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CHARACTERIZATION AND MODELING OF INDUCTORS By LongChing Yeh A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1984 ACKNOWLEDGEMENTS I would like to express my gratitude and deep appreciation to the chairman of my supervisory committee, Professor J. Kenneth Watson, for his guidance and encouragement throughout the course of this research. Also, I give thanks to Professors A.M. Meystel, S.S. Li, J.R. Smith and P.P. Kumar for their advice and support. Thanks are also extended to W.L. Wang, Robert Jen, David Talcott for their assistance. Special thanks are extended for the financial support of National Science Foundation. The author thanks his wife and his parents for their patience and encouragement throughout his graduate school career. TABLE OF CONTENTS ACKNOWLEDGEMENTS... ......... ........................ ABSTRACT............................................... CHAPTER INTRODUCTION .................................. MODELING OF INDUCTORS ......................... 2.1 Introduction ............................. 2.2 Measurements and Calculations ............ 2.3 Limitations of Legg's Model and Modified Model .................................... 2.4 A New Model for Lossy Inductors .......... 2.5 Relation of Model to Legg Equation....... 2.6 Conclusion ............................... APPLICATION OF LOSSY INDUCTOR MODEL TO Q FACTOR AND POWER LOSS OF COIL ........................ 3.1 Application of Model to Q of Coil ........ 3.2 Power Dissipation Using the Model ........ 3.3 Measurements and Calculations ............ 3.4 Conclusion ................................ NORMAL INDUCTANCE MEASUREMENTS ................ 4.1 Introduction ............................. 4.2 Measurements and Results ................. 4.3 Summary................... ............ INCREMENTAL INDUCTANCE MEASUREMENTS ........... 5.1 Incremental Inductance Measurement....... 5.2 Summary ................................. HYSTERESIS LOOP OBSERVATIONS .................. 6.1 Introduction ............................. 6.2 BH Loops ................................ 6.3 Summary......... ..... .................. ... iii 1 2 PAGE ii v 1 12 12 13 20 28 31 36 40 40 43 49 49 55 55 57 66 71 71 88 90 90 92 L00 7 DISCUSSION AND CORRELATION BETWEEN THE MEASUREMENTS.............................. 103 7.1 Correlation Among the Measurements ....... 103 7.2 A Model to Explain the Transition Regions 106 7.3 Conclusion .............................. 113 8 MODELS OF NORMAL INDUCTANCE ................... 115 8.1 Introduction.............. .............. 115 8.2 Inductance Models and Distortion.......... 116 8.2.1 "Instantaneous" Model............. 116 8.2.2 "State" Model..................... 117 8.2.3 Measurement of Harmonic Distortion 120 8.3 A Physical Model of inductance ........... 122 8.3.1 A Model For the Inductance Below H .......................... ..... 129 8.3.2 Inductance in the Region Above the Critical Field.............. 135 8.4 The Magnetization at High Fields......... 137 8.5 Summary.............. ................... 141 9 A PLAUSIBLE MODEL OF INCREMENTAL INDUCTANCE... 143 9.1 Introduction ............................. 143 9.2 Reversible Mechanisms Under Small ac Excitation.............................. 144 9.2.1 Reversible Wall Displacement...... 144 9.2.2 Domain Wall Bowing................ 146 9.3 The Model.......... ..................... 149 9.4 Measurements to Support Model............ 152 9.5 Conclusion................... ........... 153 10 CONCLUSIONS................................... 160 REFERENCES....................................... ...... 163 BIOGRAPHICAL SKETCH................................... .. 166 Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CHARACTERIZATION AND MODELING OF INDUCTORS By LONGCHING YEH December 1984 Chairman: Dr. J. Kenneth Watson Major Department: Electrical Engineering This dissertation presents the modeling and characterization of inductors wound on a ferrite core, an amorphous alloy (Metglas ) core, a permalloy 4 mil tape wound core and three permalloy 1 mil tape wound cores with different kinds of heat treatment, and an MPP core. Many kinds of measurements have been made including normal inductance measurements, incremental inductance measurements, inductance quality factor measurements, effective series resistance measurements, hysteresis loop observations and measurements of waveform distortion. All the materials studied were found to have certain common features: the normal inductance increases as signal amplitude increases but eventually passes through a maximum value. Incremental inductance, on the other hand, is independent of direct current below a critical value, and then decreases for larger values of current. The critical dc field for incremental inductance was found to have the same value as the peak ac field at which there is a change of the rate of increase of normal inductance. Both critical fields may be attributed to the same physical process, the onset of irreversible motion of domain walls. This new finding, for the first time, relates normal inductance and incremental inductance measurements. A new mathematical model for inductors is worked out using a computer curvefitting program to describe the inductance and equivalent series resistance measured with ac signal levels ranging from low to high amplitude. The model, which may be used to calculate the quality factor and the power loss of inductors, also explains Legg's equation in an extended form. The voltage waveform of an inductor was found to be somewhat distorted even when the applied current is a pure sinusoid. The measured distortion was less than is predicted by a new "instantaneous" model of permeability, but was in reasonable agreement with a domainstate interpretation of the Rayleigh model. Other findings include a critique and extension of Legg's equation, an experimental validation of a domain wall model for incremental inductance, and the recognition of a omainwall unpinning model to explain the increase of permeability with ac amplitude. CHAPTER 1 INTRODUCTION This dissertation can be separated into two parts: in the first part we study the modeling of inductors under ac performance. A modified model and a new model will be presented. In the second part we study the nonlinear proper ties of magnetic materials of inductive type with emphasis on the effects due to different magnetic reversal processes. The first part includes chapter 2 and chapter 3, the second part includes chapter 4 through chapter 9. Device modeling for inductors is very important for magnetic circuit designers, but there are only a few models which exist. Three of them,the model by Legg [Le36], the model by Jordan [Jo24], and the model by Rayleigh [Ra87], are only applicable to small signal levels. No models has been derived for large signal performances. Yet large signal applications are important because they save expense and reduce the size and weight of devices. A simple circuit model of an inductor consists of a resistance in series with a pure inductance, with impedance Z= R + julL. The equivalent series resistance may usually consist of two terms, R = R + R (11) w ac 1 2 where R is the winding resistance and R is the coreloss w ac resistance. The quality factor Q, which gives the ratio of energy stored to energy dissipated per cycle, is given by Q = (12) w ac In general applications of inductors where Q and/or power losses are of interest, the engineer may wish to know how R and how L vary with signal amplitude and frequency. ac These subjects are studied in the first part of this disser tation. In 1936, Legg derived a relation between Rac and per meability based on the classical assumptions that permeabil ity is linear and is uniform inside the material [Le36], his final result is R ac = aB +c+ef (13) pfL m where a is the hysteresis loss coefficient, c is the resi dual loss coefficient, and e is the eddy current loss coef ficient. With regard to eddy current losses this expression is derived from losses in laminations, while the hysteresis losses are obtained from the Rayleigh equation. In MKS units, e=wd 2/3p, where d is the thickness of the lamination and p is the resistivity of the material. In chapter 2 we use the left hand side format of Eq.