1/91

Magnetic Circuits

Power Electronic Systems & Chips Lab., NCTU, Taiwan

Power Electronic Systems & Chips Lab.

~

Chapter 1 Magnetic Circuits and Magnetic Materials, Fitzgerald & Kingsley's Electric Machinery, 7th Ed, S.D. Umans, McGraw-Hill Book Company, 2013.

2/91

Modeling the Stator Inductance of an IPMSM

iv iL

Rviv

Lv

Li

L

R Rv 1L

LR

1s

0( )L t

1( ) ?rL

Assume the rotor produces a sinusoidal flux distribution across the air gap, how to model the stator winding inductance as a function of the rotor position of an interior permanent magnet synchronous motor (IPMSM)?

REF: Chapter 4 Inductances, Design of Rotating Electrical Machines, Juha Pyrhonen, Tapani Jokinen, Valeria Hrabovcova, 2nd Ed., October 2013, Wiley.

Modeling of Synchronous Machine in dq-Frame

q d

PM

PM

PM

s q qi L

d di L

ss

sv

si

r

e

d

axisa

axisb q

axisc

aiavFv

bvbi

Fi

ci

cv

di1q1 fd

d1qi1 r

a

'a

sR e q qL i

dv

dLdi sR e d dL i

qv

qLqi

e PM d s d e q d PM

q e d s q q e PM

v R sL L i sv L R sL i

PMs

3 3( ) ( ) ( )2 2 2 2e d q d q d q d d PM q d q d q

P PT i L L i i i L i L L i i

Electric Equations:

Torque Equations:

mm

Lfe BdtdJTTT

Torque Characteristics of Synchronous Machines

d qL L d qL L

d

q

d

q

dq

d

q

d qL L d qL L0m

3 [( ) ( )]2 2e d d PM q q d d q

PT i L i i i L L

emT erT

0erT 0emT eem rT T eem rT T

Inductance Plays a Key Role in Motor Characteristics

1( ) ?rL

The rotor structure determines the major characteristics of a synchronous machine (SM). For SM with concentrated winding stator, the inductance of the coil of a segmented teeth can be calculated as a function its rotor position if the rotor has an anisotropic structure.

[1] I. A. Viorel, A. Banyai, C. S. Martis, B. Tataranu, and I. Vintiloiu, On the segmented rotor reluctance synchronous motor saliency ratio calculation, Advances in Electrical and Electronic Engineering, vol. 5, vo. 1-2, June, 2011.

[2] B.J. Chalmers and A. Williamson, AC Machines Electromagnetics and Design, Research Studies Press Ltd., John Wiley and sons Inc., 1991.[3] Jong-Bin Im, Wonho Kim, Kwangsoo Kim, Chang-Sung Jin, Jae-Hak Choi, and Ju Lee, Inductance calculation method of synchronous

reluctance motor including iron loss and cross magnetic saturation, IEEE Transactions on Magnetics, vol. 45, no. 6, pp. 2803-2806, 2009.

Design of Rotating Electrical Machines, 2nd Ed., [Chapter 4: Inductances] Juha Pyrhonen, Tapani Jokinen, Valeria Hrabovcova, October 2013, Wiley.

3 [ ( )]2 2e q m q d d q

PT I I I L L Generated electric torque of synchronous machine:

6/91

Basic Notations for Electromagnetism

Electric field strength E [V/m] Magnetic field strength H [A/m] Electric flux density D [C/m2] Magnetic flux density B [Vs/m2], [T] Current density J [A/m2] Electric charge density, dQ/dV [C/m3]

D E

B H

permittivity of free space (Farads/m)

permeability of free space (Henrys/m)

7/91

1865

18652020(Oliver Heaviside) (Josiah Gibbs)(Heinrich Hertz)1884

Source of Electric Field Changing Electric Field Source of Magnetic Field Changing Magnetic Field

A Students Guide to Maxwells Equations (Daniel Fleisch), , 20101018

0

E

t

BE

0 B0 0 t

EB J

Maxwell's Equations (1860s~1970s)

(Gausss Law for Electric Field)

0

encS

Qnda

E

0S

nda B

0 0 encC Sdd I ndadt

B l E

C S

dd ndadt

E l B

0

E

(Gausss Law for Magnetic Field)

(Faradays Law)

- (The Ampere-Maxwell Law)

t

BE

0 B

0 0 t

EB J

Symbols and Units of Electromagnetic Quantities

Summary of Quasi-Static Electromagnetic Equations

Electromechanical Dynamics (MIT Course Notes)

REF: Electromechanical Dynamics - Part 1 Discrete Systems (Woodson & Melcher, MIT 1968)

Electromechanical Dynamics, Discrete Systems (Part 1), Herbert H. Woodson and James R. Melcher, Wiley, 1st Ed., January 15, 1968.

( )( )

( )

s s s sr r

r sr s r r

sre s r

i L L iL i i L

dLT i id

eT s si

r ri

12/91

Basic Relations of Electrical and Magnetic Field

Faradays Law

Amperes Law

terminalcharacteristics

Corecharacteristics

( )v t ( ), ( )B t t

( ), ( )H t F t( )i t

Magnetic CircuitsElectrical Circuits

13/91

Magnetic Field

Magnetic fields are produced by electric currents, which can be macroscopic currents inwires, or microscopic currents associated with electrons in atomic orbits. The magneticfield B is defined in terms of force on moving charge in the Lorentz force law. Theinteraction of magnetic field with charge leads to many practical applications. Magneticfield sources are essentially dipolar in nature, having a north and south magnetic pole.The SI unit for magnetic field is the Tesla, which can be seen from the magnetic part of theLorentz force law Fmagnetic = qvB to be composed of (Newton x second)/(Coulomb xmeter).A smaller magnetic field unit is the Gauss (1 Tesla = 10,000 Gauss).

14/91

Right-Handed System and Left-Handed System

x

y

z

y

x

z

Right-Handed SystemLeft-Handed System

15/91

Magnetic Field of Current: Right-Handed Rule

The magnetic field lines around a long wire which carries an electric current formconcentric circles around the wire. The direction of the magnetic field isperpendicular to the wire and is in the direction the fingers of your right handwould curl if you wrapped them around the wire with your thumb in the directionof the current.

16/91

Amperes Law

(a) (b)

(a) General formulation of Amperes law. (b) Specific example of Amperes law in the case of a winding on a magnetic core

with air gap.

Direction of magnetic field due to currents Amperes Law: Magnetic field along a path

id lH

17/91

Amperes Law

B H H = magnetic field intensity (Ampere-turns/m) = magnetic permeability of material (Wb/A.m, or Henery/m)B = magnetic flux density (Tesla, Weber/m2)

r 0

= permeability of free space

074 10 H / m

r = relative permeability (between 2000-80,000 for ferromagnetic materials)

H l I d

I

Id

enclosenot doescontour if ,0 enclosescontour if ,I

lH

IlB dld

B

18/91

Permeability: Relationship Between B and H

Ampere,s Law H l I d

H = magnetic field intensity (Ampere-turns/m) = magnetic permeability of material (Wb/A.m, or Henry/m)B = magnetic flux density (Tesla, Weber/m2)

r 0

= permeability of free space

074 10 H / m

r = relative permeability (between 2000-6000 for general ferromagnetic materials used in electrical machines)

permeability = = BH

In magnetics, permeability is the ability of a material to conduct flux. The magnitude of thepermeability at a given induction is a measure of the ease with which a core material can bemagnetized to that induction. It is defined as the ratio of the flux density B to the magnetizingforce H. Manufacturers specify permeability in units of Gauss per Oersted (G/Oe).

cgs: = 1 gaussoersted oersted

0410

tesla mks: = 4 henrrymeter

0710

19/91

(Wb), (Tesla)

(SI)Wb

11 11Wb=1Vs

2-2 -1 (m 2 kgs-2A-1) = (Voltsec)

188218951948

CGS

1108[1 Wb = 108 Maxwell]

(Tesla) [1 Tesla = 1 Wb/m2]

1 oersted = 1000/4 ampere/turn = 79.57747154594 ampere/meter 80 A/m

20/91

Properties of Ferromagnetic Materials

1.4

1.2

1.0

0.8

0.6

0.4

0.2

00 200 400 600 800 1000

H, A-turn/m

B, Wb/m2

B H r 0

Ferromagnetic materials, composed of iron and alloys of iron with cobalt,tungsten, nickel, aluminum, and other metals, are by far the most commonmagnetic materials.

Transformers and electric machines are generally designed so that somesaturation occurs during normal, rated operating conditions.

DC Excitation

i

N

AB

A toroidal coil and the magnetic field inside it.

A is the cross-sectional area

21/91

B-H Curve, Permeability, and Incremental Permeability

Relation between B- and H-fields.

H

Bs

Hs

Linear region

BH

HB

HB

HB

B

H

HHB r 0

Magnetic intensity H, [A-turns/m]

Incremental PermeabilityB The B-H characteristics of acore material is high nonlinear.Depends on its averagecurrent, current ripple,switching frequency, andoperation temperature.

When measuring theinductance of a magneticcircuit, it should first toidentify its operating point.

22/91

B-H Curve of Major Materials

This is because there is a limit to the amount of flux density that can be generated by the core as all thedomains in the iron are perfectly aligned. Any further increase will have no effect on the value of M, andthe point on the graph where the flux density reaches its limit is called Magnetic Saturation also known asSaturation of the Core and in our simple example above the saturation point of the steel curve begins atabout 3000 ampere-turns per meter.

The set of magnetization curves as shown inleft figure represents an example of therelationship between B and H for soft-iron andsteel cores but every type of core material willhave its own set of magnetic hysteresis curves.You may notice that the flux density increasesin proportion to the field strength until itreaches a certain value were it can notincrease any more becoming almost level andconstant as the field strength continues toincrease.

B-H Characteristics of a Magnetic Material

Performance Tradeoffs: saturation Bs, permeability , resistivity (core loss), remanence Br, and coercivity Hc.

sH

24/91

Flux Density or B-Field

Determination of the magnetic field direction via the right-hand in (a) the general caseand (b) a specific example of a current-carrying coil wound on a toroidal core.

