models for the simulation of electronic circuits with … for the simulation of electronic circuits...

Models for the simulation of electronic circuits with hysteretic inductors

PETRU ANDREI

Department of Electrical and Computer Engineering Florida State University and Florida A&M University

2525 Pottsdamer St., Tallahassee FL 32310 USA

http://www.eng.fsu.edu/~pandrei

Abstract: - Time-dependent magnetization processes in ferritic materials are described by using a new dynamic Preisach-type model. This model is based on the mean-field approximation and has the advantage that it can be easily implemented in electronic circuit applications. The numerical implementation of the new model is discussed in detail. Sample numerical results obtained for resistor-inductor and resistor-inductor-capacitor circuits operating at different frequencies are presented and compared with experimental data. Key-Words: - hysteretic inductors, Preisach model, electrical circuits. 1 Introduction

Preisach-type models (PM) are widely used in the literature for the description of magnetic hysteresis [1-3]. These models provide a fairly accurate description of magnetization processes in magnetic materials and, for this reason, they have been often applied to the modeling and simulation of these materials. Most of the existing research related to the modeling of magnetic hysteresis has focused on the mathematical description of hysteresis phenomena in various materials. Phenomena such as accommodation, Barkhausen effects, or time and temperature dependent hysteresis have been extensively analyzed with the help of Preisach-type models. However, these models have rarely, if at all, been used to study more complex systems that contain hysteretic inductors, such as electronic circuits. The reason why this has been the case is that PM-based analyses are quite mathematically challenging and computationally expensive. In this article we develop a new dynamic PM suitable for the study of electrical circuits comprising of hysteretic inductors, transformers, or actuators. The distinctive feature of our model is that it can be easily written in differential form, which makes it convenient to implement numerically.

The basic idea of dynamic Preisach models of hysteresis [1], [4-11] is to consider that the hysteresis operator, usually denoted by , is rate-dependent. The standard approach is to assume that the Preisach function depends on the speed of output variations,

Γ̂

M dM dt= . This approach leads to complex mathematical expressions for the magnetization as a function of the applied field,

which makes it inconvenient to use in simulations of electrical circuits. Moreover, the identification problem is quite complicated and involves the evaluation of the relaxation times of first-order reversal-curves [1], which are difficult to measure experimentally. Our approach is based on the mean-field approximation and circumvents the disadvantages of the traditional dynamic PMs of hysteresis. Our model was first introduced in [12-13]. In this article we revise the initial model and also present it in a different form, which makes it easy to use in the modeling and simulation of electrical circuits. The article is organized as follows. In Section 2, the basic idea of the dynamic PM is introduced. Special emphasis is given to the analytical computation of magnetic susceptibility; it is shown that the equation of magnetic susceptibility can be regarded as an alternative definition of the dynamic PM. The numerical implementation of the dynamic PM, as well as details related to the modeling of electrical circuits containing hysteretic inductors are presented in Section 3. The experimental setup and the comparison of experimental data with numerical simulations are discussed in Section 4. Finally, conclusions are drawn in Section 5. 2 Technical discussion

In this section the dynamic Preisach model is introduced by using two approaches. The first approach presents magnetization M as the solution of an integral equation in the applied magnetic field

. This approach is suitable for direct calculations of H

( )M H characteristics, when the magnetic field

Proceedings of the 5th WSEAS Int. Conf. on Microelectronics, Nanoelectronics, Optoelectronics, Prague, Czech Republic, March 12-14, 2006 (pp86-91)

is known a-priori. The second approach presents magnetization M as the solution of a first-order differential equation; this approach is more convenient for circuit applications, as it will be shown in the next section. 2.1. Definition of the Dynamic Preisach Model

Let us consider the general definition of the Preisach model of hysteresis [1], in which the magnetization ( )M t can be written as function of

the effective magnetic field as follows: ( )effH t

( ) ( ) ( )ˆ,irr effH H

M t P H H H t dH dHα β

α β αβ α βγ>

= ∫∫

( ) ( )ˆrev effP H H t dHα α

∞

−∞

+ γ∫ α (1)

