ECE 320 Energy Conversion and Power Electronics

Dr. Tim Hogan

Chapter 2: Magnetic Circuits and Materials Chapter Objectives In this chapter you will learn the following: • How Maxwell’s equations can be simplified to solve simple practical magnetic problems • The concepts of saturation and hysteresis of magnetic materials • The characteristics of permanent magnets and how they can be used to solve simple problems • How Faraday’s law can be used in simple windings and magnetic circuits • Power loss mechanisms in magnetic materials • How force and torque is developed in magnetic fields

2.1 Ampere’s Law and Magnetic Quantities Ampere’s experiment is illustrated in Figure 1 where there is a force on a small current element

I2l when it is placed a distance, r, from a very long conductor carrying current I1 and that force is quantified as:

lIr

IF 21

2πμ

= (N) (2.1)

I1

rI2

F

Conductor1

Current Elementof Length l

Figure 1. Ampere’s experiment of forces between current carrying wires.

The magnetic flux density, B, is defined as the first portion of equation (2.1) such that:

lBIF 2= (N) (2.2)

1- 1

From (2.1) and (2.2) we see the magnetic flux density around conductor 1 is proportional to the current through conductor 1, I1, and inversely proportional to the distance from conductor 1. Looking

at the units and constants handout given in class, or from (2.2) the units of, B, are seen as ⎟⎠⎞

⎜⎝⎛

⋅mAN ,

thus µ, called permeability, has units of ⎟⎠⎞

⎜⎝⎛

2AN . More commonly, the relative permeability of a given

material is given where 0μμμ r= and 90 2

N400 10 A

μ π − ⎛= × ⎜⎝ ⎠

⎞⎟ . Since a Newton-meter is a Joule, and

a Joule is a Watt-second: ⎟⎠⎞

⎜⎝⎛ ⋅

=⎟⎠⎞

⎜⎝⎛

⋅⋅⋅

=⎟⎠⎞

⎜⎝⎛

⋅=⎟

⎠⎞

⎜⎝⎛

⋅⋅

=⎟⎠⎞

⎜⎝⎛

⋅ 2222 msV

mAsAV

mAJ

mAmN

mAN . This shows B is a

per meter squared quantity, and the (V·s) units represents the magnetic flux and is given units of Webers (Wb). This flux can be found by integrating the normal component of B over the area of a given surface:

∫ ⋅=S

dsnB ˆφ (2.3)

The magnetic field intensity is related to the magnetic flux density by the permeability of the

media in which the magnetic flux exists.

μBH ≡ (2.4)

For the system in Figure 1, r

IBHπμ 21== and have units of ⎟

⎠⎞

⎜⎝⎛

mA . If there were multiple

conductors in place of conductor 1, for example in a coil, then the units would be ampere-turns per meter. A line integration of H over a closed circular path gives the current enclosed by that path, or for the system in Figure 1:

1

2

0

1

2Idl

rIdlHH

r

C==⋅= ∫∫

π

π (2.5)

again, if the system contained multiple conductors within the enclosed path, the result would give ampere-turns. Equation (2.5) is Ampere’s circuital law.

An alternative approach as described in your textbook is to begin with Maxwell’s equations

which include Ampere’s circuital law in a more general form as shown in Table I below:

Table I. Maxwell’s equations. Name Point Form Integral Form

Faraday’s Law tB∂∂

−=×∇ E ∫ ∫ ⋅−=⋅C S

dsnBdtddl ˆE

Ampere’s Law Modified by MaxwelltDJH∂∂

+=×∇ ∫∫ ⋅⎥⎦

⎤⎢⎣

⎡∂∂

+=⋅SC

dsntDJdlH ˆ

Gauss’s Law ρ=⋅∇ D ∫∫ =⋅VS

ˆ dvdsnD ρ

Gauss’s Law for Magnetism 0=⋅∇ B 0 ˆS

=⋅∫ dsnB

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where ∫C represents the integral over a closed path, ∫S represents the integral over the surface of a

closed volume of space, is a surface normal vector, n̂ E is the spatial vector of electric field, B is the spatial vector of magnetic flux density, J is the spatial vector of the electric current density, D is

the displacement charge vector, ρ is the electric charge density, ∫V is a volume integral ×∇ is the

curl and the divergence of the vector being acted upon. ⋅∇

Some assumptions commonly used in electromechanical energy conversion include a low enough

frequency, that the displacement current, tD∂∂

, can be neglected, and the assumption of homogeneous

and isotropic media used in the magnetic circuit. Under these assumptions, Ampere’s circuital law is modified to remove the displacement current component such that

∫∫ ⋅=⋅SC

dsnJdlH ˆ (2.6)

which for the system of Figure 1 reduces to equation (2.5).

