couplings.
Although other topologies can be adopted, the folded structure is the most common
one. A general procedure to transform the coupling matrix into other topologies has
been developed by Atia and Williams [6].
1 2 3 4
7 6 5
in
out
Figure 9.21
Folded canonical
topology: a
seventh-order filter
example

9.4.2.2 Filter Topologies and Relevant Transmission Zero Locations
It is important to recognize that the location of the transmission zeros in the complex
s plane is directly related to which cross couplings are present (thus to the topology
of the filter) and to the signs of the coupling elements (whether positive or negative).
In general, both signs are necessary to obtain all possible transfer functions of a given
topology.
For a given topology, and for given signs of the coupling coefficients, one can predict
the type of filter response that can be obtained: whether symmetrical or asymmetrical
and how many zeros can be located outside the pass-band. Here, we confine our atten-
tionto the case of transmission zeros on the imaginaryaxis s = jω (attenuation poles).
The basic topologies of three, four, and six resonator filters and the corresponding
response types are illustrated in Figure 9.22. The sign of each coupling element is
indicated on the corresponding branch.
(a)
1 2
3
in
out
+
+
+
(b)
1 2
3
in
out
+
+
−
(c)
1 2
3
4
in
out
+
+
+
+
(e)
1 2
3
4
in
out
+
+
+
+
+
(f)
1 2
3
4
in
out
+
−
+
+
+
(d)
+
1 2
3
4
in
out
+
−
+
(g)
1 2 3
5
6 4
in
out
+ +
+ +
+
+ −
Figure 9.22
Filter topologies
and corresponding
locations of the
transmission
zeros:
a) third-order filter
with one
asymmetric (upper
stop-band)
transmission zero
(TZ), b) third-order
filter with one
asymmetric (lower
stop-band) TZ,
c) fourth-order
filter with no TZs,
d) fourth-order
filter with two
symmetric TZs,
e) fourth-order
filter with two
asymmetric (upper
stop-band) TZs,
f) fourth-order filter
with two
asymmetric (lower
stop-band) TZs,
and g) sixth-order
filter with four
symmetric TZs

It can be noted that in the case of third-order filters, one zero can be placed either at
the upper or lower stop-band, depending on whether the cross coupling is positive or
negative, respectively.
In the case of fourth-order filters, the cross coupling between cavities 1 (input) and
4 (output) produces two transmission zeros at either side of the pass-band. Observe,
however, that in the case of Figure 9.22c, no zeros occur on the imaginary axis.
9.4.2.3 Waveguide Filter Topologies
Folded-filter topologies best suited for waveguide realization are those with no
oblique couplings that are clearly of impractical implementation. With reference to
Figure 9.22, this corresponds to considering the configurations c), d), and g).
With this choice, the filter order N is even. The direct couplings (such as 1-2, 2-3,
. . . , (N − 1)-N) are positive, while the cross couplings (such as 1-N, 2-(N − 1), etc.
. . .) may take either positive or negative values in order to locate the transmission
zeros on the imaginary axis.. The schematic of an eighth-order folded filter is shown
in Figure 9.23.
By properly selecting the signs of the cross couplings, one can allocate up to N − 2
transmission zeros so as to obtain a high selectivity response. Such filters exhibit only
symmetrical responses with (N − 2)/2 transmission zeros at each side of the pass-
band. As a consequence, only half of the transmission zeros can be chosen arbitrarily.
Indeed, one may note that the degrees of freedom of the filter are equal to the number
of cross couplings—that is, (N/2) − 1 = (N − 2)/2.
Observe that by choosing both direct and cross couplings in such a way that they all
have the same sign, all zeros are located on the real axis so that a linear phase response
can be obtained.
9.4.2.4 Tables of Coupling Matrices for High-Selectivity
Narrow-Band Filters
The procedure outlined in Section 9.4.2 allows one to synthesize the coupling matrix
of any cross-coupled filter topology. In the case of waveguide filters with no oblique
couplings, the coupling matrices have a simple structure where only the nonoblique
couplings are non-zero. For instance, in the case of a sixth-order filter, the coupling
1 2 3 4
7
8 6 5
in
out
Figure 9.23
Eighth-order folded
canonical topology
with no oblique
couplings

matrix has the form:
[M] =











0 +M12 0 0 0 ±M16
+M12 0 +M23 0 ±M25 0
0 +M23 0 +M34 0 0
0 0 +M34 0 +M23 0
0 ±M25 0 +M23 0 +M12
±M16 0 0 0 +M12 0











The procedure used to synthesize the M-matrix [4, 5] is rather complex. In order
to facilitate the design, using an optimization procedure, we have computed the
normalized coupling matrices and load resistances for filters of order N = 4, 6, 8,
for given return loss and maximum selectivity. The values of Mij and R are quoted
in Tables 9.8, 9.9, and 9.10. Such parameters can be used to design high-selectivity
waveguide filters with pass-bands not exceeding 5 percent. The N − 2 transmission
zeros are located symmetrically on both sides of the pass-band in such a way as to
achieve maximum selectivity.
The typical responses of fourth-order filters with various out-of-band attenuations are
shown in Figure 9.24. It can be observed that the higher the selectivity, the lower is
the stop-band attenuation and vice versa.
Example:
Let us synthesize a high-selectivity filter with the following characteristics:
• Filter order: N = 6
• RL: 20dB
• Out-of-band attenuation: 50 dB
• Pass-band: 3.93–4.07 GHz , corresponding to a fractional bandwidth
w =
f2 − f1
f0
= 3.5%
The center frequency corresponding to the resonant frequency of the resonators is:
f0 = f1 f2 =
√
3.93 · 4.07 GHz = 3.999 GHz
Normalized inductances and capacitances are given by:
L = C =
1
ω0
=
1
2πf0
= 3.9796 × 10−11

