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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 Authorized
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02,2020 at 08:32:29 UTC from IEEE Xplore. Restrictions apply. 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 “. Authorized
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02,2020 at 08:32:29 UTC from IEEE Xplore. Restrictions apply. 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 Authorized
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02,2020 at 08:32:29 UTC from IEEE Xplore. Restrictions apply. 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 Authorized licensed use limited to: University of Sydney. Downloaded on September 02,2020 at 08:32:29 UTC from IEEE Xplore. Restrictions apply. 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 Authorized licensed use limited to: University of Sydney. Downloaded on September 02,2020 at 08:32:29 UTC from IEEE Xplore. Restrictions apply. 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) Authorized licensed use limited to: University of Sydney. Downloaded on September 02,2020 at 08:32:29 UTC from IEEE Xplore. Restrictions apply. 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 Authorized licensed use limited to: University of Sydney. Downloaded on September 02,2020 at 08:32:29 UTC from IEEE Xplore. Restrictions apply. 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- Authorized licensed use limited to: University of Sydney. Downloaded on September 02,2020 at 08:32:29 UTC from IEEE Xplore. Restrictions apply. 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. Authorized licensed use limited to: University of Sydney. Downloaded on September 02,2020 at 08:32:29 UTC from IEEE Xplore. Restrictions apply.