2.2 Magnetic Circuits

From Ampere’s circuital law, ∫∫ ⋅=⋅SC

dsnJdlH ˆ , we see the magnetic field intensity around a

closed contour is a result of the total electric current density passing through any surface linking that

contour. Gauss’s Law for magnetism, 0 ˆS

=⋅∫ dsnB states that there are no magnetic monopoles –

that is to say there for a closed surface there is as much magnetic field density leaving that closed surface as there is entering the closed surface. If the integration is for an area, but not a closed

surface area, then we obtain the flux or ∫ ⋅=S

dsnB ˆφ .

The permeability of free space is 1, while the permeability of magnetic steel is a few hundred thousand. Magnetic flux can be confined to the structures or paths formed by high permeability materials. In this way, magnetic circuits can be formed such as the one shown in Figure 2.

Figure 2. Simple magnetic circuit [1].

1- 3

The driving force for the magnetic field is the magnetomotive force (mmf), F, which equals the

ampere-turn product iN ⋅=F (2.7)

The analysis of a magnetic circuit is similar to the analysis of an electric circuit, and an analogy

can be made for the individual variables as shown in Table II.

Table II. Comparison of electric and magnetic circuits. Electric Circuit Magnetic Circuit

V r

I

I

a

b

c

Electrically ConductiveMaterial

V r

φ

I

a

b

c

High PermeabilityMaterial

Driving Force applied battery voltage = V applied ampere-turns = F

Response

resistance electricforce driving current =

or

RVI =

reluctance magneticforce driving flux =

or

RF

=φ

Impedance Impedance is used to indicate the impediment to the driving force in establishing a response.

AlRσ

==resistance

where l = 2πr, σ = electrical conductivity, A = cross-sectional area

Alμ

== Rreluctance

where l = 2πr, μ = permeability, A = cross-sectional area

Equivalent Circuit

V

I

R

V = IR

F

φ

R

F = φR

1- 4

1- 5

Electric Circuit Magnetic Circuit

Fields Electric Field Intensity

rV

lV

π2=≡E (V/m)

or

∫ =⋅ VdlE

Magnetic Field Intensity

rlH

π2FF

=≡ (A-t/m)

or

∫ =⋅ FdlH

Potential Electric Potential Difference

abababb

a

b

aab IRl

Al

lIl

lIRdl

lVdlV ====⋅= ∫∫ σ

E

Magnetic Potential Difference

ababababb

aab l

Al

ll

ll

ldlH RRFF φ

μφφ

====⋅= ∫Flow Densities

Current Density

( ) EE σσ

===≡A

lAl

ARV

AIJ

Flux Density

H

AlA

HlAA

B μ

μ

φ=

⎟⎠⎞⎜

⎝⎛

==≡R

F

An example magnetic circuit is shown in Figure 3, below.

Figure 3. Simple magnetic circuit with an air gap [1].

For this circuit, we will assume: • the magnetic flux density is uniform throughout the magnetic core’s cross-sectional area and is

perpendicular to the cross-sectional area • the magnetic flux remains within the core and the air gap defined by the cross-sectional area of the

core and the length of the gap (no leaking of field, no fringe fields at the gap). Then

ggccA

ABABAdBc

==⋅= ∫φ (2.8)

with Ac = Ag

HH

BB

gc

gc

μμ =

= (2.9)

Since, F = Hl

g

g

gc

c

c

gcc

lB

lB

gHlH

μμ+=

+=F (2.10)

or

( )gc

ggcc

c

Ag

Al

RR

F

+=

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

φ

μμφ

(2.11)

Thus the magnetic circuit shown in Figure 3 can be represented as

F

φ

R

R

c

g

gcgc

iNRRRR

F+⋅

=+

=φ

Figure 4. Equivalent circuit for the magnetic circuit in Figure 3.

This concept is helpful for more complex configurations such as shown in Figure 5.

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AreaA1

AreaA1

AreaA1

Area A2

AreaA3

g2

I

Area A2

g1

I21

AreaA1

Figure 5. Magnetic circuit with various cross-sectional areas and two coils.

Using the lengths defined in Figure 6, and paying attention to the direction of the magnetomotive

force from each of the coils by use of the right hand rule, the equivalent circuit can be drawn as seen in Figure 7.

Area A

Area A2

AreaA3

g2

I

g1

I21

1

Area A1

Area A1

l 1

·l221

·l321

Figure 6. Lengths for each section of the magnetic circuit of Figure 5.

