LTspice Implementation of Gyrator-Capacitor Magnetic Circuit Model Considering Losses and Magnetic Saturation for Transient Simulations of Switching Mode Power Supplies Utilizing Inductive Elements with Cores Made of Amorphous Alloys

LTspice Implementation of Gyrator-Capacitor Magnetic Circuit Model Considering Losses and Magnetic Saturation for Transient Simulations of Switching Mode Power Supplies Utilizing Inductive Elements with Cores Made of Amorphous Alloys Roman Szewczyk1(B) , Oleg Petruk1 , Michał Nowicki1 , Anna Ostaszewska-Liżewska1 , Aleksandra Kolano-Burian2 , Piotr Gazda1 , Adam Bieńkowski3 , Paweł Nowak1 , and Tomasz Charubin1 1 ŁUKASIEWICZ Research Network – Industrial Research Institute for Automation and Measurements PIAP, Al. Jerozolimskie 202, 02-486 Warsaw, Poland rszewczyk@onet.pl 2 ŁUKASIEWICZ Research Network – Institute of Non-Ferrous Metals, Sowińskiego 5, 44-100 Gliwice, Poland 3 Faculty of Mechatronics, Warsaw University of Technology, św. A. Boboli 8, 02-525 Warsaw, Poland Abstract. Modelling the transient simulation of magnetic circuits is the crucial process for development of modern switching mode power converters. Such models are especially important for newly developed magnetic materials, such as amorphous alloys. Paper presents the gyrator-capacitor model of inductive element with core made of high permeability, Co-based amorphous alloy. Parameters of the model were identified during Nedler-Mead method based optimization process. Presented results of modelling confirm that model can be used for SPICE simulations; however, the accuracy of losses modeling is limited in the wide range of frequencies. Keywords: Gyrator-capacitor model · SPICE modelling · Amorphous alloys 1 Introduction Electronic components with inductive cores, such as coils and transformers, play important role in switching mode power converters [1]. Development of switching mode power converters is the crucial part for development of electromobility systems [2], such as Polish electric car IZERA [3] or mobile robots. As a result, recently we observe increasing interest in optimization of power conversion systems and development of magnetic materials for cores of inductive components. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 R. Szewczyk et al. (Eds.): AUTOMATION 2021, AISC 1390, pp. 416–424, 2021. https://doi.org/10.1007/978-3-030-74893-7_37 LTspice Implementation of Gyrator-Capacitor Magnetic Circuit Model 417 Efficient development of power conversion systems requires efficient optimization during the development process. However, in spite of significant efforts, cores of inductive components seem to be the most difficult part from the point of view of accurate modelling. Such cores are highly nonlinear due to magnetic saturation. Moreover, magnetic hysteresis occurs in inductive cores, what makes modelling even more sophisticated. The lack of accurate models is especially visible considering newly developed soft magnetic materials, such as amorphous alloys [4]. In spite of theoretical efforts to for development of models describing magnetic hysteresis [5], the model of inductive component with core made of such material, suitable for transient simulations required in practical application, seems to be not previously presented. Paper is trying to fill this gap. For efficient modelling of inductor with core made of Co-based amorphous alloy, gyrator-capacitor [6] model was adapted. Proposed model was implemented in LTspice environment. It should be highlighted, that LTspice is considered as the most efficient, freely available modelling tool for switching mode power conversion modelling [7]. As a result, proposed model creates new possibilities in efficient and accurate modelling of inductive cores for switching mode power converters development. 2 Gyrator Concept The concept of gyrator was first introduced in 1948 by Bernard D. H. Tellegen [8]. Gyrator is linear and loss-less element, which can invert the current–voltage characteristic of electrical network. From the practical point of view, gyrators can be developed utilizing two operational amplifiers [9]. As a result, the use of gyrator enables substitution of real inductive component (such as inductor) by gyrator and the electric network including capacitor and resistance. Such circuit was proposed by David C. Hamil in 1993 [6], creating new possibilities of transient simulation of circuits including inductive components. Figure 1 presents the symbol of gyrator proposed in 1948 by Bernard D. H. Tellegen (Fig. 1a) as well as practical implementation of gyrator in LTspice environment (Fig. 1b). This implementation utilizes two voltage controlled current sources as well as two resistors with resistance 100 G. These resistors don’t influence on gyrator’s characteristics, however, they increase the numerical stability of the gyrator’s model. The operation of gyrator is determined by the following equations [10]: i1 = N · v2 (1) i2 = −N · v1 (2) where i1 , v1 are respectively the current and voltage on the left side of gyrator, and i2 , v2 are respectively the current and voltage on the right side of gyrator. 