(a) (b)

H-fieldCross-sectional area A

HHB r 0

i

iH

N

The total flux pass through the coil with N turns is called flux linkage and named as .

BA

N

N

25/91

Continuity of Flux

A1 A2

A3

1 23

dABA 0surface) (closed dABA

k

k 0

0or 0 321332211 ABABAB

26/91

Magnetic Cores

Ideal Inductor

v N ddt

dN

vdt 1

The above equation shows that the change in flux during a time interval t0-t1 isproportional to the integral of the voltage over the interval, or the volt-seconds appliedto the winding.

Negligible winding resistance Perfect coupling between windings An ideal core

v

i

N

1

0

1)()( 01t

tvdt

Ntt

27/91

Ideal Inductor [Define its Initial Conduction]

(a) Circuit model. (b) -i characteristic (or B-H curve).

(c) v is a step input; (t0) = 0. (d) v = Vm sin t ; (t0) = 0.

(e) v is a square wave; (t0) = -m. (f) v = Vm sin t ; (t0) = -m.

i

N

v

i

0

v

0

v

t0

v

t0

v

28/91

Magnetic Field Strength H of Some Configurations

long, straight wire

Toroidal Coil

Long solenoid

29/91

Inductance of Wound Magnetic Core

magnetic flux per turnwebers (Wb) [1 Wb = 108 Maxwell]

magnetic flux density webers/meter2 (teslas)

flux linkage webers

core cross-sectional area square meters

magnetic field strength ampere-turns/meter

number of turns

coil current ampere

mean length of magnetic flux path meters

permeability henrys/meter (410-7 in perfect vacuum)

inductancehenrys

B A H N i

lm

L

The inductance of a wound magnetic core is directly proportionalto the incremental permeability of the core material, which is theslope of the B-H curve.

v L didt

N ddt

ddt

L N ddi

ddi

BA H Nil m

L N Al

dBdH

N Alm m

2 2

and

N

v N

i

30/91

Inductance of a Core

slope L

(a)

(b)

2NL

lAC

1 lACC oo

1

2

21

2

NCNC

lANL

r

or

ore

1 elL

AL

The inductance L represents the capability of magnetic flux density produced by unit current of a circuit loop.

v

i

A

el

Flux saturation

N

i

31/91

Magnetic Reluctance and Permeance

Reluctance

Mean path length l Cross-sectional area A

Permeability

Al

Ni

NiHld lH

lNiH

AB

lNiH

Al

Nil

ANi

Magnetic-motive force (mmf) Ni

Permeance

1

N

i

32/91

Inductance of a Toroidal Core (Self Inductance)

Amp (I)

Weber-turns (=N)

Li

Mean path length lCross-sectional

area A

Permeability

For a magnetic circuit that has a linear relationship between and i because of material ofconstant permeability or a dominating air gap, we can define the -i relationship by the self-inductance (or inductance) L as

iN

iL AN i

l

lAN

lANi

iN

iL 2

where =N, the flux linkage, is in weber-turns. Inductance is measured in henrys or weber-turns per amp.

N

i

33/91

Flux Density Distribution of a Toroidal Core

Representing the magnetic vector potential (A), magnetic flux (B), and current density (j) fields around a toroidal inductor of circular cross section. Thicker lines indicate field lines of higher average intensity. Circles in cross section of the core represent B flux coming out of the picture. Plus signs on the other cross section of the core represent B flux going into the picture. Div A = 0 has been assumed.

A toroidal coil and the magnetic field inside it.

34/91

Energy Stored in a Core

Mean path length l Cross-sectional area Ac

Permeability

I

N: number of turns

22 A NL N

l

The energy stored in the core:

tt

L LIdiLiPdtE 02

0 21''

The energy density (energy/volume) is:2 2 2 21

22 2

2 2

0

1 12

12 2

cB

c c

r

LI N A B lA l A l l N

B B

The energy stored in the core:

coreBL VLIE 2

21

Vcore: volume of the core

Chapter 11 Inductance and Magnetic Energy of Introduction to Electricity and Magnetism, MIT 8.02 Course Notes, Sen-Ben Liao, Peter Dourmashkin, and John Belche, Prentice Hall, 2011.

35/91

Typical Energy Density of a Ferrite Core

0

2

2

re

cB

BVE

For a typical ferrite, assuming the relative permeability is about r = 2000, and the saturation flux density Bsat = 0.3 T (3000 G), we get (for most ungapped ferrite cores) a typical power density of

3J/m 9.1710420002

3.02 7

2

0

2

re

cB

BVE

2Newton/A H/m

7

70

104

104

3000G)B 2000,( J/cm 18J/m 18 satr

33 e

c

VE

(Ferrite core)18100 kHz50%(CRM)3.63610

36/91

Inductance of Air-Core Solenoid

H dl N i

Long air-core solenoid Hl Nic

2 2 27 27

0( ) 4 10 10c

c c

N Dd d BA dH N AL N N NAdi di di l l

inductance in henrysTotal number of turnscross-sectional area inside of solenoid coil in square meters ( )diameter of solenoid in meterslength of solenoid in meters

LN A

Dc lc

Dc2 4/

where

Hdl dl dl dl Nil

l

l d

l d

l d

l d

l dc

c

c

c

c

c

c

0

2

2

2 20 0 0

( ) ( ) ( )

cD

cl

clN

iH

2Newton/A H/m

7

70

104

104

37/91

Inductance of a Solenoid

This is a single purpose calculation which gives you the inductance value when you make any change in the parameters.Small inductors for electronics use may be made with air cores. For larger values of inductance and for transformers, iron isused as a core material. The relative permeability of magnetic iron is around 200.This calculation makes use of the long solenoid approximation. It will not give good values for small air-core coils, since theyare not good approximations to a long solenoid.

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/indsol.html

38/91

Inductance of a Solenoid

D=10 mm

l=50 mmN=30

WD=1.0 mmWire diameter

I

a

b c

d

I

Id

enclosenot doescontour if ,0 enclosescontour if ,I

lH

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/indsol.html

39/91

HW: Inductance of an Air-Core Solenoid

D=10 mm

l=50 mmN=30

WD=1.0 mmWire diameter

I

a

b c

d

An air-core solenoid with construction parameters as shown above, solve the followingproblems:1. Calculate the ideal equivalent inductance of the air-coil solenoid?2. Compute the equivalent inductance of the air-coil solenoid.3. Make a Maxwell simulation of the flux distribution of the air-core solenoid and compare

the simulated inductance with the the analytical result.

Simple Magnetic Circuits

Analogy between electric and magnetic circuits.

cF

c

F

g c

41/91

Electrical-Magnetic Analogy

Magnetic Circuit Electric Circuitmmf NiFlux reluctance permeability

viR1/, where =resistivity

N

i

42/91

Equivalent Electrical Circuit of a Magnetic Circuit

Reluctance

)H :(unit 1-Al

ANi

1

m

mmk

k iN

0k

k

/:law sOhm'

AlR

iv

m

mk

k vRi :law voltage sKirchhoff'

0 :lawcurrent sKirchhoff' k

ki

Magnetic Electrical

Inductance

2Ni

Ni

L

N

i

43/91

Magnetic Circuits of a Gapped Core

mean flux path in the ferromagnetic material

N1gAirgap: Hg

i1

l1 = mean path length

Core: H1

i1 in

H

(a) (b)

44/91

Modeling of a Simple Magnetic Circuit

IlH dH dl H dl Niia

b

gb

a

Hi : Magnetic field intensity in the ferromagnetic materialHg : Magnetic field intensity in the air gap

magnetic motive force (mmf)(unit: Ampere-turns)

H l H l Nii i g g

mean flux path in theferromagnetic material

v

li

ab

mean flux path in the air gapgl

N

i

45/91

Modeling of a Simple Magnetic Circuit

B H B lB

l Niii

ig

gg

B SA dFlux

The surface integral of flux density is equal to the flux.

If the flux density is uniformly distributed over the cross-sectional area, then

i i iB A g g gB A

The streamlines of the flux density are closed, therefore i g

lA

lA Ni

i

i i

g

g g

ii

ii A

l

gg

gg A

l

Nigigi )(

46/91

Modeling of the Air-Gap

gR

Ni

li

ab

lg

mean flux path in the air gap

mean flux path in theferromagnetic material

cR

In general, cg RR

v N

i

47/91

Inductance of a Slotted Ferrite Core

L NB Ai

N Al

c c c

g

2

0

a

b

~

glv

AC: Cross Section Area

N

i

The shearing of an idealized B-H loop due to an air gap.

48/91

Air-gap Fringing Fields

[1] Colonel Wm. T. McLyman, Fringing Flux and Its Side Effects, AN-115 Kg Magnetics Inc. [2] Colonel Wm. T. McLyman, Chapter 3 Magnetic Cores of Transformer and Inductor Design Handbook, Fourth Edition, CRC

Press, April 26, 2011. [3] W.A. Roshen, Fringing field formulas and winding loss due to an air gap, IEEE Transactions on Magnetics, vol. 43, no. 8, pp.

3387-3394, 2007.

gg

gg A

l

The effect of the fringing fields is to increase the effective cross-section area Ag of the air gap. Fringing flux decreases the total reluctance of the magnetic path and, therefore, increases the inductance by a factor, F, to a value greater than the one calculated.

Reluctance of the air gap:

49/91

Fringing Flux at the Gap

The effect of the fringing fields is to increase the effective cross-section area Ag of the air gap. The fringing flux effect is a function of gap dimension, the shape of the pole faces, and the shape, size and location of the winding. Its net effect is to shorten the air gap. Fringing flux decreases the total reluctance of the magnetic path and, therefore, increases the inductance by a factor, F, to a value greater than the one calculated. In most practical applications, this fringing effect can be neglected.