Table 9.8
Normalized Coupling Matrix Values for Fourth-Order Filters with High
Selectivity Symmetric Response
Fourth-Order High-Selectivity Filter–Symmetric Response–Two Transmission Zeros
RL = 16 dB
Attenuation, dB R M12 M23 M14
20 0.88723 0.74796 0.80353 −0.33001
25 0.89412 0.77819 0.76913 −0.24652
30 0.89997 0.79755 0.74290 −0.18430
35 0.90901 0.81240 0.72373 −0.13766
40 0.91028 0.82014 0.70826 −0.10323
50 0.91452 0.82952 0.68818 −0.05798
60 0.91818 0.83482 0.67717 −0.03288
RL = 20 dB
20 1.06365 0.82929 0.83874 −0.34695
25 1.12560 0.87588 0.81678 −0.25986
30 1.08301 0.88323 0.78468 −0.19333
35 1.10699 0.90609 0.77372 −0.14542
40 1.09138 0.90514 0.75204 −0.10779
50 1.10736 0.92088 0.73581 −0.06068
60 1.106126 0.92363 0.72468 −0.03438
RL = 25 dB
R M12 M23 M14
20 1.30603 0.95178 0.89906 −0.36972
25 1.31708 0.98518 0.87041 −0.27386
30 1.32534 1.00523 0.85034 −0.20940
35 1.34693 1.03042 0.84118 −0.15784
40 1.35442 1.04265 0.82941 −0.11766
50 1.38685 1.06700 0.81833 −0.06660
60 1.38237 1.07130 0.80866 −0.03635
RL = 28 dB
20 1.46391 1.03309 0.94665 −0.39268
25 1.47516 1.06674 0.91967 −0.29598
30 1.48576 1.08985 0.89774 −0.22004
35 1.50081 1.10968 0.88392 −0.16516
40 1.49430 1.11437 0.86683 −0.12103
50 1.53702 1.14824 0.86476 −0.06898
60 1.53808 1.14734 0.84912 −0.03963

Table 9.9
Normalized Coupling Matrix Values for Sixth-Order Filters with High-
Selectivity Symmetric Response
Sixth-Order High Selectivity Filter–Symmetric Response–Four Transmission Zeros
RL = 16 dB
Attenuation,
dB R M12 M23 M34 M16 M25
30 0.87646 0.77560 0.46924 0.86120 0.12798 −0.42538
40 0.86605 0.78014 0.52019 0.79059 0.06201 −0.30140
50 0.86998 0.78467 0.55415 0.72928 0.02803 −0.20480
60 0.87337 0.78881 0.57240 0.68536 0.01367 −0.14261
70 0.87954 0.79298 0.58408 0.65063 0.00621 −0.09544
80 0.87990 0.79445 0.58888 0.62872 0.00303 −0.06670
RL = 20 dB
Attenuation,
dB R M12 M23 M34 M16 M25
30 1.02728 0.83973 0.50100 0.85961 0.13025 −0.41379
40 1.04109 0.85566 0.55445 0.79583 0.06136 −0.28969
50 1.04705 0.86199 0.58511 0.74129 0.02881 −0.20019
60 1.05305 0.86669 0.60206 0.69786 0.01351 −0.13695
70 1.05695 0.87026 0.61215 0.66747 0.00635 −0.09380
80 1.06053 0.87301 0.61789 0.64611 0.00300 −0.06429
RL = 25 dB
Attenuation,
dB R M12 M23 M34 M16 M25
30 1.24767 0.94170 0.54874 0.86978 0.13319 −0.40388
40 1.26402 0.95815 0.59755 0.80963 0.06232 −0.28158
50 1.28137 0.97123 0.62827 0.76026 0.02926 −0.19441
60 1.29916 0.98300 0.64958 0.71534 0.01110 −0.12148
70 1.29893 0.98430 0.65664 0.69176 0.00570 −0.08689
80 1.33143 1.00691 0.67350 0.68761 0.00315 −0.06432
RL = 28 dB
Attenuation,
dB R M12 M23 M34 M16 M25
30 1.50162 1.06052 0.60894 0.90725 0.13214 −0.40447
40 1.41124 1.04175 0.64294 0.84943 0.06163 −0.28473
50 1.41953 1.04925 0.66957 0.80071 0.03037 −0.20098
60 1.56925 1.12219 0.71890 0.77486 0.01247 −0.13003
70 1.48309 1.09077 0.71350 0.74815 0.00672 −0.09539
80 1.48375 1.09085 0.71694 0.72584 0.00324 −0.06586

Table 9.10
Normalized Coupling Matrix Values for Eighth-Order Filters with High-Selectivity
Symmetric Response
Eighth-Order High-Selectivity Filter–Symmetric Response–Six Transmission Zeros
RL = 20 dB
Atten. R M12 M23 M34 M45 M18 M27 M36
30 dB 1.00266 0.82184 0.55133 0.32988 0.92805 −0.04565 0.21175 −0.59590
40 dB 1.00998 0.82896 0.57155 0.39411 0.88712 −0.02306 0.14000 −0.49275
50 dB 1.00289 0.83010 0.58029 0.42193 0.86789 −0.01746 0.11504 −0.44590
60 dB 1.00568 0.83335 0.58919 0.47040 0.81380 −0.00776 0.06813 −0.34641
70 dB 1.00870 0.83561 0.59325 0.50393 0.76038 −0.00352 0.03967 −0.26344
80 dB 1.01158 0.83822 0.59607 0.52322 0.72202 −0.00177 0.02500 −0.20822
RL = 25 dB
Atten. R M12 M23 M34 M45 M18 M27 M36
30 dB 1.19527 0.90874 0.59419 0.39021 0.90464 −0.03093 0.15371 −0.51580
40 dB 1.24491 0.93116 0.60512 0.40769 0.90640 −0.03085 0.15628 −0.50958
50 dB 1.21212 0.92132 0.61036 0.45177 0.86093 −0.01674 0.10333 −0.41920
60 dB 1.21608 0.92405 0.61633 0.48998 0.81389 −0.00881 0.06617 −0.33563
70 dB 1.22342 0.92887 0.62133 0.52117 0.76507 −0.00398 0.03903 −0.25771
80 dB 1.24691 0.94354 0.63148 0.54815 0.72828 −0.00170 0.02230 −0.19487
RL = 28 dB
Atten. R M12 M23 M34 M45 M18 M27 M36
30 dB 1.34265 0.97985 0.61969 0.40762 0.91729 −0.03088 0.15527 −0.51969
40 dB 1.34231 0.97994 0.62065 0.42111 0.90727 −0.03371 0.15486 −0.50466
50 dB 1.40149 1.00991 0.63933 0.47474 0.86407 −0.01815 0.10013 −0.40843
60 dB 1.37754 1.00290 0.64527 0.51150 0.82461 −0.00932 0.06480 −0.33116
70 dB 1.40650 1.02264 0.65934 0.55093 0.78316 −0.00383 0.03659 −0.25120
80 dB 1.42639 1.03715 0.66983 0.57696 0.75065 −0.00172 0.02154 −0.19319
From Table 9.9, for RL = 20 dB and attenuation of 50 dB, we obtain the following
normalized coupling matrix:
[M] =












0 0.86199 0 0 0 0.02881
0.86199 0 0.58511 0 −0.20019 0
0 0.58511 0 0.74129 0 0
0 0 0.74129 0 0.58511 0
0 −0.20019 0 0.58511 0 0.86199
0.02881 0 0 0 0.86199 0












and R = 1.04705.