In this equation ˆαβγ are elementary rectangular

hysteresis operators with Hα and Hβ as up and down switching fields and with ± 1 as saturation values. The hysteresis operators are defined by

if ( )ˆ 1H tαβγ = ( )effH t Hα> , ( )ˆαβγ 1effH t = ± if

Hα > )>( )effH t Hβ , and if

<

( )ˆ 1effH tαβγ = −

( )effH t Hβ . In equation (1) ˆαγ = ˆααγ . The Preisach function has been split into the irreversible

( ),irrP H Hα β and the reversible ( )revP Hα

components. Both components can be identified by measuring the first-order reversal hysteresis curves and by computing the second-order derivatives with respect to the magnetic field. More details about the identification technique can be found in [1].

We propose a dynamic PM in which the effective field is given by: , (2) ( ,effH H F M M= + )where is a function of magnetization F M and of its derivative with respect to time, M . Function could be determined through micromagnetic computations, but in this analysis we will take it as given. Equations

F

(1) and (2) represent the integral definition of the dynamic PM.

It should be noted that, in the framework of the moving PM, which is presented in [2], the effective magnetic field can be written in a form similar to equation (2): effH H Mα= + , (3) where α is the moving parameter. If

( ),F M M Mα= equations (2) and (3) are

equivalent, which suggests that the dynamic PM can be regarded as a generalization of the moving PM. For this reason, we call function the “generalized moving function.”

F

2.2 Magnetic susceptibility at low magnetization rates

Let us consider the magnetic susceptibility at low speed of variations of the applied magnetization. In this case:

effH H= , 0

effeff H H

dMdH

χ=

= (4)

and the dynamic Preisach model is reduced to the generalized Preisach model. By taking the derivative of (1) with respect to we obtain: H

( )( )0 H tχ =

( ) ( )

( ) ( )

2

if is increasing,

2 ,

if is decreasing,

N

N

H

irr revH

H

irr revH

P h,H dh P H

H

P H h dh P H

H

⎧ ⎡ ⎤+⎪ ⎢ ⎥

⎢ ⎥⎪ ⎣ ⎦⎪⎪⎨

⎡ ⎤⎪ +⎢ ⎥⎪⎢ ⎥⎣ ⎦⎪

⎪⎩

∫

∫ (5)

where is the last extreme value of the magnetic field. It should be noted that the integrals in equation

NH

(5) can be solved analytically for particular expressions of the Preisach function. In our analysis we assume that can be written as follows: irrP

( ) ( )2s

irr a bc i

SMP H ,HH H Hσ α βπ σ

=−

( )2

22

02 2

ln2

exp4 2

c

i c

H HH H H

H

α β

α β

σ

σ

σ

⎡ ⎤−⎛ ⎞−⎢ ⎥⎜ ⎟+⎢ ⎥⎝ ⎠× − −⎢ ⎥

⎢ ⎥⎢ ⎥⎣ ⎦

(6)

where parameters iHσ , cσ , and are determined by matching experimental results to simulated data. The reversible component of the Preisach function is given by:

0H

( ) (1 ) exp2

srev

rev rev

HS MP HH H

αα

⎛ ⎞−= ⎜

⎝ ⎠− ⎟ , (7)

where is a parameter that can be determined by using the same procedure as above.

revH


2.3 Magnetic susceptibility of the dynamic Preisach model

The magnetic susceptibility χ in the case of the dynamic PM is slightly more difficult to compute analytically. However, we can compute χ as the solution of a differential equation, which will be derived in the following. By taking the derivative of (1) with respect to the applied field we obtain: , ( ), ,H M Mχ

( )0 , 1eff M M

dM dMH M F FdH dH

χ⎡

= + +⎢⎣ ⎦

⎤⎥ (8)

where:

( ) (, , , ,dMH M M H M MdH

χ = ) , (9)

is the magnetic susceptibility and MF and MF denote the partial derivatives of function with respect to

FM and M , respectively. The derivative

dMdH

can be expressed as a function of the

derivatives of the applied magnetic field and susceptibility as follows:

( )dM H d dM dH

dH dH dH dt⎛ ⎞= ⎜ ⎟⎝ ⎠

( )d HHdH H dt

dχχ χ= = + .