418 R. Szewczyk et al. a) b) Fig. 1. Gyrator: a) symbol, b) LTspice implementation consisting two voltage dependent current sources 3 Implementation of Gyrator-Capacitor Model of Inductive Component The gyrator-capacitor model of real inductor proposed by D. C. Hamill [6] and developed by M. Eaton [11] in 1994 consists capacitor C, nonlinear voltage source V B and resistor RL . Capacitance of capacitor C is given by the following equation [6]: C= μo μr Se le (3) where μ0 is magnetic constant, µr is permeability of magnetic core, S e is effective cross-section of magnetic core and le is magnetic path length. Voltage on nonlinear voltage source V B reproduces saturation in nonlinear characteristic of inductive core. Voltage V B is dependent on potential difference on capacitor C and is given by following equation [11]: VB (Vc ) = b · (Vc )n (4) where b is parameter determining the location of saturation in core’s magnetization curve and n is the odd integer value. Magnetic core’s losses are represented by resistor RL . It should be highlighted that value of loses increases with frequency what is in line with physical processes observed in the magnetic core of inductive element. The LTspice implementation of described gyrator-capacitor model is presented in Fig. 2. Parameters of the model are stored in separate variables, what clarifies the implementation and simplifies the variable update process. Figure 3 presents the model of simple hysteresis graph enabling verification of the proposed model and identification of its parameters. Proposed hysteresis graph consists of arbitrary controlled sine wave current source, implementation of gyrator-capacitor model of coil as well as instrumental amplifier with amplification k = 1 and integrator with Laplace transmittance 1/s. As a result, for given sine wave current amplitude a with frequency f , the magnetic B(H) hysteresis loop can be plotted as dependence of variable LTspice Implementation of Gyrator-Capacitor Magnetic Circuit Model 419 Fig. 2. LTspice implementation of the gyrator-capacitor model of inductive coil considering material permeability (described by parameter “mi”) as well as both saturation (described by parameter “b”) flux density and loses (described by parameter “RL”). Two 100 G resistors added for numerical stability of the model B on variable H. Both variables are given in volts, whereas they are scaled in Tesla (T) and amperes per meter (A/m). Proposed model of inductive component may be easily integrated with OCTAVE [12], the open-source MATLAB alternative. During such integration, OCTAVE script may modify model’s parameters presented text values in.asc file describing electronic circuit. Next, LTspice can work as external solver working in text mode. Results of transient simulation may be acquired by OCTAVE using LTspice2OCTAVE.m function proposed by Marcin Mleczko [13]. 4 Accuracy of the Model Parameters of presented gyrator-capacitor model for ring-shaped core made of Co68 Fe4 B13 Si13.5 Mo1.5 amorphous alloys in as-quenched state [14] were identified during the Nedler-Mead method-based optimization process. The optimization target function was defined as: n (5) T100 = ( Bmeas (i)|100Hz − Bsim (i)|100Hz )2 i=1 where Bmeas (i)|100Hz are the results of measurements carried out at frequency f = 100 Hz, whereas Bsim (i)|100Hz were the results of simulation. The results of modelling are presented in Fig. 4. Parameters of the model obtained in optimization process are presented in Table 1. Results of simulation are in good agreement with the results of measurements. This agreement is quantitatively confirmed by coefficient of determination R2 = 0.9955. 