Finite-Element-Based Optimal Design Approach for High-Voltage ...

IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 43, NO. 6, JUNE 2015

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Finite-Element-Based Optimal Design Approach for High-Voltage Pulse Transformers Sylvain Candolfi, Member, IEEE, Philippe Viarouge, Davide Aguglia, Member, IEEE, and Jérôme Cros Abstract— This paper presents an optimal design methodology of monolithic high-voltage (HV) pulse transformers based on the direct 2-D finite-element analysis (FEA) identification of the electrical equivalent circuit parameters without any simplified analytical modeling. This method is applied to the preliminary optimal design of the monolithic HV pulse transformer for the future compact linear collider modulators under study at CERN. The feasibility of such a transformer with tight specifications is demonstrated. The predicted performance obtained with the direct 2-D FEA optimization process is validated by 3-D FEA simulation. The dimensioning model based on the direct 2-D FEA identification of the electrical equivalent circuit parameters is validated on a prototype.

Fig. 1.

Klystron modulator using a monolithic pulse transformer.

Fig. 2.

Oil-immersed HV pulse transformer structure.

Index Terms— Design optimization, high-voltage techniques, klystrons, pulse transformers.

I. I NTRODUCTION

H

IGH-VOLTAGE (HV) pulse transformers are widely used in klystron modulators. For these topologies, the quality of the modulator output voltage is strongly dependent on the pulse transformer performance. An optimal design process is a necessity, particularly when the specifications of the output voltage are tight such as in the case of the compact linear collider (CLIC) klystron modulator under study at CERN [1]. This paper presents the development of a CAD environment using a direct nonlinear optimization design procedure and a pulse transformer electromagnetic dimensioning model based on 2-D finite-element analysis (FEA) only to get rid of the usual simplified analytical modeling used in [3], [4], [10], [12], and [14]. For each iteration of the nonlinear optimization process, the transformer parameters used to compute the design objective and constraints functions are directly derived from the 2-D FEA dimensioning model. II. H IGH -VOLTAGE P ULSE T RANSFORMERS IN K LYSTRON M ODULATORS The HV pulse transformer to be designed is used in a solid-state klystron modulator presented in Fig. 1. In such a Manuscript received October 31, 2013; revised May 8, 2014, September 13, 2014, and November 7, 2014; accepted March 31, 2015. Date of publication May 1, 2015; date of current version June 8, 2015. This work was supported in part by European Organization for Nuclear Research, Geneva, Switzerland, and in part by the Conseil de Recherches en Sciences Naturelles et en Génie du Canada. S. Candolfi, P. Viarouge, and J. Cros are with the Laboratory of Electro-Technology, Power Electronics and Industrial Control, Department of Electrical and Computer Engineering, Laval University, Quebec, QC G1V 0A6, Canada (e-mail: [email protected]; [email protected]; [email protected]). D. Aguglia is with the Electric Power Converter Group, Department of Technology, European Organization for Nuclear Research, Geneva 1211, Switzerland (e-mail: [email protected]). 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/TPS.2015.2423234

topology, a capacitor bank is charged around 10 kV and this voltage is applied to the pulse transformer primary winding by use of a solid-state switch. According to the transformer ratio, a 150-kV output voltage pulse is delivered to the klystron. The bouncer is an auxiliary power converter that is compensating the capacitor voltage droop during its discharge in order to control the output pulse flat top [1], [2]. At the end of the pulse, the core magnetizing energy is absorbed by a demagnetizing circuit connected on the primary side. Several structures of such monolithic HV pulse transformers can be found in [3], [10], [12], and [14]. In this paper, the optimal design methodology using a direct 2-D FEA optimization process is applied to a conventional transformer topology using a laminated core, two primary windings placed on each core leg and connected in parallel, two secondary HV windings wound above the primary windings and connected in series [4]. It is presented in Fig. 2. An auxiliary winding supplied by a dc current source is also implemented to provide a core premagnetization with a negative flux density before the application of the voltage pulse. With such an arrangement, a bipolar magnetic flux excursion is performed during the application of the unipolar voltage pulse and the transformer core mass and volume can be minimized during the design process [3], [4].

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Fig. 3.


Fig. 5.

HV pulse transformer generic dimensioning model.

Fig. 6.

HV pulse transformer dimensioning model using 2-D FEA.

Electrical equivalent circuit of HV pulse transformer.

