2290 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 11, NOVEMBER 2007 Behavioral Thermal Modeling for Microwave Power Amplifier Design Julie Mazeau, Raphaël Sommet, Daniel Caban-Chastas, Emmanuel Gatard, Raymond Quéré, Senior Member, IEEE, and Yves Mancuso Abstract—System-level models simplify the analysis of complex RF systems, such as transmission-reception modules, by expressing global input–output relationships. However, the development of high RF power models for nonlinear subsystems requires the prediction of the distortion induced by low-frequency memory effects such as self-heating effects. In this framework, we present a new electrothermal behavioral model for power amplifiers. This global model is based on the coupling between a behavioral electrical model derived from the transistor-level description of the amplifier and a thermal reduced model. This model, implemented into a circuit simulator, allows to predict the impact of the thermal effects in pulsed RF mode thanks to an envelope transient analysis. This approach has also been validated by measurements. Index Terms—Behavioral electrothermal (BET) model, power amplifier, reduced thermal model, Ritz vector approach, systemlevel model, Volterra series. Fig. 1. Description of circuit simulation with transistor-level model. I. INTRODUCTION HE FAST development of high-performance subsystems requires the use of system-level simulations and models. However, designing tools to perform this task are limited. Either they use transistor-level description models to simulate the global performances of microwave systems (Fig. 1) or AM/AM AM/PM data. With the first approach, designers are commonly faced with very long simulation times and even with convergence problems. With the second one, dynamic nonlinear effects are not taken into account. An intermediate solution can be proposed by means of expertise of the transistor-level model [1]. Between transistor circuits and RF integrated circuits (ICs), behavioral models depict nonlinear subsystem behavior like high power amplifiers (HPAs) (Fig. 2). These models estimate the performance of an entire subsystem thanks to relatively “simple” equations able to capture the essential nonlinear behavior. This simplification allows to decrease simulation times and to accurately perform the complete analysis of microwave systems. This task remains difficult because of complex phenomena causing damages and T Manuscript received April 3, 2007; revised July 27, 2007. J. Mazeau, D. Caban-Chastas, and Y. Mancuso are with THALES Airborne Systems, 78851 Elancourt, France (e-mail: julie.mazeau@fr.thalesgroup.com; daniel.caban-chastas@fr.thalesgroup.com; yves.mancuso@fr.thalesgroup. com). R. Sommet, E. Gatard, and R. Quéré are with the XLIM Research Institute, Unité Mixte de Recherche 6172, Centre National de la Recherche Scientifique, University of Limoges, 19100 Limoges, France (e-mail: raphael. sommet@xlim.fr; emmanuel.gatard@xlim.fr; raymond.quere@xlim.fr). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2007.907715 Fig. 2. Description of system simulation with subsystem-level model. instabilities like nonlinear memory effects in power amplifiers. Some system-level models are available to predict the electrical performances, but the thermal dependence is not currently considered. They are obtained either from simulation of the circuit-level model or measurements. The major advantages of the simulation approach rely on the low cost in time or equipment and the capability to simulate the ICs’ performances before realization. Improved description of memory effects based on Volterra series [2] can be found in [3]–[8]. The goal of this study is to efficiently take into account the distortion of the RF envelope signals due to self-heating [9]. This effect presents long time constants and modifies the amplitude, as well as the phase during the pulse. The transient temperature waveform contributes significantly to the network transfer function with an unwanted modulation. This phenomenon must be considered to improve accuracy and development of electronic beam scanning radar system [10], [11]. In this study, we propose a coupled behavioral electrothermal (BET) model, as illustrated in Fig. 3. The electrical system model is based on 0018-9480/$25.00 © 2007 IEEE MAZEAU et al.: BEHAVIORAL THERMAL MODELING FOR MICROWAVE POWER AMPLIFIER DESIGN 2291 crease of the bandwidth and the convergence property. The extraction process requires only one-tone measurements or simulation providing the input signal is quasi-constant. Inconvenient is the weak long-term memory effects prediction. The nonlinear network is represented by a two-port circuit loaded by a resistor. The reflected waves are neglected. Thus, only the transmission parameter is considered [18] and is given by Fig. 3. BET model. (1) Fig. 4. Complex envelope behavioral model. a modified form of Volterra series to determine the isothermal and the instantaneous large-signal transmission parameter . In order to deduce a precise transient dissipated