Magnetic Material

UNIVERSITY OF TECHNOLOGY, SYDNEYUNIVERSITY OF TECHNOLOGY, SYDNEY

FACULTY OF ENGINEERINGFACULTY OF ENGINEERING

48531 Electromechanical Systems48531 Electromechanical Systems

Magnetic Materials andMagnetic Materials andMagnetic Circuit Anal ysisMagnetic Circuit Anal ysis

Topics to cover:

1. Magnetic Core Losses

2. Electrical Circuit Model of Magnetic Cores

3. Magnetic Circuital Laws and Circuit Analysis

4. Magnetic Circuit Model of Permanent Magnets

IntroductionIntroduction• In general, magnetic materials can be classified as magnetically

“soft” and “hard” materials. Soft magnetic materials are normallyused as the magnetic core materials for inductors, transformers,actuators and rotating electrical machines, in which the magneticfields vary frequently, whereas hard magnetic materials, orpermanent magnets, are used to generate magnetic fields in devicesfor electromechanical energy conversion, such as electricalgenerators, motors, and actuators.

• When the magnetic field varies in a magnetic core, power loss,known as magnetic core loss, occurs. This loss is caused bymagnetic hysteresis and electric eddy currents induced in the core.

Introduction Introduction (Cont.)(Cont.)• For the performance prediction of electromagnetic devices, magnetic

field analysis is required. For magnetic field analysis, there are threetype of methods: analytical field analysis by solving the Maxwell’sequations in either integral or differetial forms, magnetic circuitanalysis under certain assumptions to simplify the problem, andnumerical methods.

• Analytical field solution is difficult or impossible for practicalengineering problems, while powerful numerical methods, such asthe finite element method, require specific sophisticated computersoftware and computers of high speed large memory.

• In this chapter, we introduce magnetic circuit nalysis method. Basicmagnetic circuital laws and magnetic circuit models of both soft andhard magnetic materials will be developed.

Soft Magnetic MaterialsSoft Magnetic Materialsunder Alternating Excitationsunder Alternating Excitations

- Magnetic Core Loss- Magnetic Core LossMagnetic core losses occur inferromagnetic materials under timevarying magnetic field excitations. Inorder for the users to evaluate thecore losses, material manufacturersgenerally supply data of specific corelosses (in W/kg) under sinusoidalmagnetic flux density excitations.The diagram on the right hand sideillustrates the specific power loss ofM-36, 0.356 mm silicon steel sheet.


- Magnetic Hysteresis Loss- Magnetic Hysteresis Loss

As the external magnetic field varies at a very low rate periodically, dueto the effects of magnetic domain wall motion the B-H relationship is ahysteresis loop. The area enclosed by the loop is a power loss known asthe hysteresis loss, and can be calculated by

P dhyst = •∫H B (W/m3/cycle) or (J/m3)

For magnetic materials commonly used in electric machines an approxi-mate relation is used:P C fBhyst h p

n= (1.5 < n < 2.5) (W/kg)

where Ch is a constant of the material, f the frequency of excitation, andBp the peak value of the flux density.


- An Example of Hysteresis Loss- An Example of Hysteresis Loss

A B-H loop for a type of electricsteel sheet is shown in the diagramon the right hand side. Determineapproximately the hysteresis lossper cycle in a torus of 300 mmmean diameter and a square crosssection of 50×50 mm. The scalesfor the horizontal and vertical axesare 25 A/m/div and 0.1T/div,respectively.

Hysteresis loop of M36 steel sheet


- An Example of Hysteresis Loss (Cont.)- An Example of Hysteresis Loss (Cont.)

