Advanced analysis of propagation losses in rectangular waveguide structures using perturbation of boundary conditions

Abstract

In this paper, propagation losses effects present in rectangular waveguide structures are rigorously considered. For this purpose, a new formulation based on the perturbation of the boundary conditions on the metallic walls of the waveguides combined with an Integral-Equation (IE) analysis technique is proposed. Following this advanced technique, the drawbacks of the classical power-loss method are overcome and a complex modal propagation constant is computed. To validate this theory, we have successfully compared our results with numerical data of lossy hollow waveguides. Next, a Computed-Aided-Design (CAD) software package based on such a novel modal analysis tool has been used to predict the propagation losses effects in a 2-pole symmetric bandpass filter.

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Advanced Analysis of Propagation Losses in Rectangular Waveguide
Structures Using Perturbation of Boundary Conditions
Stephan Marini#, Michael Mattes∗, Benito Gimeno‡, Pablo Soto§and Vicente E. Boria§
#Departamento de F´ ısica, Ingenier´ ıa de Sistemas y Teor´ ıa de la Se˜ nal,
Instituto Universitario de F´ ısica Aplicada a las Ciencias y Tecnolog´ ıas, Universidad de Alicante, Spain
∗Laboratory of Electromagnetics and Acoustics, Ecole Polytechnique F´ ed´ erale de Lausanne, Switzerland
‡Departamento de F´ ısica Aplicada, Instituto de Ciencia de los Materiales,
Universidad de Valencia, Spain
§Departamento de Comunicaciones, Instituto de Telecomunicaciones y Aplicaciones Multimedia,
Universidad Polit´ ecnica de Valencia, Spain
Abstract—In this paper, propagation losses effects present in
rectangular waveguide structures are rigorously considered. For
this purpose, a new formulation based on the perturbation of
the boundary conditions on the metallic walls of the waveguides
combined with an Integral-Equation (IE) analysis technique is
proposed. Following this advanced technique, the drawbacks of
the classical power-loss method are overcome and a complex
modal propagation constant is computed. To validate this theory,
we have successfully compared our results with numerical data of
lossy hollow waveguides. Next, a Computed-Aided-Design (CAD)
software package based on such a novel modal analysis tool has
been used to predict the propagation losses effects in a 2-pole
symmetric bandpass filter.
Index Terms—Losses, Waveguide components, Dissipative fil-
ters, Complex propagation constant, Perturbation Method.
I. INTRODUCTION
New generation of telecommunication satellites are de-
signed to operate at higher frequency bands and with more
stringent filter characteristics to better exploit the available
bandwidth. At higher frequencies, such as microwave and
millimeter-wave bands, losses effects are more pronounced
and can severely distort the electrical response of space
hardware like filters and multiplexers. Therefore, the rigorous
prediction of the ohmic losses effects affecting such passive
waveguide devices is becoming more and more necessary [1]-
[3].
Propagation losses in hollow waveguides have been typi-
cally solved by means of the classical perturbative power-loss
method [4]. A very early contribution based on this method can
be found in [5], where degenerated modes in lossy waveguides
and cavities are treated. More recently rigorous consideration
of metal losses in planar junctions has been included in [6]
and [7], nevertheless losses associated with evanescent modes
are not considered.
In this paper we propose an enhanced full-wave method for
the computation of propagation losses in rectangular waveg-
uide structures. This method, first proposed in [8] and [9],
where only TM modes were treated with details, is based
on the perturbation of the boundary conditions. As it will
be demonstrated in the next sections, the proposed technique
allows to overcome all the drawbacks of the classical power-
loss method and it is valid assuming that the metallic walls
are good conductors. In order to analyze realistic metal-
based rectangular structures, we have included the proposed
technique in a CAD tool based on the Integral Equation (IE)
technique originally described in [10]. Proceeding in this way,
a very precise modal analysis tool of rectangular waveguides
filters considering propagation losses effects more accurately
has been developed.
II. THEORY
To reduce the number of subscripts, we denote the electro-
magnetic field for perfectly conducting walls by a subscript
zero, and the electromagnetic field for walls of finite conduc-
tivity value without subscript; moreover, we assume and omit
the harmonic variation ejωtthroughout this paper.