power thermal model of the amplifier, a 3-D thermal finite-element (FE) description has been performed. A reduced model based on the Ritz vector approach is then applied to extract the thermal [12]–[15]. A SPICE thermal equivalent subcirimpedance cuit describes the exact analysis of the operating temperature during an envelope transient simulation. The first part of this paper is dedicated to the modified form of the dynamic Volterra-series model equations [3], [16]–[18]. The development of the reduced thermal circuit is then applied to a power amplifier. Finally, the results of the BET model implemented into the Agilent Advanced Design System (ADS) circuit simulator are compared with measurement results. The monolithic microwave integrated circuit (MMIC) power amplifier used in our example is based on an InGaP/GaAs HBT technology delivering 8 W for -band radar applications. However, the described method is generic for any power amplifier and pulsed RF operating modes. is the input power wave and is the static transmission parameter at central frequency of the modulation band. is the first-order dynamic kernel at frequency. width is the phase. In this paper, we use the good prediction of the nonlinear short-term memory effects and the easy characterization process of this model. The thermal modulation is a long-term memory effect and will be modeled out of the electrical model by the reduced impedance model. Thus, the electrical system-level model must be able to supply the average dissipated power generated by the instantaneous self-heating to the thermal model. Moreover, the thermal dependence will be explained in . the transmission parameter B. Isothermal Static Model [19] The increase of the operating temperature of the amplifier generates a fall of the static current within the transistors. It is the dominating thermal effect. Thus, at first, the thermal effects are supposed to be independent of the spectral dispersion. Only the static term of the dynamic Volterra series depends on the is expressed by temperature. The thermal variation of as follows: a differential term (2) II. ISOTHERMAL ELECTRICAL SYSTEM-LEVEL MODEL The aim of such an approach is to provide an analytical function, which links subsystem input and output signals without describing in detail all the elements of the circuit. System modeling needs accurate formalism, particularly for nonlinear memory effects. Classical Volterra series show convergence problems when modeling strong nonlinearities. The dynamic Volterra-series approach is more suited. A. Dynamic Volterra Series This formalism is based on a limited modulation band around the carrier frequency [4], [16]–[18]. The convergence property of the dynamic series is enhanced and allows to work only with the input and output complex envelope signals. From and are real input–output signals and is the Fig. 4, carrier frequency. The first-order modified Volterra series model [3], [4] is applied to large -parameters signal description for radar application [18]. The main advantages of this description are the in- (3) is the ambient temperature. is the uniform temperature applied to the circuit without self-heating effects. term allows to perform an easier extraction of the The model by separating the thermal dependence of the nonlinear effect prediction. The static function interpolation is then more precise and the errors can be easily detected. Moreover, this implementation offers the possibility to remove the influence of self-heating effects if designers want to simulate only nonlinear or short memory behaviors. Likewise, the average dissipated power of HBT transistors can be obtained by (4) 2292 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 11, NOVEMBER 2007 Fig. 5. Characterization of the thermal static term and the bias current. Fig. 7. Neural-network structure with a single hidden layer of wavelet. D. Neural-Network Approach Several approaches allow to fit nonlinear functions [20], [21], but the high performance of neural-network methods are particularly suited to approximate all discontinuities [22]–[25]. These , to feed a methods require the databases , the input vector neural network training process, as illustrated in Fig. 7. One hidden layer of wavelets captures the nonlinear behavior. A wavelet is a nonlinear processing unit with the “sigmoid” transfer function (6). In this study, the radial wavelet function is given by Fig. 6. Characterization of dynamic kernels. (6) The collector bias current depends both on the input magtemperature. , the collector bias nitude signal and the voltage, is fixed and the base bias is neglected. The bias power . The second term is calculated thanks is only a function of to the static transmission parameter. The characterization process of and requires isothermal single-tone harmonic-balance simulations of the circuit model at central carrier frequency, as shown in Fig. 5. Indeed, an isothermal measurement is difficult to obtain. These databases are representative of the behavior of the network and set the validity