Solution:

The area of each square in the B-Hdiagram represents

(0.1T)×(25A/m)=2.5 (Wb/m2)(A/m)

= 2.5 VsA/m3 = 2.5 J/m3

If a square that is more than half within the loop is regarded as totallyenclosed, and one that is more than half outside is disregarded, then thearea of the loop is 2 × 43 × 2.5 = 215 J/m3

The volume of the torus is 0.052 × 0.3 = 2.36 × 10−3 m3

Energy loss in the torus per cycle is thus 2.36 × 10-3 × 215 = 0.507 J


- Eddy Current Loss- Eddy Current LossIf the excitation magnetic field varies quickly,the effect of currents induced in ferromagneticmaterials will become significant. Thesecurrents are known as the eddy currents andthe power loss generated is known as the eddycurrent loss. The resultant B-H or λ-i loopwill be fatter than the corresponding purehysteresis loop. With a sinusoidal magnetic

( )P C fBeddy e p=2

flux density, the average eddy current loss in a magnetic core can beexpressed by

(W/kg)


- Methods for Reducing Eddy Current Loss- Methods for Reducing Eddy Current LossSince the eddy current loss is caused bythe induced eddy currents in a magneticcore, an effective way to reduce the eddycurrent loss is to increase the resistivityof the material. This can be achieved byadding Si in steel. However, too muchsilicon would make the steel brittle.Commonly used electrical steels contain3% silicon.

Another effective way to reduce the eddy current loss is to use electricalsteel sheets coated by insulation, which breaks the eddy current path.


- Excess Loss- Excess LossThe previous formulation for eddy currentloss, which is obtained under the assumptionof global eddy currents, is incorrect formaterials with magnetic domains. When theexcitation field varies, the domain walls moveaccordingly and local eddy currents areinduced by the fluctuation of the local fluxdensitycaused by the domain wall motion.The difference between the total eddy currentloss and that predicted by the previousformulation is known as the excess loss.

H

M sM s M sH


- Excess Loss and Total Core Loss- Excess Loss and Total Core LossBy statistical analysis, it waspostulated that for most softmagnetic materials under asinusoidal excitation, theexcess loss can be predicted by

Pex/Freq

Peddy/Freq

Physt/Freq

Frequency (Hz)

Core Loss (J/kg)

0

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0 50 100 150 200 250

B = 1 T

Separation of core loss of Lycore-140 at B=1 T

( )P C fBex ex p=3 2/

(W/kg)

Therefore, the total core loss canbe calculated by

P P P Pcore hyst eddy ex= + +


- Electric Circuit Model of Magnetic Cores- Electric Circuit Model of Magnetic CoresConsider a magnetic core with a coilexcited by a sinusoidal voltage. Since theemf balances the voltage, the fluxlinkage is also sinusoidal while theexcitation current is non-sinusoidal.


- Electric Circuit Model of Magnetic Cores (Cont.)- Electric Circuit Model of Magnetic Cores (Cont.)If only the fundamental component of the excitation current isconsidered, in AC circuits, a magnetic core can be represented by anelectric circuit model of an inductor in parallel with a resistor, whichstands for the core loss.

Magnetic Circuit AnalysisMagnetic Circuit Analysis- A Simple Magnetic Circuit- A Simple Magnetic Circuit

Consider a simple structure consisting of a current carrying coil and amagnetic core. Assume that the size of the device and the operationfrequency are such that the displacement current in Maxwell’s equationsare negligible, and that the permeability of the core material is very highso that all magnetic flux will be confined within the core. By Ampere’slaw, H J• = •∫ ∫d d

C S

l a

we have H l N ic c =�

φc c c

c c c

B A

and B H

== µ

( )∴ = = φµc

c c c c

Ni

l A

F

R

Magnetic Circuit AnalysisMagnetic Circuit Analysis- Ohm’s Law- Ohm’s Law

If we take the magnetic flux φc as the “current”, the magnetomotiveforce (mmf) F=Ni as the “emf of a voltage source”, and Rc=lc/(µcAc)(known as the magnetic reluctance) as the “resistance” in the magneticcircuit, we have an analog to the Ohm’s law in electrical circuit theory.