Fig. 4. HV pulse transformer electromagnetic dimensioning model decoupled in two separate electrostatic and magnetostatic models.

III. H IGH -VOLTAGE P ULSE T RANSFORMER D IMENSIONING M ODEL A. HV Pulse Transformer Design Problem High performance klystron modulators impose tight design specifications to the monolithic HV pulse transformer [1], [4], [10]. With large flat pulsewidths and high pulse repetition rates, the pulse rise and fall times must be minimized to improve the total power consumption of the modulator. To perform such design objectives in terms of output pulse quality, it is mandatory to develop an accurate dimensioning model of the HV transformer leakage inductances and parasitic capacitances in order to control its dynamics during the optimal design process [4], [13]. B. HV Pulse Transformer Equivalent Circuit The lumped-parameter equivalent circuit of Fig. 3 is derived from a conventional transformer model with distributed capacitance [3]–[5]. If the propagation phenomenon is neglected, the transformer electromagnetic model is decoupled in two separate models: 1) an electrostatic model used to directly identify the capacitances by 2-D FEA electrostatics and 2) a magnetostatic model used to directly identify the inductances by 2-D FEA magnetostatics [4], [5]. With such a modeling approach, a 2-D FEA dimensioning model of all the electrical equivalent circuit is established if a linear-distributed voltage in windings is assumed (Fig. 4). C. HV Pulse Transformer Generic Dimensioning Model The transformer dimensioning model can be derived from the electrical equivalent circuit and the determination of

its components in terms of dimensions and material data. A generic dimensioning model of the monolithic HV pulse transformer is presented in Fig. 5. It includes the preceding electromagnetic dimensioning model, a simplified thermal modeling tool and a simple mechanical modeling tool used to evaluate the structural behavior of the transformer operated in pulsed mode. The output voltage pulse is determined by simulation of the modulator electrical modeling that is including a transformer equivalent circuit derived from the electromagnetic dimensioning model. IV. O PTIMAL D ESIGN OF H IGH -VOLTAGE P ULSE T RANSFORMER U SING 2-D F INITE -E LEMENT A NALYSIS In the case of HV transformers, it is difficult to obtain an analytical electromagnetic dimensioning model that is sufficiently accurate to perform an optimal design of the transformer in terms of dynamical performances. In a previous design approach developed by Viarouge et al. [4], field computation techniques based on 2-D FEA have already been implemented in the CAD environment but only to validate or correct the analytical electromagnetic dimensioning model. In this paper, a new efficient optimal design methodology is presented that is using a direct 2-D FEA electromagnetic dimensioning model during the transformer optimization process. The dimensioning model based on 2-D FEA only is presented in Fig. 6. The 2-D finite-element solver FEMM4.2 [7] controlled from Excel via a script is used for an automatic computation of the magnetic and electrical fields. The simulated experiment concept has been applied to perform direct FEA identification tests of the electrical equivalent circuit components. With such an approach, no simplified analytical expressions of the inductances and capacitances are used.

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Fig. 7. Description of the operation conditions of the three FEA identification tests. (a) V1 = V1n , V2 = 0 W2cc . (b) V1 = 0, V2 = V2n W1cc . (c) V1 = V1n , V2 = V2n W122 .

A. 2-D FEA Identification of Transformer Inductances The following are the two FEA identification tests based on magnetostatic field computations that are performed for the determination of the magnetizing and leakage inductances. 1) Open Secondary FEA Test: The magnetizing inductance is derived from the FEA computation of the magnetic energy stored during transformer no-load operation. 2) Short-Circuit FEA Test: The total leakage inductance is derived from the FEA computation of the magnetic energy stored during transformer short-circuit operation. B. 2-D FEA Identification of Transformer Capacitances 2-D electrostatic field computations have been performed to identify the three lumped capacitors of the electrical equivalent circuit of Fig. 3. The three lumped capacitors C11 , C22 & C12 can be derived from the electrostatic FEA simulations of three different tests presented in Fig. 7(a)–(c): 1) Secondary short circuited & grounded (V2n = 0), 2) Primary short circuited & grounded (V1n = 0), 3) Primary & secondary with V2n linear voltage distribution. One can compute the three capacitances from the three values W2cc , W1cc , and W122 of total electrical energy derived from the electrostatic FEA simulations 2W2cc 2W1cc C22 + C12 = C11 + C12 = 2 V1 V22 V12 C11 + V22 C22 + (V2 − V1 )2 C12 = 2W122 .