domain for each parameter, carrier fre, bias point, and quency, magnitude of input power wave temperature within the device. The input vector is connected to the hidden layer through a set of linear weight. Each kind of connection is defined by a and an offset . This topology is associated with vector defined in (7) as follows: an analytic function (7) and are, respectively, the translation and dilatation vectors of the wavelet base functions. In order to obtain an accurate function with reasonable CPU time, the number of wavelets and, consequently, linear weights, are minimized. Equation (7) is evaluated by optimizing the average mean square error (MSE) (8). The error between the neural-network output and the target is given by (8) C. Dynamic Kernels of the BET Model The first-order dynamic kernel of (1) allows predicting the short-memory effect and is characterized by the approach described in [17] and [18]. Thanks to the same characterization process, to take into account the frequency dependence of the , a dynamic kernel is added as follows: current is the amount of data available in the target . is then translated by a simple C program into the circuit simulator [23]. III. THERMAL REDUCED MODEL (5) This one is extracted at ambient temperature on all the bandwidth either from one-tone signal harmonic-balance simulation or pulse measurements (short duration with regard to the thermal constant, no self-heating effects), as shown in Fig. 6. The thermal behavior of a device can be predicted by a circuit using the following electrothermal analogy. A impedance temperature corresponds to a voltage and a dissipated power to a current. Thus, the coupling of a thermal model to an electrical model is possible in a circuit simulator. Moreover, the knowlprecisely leads to the operating temperature. Meaedge of surement techniques are difficult to set up because the hot area MAZEAU et al.: BEHAVIORAL THERMAL MODELING FOR MICROWAVE POWER AMPLIFIER DESIGN 2293 is small and not on the top surface of the device. 3-D FE simulation is easier to perform and has proven to be reliable [15], [26]. The thermal system is governed by the following heat equation: (9) is the thermal conductivity, is the temperature, is the volumetric heat generation, is the mass density, and is the specific heat. The FE formulation of (9) leads to the semidiscrete equation defined as Fig. 8. Equivalent volume of the power bar computing process. (10) where the mass matrix and the stiffness matrix are -by- symmetric and positive-definite matrices, is the -by- temperature vector at mesh nodes, and is the -byload vector, which takes into account the power generation and boundary conditions. , the number of nodes, is the order of the FE system. can be represented by an -by- thermal impedance matrix deduced from (10) expressed in the frequency domain (11) For an amplifier, the dimension is large, in the order of several ten thousands, which makes the direct integration of the impedance into a circuit simulator prohibitive. Moreover, it is not useful to keep temperature information for all nodes. Also, once the 3-D model is achieved, a reduction technique of the matrix system must be applied. A. Ritz Vector Approach [15] The Ritz vector approach is powerful for linear problems and assumes the thermal conductivity to be constant. The mean response mode and yield approximations are enhanced of Ritz vecwith the generation of an orthogonal basis tors . Thanks to this new projection basis, the initial problem is transformed into a smaller one. The next step consists of doing an eigendecomposition. The eigenvectors make up the new set of axes corresponding to the diagonal matrix constructed from the corresponding eigenvalues . In the frequency domain, the system becomes .. (12) . allows to pick up temperaAn -by- selection matrix ture nodes among to represent the system. Thus, the reduced thermal impedance is expressed as follows: .. . (13) Fig. 9. Thermal model of the power amplifier. B. Reduced Thermal Model of the HBT Power Amplifier The MMIC power amplifier used in this study is based on an InGaP/GaAs HBT technology with Au thermal drain, and is composed on two amplification stages. In order to apply the reduction order technique with common computation capacity, the order of the 3-D model must be minimized at the beginning of the design. Epitaxial layers and their geometries can be approximated without a significant degradation of the thermal behavior. The second stage of the amplifier mainly influences the transient thermal response. Moreover, if the distance between the amplifier stages is large enough, the thermal coupling between the first and second stage can be neglected in a first approach. Therefore, only the second stage is considered. The dissipated power is supposed to be uniform and localized under the InGaP m volume. An equivalent power emitter finger in a dissipation volume, a “power bar,” is defined for each transistor depending on the finger length , the width , and the length of an equivalent surface, as shown in Fig. 