(l3) as a basis to study the Legg model at low Bm for many inductive magnetic materials which are not limited to lamination cores. Limitations of Legg's model are pointed out and a modified form is given. A mathematical model that shows how pure inductance and series resistance vary with f and B is developed. Chapter 3 describes the applications of this mathematical model for quality factor and power losses in the inductors. The materials selected for this study are described as follows: a linear ferrite (Ferroxcube 3E2A), an amorphous alloy (Metglas 2605SC with high frequency anneal, Fe 81%, B 13.5%, Si 3.5%, C 2%)), and a permalloy 4 mil tape wound core with special anneal (Magnetic Metals: core size 1321; Ni 80.5%, Mo 4.6%, Fe and impurities 14.9%; core #6), a powder permalloy (Magnetics, MPP 160 p), and three special annealed permalloy 1 mil tape wound cores made by different annealing processes (HYMU 80, core #1, #8, #14). All of the materials are in toroidal geometry possess ing no air gap (except MPP), a geometrical shape which exposes the intrinsic material properties most simply. All these materials are of commercial importance and are interesting both for physicists and engineers [Ma80,Kr79,Ts79,Na80,Bo81]. The electrical models we derived in chapter 2 and 3 are relevant to such a broad class of materials as described 4 above; thus the models might be applicable to all inductor cores without airgap. After concluding the first part of this study inductorr electrical model, chapter 2,3), this dissertation concen trates on the characterization of inductive magnetic core materials. The following introduction is given for the study in this area. The materials mentioned above are "soft" magnetic materials of "inductive" type. The outstanding properties of these materials is the ease with which their intensity of magnetization can be varied. The term "soft" refers to a large response of the magnetization to a small applied field. The term "inductive" means that the materials have low remanence; thus their BH characteristics are predom inately sloping, so that the concept of permeability p= B/H is meaningful. From a technical point of view, the most important region of the magnetization of this kind of material is that below the "knee," as shown in Fig. 11, but it is unfor tunately just this region that is most difficult to inter pret, since the processes involved depend on small irregu larities in the material. The magnetization processes also depend on the domain arrangements as well as the factors that determine the easy directions of magnetization. The magnetic response of the material, thus, shows nonlinear and hysteresis nature. The second part of this dissertation studies in this area. approach to saturation B I knee H Fig. 11 A representative magnetization curve of an "inductive" magnetic material. 6 In a recent study [Wa81], Watson carried out some exploratory work on linear ferrites, investigating the non linear responses of ferrite cores to a variety of electrical measurements. At large signal levels, there is generally poor agreement between reversible permeability, normal per meability, and pulse permeability. In a theoretical model, he assumed that reversal takes place by rotational processes and assumed that the core saturates at a specific field, he found that actual measurements of ferrite response do not fit the model. Puzzling aspects were part motivation for these research. Domain theory plays a very important role in the study of nonlinear behavior of magnetic materials. In 1907, Weiss [We07] introduced the great concepts of domain hypothesis and the spontaneous magnetization by molecular field. For a period of nearly forty years, investigators made virtually no application of domain concepts in attempting to explain the nonlinearity or the mechanism of magnetic hysteresis. It was not until 1949, when Williams, Bozorth and Shockley [Wi49] published their work on the experimental evidence of domain structure of a real material, that domain theory became absolutely central to any discussion of nonlinearity and hysteresis of materials. Chapter 8 and 9 use the domain theory to explain the experimental results of normal induc tance and incremental inductance. 7 The study of nonlinear properties of magnetic materials in general falls into two categories of approach: physical theory and empirical modeling. The theoretical approach in general consists of two methods: one is based on arbitrary natural assumptions, like that of PreisachNeel model [Pr35]. The other method, which is theoretically more sound, is based on the micromagnetic theory, like those of Brown [Br59] and Aharoni [Ah59]. How ever, these methods do not yield a simple equation of state for a ferromagnetic. In recent study [Ji83], Jiles and Ath erton published their work on the theory of ferromagnetic hysteresis based on a mean field approximation in which each domain is assumed to interact with the field H and a weighted mean of the bulk magnetization. However in their derivation of the theory, they did not take into account how different reversal processes can influence the response of the material. Using the empirical method, there have been many attempts to fit equations to actual magnetization data [De80,Ri81,Ma73]. However, no single equation has been developed to describe all the data satisfactorily: Attempts to describe the behavior of ferromagnets have always been handicapped by their restriction to only narrow ranges of field. For example, according to the review by Cullity [Cu72], there are only three instances where the magnetiza tion curve can be explained by algebraic expressions. These are the high field magnetization curve of single crystals as 8 in the work of Williams [Wi37], the high field magnetization curves of polycrystals which are governed by the law of approach to saturation as indicated by Chikazumi [Ch641, and the low fields magnetization curves of polycrystalline specimens which exhibit Rayleigh loops [Ra87]. In this study, instead of using the above methods of research, we first characterize the materials empirically, and then seek the evidences of the effects of different reversal processes on the nonlinear properties of the materials. Three kinds of measurements were used to characterize the materials. They are normal inductance measurement, incremental inductance measurement, and hysteresis loop observation. The relationships of the nonlinear responses under these three kinds of measurements for each material will be given. We hope the relationships we have found can help us to understand the nonlinear properties of the