CHARACTERIZATION AND MODELING
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
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
TABLE OF CONTENTS
ACKNOWLEDGEMENTS... ......... ........................
MODELING OF INDUCTORS .........................
2.1 Introduction .............................
2.2 Measurements and Calculations ............
2.3 Limitations of Legg's Model and Modified
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 B-H Loops ................................
6.3 Summary......... ..... .................. ...
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 a-c
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
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 d-c field for incremental inductance was
found to have the same value as the peak a-c 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 curve-fitting program to describe the
inductance and equivalent series resistance measured with
a-c 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 domain-state
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
omain-wall un-pinning model to explain the increase of
permeability with a-c amplitude.
This dissertation can be separated into two parts: in
the first part we study the modeling of inductors under a-c
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 (1-1)
where R is the winding resistance and R is the core-loss
The quality factor Q, which gives the ratio of energy
stored to energy dissipated per cycle, is given by
Q =- (1-2)
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.
These subjects are studied in the first part of this disser-
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
ac = aB +c+ef (1-3)
where a is the hysteresis loss coefficient, c is the resi-
dual loss coefficient, and e is the eddy current loss coef-
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.(l-3) as a basis to study the Legg model at low Bm for
many inductive magnetic materials which are not limited to
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 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
The electrical models we derived in chapter 2 and 3 are
relevant to such a broad class of materials as described
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 B-H characteristics are predom-
inately sloping, so that the concept of permeability p= B/H
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. 1-1, 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
Fig. 1-1 A representative magnetization curve of an
"inductive" magnetic material.
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
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.
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 Preisach-Neel 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
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
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
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 B-H
observations. Chapter 7 investigates the relationships among
normal inductance, incremental inductance and B-H 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 V-I relation due to nonlinear B-H
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.
V = 4.44xB ANfxlO 8
B = Peak flux density in gauss
N = Turns
A = Effective core area in cm
f = Frequency in Hz
Equation (1-4), 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
H = Field strength in A/m
1 = Mean path length in meter of toroidal core
Equation (1-5) allows one to determine at what current to
excite a core for measurement at a certain field strength.
9 = B/H
B = Flux density in Tesla
H = Field strength in A/m
L = 1A (1-7)
Equation (1-6) and (1-7) 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 (1-7).
L = N2P. (1-8)
L = Inductance in Henries
P = Permeance in Henry/turns squared
Equation (1-8) defines inductance from the permeance, P.
For a toroidal core without air gap, P= uA/1 m. In Eq.(l-8),
given the P, one can either calculate the inductance or
determine the turns needed if the inductance is known.
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.
This chapter presents the results of a study on the
model of an inductor when signal level varies from low to
high. Equation (1-3) 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).
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
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.(l-4) by keeping V/f
constant. The measured data were manipulated into the for-
mat of the left side of Eq.(l-3), 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.(l-7)).
Thus we actually worked
(R-R )n A
= aB +c+ef (2-1)
for each data point as the left hand side of Eq.(1-3),
where A is the core cross section and 1m is the mean path
length. When the values of Eq. (2-1) 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 2-1 illustrates
this processes for ferrite data in which coefficient e is
found. Figure 2-2 shows the same process for Core #8. Figure
2-3 and 2-4 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 2-1.
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 2-1 were imposed
by the accuracy of determining Rac by subtraction, see
Eq.(l-l). 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. 2-1 begins to deviate from linearity
to become concave down.
Fig. 2-1 Rac/uLf vs frequency at various B for
20 40 60
Fig. 2-3 Intercepts (at f=O from Fig. 2-1) vs Bm.
0 100 200 300 400
Fig. 2-4 aBm + c vs B. of core #8 at low field range.
r- H co
~ 0 0
) -4 U r) 0
0o cn u
Or mo U)
X; 0 M
4 >4 -4
s-O to Ui
4 0 C') l
E 1- a) CM
2.3 Limitations of Legg's Model and Modified Model
When the family of curves of Fig. 2-1 or 2-2 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. 2-5. A similar variation was also
found for all the core materials, with core #8 as another
example, as shown in Fig. 2-6. Others are not shown here.