(1)

Fig. 8. Distribution of voltage in pulse transformer tank for the three identification tests. (a) V1 = V1n , V2 = 0 W2cc . (b) V1 = 0, V2 = V2n W1cc . (c) V1 = V1n , V2 = V2n W122 .

The distributions of voltage in pulse transformer tank corresponding to the three identification tests are presented in Fig. 8(a)–(c). When a linear vertical rated voltage distribution is imposed along the height of the primary and secondary coils, one obtains the distribution of Fig. 8(c). C. High-Voltage Pulse Transformer Design With Global Optimization Approach Using 2-D FEA Fig. 9 presents the structure of the final CAD environment with the global optimization approach using the previous 2-D FEA electromagnetic dimensioning model. A nonlinear constrained optimization procedure based on the generalized reduced gradient method is used to find the optimal design solution [6]. The dimensions of the structure (core and coils) and the primary turn number are the state variables of the optimization problem. They are used to launch directly the described 2-D FEA identification techniques in order to directly derive the equivalent circuit parameters of the transformer. In the analytical electromagnetic dimensioning model, these parameters were derived from explicit simplified

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TABLE I E LECTRICAL C IRCUIT PARAMETERS OF THE P ROTOTYPE I DENTIFIED BY 2-D FEA AND BY E XPERIMENTAL M EASUREMENTS

Fig. 9. HV pulse transformer design with global optimization approach using 2-D FEA.

Fig. 11.

Definition of pulse characteristics.

experimentally derive the parameters of the transformer electrical equivalent circuit. The results are listed in Table I. One can verify that the prediction model is in accordance with the measurements. Fig. 10.

Validation prototype identified by 2-D FEA and experimentally.

VI. E XAMPLE OF P ULSE T RANSFORMER O PTIMIZATION analytical expressions of the state variables [4]. In the 2-D FEA electromagnetic dimensioning model developed in this paper, these parameters are directly derived from the state variables by the FEA identification procedures. With the efficient generalized reduced gradient (GRG2) algorithm implemented in the built-in Excel solver add-in, one can get rid of the FEA mesh discretization influence on the convergence of the optimization process. Several optimization scenarios using different design objectives and constraint functions can be easily tested by the designer. The CAD environment is versatile.

The proposed methodology has been applied to investigate the preliminary pulse transformer design for the CLIC klystron modulator under study at CERN [1], [3]. The CLIC will need 300 MW of average power feeding 1638 klystron modulators with tight dynamical specifications in terms of rise time for a relatively long flat-top width (Fig. 11). The efficiency must be high enough to minimize the total power consumption of the accelerator complex. The modulators will be operated at a 150 kV/160 A (24 MW) in pulsed mode. A. Preliminary CLIC Pulse Transformer Optimal Design

V. VALIDATION OF THE 2-D FEA I DENTIFICATION M ETHOD The proposed 2-D FEA identification method has been validated on a transformer prototype with one primary and one secondary winding only placed on the same leg of a ferrites magnetic core (Fig. 10). On the one hand, the parameters of the transformer electrical equivalent circuit have been determined by the proposed 2-D FEA identification method. On the other hand, a specific experimental identification method also developed in [8] has been used to

The constraints of the optimization problem listed in Table II are the specifications of the modulator, the transformer overall dimensions, its copper losses and the characteristics of the magnetic and insulation materials used. The following are the three state variables of the transformer design optimization problem: 1) the primary turn number; 2) the height of the coils; 3) the core width.

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TABLE II S PECIFICATIONS AND C ONSTRAINTS OF CLIC P ULSE T RANSFORMER O PTIMAL D ESIGN

TABLE III C HARACTERISTICS OF O PTIMAL CLIC T RANSFORMER C ONFIGURATIONS

Fig. 12. Comparison of magnetic flux density distribution for short-circuit operation (a) 2-D FEA and (b) 3-D FEA. For 3-D results, coils on the right are occulted to show the field magnitude on vertical and horizontal planes.