8. is the number of emitter “fingers” in the transistor. Once transistor topology is simplified, the thermal model of the amplifier is computed using symmetrical properties. These power bars lay on a GaAs substrate and a baseplate with volumes, as illustrated in Fig. 9. Many epitaxial layers are neglected. Indeed, their small volume and/or conductivity close to the GaAs value allow this approximation. On the one hand, this model takes into account the InGaP volume of the transistors because its thermal conductivity is ten times smaller than GaAs conductivity and slows down the heat transfer. On the other hand, the Au thermal drain conductivity is ten times bigger than GaAs and makes the heat evacuation easier. In order to solve the heat equation during the FE simulation, two boundary conditions must be given: the dissipated power in the power bar and the baseplate temperature. To apply the reduced-order method, a constant conductivity for the material must be considered. However, for a given baseplate temperature, it is possible to minimize this error thanks to an FE simulation in which thermal conductivities are function of temperature. The solution consists of first computing the average temperature for the main volumes of the amplifier 2294 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 11, NOVEMBER 2007 Fig. 10. Reduced impedance circuit model. TABLE I COMPOSITION OF DATASETS, NUMBER OF WAVELETS, AND MSE FOR EACH FUNCTION OF THE BET MODEL Fig. 11. Dissipated power during a 96-s long pulse. using a nonlinear FE approach. Second, these temperatures are taken as reference temperatures for thermal conductivities and matrices of the linear problem can be extracted. the Generally an interpolation process is more suited for nonlinear reduced models [27]. Once the thermal matrices of the model are extracted, the Ritz vector method is applied. In this study, the reduced impedance model is defined for a baseplate temperature equal to the ambient temperature and only the maximum amplifier operating , only one output temperature). temperature is considered ( The reduced impedance circuit model is integrated in the Agilent ADS circuit simulator through a SPICE netlist (Fig. 10). is the baseplate temperature and is the increase of operating temperature resulting of the self-heating effects. Fig. 12. Operating temperature T during a 96-s-long pulse. IV. INTEGRATION AND RESULTS OF THE BET MODEL The first-order dynamic Volterra series have been interpolated by neural networks according to a description of each kernel in real-imaginary parts. The static kernels are generated from simulation datasets. The dynamic kernels are computed from simple measurements with pulse duration short enough to neglect the self-heating effect (2- s RF pulse). The number of wavelets attributed to these terms, as well as the MSE, is listed in Table I. The BET model has been implemented as a compiled circuit model. The program uses global and local functions to manage the neural model files, the thermal feedback during the envelope transient simulation, and the conversion of the power waves to the electrical voltage and current I/V parameters. A time-domain pulse generator is used as the RF source for the BET model. A. Static Results and Validation As shown in Figs. 11 and 12, the BET model allows to predict during a pulse envelope the evolution of the average dissipated , the maximal temperature , and the RF output power Fig. 13. Output RF power of the power amplifier during a 96-s-long pulse. power (Fig. 13). This information is essential to foresee the size of the cooling system, as well as the performances of the power amplifier. In order to validate the BET model results, measurements of the HBT amplifier have been performed in pulse mode. A long pulse (96 s) is applied for a carrier frequency . Data have been measured during the pulse in three input windows: at the of the pulse. beginning , at the middle , and at the end and Thus, instantaneous modeling performances at time are compared to the measurements, as illustrated by Fig. 14. MAZEAU et al.: BEHAVIORAL THERMAL MODELING FOR MICROWAVE POWER AMPLIFIER DESIGN Fig. 14. Input voltage during a 96-s-long pulse, measurement windows (F 1; F 2; F 3), and reading time (t1; t2; t3) for simulation results. 2295 Fig. 17. Phase drift: comparison between measurements and model phase S 21(F 1) ; F 3 = phase S 21(F 3) F = phase S 21(F 2) phase S 21(F 1) ; modelt2 = phase S 21stat(t2) phase S21stat(t1) and model t3 = phase S 21stat(t3) phase S 21stat(t1) . f f g g0 f f f g0 g g0 f f f g g0 g Fig. 15. Amplifier gain: comparison between measurement (F 1; F 2; F 3) and simulation of BET model (t1; t2; t3, respectively). Fig. 18. Comparison between amplifier gain measurements (dashed lines) and simulation of BET model (continuous lines), P in = 0 dBm (triangle), 10 dBm (circle), 21 dBm (square), Dfreq = freq f . 0 Fig. 16. Amplitude drift: comparison between measurements and model F 2 = S 21(F ) S 21(F 1); F 3 = S 21(F 3) S 21(F 1) and model t2 = S 21stat(t2) S 21stat(t1); t3 = S 21stat(t3) S 21stat(t1). 