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matlab代写-MARCH 1996 25

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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 11, NO. 2, MARCH 1996 25 1

A Generalized Dynamic Circuit Model of

Magnetic Cores for Low- and High-Frequency

Applications-Part 11: Circuit Model

Formulation and Implementation

S . Y. R. Hui, Senior Member, IEEE, Jian G.

Abstruct- This paper describes the formulation and imple-

mentation of a generalized dynamic magnetic core circuit model

suitable €or both low- and high-frequency applications. The

behavior of magnetic cores with any arbitrary flux waveforms

is modeled by a simple ladder network consisting of nonlinear

inductors and resistors. The nonlinear B-H loop and the hystere-

sis loss are incorporated in distributed nonideal inductors and

calculated by the Preisach scalar model of magnetic hysteresis.

The eddy current and anomalous losses are accounted for by

the generalized nonlinear equivalent resistors reported in Part I

of the paper. The transmission line modeling (TLM) method

is employed to solve the nonlinear state equations. Numerical

aspects and software implementation of the model are discussed.

The generalized model has been verified by simulations and

measurements at both low- and high-frequency operations.

I. INTRODUCTION

YNAMIC modeling of magnetic cores has long been a D difficult task owing to the nonlinear characteristics of

magnetic materials and the complicated mechanisms of core

losses. Accurate prediction of the instantaneous B-H operating

point is important because magnetic cores are often excited

with nonsinusoidal waveforms and at high frequency in most

power electronic circuits. Core loss assessment is another

important issue as the core loss increases with switching

frequency. It would be useful and practical to engineers if a

simple but accurate magnetic core model could be developed

for both low- and high-frequency applications. The model

must be simple to use, easy to formhate, based on data

supplied from manufacturers and/or obtained from simple

measurements, and able to predict all types of core losses to

a good degree of accuracy. In order to satisfy these criteria,

a circuit model of magnetic cores is preferred because of its

simplicity and time efficiency in simulation. However, a good

magnetic core circuit model, despite its inherent simplicity,

must be able to handle both the hysteresis behavior and total

eddy current losses including the anomalous loss.

Recently, the authors have proposed a dynamic discrete

circuit model for thin laminated magnetic cores for low-

Manuscript received February 14, 1995; revised October 20, 1995.

S . Y. R. Hui is with the Department of Electrical Engineering, University

J. G. Zhu and V. S . Ramsden are with the School of Electrical Engineering,

Publisher Item Identifier S 0885-8993(96)01916-3.

of Sydney, NSW 2006, Australia.

University of Technology, Sydney, NSW 2007, Australia.

I

1

Zhu, Member, IEEE, and V. S . Ramsden

frequency applications [ 11. This model includes all types

of core losses (i.e., hysteresis, classical eddy current, and

anomalous losses) and has been shown to provide significant

improvement in the accuracy of the core loss prediction

over traditional models without the anomalous loss. The

same concept was also applied successfully to a simple high-

frequency model for a solid magnetic core [2]. The results

reported indicate that the inclusion of anomalous loss into the

core loss model is achievable and practical. However, there

is so far no general approach that can be used to model both

laminated and solid magnetic cores for both low- and high-

frequency applications. Built on the two models previously

described, a generalized discrete dynamic magnetic core model

that can be readily implemented in digital computers for both

laminated and solid cores and for low- and high-frequency

applications is presented in this paper. The generalized model

is based on a simple ladder network consisting of nonlinear

inductors and resistors. The hysteresis effects are incorporated

into nonlinear inlductors and calculated by using the Preisach

theory [3]-[5]. Other core losses are included in nonlinear

resistors, the generalized expression of which is explained in

Part I of this paper. A recently introduced transmission-line

modeling (TLM) based discrete transform technique [6], [7] is

employed to develop the dynamic discrete circuit model. The

circuit model formulation and its implementation are described

in details. Criteria and factors influencing the size of the ladder

network for different core materials and different operating

frequencies are discussed. The model has been verified in

two examples, namely a laminated Lycore 140 magnetic core

at low-frequency operation and a solid ferrite core at high

frequency operaition. Both simulations and measurements are

included.

11. DYNAMIC CIRCUIT MODEL OF MAGNETIC ORES

Fig. 1 shows a simple circuit model for magnetic core. As

pointed out in Part I of this paper, this model is suitable

for low-frequency applications in which the magnetic flux

distribution in the core is close to uniform. At high-frequency

aperation, the eddy currents within the core are not identical,

md therefore, a ladder network incorporating the nonlinear

flux distribution within the core is needed. In fact, it is shown

in Section V that even for low-frequency operation, a ladder

0885-8993/96$05.00 0 1996 IEEE

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252 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 11, NO. 2, MARCH 1996

Fig. 1. Traditional single-stage magnetic core circuit model.

d ?2

oil

Nis ' > / I

I

re

~

Fig. 3. Equivalent circuit of a generalized magnetic core model.

I

Fig. 2.

paths.

Cross section of a solid magnetic core with assumed eddy current

network can provide better accuracy than the single-stage

simple circuit in Fig. 1.

A. Generalized Ladder Network for Dynamic

Magnetic Core Circuit Model

Cross sections of laminated and solid magnetic cores can

be divided into a few segments for magnetic circuit analysis.

To calculate eddy currents, eddy current paths, and their

associated magnetic flux paths are assumed in each segment,

as shown in Fig. 2.

These eddy current paths are divided so that the reaction of

eddy currents on the distribution of flux density is negligible,

that is, the flux density in each path is assumed uniform. In

general, let's assume that there are n eddy current loops in the

core. The generalized circuit equations are

Coil:

(1)

d(@1 + @2 + . . . + an) - Vs

d t N

--

Core:

where R, is the winding ac resistance, N the number of

turns of the winding, is the current in the winding, @I, ( k =

1,2, . . . , n) the flux within each path, V, the terminal voltage

across the winding, and RI, ( k = 1,2, . . . , n) the equivalent

resistance of each assumed eddy current path representing

eddy current and anomalous losses.

The generalized set of (1)-(4) can be represented as an

equivalent circuit in the form of a ladder network, as shown

in Fig. 3.

B. Discrete Mathematical Description

of Dynamic Magnetic Core Model

be written in matrix form as

In terms of the mesh currents, these circuit equations can

The relationship between the assumed eddy currents i ~ ,

(k = 1,2, . . . ,n) in Fig. 2 and the mesh currents ii

( k = 1,2, . . . , n) in the ladder network (Fig. 3) is as follows:

Equations (5) and (6) give the mathematical description of the

generalized magnetic core model in a continuous form. In the

ladder network, while resistor RI, (k = 1 , 2 , . . . , n) represents

the core loss equivalent resistance in the kth eddy current path,

the hysteresis effects can be included in the terms in (5).