0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04
−80
−60
−40
−20
0
20
log
10
(|s
21
|)
Normalized angular frequency, rad/s
−60
−40
−20
0
20
←s21
s11→
20
log
10
(|s
11
|)
Figure 9.24
Normalized
responses of
fourth-order,
high-selectivity
filters for different
out-of-band
attenuations (20,
30, 40, 50, 60 dB);
RL = −25 dB,
fractional
bandwidth
w = 1%,
ω0 = ω0 = 1
Multiplying these values by w = 0.035, we obtain the denormalized values:
Rd = Rw = 0.03665
K12 = M12w = 0.03017
K23 = M23w = 0.02048
K34 = M34w = 0.02594
K16 = M16w = 0.00100
K25 = M25w = −0.00700
The response of the filter in Figure 9.20 synthesized as shown in Figure 9.25.
3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4
−80
−60
−40
−20
0
20
log
10
(|s
21
|)
Frequency, GHz
−30
−20
−10
0
10
←s21
s11→
20
log
10
(|s
11
|)
Figure 9.25
Equivalent circuit
simulation (MWO)
for the example
filter

2
1
3
5
6
4
(a)
1
2
3
6
5
4
(b)
Figure 9.26
Six-pole
cross-coupled
filters folded along
the H-plane (a) or
E-plane (b)
9.4.3 Rectangular Waveguide Realization
cross couplings in a rectangular waveguide filter can easily be realized by folding
the conventional direct-coupled configuration either along the E-plane or the H-plane
and opening windows in the side walls of adjacent cavities. Sketches of six-pole
filters folded along the H-plane and E plane are shown in Figures 9.26 a and 9.26b,
respectively. Such geometries realize the cross-coupled structure without oblique
coupling.
The H-plane folded geometry, however, is such that the cross couplings are all positive
so that, as shown in Figure 9.22c for a fourth-order filter, linear phase responses [7]
with no transmission zero can be obtained.8
On the contrary, with the E-plane folded geometry of Figure 9.26b, both positive
and negative couplings can be realized. Apertures cut at the center of the bottom
cavity wall, where the electric field has a maximum (magnetic field has a minimum),
give rise to negative couplings (such as between cavities 2 and 5 of Figure 9.26b),
while pairs of apertures close to the side walls, where the magnetic field has a max-
imum (electric field has a minimum), give rise to positive couplings (see cavities
1 and 6 of Figure 9.26b). The E-plane folded configuration, therefore, lends itself
to the realization of high-selectivity filters with symmetrical responses and N − 2
transmission zeros located at the outer edges of the pass-band.
8Transmission zeros can be realized by introducing stop-band cavities in the form of H- or E-plane stubs
in a direct-coupled filter. This leads to the so-called extracted pole filters [12], particularly useful for
realizing asymmetrical responses.

9.4.4 Design Procedure of H-Plane and E-Plane
Folded Filters
The design of folded filters in waveguide technology consists of converting the
coupling matrix, as discussed in the previous sections, into a waveguide structure
of the type sketched in Figure 9.26. The irises corresponding to direct coupled
cavities are assumed to be symmetrical with width di (i = 1, 2, N). Observe that the
first (i = 1) and last (i = N) irises correspond to the coupling between the connecting
lines and the filter. The windows corresponding to cross couplings are assumed to be
rectangular: one side can be determined based on practical considerations; the other
one, dci (i = 1, 2, N/2) is to be determined by the design procedure described next.
This procedure employs a full-wave analysis approach in order to achieve experi-
mental results that are in close agreement with the predictions. The method described
here is similar to what has been proposed in [8].
The design procedure is based on the knowledge of following input parameters:
• N: the order of the filter and number of resonators. There is no general procedure
to determine N. The selection is done empirically, based on the designer’s intuition
and experience.
• Fractional bandwidth w and center frequency f0. This information is used to deter-
mine the waveguide width a.
• Mij and R: the coupling matrix and load resistance. They are computed according
to the procedure described in previous sections or taken from Tables 9.8 through
9.10. Using the fractional bandwidth w and Equation 9.50, the corresponding
denormalized parameters Kij e Rd are computed.
The dimensioning of the waveguide structure requires determining the following
parameters:
1. l: lengths of the cavities. This quantity is determined by the condition that the
resonant frequency is equal to the center frequency f0. Since the resonant mode is
the TE101, such a condition yields:
=
a
2af0
c
2
− 1
(9.52)
where c = 1/
√
µε is the phase velocity in the medium filling the waveguide.
2. di (i = 2, . . . , N−1): widthofthe i-th innerwindow.Todetermine thisquantity,let
usfirst observe that two identical coupled LC resonators resonate at the frequencies
ωe = 1/ (L + M)C, ωo = 1/ (L − M)C (9.53)
where the subscripts e, o refer to the even or odd resonance, respectively. In the
waveguide structure, the even and odd resonances are obtained by replacing the