0 0 0 0 A comparison for the amplitude gain, amplitude drift, and phase drift is presented, respectively, in Figs. 15–17. We can observe a good agreement between the measurements and the transient envelope simulation using the BET model. These results validate the performances of the model. Other long-term memory effects such as the modulation of the bias point can influence the measurement, but the application of a pulse length with regard to the duration of thermal effect establishment supposes that their influence is dominating. As a is considered matter of fact, if the beginning of the pulse without thermal effects, the amplitude drift (Fig. 16) and the phase drift (Fig. 17) can be attributed to transient self-heating. It is important to notice that it is well rendered by the BET model. Fig. 19. Comparison between bias collector current measurements (dashed lines) and simulation of BET model (continuous lines), P in = 0 dBm (triangle), 10 dBm (circle), 21 dBm (square), D freq = freq f . 0 Moreover, the BET model is robust and requires small computational resources: 40 s for three input powers and 11 points in the time domain. B. Dynamic Results and Validation A comparison of the amplitude gain and bias collector current is shown in Figs. 18 and 19 for several carrier frequencies. 2296 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 11, NOVEMBER 2007 A good agreement between measurements and the transient envelope simulation using the BET model can be observed. These results validate the frequency modeling performances. The amplitude difference is mainly due to the static dataset generated from the circuit-level model. Moreover, the BET model is robust and makes use of small computational resources: 90 s for three input powers, five frequencies, and 11 points in the time domain. V. CONCLUSION A BET system-level model for power amplifiers has been presented. The coupling behavioral electrical neural network model with reduced thermal impedances has been implemented into a common circuit simulator (ADS). An envelope transient simulation including the thermal transient feedback has been performed with good numerical convergence and only small computational resources. The method to obtain the BET model is simple because the extraction of the thermal static model only needs simulation datasets. This model includes the dynamic term of the dynamic modified Volterra series and allows to predict the long-term memory effects. The convergence of the BET model is obtained if the characterization of the isothermal . Good behavioral model allows to reach the maximum of agreement between simulations and pulsed measurements at ambient temperature validates this approach and its use for radar applications. Further studies will be dedicated to the baseplate temperature dependency [27] and intermodulation study. The BET approach has been illustrated with an HBT MMIC, but this approach can be applied to any other amplifier technology. ACKNOWLEDGMENT The authors wish to acknowledge AMCAD Engineering, Limoges, France, for measuring the circuit and Agilent Technologies, Massy, France, for providing technical support. REFERENCES [1] O. Jardel, R. Quéré, S. Heckmann, H. Bousbia, D. Barataud, E. Chartier, and D. Floriot, “An electrothermal model for GaInP/GaAs power HBTs with enhanced convergence capabilities,” in 1st Eur. Microw. Integrated Circuits Conf., Manchester, U.K., Sep. 2006, pp. 296–299. [2] M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems, reprint ed. Melbourne, FL: Krieger, 1989. [3] D. Mirri, F. Filicori, G. Iuculano, and G. Pasini, “A new linear dynamic model for performance analysis of large signal amplifiers in communication systems,” in IEEE Instrum. Meas. Conf. Tech. Dig., Venice, Italy, May 1999, pp. 193–197. [4] E. Ngoya, N. Le Gallou, J. M. Nebus, H. Buret, and P. Reig, “Accurate RF and microwave system level modeling of wideband nonlinear circuits,” in IEEE MTT-S Int. Microw. Symp. Dig., Boston, MA, Jun. 2000, vol. 1, pp. 79–82. [5] C. Maziere, T. Reveyrand, S. Mons, D. Barataud, J. M. Nebus, R. Quéré, A. Mallet, L. Lapierre, and J. Sombrin, “A novel behavioral model of power amplifier based on a dynamic envelope gain approach for the system level simulation and design,” in IEEE MTT-S Int. Microw. Symp. Dig., Philadelphia, PA, Jun. 2003, vol. 2, pp. 769–772. [6] C. Florian, F. Filicori, D. Mirri, T. Brazil, and M. 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Paris, France: Sodipe, 1981. [11] H. Wilden, “Microwave tests on prototype-T/R-modules,” in Proc. Radar Syst., Edinburgh, U.K., Oct. 1997, pp. 517–521, Pub. 449. [12] K. J. Joo, E. L. Wilson, and P. Leger, “Ritz vectors and generation criteria for mode superposition analysis,” Earthquake Eng. Struct. Dynam., vol. 18, pp. 149–167, 1989. [13] E. L. Wilson and M. W. Yuan, “Dynamic analysis by direct superposition of Ritz vectors,” Earthquake Eng. Struct. Dynam., vol. 10, no. 6, pp. 813–821, Nov. 1982. [14] J. T. Hsu and L. Vu-Quoc, “A rational formulation of thermal circuit models for electrothermal simulation—Part I: Finite element method,” IEEE Trans. Circuits Syst., vol. 43, no. 9, pp. 721–732, Sep. 1996. [15] R. Sommet, D. Lopez, and R. Quéré, “From 3-D thermal simulation of HBT devices to their thermal model integration into circuit simulators via Ritz vectors reduction technique,” in 8th Intersoc. Thermal Thermomech. Phenomena Electron. Syst. 