For simulations in digital computers, a iscrete circuit

model can be derived using the TLM-based discrete transform

".

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HUI et al.: A GENERALIZED DYNAMIC CIRCUIT MODEL OF MAGNETIC CORES-PART 11 253

technique [6], [7]. First, &% is replaced by the corresponding

differential inductance L(i,) in the kth loop, where di;

(7)

These differential inductances at any time instant can be

determined by the Preisach theory as described in Section III-

A. Second, express the 3 ( I C = 1,2 , e . . , n) terms in ( 5 )

as

using the TLM-based discrete transform technique, where

Zu = 2/Tp, Tp the propagation time (which equals the time

step used in the numerical solution), Vik’ is the incident

voltage pulse on inductor L(iL) used in the TLM method, and

V i is equal to the magnitude of d iL /d t or numerically the

voltage across an inductor of 1 Henry with the same &’,/at.

Hence, using (7) and (8), ( 5 ) can be rewritten as (9a) (see

equation at the bottom of the page), or

where L( i i ) ( k = 1,2 , . . . , n), are nonideal differential induc-

tors when magnetic hysteresis is considered and [V”] i s the

voltage vector containing the incident pulses. Equation (9) is

the generalized discrete-time model of magnetic cores and is

suitable for direct implementation on digital computers.

At the beginning of each time stepping procedure, incident

pulses V$’ and the instantaneous differential inductances

L( i i ) calculated from the ii ( I C = 1,2, . e . , n) in the previous

time step are substituted into (9) in order to determine the

new current vector [i’]. The differential inductances L(zL) can

then be updated according to the new iL and the history of

magnetization using the Preisach theory.

According to the TLM discrete transforms, it can be shown

that the incident pulses (which are required for each time step)

for the next time step can be obtained as follows:

where a‘, = ii, ih, . . . and ik, and the bracketed subscripts

(m) and (m + 1) denote the mth and (m + 1)th time steps,

respectively. From (9) and (lo), it can be seen that the discrete

algorithms of the dynamic core model are recursive and that

the discrete transform technique allows simple discrete circuit

formulation and easy handling of nonlinearities. Therefore,

a nonlinear magnetic core can be modeled by a tridiagonal

system of nonlinear algebraic equations which can be easily

solved.

C. A Criterion for the Size of the Ladder Network for Low-

and High-Frequency Applications

For solid or laminated cores, the flux distribution within

the core becomes less uniform as the operating frequency

increases. The number of stages of the ladder network (which

represents the number of assumed eddy current paths) depends

on the operating frequency, conductivity and permeability of

the core material:;. Generally, the width of each eddy current

path (we) should be smaller than the skin depth 6 at the

fundamental excitation frequency f , i.e.,

we 5 6 = /= (1 1)

where CT is the conductivity, w = 2r f the angular frequency,

po = 47~ x the permeability of a vacuum in SI units,

and pr the relative permeability. Equation (11) shows that the

skin depth decreases with increasing frequency. Thus more

stages are required in the ladder network for a high-frequency

model. For low-frequency applications, the flux distribution

within the core is fairly uniform so that only one or two eddy

current paths are required. The generalized model reduces to

the low-frequency model developed in [ 13 when only one stage

of the ladder network is used. However, it should be noted

that (11) gives the minimum criterion in choosing the size

(i.e., the number of stages) of the ladder network. In general,

a larger ladder network would give more accurate results than

a smaller one.

UWPr Po

111. PARAMETERS AND LOSSES

All parameters of the model outlined in Section I1 can be

determined from the data sheet provided by the magnetic

material manufacturers.

A. Hysteresis Model

To model the nonlinear magnetization property and the hys-

teresis loss, the F’reisach model of hysteresis is adopted for its

easy parameter identification and considerable accuracy. Since

details of Preisach theory can be obtained in the literature

[3]-[5], this section describes its fundamental concepts only.

The Preisach imodel describes the hysteresis of a magnetic

material via an infinite set of magnetic dipoles, which have

rectangular hysteresis loops, as shown in Fig. 4(a). The flux

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254 E E E TRANSACTIONS ON POWER ELECTRONICS, VOL. 11, NO. 2, MARCH 1996

density. Function T(a , P) , the area integration over the right

triangle of vertex (a , p) in the (a , p) plane, is calculated by

(17)

where B d and B, are the flux densities on the downward and

upward trajectories of the limiting B-H loop, and

F ( a ) = B d ( a ) - Bu(a) ( a 0) (18)

Bu(a) - Bd(P) + F ( a ) F ( - P )

2 T(a,P) =

2 d m

(a) (b)

Fig. 4.

diagram.

(a) Rectangular hysteresis loop of dipoles. @) Typical Preisach

density B due to the magnetic field H is expressed as

a = pL(a, P)%P(H) da dP

where 5’ is the triangular region Hsat 2 a 2 P 2 -Hsat

on the (a,P) plane shown in Fig. 4(b), Hsat the saturation

magnetic field strength, a and P are the magnetic field strength

in the increasing positive and negative directions respectively,

p(a, p ) the distribution function of the dipoles, ~ ~ a p ( H ) = 1

on enclosed area S+, and y,p(N) = -1 on enclosed area

S-. The interface between S+ and S- is determined by the

history and the present state of magnetization. Through certain

function transforms, the area integration can be related to the

limiting hysteresis loop, which is also the only information

required in the Preisach modeling.

When a magnetic material is magnetized from the initial

unmagnetized state, the magnetic flux density can be calculated

~31% [51 by

Bt = l+ P(Q, P ) d a dP - k- A@, P ) da @

= T ( H , - H ) (13)

where

T ( a , PI = la P ( X , Y) dx dY

P Y

is the area integration over the right triangle of vertex

T(a , a ) = 0, and subscript i denotes the initial B-H curve.

For general situation of Preisach diagram, as shown in

Fig. 4(b), where the magnetization curve has m local extrema,

namely, m reversal points, the flux density can be derived as

( % P I , T(a,P) = T(-W -P) since P(Q,P) = P(-& -a>,

B(H) = B(H,) - 2T(H,, H ) (15)

on a downward trajectory, and

B ( H ) = B ( H m ) + 2T(H, Hm) (16)

on an upward trajectory, where H , is the magnetic field of the

mth (last) reversal point, and B(H,) is the corresponding flux

or

F ( a ) = Jm ( a < 0). (19)

The new mathematical function F ( a ) in (18) and (19) now

provide information for the determination of T ( a , p) without

considering the distribution function p(a, p). It should be

noted that F ( a ) is expressed only in terms of flux density

values of the B-H limiting loop (i.e., B d and Bu). Thus, only

the B-H limiting loop is needed for the implementation of

the hysteresis model.

