Characterization and modeling of inductors

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Title:
Characterization and modeling of inductors
Physical Description:
vi, 166 leaves : ill. ; 28 cm.
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English
Creator:
Yeh, Long-Ching, 1949-
Publication Date:

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Subjects / Keywords:
Electric inductors -- Mathematical models   ( lcsh )
Inductance   ( lcsh )
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bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1984.
Bibliography:
Includes bibliographical references (leaves 163-165).
Statement of Responsibility:
by Long-Ching Yeh.
General Note:
Typescript.
General Note:
Vita.

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University of Florida
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All applicable rights reserved by the source institution and holding location.
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oclc - 12015718
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CHARACTERIZATION AND MODELING
OF INDUCTORS






By

Long-Ching 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 B-H 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 a-c
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

LONG-CHING 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 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.













CHAPTER 1
INTRODUCTION


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)
w ac


-1-




-2-


where R is the winding resistance and R is the core-loss
w ac
resistance.

The quality factor Q, which gives the ratio of energy

stored to energy dissipated per cycle, is given by



Q =- (1-2)
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 (1-3)
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.(l-3) 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 B-H 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. 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
B

I knee












H



Fig. 1-1 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 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

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 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.





-10-


-8
V = 4.44xB ANfxlO 8
rms m


where


(1-4)


B = Peak flux density in gauss


N = Turns

2
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
*m


(1-5)


where


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


(1-6)


where


B = Flux density in Tesla


H = Field strength in A/m





-11-


L = 1A (1-7)
m

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)

where

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.













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 (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).


-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.(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)
fL21 m
m





-14-


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.




-15-


lk2k 4k


10k
FREQUENCY (Hz)


LUK


Fig. 2-1 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. 2-3 Intercepts (at f=O from Fig. 2-1) vs Bm.
(ferrite)




-18-


20





15--




10





5






0 100 200 300 400
Bm (gauss)


Fig. 2-4 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
4-4
vi


C)








r-i
-4




r-4


I
-4


(N


I
CO

-4
X
CD
o\





L-4
x


H-


CO



-4
I I


1-4

) 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
s-O to Ui
4 0 C') l
044)C
E 1- a) CM


0

vl






-o

VI
0










Vo


C)
x*


-4
0
1-4




x
in


0
-4
X


CD




0

0 I

SIn

pu
fcl LC


>1
0
r-4
-H

P-4V
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. 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
m m
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:


R
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
-6
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




-21-


Bm (Gauss)


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 *


e(10-4 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 2-1,

the value of (a B + c) is found to deviate from a linear
m
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
m m
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.




-24-


0 150 300 450 600 750
Bm (Gauss)


Fig. 2-7 (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. 2-9 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 ) (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

Eq.(l-1) becomes



R = r0+r f+r f2 (2-4)




-29-





































0
0
0

0
0
0

0

0
0
0o







<< o


-30-


4-D
r-4
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 curve-fitting computer program was used to process
ac
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

represented as



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

not listed.


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
m analysis:
analysis:




-32-


15 1


10


R (ohms)


