Models for the simulation of electronic circuits with hysteretic inductors

Transcription

1 Proeedings of the 5th WSEAS Int. Conf. on Miroeletronis, Nanoeletronis, Optoeletronis, Prague, Czeh Republi, Marh 12-14, 26 (pp86-91) Models for the simulation of eletroni iruits with hystereti indutors PETRU ANDREI Department of Eletrial and Computer Engineering Florida State University and Florida A&M University 2525 Pottsdamer St., Tallahassee FL 3231 USA Abstrat: - Time-dependent magnetization proesses in ferriti materials are desribed by using a new dynami Preisah-type model. This model is based on the mean-field approximation and has the advantage that it an be easily implemented in eletroni iruit appliations. The numerial implementation of the new model is disussed in detail. Sample numerial results obtained for resistor-indutor and resistor-indutorapaitor iruits operating at different frequenies are presented and ompared with experimental data. Key-Words: - hystereti indutors, Preisah model, eletrial iruits. 1 Introdution Preisah-type models (PM) are widely used in the literature for the desription of magneti hysteresis [1-3]. These models provide a fairly aurate desription of magnetization proesses in magneti materials and, for this reason, they have been often applied to the modeling and simulation of these materials. Most of the existing researh related to the modeling of magneti hysteresis has foused on the mathematial desription of hysteresis phenomena in various materials. Phenomena suh as aommodation, Barkhausen effets, or time and temperature dependent hysteresis have been extensively analyzed with the help of Preisah-type models. owever, these models have rarely, if at all, been used to study more omplex systems that ontain hystereti indutors, suh as eletroni iruits. The reason why this has been the ase is that PM-based analyses are quite mathematially hallenging and omputationally expensive. In this artile we develop a new dynami PM suitable for the study of eletrial iruits omprising of hystereti indutors, transformers, or atuators. The distintive feature of our model is that it an be easily written in differential form, whih makes it onvenient to implement numerially. The basi idea of dynami Preisah models of hysteresis [1], [4-11] is to onsider that the hysteresis operator, usually denoted by ˆΓ, is ratedependent. The standard approah is to assume that the Preisah funtion depends on the speed of output variations, = dm dt. This approah leads to omplex mathematial expressions for the magnetization as a funtion of the applied field, whih makes it inonvenient to use in simulations of eletrial iruits. Moreover, the identifiation problem is quite ompliated and involves the evaluation of the relaxation times of first-order reversal-urves [1], whih are diffiult to measure experimentally. Our approah is based on the meanfield approximation and irumvents the disadvantages of the traditional dynami PMs of hysteresis. Our model was first introdued in [12-13]. In this artile we revise the initial model and also present it in a different form, whih makes it easy to use in the modeling and simulation of eletrial iruits. The artile is organized as follows. In Setion 2, the basi idea of the dynami PM is introdued. Speial emphasis is given to the analytial omputation of magneti suseptibility; it is shown that the equation of magneti suseptibility an be regarded as an alternative definition of the dynami PM. The numerial implementation of the dynami PM, as well as details related to the modeling of eletrial iruits ontaining hystereti indutors are presented in Setion 3. The experimental setup and the omparison of experimental data with numerial simulations are disussed in Setion 4. Finally, onlusions are drawn in Setion 5. 2 Tehnial disussion In this setion the dynami Preisah model is introdued by using two approahes. The first approah presents magnetization M as the solution of an integral equation in the applied magneti field. This approah is suitable for diret alulations of M harateristis, when the magneti field