P
. M.C. orP.E.C.
Figure 9.27
Computation of
even and odd
resonant modes of
coupled cavities.
Observe that the
wall finite thickness
is taken into
account: The
symmetry plane is
to be replaced by a
perfect magnetic
conductor (p.m.c)
or perfect electric
conductor (p.e.c.)
wall for computing
the even or odd
resonances,
respectively
symmetry plane with a magnetic or electric wall, respectively. Using Equation
9.53, one easily finds:
k =
M
L
=
ω2
o − ω2
e
ω2
o + ω2
e
(9.54)
This formula allows us to compute the coupling between two cavities in terms
of their odd and even resonant frequencies. The structure for the computation is
shown in Figure 9.27. Using a full-wave simulator to compute the even and odd
resonant frequencies, the i-th iris width between cavities i − 1 and i is determined
by imposing that the coupling (Equation 9.54) equals the prescribed coupling
Ki−1,i .
3. dci: size of the i-th square window providing the cross coupling between cavities
i and (N − i). The procedure is exactly the same as described previously, except
the opening is in the bottom wall rather than in the side walls.
4. d1 = dN : width of the input and output irises. In contrast with the inner irises,
the input/output irises have to provide the load Rd to the filter. In other words, the
matching between the load Rd and the reference impedance is:
Z0 =
2
π
λ2
0
λ2
g0
L
C
as discussed in Section 9.3.1 (Figure 9.10). To this end, the K-inverter of Figure 9.9
can be adopted with K =
√
Rd Z0. The circuit can be realized in practice in the
form of an (either thin or thick) iris comprised of two waveguide sections. The
width of the iris and the length of the two waveguide sections can be determined
using a full-wave simulator to compute the scattering matrix of the iris, adopting
a reference plane at the center of the thickness. The iris width d is determined first
by the condition that
|s11| =
Rd − Z0
Rd + Z0
=
Rd
Z0
− 1
Rd
Z0
+ 1
=
π Rd λ2
g0
2λ2
0
− 1
π Rd λ2
g0
2λ2
0
+ 1
(9.55)
where we have used the condition L
C
= 1.

Once this condition has been satisfied, the reference plane must be shifted L/2 away
from the center of the iris in such a way that the reflection coefficient becomes real.
Using such condition, one easily finds:
L =
λg0
2π
θ11 − nπ
2
(9.56)
where θ11 is the phase of s11, n is an integer number (0, ±1, ±2, . . .), and λg0 is
the guided wavelength at the center frequency f0. In practical cases, L is negative,
implying that such a length must actually be subtracted to the lengths of the input and
output cavities.
Since a number of approximations have been involved in this design procedure, a
final optimization based on full-wave simulations is necessary in order to avoid the
experimental tuning of the filter.
9.4.5 Design Examples
9.4.5.1 Four-Pole Filter with Two Symmetrically Located
Transmission Zeros
Design a cross-coupled cavity filter, with the following specifications:
• Symmetric generalized Chebyshev response with two transmission zeros
• Filter order: 4
• Pass-band ripple RP = 0.01 dB (pass-band return loss RL ∼
= 26.5 dB)
• Pass-band limits f1 = 9.9 GHz, f2 = 10.1 GHz (w = 0.02 = 2%)
• Minimum stop-band attenuation L = 40 dB at fa = 10.37 GHz
• Waveguide: WR90 (a = 22.86 mm, b = 10.16 mm); thickness of waveguide
walls: t = 1.5 mm
As shown in Section 9.4.3, these specifications on transmission zeros can be
matched by the E-plane folded configuration of Figure 9.26b, which provides both
positive and negative coupling coefficients.9
The coupling matrix and load resistance of the prototype in Figure 9.20 satisfy the
specifications on in-band RL (28 dB) and stop-band attenuation (40 dB), and are
obtained from Table 9.8:
[M] =





0 1.11437 0 −0.12103
1.11437 0 0.86683 0
0 0.86683 0 1.11437
−0.12103 0 1.11437 0





R = 1.49430
9It is worth noting that the fourth-order direct-coupled cavity filter described in Section 8.4.2 cannot
match these specifications: Its attenuation at the frequency of 10.37 GHz is lower than 40 dB.

9.4 9.6 9.8 10.0 10.2 10.4 10.6
−80
−60
−40
−20
0
20
log
10
(|s
21
|)
Frequency, GHz
−60
−40
−20
0
20
←s21
s11→
20
log
10
(|s
11
|)
Figure 9.28
Response of the
N = 4-order
cross-coupled filter
folded in the
E-plane
Observe that the coupling coefficient between cavities 1 and 4 is negative, while the
remaining couplings are positive.
As shown in Figure 9.28, the filter response after frequency denormalization is fully
compliant with the specifications.
The geometry of the waveguide filter is sketched in Figure 9.29. The cross coupling
between cavities 1 and 4 must be realized by a window in the center of the common
wall in order to provide a negative coupling. All other couplings are positive and
are realized as symmetrical inductive windows for couplings 1–2 and 3–4, and for
cavities 2–3, as a window close to the terminal wall. The distance from such wall to
the window edge has to be chosen near to the end wall—in this case, it has been set
1.5 mm of distance. Moreover, it is useful to also fix one of the two dimensions of the
aperture and then act on the other to vary the coupling value. In this case, the width
of the aperture has been imposed as 3.5 mm.
In order to compute the geometrical parameters of the waveguide filter in Figure 9.29,
let us first denormalize the filter parameters:
Rd = Rw = 0.0298
K12 = M12w = 0.0222
K23 = M23w = 0.0173
K14 = M14w = −0.0024
1
2
4
3
Figure 9.29
Fourth-order
E-plane folded
cavity filter

Let us then follow the procedure described in the previous section.
The length of the cavities is determined by Equation 9.52:
=
a
2af0
c
2
− 1
= 19.84 mm
To determine the width of irises 1–2 and 3–4, we compute the even and odd resonant
frequencies of the coupled cavities (see Figure 9.27). With a width d = 6.62 mm, we
find:10
fe = 9.6946 GHz, fo = 9.9169 GHz
Using Equation 9.54, we obtain:
k12 = k34 =
f 2
e − f 2
m
f 2
e + f 2
m
= 0.02267
which is close to the nominal value (k12 = 0.0222).
Using the same procedure for the other pairs of cavities, we obtain:
• aperture 1−4: 4.98 × 4.98 mm ( fo = 10.0280 GHz, fe = 10.0523 GHz)
• aperture 2−3: 8.35 × 3.5 mm ( fo = 9.9824 GHz, fe = 9.8074 GHz)
As far as the input and output irises are concerned, the full-wave simulator is used to
determine their width in such a way that condition (9.55) is satisfied—that is:
|s11| =
πRd λ2
g0
2λ2
0
− 1
πRd λ2
g0
2λ2
0
+ 1
=
0.082 − 1
0.082 + 1
= 0.8486
We obtain d = 10.73 mm. With such an iris in the WR90, the phase of s11 is found
to be θ11 = 2.65 rad.
The length L to be subtracted from the length of the input and output cavities is finally
determined using Equation 9.56:
L =
λg0
2π
θ11 − π
2
= 1.6 mm
At this point, all dimensions of the waveguide filter have been determined
as outlined in Table 9.11. The corresponding response, computed using a full-wave
simulator (CST), is shown in Figure 9.30. Although the general behavior agrees with
the expectations—in particular, having two transmission zeros close to the outer band
10Full-wave computations have been made using CST-eigenvalue solver.