420 R. Szewczyk et al. Fig. 3. LTspice implementation of the gyrator-capacitor model of inductive coil connected to a model of simple hysteresis graph consisting of current source, instrumental amplifier (amplification k = 1) and integrator (with transmittance L(s)= 1/s) LTspice Implementation of Gyrator-Capacitor Magnetic Circuit Model 421 Fig. 4. The results of modelling of B(H) characteristics of ring-shaped core made of Co68 Fe4 B13 Si13.5 Mo1.5 amorphous alloys in as-quenched state, at frequency f = 100 Hz (red line) in comparison to the results of measurements at the same frequency (black line) On the other hand, the quality of modelling decreases for modelling in significantly different frequency. To minimize this unfavorable effect, the optimization was carried out for modified target function: n T100,1000 = ( Bmeas (i)|100Hz − Bsim (i)|100Hz )2 + . . . i=1 (6) n + ( Bmeas (i)|1000Hz − Bsim (i)|1000Hz )2 i=1 where Bmeas (i)|1000Hz and Bsim (i)|1000Hz are the results of measurements and simulation carried out at frequency f = 1000 Hz. The results of such enhanced modelling are presented in Fig. 5. Also, for a simultaneous simulation for two frequencies, results of modelling are in moderately good agreement with the results of measurements confirmed by coefficient of determination R2 = 0. 9456 and R2 = 0.9562 for frequencies f = 100 Hz and f = 1000 Hz respectively. The limited quality of modelling over the different frequencies clearly indicates that the magnetic core losses model described by single resistor is not sufficient. Core loss model should take into account eddy current losses, anomalous loses and losses due to magnetic hysteresis. These losses, different in their nature, should be described by different loss elements in gyrator-capacitor circuit. 422 R. Szewczyk et al. a) b) Fig. 5. The results of modelling of B(H) characteristics of ring-shaped core made of Co68 Fe4 B13 Si13.5 Mo1.5 amorphous alloys in as-quenched state (red line) in comparison to the results of measurements (black line) at frequencies: a) f = 100 Hz, b) f = 1000 Hz Table 1. Parameters of the gyrator-capacitor model determined in Nedler-Mead method-based optimization process Parameter Description for f = 100 Hz for f = 100 Hz and f = 1000 Hz μ Represents magnetic permeability μ of the core 17 743 12 933 b Describe saturation of the magnetic core n 3 522 124.5 11 11 RL (Ω ) Describes losses in the core 58.53 14.58 R2 f = 100 Hz Coefficient of determination 0.9955 0.9456 R2 f = 1000 Hz Coefficient of determination n.a 0.9562 LTspice Implementation of Gyrator-Capacitor Magnetic Circuit Model 423 5 Conclusions Results of modelling presented in the paper clearly indicate that gyrator-capacitor model has a great potential for modelling the transient states in switching mode power supplies. Proposed solution reproduces moderately well the magnetic characteristics of inductive elements with core made of Co68 Fe4 B13 Si13.5 Mo1.5 amorphous alloys in as-quenched state. The quality of modelling is confirmed by coefficient of determination exceeding 0.94 in all cases. Such quality is sufficient for engineering modelling of switching mode power supplies with inductive elements with cores made of modern soft magnetic materials. On the other hand presented results clearly indicate the need of further development of presented gyrator-capacitor model. This development should be focused on improvement of nonlinearity description in the model (considering recent works on development of models of anhysteretic magnetization curves [15]) as well as on taking into account the physical background of losses occurring in cores of inductive elements, such as eddy current losses, anomalous loses and losses due to magnetic hysteresis. This work was partially realized within research project No. 2/Ł-IMN/CŁ/2020 entitled “Specialized high-performance energy conversion systems for electromobility and robotics applications” (SPECKON) co-financed by the targeted grant of the Łukasiewicz Research Network. References 1. Tse, C.: Complex Behavior of Switching Power Converters. CRC Press, Boca Raton (2004) 2. Hayes, J., Goodarzi, G.: Electric Powertrain : Energy Systems, Power Electronics and Drives for Hybrid, Electric and Fuel Cell Vehicles. John Wiley & Sons, Incorporated, Newark (2017) 3. https://izera.com/ 4. Hasegawa, R.: Applications of amorphous magnetic alloys in electronic devices. J. Non-Cryst. Solids 287, 405–412 (2001). https://doi.org/10.1016/S0022-3093(01)00633-0 5. Liorzou, F., Phelps, B., Atherton, D.: Macroscopic models of magnetization. IEEE Trans. Magn. 36(2), 418–428 (2000). https://doi.org/10.1109/20.825802 6. Hamill, D.: Gyrator-capacitor modeling: a better way of understanding magnetic components. 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