2 Proeedings of the 5th WSEAS Int. Conf. on Miroeletronis, Nanoeletronis, Optoeletronis, Prague, Czeh Republi, Marh 12-14, 26 (pp86-91) is known a-priori. The seond approah presents magnetization M as the solution of a first-order differential equation; this approah is more onvenient for iruit appliations, as it will be shown in the next setion Definition of the Dynami Preisah Model Let us onsider the general definition of the Preisah model of hysteresis [1], in whih the magnetization M () t an be written as funtion of the effetive magneti field () t () = (, ) ˆ irr γ eff () as follows: M t P t d d α β αβ α β α> β + P ˆ rev α γα ( t) dα (1) In this equation γ ˆαβ are elementary retangular hysteresis operators with α and β as up and down swithing fields and with ± 1 as saturation values. The hysteresis operators are defined by ˆ γ αβ () t = 1 if () t > α, γˆαβ () t = ± 1 if α > () t )> β, and ˆ γ () t αβ eff = 1 if () t < β. In equation (1) γ ˆα = γ ˆαα. The Preisah funtion has been split into the irreversible Pirr ( α, β ) and the reversible Prev ( α ) omponents. Both omponents an be identified by measuring the first-order reversal hysteresis urves and by omputing the seond-order derivatives with respet to the magneti field. More details about the identifiation tehnique an be found in [1]. We propose a dynami PM in whih the effetive field is given by: = + F( M, ), (2) where F is a funtion of magnetization M and of its derivative with respet to time, M. Funtion F ould be determined through miromagneti omputations, but in this analysis we will take it as given. Equations (1) and (2) represent the integral definition of the dynami PM. It should be noted that, in the framework of the moving PM, whih is presented in [2], the effetive magneti field an be written in a form similar to equation (2): = + αm, (3) where α is the moving parameter. If F M, = αm equations (2) and (3) are equivalent, whih suggests that the dynami PM an be regarded as a generalization of the moving PM. For this reason, we all funtion F the generalized moving funtion. 2.2 Magneti suseptibility at low magnetization rates Let us onsider the magneti suseptibility at low speed of variations of the applied magnetization. In this ase: dm =, χ = (4) d eff = and the dynami Preisah model is redued to the generalized Preisah model. By taking the derivative of (1) with respet to we obtain: χ ( ( t) ) = N 2 Pirr ( h, ) dh+ Prev ( ) if is inreasing, (5) 2 P irr (, h) dh+ Prev ( ) N if is dereasing, where is the last extreme value of the magneti N field. It should be noted that the integrals in equation (5) an be solved analytially for partiular expressions of the Preisah funtion. In our analysis we assume that P irr an be written as follows: SM s Pirr ( a,b ) = 2πσ σi α β 2 α β 2 2 ln σ ( α + β) 2 exp (6) 2 2 4σi 2σ where parameters σ i, σ, and are determined by mathing experimental results to simulated data. The reversible omponent of the Preisah funtion is given by: (1 SM ) s Prev ( ) exp α α = 2rev, (7) rev where is a parameter that an be determined rev by using the same proedure as above.

3 Proeedings of the 5th WSEAS Int. Conf. on Miroeletronis, Nanoeletronis, Optoeletronis, Prague, Czeh Republi, Marh 12-14, 26 (pp86-91) 2.3 Magneti suseptibility of the dynami Preisah model The magneti suseptibility χ in the ase of the dynami PM is slightly more diffiult to ompute analytially. owever, we an ompute χ as the solution of a differential equation, whih will be derived in the following. By taking the derivative of (1) with respet to the applied field we obtain: χ (, M, ), dm d = χ (, M) 1+ FM + F d d (8) where: dm χ (, M, ) = (, M, ), (9) d is the magneti suseptibility and F M and F denote the partial derivatives of funtion F with respet to M and M, respetively. The derivative d an be expressed as a funtion of the d derivatives of the applied magneti field and suseptibility as follows: d ( ) d dm d = d d d dt d dχ = ( χ ) = χ +. d dt (1) By assuming F and by substituting the last result into equation (8) we find the following firstorder differential equation: dχ 1 1 = χ FM 1 χ, (11) dt F χ where denotes the seond-order derivative of the applied magneti field with respet to time. Equations (9) and (11) should be subjet to appropriate boundary onditions and solved selfonsistently to ompute magnetization M ( t ). It is apparent from our analysis that these equations represent an alternative definition of the dynami Preisah model given by equations (1) and (12). In the following setion we apply the dynami Preisah model to the desription of eletrial iruits with hystereti indutors. R Power Amplifier Wave Generator L Osillosope Data A Data Aquisition Data A Personal Computer Fig. 1. Experimental setups used to simulate RL iruits with hystereti magneti ore. K Y 1 Y 2 R C D.C. Generator Personal Computer L Osillosope Data A Data Aquisition Y 1 Y 2 Fig. 2. Experimental setups used to simulate RLC eletroni iruits with hystereti magneti ore. 3 Appliations to nonlinear eletrial iruits Ciruits ontaining basi eletroni omponents suh as resistors, hystereti indutors, and apaitors an be modeled by solving differential systems of equations that originate from Kirhhoff s laws oupled with equations (9) and (11). In the following we present two examples that are often used in iruit appliations: the resistor-indutor iruit (RL) and the resistor-indutor-apaitor iruit (RLC). 3.1 Resistor-indutor iruits Let us onsider an eletroni iruit in whih a resistor R and an indutor L made by using a hystereti ore are onneted in series with a voltage generator Vext ( t ). It an be shown that the magneti field generated by the indutor satisfies the following differential equation: L χ + + R = nv t, (13) ( 1) ext