Table 9.11
Dimensions of the Filter after the Design Procedure and
after Optimization
After Design Procedure After Optimization
Length of cavities 1,4 18.24 mm 16.87 mm
Length of cavities 2,3 19.84 mm 18.7 mm
Width input/output iris 10.73 mm 10.88 mm
Width irises 1-2 6.62 mm 6.67 mm
Width irises 1-4 4.98 × 4.98 mm 4.98 × 4.98 mm
Width irises 2-3 8.35 × 3.5 mm 8.42 × 3.5 mm
edges—a shift toward lower frequencies is observed, along with a slight degradation
of the pass-band.
A final optimization based on a full-wave simulator is thus to be applied for the fine
tuning of the filter.11
The optimized dimensions are listed in the third column of
Table 9.11. The good accuracy of the first dimensioning can be observed, particularly
of the irises. The corresponding simulated and measured responses are shown in
Figure 9.31. Figure 9.32 shows a photograph of the filter fabricated. Two 90◦
bends
have been added at both ends of the filter in order to make it possible to measure the
filter by spacing apart the connecting waveguides. The component has been fabricated
in two symmetrical halves milled from aluminum blocks.
The residual discrepancy between theory and experiments is to be ascribed to manu-
facturing tolerances.
9.0 9.5 10.0 10.5 11.0
−80
−60
−40
−20
0
20
log
10
(|s
21
|)
Frequency, GHz
−30
−20
−10
0
10
←s21
s11→
20
log
10
(|s
11
|)
Figure 9.30
Full-wave (CST
with AR-filter
estimation)
simulated
response of the
designed
waveguide filter
11Prior to optimization, all cavity lengths could be reduced by the same amount of about 1.1 mm. In this
manner, the filter pass-band is correctly centered on 10 GHz and the ensuing optimization procedure is
speeded up.

9.2 9.4 9.6 9.8 10.0 10.2 10.4 10.6 10.8
−80
−60
−40
−20
0
simulated
measured
s11→
←s21
20
log
10
(|s
21
|)
Frequency, GHz
− 60
− 40
− 20
0
20
20
log
10
(|s
11
|)
Figure 9.31
Four-pole E-plane
folded cavity filter:
comparison
between the full
wave simulation
and experimental
results
9.4.5.2 Six-Pole Filter with Four Symmetrically Located
Transmission Zeros
An example of a sixth-order filter is illustrated here based on the following speci-
fications:
• Symmetric generalized Chebyshev response with four transmission zeros
• Filter order: N = 6
• Pass-band ripple: RP = 0.01 dB (pass-band return loss ∼
= 26.5 dB )
• Pass-band limits: f1 = 9.9 GHz, f2 = 10.1 GHz, thus w = 0.02
• Minimum stop-band attenuation: L = 40 dB at fa = 10.17 GHz
• Waveguide: WR90 (a = 22.86 mm, b = 10.16 mm); thickness of waveguide
walls: t = 1.5 mm
Figure 9.32
Photograph of the
four-pole E-plane
folded cavity filter

This filter possess a higher selectivity than the previous one (one pair of transmission
zeros are located at each sides of the pass-band); the same design procedure can be
applied, so it will not be repeated in detail here.
From Table 9.9, with RL = 28 dB and attenuation 40 dB, we obtain the coupling
matrix and the load resistance:
[M] =











0 1.04175 0 0 0 0.06163
1.04175 0 0.64294 0 −0.28473 0
0 0.64294 0 0.84943 0 0
0 0 0.84943 0 0.64294 0
0 −0.28473 0 0.64294 0 1.04175
0.06163 0 0 0 1.04175 0











R = 1.41124
Observe that the only negative coupling coefficient is that between cavities 2 and 5.
Denormalizing using w = 0.02:
Rd = Rw = 0.02822
K12 = M12w = 0.02083
K23 = M23w = 0.01286
K34 = M34w = 0.01698
K16 = M16w = −0.00123
K25 = M25w = −0.00570
The theoretical response of the sixth-order prototype is shown in Figure 9.33.
9.4 9.6 9.8 10.0 10.2 10.4 10.6
−80
−60
−40
−20
0
20
log
10
(|s
21
|)
Frequency, GHz
−60
−40
−20
0
20
20
log
10
(|s
11
|)
←s21
s11→
Figure 9.33
Six-pole elliptic
filter equivalent
circuit response:
circuit simulation
(MWO)

The geometry of the waveguide filter is illustrated in Figure 9.26b. As in the previous
example, the length of all cavities is fixed at l = 19.84 mm, corresponding to the
resonant frequency of 10 GHz.
The couplings between cavities lying in the same plane (1–2, 2–3 and 4–5, 5–6) are
realized by inductive windows. Cavities 3–6 are coupled by two rectangular apertures,
with edges 2 mm apart from the side walls in regions of high magnetic field so as
to realize a positive coupling. Similarly, cavities 3–4 are coupled by a rectangular
aperture close to the end wall at a distance of 1.5 mm, just as in the previous example.
Finally, the negative coupling between cavities 2–5 is realized by a rectangular square
aperture at the center of the common wall in a region of high electric field.12
Using the same procedure described in the previous example, the dimensions of all
apertures are determined based on the even and odd resonances of the coupled pairs
of cavities:
• Irises 1–2 and 5–6: w = 6.46 mm ( fo = 9.9198 GHz, fe = 9.7154 GHz)
• Irises 2–3 and 4–5: w = 5.49 mm ( fo = 9.9333 GHz, fe = 9.8073 GHz)
• Apertures 3–4: 8.25 × 3.5 mm ( fo = 9.9827 GHz, fe = 9.8149 GHz)
• Apertures 1–6: 3.72 × 2.1 mm each ( fo = 9.99 GHz 30, fe = 9.9785 GHz)
• Aperture 2–5: 6.10 × 6.10 mm ( fo = 10.0504 GHz, fe = 10.1084 GHz)
The width of the input/output iris has been found to be d = 10.69 mm, while the first
and last cavities must be shortened by a length L = 1.53 mm.
The full-wave simulation of the filter so dimensioned is shown in Figure 9.34.
Although the relative bandwidth is about as expected, a slight frequency shift can
be observed and the in-band matching is lower than it should be. This has to be
corrected primarily by modifying the end irises and the end cavity lengths, while the
frequency shift canbe corrected by shortening all cavity lengths of about0.85 mm (see
footnote [5]). This is an excellent starting point for the final full-wave optimization.
Some useful remarks are at hand in practical cases to further improve the filter
response before the final optimization. We refer to this procedure as “manual tuning,”
in contrast with the final optimization, which is made automatically.
The unsymmetrical response observed in Figure 9.34 is due to the detuning of the
three pairs of identical cavities. In particular, as a simple simulation of the equivalent
circuit of the filter would show, the first and last cavities appear to resonate at a lower
frequency than predicted. By simply shortening their length by 0.5 mm, we equalize
the two attenuation maxima closest to the pass-band edges. Similarly, the other pair
12Observe that for each aperture, only its coupling parameter is specified. As a consequence, only one side
of the rectangle can be determined; the other is left to the freedom of the designer and is to be determined
by practical considerations.

