Download Microwave Breakdown Prediction in Rectangular Waveguide Based

Microwave Breakdown Prediction in Rectangular
Waveguide Based Components
C. Vicente1, M. Mattes2 , D. Wolk3 , B. Mottet1 , H. L. Hartnagel1 , J. R. Mosig2 , D. Raboso4
1
Technical University of Darmstadt, Institut für Hochfrequenztechnik, Germany,
Email: [email protected]
2
Laboratoire d’Electromagnétisme et Acoustique (LEMA), Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland,
Email: [email protected]
3
TESAT-Spacecom GmbH & Co. KG, Gerberstr. 49, 71522 Backnang, Germany,
Email: [email protected]
4
ESA-ESTEC, Keplerlaan 1, Postbus 299, 2200 AG Noordwijk, The Netherlands,
Email: [email protected]
Abstract — This paper describes a new software tool capable
to predict the corona discharge threshold in microwave devices. The two necessary calculations when investigating such
a phenomenon have been performed: computation of the electromagnetic fields inside the structure and determination of the
corona breakdown onset itself. The corona discharge routine is
fully integrated into the electromagnetic tool providing accuracy
when using the electromagnetic fields as well as efficiency.
The software has been tested for a particular component for
which experimental measurements have been also performed.
The agreement between the predicted breakdown values and the
experimental results fully shows the validity of the approach
followed.
I. I NTRODUCTION
The phenomenal growth in the telecommunication industry in
recent years has brought significant advances in filter technology as new communication systems emerged, demanding
more stringent filter characteristics. In particular, the growth
of the wireless communication industry has spurred tremendous activity in the area of microwave filter miniaturisation.
Today, telecommunication systems demand for a higher component integration and an increase of services implying larger
bandwidths, thus higher frequency ranges. To achieve these
objectives the size of microwave devices on the one hand
incessantly decreases whereas at the same time the power
levels increase. Both trends lead to a higher electric field
density inside the components. This development in turn leads
to serious problems with respect to RF breakdown due to
corona discharges.
The term “gas discharge” originates with the process of
discharge of a capacitor into a circuit incorporating a gap
between electrodes. If the voltage is sufficiently high, an
electric breakdown occurs in the gas and an ionised state is
formed. The circuit is closed and the capacitor discharges.
Later the term “discharge” was applied to any flow of electric
current through ionised gas, and to any process of ionisation
of the gas by the applied electric field [1]. One of the pioneers
to investigate microwave breakdown in gases was MacDonald
[2].
Due to the increasing problem of corona discharge in microwave devices a big effort has been dedicated in the past to
understand, model, and predict the occurrence of discharges.
For example, the authors of [3] showed that the thresholds
in microwave systems can be much lower than the theoretically predicted ones due to the presence of absorbing
inhomogeneities in an air-filled microwave waveguide. In [4]
the case of output multiplexer filters is studied. A theoretical and experimental investigation of microwave breakdown
in air-filled resonators is presented in [5]. The effects on
the breakdown threshold of air pressure, pulse length in a
multi-carrier operation, and the strong inhomogeneity of the
electromagnetic field of resonators are analysed. In a recent
publication [6] several aspects of microwave breakdown in
resonators are discussed and approximate analytical criteria are
formulated for some illustrative model geometries like circular
symmetric resonators.
To predict the breakdown threshold inside microwave devices
two problems need to be solved: First of all, the spatial electric
field distribution has to be computed. Afterwards, a diffusion
type problem for the evolution of the electron density must be
solved using the afore computed electric field distribution.
Most of the publications in recent years studying the problem
of corona discharge in microwave devices deal with special
cases or simplified geometries to obtain the electromagnetic
field distribution. Complex geometries or complete waveguide
devices, e. g. filters, are not considered or a parallel plate
configuration is assumed.