50/91

A Simple Wound-Rotor Synchronous Machine

The magnetic structure of a synchronous machine is shown schematically in the right figure. Assuming that rotor and stator iron have infinite permeability ( ), find the air-gap flux and flux density Bg. For this example I = 10 A, N = 1000 turns, g = 1 cm, and Ag = 2000 cm2.

Calculate the air-gap flux density Bg

2

7

2 2 104 10 0.2

gg

g g

lA

1000 10 0.13 Wbg g

F

0.13 0.65 T0.2g g

BA

51/91

Flux linkage, Inductance, and Energy

Faradays Law When magnetic field varies in time an electric field is produced in space as

determined by Faradays Law:

C S

ddt

E ds B da

( ) d dv t Ndt dt

Line integral of the electric field intensity E around a closed contour C is equal to the time rate of the magnetic flux linking that contour.

Since the winding (and hence the contour C) links the core flux N times then above equation reduces

The induced voltage is usually referred as electromotive force to represent the voltage due to a time-varying flux linkage.

( ) dv tdt

52/91

Direction of EMF

The direction of emf: If the winding terminals were short-circuited a current would flow in such a direction as to oppose the change of flux linkage.

tBAtt c sinsin)( maxmax

tEtNte coscos)( maxmax

maxmaxmax 2 BNAfNE c e(t) N

max2 BNAfE crms

53/91

Example: Estimate the Inductance of a Gapped Core

The magnetic circuit of Fig. (a) consists of an N-turn winding on a magnetic core of infinite permeability with two parallel air gaps of lengths g1 and g2 and areas A1 and A2, respectively.Find (a) the inductance of the winding and (b) the flux density Bl in gap 1 when the winding is carrying a current i. Neglect fringing effects at the air gap.

54/91

Example: Plot the Inductance as a Function of Relative Permeability

The magnetic circuit as shown below has dimensions Ac = Ag = 9 cm2, g = 0.050 cm, lc = 30 cm, and N = 500 tums. With the given magnetic circuit, using MATLAB to plot the inductance as a function of core relative permeability over the range 100 r 100,000.

(a)

(b)

55/91

Example: Plot the Inductance as a Function of the Air Gap Length

The magnetic circuit as shown below has dimensions Ac = Ag = 9 cm2, lc = 30 cm, and N = 500 tums. With the given magnetic circuit, r = 70,000. , using MATLAB to plot the inductance as a function of the air gap length over the range 0.01 cm g 0.1 cm.

(a)

56/91

Example: Magnetic circuit with two windings

The following figure shows a magnetic circuit with an air gap and two windings. In this case note that the mmf acting on the magnetic circuit is given by the total ampere-turns acting on the magnetic circuit (i.e., the net ampere turns of both windings) and that the reference directions for the currents have been chosen to produce flux in the same direction.

2 0 01 1 1 1 1 2 2

c cA AN N i N N ig g

57/91

Example: Analysis of a Switching Inductor

Calculate the current ripple (peak-to-peak) of the inductor current.

dcV

L

A switching inductor can be used as a fundamental energy storage cell with a switching power converting system. Assume components in the following circuit are all ideal, make an analysis of the given problems. Assume the duty ratio for the MOSFET switch is 20%.

D

S

Li

SiDi

48 V20 kHz

5 mH10

DC

s

VfLR

R

58/91

Recommended Books

Electricity and Magnetism, W. N. Cottingham and D. A. Greenwood, Cambridge University Press, 1st Ed., November 29, 1991.

Introduction to Electricity and Magnetism, MIT 8.02 Course Notes, Sen-Ben Liao, Peter Dourmashkin, and John Belche, Prentice Hall, 2011.

A Students Guide to Maxwells Equations (Daniel Fleisch), , 20101018

A Student's Guide to Vectors and Tensors, Daniel A. Fleisch, Cambridge University Press, 1st Ed., November 14, 2011.

59/91

References: Magnetic Circuits

[1] G. K. Dubey, Fundamentals of Electrical Drives, Alpha Science International, Ltd, March 30th 2001. [2] Chapter 1: Magnetic Circuits and Magnetic Materials, Fitzgerald & Kingsley's Electric Machinery, S.D. Umans, 7th Ed, McGraw-Hill

Book Company, 2013. [3] Chapter 11 Inductance and Magnetic Energy, Introduction to Electricity and Magnetism, MIT 8.02 Course Notes, Sen-Ben Liao, Peter

Dourmashkin, and John Belche, Prentice Hall, 2011. [4] Chapter 4 Inductances, Design of Rotating Electrical Machines, Juha Pyrhonen, Tapani Jokinen, Valeria Hrabovcova, 2nd Ed., October

2013, Wiley.

Introduction to Electrodynamics, David Griffiths, 4th Ed., Addison-Wesley, October 6, 2012.

60/91

Power Electronic Systems & Chips Lab., NCTU, Taiwan

Modeling of Practical Inductors

Power Electronic Systems & Chips Lab.

~

61/91

Modeling of Practical Inductors

Ideal impedance model is for a simple linear relationship between frequency and impedance.

Not true across the whole frequency range for real components! For practical capacitors and inductors with nonlinear characteristics, its frequency responses

are only valid for small signal perturbation around its operating point this operating point are generally highly dependent on its dc value, frequency, and temperature.

( )Z j

L

01LC

R

1C

(a) Examples of inductor (b) Equivalent circuit

RAC

C

L

RDC

RC

(c) Frequency response

62/91

Building a Model of a Real Inductor

Ideal inductor Perfectly conducting wire Core of ideal magnetic

material

Practical inductor Real wires have small DC resistance Real wire resistance is frequency dependent Real inductor may saturate Real magnetic materials for inductors are both

frequency and temperature dependent! Parasitic capacitance exists between turns of

the coil, between layers if wound in layers Parasitic lead capacitance

dtdiLv LLv

Li

Li

( )R f ( )L f

C

63/91

Simple Model for Real Inductors

The inductor is modeled as a constant inductance with a series connected resistance (RESR).

As frequency increases, the inductive impedance increases This model does not resonate (no capacitance) There is a corner frequency where the inductive impedance begins

to dominate

ESRR L

64/91

Simple Model for Real Inductors

Example Parameter: L=100 nH, R=2

R

L

( )Z j j L R

LR

Time Constant [sec]

31/2 2dB

RfL

Corner frequency [Hz] 3 92 3.18

2 2 100 10 MHzdB

RfL

Improved Model for Practical Inductors

Parallel resonant circuit Resonant frequency is

R = series resistanceC = parallel capacitance

2( ) 1R j LZ j

j RC LC

R L

C

01LC

1( ) ( ) ||Z j R j Lj C

20 1r

LCQR

1

2Q

2

2

14rR

LC L

In general, R and C are quite small, and the resonant frequency can be approximated to the undamped natural frequency 0:

2L CR 01

r LC

[1] R. L. Boylestad, Introductory Circuit Analysis, 12th Edition, Prentice Hall, 2010.[2] Electromagnetic Compatibility Handbook, Kenneth L. Kaiser, CRC Press, 2005. [3] Cartwright, K., E. Joseph, and E. Kaminsky, Finding the Exact Maximum Impedance Resonant Frequency of a Practical Parallel

Resonant Circuit without Calculus, Technology Interface Internat. J., vol.11, no. 1, Fall/Winter 2010, pp. 26-36.

66/91

Improved Model for Practical Inductors

Example Parameter: L=100 nH, R=2 C=10 pF

R

LC

1( ) ( ) ||Z j R j Lj C

67/91

More Complex Models for HF Inductor

Fairly accurate model for SMT chip inductor

68/91

Simple Electro-Magnetic Circuits

Toroidal Inductance

Block Diagram

Equivalent Circuit

69/91

Transient Response of Inductance

dcV

( )u t

If the above PWM voltage is applied to an ideal inductor, what will be the current waveform? What about a practical inductor?

70/91

Inductor with Resistance

Equivalent circuit of a linear inductor with coil resistance

Block Diagram

71/91

Magnetic Saturation

Amp (I)

Weber-turns (=N)

iL

72/91

AC Excitation of Ferromagnetic Materials

H (or F)

B (or )

Hysteresis Loopt

i(t)

a

b

c

d

e

N

i

73/91

Magnetic Domains

Magnetic domains oriented randomly. Magnetic domains lined up in the presenceof an external magnetic field.

magnetic moment (dipole) magnetic domain

74/91

Hysteresis Curves of a Ferromagnetic Core in AC Excitation

H

B

Hysteresis Loop

H

B

Br

-Hc

Residual Flux Density

Coercive Force

Magnetization or B-H Curve

area hysteresis loss

saturation

~

75/91

AC Excitation of a Magnetic Circuit

Ac: cross-section surface area

mean flux path in theferromagnetic material

From Faradays law, the voltage induced in the N-turn winding is

Assume a sinusoidal variation of the core flux (t); thus

(t)= maxsint=Ac Bmax sint

maxmax 2)( BfNANwdtdtv c

where and

amplitude of the flux density

v v fNA Brms c 12

2max max maxmaxmax 2 BfNANv c

( )v t

cl

N

i

76/91

AC Excitation of a Magnetic Circuit

Excitation phenomena. (a) Voltage, flux, and exciting current; (b) corresponding hysteresis loop.

77/91

AC Excitation Phenomena of a Magnetic Circuit

To produce the magnetic field in the core requires current in the excitation winding know as the excitation current iFor the given , we can obtain the corresponding i from the B-H hysteresis loop. Because = BcAc , and i = HcBLc /NThe saturated hystersis loop will result peakly excitation current with sinusoidal flux variation

(a) Voltage, flux, and excitation current; (b) corresponding hysteresis loop.

(a) (b)

t

iii i

i

i

t t

CH

sv

sv

N

i

78/91

Core Saturation Due to Over-Excitation

sv

)()(1)( 011

0

tdttvN

tt

t s

0( ) 0t

N

i

79/91

Core Saturation Due to Over-Excitation

sv

N

i

80/91

Core Saturation Due to Residual Flux

Power transformer inrush current caused by residual flux at switching instant; flux (green), iron core's magnetic characteristics (red) and magnetizing current (blue).

sv

)()(1)( 011

0

tdttvN

tt

t s

0( ) 0t

N

i

81/91

Inrush Electric Current in a Transformer

sv

What you assume

What really happen

N

i

82/91

Typical Waveform of Magnetizing Inrush Current

In practical applications, the winding resistance and losses of the core will decay residual flux and the dc offset due to initial volts-sec integration.