B. Differential Inductance and Hysteresis Loss

When the flux densities in two consecutive time steps are

obtained, the differential inductance in the kth stage of the

ladder network can be calculated as following

fork = 1 ,2 , . . . , n (20)

where A k is the cross-sectional area of the kth eddy current

path, and subscripts (m) and (m - I) refer to the mth and

(m - 1)th steps, respectively.

When the effect of hysteresis is considered, the differential

inductance L(iL) is nonideal. The average hysteresis loss

dissipated in the kth inductor can be calculated by

The total average hysteresis loss ph is therefore equal to

n di l

d t

iLL(iL)A dt . (21b)

k = l k=l

C. Equivalent Resistance and Eddy Current

and Anomalous Losses

It has been shown in Part I of this paper that the equivalent

resistance accounting for the classical eddy current loss P, and

anomalous loss Pa in the kth eddy current path is

(22) R k - v$ - RDck

P e a k T J k r c k

where v k is the voltage across the resistor R k , Peak =

( P e + P a ) k the sum of the classical eddy current and anomalous

losses, R D C ~ the dc resistance of the kth eddy current path,

r J k = 5 [I +- (I + +)+I the correction factor for the skin

effect, J k = U (g) kay + J k - 1 is the maximum eddy current

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HUI et al.: A GENERALIZED DYNAMIC CIRCUIT MODEL OF MAGNETIC CORES-PART I1 255

I F a jc = 1 2 , ... ,Nc

Hystcrrsis loso PN = 0, and eady C l t m n t

Fig. 5. Flow chart of dynamic circuit model.

density in the path, Jo = 0 Nm2, Ay the thickness of the

eddy current path, r,k = 1 + c, & the correction factor to

include the anomalous loss, and C, and C, are the eddy current

and anomalous loss coefficients, respectively, determined by

the standard core loss separation procedure as explained in [l]

and [2] .

In terms of the equivalent resistance, the sum of the classical

eddy Current and the anomalous losses in the kth eddy current

d t k

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256 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 11, NO. 2, MARCH 1996

r - - - - - - - - -

1 Scalar Preisach Model

I

I

I

I

Stackempty? 7

H going up? +

I

I

I

I

Calculate B on an upward

trajec%oq by Eqns. (16) aod (17)

Calculae B on the initial curve

Fig. 6. Flow chart of the Preisach model subroutine.

path is equal to where V, = K / N - R,(Ni,) is the voltage across resistor

&, and Pea and Ph are the sum of the eddy current and

anomalous losses and the hysteresis loss, respectively. The

term on the left hand side of (24) is the total power fed into the

T

P e a k 1 1 R k i z d t . ~ 3 ~ )

" 1 sum of the classical eddy current and anomalous losses, and the

second term on the right hand side is the total hysteresis loss

dissipated in the nonideal inductors L(iL), ( I C = 1 , 2 , . . . , n),

for an n-stage ladder network.

'ea = 2 peak = T 1 R k i E dt. (23b)

k=l k=l

D. Total Core Loss Calculation

The general core loss expression of the magnetic core model Iv. SOFTWARE IMPLEMENTATION

can be described, based on the law of energy conservation, as

Fig. 5 illustrates the flow chart of the operation of the

magnetic core model. The program can be used to run the

core model routine N, times. User-chosen N, points can

be obtained within each switching period. Depending on the

nature of the excitation, a variable time step TLM technique [8]

has been implemented in the routine so that any nonlinear ex-

citation waveforms with transients can be incorporated without

$ LT VRn(Nis) d t

T T

R k i i d t + 2 $1 iLL(ii) (5) dt

k=l k=l

= pea + ph (24)

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257 HUI ef al.: A GENERALIZED DYNAMIC CIRCUIT MODEL OF MAGNETIC CORES-PART 11

I Generator H Amplifier f - I ' I I

I I I IQlannel2 I I

I I 1 1

Data Precision

6OOO Signal

voltage (v)

Current (A)

1.5

1

0.5

0

-0.5

-1

2

Fig. 7. Experimental setup of annular ring example.

TABLE I

PARAMETERS AND DIMENSIONS OF THE LYCORE-140 CORE EXPERIMENTAL SETUP

losing transient information. This is particularly important in

the study of magnetic core behavior in many modern switched

mode power supplies because the excitation waveforms are

normally highly nonlinear with rapidly changing edges.

The flow chart of the Preisach hysteresis model routine is

shown in Fig. 6. The input data are the limiting hysteresis

loop, and the specified magnetic field for simulation. The stack

keeps the flux density B and magnetic field H values of each

reversal point (local extremum) and is initialized empty (start

from the initial curve) before start. In checking the stack, a

pair of reversal points is popped out of the stack whenever

the specified H exceeds the value kept in the stack. This is to

wipe out a completed minor loop since it will not have any

effect on the future state of magnetization.

V . EXPERIMENTAL VERIFICATION

A. Annular Ring of Silicon Iron with Square

Wave Voltage Excitation

Fig. 7 is an experimental setup for measuring the response

of an annular ring of silicon iron Lycore-140 0.35 mm lam-

ination, under square wave voltage excitation. The primary

and the secondary coils are identical. The primary voltage,

primary current and the secondary voltage are recorded via

-lS -2 f

Hyst.+Eddy+Anom. I

Hyst.+Eddy+Anom. I

-1.51

(b)

Fig. 8. Results of discrete modeling under 100 Hz square wave voltage

excitation. (a) Comparison of voltage and current waveforms; - - - measured,

- calculated. The ideal value of V, used in the calculation is slightly

different from the actual signal because of the nonideal characteristcs of the

power amplifier. (b), Comparison of equivalent hysteresis loops.

three sepatate channels in the signal processing system. The

parameters and the dimensions of the setup are listed in Table I.

simulated

and measured current and equivalent hysteresis loops at 100

Hz are shown in Fig. 8. Since the skin depth (0.5 mm) is

greater than the steel sheet thickness (0.35 mm), only one

layer of eddy current path is assumed, which is identical to

the model used in [l]. The time step used to obtain the

result is 10 ,us. The total calculated core loss is 0.0691

W, and the measured 0.0726 W. The percentage error is

4.82%. Comparing the various B-H loops in Fig. 8(b), the

accuracy improvement in incorporating the anomalous loss in

the magnetic core model is obvious.