In this paper we describe some preliminary results of the
ESA/ESTEC activity “AO-4026 ITT ESA - Multipactor and
Corona Discharge: Simulation and Design in Microwave Components” (Contract No. 16827/02/NL/EC) that is intended
to create a software tool capable to predict the microwave
breakdown onset inside complex geometries, such as filters,
based on rectangular waveguide technology. For this, first of
all, the software tool computes the electric field density inside
the microwave device. Afterwards, this field distribution is
used to determine the breakdown threshold. The organisation
of the paper is as follows: We first briefly describe the software
tool to compute the electromagnetic field distribution inside
waveguide filters. Then, the approach chosen to predict the
breakdown threshold for corona discharge is outlined. The
predicted results are compared to measurements showing a
good agreement between simulation and measured data.
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II. C OMPUTATION OF ELECTROMAGNETIC FIELD
DISTRIBUTION
For an accurate corona discharge analysis, it is indispensable
to know the electromagnetic field distribution inside the microwave device, e. g. a filter. For this, the software tool FEST
(Full-wave Electromagnetic Simulation Tool) is used that has
been recently developed by ESA/ESTEC [8]. This software
uses a full-wave approach, which assures a high accuracy and
efficiency, also for complex geometries like microwave filters,
and is based on the integral equation technique [9], [10] and
microwave network theory. An analysis example is shown in
Fig. 1. It compares measurements and simulation of return and
insertion loss of a diplexer, taken from [11], demonstrating a
good agreement between theory and measurements.
its full-wave approach, making it therefore easy for a design
engineer to quickly verify the field distribution, which then
can be further used to compute the breakdown threshold.
(a) Surface current.
vertical electric field (V/m)
8000
(a) Geometry.
0
|sij |/dB
-20
-40
-60
7000
6000
5000
4000
3000
2000
1000
0
25
50
75
100
125
150
175
z/mm
-80
-100
(b) Vertical electric field along centre axis.
-120
9
9.2 9.4 9.6 9.8 10 10.2 10.4 10.6 10.8 11
f /GHz
(b) Reflection and transmission coefficients at the common port.
Fig. 2. Structure used for testing the theoretical approach to predict corona
discharge. Frequency: 7.4 GHz. Input power: 1 W rms. Figure (a) represents
the surface current and (b) the vertical electric field along the centre axis of
the structure. The circles represent the field values actually computed with
FEST. Their straight line interpolation is represented by the solid line. The
R
results of FEST have been compared to Microwave Studio
, represented by
rectangles respectively the dashed line, showing a very good agreement fully
validating the results of FEST.
Fig. 1. Geometry and frequency response of a diplexer. Fig. (a) shows the
reflection and transmission coefficients at the common port. Measurements
are drawn in solid lines, simulation results in dashed lines. Measurements
have been obtained applying an averaging over 4096 samples per frequency
point for noise reduction.
Fig. 2 presents the results of the electromagnetic field computation inside a transformer based on rectangular waveguides,
that has been used to test the theoretical model to predict
the threshold of corona discharge. High field values are
represented by light, low values by dark grey tones. The
vertical electric field along the centre line has been compared
R
showing a good agreement between
to Microwave Studio
both simulation tools, fully validating the field computation
of FEST. Whereas the analysis of such a structure can take
R
, it is only a
up to several hours with Microwave Studio
matter of seconds, at the worst of minutes, with FEST due to
III. C ORONA BREAKDOWN : CALCULATION AND
COMPARISON WITH MEASURED DATA .
In order to calculate the corona discharge threshold, one has
to compute the free electron density originated by ionisation
of the gas molecules inside the waveguide structure. To do
this, the continuity equation for the evolution of the electron
density must be solved:
∇2 (D · n) + (νi − νa )n =
∂n
,
∂t
where:
n is the electron current density,
D is the Diffusion coefficient,
νi is the ionisation rate,
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(1)
This equation shows the basic physics of the corona discharge
phenomenon. On the one hand ionisation tends to increase the
electron density, whereas diffusion and attachment inhibit the
process. Diffusion is due to the motion of electrons from high
field regions to low field ones. On the other hand, attachment
is the process by which electrons are ”absorbed” by neutral
molecules forming ions which are too massive for taking part
in the discharge. Other mechanisms like recombination can be
neglected for the calculation of the threshold [2].