materials in application. It is well known that there are three important types of reversal processes that account for the magnetization of a material by domain theory: (1) reversible boundary dis placement, (2) irreversible boundary displacement, and (3) reversible rotation. To identify the effect of different reversal processes, one needs to know the critical field at which different reversal processes begin to involve. In addition, device geometry can cause nonlinear response in application [Wa80]. Thus to characterize the intrinsic properties of the magnetic material, one needs to take into account the geometry effect. This effect is con sidered when we explain the transition regions around the critical fields of normal inductance and incremental induc tance in chapter 7. In characterization of the material, since our purpose is to study the intrinsic properties of the material, we selected a set of measurements which are made at low fre quency. These measurements exclude the errors due to parasi tic capacitance and to the eddy current shielding effect in which the applied field can not penetrate completely through the whole magnetic material. In chapter 4 we present the data of normal inductance; chapter 5 presents results for incremental inductance meas urements. In chapter 6, we characterize the materials by BH observations. Chapter 7 investigates the relationships among normal inductance, incremental inductance and BH loops for each core. A model that accounts for the change of mode of reversal around the threshold fields is presented. Chapter 8 and chapter 9 proposes physical mechanisms to explain the experimental results. In the beginning of chapter 8, the distortion mechanisms of VI relation due to nonlinear BH relation are studied first in order to lay a foundation for a proposed physical model for normal magnetization. Chapter 10 gives conclusions of this research. The following formulae are used for our core testing, which are usually used in electrical engineering. 10 8 V = 4.44xB ANfxlO 8 rms m where (14) B = Peak flux density in gauss N = Turns 2 A = Effective core area in cm f = Frequency in Hz Equation (14), Faraday's law, allows one to determine at what voltage to excite a core for measurement at a certain flux density. This equation is valid only when the signal waveform is sinusoidal and is usually used when the signal level is low. H = NI/1 *m (15) where H = Field strength in A/m 1 = Mean path length in meter of toroidal core Equation (15) allows one to determine at what current to excite a core for measurement at a certain field strength. 9 = B/H (16) where B = Flux density in Tesla H = Field strength in A/m 11 L = 1A (17) m Equation (16) and (17) determine amplitude permeability, ,' (U=0Qor' ;r is relative permeability). The amplitude permeability is a parameter which is determined by measuring inductance; permeability is calculated using equation (17). L = N2P. (18) where L = Inductance in Henries P = Permeance in Henry/turns squared Equation (18) defines inductance from the permeance, P. For a toroidal core without air gap, P= uA/1 m. In Eq.(l8), given the P, one can either calculate the inductance or determine the turns needed if the inductance is known. CHAPTER 2 MODELING OF INDUCTORS In this chapter, measurements of inductance, L, series resistance, R, and quality factor, Q, are made for five dif ferent magnetic cores. Data are analyzed as a function of B and frequency. The Legg's coefficients for each core are presented. The limitations of Legg's equation are indicated; Legg's equation is modified to extend its vadility of application. An empirical mathematical model of pure inductance and loss resistance as a function of frequency and Bm is given for higher Bm application. The relation of this model to Legg's model is shown. 2.1 Introduction This chapter presents the results of a study on the model of an inductor when signal level varies from low to high. Equation (13) is used as a basis for an empirical determination of the coefficients for five different mag netic materials that are suitable for inductors. They are a ferrite, an amorphous Metglas with high frequency anneal, a 4 mil tape wound core with special anneal (core #6), an MPP core, and a permalloy 1 mil tape wound core with tranverse magnetic anneal (core #8). 12 13 Experimental values for the Legg coefficients calcu lated for each core material are given. The limitations of the Legg coefficients to the modeling of the five magnetic materials are also examined. A mathematical model is used to describe how the Legg coefficients vary with Bm and frequency. This model is based on the observation that inductance was found initially to increase as B increases, and that R was found to m ac increase with frequency and B m. The parameters of the model were derived with a computer curve fitting program. This model also lays the foundation for the analytic expression of Q factor and power losses in an inductor. 2.2 Measurements and Calculations Measurements of inductance, R, and Q were made using an HP 4274A LCR meter in the frequency range from 100 Hz to as high as 100 KHz in some cases. Measurements were made for specified values of Bm according to Eq.(l4) by keeping V/f constant. The measured data were manipulated into the for mat of the left side of Eq.(l3), and then were analyzed by Bm and frequency in order to determine the Legg coeffi cients. In the analysis, permeability was allowed to vary with B and with f (permeability is defined by Eq.(l7)). Thus we actually worked (RR )n A = aB +c+ef (21) fL21 m m 14 for each data point as the left hand side of Eq.(13), where A is the core cross section and 1m is the mean path length. When the values of Eq. (21) are plotted against frequency, the slope is interpreted as coefficient e; extra polation to zero frequency gives intercept aB m+ c. Con stants a, c can be determined from the family of intercept values for different values of B m. Figure 21 illustrates this processes for ferrite data in which coefficient e is found. Figure 22 shows the same process for Core #8. Figure 23 and 24 shows the processes to determine the coeffi cients a, c for ferrite and core #8 separately. The same procedure was carried out for each of the other four core materials that were studied, with the results summarized as Legg coefficients a, c, in Table 21. Coefficient e0 will be explained later. The table also gives the range of frequency and of B m. The value of Bm indicates where the intercept (aBm + c) begins to depart from linear dependence on Bm The low frequency limitations of Table 21 were imposed by the accuracy of determining Rac by subtraction, see Eq.