2 ) Lycore-140 Core-I kHz Operation with Width of Eddy

Current Path Larger Than Skin Depth: Fig. 9 shows the

simulated and measured results of Lycore-140 core at 1 kHz

I ) Lycore 140 Core-I00 Hz Operation: The

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258

Voltage (VI

IEI

Fig. 9.

network model; - - - measured, __ calculated.

Simulated and measured results at 1 kHz with a one-stage ladder

Voltage gr) Cur" {A)

4 T T 0'4

41 I -0.4

Fig. 10.

network model; - - - measured, ~ calculated.

Simulated and measured results at 1 kHz with a two-stage ladder

excitation, assuming only one eddy current loop. At this

frequency, the skin depth (6) for the core is 0.134 mm. The

width of each lamination in the core is 0.35 mm, giving the

width of each eddy current path (we) to be 0.175 mm. Thls

means that the condition set in (11) is not satisfied. It can

be seen that the calculated current is lower and flatter than

measured. The calculated specific core loss per Hertz is 0.036

W/kg/Hz, and the measured core loss is 0.0336 WkgMz. The

percentage error is 7.2%. This inaccuracy can be attributed to

the reaction of eddy current on the distribution of flux density

in the core [I], 121.

-with Width of Eddy Current Path Smaller Than Skin

Depth: Applying the condition set in (1 l), the cross section

of the core is divided into two layers of eddy current paths,

resulting in the width of each layer being 0.0875 mm. Width of

the eddy current path (we) is now smaller than the skin depth

(6). The core model with two assumed eddy current paths

is represented by a two-stage ladder network (i.e., n = 2).

Fig. 10 shows the simulated results using a two-stage ladder

network. The simulated current has better agreement with the

measured current than that in Fig. 9. The calculated specific

core loss per Hz is 0.0346 W/kg/Hz, and the percentage error

is reduced to 2.98%.

B. A Ferrite Toroid in a Switching Circuit-100-kHz Operation

A SiemensMatsushita N47 soft ferrite R16 toroid core in

a high-frequency switching circuit, as shown in Fig. 11 was

3E TRANSACTIONS ON POWER ELECTRONICS, VOL. 11, NO. 2, MARCH 1996

4-0

1 I

I I

2

Fig. 11. Schematic of a high-frequency switching circuit.

TABLE I1

DIMENSIONS AND PARAMETERS OF THE N47 FERRITE CORE AND WINDINGS

Outer diameter of the CQE 16.33 mm

also simulated. Table I1 lists the dimensions and parameters

of the core and windings.

Fig. 12 shows the simulated and measured current wave-

forms and the 23-H loops of the core at 100 kHz switching

frequency. The cross section of the core was divided into five

paths. The time step is 50 ns. The calculated specific core loss

per Hz, corresponding to the area enclosed by the calculated

B-H loop in Fig. 12(b), is 2.592 x W/kg/Hz, and the

measured specific core loss, corresponding to the area enclosed

by the measured B-H loop in Fig. 12(b), is 2.566 X

WkgEIz. The percentage error is about 1%.

VI. CONCLUSION

A generalized dynamic discrete circuit model of magnetic

core for both low- and high-frequency applications has been

presented. The dynamic circuit core model takes all sources of

nonlinearity and core losses into consideration. The nonlinear

B-H loop and the hysteresis loss are incorporated in the

distributed nonideal inductors and calculated by the Preisach

scalar model of hysteresis. The classical eddy current and

anomalous losses are included in generalized nonlinear resis-

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HUI ef al.: A GENERALIZED DYNAMIC CIRCUIT MODEL OF MAGNETIC CORES-PART I1 259

Flux Density 0

0.2

0.15

0.1

0.05

0

-0.05

0 . 1

-0.15

-0.2

-150 -100 -50 0 50 100 150

Field Strength (Nm)

(b)

Fig. 12. Comparison of simulated and measured. (a) Current waveforms. (b)

B-H loops.

tors. All model parameters, such as limiting hysteresis loop

and specific core loss coefficients, can be obtained from the

manufacturers’ data sheets.

The discrete transform technique together with the Preisach

hysteresis model provides a useful means to handle hysteresis

nonlinearity. The resultant tridiagonal nonlinear system model

equation features easy programming and fast computation. A

criterion for choosing the number of stages of the ladder net-

work is discussed and demonstrated with numerical examples.

Comparisons of the simulations for two types of cores and

different frequencies confirm the flexibility and accuracy of

the generalized model.

REFERENCES

J. G. Zhu, S. Y. R. Hui, and V. S. Ramsden, “Discrete modeling

of magnetic coires including hysteresis, eddy current, and anomalous

losses,” ZEE Proc.-A, vol. 140, no. 4, pp. 317-322, 1993.

-, “A dynamic circuit model for solid magnetic cores,” in ZEEE

Power Electron. Spec. Con$ (PESC ’93) Rec., Seattle, WA, June 20-24,

1993, pp. 1111-1115.

I. D. Mayergoyz, “Mathematical models of hysteresis,” IEEE MAG-22,

no. 5 , pp. 603408, 1986.

S. R. Naidu, “Simulation of the hysteresis phenomenon using Preisach’s

theory,” IEE Proc.-A, vol. 137, no. 2, pp. 73-79, 1990.

S. Y. R. Hui and J. G. Zhu, “Numerical modeling and simulation of

hysteresis effecls in magnetic cores using the transmission line modeling

and F’reisach thieory,” in ZEE Proc.-Sei. Meas. Technol., 1995, vol. 142,

no. 1, pp. 57-62.

S. Y. R. Hui and C. Christopolous, “Discrete transform technique for

solving coupled integro-differential equations in digital computers,” ZEE

Proc.-A, vol. 138, no. 5, pp. 273-280. 1991.

__ , “Discrete transform technique for solving nonlinear circuits and

equations,” ZEE Proc.-A, vol. 139, no. 6, pp. 321-328, 1992.

S. Y. R. Hui, K. K. Fnng, M. Q. Zhang, and C. Christopolous, “Variable

time step technique for transmission line modeling,” ZEE Proc. -A, vol.

140, no. 4, pp. 299-302, 1993.

S. Y. R. (Ron) Hui (SM’94) for a photograph and biography, see this issue,

p. 250.

Jian G. Zhu (S’93--M’93) for a photograph and biography, see this issue,

p. 250.