In the continuous wave regime, the corona threshold condition
is just ∂n
∂t = 0, i.e., it is only defined by the moment in which
the electron density starts to grow with time. This results in
∇2 (D · n) + (νi − νa )n = 0 .
required precision is achieved, the input power is taken as the
breakdown power of the device. Fig. 3 shows the corona onset
obtained for the structure of Fig. 2.
280
270
Breakdown power / W
νa is the attachment rate.
250
240
230
220
(2)
210
This equation defines the problem: once ionisation is capable
to overcome the losses due to diffusion and attachment, the
plasma develops.
The relevant parameters νi , νa and D depend on the microwave electric field (Erms ), the gas pressure (p) and the
frequency (f = ω/2π). In [12], expressions for these coefficients for dry air are given:
200
νi /p ≈
106
cm2 s−1 ,
p
5.14 × 1011 exp(−73α−0.44 )s−1 ,
νa /p ≈
7.6 × 10−4 α2 (α + 218)2 s−1 ,
D
≈
(3)
(4)
(5)
where:
Eef f =
Erms
,
(1+(ω/νc )2 )1/2
Eef f
p ,
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α=
νc ≈ 5 · 10 p.
The effective field Eef f is a measure of the capability of the
microwave field of transferring energy to the surrounding gas.
The higher the pressure of a gas the most efficient the RF field
transfers energy. At very high pressures, the RF effective field
indeed behaves like a DC field.
The solution to (2) has to be found using numerical methods
if the particular device is complex and therefore, the electric
field highly inhomogeneous. However, if there are zones with
a high and homogeneous electric field, an analytical approach
to (2) can be used [2]:
π 2 ν − ν
i
a
=0
(6)
+
−
b
D
where b is the distance between the plates, in our case, the
waveguide height.
When the gap is small compared to the rest of the waveguide dimensions (parallel plate approximation case), (6) gives
accurate results. In the cases where this approach can not be
applied, resort must be taken to numerical analysis. In order
to solve (2), a finite difference scheme has been used. The
solution is searched iteratively assuming an initial solution
for the electron density n. If, for a given applied power, the
electron density is sufficiently higher than the initial value,
breakdown occurs. On the contrary, if the electron density falls
below the initial solution, the threshold has not been reached.
The input power is varied regarding this and finally, when the
Analytical approach
Numerical simulation
Measured
260
6
7
8
9
10
11
Pressure / mBar
12
13
14
Fig. 3. Corona discharge threshold for the structure of Fig. 2 at 7.4 GHz.
All data are for ambient temperature.
Two main results are presented: on the one hand, both the
analytic (solution to 6)) and the numeric calculation (solution
to (2)). On the other hand, the experimental breakdown values.
For the analytic result, the maximum field inside the structure
for each waveguide element has been used in the determination
of νi and νa in (6). The breakdown power is therefore
determined for each element and the lowest of all of them
is taken as the breakdown threshold of the device. Employing
the maximum field, one ensures that this is the worst possible
case (i.e. lowest threshold).
For the experimental detection of corona, the pressure was
fixed and the input power was slowly increased. Two detection
methods have been used to ensure that corona occurs: third
harmonic generation and I/P reflection nulling measurements.
The results show a very good agreement between measurements and simulation (the largest error is only around 0.2 dB).
It is remarkable the fact that the analytical solution approaches
to the measurements and to the numerical result as the pressure
increases. This behaviour occurs because the diffusion losses
become less important and therefore the field inhomogeneities
is less critical in the corona onset determination (see (2)) since
diffusion is highly field dependent. The result of the decrease
in the diffusion losses is that the discharge becomes more and
more local and thus the maximum field represents better the
final result. This also explains why the analytical approach
improves with the pressure.
Apart from the breakdown onset value it is also interesting
from the microwave component designer to locate the place
where the discharge occurs. Fig. 4 shows a plot of the electron
density for a power just above the breakdown threshold. It
is seen how the discharge (peak) occurs in the transformer
at the right of the inner gap. The point where the electron
density is maximum is in fact z = 128.3 mm. It is worthwhile
to underline that despite the fact that the field is maximum
in the centre of the structure, the breakdown does not occur
there. The explanation to this has to do with the diffusion
losses which are more important for smaller gaps. Thus, the
ionisation is maximum in the centre but the diffusion losses
are also maximum there. However, in the transformer located
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at the right of the inner gap the field is also quite high and
the losses due to diffusion are less important due to its higher
size.