Then,

( ) 121111 11φφφ glIN RRF ++=⋅= (2.12)

and ( )

2231 21222 lglgIN RRRRF ++−=⋅= φφ (2.13) For a system with the dimensions, number of turns, current, and permeability known, then

equations (2.12) and (2.13) give two equations with two unknowns such that φ1 and φ2 can be found.

1- 7

F

Rg

Rg

F2

11

2

R l1

Rl2

Rl3

φ 1

φ2

Figure 7. Equivalent circuit for the configuration of Figure 5.

Assuming the gaps are air gaps, the value of each reluctance can be found as:

1

11 A

ll μ=R

10

11 A

gg μ=R

2

22 A

ll μ=R

3

33 A

ll μ=R

20

22 A

gg μ=R

Then we can find the magnetic flux density for each gap as:

2

2

1

1

2

1

AB

AB

g

g

φ

φ

=

= (2.14)

Note that the permeability of the core material can be much larger than the permeability of the gaps such that the total reluctance is dominated by the reluctances of the gaps.

2.3 Inductance From circuit theory we recall that the voltage across an inductor is proportional to the time rate of

change of the current through the inductor.

( ) ( )dt

tdiLtv LL = (2.15)

and while the power of an inductor can be positive or negative, the energy is always positive as

( ) 2

21

LL iLtw ⋅= (2.16)

In Table I, Faraday’s law is ∫ ∫ ⋅−=⋅C S

dsnBdtddl ˆE or the electric field intensity around a

closed contour C is equal to the time rate of change of the magnetic flux linking that contour. Integrating over the closed contour of the coil itself gives us the negative of the voltage at the terminals of the coil. On the right side of Faraday’s law we then must integrate over the surface of the full coil, thus including the N turns of the coil. Then Faraday’s law gives

1- 8

( ) ( )dt

tdNtvLφ

= (2.17)

Comparing equation (2.15) and (2.17) gives

didNL φ

= (2.18)

For linear inductors, the flux φ is directly proportional to current, i, for all values such that

i

NL φ= (2.19)

with units of (Weber-turns per ampere), or Henry’s (H). The flux linkage, λ, is defined as φλ N=

and combining this with the relationship between flux and total circuit reluctance tottot

NiRR

F==φ

along with (2.19) gives

tot

NLR

2

= (2.20)

Thus inductance can be increased by increasing the number of turns, using a metal core with a higher permeability, reducing the length of the metal core, and by increasing the cross-sectional area of the metal core. This is information that is not readily seen by either the circuit laws for inductors, or through Faraday’s equation alone.

Mutual Inductance

For a magnetic circuit containing two coils and an air gap such as in Figure 8, with each coil wound such that the flux is additive, then the total magnetomotive force is given by the sum of contributions from the two coils as 2211 iNiN +=F (2.21)

i i21

Area Ac lc

φ

g

λ 1 λ 2

Figure 8. Mutual inductance magnetic circuit.

The equivalent circuit is shown in Figure 9.

1- 9

F2

Rg

R lc

F1

φ

Figure 9. Equivalent circuit for Figure 8.

If the permeability of the core is large such that glc RR << and the cross-sectional area of the gap

is assumed equal to the cross-sectional area of the core (Ag = Ac), then

( )gAiNiN c0

2211μφ +≈ (2.22)

The flux linkage, λ1, in Figure 8 is φλ 11 N= , or

212111

20

21102

111

iLiL

igANNi

gANN cc

+=

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛==

μμφλ (2.23)

where

⎟⎟⎠

⎞⎜⎜⎝

⎛=

gANL c02

111μ (2.24)

is the self-inductance of coil 1 and L11i1 is the flux linkage of coil 1 due to its own current i1. The mutual inductance between coils 1 and 2 is

⎟⎟⎠

⎞⎜⎜⎝

⎛=

gANNL c0

2112μ (2.25)

and L12i2 is the flux linkage of coil 1 due to current i2 in the other coil. Similarly, for coil 2

222121

202

210

2122

iLiL

igANi

gANNN cc

+=

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛==

μμφλ (2.26)

where L21 = L12 is the mutual inductance and L22 is the self-inductance of coil 2.

⎟⎟⎠

⎞⎜⎜⎝

⎛=

gANL c02

222μ (2.27)

1- 10

2.4 Magnetic Material Properties A simplification we have used is that the permeability of a given material is constant for different

applied magnetic fields. This is true for air, but not for magnetic materials. Materials that have a relatively large permeability are ferromagnetic materials in which the magnetic moments of the atoms can align in the same direction within domains of the material when and external field is applied. As more of these domains align, saturation is reached when there is no further increase in flux density of that of free space for further increases in the magnetizing force. This leads to a changing permeability of the material and a nonlinear B vs. H relationship as shown in Figure 10.