9.4 9.6 9.8 10.0 10.2 10.4 10.6
−60
−50
−40
−30
−20
−10
0
20
log
10
(|s
21
|)
Frequency, GHz
−40
−30
−20
−10
0
10
20
←s21
s11→
20
log
10
(|s
11
|)
Figure 9.34
Full-wave (CST
with AR-filter
estimation)
simulation of the
sixth-order filter
designed prior to
optimization
of attenuation maxima are equalized by slightly lengthening (just 0.04 mm!) the third
and fourth cavities. With these changes, the filter response is notably improved, both
in the stop-band and in the pass-band (see Figure 9.35). In particular, the symmetrical
response behavior has been recovered by just cavity tuning without modifying the
coupling apertures.
After full-wave optimization using CST Microwave Studio, the final response of
Figure 9.36 is obtained. The filter dimensions after the design procedure, after the
first manual tuning, and after final optimization are shown in Table 9.12, columns 2,
3, and 4, respectively.
9.4 9.6 9.8 10.0 10.2 10.4 10.6
−60
−50
−40
−30
−20
−10
0
20
log
10
(|s
21
|)
Frequency, GHz
−40
−30
−20
−10
0
10
20
s11→
20
log
10
(|s
11
|)
←s21
Figure 9.35
Full-wave
simulation (CST
with AR-filter
estimation) of the
filter after manual
tuning of cavities
(dimensions listed
in Table 9.12,
column 3)

9.4 9.6 9.8 10.0 10.2 10.4 10.6
−60
−50
−40
−30
−20
−10
0
20
log
10
(|s
21
|)
Frequency, GHz
−40
−30
−20
−10
0
10
20
←s21
s11→
20
log
10
(|s
11
|)
Figure 9.36
Full-wave (CST)
simulation of the
filter optimized
(dimensions listed
in Table 9.12,
column 4)
9.5 DUAL-MODE CAVITY FILTERS
While the filter performance can be improved by increasing its order N, this has also
the side effect of increasing the in-band loss and, more importantly, the size and mass
of the filter. The latter is a critical issue in space applications. Dual-mode filters can
be adopted to alleviate this problem. In dual-mode filters, in fact, two orthogonal
degenerate modes resonate in each cavity so as to halve the size of the filter.
In this section, we describe the basics of the most common dual-mode filter, along
with some design examples. The corresponding CST files are provided in the attached
CD-ROM.
Table 9.12
Filter Dimensions (mm) of the Sixth-Order Filter Designed, after
“Manual” Tuning, and after Full-Wave Optimization
After After
Design “Manual” After
Procedure Tuning Optimization
Length of cavities 1,6 18.31 16.96 16.79
Width input/output iris 10.69 10.69 10.75
Width irises 1-2 6.46 6.46 6.476
Width irises 2-3 5.49 5.49 5.535
Width irises 1-6 3.72 × 2.10 3.72 × 2.10 3.72 × 2.10
Width irises 3-4 8.25 × 3.5 8.25 × 3.5 8.22 × 3.5
Width irises 2-5 6.10 × 6.10 6.10 × 6.10 6.11 × 6.11

y
x z
45°
Figure 9.37
Typical dual-mode
circular cavity: two
TE111 modes with
electric ﬁeld
rotated 90◦
with
respect to each
other
9.5.1 Dual-Mode Circular and Rectangular
Cavity Filters
Although square waveguides can be used as well, the most common dual-mode cav-
ities use circular waveguide sections. This is due to the dominant mode TE111 of the
latter exhibiting lower loss than the dominant TE101 mode of the former.
A circular cylinder (see Figure 9.37) supports two degenerate TE111 modes with
orthogonal polarizations. The coupling between them is provided by a screw inserted
into the waveguide wall and inclined 45◦
with respect to both polarizations. Two
additional screws, one for each polarization, are used to individually tune the resonant
modes.
The same concepts can be implemented with rectangular shaped cavities, as shown
in Figure 9.38.
In contrast with the circular cavity, the independent tuning of the two resonant modes
can be obtained by varying one of the sides of the cross-section of a rectangular cavity,
y
x z
45°
Figure 9.38
Dual-mode
rectangular cavity
with quasi-square
cross-section

4
3
6
5
8
7
2
1
Figure 9.39
Propagating
dual-mode filter:
Typical waveguide
realization of an
eighth-order
dual-mode filter
with input/output
couplings located
at the opposite
filter ends
thus obtaining a quasi-square cross-section. In this manner, only the coupling screw
is to be used.
An exhaustive theory of such dual-mode filters, employing either circular or square
cavities, has been presented by Atia and Williams [9]. Such filters consist of cascading
dual-mode cavities coupled through cross-shaped irises that allow for the independent
coupling of the two pairs of modes resonating in each cavity. The input/output
transition is designed in such a way as to excite only one resonant mode.
A typical example of a circular eighth-order dual-mode filter with four cavities is
shown in Figure 9.39. This filter is named “propagating” because the signal travels
through its length from input to output, in contrast with other geometries that will be
discussed later on.
Depending on the position of the coupling screws, either positive or negative coupling
coefficients can be obtained.
It should be noted, however, that only a restricted class of cavity couplings can be
realized with a structure like that in Figure 9.39. Except for the case N = 4, the
dual-mode filter topology allows one to obtain only a subset of the responses of
the cross-coupled cavity filters described in Section 9.4.
With reference to Figure 9.40, it is seen that the minimum path from input to output
is 1–4–5–8, bypassing only four cavities (2, 3, 6, 7). As a consequence, for an eighth-
order filter, only four zeros instead of six can be allocated.
1
2
4 5 8
3 6 7
in out
Figure 9.40
Topology of the
eighth-order
dual-mode filter of
Figure 9.39