TABLE IV C HARACTERISTICS OF O PTIMAL CLIC T RANSFORMER C ONFIGURATION W ITH C ONSTANT S PACE B ETWEEN P RIMARY AND

S ECONDARY C OILS

Some parameters are fixed by the material physical properties and characteristics. They can be eventually changed in other optimization scenarios to perform a sensitivity analysis of their influence on the optimal design. The different pulse specifications, the maximal saturation flux density in the core and the acceptable limits of the overall transformer dimensions are set as inequality constraints of the optimization problem. The objective function to minimize is the total weight of the transformer active parts excluding the oil in the tank. Two transformer configurations have been optimized one configuration with: 1) a constant space between the primary and secondary coils and 2) a conical shape of the secondary windings. The optimization procedure was operated in multistart mode to avoid the convergence on local minima of the objective function in the space of feasible solutions bounded by the constraints. The feasibility of the CLIC monolithic pulse transformer with the imposed specifications has been demonstrated. All the constraints including the voltage rise time and voltage overshoot specifications are respected for the optimal solutions found for both configurations under study. The optimal transformer weights are listed in Table III. The conical winding configuration complicates the transformer production process, but it does not significatively increase the performance.

Fig. 13. Comparison of electrical field density distribution for the case V1 = 0 and V2 = 150 kV (a) 2-D FEA and (b) 3-D FEA. For 3-D results, coils on the right are occulted to show the field magnitude on vertical and horizontal planes.

B. Validation of the Optimal Design by 3-D FEA Simulation The optimal solution of the first transformer configuration was simulated with a 3-D FEA code to further investigate the influence of the 2-D FEA assumptions used in the proposed design methodology. The results of the 2-D and 3-D FEA simulations in terms of equivalent circuit parameters and voltage pulse characteristics can be compared in Table IV. One can notice that the difference on the pulse rise time and overshoot are 3% and 0% only. This is an important result for the validation of the use of the 2-D assumption in the proposed transformer optimization process because time consuming 3-D FEA simulations are not well adapted to an iterative process. The error on the rise time is low despite the fact that the most important errors related to the 2-D assumption

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occur on the capacitance values. The rise time depends on both the capacitance and inductance values and in the absence of overshoot imposed by the specifications constraints, the errors on the inductance and capacitance values partially compensate each other in the computation of the rise time. Both 3-D and 2-D models are then in accordance when no pulse overshoot is allowed. The magnetic flux density distribution for short-circuit operation (Fig. 12) and the electrical field density distribution for the test V1 = 0 and V2 = 150 kV (Fig. 13) with 2-D FEA and 3-D FEA can be compared. On the plane where 2-D FEA is performed, one can notice that the results obtained by 2-D and 3-D FEA are in accordance. The differences between 2-D and 3-D FEA mainly occur in the windings corners where one can see that the magnetic field is more homogeneous than the electrostatic field. Consequently, the error on the capacitances is larger than the error on the leakage inductance. There is no significative difference between 2-D and 3-D approach for the magnetizing inductance estimation.

[11] M. Akemoto, S. Gold, A. Krasnykh, and R. Koontz, “Development of the pulse transformer for NLC klystron pulse modulator,” in Proc. Particle Accel. Conf., vol. 1. May 1997, pp. 1322–1324. [12] D. Habibinia and M. R. Feyzi, “Optimal winding design of a pulse transformer considering parasitic capacitance effect to reach best rise time and overshoot,” IEEE Trans. Dielectr. Electr. Insul., vol. 21, no. 3, pp. 1350–1359, Jun. 2014. [13] D. Bortis, J. Biela, and J. W. Kolar, “Transient behaviour of solid state modulators with matrix transformers,” in Proc. IEEE PPC, Washington, DC, USA, Jun./Jul. 2009, pp. 1396–1401. [14] Y. Wang, M. Li, K. Li, and G. Zhang, “Optimal design and experimental study of pulse transformers with fast rise time and large pulse duration,” IEEE Trans. Plasma Sci., vol. 42, no. 2, pp. 300–306, Feb. 2014.

VII. C ONCLUSION

Philippe Viarouge was born in périgueux, France, in 1954. He received the Engineering, D.Eng., and H.D.R. degrees from the Institut National Polytechnique de Toulouse, Toulouse, France, in 1976, 1979, and 1992, respectively. He was a Project Associate with CERN, Geneva, Switzerland, from 2009 to 2010. He has been a Professor with the Department of Electrical Engineering, Laval University, Quebec, QC, Canada, since 1979. His current research interests include the design and modeling of electrical machines and medium frequency magnetic components, ac drives, and power electronics.