(ll). At low frequencies where core losses are small, the measured R hardly exceeds the winding resistance espe cially for ferrite and for MPP cores. The high frequency limit indicates the approximate value of frequency at which a curve such as in Fig. 21 begins to deviate from linearity to become concave down. 15 lk2k 4k 10k FREQUENCY (Hz) LUK Fig. 21 Rac/uLf vs frequency at various B for ferrite. Rac ~Lf 40 30 20 10 70.4 G 52.8 G 35.2 G 17.6 G 16 co 0 S 0 CO 0 CD P co C) CH Cd (\l ^ ^ bfl 'H 17 25 20 aBm+C(g() 15 10 5 20 40 60 Bm (Gauss) Fig. 23 Intercepts (at f=O from Fig. 21) vs Bm. (ferrite) 18 20 15 10 5 0 100 200 300 400 Bm (gauss) Fig. 24 aBm + c vs B. of core #8 at low field range. >1 r H co ~ 0 0 ) 4 U r) 0 19 CD VI vl 441 o VI 0 vi 44 vi C) ri 4 r4 I 4 (N I CO 4 X CD o\ L4 x H CO 4 I I 14 ) I1 4 I S4 00 km o0 VI o v CO x m * Lin 0o cn u Or mo U) X; 0 M 4 >4 4 sO to Ui 4 0 C') l 044)C E 1 a) CM 0 vl o VI 0 Vo C) x* 4 0 14 x in 0 4 X CD 0 0 I SIn pu fcl LC >1 0 r4 H P4V U ) OH *4 4 H C u a) me 20 2.3 Limitations of Legg's Model and Modified Model When the family of curves of Fig. 21 or 22 is exam ined carefully, it may be observed that the lines are not exactly parallel. The slopes are found to increase slightly with Bm, as shown in Fig. 25. A similar variation was also found for all the core materials, with core #8 as another example, as shown in Fig. 26. Others are not shown here. The intercept at B =0 for the figure of e versus B is the m m coefficient e0, listed in Table 21. In the data range of study, this variation of slope was found significant for ferrite, core #6, core #8 and MPP core. For this reason, data for these three materials may be modeled more accu rately by modifying the e term of the right side of Eq. (1 3). Thus we modify the Legg's equation as follows: R ac = aB +c+e(l+dB )f (22) pfL m m For the geometries and turns used, coefficient d was found to be 1.37 x 102 for ferrite and d = 2.33 x 103 for core #6, 2.52xl04 for core #8 and 2.13 x 103 for MPP. Only negligible dependence was observed for Metglas in the range of the study. It is of interest to compare the results of Legg coef ficients we worked out for MPP core with catalog data, wuich 6 data were found only for the MPP core: a = 0.9 x 10 c = 2 5x 15xl 2.5 x 105, e = 1.7 x108 for a = 160 (suitable below 200 g, frequency limit unspecified). Our results agree within 21 Bm (Gauss) Fig. 25 Slopes (of lines from Fig. 21) vs Bm interpreted as e. The line is a linear regression with eo = 3.4 x 104 P0 d = 4.67 x 106 p0 * e(104 o) 22 0 o 0 a, 0 O 0 CC c00 0 o *O 1O 0 00 0 CI SCH 0 0 0 0 C' o o 'C4 1:1.4 I 23 about 5 percent for e, c but are about 3 times higher for the hysteresis coefficient, a. The more serious limitation of the Legg data is the low range of allowed induction, B m. When the measuring induc tion Bm is increased beyond the listed value of Table 21, the value of (a B + c) is found to deviate from a linear m relation with B The data points of Fig. 27 and of Fig. 28 illustrate this effect and thereby raise the question whether the value of coefficient a is valid. The only approximate agreement of Fig. 27 with the data of Fig. 23 raises the additional question whether our procedure Eq.(2 1) leaves the term a entirely independent of the number of turns as it should be. For the highfield measurements of Fig. 27, n =40, whereas more turns (n =100) were used for the low field measurement of Fig. 23. The values of e versus Bm also show two different slopes at two different numbers of turns, as can be seen from Fig. 29 for ferrite as an example. Figure 210 shows another example the effect of number of turns on the value of aBm + c for Metglas, thus the Legg coefficient a and coefficient d seem to be depen dent on the number of turns of winding. It interesting to note that although both the slope of aB + c and the slope of e as a function of B increase as n m m increases, as has been shown in Figs. 27, 29 and 210, their intercepts at Bm seem to be independent of the number of turns. This suggests that the coefficients c and e0 might be independent of the number of turns. 24 0 150 300 450 600 750 Bm (Gauss) Fig. 27 (aBm + c) vs Bm. aBm +C 800 (ferrite) 25 I I 1 I i I I I I I I 1 1 i I I 26 12.0 10.5 9.0 e(104po) 7.5 6.0 4.5 3.0 0 150 300 450 600 750 Bm (Gauss) Fig. 29 Coefficient e versus Bm. 900 (ferrite) 27 00 O o Ca 3 M '4 * C U) 0 0d 0) + .el *H <H F.L 28 In the next section, presentation is made on an ana lytic model (results from a rather preliminary model) for lossy inductors that seems to be valid over a wider range of Bm as compared to the Legg coefficients. The model gives an interpretation of the Legg coefficients and also sheds light on the frequency and Bm dependence of quality factor, Q, and of power loss, P, which will be described in chapter 3. 2.4 A New Mathematical Model for Lossy Inductors A new model is now proposed to represent a lossy induc tor. Normal inductance typically increases with the ampli tude of the measuring signal, and is modeled here as L = L 0(+bB ) (23) where L is initial inductance and where coefficient b is the model parameter of interest. This trend was observed for all five materials of the present study: Figure 211 and Fig. 212 show typical results for ferrite and Metglas core as examples; others are not shown here. The effective series resistance of an inductor may be represented as a series in powers of frequency, so that Eq.(l1) becomes R = r0+r f+r f2 (24) 29 0 0 0 0 0 0 0 0 0 0o << o 30 4D r4 0 0 2 @0 cc  .+ 31 where r0 is identified as the winding resistance Rw and thus the two higher order terms define the core loss parameter R A curvefitting computer program was used to process ac our measured data for R using the format of Eq. (24), which turned out to match with our data very well. Figure 213 shows a calculation compared with data for the ferrite core inductor at Bm = 44 G, plotted vs frequency. The coefficients resulting from the Eq.(24) analysis were found to vary with Bm (in units of gauss) and can be represented as r = r 0+r B +r B2 (25) 1 10 11 m 12 m r2 = r20+r 21Bm (26) where again the coefficients were identified using the com puter curvefitting program, as shown in Figs. 214, 215. In summary, the model consists of Eqs. (23) through (26) for which the resulting coefficients are listed in Table 22 for three of the five materials. Due to the frequency limi tation of the instrument, data for core #6 and Core #8 are not listed. 25 Relation of Model to Legg Equation When model Eqs.(23) through (26) are substituted into Eq.(l3), the Legg equations (13) and (21) take the form, with k=n A/i and omitting B terms to simplify our m analysis: analysis: 32 15 1 10 R (ohms) Fig. 213 ~~~1 / I, / I .. /  FREQUENCY IFI I i I I I f *Rac vs f for ferrite inductor. Dot points are measured data. curve is calculated with ri = 5.033 x 10 and r2= 1.675 x 108. . Y (kHz) _ i i i i I I I t I I I t  l  i I i 33 S 1 I I I I I ! .3 .6 .9 Bm kGauss Fig. 214 Parameter rI vs B for ferrite calculated by model. .705 r2(10'8) .47  .235 core. Curve is .3 .6 .9 Bm kGauss Fig. 215 Parameter r2 vs B for ferrite core. Curve is calculated by model. ri( 103) 34 Table 22 Model parameters for three cores. Parameter b rO rl 11 r 12 r 20 21 k Ferrite 1.7x10 4 4 2.8xl04 5.7xl06 3.0xl09 8 1.4xl0 8 6.4x108 6.4xlO11 6.94x101 Material Metglas0 2.1x 4 1.2xl04 1.8xl06 0 7 1.6xl07 1.lxl10 6.37xl0~1 6.37xi0 MPP Core 6.3x105 5 2.1xl05 6 1.8x106 0 8 3.0xl08 0 48.04 e 35 R r +rf ac= 2k= (27) L L (1+bB ) (rl0+rllBm)+(r20+r21B m)f L2(1+2bB +b2B2)/k 0 m m After some manipulation and assuming 2 b Bm << 1 the right side of Eq. (27) takes the form (a +aaB a B2)+(e +e B e B )f (28) 0 1 m 2 m 0 1 m 2 m which is a generalization of Eqs.(21) in which the actual Legg coefficients may be identified as c = ao, a=(al2a2Bm), e 0, d=(el2e2Bm)/e0. (29) The a and e parameters may be found from Eq. (27) as below, if each right side is multiplied by k/L02 a0 = rl0, e0 = r20 (210a) a1 = (r112brl0), e1 = (r212br20) (210b) a2 = 2br11, e2 = 2br21 (210c) In Fig. 27, the smooth curve identified as the mathematical model is Eq.(28) plotted for f = 0, which is the aseries part of Eq. (28). The eseries multiplier of f in Eq.