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Zhang, “Neural-based dynamic modeling of nonlinear microwave circuits,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2769–2780, Dec. 2002. [25] B. O’Brien, J. Dooley, and J. Brazil, “RF power amplifier behavioral modeling using a globally recurrent neural network,” in IEEE MTT-S Int. Microw. Symp. Dig., San Francisco, CA, Jun. 2006, pp. 1089–1092. [Online]. Available: http:hertz.ucd.ie/publications/IMS2006_OBrien.pdf [26] R. Sommet, C. Chang, P. Duéme, and R. Quéré, “Electrothermal models of transistors based on finite element analysis for radar applications,” in 9th Intersoc. Thermal Thermomech. Phenomena Electron. Syst. Conf., Las Vegas, NV, Jun. 2004, pp. 515–522. [27] E. Gatard, R. Sommet, and R. Quéré, “Nonlinear thermal reduced model for power semiconductor devices,” in Proc. 10th Intersoc. Thermal Thermomech. Phenomena Electron. Syst. Conf., San Diego, CA, May 2006, pp. 638–644. MAZEAU et al.: BEHAVIORAL THERMAL MODELING FOR MICROWAVE POWER AMPLIFIER DESIGN Julie Mazeau received the Master’s degree in high-frequency and optical telecommunications from the University of Limoges, Limoges, France, in 2003, and is currently working toward the Ph.D. degree at the Research Institute XLIM, University of Limoges (in collaboration with THALES Airborne Systems, Elancourt, France). Her research interests are dedicated to electrothermal system-level models of power amplifiers for radar applications. Raphaeël Sommet received the French Aggregation in Applied Physics degree and Ph.D. degree from the University of Limoges, Limoges, France, in 1991 and 1996, respectively. Since 1997, he has been a Permanent Researcher with the C2S2 team “Nonlinear Microwave Circuits and Subsystems,” XLIM Research Institute, Centre National de la Recherche Scientifique (CNRS), University of Limoges. His research interests concern HBT device simulation, 3-D thermal FE simulation, model-order reduction, microwave circuit simulation, and generally the coupling of all physics-based simulation with circuit simulation. Daniel Caban-Chastas received the Diploma degree in engineering of electronics and microwaves systems from the University of Pierre and Marie Curie, Paris, France, in 2001. In 2002, he joined the Advanced Technologies Team, THALES Airborne Systems, Elancourt, France, where he is an Electromagnetic Simulation Expert currently in charge of microwave design for active antenna transmit/receive (T/R) modules. He is also involved in microwave packaging and new microwave development. View publication stats 2297 Emmanuel Gatard received the Electronics and Telecommunications Engineering degree from Ecole Nationale Supérieure d’Ingénieurs de Limoges (ENSIL), Limoges, France, in 2003, and the Ph.D. degree in electrical engineering from the University of Limoges, Limoges, France, in 2006. He is currently with the XLIM Research Institute, Centre National de la Recherche Scientifique (CNRS), University of Limoges. His research interests concern electrothermal modeling of power semiconductor devices dedicated to circuit simulations, physics-based electron device simulations, and nonlinear thermal modeling. Raymond Quéré (M’88–SM’99) received the Electrical Engineering degree and French Aggregation degree in physics from ENSEEIHT–Toulouse, Toulouse, France, in 1976 and 1978, respectively, and the Ph.D. degree in electrical engineering from the University of Limoges, Limoges, France in 1989. In 1992, he became a Full Professor with the University of Limoges, where he currently heads the research group on high-frequency nonlinear circuits and systems with the XLIM Research Institute, Centre National de la Recherche Scientifique (CNRS), University of Limoges. He is mainly involved in nonlinear stability analysis of microwave circuits. Prof. Quéré is a member of Technical Program Committee for several conferences. He was the chairman of European Microwave Week, Paris, France, 2005. Yves Mancuso received the Diploma of engineer degree from the Ecole Nationale Supérieure de Génie Physique, Grenoble, France, in 1979. In 1981, he joined THALES, where he was in charge of different technological developments for T/R modules, MMICs, packaging, and test benches. From 1993 to 1997, he was mainly in charge of the T/R module for two major phased-array programs: a European one, AMSAR, and a French space program. He is currently involved with phased-array antennas and T/R modules design authority for THALES Airborne Systems, Elancourt, France, including airborne and space, radar and electronic warfare applications and new microwave developments including microwave components, circuits, and technology.