4 Proeedings of the 5th WSEAS Int. Conf. on Miroeletronis, Nanoeletronis, Optoeletronis, Prague, Czeh Republi, Marh 12-14, 26 (pp86-91) Magneti Field (ka/m) Magnetization (ka/m) (a) Experiment (25 V) (b) Experiment (25 V) Fig. 3. Measured and simulated magneti field (a) and magnetization (b) in an RL iruit with R =.1 Ω and V ext (t) = 25 sin(125 t) (V). L where is the indutane of the indutor without magneti ore and n is a parameter that depends on the geometri harateristis of the oil. For toroidal indutors, n is the number of turns per unit length of indutor oil. The seond derivative of the magneti field an be omputed by differentiating equation (13) with respet to time. It follows that an be expressed as: 2 d dχ = a 2 (, χ ), (14) dt χ + 1 dt where a is a funtion defined as follows: nv ext () t R a(, χ ) =. (15) L ( χ + 1) By ombining equations (11) and (14) we obtain the following system of nonlinear differential equations: dm d = χ, (16) dt dt d nvext () t R =, (17) dt L χ + 1 d χ 1 χ = ( χ + 1) χfm 1 dt F χ a(, χ ) χ. (18) This system should be solved for M and as funtions of time. 3.2 Resistor-indutor-apaitor iruits Now let us look at the resistor-indutor-apaitor iruit onneted in series with external voltage generator Vext ( t ). By using the same line of reasoning as in the previous subsetion, we an obtain the following system of equations: dm d = χ, dt dt (19) d nvext ( t) nuc R =, dt L ( χ + 1) (2) duc =, dt nc (21) (22) d χ 1 χ = ( χ + 1) χfm 1, dt F χ a(,, χ ) χ (23) where U is the voltage aross C and funtion a is C given by the following equation: nvext ( t) R C a(,, χ ) =. (24) L χ + 1 Again, equations (21)-(24) should be subjet to appropriate boundary onditions and solved for the magnetization, magneti filed, suseptibility, and voltage aross the apaitor. 4 Simulation results and experiments The dynami PM presented in the previous setion has been numerially implemented and tested on RL and RLC iruits. The experimental setup of the RL iruit is presented in Figure 1. The power supply onsists of a wave generator and a power amplifier and is used to generate a sinusoidal voltage aross the RL iruit. The urrent through the iruit, whih is proportional to the magneti field is alulated by measuring the voltage aross

5 Proeedings of the 5th WSEAS Int. Conf. on Miroeletronis, Nanoeletronis, Optoeletronis, Prague, Czeh Republi, Marh 12-14, 26 (pp86-91) Magnetization (ka/m) Magnetization (ka/m) (a) Experiment (25 V) (b) Magneti Field (ka/m) Experiment (6 V) Magneti Field (ka/m) Fig. 4. Measured and simulated magneti hysteresis loops in the RL iruit with R =.1 Ω and V ext (t) = 25 sin(125 t) (a ) and V ext (t) = 6 sin(125 t) (V) (b ) the resistor. The magneti indution is found by integrating the potential measured at the terminals of a seondary oil with respet to time. All eletrial signals are olleted by a data aquisition system onsisting of a digital osillosope onneted to a personal omputer by using a data aquisition ard. In the RLC experimental setup presented in Figure 2, the resistor, the indutor and, the apaitor are onneted in series. First the apaitor is harged at some potential V C and then it is disharged in the RL iruit by using a swith K. Sine the value of resistane R is relatively small, most of the initial eletrial energy stored in the apaitor is dissipated through magneti losses in the indutor. More details related to the experimental setup and data aquisition tehnique an be found in Refs. [15-17]. The parameters of the dynami PM have been alulated by fitting the stati magneti hysteresis loop (i.e. measured at low frequenies) to the simulated data. By using this proedure we found: M s = 3x1 5 A/m, S =.6, rev = 6 A/m, σ =.75, σ = 9 A/m, = 1 A/m, and α = 4x1-4. The generalized moving funtion is approximated in our analysis by: F M, = αm sgn β, (25) where β is a parameter that desribes the delay of the total magnetization with respet to the applied magneti field. By mathing the measured magnetization to the simulated magnetization at high operating frequenies we found: β = 8.5x1-4 s 1/2 A 1/2 m -1/2. Voltage aross apaitor (V) Voltage aross apaitor (V) Voltage aross apaitor (V) Dynami Preisah Model Experiment (C = 12 μf) (a) Dynami Preisah Model Experiment (C = 24 μf) (b) Dynami Preisah Model Experiment (C = 48 μf) () Fig. 5. Measured and simulated voltage aross the apaitor of an RLC iruit for three values of the apaitane: (a) C = 12 μf, (b) C = 12 μf, and () C = 48 μf. Figures 3(a) and 3(b) present the omparison between the measured and the simulated magneti field and magnetization, respetively. The experimental results are represented by symbols, while the simulated data are represented by