V. S. Ramsden photograph and biography not available at the time of

publication.

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学霸联盟

A Generalized Dynamic Circuit Model of

Magnetic Cores for Low- and High-Frequency

Applications-Part 11: Circuit Model

Formulation and Implementation

S . Y. R. Hui, Senior Member, IEEE, Jian G.

Abstruct- This paper describes the formulation and imple-

mentation of a generalized dynamic magnetic core circuit model

suitable €or both low- and high-frequency applications. The

behavior of magnetic cores with any arbitrary flux waveforms

is modeled by a simple ladder network consisting of nonlinear

inductors and resistors. The nonlinear B-H loop and the hystere-

sis loss are incorporated in distributed nonideal inductors and

calculated by the Preisach scalar model of magnetic hysteresis.

The eddy current and anomalous losses are accounted for by

the generalized nonlinear equivalent resistors reported in Part I

of the paper. The transmission line modeling (TLM) method

is employed to solve the nonlinear state equations. Numerical

aspects and software implementation of the model are discussed.

The generalized model has been verified by simulations and

measurements at both low- and high-frequency operations.

I. INTRODUCTION

YNAMIC modeling of magnetic cores has long been a D difficult task owing to the nonlinear characteristics of

magnetic materials and the complicated mechanisms of core

losses. Accurate prediction of the instantaneous B-H operating

point is important because magnetic cores are often excited

with nonsinusoidal waveforms and at high frequency in most

power electronic circuits. Core loss assessment is another

important issue as the core loss increases with switching

frequency. It would be useful and practical to engineers if a

simple but accurate magnetic core model could be developed

for both low- and high-frequency applications. The model

must be simple to use, easy to formhate, based on data

supplied from manufacturers and/or obtained from simple

measurements, and able to predict all types of core losses to

a good degree of accuracy. In order to satisfy these criteria,

a circuit model of magnetic cores is preferred because of its

simplicity and time efficiency in simulation. However, a good

magnetic core circuit model, despite its inherent simplicity,

must be able to handle both the hysteresis behavior and total

eddy current losses including the anomalous loss.

Recently, the authors have proposed a dynamic discrete

circuit model for thin laminated magnetic cores for low-

Manuscript received February 14, 1995; revised October 20, 1995.

S . Y. R. Hui is with the Department of Electrical Engineering, University

J. G. Zhu and V. S . Ramsden are with the School of Electrical Engineering,

Publisher Item Identifier S 0885-8993(96)01916-3.

of Sydney, NSW 2006, Australia.

University of Technology, Sydney, NSW 2007, Australia.

I

1

Zhu, Member, IEEE, and V. S . Ramsden

frequency applications [ 11. This model includes all types

of core losses (i.e., hysteresis, classical eddy current, and

anomalous losses) and has been shown to provide significant

improvement in the accuracy of the core loss prediction

over traditional models without the anomalous loss. The

same concept was also applied successfully to a simple high-

frequency model for a solid magnetic core [2]. The results

reported indicate that the inclusion of anomalous loss into the

core loss model is achievable and practical. However, there

is so far no general approach that can be used to model both

laminated and solid magnetic cores for both low- and high-

frequency applications. Built on the two models previously

described, a generalized discrete dynamic magnetic core model

that can be readily implemented in digital computers for both

laminated and solid cores and for low- and high-frequency

applications is presented in this paper. The generalized model

is based on a simple ladder network consisting of nonlinear

inductors and resistors. The hysteresis effects are incorporated

into nonlinear inlductors and calculated by using the Preisach

theory [3]-[5]. Other core losses are included in nonlinear

resistors, the generalized expression of which is explained in

Part I of this paper. A recently introduced transmission-line

modeling (TLM) based discrete transform technique [6], [7] is

employed to develop the dynamic discrete circuit model. The

circuit model formulation and its implementation are described

in details. Criteria and factors influencing the size of the ladder

network for different core materials and different operating

frequencies are discussed. The model has been verified in

two examples, namely a laminated Lycore 140 magnetic core

at low-frequency operation and a solid ferrite core at high

frequency operaition. Both simulations and measurements are

included.

11. DYNAMIC CIRCUIT MODEL OF MAGNETIC ORES

Fig. 1 shows a simple circuit model for magnetic core. As

pointed out in Part I of this paper, this model is suitable

for low-frequency applications in which the magnetic flux

distribution in the core is close to uniform. At high-frequency

aperation, the eddy currents within the core are not identical,

md therefore, a ladder network incorporating the nonlinear

flux distribution within the core is needed. In fact, it is shown

in Section V that even for low-frequency operation, a ladder

0885-8993/96$05.00 0 1996 IEEE

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252 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 11, NO. 2, MARCH 1996

Fig. 1. Traditional single-stage magnetic core circuit model.

d ?2

oil

Nis ' > / I

I

re

~

Fig. 3. Equivalent circuit of a generalized magnetic core model.

I

Fig. 2.

paths.

Cross section of a solid magnetic core with assumed eddy current

network can provide better accuracy than the single-stage

simple circuit in Fig. 1.

A. Generalized Ladder Network for Dynamic

Magnetic Core Circuit Model

Cross sections of laminated and solid magnetic cores can

be divided into a few segments for magnetic circuit analysis.

To calculate eddy currents, eddy current paths, and their

associated magnetic flux paths are assumed in each segment,

as shown in Fig. 2.

These eddy current paths are divided so that the reaction of

eddy currents on the distribution of flux density is negligible,

that is, the flux density in each path is assumed uniform. In

general, let's assume that there are n eddy current loops in the

core. The generalized circuit equations are

Coil:

(1)

d(@1 + @2 + . . . + an) - Vs

d t N

--

Core:

where R, is the winding ac resistance, N the number of

turns of the winding, is the current in the winding, @I, ( k =

1,2, . . . , n) the flux within each path, V, the terminal voltage

across the winding, and RI, ( k = 1,2, . . . , n) the equivalent

resistance of each assumed eddy current path representing

eddy current and anomalous losses.

The generalized set of (1)-(4) can be represented as an

equivalent circuit in the form of a ladder network, as shown

in Fig. 3.

B. Discrete Mathematical Description

of Dynamic Magnetic Core Model

be written in matrix form as

In terms of the mesh currents, these circuit equations can

The relationship between the assumed eddy currents i ~ ,

(k = 1,2, . . . ,n) in Fig. 2 and the mesh currents ii

( k = 1,2, . . . , n) in the ladder network (Fig. 3) is as follows:

Equations (5) and (6) give the mathematical description of the

generalized magnetic core model in a continuous form. In the

ladder network, while resistor RI, (k = 1 , 2 , . . . , n) represents

the core loss equivalent resistance in the kth eddy current path,

the hysteresis effects can be included in the terms in (5).