(28) is shown in Fig. 29 as the smooth curve. The new model is therefore consistent with a generalized form of the 36 Legg equation and is evidently applicable over a wider range of B However, we have not addressed the problem of presenting the model in a form that is assured of being independent of geometry and of turns. It is interesting to compare the data of e and the data of aB + c as a function of B of the five materials, which m m are plotted as a function of f in loglog scale in Fig. 216 and Fig. 217. Core #6 has the highest e value; ferrite has the lowest e of the five; however, the e value of ferrite as a function of Bm increases sharply as Bm increases. Figure 217 shows that MPP core has the highest aB +c loss, while core #8 has the lowest aB +c loss of the five. The residual m losses of core #6 and core #8 are very small. 2.6 Conclusion An experimental investigation has been carried out on inductors wound on five different kinds of magnetic core materials, yielding values for their Legg coefficients that are listed in Table 21. These data may be interesting in their own right although they were found to be limited to low values of B m. Legg's equation can be modified by adding a cross product term to the right side of Eq.(l3) to increase its accuracy of application even when Bm is low. A mathematical model of inductance and of R was evaluated for three of the five inductors, using a curve fitting computer program to find coefficients to match the 37 *H 0 *H Cl 0 r. 0 *H 0 a 38 39 experimantal data. The mathematical model was then manipu lated into the form of a somewhat generalized Legg equation which seems applicable to a higher B m. Frequency variations of inductance due to capacitance or other effects were not considered. The problem of making the model independent of geometry and turns has not yet been studied; so the coeffi cients listed in Table 21 may be specific to these specific inductors. CHAPTER 3 APPLICATION OF LOSSY INDUCTOR MODEL TO Q FACTOR AND POWER LOSS OF COIL This chapter describes how the lossy inductor model derived in chapter 2 can be applied to the modeling of Q factor and of power losses of an inductor. 3.1 Application of Model to Q of Coil The quality factor Q of an inductor is defined as the ratio of energy stored to energy dissipated per cycle, as shown in Eq.(l2), repeated here as Eq.(31): = L (31) SR +R w ac A representative curve of the Q factor of an inductor may be shown as in Fig. 31. When Eq.(31) is rewritten by substituting Eq.(25) for (31) in the denominator, the result shows that 2irfL _ Q = 2 (32) r0+rlf+r2f Assuming the effective L is invariant with frequency, at low frequency end, the r2f2 and the rlf terms are negli gible comparing with r0, the asymptote for Q at low frequency is 40 41 CLW R oL Rac log f Fig. 31 A representative Qfactor for an inductor plotted against frequency. log Q 42 Q = (33) r0 Thus, Q increases as f increases. At the high frequency end, both the terms r0 and r1 f are smaller than r2 f2 thus Q = 2fL (34) 2 rf This analysis supports the concavedown trend of Fig. 31. The maximum Q occurs at the frequency where dQ/df = 0, which is readily shown to satisfy the condition r0 = r2f 2, namely Ir0 f = I (35) \ r2 The Q factor at this frequency is Qmax = (36) 2\ r0r2+rl A more general form of Eqs. (35) and (36) may be obtained by allowing variations of B according to Eqs. (23), (25) m and (26). The results are I r0 f (r +r B (37) \I'20 21 m The maximum Q of the inductor is then 2WTfL0(l+bB) (38) ax = ( +r 2Bm)+ 38)rl Bm 2\ r0(r20+r21Bm)+(rl0+rllm) 43 The frequencies at which Q has half value of Qmax are the frequencies that satisfy the following equation: (r0+rlf+r2f2) = 2 rr2 +r (39) which are shown to be rl \ rl4r2(r04\r2r02rl) f =1 2 1 (310) 2r2 2 The "bandwidth" (the width of the Q curve with Q>1/2 Qmax) of the Q figure is then 12 \Irl 4r2(r04\r2r02rl) 1 = 22 (311) r2 This equation shows that the larger the r2, the narrower the "bandwidth." Figure 32 shows values of Q of Metglas core measured as a function of frequency, and calculated points for Qmax' using Eq.(38). The solid lines represent the calculated value of Q using Eq.(32). The good agreement between the measured Q and the calculated Q supports the consistency of the model and of the data analysis. 3.2 Power Dissipation Using the Model We described in chapter 2 that a lossy inductor can be represented as a pure inductor in series with a resistance(series model). This model, however, can be 1 * 0 0 0 4' 0 o 0 0 C'\ 4 I , *H" 0 ? 45 transformed into a parallel model which is an inductor L in p parallel with a resistance R as shown in Fig. 33. The relation between this two models is R = R (l+Q2) (312) p s L = L (1+) (313) with the Q value UL R Q s a (314) R u=L s p For the series model, the power losses of the inductor is P = 2 R (315) rms s where I is the rms value of current flow in the inductor. For the parallel model, V2 P rms (316) R p where Vrms is the voltage (rms value) across the inductor. 2 Since R = R (1+Q ), and for a sinusoidal voltage, B relates to Vrms according to Eq.(l5), thus 2 A2N2B2 = (4.44) x m(317) 1016 Rs (+Q2 s For a high Q inductor, Q>>1, thus 46 ip p (b) Fig. 33 Inductor Models (a) series model (b) Parallel model. 47 22 2 2 2 P = 1.97 x 1015msA = .97x1015 A2N2B (r+(r +r B )f+(r20+r21B )f 4w L (1+bB ) (318) 40 m Equation (318) shows that power loss is a complicated func tion of frequency and B This equation, however, can be simplified under some circumstances. For example, if f2 loss is dominant, Eq.(317) in logarithmic form becomes B2 log P = c + log m 2log Q. (319) 0 (r +r2 B 3) 20 21 m where c0 is a lumped constant that includes geometry A and number of turns N. In the following equations, all the con tants c.'s are lumped constant. Equation (319) shows that at a specified Bm, log P = cI 2 log Q. For a typical inductor where Q is a monotoni cally decreasing function of frequency at high frequency end, as shown in Fig. 31, the functional dependence of Q on f may be written as log Q = c2 c(log f (320) thus, log P = c3 + 2dllogf, P = c4f = c4f (321) On the other hand, if f loss term is dominant, Q is nearly a constant, then at a specified Bm, 48 log P = c5 + log f, (322) which is a typical result of hysteresis loss. Similarly, at a specified frequency, Eq. (318) can be simplified as 15 B2(R +R Bm) P= 1.97x1015 m 0 m (323) L0 (l+bBm) 0 m 2 where R0 is the sum of r0,r10f and r20f and R1 is the sum of r11f and r21B Thus, log P =c6+21ogB m+21og(R+R B m)21og(l+bB ) (324) In certain Bm range, and if the last term is neglected, the superposition of the second and the third terms in log Plog Bm plot shows that P may take the form P = c 7B (325) with p >2. In summary, the analysis with our model shows that power loss of an inductor depends on the frequency and Bm in such a way that power loss is proportional to the c(th order of f and pth order of B m. The value of c( depends on the relative magnitude of the three components of loss. If f loss is dominant, c( is close to 1, if f2 loss dominate, d is close to 2. The value of p value depends on the range of Bm of the inductor in application, it is not a constant. 49 3.3 Measurements and Calculations The ferrite core, the Metglas core, and the MPP core are selected to verify our analysis. Measurements of L ESR and Q in series mode and L , R in parallel mode were made using HP 4274A LCR meter at various Bm and at various frequencies ranges; data are for mulated into Eq.