In a good conductor, the fields E and H are tangential to
the surface and propagate normal to it. The small tangential
electric field can be obtained using the Leontovich condition
E?? Zs(ˆ n × H0?) =(1 + j)
where ˆ n is the unit normal vector outward from the conductor,
Zsis the surface impedance, σ is the metallic conductivity, δ =
?2/(µcωσ) is the skin depth, and µcthe conductor magnetic
of electric field, in addition to the normal electric field and
tangential magnetic field, generates a power flow and ohmic
loss along the walls of an uniform metallic waveguide with
finite but large conductivity values
σδ
(ˆ n × H0?)
(1)
permeability. The existence of this small tangential component
dP
dz= −1
2Re[ˆ n · E × H∗] =
1
2σδ
?
|ˆ n × H0?|2
(2)
2011 IEEE MTT-S International Microwave Workshop Series on Millimeter Wave Integration Technologies
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Equation (2) together with the conservation of energy are
normally used in the classical pertubative power-loss method
for the computation of the attenuation constant [4]
α = −1
2P
dP
dz
(3)
where −dP
length of the waveguide, and P is the total power flow at
z. This classical method has some disadvantages: it gives
physically meaningless results at cut-off, where the attenu-
ation constant tends to infinity, and losses associated with
the evanescent modes are not taken into account. All these
drawbacks can be overcome using the technique developed and
called perturbation of boundary conditions [8], [9]. With this
enhanced method at each analysis frequency, even at the modal
cut-off frequency, both attenuation α and phase constant β are
computed taking positive real values (e−γz= e−αze−jβz, be-
ing γ the complex propagation constant), and losses associated
to the evanescent modes are rigorously accounted for.
Next, we proceed to present the theoretical basics re-
lated to the proposed analysis technique considering first
the TM modal problem. For finite but large conductivity
values, the TM boundary condition on the waveguide contour
Ez0|∂S= 0 is not valid, because Ezis not zero on the walls.
From (1) and expressing the transverse magnetic field in terms
of the longitudinal electric field [8], we have
dzis the power dissipated in ohmic loss per unit
Ez|∂S?(1 + j)
σδ
(1 − j)
2
H0l|∂S=(1 + j)
?ω
σδ
?2∂Ez0
−jω?
k2
c0
∂Ez0
∂n
????
∂S
=
µc
µδωc0
∂n
????
∂S
(4)
where kc0= ωc02µ? is the modal lossless cut-off wavenumber,
and µ the magnetic permeability of the medium which homo-
geneously fills the waveguide. So, the new proposed perturbed
TM problem considering losses is thus
(∇2
t+ k2
c)Ez= 0
∂Ez0
Ez|∂S? fTM
?
∂n
????
∂S
(5)
with fTM=(1−j)
In order to obtain the new complex propagation constant,
the Green theorem in two dimensions can be employed:
?
(k2
S
2
µc
µδ
ω
ωc0
?2
.
S
?Ez0∇2
c)
tEz− Ez∇2
tEz0
?
?dS =
Ez∂Ez0
c0− k2
?
Ez0EzdS =
∂S
?
∂n
?
dl
(6)
Since fTM is assumed to be a parameter with small value,
it is consistent to approximate Ez in the integral on the left
hand size by its unperturbed value Ez0, thus obtaining
γ2− γ2
0= k2
c0− k2
c? fTM
?
∂S
?
??∂Ez0
∂n
??2dl
s|Ez0|2dS
(7)
where γ0= ω2µε − k2
c0. Introducing the following parameter
?
ξTM=
fTM
2(1 − j)γ0
∂S
?
??∂Ez0
∂n
??2dl
s|Ez0|2dS
(8)
the new complex propagation constant γ considering losses is
finally given by
?
which is also valid for TE modes as it will be shortly
demonstrated.
For the TE perturbed problem, we need to consider that
an axial electric field component also appears on the surface,
from (1) we have
?
Next, the normal component of the magnetic field associated to
this axial electric field can be obtained by using the Maxwell
equations [8]:
????
TE problem considering losses is thus
(∇2
∂Hz
∂n
∂S
The last step in the formulation is to use the Green theorem
in two dimensions, obtaining the following parameter for TE
modes
?