A a soft start procedure can be used to reduce this effect and a balance control loop can be used to eliminate this dc offset in application to inverters.

83/91

Saturation of an Inductor (Incremental Inductance)

Amp (I)

Weber-turns (=N)

( )x

xi I

L Ii

Mean path length l

Permeability

Cross-sectional area Ac

Ni

A practical inductor will saturate as the current is increased. The incremental inductance is defined as the inductance at a specified current

with small signal perturbation, this is equivalent to a linear inductance for currentaround this operating point.

Note: In the given example, the current source as a perturbation source.

xI

84/91

Measuring the Incremental Inductance at Specific Operating Points (1/3)

N

Amp (I)

Weber-turns (=N)

( )x

xi I

L Ii

Practical inductors are nonlinear and its incremental inductance (small-signal inductance) is highly dependent on its operating point, such as its average current, the magnitude of current ripples, the switching frequency, and the core temperature, etc.

The winding inductance of a synchronous machine is nonlinear, especially for an interior PMSM. This characteristics is useful for the detection of its rotor pole position under sensorless control. Devise a scheme to measure the incremental inductance for different operation points (A, B, and C)?

dcV

3S

4S

1S

2S

A

B

C

85/91

Measuring the Incremental Inductance at Specific Operating Points (2/3)

Design an inductor with 0.1 mH, average current from 1 A to 6A, and operating withswitching frequency of 20 kHz. An illustrated design example can be found in [1].

Devise an incremental inductance measurement scheme for operation points of A (0A), B(4A), and C (8A) with a current ripple of 20 kHz, 2 A (peak-to-peak).

Make a simulation study in consideration of the RDS(ON) and the diode forward voltage drop. Make experimental verifications for the proposed scheme.

dcV

3S

4S

1S

2S

0.1mH, IL(pp)=2A, 20 kHz inductor

8 Ohm, 50W, Cement Resistors

oR

L ESRR

ABv

A

B

( )2e

ESR o DS ON

L LR R R R

REF: [1] Inductor Design in Switching Regulators (Technical Bulletin SR-1A, Magnetics).pdf

ABv

s eT

86/91

Compute the Inductance of a Toroidal Ferrite Core

[1] Rosa Ana Salas and Jorge Pleite, Simple procedure to compute the inductance of a toroidal ferrite core from the linear to the saturation regions, Materials, no. 6, pp. 2452-2463, 2013.

60-turn toroidal inductor with the TN23/14/7 ferrite core

TN23/14/7 Ring Core

TN23147-3R1 - FerroxcubeEffective Core Parameters

Permeability as a Function of Frequency of Different Materials

89/91

B and H Magnetic Fields Inside the Toroidal Core

Moduli of the B and H magnetic fields as a function of the distance from the center of the inductor core (x = 30 mm) obtained by 2D (red dashed line) and 3D (black solid line) simulations, for (a,b) I = 0.0057 A (linear region); (c,d) I = 0.16 A (intermediate region); (e,f) I = 3 A (saturation region).

90/91

Winding Inductances of an IPMSM

a-bc

3 cos22 2 2

d q d qL L L LL

S N

S

N

SN

S

N

S N

Rotor pole position ()0 3/2 2/2

1.5 qL

1.5 dL

IPMSM stator coil

a-bcLa

b c

a-bcL

91/91

Mid-Term Report (April 15, 2016)Modeling of the Stator Winding Inductance

1( ) ?rL

1. Define the stator structure, mechanical dimensions, windingmechanism, and material parameters of the segmented motor.

2. Construct an equivalent circuit for a single segment of the statorteeth and calculate its inductance. Make a Maxwell simulation toverify the calculation.

3. Put the segmented teeth into the stator but without the rotor, makea Maxwell simulation to calculate the inductance of a single statorsegment.

4. Define the rotor structure and material parameters and make aMaxwell simulation to calculate the inductance of a single statorsegment as a function of the rotor pole position.

1 ?L

UNITS FOR MAGNETIC PROPERTIES

Quantity

Magnetic flux density,magnetic induction

Magnetic flux

Magnetic potential difference,ma~netomotive force

Magnetic field strength,,magnetizing force

(Volume) magnetization g

(Volume) magnetization

Magnetic polarization,intensity of magnetization

(Mass) magnetization

Magnetic moment

Magnetic dipole moment

(V01ume) susceptibility

Symbol

B

U,F

H

M

477M

1,1

(F, M

m

j

Gaussian & cgs emu a

gauss (0) d

maxwell (Mx), Gcm2

gilbert (Gb)

oersted (Oe),e Gb/cm

emu/cm3 h

G

emu/cm3

emu/g

emu, erg/G

emu, erg/G

dimensionless, emu/cm3

Conversionfactor, C b

10-4

10/477

477 X 10-4

1477 X 10-7

10-3

477 X 10- 10

SI & rationalized mks C

tesla (T), Wb/m2

weber (Wb), volt second (Vs)

ampere (A)

A/m!

A/m

A/m

T, Wb/m2i

A.m2/kgWbm/kg

A.m2, joule per tesla (l/T)

Wbm'

dimensionlesshenry per meter (H/m), Wb/(A.m)

(Mass) susceptibility

(Molar) susceptibility

Permeability

Relative permeability j

(Volume) energy density,energy product k

Demagnetization factor

cm3/g, emu/g477 X 10-3 m3/kg

Xp, K p (477)2 X 10- 10 H.m2/kg

cm3/mol, emu/mol477 X 10-6 m3/mol

Xmo}, K mo] (477)2 X 10- l3 Hm2/mol

1J- dimensionless 477 X 10-7 H/m, Wb/(Am)

1J-r not defined dimensionless

W erg/cm3 10- l J/m3

D,N dimensionless 1/477 dimensionless

a. Gaussian units and cgs emu are the same for magnetic properties. The defining relation is B =H +477M.b. Multiply a number in Gaussian units by C to convert it to SI (e.g., 1 G X 10-4 T/G = 10-4 T).c. SI (Systeme International d'Unites) has been adopted by the National Bureau of Standards. Where two conversion factors are

given, the upper one is recognized under, or consistent with, SI and is based on the definition B = 1J-o(H +M), where1J-o = 477 X 10- 7 H/m. The lower one is not recognized under SI and is based on the definition B =1J-oll +J, where the symbolI is often used in place of J.

d. 1gauss = 105 gamma (1').e. Both oersted and gauss are expressed as em -1I2.g1l2.S-1 in terms of base units./. A/m was often expressed as "ampere-turn per meter" when used for magnetic field strength.g. Magnetic moment per unit volume.h. The designation "emu" is not a unit.i. Recognized under SI, even though based on the definition B = 1J-oll +J. See footnote c.j. 1J-r = 1J-/1J-o = 1+X, all in SI. 1J-r is equal to Gaussian 1J-.k. BH and 1J-oMH have SI units J/m3; MH and BH /477 have Gaussian units erg/cm3.

R. B. Goldfarb and F. R. Fickett, U.S. De~artment of Commerce, National Bureau of Standards, Boulder, Colorado 80303, March 1985NBS Special Publication 696 For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402

CHAPTER

agnetic Circuits and agnetic Materials

T he objective of this book is to study the devices used in the interconversion of electric and mechanical energy. Emphasis is placed on electromagnetic ro-tating machinery, by means of which the bulk of this energy conversion takes place. However, the techniques developed are generally applicable to a wide range of additional devices including linear machines, actuators, and sensors.

Although not an electromechanical-energy-conversion device, the transformer is an important component of the overall energy-conversion process and is discussed in Chapter 2. As with the majority of electromechanical-energy-conversion devices discussed in this book, magnetically coupled windings are at the heart of transformer performance. Hence, the techniques developed for transformer analysis form the basis for the ensuing discussion of electric machinery.

Practically all transformers and electric machinery use ferro-magnetic material for shaping and directing the magnetic fields which act as the medium for trans-ferring and converting energy. Permanent-magnet materials are also widely used in electric machinery. Without these materials, practical implementations of most famil-iar electromechanical-energy-conversion devices would not be possible. The ability to analyze and describe systems containing these materials is essential for designing and understanding these devices.

This chapter will develop some basic tools for the analysis of magnetic field systems and will provide a brief introduction to the properties of practical magnetic materials. In Chapter 2, these techniques will be applied to the analysis of transform-ers. In later chapters they will be used in the analysis of rotating machinery.

In this book it is assumed that the reader has basic knowledge of magnetic and electric field theory such as is found in a basic physics course for engineering students. Some readers may have had a course on electromagnetic field theory based on Maxwell's equations, but an in-depth understanding of Maxwell's equations is not a prerequisite for mastery of the material of this book. The techniques of magnetic-circuit analysis which provide algebraic approximations to exact field-theory solutions

1

1

YY

YYFitzgerald & Kingsley's Electric Machinery, Stephen Umans, McGraw-Hill Education, 7th Ed., Jan. 28, 2013.

YY

YY

YY

YY

YY

YY

2 CHAPTER 1 Magnetic Circuits and Magnetic Materials

are widely used in the study of electromechanical-energy-conversion devices and form the basis for most of the analyses presented here.

1.1

The complete, detailed solution for magnetic fields in most situations of practical engineering interest involves the solution of Maxwell's equations and requires a set of constitutive relationships to describe material properties. Although in practice exact solutions are often unattainable, various simplifying assumptions permit the attainment of useful engineering solutions. 1

We begin with the assumption that, for the systems treated in this book, the fre-quencies and sizes involved are such that the displacement-current term in Maxwell's equations can be neglected. This term accounts for magnetic fields being produced in space by time-varying electric fields and is associated with electromagnetic radi-ation. Neglecting this term results in the magneto-quasi-static form of the relevant Maxwell's equations which relate magnetic fields to the currents which produce them.