For simulations in digital computers, a iscrete circuit

model can be derived using the TLM-based discrete transform

".

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HUI et al.: A GENERALIZED DYNAMIC CIRCUIT MODEL OF MAGNETIC CORES-PART 11 253

technique [6], [7]. First, &% is replaced by the corresponding

differential inductance L(i,) in the kth loop, where di;

(7)

These differential inductances at any time instant can be

determined by the Preisach theory as described in Section III-

A. Second, express the 3 ( I C = 1,2 , e . . , n) terms in ( 5 )

as

using the TLM-based discrete transform technique, where

Zu = 2/Tp, Tp the propagation time (which equals the time

step used in the numerical solution), Vik’ is the incident

voltage pulse on inductor L(iL) used in the TLM method, and

V i is equal to the magnitude of d iL /d t or numerically the

voltage across an inductor of 1 Henry with the same &’,/at.

Hence, using (7) and (8), ( 5 ) can be rewritten as (9a) (see

equation at the bottom of the page), or

where L( i i ) ( k = 1,2 , . . . , n), are nonideal differential induc-

tors when magnetic hysteresis is considered and [V”] i s the

voltage vector containing the incident pulses. Equation (9) is

the generalized discrete-time model of magnetic cores and is

suitable for direct implementation on digital computers.

At the beginning of each time stepping procedure, incident

pulses V$’ and the instantaneous differential inductances

L( i i ) calculated from the ii ( I C = 1,2, . e . , n) in the previous

time step are substituted into (9) in order to determine the

new current vector [i’]. The differential inductances L(zL) can

then be updated according to the new iL and the history of

magnetization using the Preisach theory.

According to the TLM discrete transforms, it can be shown

that the incident pulses (which are required for each time step)

for the next time step can be obtained as follows:

where a‘, = ii, ih, . . . and ik, and the bracketed subscripts

(m) and (m + 1) denote the mth and (m + 1)th time steps,

respectively. From (9) and (lo), it can be seen that the discrete

algorithms of the dynamic core model are recursive and that

the discrete transform technique allows simple discrete circuit

formulation and easy handling of nonlinearities. Therefore,

a nonlinear magnetic core can be modeled by a tridiagonal

system of nonlinear algebraic equations which can be easily

solved.

C. A Criterion for the Size of the Ladder Network for Low-

and High-Frequency Applications

For solid or laminated cores, the flux distribution within

the core becomes less uniform as the operating frequency

increases. The number of stages of the ladder network (which

represents the number of assumed eddy current paths) depends

on the operating frequency, conductivity and permeability of

the core material:;. Generally, the width of each eddy current

path (we) should be smaller than the skin depth 6 at the

fundamental excitation frequency f , i.e.,

we 5 6 = /= (1 1)

where CT is the conductivity, w = 2r f the angular frequency,

po = 47~ x the permeability of a vacuum in SI units,

and pr the relative permeability. Equation (11) shows that the

skin depth decreases with increasing frequency. Thus more

stages are required in the ladder network for a high-frequency

model. For low-frequency applications, the flux distribution

within the core is fairly uniform so that only one or two eddy

current paths are required. The generalized model reduces to

the low-frequency model developed in [ 13 when only one stage

of the ladder network is used. However, it should be noted

that (11) gives the minimum criterion in choosing the size

(i.e., the number of stages) of the ladder network. In general,

a larger ladder network would give more accurate results than

a smaller one.

UWPr Po

111. PARAMETERS AND LOSSES

All parameters of the model outlined in Section I1 can be

determined from the data sheet provided by the magnetic

material manufacturers.

A. Hysteresis Model

To model the nonlinear magnetization property and the hys-

teresis loss, the F’reisach model of hysteresis is adopted for its

easy parameter identification and considerable accuracy. Since

details of Preisach theory can be obtained in the literature

[3]-[5], this section describes its fundamental concepts only.

The Preisach imodel describes the hysteresis of a magnetic

material via an infinite set of magnetic dipoles, which have

rectangular hysteresis loops, as shown in Fig. 4(a). The flux

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254 E E E TRANSACTIONS ON POWER ELECTRONICS, VOL. 11, NO. 2, MARCH 1996

density. Function T(a , P) , the area integration over the right

triangle of vertex (a , p) in the (a , p) plane, is calculated by

(17)

where B d and B, are the flux densities on the downward and

upward trajectories of the limiting B-H loop, and

F ( a ) = B d ( a ) - Bu(a) ( a 0) (18)

Bu(a) - Bd(P) + F ( a ) F ( - P )

2 T(a,P) =

2 d m

(a) (b)

Fig. 4.

diagram.

(a) Rectangular hysteresis loop of dipoles. @) Typical Preisach

density B due to the magnetic field H is expressed as

a = pL(a, P)%P(H) da dP

where 5’ is the triangular region Hsat 2 a 2 P 2 -Hsat

on the (a,P) plane shown in Fig. 4(b), Hsat the saturation

magnetic field strength, a and P are the magnetic field strength

in the increasing positive and negative directions respectively,

p(a, p ) the distribution function of the dipoles, ~ ~ a p ( H ) = 1

on enclosed area S+, and y,p(N) = -1 on enclosed area

S-. The interface between S+ and S- is determined by the

history and the present state of magnetization. Through certain

function transforms, the area integration can be related to the

limiting hysteresis loop, which is also the only information

required in the Preisach modeling.

When a magnetic material is magnetized from the initial

unmagnetized state, the magnetic flux density can be calculated

~31% [51 by

Bt = l+ P(Q, P ) d a dP - k- A@, P ) da @

= T ( H , - H ) (13)

where

T ( a , PI = la P ( X , Y) dx dY

P Y

is the area integration over the right triangle of vertex

T(a , a ) = 0, and subscript i denotes the initial B-H curve.

For general situation of Preisach diagram, as shown in

Fig. 4(b), where the magnetization curve has m local extrema,

namely, m reversal points, the flux density can be derived as

( % P I , T(a,P) = T(-W -P) since P(Q,P) = P(-& -a>,

B(H) = B(H,) - 2T(H,, H ) (15)

on a downward trajectory, and

B ( H ) = B ( H m ) + 2T(H, Hm) (16)

on an upward trajectory, where H , is the magnetic field of the

mth (last) reversal point, and B(H,) is the corresponding flux

or

F ( a ) = Jm ( a < 0). (19)

The new mathematical function F ( a ) in (18) and (19) now

provide information for the determination of T ( a , p) without

considering the distribution function p(a, p). It should be

noted that F ( a ) is expressed only in terms of flux density

values of the B-H limiting loop (i.e., B d and Bu). Thus, only

the B-H limiting loop is needed for the implementation of

the hysteresis model.