2.5
2.5
2
1.5
2
1
1.5
0.5
1
[9] G. Gerini, M. Guglielmi, and G. Lastoria, “Efficient Integral Equation
Formulations for Admittance or Impedance Representation of Planar
Waveguide Junctions,” IEEE MTT-S International Microwave Symposium Digest, vol. 3, pp. 1747–1750, 1998.
[10] M. Mattes, “Contribution to the Electromagnetic Modelling and Simulation of Waveguide Networks Using Integral Equations and Adaptive
Sampling,” Ph.D. dissertation, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland, 2003, thesis No. 2693.
[11] A. Alvarez-Melcón, M. Mattes, and J. R. Mosig, “A Simple Waveguide
Diplexer for Satellite Communications,” in PIERS 2002 Conference
Proceedings, Cambridge, MA, USA, 2002, p. 613.
[12] W. Woo and J. DeGroot, “Microwave absortion and plasma heating due
to microwave breakdown in the atmosphere,” IEEE Physical Fluids,
vol. 27, no. 2, pp. 475–487, 1984.
0
0.5
0
-1
-0.5
x/cm
0
0.5
1
0
2
4
6
8
10
12
14
16
18
20
z/cm
Fig. 4. Electron density inside the structure for a pressure of 10 mBar and
a power of 234 W. The location of the discharge (peak) is clearly shown.
IV. C ONCLUSIONS
In this paper we have presented some preliminary results of
an ESA/ESTEC activity towards a simulation tool for the
prediction of corona discharge inside arbitrary geometries,
such as filters, based on rectangular waveguide technology.
Theoretical results and measurements have been provided for
a test structure showing good agreement between predicted
and measured data.
V. ACKNOWLEDGEMENT
The authors would like to thank TESAT-Spacecom GmbH &
Co. KG, Gerberstr. 49, 71522 Backnang, Germany, for providing the test structure Envisat as well as the measurements
for corona discharge breakdown that have been used to verify
the simulation tool.
R EFERENCES
[1] Y. P. Raizer, Gas Discharge Physics. Berlin, Heidelberg, New York,
London, Paris: Springer-Verlag, 1991.
[2] A. D. M. Donald, Microwave breakdown in gases. John Wiley & Sons,
1966.
[3] D. G. Anderson, M. Lisak, and P. T. Lewin, “Thermal lowering
of the threshold for microwave breakdown in air-filled waveguides,”
IEEE Transactions on Microwave Theory and Techniques, vol. 35, no. 7,
pp. 653–656, July 1987.
[4] C. Boussavie, D. Baillargeat, M. Aubourg, S. Verdeyme, P. G. A.
Catherinot, S. Vigneron, and B. Theron, “Microwave breakdown in
output multiplexer filters,” IEEE MTT-S International Microwave Symposium Digest, vol. 2, pp. 1185–1188, June 2000.
[5] T. Olsson, D. Andersson, J. Jordan, M. Lisak, V. Semenov,
and M. Ahlander, “Microwave breakdown in air-filled resonators,”
IEEE MTT-S International Microwave Symposium Digest, vol. 3, pp.
915–918, June 1999.
[6] D. Anderson, U. Jordon, M. Lisak, T. Olsson, and M. Ahlander,
“Microwave breakdown in resonators and filters,” IEEE Transactions
on Microwave Theory and Techniques, vol. 47, no. 12, pp. 2547–2556,
Dec. 1999.
[7] D. Anderson, , D. S. Dorozhkina, U. Jordan, M. Lisak, T. Olsson,
J. Puech, and V. E. Semenov, “Microwave breakdown around a metal
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(MULCOPIM), ESTEC, 2200 Noordwijk, The Netherlands, Sept. 2003.
[8] M. Mattes and J. R. Mosig, “Integrated CAD tool for waveguide
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