(1.1)

i B-da=O (1.2) Equation 1.1, frequently referred to as Ampere's Law, states that the line integral

of the tangential component of the magnetic field intensity H around a closed contour C is equal to the total current passing through any surface S linking that contour. From Eq. 1.1 we see that the source of His the current density J. Eq. 1.2, frequently referred to as Gauss' Law for magnetic fields, states that magnetic flux density B is conserved, i.e., that no net flux enters or leaves a closed surface (this is equivalent to saying that there exist no monopolar sources of magnetic fields). From these equations we see that the magnetic field quantities can be determined solely from the instantaneous values of the source currents and hence that time variations of the magnetic fields follow directly from time variations of the sources.

A second simplifying assumption involves the concept of a magnetic circuit. It is extremely difficult to obtain the general solution for the magnetic field intensity Hand the magnetic flux density B in a structure of complex geometry. However, in many practical applications, including the analysis of many types of electric machines, a thtee-dimensional field problem can often be approximated by what is essentially

1 Computer-based numerical solutions based upon the finite-element method form the basis for a number of commercial programs and have become indispensable tools for analysis and design. Such tools are typically best used to refine initial analyses based upon analytical techniques such as are found in this book. Because such techniques contribute little to a fundamental understanding of the principles and basic performance of electric machines, they are not discussed in this book.

1.1 Introduction to Magnetic Circuits

Mean core length lc

Cross-sectional areaAc

Magnetic core permeability f1,

Figure 1.1 Simple magnetic circuit. )._ is the winding flux linkage as defined in Section 1.2.

a one-dimensional circuit equivalent, yielding solutions of acceptable engineering accuracy.

A magnetic circuit consists of a structure composed for the most part of high-permeability magnetic material. 2 The presence of high-permeability material tends to cause magnetic flux to be confined to the paths defined by the structure, much as currents are confined to the conductors of an electric circuit. Use of this concept of the magnetic circuit is illustrated in this section and will be seen to apply quite well to many situations in this book.3

A simple example of a magnetic circuit is shown in Fig. 1.1. The core is assumed to be composed of magnetic material whose magnetic permeability fJ., is much greater than that of the surrounding air (JJ., >> tJvo) where /Jvo = 4n x 1 o-7 Him is the magnetic permeability of free space. The core is of uniform cross section and is excited by a winding of N turns carrying a current of i amperes. This winding produces a magnetic field in the core, as shown in the figure.

Because of the high permeability of the magnetic core, an exact solution would show that the magnetiC flux is confined almost entirely to the core, with the field lines following the path defined by the core, and that the flux density is essentially uniform over a cross section because the cross-sectional area is uniform. The magnetic field can be visualized in terms of flux lines which form closed loops interlinked with the winding.

As applied to the magnetic circuit of Fig. 1.1, the source of the magnetic field in the core is the ampere-turn product N i. In magnetic circuit terminology N i is the magnetonwtive force (mmf) F acting on the magnetic circuit. Although Fig. 1.1 shows only a single winding, transformers and most rotating machines typically have at least two windings, and N i must be replaced by the algebraic sum of the ampere-turns of all the windings.

2 In its simplest definition, magnetic permeability can be thought of as the ratio of the magnitude of the magnetic flux density B to the magnetic field intensity H. 3 For a more extensive treatment of magnetic circuits see A.E. Fitzgerald, D.E. Higgenbotham, and A. Grabel, Basic Electrical Engineering, 5th ed., McGraw-Hill, 1981, chap. 13; also E.E. Staff, M.I.T., Magnetic Circuits and Transformers, M.I.T. Press, 1965, chaps. 1 to 3.

3

4 CHAPTER 1 Magnetic Circuits and Magnetic Materials

The net magnetic flux crossing a surface S is the surface integral of the normal component of B; thus

= 1 Bda In SI units, the unit of is the weber (Wb).

(1.3)

Equation 1.2 states that the net magnetic flux entering or leaving a closed surface (equal to the surface integral of B over that closed surface) is zero. This is equivalent to saying that all the flux which enters the surface enclosing a volume must leave that volume over some other portion of that surface because magnetic flux lines form closed loops. Because little flux "leaks" out the sides of the magnetic circuit of Fig. 1.1, this result shows that the net flux is the same through each cross section of the core.

For a magnetic circuit of this type, it is common to assume that the magnetic flux density (and correspondingly the magnetic field intensity) is uniform across the cross section and throughout the core. In this case Eq. 1.3 reduces to the simple scalar equation

(1.4)

where

c/Je =core flux

Be = core flux density

Ae = core cross-sectional area

From Eq. 1.1, the relationship between the mmf acting on a magnetic circuit and the magnetic field intensity in that circuit is.4

(1.5)

The core dimensions are such that the path length of any flux line is close to the mean core length le. As a result, the line integral of Eq. 1.5 becomes simply the scalar product Hele of the magnitude of H and the mean flux path length le. Thus, the relationship between the mmf and the magnetic field intensity can be written in magnetic circuit terminology as

(1.6)

where He is average magnitude of H in the core. The direction of He in the core can be found from the right-hand rule, which can

be stated in two equivalent ways. (1) Imagine a current-carrying conductor held in the right hand with the thumb pointing in the direction of current flow; the fingers then point in the direction of the magnetic field created by that current. (2) Equivalently, if the coil in Fig. 1.1 is grasped in the right hand (figuratively speaking) with the fingers

4 In general, the mmf drop across any segment of a magnetic circuit can be calculated as J Hdl over that portion of the magnetic circuit.

1.1 Introduction to Magnetic Circuits

pointing in the direction of the current, the thumb will point in the direction of the magnetic fields.

The relationship between the magnetic field intensity H and the magnetic flux density B is a property of the material in which the field exists. It is common to assume a linear relationship; thus

(1.7)

where f.1, is the material's magnetic permeability. In SI units, His measured in units of amperes per meter, B is in webers per square meter, also known as teslas (T), and JL is in webers per ampere-turn-meter, or equivalently henrys per 1neter. In SI units the permeability of free space is /Lo = 4Jr X 1 o-7 henrys per meter. The permeability of linear magnetic material can be expressed in terms of its relative permeability JLr, its value relative to that of free space; JL = /Lr/LO Typical values of /Lr range from 2,000 to 80,000 for materials used in transformers and rotating machines. The characteristics of ferromagnetic materials are described in Sections 1.3 and 1.4. For the present we assume that J.l,r is a known constant, although it actually varies appreciably with the magnitude of the magnetic flux density.

Transformers are wound on closed cores like that of Fig. 1.1. However, energy conversion devices which incorporate a moving element must have air gaps in their magnetic circuits. A magnetic circuit with an air gap is shown in Fig. 1.2. When the air-gap length g is much smaller than the dimensions of the adjacent core faces, the core flux c/Jc will follow the path defined by the core and the air gap and the techniques of magnetic-circuit analysis can be used. If the air-gap length becomes excessively large, the flux will be observed to "leak out" of the sides of the air gap and the techniques of magnetic-circuit analysis will no longer be strictly applicable.

Thus, provided the air-gap length g is sufficiently small, the configuration of Fig. 1.2 can be analyzed as a magnetic circuit with two series components both carrying the same flux: a magnetic core of permeability f.l,, cross-sectional area Ac and mean length lc, and an air gap of permeability fLo, cross-sectional area Ag and length g. In the core

i + ---+-

Mean core length lc

+--Air gap, permeability fl-o, AreaAg

Magnetic core permeability fl-, AreaAc

Figure 1.2 Magnetic circuit with air gap.

(1.8)

5

6 CHAPTER 1 Magnetic Circuits and Magnetic Materials

and in the air gap

Ba=-

1:> Ac

Application of Eq. 1.5 to this magnetic circuit yields

:F = Hclc + Hgg and using the linear B-H relationship of Eq. 1.7 gives

B + ___! g JLo

(1.9)

(1.10)

(1.11)

Here the :F = Ni is the mmf applied to the magnetic circuit. From Eq. 1.10 we see that a portion of the mmf, Fe = Hclc, is required to produce magnetic field in the core while the remainder, :Fg = Hgg produces magnetic field in the air gap.

For practical magnetic materials (as is discussed in Sections 1.3 and 1.4), Be and He are not simply related by a known constant permeability JL as described by Eq. 1.7. In fact, Be is often a nonlinear, multi-valued function of He. Thus, although Eq. 1.10 continues to hold, it does not lead directly to a simple expression relating the mmf and the flux densities, such as that of Eq. 1.11. Instead the specifics of the nonlinear Be-He relation must be used, either graphically or analytically. However, in many cases, the concept of constant material permeability gives results of acceptable engineering accuracy and is frequently used.

From Eqs. 1.8 and 1.9, Eq. 1.11 can be rewritten in terms of the flux cas

:F = (__!_:__ + _g_) J-LAc JLoAg

(1.12)

The terms that multiply the flux in this equation are known as the reluctance (R) of the core and air gap, respectively,

g Ra=--

1:> JLoAg

and thus

or

"Finally, Eq. 1.15 can be inverted to solve for the flux

:F =---

Rc+Rg

:F = ---;---

+-g-!LoAg

(1.13)

(1.14)

(1.15)

(1.16)

(1.17)

1.1 Introduction to Magnetic Circuits

I ~ ---+-

Rl nc + +

v :F

R2 ng

I---v- = :F

- (R1 +R2) ('Rc + 'Rg)

(a) (b)

Figure 1.3 Analogy between electric and magnetic circuits. (a) Electric circuit. (b) Magnetic circuit.