(10) By assuming and by substituting the last result into equation

0MF ≠(8) we find the following first-

order differential equation:

0

1 1 1MM

d Fdt F Hχ χ

χ⎡ ⎤⎛ ⎞

= − − −⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

Hχ , (11)

where denotes the second-order derivative of the applied magnetic field with respect to time.

H

Equations (9) and (11) should be subject to appropriate boundary conditions and solved self-consistently to compute magnetization ( )M t . It is apparent from our analysis that these equations represent an alternative definition of the dynamic Preisach model given by equations (1) and (12). In the following section we apply the dynamic Preisach model to the description of electrical circuits with hysteretic inductors.

Oscilloscope

Data A Data Aquisition

Wave Generator

Power Amplifier

L

R

Data A Personal Computer

Fig. 1. Experimental setups used to simulate RL circuits with hysteretic magnetic core.

Oscilloscope

Data A Data Aquisition

D.C. Generator

L R

Y2

Personal Computer

C

Y1 Y2

Y1

K

Fig. 2. Experimental setups used to simulate RLC electronic circuits with hysteretic magnetic core. 3 Applications to nonlinear electrical circuits

Circuits containing basic electronic components such as resistors, hysteretic inductors, and capacitors can be modeled by solving differential systems of equations that originate from Kirchhoff’s laws coupled with equations (9) and (11). In the following we present two examples that are often used in circuit applications: the resistor-inductor circuit (RL) and the resistor-inductor-capacitor circuit (RLC). 3.1 Resistor-inductor circuits

Let us consider an electronic circuit in which a resistor R and an inductor made by using a hysteretic core are connected in series with a voltage generator

L

( )extV t . It can be shown that the magnetic field generated by the inductor satisfies the following differential equation: ( )0 1 ext ( )L H RH nV tχ + + = , (13)


0 2 4 6 8 10

-1.0

-0.5

0.0

0.5

1.0

(a)

Mag

netic

Fie

ld (k

A/m

)

Time (ms)

DPM Experiment (25 V)

0 2 4 6 8 10-400

-200

0

200

400

(b)

Mag

netiz

atio

n (k

A/m

)

Time (ms)


Fig. 3. Measured and simulated magnetic field (a) and magnetization (b) in an RL circuit with R = 0.1 Ω and Vext(t) = 25 sin(125 t) (V). where is the inductance of the inductor without magnetic core and is a parameter that depends on the geometric characteristics of the coil. For toroidal inductors, is the number of turns per unit length of inductor coil. The second derivative of the magnetic field can be computed by differentiating equation

0Ln

n

(13) with respect to time. It follows that can be expressed as: H

( )2

2 ,1

d H H da Hdt dt

χχχ

= −+

, (14)

where is a function defined as follows: a

( ) ( )( )0

,1

extnV t RHa H

Lχ

χ−

=+

. (15)

By combining equations (11) and (14) we obtain the following system of nonlinear differential equations:

dM dHdt dt

χ= , (16)

( )( )0 1

extnV t RHdHdt L χ

−=

+, (17)

( )0

11 1MM

d Fdt Fχ χχ χ

χ⎡ ⎛ ⎞

= + −⎢ ⎜ ⎟⎝ ⎠⎣

−

( ),a HHχ χ ⎤

− ⎥⎥⎦

. (18)

This system should be solved for M and as functions of time.

H

3.2 Resistor-inductor-capacitor circuits

Now let us look at the resistor-inductor-capacitor circuit connected in series with external voltage generator ( )extV t . By using the same line of reasoning as in the previous subsection, we can obtain the following system of equations:

dM dHdt dt

χ= , (19)

( )( )0 1

ext CnV t nU RHdHdt L χ

− −=

+, (20)

CdU Hdt nC

= , (21) (22)

( )0

11 1MM

d Fdt Fχ χχ χ

χ⎡ ⎛ ⎞

= + −⎢ ⎜ ⎟⎝ ⎠⎣

− ,

( ), ,a H HH

χ χ ⎤− ⎥

⎥⎦ (23)

where is the voltage across and function is given by the following equation:

CU C a

( ) ( )( )0

, ,1

extnV t RH H Ca H H

Lχ

χ− −

=+

. (24)