7
2
6
3
5
4
8
1
Figure 9.41
Reflective
dual-mode filter:
Waveguide
realization of an
eighth-order
dual-mode filter
with input and
output couplings
located at the
same side
Such a limitation can be overcome by putting both input and output at the same
end of the filter structure, as shown in Figure 9.41. The input excites one of the two
degenerate modes, while the output is coupled to the other one. This geometry makes
it possible to realize the folded configuration shown in Figure 9.23. Nonetheless, this
geometry is not commonly used because of the reduced input-output isolation, which
degrades the filter performance, particularly in the stop band. In practice, depending
on the type of transitions used to excite the first cavity, it is difficult to obtain isolations
better than 25–30 dB.
In order to combine the advantages of both geometries in such a way as to fully
exploit the potential N − 2 transmission zeros, having, at the same time, input and
output ports at different cavities, hybrid solutions can be devised, as in the example of
Figure9.42. In thiseighth-order filter,the input is realized as a coaxialcable connected
to the second cavity, while the output is a waveguide connected by an iris to the first
cavity. The topology of the filter, as shown in Figure 9.43, is such as to produce six
transmission zeros.
7
8
6
1
5
2
4
3
Figure 9.42
Eighth-order
dual-mode filter:
Input/output ports
are applied to
different cavities;
all of the six
available
transmission zeros
are located in the
complex plane

8
7
1 2 3
6 5 4
out
in
Figure 9.43
Equivalent
topology of the
filter shown in
Figure 9.42
9.5.1.1 Fourth-Order Rectangular Dual-Mode Filter
The design of dual-mode filters is not easy and does not usually lead to accurate
results. A design example is reported in [10]. In practice, a tuning phase of the filter
based on the three screws for each cavity is necessary in order to make the filter
behave properly.
The EM modeling of the discontinuities represented by the tuning and coupling
screws is rather cumbersome and computer-intensive and is, therefore, not easy to
take into account in the design. The usual design procedure, therefore, consists of
dimensioning the input and output apertures and the cross-shaped irises between
cavities using the approximate Bethe’s formulas [1]. As an alternative, such apertures
can be dimensioned using the same procedure presented in Section 9.4.4.13
As an example, a fourth-order dual-mode filter based on quasi-square cavities has
been designed. The negative coupling is realized by placing the two coupling screws
90◦
to one another so as to realize two symmetrical transmission zeros according to
the topology of Figure 9.22d.
The geometry of the filter is sketched in Figure 9.44, the dimensions being listed in
Table 9.13. The CST simulations are shown in Figure 9.45.
2
1
4
3
Figure 9.44
Fourth-order
dual-mode
rectangular cavity
filter
13In a first approximation, adjacent cavities can be considered as identical, although they are not because
of the presence of the screws.

Table 9.13
Dimensions of the Filter in Figure 9.44
Dimension Value, mm
First (second) cavity length 18.33
First (second) cavity width 22.79
First (second) cavity height 24.07
Input/output iris width 11.7
Input/output iris height 6
Crosswise iris horizontal dimension 8.3
Crosswise iris vertical dimension 11.03
Crosswise iris width 1
Screw diameters 2
Screw oblique penetration 11.13
Apertures thickness 1.5
Waveguide feed width 22.86
Waveguide feed height 10.16
9.5 10.0 10.5
−60
−50
−40
−30
−20
−10
0
20
log
10
(|s
21
|)
Frequency, GHz
−50
−40
−30
−20
−10
0
10
s11→
20
log
10
(|s
11
|)
←s21
Figure 9.45
Full-wave
simulation (CST)
of the fourth-order
dual-mode
rectangular cavity
filter
9.6 LOW-PASS FILTERS
Waveguide low-pass filters are usually employed in antenna feed systems to reject
spurious harmonics from the microwave high-power transmitter. Such filters are,
therefore, often classified as harmonics reject filters.
Because of the propagation properties of the waveguides and the existence of a
lower cutoff frequency, such filters may be considered low-pass, provided that their
response is plotted with respect to the waveguide electrical length. In practice,

z
y
b b
Figure 9.46
E-plane
longitudinal
section of
corrugated
waveguide
low-pass filter
with input/output
stepped
impedance
transformers
low-pass waveguide filters are band-pass filters with stop-bands much wider than
the conventional filters treated in the previous sections.
9.6.1 Tapered Corrugated Waveguide Filters
Historically, the first design method is based on the image parameter technique dis-
cussed extensively in Chapter 2. Such design results in periodic structures are clas-
sified as corrugated waveguides [1], in which the waveguide height is periodically
stepped within the filter to create a cascade of low-impedance capacitive and high-
impedance inductive sections with small and large waveguide heights, respectively.
The steps are quite near to each other, much closer than λg/4 at the filter cutoff fre-
quency. With regards to the broader waveguide dimension, the structure is uniform
within the filter.
In [1], several normalized graphs and tables are provided to facilitate the design,
which could otherwise be rather complex. The image parameter method, however,
produces results that are not all that accurate, so the synthesized filter usually needs
several adjustments using a full-wave simulator.
Another characteristic is that the filter structure, as illustrated in Figure 9.46, needs
to be loaded at its ends by an impedance that is generally much smaller than the
impedance level of the inductive sections—stepped impedance transformers need,
therefore, to be inserted at the input and output ports. Such transformers make the
filter rather bulky, as they could be even larger than the filter itself.
An advanced design method has been proposed by Levy [17, 18]. The method is
based on the distributed low-pass prototype filter shown in Figure 9.16 and already
introduced in Section 9.3.3 as a band-pass filter. Because of the periodic nature of the
distributedfilter, in fact, such a prototypecan also beused as alow-pass filter,as shown
in Figure 9.17, where the distributed filter response with Chebyshev approximation
[15] is plotted versus the electrical length θ, all line lengths being λ0 at the cutoff
frequency.
By introducing generalized impedance inverters Ki between Zi and Zi+1, the
distributed low-pass filter of Figure 9.16 is then transformed into the generalized
distributed low-pass filter of Figure 9.47.
The design method that will be described in the following section leads to a tapered
corrugated waveguide structure, shown in Figure 9.48. This structure can be seen as
an extension of the conventional corrugated structure of Figure 9.46, with the external
impedance tapering incorporated within the filter so as to eliminate the need for the
terminating impedance transformers. In contrast with the conventional structure, the