In general, for any magnetic circuit of total reluctance Rtob the flux can be found as

= _!__ (1.18) Rtot

The term which multiplies the mmf is known as the permeance P and is the inverse of the reluctance; thus, for example, the total permeance of a magnetic circuit is

1 Ptot = rn

/'\-tot (1.19)

Note that Eqs. 1.15 and 1.16 are analogous to the relationships between the cur-rent and voltage in an electric circuit. This analogy is illustrated in Fig. 1.3. Figure 1.3a shows an electric circuit in which a voltage V drives a current I through resistors R1 and R2 Figure 1.3b shows the schematic equivalent representation of the magnet~c circuit of Fig. 1.2 . Here we see that the mmf :F (analogous to voltage in the electric circuit) drives a flux (analogous to the current in the electric circuit) through the combination of the reluctances of the core Rc and the air gap Rg. This analogy be-tween the solution of electric and magnetic circuits can often be exploited to produce simple solutions for the fluxes in magnetic circuits of considerable complexity.

The fraction of the mmf required to drive flux through each portion of the magnetic circuit, commonly referred to as the mnif drop across that portion of the magnetic circuit, varies in proportion to its reluctance (directly analogous to the voltage drop across a resistive element in an electric circuit). Consider the magnetic circuit of Fig. 1.2. From Eq. 1.13 we see that high material permeability can result in low core reluctance, which can often be made much smaller than that of the air gap; i.e., for (fl,Ac/lc) >> (f.loAg/ g), Rc

8 CHAPTER 1 Magnetic Circuits and Magnetic Materials

Fringing fields

++--+---+- Air gap H-+++-t-1-HI-H--+-H-+H

Figure 1.4 Air-gap fringing fields.

As will be seen in Section 1.3, practical magnetic materials have permeabilities which are not constant but vary with the flux level. From Eqs. 1.13 to 1.16 we see that as long as this permeability remains sufficiently large, its variation will not significantly affect the performance of a magnetic circuit in which the dominant reluctance is that of an air gap.

In practical systems, the magnetic field lines "fringe" outward somewhat as they cross the air gap, as illustrated in Fig. 1.4. Provided this fringing effect is not excessive, the magnetic-circuit concept remains applicable. The effect of these fringing fields is to increase the effective cross-sectional area Ag of the air gap. Various empirical methods have been developed to account for this effect. A correction for such fringing fields in short air gaps can be made by adding the gap length to each of the two dimensions making up its cross-sectional area. In this book the effect of fringing fields is usually ignored. If fringing is neglected, Ag = A c.

In general, magnetic circuits can consist of multiple elements in series and parallel. To. complete the analogy between electric and magnetic circuits, we can generalize Eq. 1.5 as

(1.21)

where F is the mmf (total ampere-turns) acting to drive flux through a closed loop of a magnetic circuit, and Fk = Hklk is the mmf drop across the k'th element of that loop. This is directly analogous to Kirchoff's voltage law for electric circuits consisting of voltage sources and resistors

(1.22)

where V is the source voltage driving current around a loop and Rkik is the voltage drop across the k'th resistive element of that loop.

1.1 Introduction to Magnetic Circuits

Similarly, the analogy to Kirchoff's current law

(1.23) n

which says that the net current, i.e. the sum of the currents, into a node in an electric circuit equals zero is

(1.24) 1l

which states that the net flux into a node in a magnetic circuit is zero. We have now described the basic principles for reducing a magneto-quasi-static

field problem with simple geometry to a magnetic circuit model. Our limited purpose in this section is to introduce some of the concepts and terminology used by engineers in solving practical design problems. We must emphasize that this type of thinking depends quite heavily on engineering judgment and intuition. For example, we have tacitly assumed that the permeability of the "iron" parts of the magnetic circuit is a constant known quantity, although this is not true in general (see Section 1.3), and that the magnetic field is confined solely to the core and its air gaps. Although this is a good assumption in many situations, it is also true that the winding currents produce magnetic fields outside the core. As we shall see, when two or more windings are placed on a magnetic circuit, as happens in the case of both transformers and rotating machines, these fields outside the core, referred to as leakage fields, cannot be ignored and may significantly affect the performance of the device.

The magnetic circuit shown in Fig. 1.2 has dimensions Ac = Ag = 9 cm2, g = 0.050 em, lc = 30 em, and N = 500 turns. Assume the value Mr = 70,000 for core material. (a) Find the reluctances Rc and Rg. For the condition that the magnetic circuit is operating with Be= 1.0 T, find (b) the flux and (c) the current i.

II Solution

a. The reluctances can be found from Eqs. 1.13 and 1.14:

lc 0.3 3 A turns Rc = -- = = 3.79 X 10 fJ.-rfJ.-oAc 70,000 (4n X 10-7)(9 X 10-4) Wb

8 5 x 10-4 Rg = -- = -----'---- = 4.42 X 105 fJ.,oAg (4n X 10-7)(9 X

b. From Eq. 1.4,

c. From Eqs. 1.6 and 1.15,

. F l=

N

A turns

Wb

9

10 CHAPTER 1 Magnetic Circuits and Magnetic Materials

Find the flux cjJ and ctment for Example 1.1 if (a) the number of turns is doubled to N = 1000 turns while the circuit dimensions remain the same and (b) if the number of turns is equal to N = 500 and the gap is reduced to 0.040 em. Solution

a. cjJ = 9 x IQ-4 Wb and i = 0.40 A b. dJ = 9 X IQ-4 Wb and i = 0.64 A

The magnetic structure of a synchronous machine is shown schematically in Fig. 1.5. Assuming that rotor and stator iron have infinite permeability (J.L -7 oo ), find the air-gap flux cjJ and flux density Bg. For this example I= 10 A, N = 1,000 turns, g = 1 em, and Ag = 200 cm2

II Solution Notice that there are two air gaps in series, of total length 2g, and that by symmetry the\ flux density in each is equal. Since the iron permeability is assumed to be infinite, its reluctance is negligible and Eq. 1.20 (with g replaced by the total gap length 2g) can be used to find the flux

- N I J.LoAg - 1000(10)(4n X 10-7)(0.02) - 12 6 Wb - - - . m

2g 0.02

and

cjJ 0.0126 Ba = - = = 0.630 T

" Ag 0.02

Figure 1.5 Simple synchronous machine.

1.2 Flux Linkage, Inductance, and Energy

For the magnetic structure of Fig. 1.5 with the dimensions as given in Example 1.2, the air-gap flux density is observed to be Bg = 0.9 T. Find the air-gap flux and, for a coil of N = 500 turns, the current required to produce this level of air-gap flux.

Solution

= 0.018 Wb and i = 28.6 A.

1.2 FLUX LINKAGE, INDUCTANCE, AND ENERGY

When a magnetic field varies with time, an electric field is produced in space as determined by another of Maxwell's equations refened to as Faraday's law:

i Eds= d { B da dt Js

(1.25)

Equation 1.25 states that the line integral of the electric field intensity E around a closed contour C is equal to the time rate of change of the magnetic flux linking (i.e., passing through) that contour. In magnetic structures with windings of high electrical conductivity, such as in Fig. 1.2, it can be shown that the E field in the wire is extremely small and can be neglected, so that the left-hand side ofEq. 1.25 reduces to the negative of the induced voltage5 e at the winding terminals. In addition, the flux on the right-hand side ofEq. 1.25 is dominated by the core flux. Since the winding (and hence the contour C) links the core flux N times, Eq. 1.25 reduces to

e=Nd

12 CHAPTER 1 Magnetic Circuits and Magnetic Materials

and i will be linear and we can define the inductance L as

A. L=-

i

Substitution ofEqs. 1.5, 1.18 and 1.27 into Eq. 1.28 gives

N2 L=

Rtot

(1.28)

(1.29)

from which we see that the inductance of a winding in a magnetic circuit is proportional to the square of the turns and inversely proportional to the reluctance of the magnetic circuit associated with that winding.

For example, from Eq. 1.20, under the assumption that the reluctance of the core is negligible as compared to that of the air gap, the inductance of the winding in Fig. 1.2 is equal to

(1.30)

Inductance is measured in henrys (H) or weber-turns per ampere. Equation 1.30 shows the dimensional form of expressions for inductance; inductance is proportional to the square of the number of turns, to a magnetic permeability and to a cross-sectional area and is inversely proportional to a length. It must be emphasized that strictly speaking, the concept of inductance requires a linear relationship between flux and mmf. Thus, it cannot be rigorously applied in situations where the non-linear characteristics of magnetic materials, as is discussed in Sections 1.3 and 1.4, dominate the performance of the magnetic system. However, in many situations of practical interest, the reluctance of the system is dominated by that of an air gap (which is of course linear) and the non-linear effects of the magnetic material can be ignored. In other cases it may be perfectly acceptable to assume an average value of magnetic permeability for the core material and to calculate a corresponding average inductance which can be used for calculations of reasonable engineering accuracy. Example 1 ~3 illustrates the former situation and Example 1.4 the latter.

The magnetic circuit of Fig. 1.6a consists of an N -turn winding on a magnetic core of infinite permeability with two parallel air gaps of lengths g1 and g2 and areas A1 and A2, respectively.

Find (a) the inductance of the winding and (b) the flux density B1 in gap I when the winding is carrying a current i. Neglect fringing effects at the air gap.

Solution

a. The equivalent circuit of Fig. 1.6b shows that the total reluctance is equal to the parallel ~ combination of the two gap reluctances. Thus

(a)

1.2 Flux Linkage, Inductance, and Energy

+ Ni

(b)

Figure 1.6 (a) Magnetic circuit and (b) equivalent circuit for Example 1.3.

where

From Eq. 1.28,

). L=

N

b. From the equivalent circuit, one can see that

Ni JJ,0A1Ni 1=-=--

nl gl

and thus

In Example 1.1, the relative permeability of the core material for the magnetic circuit of Fig. 1.2 is assumed to be Jl,r = 70, 000 at a flux density of 1.0 T.

a. In a practical device, the core would be constructed from electrical steel such as M-5 electrical steel which is discussed in Section 1.3. This material is highly nonlinear and its relative permeability (defined for the purposes of this example as the ratio B I H) varies from a value of approximately Jl,r = 72,300 at a flux density of B = 1.0 T to a value of on the order of Jl,r = 2,900 as the flux density is raised to 1.8 T. Calculate the inductance under the assumption that the relative permeability of the core steel is 72,300.

b. Calculate the inductance under the assumption that the relative permeability is equal to 2,900.

