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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. 17 GeMiC 2005 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, 18 GeMiC 2005 (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 , 9 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 19 GeMiC 2005 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 wedge,” in Proceedings of the 4th International Workshop of Multipactor, Corona and Passive Intermodulation in Space RF Hardware (MULCOPIM), ESTEC, 2200 Noordwijk, The Netherlands, Sept. 2003. [8] M. Mattes and J. R. Mosig, “Integrated CAD tool for waveguide components, Final Report,” Dec. 2001, contract ESA/ESTEC No. 12465/97/NL/NB. 20 GeMiC 2005