Fig. 2-13


~~~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 10-8.


. 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. 2-14 Parameter rI vs B for ferrite
calculated by model.





.705

r2(10'8)


.47 -




.235-


core. Curve is


.3 .6 .9
Bm kGauss


Fig. 2-15 Parameter r2 vs B for ferrite core. Curve is
calculated by model.


ri( 103)




-34-


Table 2-2 Model parameters for three cores.


Parameter


b

rO

rl
11
r
12
r
20

21
k


Ferrite

1.7x10 4
-4
2.8xl04

5.7xl0-6

3.0xl09
-8
1.4xl0 8
6.4x108

6.4xlO-11

6.94x10-1


Material

Metglas0

2.1x
-4
1.2xl04

1.8xl0-6

0
-7
1.6xl07

1.lxl-10

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= (2-7)
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. (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




-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 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
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 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





-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 2-1 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.(l-2), repeated here as Eq.(3-1):




= L (3-1)
SR +R
w ac

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


2irfL _
Q = 2 (3-2)
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. 3-1


A representative Q-factor for an inductor
plotted against frequency.


log Q




-42-


Q = (3-3)
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 (3-4)
2
rf


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


Ir0
f = I- (3-5)
\ r2
The Q factor at this frequency is



Qmax = (3-6)
2\ r0r2+rl


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)
m
and (2-6). The results are


I r0
f (r +r B (3-7)
\I'20 21 m

The maximum Q of the inductor is then


2WTfL0(l+bB) (3-8)
ax = ( +r 2Bm)+ 3-8)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 (3-9)


which are shown to be



-rl \ rl-4r2(r0-4\r2r0-2rl)
f =1 2 1 (3-10)
2r2
-2
The "bandwidth" (the width of the Q curve with Q>1/2 Qmax)

of the Q figure is then



12
\Irl- 4r2(r0-4\r2r0-2rl)
1 = 22-- (3-11)
r2


This equation shows that the larger the r2, the narrower the

"bandwidth."

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















-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. 3-3. The

relation between this two models is



R = R (l+Q2) (3-12)
p s
L = L (1+) (3-13)

with the Q value


UL R
Q s a (3-14)
R u=L
s p

For the series model, the power losses of the inductor is



P = 2 R (3-15)
rms s

where I is the rms value of current flow in the inductor.

For the parallel model,



V2
P rms (3-16)
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.(l-5), thus


2 A2N2B2
= (4.44) x m(3-17)
1016 Rs (+Q2
s


For a high Q inductor, Q>>1, thus




-46-


ip -p




(b)


Fig. 3-3 Inductor Models (a) series model
(b) Parallel model.





-47-


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)
40 m

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



B2
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)

thus,


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,




-48-


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)
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 ) (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.





-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.(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

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. 3-4 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. 3-5








170-3 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. 3-6 Power loss as a function of Bm at f=1000 Hz.
(Metglas)




-53-


Table 3-1. 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.(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.












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 a-c magnetic

state.

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

total B/H.

By means of hysteresis loop, one also can define a

large number of permeabilities (0166). The initial permea-

bility




= lim (4-1)
H->0H


-55-




-56-


This is the permeability at the origin of the curve at the

first magnetization. The amplitude permeability

B
a = B (4-2)
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 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-

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-


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

Eq.(1-5).

















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 r-4 L A c








0 N C4 (U N
0 H NgU


4
0


i-l

.-i
t)




4)
4-

*rz4




.H
C)
&4













41




-59-


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

region



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




-60-


0

0

o




o a
O <8<
0 ,- --
0





0 Eo -1

O 0-0--- 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 -











II1-1-.- 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




4-1
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. 4-6 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. 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.


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
ct-I


E5 R/


<
E C"-


0
I


S
~bk I*I V a


I I I


--


v


co
-p



4--I







C




o
4-4
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. 4-8


Q factor as a function of Hm
f = 1000 Hz.


of Metglas at


*
- 9
S*


2 4




-69-


4c c v ko
M C r-4 r-l
(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


1-1l
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












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- r-q
(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 *r-li r-I r-1 r-4








d4 x. x x
m C; Co



e n r- ^o oo inco
E-i
(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 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,
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 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




inc (5-1)

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


-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. 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
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. 5-2. 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. 5-1 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 r-i


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
C-1
0





+)
0
SCO







0
3
C3





SE
0H


ci
4.)


0
0 '

8
0- ---,s




-79-


o o


0 <



I "0
-J-4
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. 5-3

and 5-6.

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

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

4-1



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. 5-11


2 3 4


Incremental inductance of core #1 on log-log
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 0-275
L.1275


I ,


I II


S4 5


Idc


20 30 40 50


100


(mA)


Fig. 5-12 Incremental inductance of core #8 on log-log
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

i-4





-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 1-1
N 10 Co

,-.-il


) -- (N
eN I
I 0


on > m


U- LA

* N
U-n (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.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.


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 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.














CHAPTER 6
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

inductance measurements.


6.1 Introduction


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.


-90-




-91-


a-c I+ 711 + C t+
V I nn n v2 C V

Variable autotransformer x x
Filament transformer




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.





-92-


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
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. 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.
mm





-93-


(a)















(b)


Fig. 6-2


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.
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 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

here.

The magnetization curves at high fields (Fig. 6-7) can

be modeled with the Frolich-Kennelly equation [Bo51, p.476],



1 Bm
S= a + b Hm, m (6-1)
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 = (6-2)
s H->- m H-> m b