(318) to calculate power losses of the inductor; Figure 34 shows the results of power loss versus frequency at three Bm values with Metglas as an example; solid lines are those calculated from Eq.(318). Figure 35 shows the example of P versus Bm at three frequencies for ferrite. Figure 36 shows the example of P vs Bm at f=1k for Metglas. Model calculation and experiment result are in very good agreement. For the three materials, in our data range, power losses are fairly well described by Eq.(321) and Eq.(325), and the parameters d and p are listed in Table 31. Data regions are also indicated for each core in the table. 3.4 Conclusion Power losses of inductors wound on ferrite, Metglas and MPP core materials have been studied using an equivalent electrical circuit model. Important elements of the circuit like pure inductance and ESR, which were modeled in a mathematical form in chapter 2, were manipulated into the form of Q factor and power loss to describe their general features as a function of Bm and frequency. A mathematical 50 ( Me;glas ) P (watts) Points: measured da Solid lines are caV rorm model. Bm=40o G Fig. 34 Power loss as a function of frequency at various Bm. Solid lines are calculated from model. (Metglas) 10 51 1000 Power loss vs Bm at various frequency. (ferrite) Fig. 35 1703 I 1 i i i Fl P (watts) Solid L  10 V I ii ti~~ ine : del 0 : Data Pints f ik Hz (Gausses) 100 i I 1 I 1 I\ I IC t Fig. 36 Power loss as a function of Bm at f=1000 Hz. (Metglas) 53 Table 31. Power Loss Parameters of Three Materials. Material I c( IFerrite I 1.08 I 2.39 Metglas 1.70 2.0 MPI I I Data Region 4k < f < 40k, B < 500 I S m I 4k < f <40k, B < 600 4k < f < 40k, Bm 50 I I 54 expression for Q of a coil has been derived. The frequency at which Q has a maximum, and the maximum Q value of the inductor can be calculated from our model. Power loss of an inductor as a function of frequency and Bm are derived; power losses were found to vary with frequency in the form of Eq.(321) and with Bm in the form of Eq.(325) of our model. Parameters of d and p of the three cores are listed in Table 31. CHAPTER 4 NORMAL INDUCTANCE MEASUREMENTS This chapter reports measurements of normal inductance for six inductors wound on the following materials in toroidal form: a ferrite (Ferroxcube, 3E2A), an amorphous alloy (Metglas 2605SC with high frequency anneal), a per malloy 4 mil tape wound core (core #6), and three special anneal 1 mil tape wound permalloy cores with different kinds of heat treatments. The data of this chapter are inter preted in chapters 7 and 8. 4.1 Introduction Normal inductance, often referred to simply as induc tance, is measured when the specimen is in the ac magnetic state. Normal inductance is related to amplitude permeability, a', by Eq.(l7), where pa is defined by the slope of a line drawn in Fig. 11 from the origin to a point defined by the total B/H. By means of hysteresis loop, one also can define a large number of permeabilities (0166). The initial permea bility = lim (41) H>0H 55 56 This is the permeability at the origin of the curve at the first magnetization. The amplitude permeability B a = B (42) a ' This is the permeability for a stated value of the field strength (or induction) when the field strength varies periodically with time. Many amplitude permeabilities are defined due to the fact that BH relation of a magnetic material is nonlinear during the ac cycle (i.e., hysteresis effect). Thus when deal with permeability, one must state which curve form for the independent variable is chosen (e.g.if H is sinusoidal then B is not sinusoidal). Furthermore, we can select the relevant B and H or the separate B and H e.g. ap = (B / m m ap m H m) is the peakto peak amplitude permeability often named normal permeability when it is derived from the magnetiza tion curve. The definitions of each of the different amplitude permeabilities can be seen from the book by Olsen [0166]. It should noted that at a specified signal level (B or H ), m m and when the signal level is low, the differences among the various amplitude permeabilities can be ignored. The inductance of an inductor wound on magnetic core is found to depend on the level of testing signal, and is of interestt here. The detailed physical and chemical properties of the ferrite and Metglas cores and core #6 have been described in chapter 1. The properties of the three special anneal, 1 mil 57 tapewound permalloy cores (core #1, core#8 and core #14) are described as follows: all the materials were HYMU 80. Core #14 was annealed to maximize the initial permeability by adjusting the order/disorder in the alloy so that aniso tropy constant kI = 0. Core #1 was heat treated to provide more disorder and more vacancies in the crystal structure. Core #8 was heat treated in an axial magnetic field to introduce a uniaxial anisotropy in the axial direction, per pendicular to the direction of magnetization, in order to reduce remanent magnetization for inductor applications. All the cores are toroidal with mean path length of 8.98 cm. The effective cross section area for each core is based on its core weight. The dimensions of all the above mentioned six cores are listed in Table 41. 4.2 Measurements and Results Measurements of normal inductance and Q factor were made using an HP 4274A digital LCR meter. Before the meas urements, materials were demagnetized by applying an alter nating field with an amplitude high enough to cause the inductance to approach saturation, then slowly reducing the amplitude to zero. The signal level of measurement was adjusted by con trolling the current (in rms value) through the coil, and the peak field strength Hm was calculated according to Eq.(15). co 0 U 0 u O 58 COO O n O I o m  L m Ln M Lr CM L A m CN MA IV N r N O N n 'I N 0 N co m c CN C S C L * *N c O  O r4 L A c 0 N C4 (U N 0 H NgU 4 0 il .i t) 4) 4 *rz4 .H C) &4 41 59 Measured normal inductance is plotted against current I and Hm as shown in Figs. 41 to 46, for the six materials. The number of turns of winding on each core is given in the legend. In each case, the curve rises from a point on the L axis above the origin (the initial inductance is nonzero) to a maximum (the maximum inductance) and falls off rapidly and then more slowly as Irms increases. When the curves of L versus Hm are examined carefully, many interesting features may be observed: The figures sug gest that there are three regions of operation: a low field region in which L increases linearly with Hm with slope a1 (except at very low fields), (a1 is defined as (1/L0)dL/dH, L0 is initial inductance), a medium Hm region where L increases linearly with another slope a2 (defined as (1/L2)dL/dH, where L2 is the intercept at H=0 of second linear regression line), and a high field region where L decreases from its maximun value at H mm. Thus, in the first region L = L0(l+alHm) (43) and in the second region L = L2(1+a2Hm) (44) At the end of second region and before the high field region, the inductance increases at a slower rate and then reaches a maximum. Different materials have different values of aI and a2.* The separation point of first and 60 0 0 o o a O <8< 0 ,  0 0 Eo 1 O 00 e 0 *r 0 o o   1 o 8o a= 8 Q S0 0 1 I  b b.  61 0 4, 0 0 0 _. *r 7 r0  II11. 0 C0) 3' ~ 0 0 0\ 62 0 0 O O O, 0 a: v~  '1 I  0 0 l7: 4) 0 H o3 CD ,r. K 63 4 L' 0 O O II I *0 *0 E * * a 0 \ i I I **ml 0 J 0 0 41 0 4  64 S00 E \ p 0 0 C\P ,0 0 0 0 2! S r^c i 65  'mm 70 60 L 50 40 30 20 O n=45 f=100 Hz T (mA) rms 0 1 2 3 (A/m) Fig. 46 Normal inductance * * * * O 0 0 A 0 0 0 ;, g Hfc * of core #14. 