The intercept at B =0 for the figure of e versus B is the
coefficient e0, listed in Table 2-1. 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:
ac = aB +c+e(l+dB )f (2-2)
pfL m m
For the geometries and turns used, coefficient d was
found to be 1.37 x 10-2 for ferrite and d = 2.33 x 10-3 for
core #6, 2.52xl0-4 for core #8 and 2.13 x 10-3 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
data were found only for the MPP core: a = 0.9 x 10 c =
2 5x 1-5xl-
2.5 x 10-5, e = 1.7 x108 for a = 160 (suitable below 200 g,
frequency limit unspecified). Our results agree within
Fig. 2-5 Slopes (of lines from Fig. 2-1) vs Bm
interpreted as e. The line is a linear
regression with eo = 3.4 x 10-4 P0
d = 4.67 x 10-6 p0 *
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 2-1,
the value of (a B + c) is found to deviate from a linear
relation with B The data points of Fig. 2-7 and of Fig.
2-8 illustrate this effect and thereby raise the question
whether the value of coefficient a is valid. The only
approximate agreement of Fig. 2-7 with the data of Fig. 2-3
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 high-field measurements of
Fig. 2-7, n =40, whereas more turns (n =100) were used for
the low field measurement of Fig. 2-3. The values of e
versus Bm also show two different slopes at two different
numbers of turns, as can be seen from Fig. 2-9 for ferrite
as an example. Figure 2-10 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
increases, as has been shown in Figs. 2-7, 2-9 and 2-10,
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.
0 150 300 450 600 750
Fig. 2-7 (aBm + c) vs Bm.
I I 1
I I I I
i I I
0 150 300 450 600 750
Fig. 2-9 Coefficient e versus Bm.
00 O o
Ca 3 M '-4
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-
Normal inductance typically increases with the ampli-
tude of the measuring signal, and is modeled here as
L = L 0(+bB ) (2-3)
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 2-11 and
Fig. 2-12 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
R = r0+r f+r f2 (2-4)
where r0 is identified as the winding resistance Rw and thus
the two higher order terms define the core loss parameter
R A curve-fitting computer program was used to process
our measured data for R using the format of Eq. (2-4), which
turned out to match with our data very well. Figure 2-13
shows a calculation compared with data for the ferrite core
inductor at Bm = 44 G, plotted vs frequency.
The coefficients resulting from the Eq.(2-4) analysis
were found to vary with Bm (in units of gauss) and can be
r = r 0+r B +r B2 (2-5)
1 10 11 m 12 m
r2 = r20+r 21Bm (2-6)
where again the coefficients were identified using the com-
puter curve-fitting program, as shown in Figs. 2-14, 2-15.
In summary, the model consists of Eqs. (2-3) through (2-6)
for which the resulting coefficients are listed in Table 2-2
for three of the five materials. Due to the frequency limi-
tation of the instrument, data for core #6 and Core #8 are
2-5 Relation of Model to Legg Equation
When model Eqs.(2-3) through (2-6) are substituted into
Eq.(l-3), the Legg equations (1-3) and (2-1) take the form,
with k=n A/i and omitting B terms to simplify our
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 10-8.
. Y (kHz)
_ i i i i
I I I t I
I I t
-- l | i I
S 1 I I I I I !
.3 .6 .9
Fig. 2-14 Parameter rI vs B for ferrite
calculated by model.
core. Curve is
.3 .6 .9
Fig. 2-15 Parameter r2 vs B for ferrite core. Curve is
calculated by model.
Table 2-2 Model parameters for three cores.
R r +rf
ac= 2k= (2-7)
L L (1+bB )
0 m m
After some manipulation and assuming 2 b Bm << 1 the right
side of Eq. (2-7) takes the form
(a +aaB -a B2)+(e +e B -e B )f (2-8)
0 1 m 2 m 0 1 m 2 m
which is a generalization of Eqs.(2-1) in which the actual
Legg coefficients may be identified as
c = ao, a=(al-2a2Bm), e 0, d=(el-2e2Bm)/e0. (2-9)
The a and e parameters may be found from Eq. (2-7) as below,
if each right side is multiplied by k/L02
a0 = rl0, e0 = r20 (2-10a)
a1 = (r11-2brl0), e1 = (r21-2br20) (2-10b)
a2 = 2br11, e2 = 2br21 (2-10c)
In Fig. 2-7, the smooth curve identified as the mathematical
model is Eq.(2-8) plotted for f = 0, which is the a-series
part of Eq. (2-8). The e-series multiplier of f in
Eq.(2-8) is shown in Fig. 2-9 as the smooth curve. The new
model is therefore consistent with a generalized form of the
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
are plotted as a function of f in log-log scale in Fig. 2-16
and Fig. 2-17. 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
2-17 shows that MPP core has the highest aB +c loss, while
core #8 has the lowest aB +c loss of the five. The residual
losses of core #6 and core #8 are very small.