?
γ =
γ0(γ0+ 2ξn) − j2ξnγ0;
n = TM,TE;
(9)
E?|∂S?
Zs−jγ
k2
c0
∂Hz0
∂l
?????
∂S
ˆ z − ZsHz0|∂Sˆl
(10)
Hn|∂S=
j
µcω
∂Ez
∂l
∂S
=
Zsγ
µcωk2
c0
∂2Hz0
∂l2
????
∂S
(11)
Then, defining fTE= −(1−j)δ
2, we found that the perturbed
t+ k2
ω2µ?µc
c)Hz= 0
µHz0−γ2
????
? fTE
?
0
k2
c0
∂2Hz0
∂l2
?????
∂S
(12)
ξTE=
−fTE
2(1 − j)γ0
δ
4γ0
∂S
?
ω2µ?µc
µ|Hz0|2−
?
k2
γ2
k2
c0
0
∂2Hz0
∂l2
?
dl
s|Hz0|2dS
γ2
0
=
∂S
?
ω2µ?µc
µ|Hz0|2+
?
c0
??∂Hz0
∂l
??2?
dl
s|Hz0|2dS
(13)
where the last expression is obtained after some algebraic
manipulation. Finally, the TE complex propagation constant
can be obtained using again (9).
III. RESULTS
Note that in all the next figures we indicate with method 1
the results obtained by the classical power-loss method, and
with method 2 the results obtained by the proposed technique.
First of all, in order to validate our theory, we have performed
the modal analysis of a WR-75 rectangular waveguide (a =
19.05 mm and b = 9.525 mm) with a finite conductivity value
of σ = 5.8·107S/m. Fig. 1(a) shows the attenuation constant
of the first WR-75 waveguide modes as function of frequency
computed with method 1, with method 2 and the commercial
software HFSS1. Note that method 2 does not fail at the cut-off
frequency in contrast with classical method 1 (infinite value
1www.ansoft.com/products/hf/hfss12
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510152025
10−2
10−1
100
101
102
103
Frequency (GHz)
α (rad/m)
TE10
TE01
TE20
TE11
TM11
TM21
(a)
5 10 1520 25
10−2
10−1
100
101
102
103
Frequency (GHz)
β (rad/m)
TE10
TE01
TE20
TE11
TM11
TM21
(b)
Fig. 1.
75 rectangular modes computed with method 1 (dashed lines, only in
(a)), method 2 (solid line) and HFSS data (dashdot line). The vertical
dotted lines indicate the cut-off frequency of the different modes in
the lossless case.
Attenuation (a) and phase constants (b) for the first WR-
for α). In Fig. 1(b) we also present the phase constant for the
same modes computed with method 2 and HFSS data. A very
good agreement between our results and those from Ansoft
can be observed.
With the aim of verifying the accuracy of the previous
results, Fig. 2 successfully compares the attenuation versus
phase constant of the first WR-28 (a = 7.112 mm and
b = 3.556 mm) rectangular mode for a conductivity range
between 105and 5.8·107S/m, computed at frequencies close
to the cut-off (21.076 GHz). The points are also computed
with HFSS.
Once the proposed theory has been validated with several
0 1020
α (rad/m)
3040
0
10
20
30
40
β (rad/m)
σ=5.8x107 (S/m)
σ=106 (S/m)
σ=105 (S/m)
Fig. 2.
rectangular mode at frequencies close to cut-off (21.076 GHz), for
different conductivities and computed with method 2 (solid lines) and
with HFSS (points).
Phase constant versus attenuation of the first WR-28
3333.53434.53535.5 36
-40
-20
Frequency (GHz)
-0.2
-0.195
-0.19
-0.185
method 2
method 1
HFSS
lossless
-0.18
|S11| (dB)
|S21| (dB)
Fig. 3.
(a = 7.112 mm and b = 3.556 mm) coupled with a central iris of
dimensions: a = 5.69 mm, b = 2.845 mm and length l = 2 mm.
Scattering parameters of an input/output WR-28 waveguide
benchmark tests, its direct application to the analysis of passive
rectangular waveguide structure is faced. For this purpose, we
have integrated the enhanced technique within a CAD software
package based on the Integral-Equation method (IE) [10].