66 second region is indicated as Ht, found from the intersec tion of the two linear regression lines. The measured Q factor at f= 1 kHz is plotted as log(Q) versus H scale for the ferrite and Metglas in Fig. 47 and Fig. 48; the field Ht is indicated in these figures. It is interesting to note that the slope of Log(Q) as a function of Hm changes around Ht. It is worthwhile to mention that the normal inductance measurements of the three 1 mil tape wound cores with dif ferent kinds of heat treatment have different features: as shown in Figs. 44, 45, and 46. Each core shows two region of linearly increasing permeability as a function of field strength. For core #1, a1 is smaller than a2, the critical field Ht is 0.88 A/m. For core #8 with special anneal, a1 is larger than a2, the critical fields Ht is 2.2 A/m. For core #14, a is smaller than a2, and the critical fields Ht is 0.78 A/m. Core #1 has the highest initial per meability, and core #8 has the lowest initial permeability of the three. The values can be found from Table 42. 4.3 Summary This chapter investigated the normal inductance of inductors wound on toroidal cores of six different magnetic materials: a ferrite, an amorphous alloy, a permalloy 4 mil tape wound core, and three permalloy 1 mil tape wound cores with different heat treatments. 67 IT~ I I I ill I * ir 0 0 0 0 .~ ~ II II ctI E5 R/ < E C" 0 I S ~bk I*I V a I I I  v co p 4I C o 44 0 < o 0 *t Uo a, *H CH LH r 68 : I I I 100 Q 50 40 Ht 0 *0 S 0 I (rms) (mA) 5 10 15 20 6 8 Hmm 10 12 14 16 13 (A/m) Fig. 48 Q factor as a function of Hm f = 1000 Hz. of Metglas at *  9 S* 2 4 69 4c c v ko M C r4 rl (1)~* I n 0 0 uI HH H O r a O 0 r U * S 0 H N N H H H 00I I =e0 0 0 'V H H H 0 (U X X X H 0'0 C 0 r~ C4 4 y C4\ Uo C. * (N H N N' N In 11l r I :T Iv 0 0 0 il iI r1 x x x H ) X X X IV m co cq m H H 0 I N 0 NN IV cO 0 0 0 U 0 .H u . 4.N O IN I 0 C) C)) C= In N 0tN '. v IV '0 H r o L n3 0^ co ) m C: '^< C O u X N i1l    m I Z 0 1 0 N S H r rq (u7'i x x x 0 4J mr in m 0)i in c14 *H N I N m    44 CN N C 4 'u0 0 CD C0 M *rli rI r1 r4 d4 x. x x m C; Co e n r ^o oo inco Ei (0 I  ____  C%4 N  70 Measurements of normal inductance, L, as a function of peak field strength, Hm, show two regions of nonlinearity with different slopes, a1 and a2, in which permeability increases as the field strength Hm increases. For ferrite and core #6, a1 is greater than a2 while for Metglas, a2 is greater than a1. For magnetically annealled core #8, the value a1 is greater than a2 opposite is found for core #1 and core #14 without magnetic annealing. The important parameters of normal inductance measure ments made on the six cores are listed in Table 42 for each. These include initial permeability, i', maximum per meability, m', the field Ht where permeability change its slope of increasing with field strength, slopes a1 and a2, and the field Hmm where permeability has its maximum value, mmm Pm The similar trends of normal inductance, observed for cores of such varying physical and chemical properties, raises the question whether these properties may be general for inductive materials without airgaps. The results of normal inductance are compared with the results of incremental inductance in chapter 7, and a plau sible physical interpretation is given in chapter 8. CHAPTER 5 INCREMENTAL INDUCTANCE MEASUREMENTS This chapter gives results of measurements of incremen tal inductance of the same six inductors as selected in chapter 4. Analysis of these data is in chapters seven and nine. 5.1 Incremental Inductance Measurement Incremental inductance is understood to mean the induc tance when a dc bias current acting on a specimen is main tained constant and an additional ac current is alternated cyclically between two limiting values. Correspondingly, the incremental permeability is obtained when we deal with a combination of a static and an alternating field inc (51) In general, the amplitude of field excursions is without restriction. When the ac signal is small, the measured inductance is referred to as reversible inductance, and the corresponding permeability is reversible permeability, which is is probably applicable to the following results, but this dissertation will use the more general terms incremental inductance and incremental permeability in order to avoid 71 72 overuse of the letter r subscript that might be confused with relative permeability. Measurements have been made of core incremental induc tance as a function of direct current applied to a second winding of the core. The direct current was controlled by a transistor constant current circuit, as shown in Fig. 51, in order not to affect the inductance measurement at the primary winding. The small ac excitation level can be adjusted by keeping I or Vrms at a specified value according to Eq.(l4) and Eq.(l5). Results of the incre mental inductance, Lin as a function of dc bias for the six cores are shown in Figs. 52 through 57. For each core, a graph of L. versus Hdc can be inc dc divided into three regions of operation: a low field region where the inductance is nearly a constant; a medium field region where L. decreases with increasing H dc; and a high Hdc region, where L. decreases at a slower rate. The dc inc inductance in the first region is referred to as L . Transition regions in the figures will be explained in chapter seven. In the first region, pinc was found to be identical to the initial permeability, pi of the material. In the second region, L. was found to decrease linearly with slope g as Hdc increases, where g is defined as (1/Ld)dLinc/dHdc, where Ld is the intercept at Idc=0 of the linear regression line, shown only in Fig. 52. The value Hd is the critical field that separate the first and the second region. The value of 73 V cc T2D LOR ME'E Rb R Fig. 51 Circuit for measurement of incremental inductance. 74 0 0 0 co 0 0 0 O .O 0 a  0 c C. i C> C) 0 H Fr> 75 ccO \0 t C 0 C\ o 0 1<4 ri 0 CO 0 s ,I o CH 0 co  76 2 0 00 C~' D 0 'I 0  I /2 /0 0 0 0 0 :1 3 ' " L1 77 I I I I. KK E   0 00 0 O E 11 11 11 H 0 S =  ', C O < oo S ~ ." 00 0 0 " 0 I o 0 0 T3 aa 0 @00 o;   * * * .^ * I I I I 0 0 '~' o 0 o *H E ^ ( 78 0/ o/ o/ C C ~ 0 c~,C\0o 0 IL II II ~ C C C II ,DC~ C~ . ~rn C ^ 1 I I I 'I 0  0 O C1 0 +) 0 SCO 0 3 C3 SE 0H ci 4.) 0 0 ' 8 0 ,s 79 o o 0 < I "0 J4 0 0 000 i II O S< '02 0 "C. 0 S0 ^  0 0 oo 0 0 0 H 0  O N 0 . I 0O 0 0 0 o 0 0o 0 80 Hd is found from the intersection of two linear regression lines. Beyond the second region, for Metglas and core #8, there is an extra region where incremental inductance decreases linearly with another slope. For this reason, the decreasing slopes for these two regions are indicated as gl' g2 separately and the demarcation point of these two linear decreasing regions is indicated as H shown in Figs. 53 and 56. When the incremental inductance,Linc ,is plotted as Log(Linc ) versus Log(Idc), as shown in Figs. 58 through 5 13, the inductance in the third region is found to decrease in proportion to the kth power of Idc (or of Hdc, since Hdc is proportional to Idc). Each different material was found to have a different k value. The separation point between this region and the linear decreasing region is indicated as H e. (this point is calculated by averaging the field at which the linear decreasing region ends and the field where the third region begins). The demarcation of these three regions as a function of field strength depends on the type of material been con sidered. Demarcation points between one region and another are not sharp. The values of Hd, g (g, and g2 for Metglas and core #8), k, Hd and H of incremental inductance are listed in Table 51 for each of the six materials. The critical field Hd will be studied in chapter 7, a plausible physical model of incremental inductance is given in chapter 9. 