Z0 = 1
Z1
θ0
Z2
θ0
ZN
θ0
ZN+1 = 1
K1 KN KN+1
K2
Figure 9.47
Generalized distributed low-pass filter
tapered structure is non-periodic, though it is longitudinally symmetrical with respect
to its center.
The tapered corrugated waveguide filter has the advantage of being compact, but it
suffers from the return loss, which tends to worsen at lower frequencies [18].
Regarding the stop-band characteristics, the stop-band extends up to 3fc, where fc is
the filter cutoff frequency. The tapered corrugated waveguide filter is mainly applied
in the second harmonic rejection.
The filter response in the stop-band might be affected by spikes due to the spurious
excitation of TEn0 modes. Such spikes, however, are well below −30 dB of trans-
mission. A way to suppress the spurious resonance due to these higher-order modes
is to cut small longitudinal slots along the filter’s upper wall, resulting in the so-called
waffle iron filter [1].
9.6.1.1 Design Procedure for Tapered Corrugated Waveguide Filters
The tapered corrugated filter of Figure 9.48 can be obtained from the prototype of
Figure 9.47. The impedance inverters are realized by thick capacitive irises, spaced
by waveguide lengths with heights corresponding to the prescribed impedance levels.
It can be noted that the impedance inverters employed in the prototype are non-
symmetrical two-ports, where the impedance level at one port is inverted and scaled
by K2
at the other port.
It is worth adding here that any loss-less two-port network can be transformed into an
impedance inverter simply by cascading at each port a (loss-less) transmission line
z
y
b b
z
y
b b
Figure 9.48
Longitudinal
E-plane section of
the tapered
corrugated
waveguide
low-pass filter

LOSS
LESS
2 PORT
NET
ZA
ZA
ZA ZB
ΦA
ZB
ΦB
K
Figure 9.49
Generalized
impedance
inverter obtained
from a loss-less
two-port network
with cascaded line
lengths A and B
section, as indicated in Figure 9.49. With the notation of Figure 9.49, it can be easily
demonstrated [17] that by properly choosing the electrical line lengths A and B,
the overall network acts as an impedance inverter with:
K
√
ZA ZB
=
1
√
VSWR
(9.57)
The synthesis procedure starts with the following specifications:
• Return loss (or, equivalently, the VSWR pass-band ripple)
• Number of sections N
• Cutoff frequency fc of the filter and the corresponding angle θ0
It is worth observing that the angle θ0 is related, on the one hand, to the fractional
bandwidth of the filter (referred to as its first pass-band—see Figure 9.17) and, on
the other hand, to the length of the transmission line sections interposed between the
irises.14
Although its choice is somewhat arbitrary, one has, nevertheless, to keep in
mind that θ0 should be small enough to yield a reasonably wide stop-band and, at the
same time, θ0 should not be too small in order not to yield too short line lengths that
would either compromise the filter performance or make it physically unrealizable.
The synthesis procedure can be summarized as follows:
1. Synthesis of the distributed low-pass prototype of Figure 9.16. The impedances
Zi of the prototype are obtained from the tables quoted in [15]. To this end, the
fractional bandwidth wfilter of the half-wave filter has to be chosen. In [15], the
fractional bandwidth is defined as wtransf = 2wfilter = 4θ0
π
, where θ0 is the electrical
length corresponding to the cutoff frequency fc, as shown in Figure 9.17.
Once the distributed low-pass filter prototype has been synthesized, all the
VSWR at the impedance step junctions are calculated from VSWRi = Zi
Zi−1
for
i = 1, 2, . . . , N + 1.
14Recall that the following relation holds θ0 = βTE10l.

2. Synthesis of the generalized distributed low-pass filter of Figure 9.47. The
generalized impedance inverters Ki (i = 1, . . . , N) are inserted into the distributed
low-pass prototype filter and arbitrary impedance levels Zi are set for the sections
so that Ki
√
Zi−1 Zi
= 1
√
VSWRi
for i = 1, 2, . . . , N + 1.
The only constraint on Zi is that the impedance level must gradually decrease step
by step from the input to the center of the filter, giving rise to a filter tapering of
the inductive sections, as shown in Figure 9.48. To achieve the best results, the
tapering should be smooth, even in cases when the waveguide’s narrow side has
to be reduced substantially to the center of the filter. It is not unusual to produce
more than one design in order to achieve a fully satisfactory result.
3. Calculation of the waveguide’s narrow sides of the inductive sections. The narrow
side of the waveguide’s inductive sections can now be determined from the
impedance levels Zi . The height b is calculated from Equation 9.12 as:
bi =
Zi a
2 · ηTE
(9.58)
where ηTE is the TE10 mode wave impedance as defined in Equation 9.10—such
impedance has to be evaluated at the filter cutoff frequency fc.
4. Dimensioning the capacitive irises. To achieve the best results, a full-wave
simulator is usually required to compute the dimensions of the capacitive irises—
that is, the gap gi and the thickness ti (i = 1, . . . , N). The i-th iris is sized so
as to produce the prescribed VSWRi calculated in step 1, taking into account the
tapering imposed in step 3. In other words, the amplitude of the reflection co-
efficients, looking at either side of the i-th iris, has to be:
s(i)
11 = s(i)
22 =
Zi − Zi−1
Zi + Zi−1
=
VSWRi − 1
VSWRi + 1
(9.59)
Such a condition must hold at the cutoff frequency fc of the filter.
As can be expected, several pairs of gi and ti can be chosen that produce the same
VSWRi at the discontinuity. In making such a choice, one should keep in mind
that the iris gap should not be too narrow in order not to limit the power handling
capability of the filter.
Once the dimensions gi and ti have been set by imposing the condition (9.59),
the corresponding reference planes must be shifted away from the iris center to
achieve the impedance inverter—this is verified when both s11 and s22 are real.
In contrast with the inductive irises (see Section 9.3), for the capacitive irises, it
is usually necessary to add positive line lengths. The two waveguide lengths Li1
and Li2 (at the larger and smaller waveguide ports, respectively) to be added are
calculated from the conditions:
Li1 =
λgc
2π
θ11i + nπ
2
, Li2 =
λgc
2π
θ22i + nπ
2
(9.60)

Chapter 9

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Chapter 9