B. Differential Inductance and Hysteresis Loss

When the flux densities in two consecutive time steps are

obtained, the differential inductance in the kth stage of the

ladder network can be calculated as following

fork = 1 ,2 , . . . , n (20)

where A k is the cross-sectional area of the kth eddy current

path, and subscripts (m) and (m - I) refer to the mth and

(m - 1)th steps, respectively.

When the effect of hysteresis is considered, the differential

inductance L(iL) is nonideal. The average hysteresis loss

dissipated in the kth inductor can be calculated by

The total average hysteresis loss ph is therefore equal to

n di l

d t

iLL(iL)A dt . (21b)

k = l k=l

C. Equivalent Resistance and Eddy Current

and Anomalous Losses

It has been shown in Part I of this paper that the equivalent

resistance accounting for the classical eddy current loss P, and

anomalous loss Pa in the kth eddy current path is

(22) R k - v$ - RDck

P e a k T J k r c k

where v k is the voltage across the resistor R k , Peak =

( P e + P a ) k the sum of the classical eddy current and anomalous

losses, R D C ~ the dc resistance of the kth eddy current path,

r J k = 5 [I +- (I + +)+I the correction factor for the skin

effect, J k = U (g) kay + J k - 1 is the maximum eddy current

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HUI et al.: A GENERALIZED DYNAMIC CIRCUIT MODEL OF MAGNETIC CORES-PART I1 255

I F a jc = 1 2 , ... ,Nc

Hystcrrsis loso PN = 0, and eady C l t m n t

Fig. 5. Flow chart of dynamic circuit model.

density in the path, Jo = 0 Nm2, Ay the thickness of the

eddy current path, r,k = 1 + c, & the correction factor to

include the anomalous loss, and C, and C, are the eddy current

and anomalous loss coefficients, respectively, determined by

the standard core loss separation procedure as explained in [l]

and [2] .

In terms of the equivalent resistance, the sum of the classical

eddy Current and the anomalous losses in the kth eddy current

d t k

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256 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 11, NO. 2, MARCH 1996

r - - - - - - - - -

1 Scalar Preisach Model

I

I

I

I

Stackempty? 7

H going up? +

I

I

I

I

Calculate B on an upward

trajec%oq by Eqns. (16) aod (17)

Calculae B on the initial curve

Fig. 6. Flow chart of the Preisach model subroutine.

path is equal to where V, = K / N - R,(Ni,) is the voltage across resistor

&, and Pea and Ph are the sum of the eddy current and

anomalous losses and the hysteresis loss, respectively. The

term on the left hand side of (24) is the total power fed into the

T

P e a k 1 1 R k i z d t . ~ 3 ~ )

" 1 sum of the classical eddy current and anomalous losses, and the

second term on the right hand side is the total hysteresis loss

dissipated in the nonideal inductors L(iL), ( I C = 1 , 2 , . . . , n),

for an n-stage ladder network.

'ea = 2 peak = T 1 R k i E dt. (23b)

k=l k=l

D. Total Core Loss Calculation

The general core loss expression of the magnetic core model Iv. SOFTWARE IMPLEMENTATION

can be described, based on the law of energy conservation, as

Fig. 5 illustrates the flow chart of the operation of the

magnetic core model. The program can be used to run the

core model routine N, times. User-chosen N, points can

be obtained within each switching period. Depending on the

nature of the excitation, a variable time step TLM technique [8]

has been implemented in the routine so that any nonlinear ex-

citation waveforms with transients can be incorporated without

$ LT VRn(Nis) d t

T T

R k i i d t + 2 $1 iLL(ii) (5) dt

k=l k=l

= pea + ph (24)

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257 HUI ef al.: A GENERALIZED DYNAMIC CIRCUIT MODEL OF MAGNETIC CORES-PART 11

I Generator H Amplifier f - I ' I I

I I I IQlannel2 I I

I I 1 1

Data Precision

6OOO Signal

voltage (v)

Current (A)

1.5

1

0.5

0

-0.5

-1

2

Fig. 7. Experimental setup of annular ring example.

TABLE I

PARAMETERS AND DIMENSIONS OF THE LYCORE-140 CORE EXPERIMENTAL SETUP

losing transient information. This is particularly important in

the study of magnetic core behavior in many modern switched

mode power supplies because the excitation waveforms are

normally highly nonlinear with rapidly changing edges.

The flow chart of the Preisach hysteresis model routine is

shown in Fig. 6. The input data are the limiting hysteresis

loop, and the specified magnetic field for simulation. The stack

keeps the flux density B and magnetic field H values of each

reversal point (local extremum) and is initialized empty (start

from the initial curve) before start. In checking the stack, a

pair of reversal points is popped out of the stack whenever

the specified H exceeds the value kept in the stack. This is to

wipe out a completed minor loop since it will not have any

effect on the future state of magnetization.

V . EXPERIMENTAL VERIFICATION

A. Annular Ring of Silicon Iron with Square

Wave Voltage Excitation

Fig. 7 is an experimental setup for measuring the response

of an annular ring of silicon iron Lycore-140 0.35 mm lam-

ination, under square wave voltage excitation. The primary

and the secondary coils are identical. The primary voltage,

primary current and the secondary voltage are recorded via

-lS -2 f

Hyst.+Eddy+Anom. I

Hyst.+Eddy+Anom. I

-1.51

(b)

Fig. 8. Results of discrete modeling under 100 Hz square wave voltage

excitation. (a) Comparison of voltage and current waveforms; - - - measured,

- calculated. The ideal value of V, used in the calculation is slightly

different from the actual signal because of the nonideal characteristcs of the

power amplifier. (b), Comparison of equivalent hysteresis loops.

three sepatate channels in the signal processing system. The

parameters and the dimensions of the setup are listed in Table I.

simulated

and measured current and equivalent hysteresis loops at 100

Hz are shown in Fig. 8. Since the skin depth (0.5 mm) is

greater than the steel sheet thickness (0.35 mm), only one

layer of eddy current path is assumed, which is identical to

the model used in [l]. The time step used to obtain the

result is 10 ,us. The total calculated core loss is 0.0691

W, and the measured 0.0726 W. The percentage error is

4.82%. Comparing the various B-H loops in Fig. 8(b), the

accuracy improvement in incorporating the anomalous loss in

the magnetic core model is obvious.