81 I i I i I iI ui i I 0* 0 0 0 0 0 0 0 0 0  0 l I I I I I c~) .~   ~ < iiI~1l 1 1 0 C ! ! . 0 0  cN 0* 00 0 C\2  I CM _ _I I_ I I 82 I I I i i r ._. SS0 S.* C  0H * C.' 4 * )  C  O  0  0 O I I 8 r  I. 83 p a 0 o/ * * * * I I i I~ lil i i i 0 0 0 I 0 0 0 0 o 0 o j o3 o Co " * 0 0 0 H O 41 C0 ) H O C 'I a 84 I i I I I I I I I i I I I I  @ @0 @00% . * 0 1e4 1 nA  0.44 A/m _ (m A I I I I I ii Idc ~~I I I I 11 9 .2 .3 4 Fig. 511 2 3 4 Incremental inductance of core #1 on loglog scale. i:CW ,\ _ __ I I I I I I I I III I I I I I I 85 I I I I II1 i I I' 0* * * * L 0275 L.1275 I , I II S4 5 Idc 20 30 40 50 100 (mA) Fig. 512 Incremental inductance of core #8 on loglog scale. 100  * 50 LO0 30 20 10 5 4 1 1 1 1 1 1 1 1 I I I I I I mI 86 9** V0 II i I I I 0 *0 0 0 c H i4 4 ^ C) 4' i ' I o o 0 '~'0 ~ (f' C~j 4 0~ g* a I i 0 O0 0 ,4 0 OU 0 0 CM  N I I o o x x I I .n in 0 1 ONN *o I CD CM 0 ,4 in 11 N 10 Co ,.il )  (N eN I I 0 on > m U LA * N Un (N " Q ) 87 Un r Lln * * 0 O U) (U p4 88 It is interesting to compare among the results of the three 1 mil permalloy cores. The plot of incremental induc tance of core #8 (Fig.56) shows two regions of linearly decreasing incremental inductance, while for core #1 (Fig.55) and core #14 (Fig.57) show only one region of linear decrease. The critical field Hd of the three cores are different: for core #1, it is 0.89; for core #8, it is 2.0 and for core #14, it is 0.75 all in units of A/m. The k value for core #8 is 2.75, while for core #1 it is 0.8, and for core #14, it is close to 1. The characteristic of the core is different for cores with different heat treat ments. This observation is recommended for future study. 5.2 Summary This chapter investigates the incremental inductance of inductors wound on toroidal cores of six materials. It is convenient to divide the incremental inductance measurements into three regions of performance for each core: a low field region with a nearly constant incremental permeability, a medium field region in which incremental permeability decreases linearly with increasing field in a manner which depends on the material, and a high field region in which incremental permeability further decreases proportional to the (kth) power of Hdc. Each different material has different k value. Between the second region and the third region, for Metglas and core #8, an extra 89 region was found where incremental inductance decreases linearly with different slope as Hdc increases. The important parameters of incremental inductance made on six cores are listed in Table 51 for each, these include the critical field Hd, decreasing slope g, and k. The first region and the second region of incremental inductance are of particular interest to this investigation and will be examined more closely in chapter 7. They will be correlated to power loss measurements at small field and at small induction as a function of dc bias in chapter 9. A plausible physical process is also proposed to explain the incremental inductance in chapter 9. CHAPTER 6 HYSTERESIS LOOP OBSERVATIONS In this chapter, measurements are reported of the BH loops of six magnetic materials. The six are the same as previously selected for normal inductance and incremental inductance measurements. 6.1 Introduction The most outstanding characteristic of a magnetic material is its hysteresis loop, also called the BH curve. To explain the observation of a hysteresis loop, con sider a toroidal core with two windings. In one of the windings the magnetizing current flows which creates the magnetic field H. The other winding is connected to a measuring instrument from which the resulting flux density may be read. Figure 61 shows a typical measuring circuit: The voltage across xx is proportional to the magnetizing current, (Vx = ir), and v3 which is intended to be propor tional to f V2 dt (=NB mA), is used to calculate the result ing induction, B, caused by the magnetic field strength, H. By connecting the xx ends to the horizontal input and yy ends to the vertical input of the oscilloscope, the BH relation can be observed. 90 91 ac I+ 711 + C t+ V I nn n v2 C V Variable autotransformer x x Filament transformer Fig. 61 Circuit for measuring the hysteresis loop of small toroidal cores. The hysteresis loop may also be measured without the transformers by connecting a signal generator at points A. 92 6.2 BH Loops A representative series of normal hysteresis loops is shown in Fig. 62a, with ferrite as an example. All the loops are symmetrical about the origin. The value of H for which B=0 is called the coercive force Hc and is often used as a measure of quality of the material. The value of induction for H=0 is the residual induction B When the field strength has been sufficient to magnetize the material practically to saturation, the coercive force and residual induction become the "coer civity" and retentivityy". The values of H and B at the tips of a loop are usually called H and B For an ideal m m "inductive" material, both H and B are zero. c r When the field strength Hm is very small, the inner loop of Fig. 62a shows that the BH relation has very small hysteresis. If define the slope of the line that connect the points (Bm ,Hm) and the origin (0, 0) as t, the value t was found to increase as H increases, and the BH loop opens up slowly as Hm increases. When the field is increased above a certain value, which was found to be close to Ht of normal inductance, the loop begins to open up very fast. The loop is lensshaped with its sides parabolic. The lens becomes thicker as the field strength increases. The BH loops begin to deviate from a lensshape and show bent down tips in the first and third quadrants as the field increases beyond a field which was found to close to H of normal inductance. mm 93 (a) (b) Fig. 62 BH loops of (a) ferrite, f=1000 Hz, X=8.44 (A/m)/div., Y=.115 T/div. (b) Metglas, f=200 Hz, X=36.4 (A/m)/div. Y=.57T/div. 94 When H is increased further the horizontal width of m the loop tends to become nearly the same at every value of Bm, and at the same time the loop becomes somewhat Sshaped; when Hm is still greater, the branches of the loop converge to a tail, and the shape of the loop changes only by the addition of "tails" in the first and third quadrants. The BH loops of the other five materials are shown in Fig.62b through Fig.64. All of them show the same trend. However, each material shows its own characteristics. The curve that connects all the tips of the loops of the family is referred to as the normal magnetization curve. Figs. 65 through 66 show examples of the curves for fer rite, Metglas, core #6 and core #8, others are not shown here. The magnetization curves at high fields (Fig. 67) can be modeled with the FrolichKennelly equation [Bo51, p.476], 1 Bm S= a + b Hm, m (61) m H m i.e., the reciprocal of amplitude permeability varies linearly with H Assuming this relation is still valid at very high fields for the material, the saturation induction is H B = lim pH = lim m = (62) s H> m H> m b 