An efficient optimal design methodology of monolithic HV pulse transformers based on the direct 2-D FEA identification of the electrical equivalent circuit parameters has been presented. It can be used for a wide range of pulse transformer applications. This method has been validated on a prototype of ferrites pulse transformer and successfully applied to the preliminary optimal design of the monolithic pulse transformer for the future CLIC modulators under study at CERN. The feasibility of such a transformer with tight specifications has been demonstrated. The predicted performances obtained with the direct 2-D FEA optimization process have been compared and validated by 3-D FEA simulation. R EFERENCES [1] D. Aguglia et al., “Klystron modulator technology challenges for the compact linear collider (CLIC),” in Proc. IEEE Pulsed Power Conf. (PPC), Chicago, IL, USA, Jun. 2011, pp. 1413–1421. [2] C. A. Martins, F. Bordry, and G. Simonet, “A solid state 100 kV long pulse generator for klystrons power supply,” in Proc. 13th Eur. Conf. Power Electron. Appl. (EPE), Barcelona, Spain, Sep. 2009, pp. 1–10. [3] G. N. Glasoe and J. V. Lebacqz, Pulse Generators. New York, NY, USA: Dover, 1965. [4] P. Viarouge, D. Aguglia, C. A. Martins, and J. Cros, “Modeling and dimensioning of high voltage pulse transformers for klystron modulators,” in Proc. 20th Int. Conf. Elect. Mach. (ICEM), Sep. 2012, pp. 2332–2338. [5] D. Aguglia, P. Viarouge, and C. A. Martins, “Frequency domain non-linear identification method for high voltage pulse transformers,” in Proc. 20th Int. Conf. Elect. Mach. (ICEM), Sep. 2012, pp. 1977–1983. [6] L. S. Lasdon, A. D. Waren, A. Jain, and M. Ratner, “Design and testing of a generalized reduced gradient code for nonlinear programming,” ACM Trans. Math. Softw., vol. 4, no. 1, pp. 34–49, 1978. [7] D. Meeker. (Oct. 1, 2011). Finite Element Method Magnetics, Version 4.2. [Online]. Available: http://www.femm.info [8] S. Candolfi, D. Aguglia, P. Viarouge, and J. Cros, “Efficient parametric identification method for high voltage pulse transformers,” in Proc. 19th IEEE Pulsed Power Conf. (PPC), Jun. 2013, pp. 1–6. [9] E. Accomando et al., “Physics at the CLIC multi-TeV linear collider,” CERN, Meyrin, Switzerland, Tech. Rep. CERN-2004-005, 2004. [10] D. Bortis, J. Biela, G. Ortiz, and J. W. Kolar, “Design procedure for compact pulse transformers with rectangular pulse shape and fast rise times,” IEEE Trans. Dielectr. Electr. Insul., vol. 18, no. 4, pp. 1171–1180, Aug. 2011.

Sylvain Candolfi (M’13) was born in Switzerland in 1988. He received the B.Sc. and M.Sc degrees from the École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, in 2009 and 2012, respectively. He is currently pursuing the Ph.D. degree with Laval University, Quebec, QC, Canada, with a focus on pulse transformer design for klystron modulators in collaboration with CERN, Geneva, Switzerland.

Davide Aguglia (S’06–M’08) was born in Switzerland in 1979. He received the Diploma degree from the University of Applied Sciences of Western Switzerland, Fribourg, Switzerland, in 2002, and the M.Sc. and Ph.D. degrees from Laval University, Quebec, QC, Canada, in 2004 and 2010, respectively, all in electrical engineering. He was an External Consultant for a private company in magnetic components design from 2009 to 2010. He has been with the Electric Power Converter Group, Department of Technology, European Organization for Nuclear Research (CERN), Geneva, Switzerland, since 2008, where he has been responsible for fast pulsed converter design and projects management. He has also been an Associate Professor with Laval University since 2011, where he is co-directing Ph.D. students. He is currently the Head of the Fast Pulsed Converter Section with the Electric Power Converter Group, CERN. He leads the International Research and Development Program on klystron modulators design for the next-generation of particle accelerators. His current research interests include power electronics systems design and high-voltage engineering for particle accelerators, and electrical machines and renewable energy systems design.

Jérôme Cros was born in Millau, France, in 1964. He received the Engineering and D.Eng. degrees from the Institut National Polytechnique de Toulouse, Toulouse, France, in 1992. He has been a Professor with the Department of Electrical Engineering, Laval University, Quebec, QC, Canada, since 1995. His current research interests include field calculation, ac drives, and electrical machine design.