6 Proeedings of the 5th WSEAS Int. Conf. on Miroeletronis, Nanoeletronis, Optoeletronis, Prague, Czeh Republi, Marh 12-14, 26 (pp86-91) ontinuous lines. The operating frequeny in this experiment is 2 z and the amplitude of the sinusoidal signal generated by the soure is 25 V. By using the omputed values for M and we have plotted the magneti hysteresis loops for two values of the amplitude of the external voltage generator: 6 V and 25 V. The results of these omputations are represented in figures 4(a) and 4(b), respetively. Although the speed of variation of magnetization has hanged substantially, the agreement between the theoretial results and the experimental data is remarkable, whih suggests that the dynami PM desribed in the previous setion is aurate. A very good agreement between the experimental and simulated data is also observed in the ase of the RLC iruit. The apaitor is initially harged at 42 V and then disharged on the RL iruit, whih indues damped osillations. Due to the strong nonlinearity of the indutor, the damped osillations are haraterized by fast speed of variation of the magnetization and magneti field, whih makes the numerial modeling of these iruits umbersome. Figures 5(a)-() present the omparison between the measured and simulated values of the eletri potential aross the apaitor for three values of the apaitane: C = 12 μ F, C = 12 μ F, and C = 48 μ F. 5 Conlusions A new dynami Preisah model based on the mean-field approximation is proposed and used to simulate various magnetization proesses in ferromagneti materials. The model has been tested on different eletroni iruits and under a variety of operational onditions. A very good agreement between the measured and the simulated data was observed. We onlude that this model an be suessfully applied to the analysis and modeling of eletroni iruits ontaining hystereti indutors. Referenes [1] I. D. Mayergoyz, Mathematial Models of ysteresis, Springer-Verlag, New York, [2] G. Bertotti, ysteresis in Magnetism for Physiists, Material Sientists, and Engineers, Aademi Press, [3] A. Ivanyi, ysteresis Models is Magneti Computation, Akadémiai Kiadó, Budapest, [4] M. Pasquale and G. Bertotti, "Appliation of the dynami Preisah model to the simulation of iruits oupled by soft magneti ores," IEEE Trans. on Magn., vol. 32, no. 5, pp , [5] G. Bertotti, F. Fiorillo, and M. Pasquale, "Measurement and predition of dynami loop shapes and power losses in soft magneti materials," IEEE Trans. on Magn., vol. 29, no. 6, pp , [6] N. Shmidt and. Güldner, "A simple method to determine dynami hysteresis loops of soft magneti materials," IEEE Trans. on Magn., vol. 33, no. 2, pp , [7] J..B. Deane, "Modeling the dynamis of nonlinear indutor iruits," IEEE Trans. on Magn., vol. 3, no. 5, pp , [8] V. Basso, G. Bertotti, F. Fiorillo, and M. Pasquale, Dynami Preisah model interpretation of power losses in rapidly quenhed 6.5% SiFe,," IEEE Trans. on Magn., vol. 3, no. 6, pp , [9] D. A. Philips and L. R. Dupre, Marosopi fields in ferromagneti laminations taking into aount hysteresis and eddy urrent effets, J. Magn. Mag. Mat., vol. 16, pp. 5-1, [1] L. R. Dupre, G. Bertotti, and A. A. A. Melkebeek, Dynami Preisah model and energy dissipation in soft magneti materials.," IEEE Trans. on Magn., vol. 34, no. 4, pp , [11] J. Fuzi and A. Ivanyi, Features of two ratedependent hysteresis models, Physia B, in press. [12] P. Andrei, O. Caltun, and A. Stanu, Differential Phenomenologial Models for the Magnetization Proesses in Soft MnZn Ferrites, IEEE Trans. on Magn., vol. 34, no. 1, pp , [13] P. Andrei, O. Caltun, and A. Stanu, Differential Preisah model for the desription of dynami magnetization proesses J. Appl. Phys., vol. 83, no. 11, pp , [14] [15] Al. Stanu, O. Caltun, P. Andrei, Models of hysteresis in magneti ores, J.Phys. IV Frane 7, pp , [16] O. Caltun, C. Papusoi, Al. Stanu, P. Andrei, W. Kappel, "Magneti ores diagnosis", IOS Series "Studies in Applied Eletromagnetis and Mehanis", editors V. Kose and J. Sievert, pp , [17] P. Andrei, O. Caltun, C. Papusoi, A. Stanu, M. Feder, Losses and magneti properties of Bi23 doped MnZn ferrites, J. Mag. Magn. Mat., vol , pp , 1999.