13

14 CHAPTER 1 Magnetic Circuits and Magnetic Materials

Solution

a. From Eqs. 1.13 and 1.14 and based upon the dimensions given in Example 1.1,

Rc = _lc_ _ = 0.3 = 3.67 X 103 A turns f.Lrf.LoAc 72,300 (4n X 10-7)(9 X Wb

while Rg remains unchanged from the value calculated in Example 1.1 as Rg = 4.42 x 105 Aturns/Wb.

Thus the total reluctance of the core and gap is

and hence from Eq. 1.29

5 A turns Rtot = Rc + Ra = 4.46 X 10 ---o Wb

N2 5002 L = - = = 0.561 H

Rtot 4.46 X 105

b. For f.Lr = 2,900, the reluctance of the core increases from a value of 3.79 x 103 A turns I Wb to a value of

Rc = __ Zc_ = _____ 0_.3 _____ = 9.15 X 104 _A_ _tu_rn_s f.Lrf.LoAc 2,900 (4n X 10-7)(9 X Wb

and hence the total reluctance increases from 4.46 x 105 A turns I Wb to 5.34 x 105 A turns I Wb. Thus from Eq. 1.29 the inductance decreases from 0.561 H to

N2 5002 L = - =

5 = 0.468 H

Rtot 5.34 X 10

This example illustrates the linearizing effect of a dominating air gap in a magnetic circuit. In spite of a reduction in the permeablity of the iron by a factor of 72,300/2,900 = 25, the inductance decreases only by a factor of 0.46810.561 = 0.83 simply because the reluctance of the air gap is significantly larger than that of the core. In many situations, it is common to assume the inductance to be constant at a value corresponding to a finite, constant value of core permeability (or in many cases it is assumed simply that f.Lr ~ oo ). Analyses based upon such a representation for the inductor will often lead to results which are well within the range of acceptable engineering accuracy and which avoid the immense complication associated with modeling the non-linearity of the core material.

Repeat the inductance calculation of Example 1.4 for a relative permeability f.Lr = 30,000.

Solution L = 0.554H

Using MATLAB,6 plot the inductance of the magnetic circuit of Example 1.1 and Fig. 1.2 as ; function of core permeability over the range 100 :S f.Lr :S 100,000.

6 "MATLAB" ia a registered trademarks of The Math Works, Inc., 3 Apple Hill Drive, Natick, MA 01760, http://www.mathworks.com. A student edition ofMatlab is available.

1.2 Flux Linkage, Inductance, and Energy

II Solution Here is the MATLAB script:

clc clear

% Permeability of free space muO = pi*4.e-7;

%All dimensions expressed in meters Ac = 9e-4; Ag = 9e-4; g Se-4; lc = 0.3; N 500;

%Reluctance of air gap Rg = g/(muO*Ag);

mur = 1:100:100000; Rc = lc./(mur*muO*Ac); Rtot = Rg+Rc; L = W'2. /Rtot;

plot(mur,L) xlabel('Core relative permeability') ylabel('Inductance [H] ')

The resultant plot is shown in Fig. 1.7. Note that the figure clearly confirms that, for the magnetic circuit of this example, the inductance is quite insensitive to relative permeability

0.6

0.5

53' '";; 0.4 u . 0.3 ,s

0.2

0.1

2 3 4 5 6 7 8 9 10 Core relative permeability X 104

Figure 1. 7 MATLAB plot of inductance vs. relative permeability for Example 1.5.

15

16 CHAPTER 1 Magnetic Circuits and Magnetic Materials

until the relative permeability drops to on the order of 1,000. Thus, as long as the effective relative permeability of the core is "large" (in this case greater than 1,000), any non-linearities in the properties of the core material will have little effect on the terminal properties of the inductor.

Write a MATLAB script to plot the inductance of the magnetic circuit of Example 1.1 with f.Lr = 70,000 as a function of air-gap length as the the air gap is varied from 0.01 em to 0.10 em.

Figure 1.8 shows a magnetic circuit with an air gap and two windings. In this case note that the mmf acting on the magnetic circuit is given by the total ampere-turns acting on the magnetic circuit (i.e., the net ampere-turns of both windings) and that the reference directions for the currents have been chosen to produce flux in the same direction. The total mmf is therefore

(1.31)

and from Eq. 1.20, with the reluctance of the core neglected and assuming that Ac = Ag, the core flux is

, (N . N . )JloAc 'f' = JZJ + 2Z2 --

g (1.32)

In Eq. 1.32, is the resultant core flux produced by the total mmf of the two windings. It is this resultant which determines the operating point of the core material.

If Eq. 1.32 is broken up into terms attributable to the individual currents, the resultant flux linkages of coil 1 can be expressed as

2 ( JloAc) ( JloAc) )q = NI = NI -g- i! + NtN2 -g- i2

which can be written

Magnetic core permeability f.L, mean core length lc, cross-sectional area Ac

Figure 1.8 Magnetic circuit with two windings.

(1.33)

(1.34)

where

1.2 Flux Linkage, Inductance, and Energy

L N 2J.LoAc

u= 1--g

(1.35)

is the self-inductance of coil 1 and L IIi I is the flux linkage of coil 1 due to its own current iz. The mutual inductance between coils 1 and 2 is

J.LoAc L12 = N1Nz-- (1.36)

g

and L12i2 is the flux linkage of coill due to curr-ent i2 in the other coil. Similarly, the flux linkage of coil 2 is

( J.LoAc) 2 ( ~toAc) A2 = Nz = N1N2 -g- i1 + N2 -g- i2 (1.37)

or

Az = L21i1 + L2ziz (1.38) where L2I = L 12 is the mutual inductance and

is the self-inductance of coil 2.

L N 21LoAc 22 = 2 --g

(1.39)

It is important to note that the resolution of the resultant flux linkages into the components produced by iz and i2 is based on superposition of the individual effects and therefore implies a linear flux-mmf relationship (characteristic of materials of constant permeability).

Substitution of Eq. 1.28 in Eq. 1.26 yields

d e = dt (Li) (1.40)

for a magnetic circuit with a single winding. For a static magnetic circuit, the induc-tance is fixed (assuming that material nonlinearities do not cause the inductance to vary), and this equation reduces to the familiar circuit-theory form

di e = L- (1.41)

dt However, in electromechanical energy conversion devices, inductances are often time-varying, and Eq. 1.40 must be written as

di dL e = L- + i- (1.42)

dt dt Note that in situations with multiple windings, the total flux linkage of each

winding must be used in Eq. 1.26 to find the winding-terminal voltage. The power at the terminals of a winding on a magnetic circuit is a measure of the

rate of energy flow into the circuit through that particular winding. The power, p, is determined from the product of the voltage and the current

dA. p = ie = i- (1.43)

dt

17

18 CHAPTER 1 Magnetic Circuits and Magnetic Materials

and its unit is watts (W), or joules per second. Thus the change in magnetic stored energy ~ W in the magnetic circuit in the time interval t1 to t2 is

(1.44)

In SI units, the magnetic stored energy W is measured in joules (J). For a single-winding system of constant inductance, the change in magnetic

stored energy as the flux level is changed from A 1 to A2 can be written as

1 A.2 1 A.2 A 1 ~ w = idA= - dA =-(A~- AI) A., A., L 2L (1.45) The total magnetic stored energy at any given value of A can be found from

_setting A 1 equal to zero:

(1.46)

For the magnetic circuit of Example 1.1 (Fig. 1.2), find (a) the inductance L, (b) the magnetic stored energy W for Be = 1.0 T, and (c) the induced voltage e for a 60-Hz time-varying core flux of the form Be= 1.0 sinwt T where w = (2n)(60) = 377.

Ill Solution

a. From Eqs. 1.16 and 1.28 and Example 1.1,

L- ~- N- N2 - i - i - Re +Rg

5002

4.46 X 105 = 0.56 H

Note that the core reluctance is much smaller than that of the gap (Rc Rg). Thus to a good approximation the inductance is dominated by the gap reluctance, i.e.,

N2 L ~ = 0.57H

Rg

b. In Example 1.1 we found that when Be= 1.0 T, i = 0.80 A. Thus from Eq. 1.46,

1 2 1 2 W = 2Lz = 2:(0.56)(0.80) = 0.18 J

c. From Eq. 1.26 and Example 1.1,

d). d

1.3 Properties of Magnetic Materials 19

Practice Problem 1.5

Repeat Example 1.6 for Be = 0.8 T and assuming the core flux varies at 50 Hz instead of 60Hz.

Solution

a. The inductance L is unchanged. b. w = 0.115 J c. e = 113 cos (314t) V

1.3 PROPERTIES OF MAGNETIC MATERIALS In the context of electromechanical-energy-conversion devices, the importance of magnetic materials is twofold. Through their use it is possible to obtain large magnetic flux densities with relatively low levels of magnetizing force. Since magnetic forces and energy density increase with increasing flux density, this effect plays a large role in the performance of energy-conversion devices.

In addition, magnetic materials can be used to constrain and direct magnetic fields in well-defined paths. In a transformer they are used to maximize the cou-pling between the windings as well as to lower the excitation current required for transformer operation. In electric machinery magnetic materials are used to shape the fields to obtain desired torque-production and electrical terminal characteristics. Thus a knowledgeable designer can use magnetic materials to achieve specific desirable device characteristics.

Ferromagnetic materials, typically composed of iron and alloys of iron with cobalt, tungsten, nickel, aluminum, and other metals, are by far the most common mag-netic materials. Although these materials are characterized by a wide range of prop-erties, the basic phenomena responsible for their properties are common to them all.

Ferromagnetic materials are found to be composed of a large number of domains, i.e., regions in which the magnetic moments of all the atoms are parallel, giving rise to a net magnetic moment for that domain. In an unmagnetized sample of material, the domain magnetic moments are randomly oriented, and the net resulting magnetic flux in the material is zero.