2 ) Lycore-140 Core-I kHz Operation with Width of Eddy

Current Path Larger Than Skin Depth: Fig. 9 shows the

simulated and measured results of Lycore-140 core at 1 kHz

I ) Lycore 140 Core-I00 Hz Operation: The

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258

Voltage (VI

IEI

Fig. 9.

network model; - - - measured, __ calculated.

Simulated and measured results at 1 kHz with a one-stage ladder

Voltage gr) Cur" {A)

4 T T 0'4

41 I -0.4

Fig. 10.

network model; - - - measured, ~ calculated.

Simulated and measured results at 1 kHz with a two-stage ladder

excitation, assuming only one eddy current loop. At this

frequency, the skin depth (6) for the core is 0.134 mm. The

width of each lamination in the core is 0.35 mm, giving the

width of each eddy current path (we) to be 0.175 mm. Thls

means that the condition set in (11) is not satisfied. It can

be seen that the calculated current is lower and flatter than

measured. The calculated specific core loss per Hertz is 0.036

W/kg/Hz, and the measured core loss is 0.0336 WkgMz. The

percentage error is 7.2%. This inaccuracy can be attributed to

the reaction of eddy current on the distribution of flux density

in the core [I], 121.

-with Width of Eddy Current Path Smaller Than Skin

Depth: Applying the condition set in (1 l), the cross section

of the core is divided into two layers of eddy current paths,

resulting in the width of each layer being 0.0875 mm. Width of

the eddy current path (we) is now smaller than the skin depth

(6). The core model with two assumed eddy current paths

is represented by a two-stage ladder network (i.e., n = 2).

Fig. 10 shows the simulated results using a two-stage ladder

network. The simulated current has better agreement with the

measured current than that in Fig. 9. The calculated specific

core loss per Hz is 0.0346 W/kg/Hz, and the percentage error

is reduced to 2.98%.

B. A Ferrite Toroid in a Switching Circuit-100-kHz Operation

A SiemensMatsushita N47 soft ferrite R16 toroid core in

a high-frequency switching circuit, as shown in Fig. 11 was

3E TRANSACTIONS ON POWER ELECTRONICS, VOL. 11, NO. 2, MARCH 1996

4-0

1 I

I I

2

Fig. 11. Schematic of a high-frequency switching circuit.

TABLE I1

DIMENSIONS AND PARAMETERS OF THE N47 FERRITE CORE AND WINDINGS

Outer diameter of the CQE 16.33 mm

also simulated. Table I1 lists the dimensions and parameters

of the core and windings.

Fig. 12 shows the simulated and measured current wave-

forms and the 23-H loops of the core at 100 kHz switching

frequency. The cross section of the core was divided into five

paths. The time step is 50 ns. The calculated specific core loss

per Hz, corresponding to the area enclosed by the calculated

B-H loop in Fig. 12(b), is 2.592 x W/kg/Hz, and the

measured specific core loss, corresponding to the area enclosed

by the measured B-H loop in Fig. 12(b), is 2.566 X

WkgEIz. The percentage error is about 1%.

VI. CONCLUSION

A generalized dynamic discrete circuit model of magnetic

core for both low- and high-frequency applications has been

presented. The dynamic circuit core model takes all sources of

nonlinearity and core losses into consideration. The nonlinear

B-H loop and the hysteresis loss are incorporated in the

distributed nonideal inductors and calculated by the Preisach

scalar model of hysteresis. The classical eddy current and

anomalous losses are included in generalized nonlinear resis-

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HUI ef al.: A GENERALIZED DYNAMIC CIRCUIT MODEL OF MAGNETIC CORES-PART I1 259

Flux Density 0

0.2

0.15

0.1

0.05

0

-0.05

0 . 1

-0.15

-0.2

-150 -100 -50 0 50 100 150

Field Strength (Nm)

(b)

Fig. 12. Comparison of simulated and measured. (a) Current waveforms. (b)

B-H loops.

tors. All model parameters, such as limiting hysteresis loop

and specific core loss coefficients, can be obtained from the

manufacturers’ data sheets.

The discrete transform technique together with the Preisach

hysteresis model provides a useful means to handle hysteresis

nonlinearity. The resultant tridiagonal nonlinear system model

equation features easy programming and fast computation. A

criterion for choosing the number of stages of the ladder net-

work is discussed and demonstrated with numerical examples.

Comparisons of the simulations for two types of cores and

different frequencies confirm the flexibility and accuracy of

the generalized model.

REFERENCES

J. G. Zhu, S. Y. R. Hui, and V. S. Ramsden, “Discrete modeling

of magnetic coires including hysteresis, eddy current, and anomalous

losses,” ZEE Proc.-A, vol. 140, no. 4, pp. 317-322, 1993.

-, “A dynamic circuit model for solid magnetic cores,” in ZEEE

Power Electron. Spec. Con$ (PESC ’93) Rec., Seattle, WA, June 20-24,

1993, pp. 1111-1115.

I. D. Mayergoyz, “Mathematical models of hysteresis,” IEEE MAG-22,

no. 5 , pp. 603408, 1986.

S. R. Naidu, “Simulation of the hysteresis phenomenon using Preisach’s

theory,” IEE Proc.-A, vol. 137, no. 2, pp. 73-79, 1990.

S. Y. R. Hui and J. G. Zhu, “Numerical modeling and simulation of

hysteresis effecls in magnetic cores using the transmission line modeling

and F’reisach thieory,” in ZEE Proc.-Sei. Meas. Technol., 1995, vol. 142,

no. 1, pp. 57-62.

S. Y. R. Hui and C. Christopolous, “Discrete transform technique for

solving coupled integro-differential equations in digital computers,” ZEE

Proc.-A, vol. 138, no. 5, pp. 273-280. 1991.

__ , “Discrete transform technique for solving nonlinear circuits and

equations,” ZEE Proc.-A, vol. 139, no. 6, pp. 321-328, 1992.

S. Y. R. Hui, K. K. Fnng, M. Q. Zhang, and C. Christopolous, “Variable

time step technique for transmission line modeling,” ZEE Proc. -A, vol.

140, no. 4, pp. 299-302, 1993.

S. Y. R. (Ron) Hui (SM’94) for a photograph and biography, see this issue,

p. 250.

Jian G. Zhu (S’93--M’93) for a photograph and biography, see this issue,

p. 250.

V. S. Ramsden photograph and biography not available at the time of

publication.

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