E R

I

F Rc

φc

IE

R= φ c

c

F

R=

Electric Circuit Magnetic Circuit

Magnetic Circuit AnalysisMagnetic Circuit Analysis- A Magnetic Circuit with Airgap- A Magnetic Circuit with Airgap

Consider a magnetic circuit with an air gap. As they cross the air gap, themagnetic flux lines bulge outward somewhat. The effect of the fringingfield is to increase the effective cross sectional area Ag of the air gap.

Magnetic Circuit AnalysisMagnetic Circuit Analysis- A Magnetic Circuit with Airgap (Cont.)- A Magnetic Circuit with Airgap (Cont.)

By Ampere’s law, we can write

F Ni H l H lc c g g= = +

H lB

lA

l Rc cc

cc

c

c cc c c= = =

µφ

µφ

where

H lB

lA

l Rg g

g

og

g

o gg g g= = =

µφ

µφand

According to Gauss’ law in magnetics,

B • =∫ dS

a 0

we have φ φ φc g= =

( )F R Rc g= + φTherefore,

FRc

φ

Rg

Magnetic Circuit AnalysisMagnetic Circuit Analysis- Kirchhoff’s Voltage and Current Laws- Kirchhoff’s Voltage and Current Laws

If we regard Hclc and Hglg as the “voltage drops” across the reluctance ofthe core and airgap respectively, the above equation from Ampere’s lawcan be interpreted as an analog to the Kirchhoff’s voltage law (KVL) inelectric circuit theory, or

i1 i2

N1 N2

g1 g

g3

µ

⇒φ1

⇓

⇒φ2

Gaussian surface2

φ3R Fk k kφ = ∑∑

The Kirchhoff’s current law (KCL) canbe derived from the Gauss’ law inmagnetics. Applying the Gauss’ law to theT joint in the circuit on the right, we have

φ kk =∑ =

1

3

0 φ kk

n

=∑ =

1

0or, in general,

Magnetic Circuit AnalysisMagnetic Circuit Analysis- Magnetic Circuit Model of Permanent Magnet- Magnetic Circuit Model of Permanent Magnet

The characteristics of permanent magnets aredescribed by demagnetization curves as shownbelow.

Magnetic Circuit AnalysisMagnetic Circuit Analysis- Magnetic Circuit Model of Permanent Magnet (Cont.)- Magnetic Circuit Model of Permanent Magnet (Cont.)

Consider a piece of permanent magnet of uniform cross section. Thedemagnetization curve of the magnet is a straight line as shown below.The demagnetization curve can be expressed analytically as

( ) ( )BB

HH H H Hm

r

cm c m m c= + = +µ

where µm=Br/Hc is the permeability of the permanent magnet, which isvery close to µo, the permeability of free space. The magnetic “voltagedrop” across the magnet can be expressed as

H lB

H ll

AH l R Fm m

m

mc m

m

m mm c m m m m= −

= − = −

µ µφ φ

Rl

Amm

m m

=µ

where

is the reluctance, and Fm=Hclm the mmf source ofthe magnet.


Therefore, we obtain the magnetic circuit model for a permanentmagnet of linear demagnetization curve. It should be noted that inthe magnet, Bm and Hm are in opposite directions.

Am

Bm

l m

φ m

µm

l mRm= Am

Hml m

Hc l mFm=

B

HHc−

Br

0

Bm

Hm


For a magnet with a nonlinear demagnetization curve, the abovemagnetic circuit model is still valid, except that the magneticpermeability becomes

µmm

m c

B

H H=

+

Am

Bm

lm

φm

µm

lmRm= Am

Hml m

Hc l mFm=

B

HHc−

Br

0 Hc

Bm

Hm Hm Hc

which is a function of the magnetic field in the magnet.


It should also be understood that the operating point (Hm,Bm) will notmove along the nonlinear demagnetization curve if a small (such thatthe magnet will not be demagnetized) periodic external magnetic field

B

HHc−

Br

0 Hc

Bm

Hm Hm Hc

Hex Hex

is applied to the magnet.Instead, the operating pointwill move along a minorloop or simply a straightline (center line of theminor loop) as illustrated inthe diagram on the righthand side.

Magnetic Material

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