Again, equations (21)-(24) should be subject to appropriate boundary conditions and solved for the magnetization, magnetic filed, susceptibility, and voltage across the capacitor. 4 Simulation results and experiments

The dynamic PM presented in the previous section has been numerically implemented and tested on RL and RLC circuits. The experimental setup of the RL circuit is presented in Figure 1. The power supply consists of a wave generator and a power amplifier and is used to generate a sinusoidal voltage across the RL circuit. The current through the circuit, which is proportional to the magnetic field is calculated by measuring the voltage across


-0.4 -0.2 0.0 0.2 0.4-400

-200

0

200

400

(a)

Mag

netiz

atio

n (k

A/m

)

Magnetic Field (kA/m)


-0.4 -0.2 0.0 0.2 0.4-400

-200

0

200

400

(b)

Mag

netiz

atio

n (k

A/m

)

Magnetic Field (kA/m)


Fig. 4. Measured and simulated magnetic hysteresis loops in the RL circuit with R = 0.1 Ω and Vext(t) = 25 sin(125 t) (a ) and Vext(t) = 6 sin(125 t) (V) (b )

the resistor. The magnetic induction is found by integrating the potential measured at the terminals of a secondary coil with respect to time. All electrical signals are collected by a data acquisition system consisting of a digital oscilloscope connected to a personal computer by using a data acquisition card.

In the RLC experimental setup presented in Figure 2, the resistor, the inductor and, the capacitor are connected in series. First the capacitor is charged at some potential and then it is discharged in the RL circuit by using a switch . Since the value of resistance

CVK

R is relatively small, most of the initial electrical energy stored in the capacitor is dissipated through magnetic losses in the inductor. More details related to the experimental setup and data acquisition technique can be found in Refs. [15-17].

The parameters of the dynamic PM have been calculated by fitting the “static” magnetic hysteresis loop (i.e. measured at low frequencies) to the simulated data. By using this procedure we found:

sM = 3x105 A/m, =0.6, = 6 A/m, S revH cσ =0.75,

cHσ = 90 A/m, = 10 A/m, and 0H α = 4x10-4. The generalized moving function is approximated in our analysis by:

( ) ( ), sgnF M M M M Mα β= −

where β is a parameter that describes the delay of the total magnetization with respect to the applied magnetic field. By matching the measured magnetization to the simulated magnetization at high operating frequencies we found: β = 8.5x10-4 s1/2A1/2m-1/2.

0.0 0.2 0.4 0.6 0.8 1.0-400

-200

0

200

400

(a)

Vol

tage

acr

oss c

apac

itor (

V)

Time (ms)

Dynamic Preisach Model Experiment (C = 12 μF)

0.0 0.2 0.4 0.6 0.8 1.0-400

-200

0

200

400

(b)

Vol

tage

acr

oss c

apac

itor (

V)

Time (ms)


0.0 0.2 0.4 0.6 0.8 1.0

-200

0

200

400

(c)

Vol

tage

acr

oss c

apac

itor (

V)

Time (ms)


Fig. 5. Measured and simulated voltage across the capacitor of an RLC circuit for three values of the capacitance: (a) C = 12 μF, (b) C = 12 μF, and (c) C = 48 μF.

, (25)

Figures 3(a) and 3(b) present the comparison between the measured and the simulated magnetic field and magnetization, respectively. The experimental results are represented by symbols, while the simulated data are represented by


continuous lines. The operating frequency in this experiment is 20 Hz and the amplitude of the sinusoidal signal generated by the source is 25 V. By using the computed values for M and we have plotted the magnetic hysteresis loops for two values of the amplitude of the external voltage generator: 6 V and 25 V. The results of these computations are represented in figures 4(a) and 4(b), respectively. Although the speed of variation of magnetization has changed substantially, the agreement between the theoretical results and the experimental data is remarkable, which suggests that the dynamic PM described in the previous section is accurate.