As first example, we have simulated a very simple case: an
input/output WR-28 waveguide coupled with a central iris
of dimensions a = 5.69 mm, b = 2.845 mm and length
l = 2 mm. We have assumed a low finite conductivity value
of σ = 105S/m. In Fig. 3 the simulated reflection and
transmission coefficients of this simple structure are compared
with the numerical data also provided by HFSS, where also
only propagation losses effects (longitudinal walls) have been
considered. It can be seen clearly, due to the low conductivity
value, a phase shift of the magnitude of S11 appears. This
shift, caused by a variation of the value of β, is not predicted
by the classical method 1.
Finally, making use of the CAD software package imple-
mented, we have designed a 2-pole symmetric rectangular
bandpass filter for operation at 30 GHz and with a bandwidth
of 600 MHz. Input/output filter ports and cavities are in WR-
28 standard rectangular waveguides. To study the propagation
loss effect, in this case a finite conductivity value of 106S/m
has been assumed. The simulated scattering parameters of such
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2929.53030.5 31
−50
−40
−30
−20
−10
0
Frequency (GHz)
Scattering Parameters (dB)
method 1
method 2
HFSS
lossless
Fig. 4.
filter. Dimensions: input/output ports and cavities are in WR-28
waveguides; the cavity length is 11.457 mm. Dimensions of the
input/output coupling irises are a1 = 4.9 mm, b1 = 2.85 mm; central
iris with a2 = 4.442 mm, b2 = 1.259 mm. All irises of length 2 mm.
Scattering parameters of a 2-pole symmetric bandpass
29.53030.5 31
−2.5
−2
−1.5
−1
−0.5
0
Frequency (GHz)
|S21| (dB)
method 1
method 2
HFSS
lossless
Fig. 5. Detailed view of the magnitude of the transmission coefficient
(S21) for the 2-pole symmetric bandpass filter.
a structure are shown in Fig. 4, where they are successfully
compared with the numerical data provided by HFSS software
(dashdot line). Dashed and dotted lines represent, respectively,
the results obtained with method 1 and considering perfect
conductor walls. As can be seen from Fig. 4, a small deviation
of the filter response appears with respect to the ideal case.
This shift, which can be critical for narrow-band applications,
is not considered either by the classical method 1. In Fig. 5 a
detailed view of the magnitude of the transmission coefficient
(S21) is shown. An identical level of 0.5 dB insertion loss
is computed by our theory and HFSS results (where a value
of 0.001 maximum delta S has been used as convergence
criterion). With method 1 this level is higher because all
modes in the rectangular irises are under cut-off, so losses
in all these irises are not taken into account. Such results (for
both methods) were obtained using 20 accessible modes, 100
basis functions, and 400 kernel terms in the IEs related to the
solution of each discontinuity [10].
IV. CONCLUSION
In this paper, a novel full-wave method for the computation
of propagation losses in metallic waveguide structures is pre-
sented. Following this technique, the propagation losses effects
are accurately predicted using a perturbation of the bound-
ary conditions on the waveguide contour combined with an
integral-equation analysis technique. With this new approach
the drawbacks of the power method can be overcome, and
losses associated to the evanescent modes can also be taken
into account. The new proposed theory has been extensively
verified through several examples of practical interest. First,
the complex propagation constants of some rectangular waveg-
uide modes have been successfully computed. Next a simple
input/output WR-28 waveguide coupled by a central iris has
been considered, and the scattering parameters obtained using
the developed method and numerical data provided by a com-
mercial software have been successfully compared. Finally,
the new modal analysis technique has been used together
with a CAD software package for design purposes. Successful
results for a 2-pole symmetric bandpass filter used for Ka-
band applications, have proved the accuracy of the proposed
method.
ACKNOWLEDGEMENT
This work has been supported by Ministerio de Educaci´ on
y Ciencia, Spanish Government, under the Grant JC2009-
0221 and the coordinated R&D project TEC2010-21520-C04,
by University of Alicante under the project GRE10-22, as
well as by the European COST Office under Action IC-0803
“RF/Microwave Communication Subsystems for Emerging
Wireless Technologies (RFCSET)”.
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