When an external magnetizing force is applied to this material, the domain mag-netic moments tend to align with the applied magnetic field. As a result, the do-main magnetic moments add to the applied field, producing a much larger value of flux density than would exist due to the magnetizing force alone. Thus the effective permeability J.L, equal to the ratio of the total magnetic flux density to the applied magnetic-field intensity, is large compared with the permeability of free space fLo. As the magnetizing force is increased, this behavior continues until all the magnetic moments are aligned with the applied field; at this point they can no longer contribute to increasing the magnetic flux density, and the material is said to be fully saturated.

In the absence of an externally applied magnetizing force, the domain magnetic moments naturally align along certain directions associated with the crystal structure of the domain, known as axes of easy magnetization. Thus if the applied magnetizing

20 CHAPTER 1 Magnetic Circuits and Magnetic Materials

1.8

I I - ~f-- ..,..,..... ~ --- ~ ~[.." 1.6

1.4

1.2

('IE 1.0 ::0 ~ t:4 0.8

0.6

0.4

0.2

0 -10

-----~ ~ I / I ~ /; I I ~

r / ;

I

I I ~ bJ}

c:, ~I u,

I ~,

I ~I Cll 0 10 20 30 40 50 70 90 110 130 150 170

H, A turns/m

1.9 B-H loops for M-5 grain-oriented electrical steel 0.012 in thick. Only the top halves of the loops are shown here. (Armco Inc)

force is reduced, the domain magnetic moments relax to the direction of easy mag-netism nearest to that of the applied field. As a result, when the applied field is reduced to zero, although they will tend to relax towards their initial orientation, the magnetic dipole moments will no longer be totally random in their orientation; they will retain a net magnetization component along the applied field direction. It is this effect which is responsible for the phenomenon known as magnetic hysteresis.

Due to this hysteresis effect, the relationship between B and H for a ferromagnetic material is both nonlinear and multi valued. In general, the characteristics of the mate-rial cannot be described analytically. They are commonly presented in graphical form as a set of empirically determined curves based on test samples of the material using methods prescribed by the American Society for Testing and Materials (ASTM).7

The most common curve used to describe a magnetic material is the B-H curve or hysteresis loop. The first and second quadrants (corresponding to B 2: 0) of a set of hysteresis loops are shown in Fig. 1.9 for M-5 steel, a typical grain-oriented

7 ~umerical data on a wide variety of magnetic materials are available from material manufacturers. One problem in using such data arises from the various systems of units employed. For example, magnetization may be given in oersteds or in ampere-turns per meter and the magnetic flux density in gauss, kilogauss, or teslas. A few useful conversion factors are given in Appendix D. The reader is reminded that the equations in this book are based upon SI units.

2.4

2.2

2.0

1.8

1.6

': 1.4 ~ 1.2

~ 1.0

018

0.6

0.4

0.2

0 1

~

-1----"v

v' /

J

I 1/

10

1.3 Properties of Magnetic Materials

~---

~ f.- -:-

v ~,.-I-

I

I 100 1000 10,000 100,000

H, A turns/m

Figure 1.10 De magnetization curve for M-5 grain-oriented electrical steel 0.012 in. thick. (Armco Inc.)

electrical steel used in electric equipment. These loops show the relationship between the magnetic flux density B and the magnetizing force H. Each curve is obtained while cyclically varying the applied magnetizing force between equal positive and negative values of fixed magnitude. Hysteresis causes these curves to be multi valued. After several cycles the B-H curves form closed loops as shown. The arrows show the paths followed by B with increasing and decreasing H. Notice that with increasing magnitude of H the curves begin to flatten out as the material tends toward saturation. At a flux density of about 1. 7 T this material can be seen to be heavily saturated.

Notice also that as H is decreased from its maximum value to zero, the flux density decreases but not to zero. This is the result of the relaxation of the orientation of the magnetic moments of the domains as described above. The result is that there remains a remanent magnetization when His zero.

Fortunately, for many engineering applications, it is sufficient to describe the material by a single-valued curve obtained by plotting the locus of the rriaximum values of B and H at the tips of the hysteresis loops; this is known as a de or normal magnetization curve. A de magnetization curve for M-5 grain-oriented electrical steel is shown in Fig. 1.1 0. The de magnetization curve neglects the hysteretic nature of the material but clearly displays its nonlinear characteristics.

Assume that the core material in Example 1.1 is M-5 electrical steel, which has the de magne-tization curve of Fig. 1.1 0. Find the current i required to produce Be = 1 T.

21

22 CHAPTER 1 Magnetic Circuits and Magnetic Materials

Solution The value of He for Be = 1 T is read from Fig. 1.10 as

He= 11 A turns/m

The mmf drop for the core path is

Fe = Hele = 11 (0.3) = 3.3 A turns

Neglecting fringing, Bg = Be and the mmf drop across the air gap is

Bgg 1 (5 x w-4) Fg = Hgg = - = = 396 A turns

/Lo 4.rr x IQ-7

The required current is

i = _Fe_+___c:. = _39_9 = 0.80 A N 500

Repeat Example 1.7 but find the current i for Be= 1.6 T. By what factor does the current have to be increased to result in this factor of 1.6 increase in flux density?

Solution The current i can be shown to be 1.302 A. Thus, the current must be increased by a factor of 1.302/0.8 = 1.63. Because of the dominance of the air-gap reluctance, this is just slightly in excess of the fractional increase in flux density in spite of the fact that the core is beginning to significantly saturate at a flux density of 1.6 T.

1 In ac power systems, the waveforms of voltage and flux closely approximate sinusoidal functions of time. This section describes the excitation characteristics and losses associated with steady-state ac operation of magnetic materials under such operating conditions. We use as our model a closed-core magnetic circuit, i.e., with no air gap, such as that shown in Fig. 1.1. The magnetic path length is lc, and the cross-sectional area is Ac throughout the length of the core. We further assume a sinusoidal variation of the core flux cp ( t); thus

where

cp(t) =max sin wt = AcBmax sin wt

max =amplitude of core flux cp in webers

Bmax = amplitude of flux density Be in teslas

w = angular frequency = 2n f f = frequency in Hz

(1.47)

From Eq. 1.26, the voltage induced in theN-turn winding is

e(t) = wN

24 CHAPTER 1 Magnetic Circuits and Magnetic Materials

(a) (b)

Figure 1.11 Excitation phenomena. (a) Voltage, flux, and exciting current; (b) corresponding hysteresis loop.

behind this representation can be explained by combining Eqs. 1.51 and 1.52. From Eqs. 1.51 and 1.52, the rms voltamperes required to excite the core of Fig. 1.1 to a specified flux density is equal to

~ lcHrms Ermsl

2.2

2.0

1.8

1.6

('I 1.4

_ 1.2 ..0

~ ~ 1.0

J 0.8

0.6

0.4

0.2

0 0.001

....

~ ~ --

0.01

1.4 AC Excitation

-I--' --/ ~ .... ~---

/ I

/ /

I /

/ /

0.1 10 100 Sa, rms VA/kg

Figure 1.12 Exciting rms voltamperes per kilogram at 60 Hz for M-5 grain-oriented electrical steel 0.012 in. thick. (Armco Inc.)

energy is dissipated as losses and results in heating of the core. The rest appears as . reactive power associated with energy storage in the magnetic field. This reactive

power is not dissipated in the core; it is cyclically supplied and absorbed by the excitation source.

Two loss mechanisms are associated with time-varying fluxes in magnetic ma-terials. The first is due to the hysteretic nature of magnetic material. As has been discussed, in a magnetic circuit like that of Fig. 1.1, a time-varying excitation will cause the magnetic material to undergo a cyclic variation described by a hysteresis loop such as that shown in Fig. 1.13.

Equation 1.44 can be used to calculate the energy input W to the magnetic core of Fig. 1.1 as the material undergoes a single cycle

W = f i d/.. = f ( i:') (A,N dB,)= A,l, f H, dB, (1.55) Recognizing that Aclc is the volume of the core and that the integral is the area of the ac hysteresis loop, we see that each time the magnetic material undergoes a cycle, there is a net energy input into the material. This energy is required to move around the magnetic dipoles in the material and is dissipated as heat in the material. Thus for a given flux level, the corresponding hysteresis losses are proportional to the area of the hysteresis loop and to the total volume of material. Since there is an energy loss per cycle, hysteresis power loss is proportional to the frequency of the applied excitation.

The second loss mechanism is ohmic heating, associated with induced currents in the core material. From Faraday's law (Eq. 1.25) we see that time-varying magnetic

25

26 CHAPTER 1 Magnetic Circuits and Magnetic Materials

B

Figure 1.13 Hysteresis loop; hysteresis loss is proportional to the loop area (shaded).

H

fields give rise to electric fields. In magnetic materials these electric fields result in induced currents, commonly referred to as eddy currents, which circulate in the core material and oppose changes in flux density in the material. To counteract the corresponding demagnetizing effect, the current in the exciting winding must increase. Thus the resultant "dynamic" B-H loop under ac operation is somewhat "fatter" than the hysteresis loop for slowly varying conditions and this effect increases as the excitation frequency is increased. It is for this reason that the charactersitics of electrical steels vary with frequency and hence manufacturers typically supply characteristics over the expected operating frequency range of a particular electrical steel. Note for example that the exciting rms voltamperes of Fig. 1.12 are specified at a frequency of 60 Hz.

To reduce the effects of eddy currents, magnetic structures are usually built with thin sheets or laminations of magnetic material. These laminations, which are aligned in the direction of the field lines, are insulated from each other by an oxide layer on their surfaces or by a thin coat of insulating enamel or varnish. This greatly reduces the magnitude of the eddy currents since the layers of insulation interrupt the current paths; the thinner the laminations, the lower the losses. In general, as a first approximation, eddy-current loss can be considered to increase as the square of the excitation frequency and also as the square of the peak flux density.

In general, core losses depend on the metallurgy of the material as well as the flux density and frequency. Information on core loss is typically presented in graphical form. It is plotted in terms of watts per unit mass as a function of flux density; often a family of curves for different frequenc