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 2-1. 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.(l-3) 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
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 2-1 may be specific to these specific
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.(l-2), repeated here as Eq.(3-1):
= L (3-1)
A representative curve of the Q factor of an inductor may be
shown as in Fig. 3-1.
When Eq.(3-1) is rewritten by substituting Eq.(2-5)
for (3-1) in the denominator, the result shows that
Q = 2 (3-2)
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-
A representative Q-factor for an inductor
plotted against frequency.
Q = (3-3)
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 (3-4)
This analysis supports the concave-down trend of Fig. 3-1.
The maximum Q occurs at the frequency where dQ/df = 0, which
is readily shown to satisfy the condition r0 = r2f 2, namely
f = I- (3-5)
The Q factor at this frequency is
Qmax = (3-6)
A more general form of Eqs. (3-5) and (3-6) may be obtained
by allowing variations of B according to Eqs. (2-3), (2-5)
and (2-6). The results are
f (r +r B (3-7)
\I'20 21 m
The maximum Q of the inductor is then
ax = ( +r 2Bm)+ 3-8)rl Bm
The frequencies at which Q has half value of Qmax are the
frequencies that satisfy the following equation:
(r0+rlf+r2f2) = 2 rr2 +r (3-9)
which are shown to be
-rl \ rl-4r2(r0-4\r2r0-2rl)
f =1 2 1 (3-10)
The "bandwidth" (the width of the Q curve with Q>1/2 Qmax)
of the Q figure is then
1 = 22-- (3-11)
This equation shows that the larger the r2, the narrower the
Figure 3-2 shows values of Q of Metglas core measured
as a function of frequency, and calculated points for Qmax'
using Eq.(3-8). The solid lines represent the calculated
value of Q using Eq.(3-2). 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
transformed into a parallel model which is an inductor L in
parallel with a resistance R as shown in Fig. 3-3. The
relation between this two models is
R = R (l+Q2) (3-12)
L = L (1+) (3-13)
with the Q value
Q s a (3-14)
For the series model, the power losses of the inductor is
P = 2 R (3-15)
where I is the rms value of current flow in the inductor.
For the parallel model,
P rms (3-16)
where Vrms is the voltage (rms value) across the inductor.
Since R = R (1+Q ), and for a sinusoidal voltage, B relates
to Vrms according to Eq.(l-5), thus
= (4.44) x m(3-17)
1016 Rs (+Q2
For a high Q inductor, Q>>1, thus
Fig. 3-3 Inductor Models (a) series model
(b) Parallel model.
22 2 2 2
P = 1.97 x 10-15-msA- =
.97x10-15 A2N2B (r+(r +r B )f+(r20+r21B )f
4w L (1+bB ) (3-18)
Equation (3-18) 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.(3-17) in logarithmic form becomes
log P = c + log m 2log Q. (3-19)
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 (3-19) 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. 3-1, the functional dependence of Q on
f may be written as
log Q = c2 c(log f (3-20)
log P = c3 + 2dllogf, P = c4f = c4f (3-21)
On the other hand, if f loss term is dominant, Q is nearly a
constant, then at a specified Bm,
log P = c5 + log f, (3-22)
which is a typical result of hysteresis loss. Similarly, at
a specified frequency, Eq. (3-18) can be simplified as
-15 B2(R +R Bm)
P= 1.97x1015 m 0 m (3-23)
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 ) (3-24)
In certain Bm range, and if the last term is neglected, the
superposition of the second and the third terms in log P-log
Bm plot shows that P may take the form
P = c 7B (3-25)
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.