H

A very good agreement between the experimental and simulated data is also observed in the case of the RLC circuit. The capacitor is initially charged at 420 V and then discharged on the RL circuit, which induces damped oscillations. Due to the strong nonlinearity of the inductor, the damped oscillations are characterized by fast speed of variation of the magnetization and magnetic field, which makes the numerical modeling of these circuits cumbersome. Figures 5(a)-(c) present the comparison between the measured and simulated values of the electric potential across the capacitor for three values of the capacitance: C = 12 Fμ ,

= 12 C Fμ , and C = 48 Fμ . 5 Conclusions

A new dynamic Preisach model based on the mean-field approximation is proposed and used to simulate various magnetization processes in ferromagnetic materials. The model has been tested on different electronic circuits and under a variety of operational conditions. A very good agreement between the measured and the simulated data was observed. We conclude that this model can be successfully applied to the analysis and modeling of electronic circuits containing hysteretic inductors. References [1] I. D. Mayergoyz, Mathematical Models of

Hysteresis, Springer-Verlag, New York, 1991. [2] G. Bertotti, Hysteresis in Magnetism for

Physicists, Material Scientists, and Engineers, Academic Press, 1998.

[3] A. Ivanyi, Hysteresis Models is Magnetic Computation, Akadémiai Kiadó, Budapest, 1997.

[4] M. Pasquale and G. Bertotti, "Application of the dynamic Preisach model to the simulation of circuits coupled by soft magnetic cores," IEEE Trans. on Magn., vol. 32, no. 5, pp. 4231-4233,

1996. [5] G. Bertotti, F. Fiorillo, and M. Pasquale,

"Measurement and prediction of dynamic loop shapes and power losses in soft magnetic materials," IEEE Trans. on Magn., vol. 29, no. 6, pp. 3496-3498, 1993.

[6] N. Schmidt and H. Güldner, "A simple method to determine dynamic hysteresis loops of soft magnetic materials," IEEE Trans. on Magn., vol. 33, no. 2, pp. 489-496, 1996.

[7] J.H.B. Deane, "Modeling the dynamics of nonlinear inductor circuits," IEEE Trans. on Magn., vol. 30, no. 5, pp. 2795-2901, 1994.

[8] V. Basso, G. Bertotti, F. Fiorillo, and M. Pasquale, “Dynamic Preisach model interpretation of power losses in rapidly quenched 6.5% SiFe,” ," IEEE Trans. on Magn., vol. 30, no. 6, pp. 4893-4995, 1994.

[9] D. A. Philips and L. R. Dupre, “Macroscopic fields in ferromagnetic laminations taking into account hysteresis and eddy current effects,” J. Magn. Mag. Mat., vol. 160, pp. 5-10, 1996.

[10] L. R. Dupre, G. Bertotti, and A. A. A. Melkebeek, “Dynamic Preisach model and energy dissipation in soft magnetic materials.” ," IEEE Trans. on Magn., vol. 34, no. 4, pp. 1138-1170, 1998.

[11] J. Fuzi and A. Ivanyi, “Features of two rate-dependent hysteresis models,” Physica B, in press.

[12] P. Andrei, O. Caltun, and A. Stancu, “Differential Phenomenological Models for the Magnetization Processes in Soft MnZn Ferrites,” IEEE Trans. on Magn., vol. 34, no. 1, pp. 231-241, 1998.

[13] P. Andrei, O. Caltun, and A. Stancu, “Differential Preisach model for the description of dynamic magnetization processes” J. Appl. Phys., vol. 83, no. 11, pp. 6359-6361, 1998.

[14] http:\\www.eng.fsu.edu\~pandrei\HysterSoft\ [15] Al. Stancu, O. Caltun, P. Andrei, “Models of

hysteresis in magnetic cores”, J.Phys. IV France 7, pp. 209 –210, 1997.

[16] O. Caltun, C. Papusoi, Al. Stancu, P. Andrei, W. Kappel, "Magnetic cores diagnosis", IOS Series "Studies in Applied Electromagnetics and Mechanics", editors V. Kose and J. Sievert, pp. 594-597, 1998.

[17] P. Andrei, O. Caltun, C. Papusoi, A. Stancu, M. Feder, ‘Losses and magnetic properties of Bi203 doped MnZn ferrites,” J. Mag. Magn. Mat., vol. 196-197, pp. 362-364, 1999.


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