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.(3-18) to calculate power losses of the
inductor; Figure 3-4 shows the results of power loss versus
frequency at three Bm values with Metglas as an example;
solid lines are those calculated from Eq.(3-18). Figure 3-5
shows the example of P versus Bm at three frequencies for
ferrite. Figure 3-6 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.(3-21)
and Eq.(3-25), and the parameters d and p are listed in
Table 3-1. Data regions are also indicated for each core in
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
( Me;glas )
Points: measured da
Solid lines are caV
Fig. 3-4 Power loss as a function of frequency at various
Bm. Solid lines are calculated from model. (Metglas)
Power loss vs Bm at various frequency. (ferrite)
170-3 I 1 i i i Fl
V I ii ti~~
ine : del
0 : Data Pints
f ik Hz
i I 1 I 1 I\ I
Fig. 3-6 Power loss as a function of Bm at f=1000 Hz.
Table 3-1. Power Loss Parameters of Three Materials.
Material I c(
IFerrite I 1.08 I 2.39
Metglas 1.70 2.0
I Data Region
4k < f < 40k, B < 500 I
I 4k < f <40k, B < 600
4k < f < 40k, Bm 50
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.(3-21) and with Bm in the form of Eq.(3-25) of our
model. Parameters of d and p of the three cores are listed
in Table 3-1.
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.
Normal inductance, often referred to simply as induc-
tance, is measured when the specimen is in the a-c magnetic
Normal inductance is related to amplitude permeability,
a', by Eq.(l-7), where pa is defined by the slope of a line
drawn in Fig. 1-1 from the origin to a point defined by the
By means of hysteresis loop, one also can define a
large number of permeabilities (0166). The initial permea-
= lim (4-1)
This is the permeability at the origin of the curve at the
first magnetization. The amplitude permeability
a = B (4-2)
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 B-H relation of a magnetic material is nonlinear
during the a-c 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 peak-to peak amplitude permeability often named
normal permeability when it is derived from the magnetiza-
The definitions of each of the different amplitude
permeabilities can be seen from the book by Olsen .
It should noted that at a specified signal level (B or H ),
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
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
tape-wound 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 4-1.
4.2 Measurements and Results
Measurements of normal inductance and Q factor were
made using an HP 4274-A 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
n O I o m
- L m Ln M
Lr CM L A m
CN MA IV N r-
N n 'I N 0 N
co m c CN C
S C L
* *N c O -
O r-4 L A c
0 N C4 (U N
0 H NgU
Measured normal inductance is plotted against current
I and Hm as shown in Figs. 4-1 to 4-6, 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 non-zero)
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
L = L0(l+alHm) (4-3)
and in the second region
L = L2(1+a2Hm) (4-4)
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
0 ,- --
0 Eo -1
O 0-0--- e
o | -- 1
o 8o- a=
0 1 I -- b
0 0 0 _.
7 r0 -
C0-) 3' ~-
O O O, 0
i I I **ml
0 -J 0
\ p 0
0 0 0
2! S r^c --i
0 1 2 3
Fig. 4-6 Normal inductance
of core #14.
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. 4-7 and
Fig. 4-8; 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. 4-4, 4-5, and 4-6. 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 4-2.
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.
IT~-- I I I
~bk I*I V a
I I I
-: I I I
5 10 15 20
10 12 14 16 13
Q factor as a function of Hm
f = 1000 Hz.
of Metglas at
4c c v ko
M C r-4 r-l
(1)~* I n
O r- a O 0 r
S 0 H N N H
=e0 0 0 'V
H- H- H- 0
(U X X X H-
0 r~ C4 4- y C4\ Uo
(N H N N' N In
r- I :T Iv
0 0 0
i-l i-I r-1
x x x
H ) X X X
IV m co cq m
0 I N 0 NN IV cO
0 0 0
.H u .
O IN I
0 C) C)) C= In N
0tN '. v IV '0 H r o L
n3 0^ co ) m C: '^< C
i1l -- -- -
Z 0 1 0 N
S H r- r-q
(u7'i- x x x
0 4J mr in m 0)i in c14
N I N
m ---------- -- --
44 CN N C 4
'u0 0 CD C0
M *r-li r-I r-1 r-4
d4 x. x x
m C; Co
e n r- ^o oo inco
(0 -I----- -- -------___-_- -
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 4-2 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,
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.
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
5.1 Incremental Inductance Measurement
Incremental inductance is understood to mean the induc-
tance when a d-c bias current acting on a specimen is main-
tained constant and an additional a-c 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
In general, the amplitude of field excursions is without
restriction. When the a-c 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
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. 5-1,
in order not to affect the inductance measurement at the
primary winding. The small a-c excitation level can be
adjusted by keeping I or Vrms at a specified value
according to Eq.(l-4) and Eq.(l-5). Results of the incre-
mental inductance, Lin as a function of d-c bias for the
six cores are shown in Figs. 5-2 through 5-7.
For each core, a graph of L. versus Hdc can be
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
inductance in the first region is referred to as L .
Transition regions in the figures will be explained in
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. 5-2. The value Hd is the critical field
that separate the first and the second region. The value of
T2D LOR ME'E
Fig. 5-1 Circuit for measurement of incremental
- 0 c-
ccO \0 t C 0
C\ o 0
C~' D 0
I I I I.
0 O E
11 11 11 H
S = -
'-, C O
S -~ ."
00 0 0 "- 0 I
* I I I -I
0 0 '~' o 0 o
IL II II -~
C C C II
^ 1 I I I
i II O
S0 -^- -
- O -N
0 o 0 0o 0
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. 5-3
When the incremental inductance,Linc ,is plotted as
Log(Linc ) versus Log(Idc), as shown in Figs. 5-8 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 5-1 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
I i I i I
iI ui i I
l I I I
< iiI~1l 1 1
_ _I I_
--I I I i i r ._.
- 0 -
I I i
lil i i i
0 0 0 I 0 0 0
0 o 0 o- j
o3 o- Co "
I i I I I I I
I I i I I I I
- @ @0 @00% .
1 nA ---- 0.44 A/m _
I I I I I ii
~~I I I I 11 9
.2 .3 4
2 3 4
Incremental inductance of core #1 on log-log
I I I I I I I I III I I I I I I
I I I I II1 i
20 30 40 50
Fig. 5-12 Incremental inductance of core #8 on log-log
1 1 1 1 1 1 1 1 I I I I I I mI
II i I I I
o o 0 '~'0 ~ (-f-' C~j
I -I .n in
N 10 Co
) -- (N
on > m
" Q )
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.5-6) shows two regions of linearly
decreasing incremental inductance, while for core #1
(Fig.5-5) and core #14 (Fig.5-7) 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.
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
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 5-1 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 d-c bias in chapter 9.
A plausible physical process is also proposed to explain the
incremental inductance in chapter 9.
HYSTERESIS LOOP OBSERVATIONS
In this chapter, measurements are reported of the B-H
loops of six magnetic materials. The six are the same as
previously selected for normal inductance and incremental
The most outstanding characteristic of a magnetic
material is its hysteresis loop, also called the B-H 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 6-1 shows a typical measuring circuit:
The voltage across x-x 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 x-x ends to the horizontal input and y-y
ends to the vertical input of the oscilloscope, the B-H
relation can be observed.
a-c I+ 711 + C t+
V I nn n v2 C V
Variable autotransformer x x
Fig. 6-1 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.
6.2 B-H Loops
A representative series of normal hysteresis loops is
shown in Fig. 6-2a, 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
"inductive" material, both H and B are zero.
When the field strength Hm is very small, the inner
loop of Fig. 6-2a shows that the B-H 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 B-H 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 lens-shaped
with its sides parabolic. The lens becomes thicker as the
field strength increases. The B-H loops begin to deviate
from a lens-shape 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.
B-H 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.
When H is increased further the horizontal width of
the loop tends to become nearly the same at every value of
Bm, and at the same time the loop becomes somewhat S-shaped;
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
B-H loops of the other five materials are shown in Fig.6-2b
through Fig.6-4. 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. 6-5 through 6-6 show examples of the curves for fer-
rite, Metglas, core #6 and core #8, others are not shown
The magnetization curves at high fields (Fig. 6-7) can
be modeled with the Frolich-Kennelly equation [Bo51, p.476],
S= a + b Hm, m (6-1)
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
B = lim pH = lim m = (6-2)
s H->- m H-> m b