H

B

Figure 10. Nonlinear B vs H normal magnetization curve.

When the field intensity is increased to some value and is then decreased, it does not follow the curve shown in Figure 10, but exhibits hysteresis as shown by the abcdea loop in Figure 11. The deviation from the normal magnetization curve is caused by some of the domains remaining oriented in the direction of the originally applied field. The value of B that remains after the field intensity H is removed is called residual flux density. Its value varies with the extent to which the material is magnetized. The maximum possible value of the residual flux density is called retentivity and results whenever values of H are used that cause complete saturation. When the applied magnetic field is cyclically applied so as to form the hysteresis loop such as abcdea in Figure 11, the field intensity required to reduce the residual flux density to zero is called the coercive force. The maximum value of the coercive force is called the coercivity.

The delayed reorientation of the domains leads to the hysteresis loops. The units of BH is

32 mJ

mN

meter-amperenewtons

meteramperes of units ==×=HB (2.28)

1- 11

or an energy density.

H (A·t/m)

B (Wb/m2)

O

a

b

c

d

e

f

Residual flux

Retentivity

Coercive force

Coercivity

Figure 11. Hysteresis loops. The normal magnetization curve is in bold.

H

B

O

a

b

c

d

e

f

Figure 12. Energy relationship for hysteresis loop per half-cycle.

1- 12

The full shaded area in Figure 12 outlined by eafe represents the energy stored in the magnetic field during the positive half cycle of H. The hatched area eabe represents the hysteresis loss per half cycle. This energy is what is required to move around the magnetic dipoles and is dissipated as heat. The energy released by the magnetic field during the positive half cycle of they hysteresis loop is given by the cross-hatched area outlined by bafb and is energy that is returned to the source.

The power loss due to hysteresis is given by the area of the hysteresis loop times the volume of the ferromagnetic material times the frequency of variation of H. This power loss is empirically given as (2.29) n

hh fBkP maxν=

where n lies in the range 1.5 ≤ n ≤ 2.5 depending on the material used, ν is the volume of the ferromagnetic material, and the value of the constant, kh, also depends on the material used. Some typical values for kh are: cast steel 0.025, silicon sheet steel 0.001, and permalloy 0.0001.

In addition to the hysteresis power loss, eddy current losses also exist for time-varying magnetic fluxes. Circulating currents within the ferromagnetic material follow from the induced voltages described by Faraday’s law. To reduce these eddy current losses, thin laminations (typically 14-25 mils thick) are commonly used where the magnetic material is composed of stacked layers with an insulating varnish or oxide between the thin layers. An empirical equation for the eddy-current loss is (2.30) 2

max22 BfkP ee ντ=

where ke = constant dependent on the material f = frequency of the variation of flux BBmax = maximum flux density τ = lamination thickness ν = total volume of the material

The total magnetic core loss is the sum of the hysteresis and eddy current losses.

If the value of H, when increasing towards some maximum, Hmax, does not increase continuously, but at some point, H1, decreases to H = 0 then increases again to it maximum value of Hmax, then a minor hysteresis loop is created as shown in Figure 13. The energy loss in one cycle includes these additional minor loop surfaces.

1- 13

H

B

O H1

Hmax

Figure 13. Minor hysteresis loops.

2.5 Permanent Magnets For a ring of iron with a uniform cross-section and hysteresis curve shown in Figure 15, the

magnetic field is zero when there is a nonzero flux density, BBr called the remnant flux density. To achieve a zero flux density, we could wind a coil around a section of the iron, and send current through the coil to reach a field intensity of –Hc (the coercive field). In practice a permanent magnet operates on a minor loop as shown in that can be approximated as a straight line, recoil line, such that

Figure 13

rm m

c

BrB H B

H= + (2.31)

Magnetization curves for some important permanent-magnet materials is shown in figure 1.19 from your textbook and shown below in Figure 14.

Example In the magnetic circuit shown with the length of the magnet, lm = 1cm, the length of the air gap is g = 1mm and the length of the iron is li = 20cm. For the magnet B

½·li

½·li

glm

Br = 1.1 (T), Hc = 750 (kA/m), what is the flux density in the air gap if the iron is assumed to have an infinite permeability and the cross-section is uniform? Since the cross-section is uniform, and there is no current: Hi[0.2(m)] + Hg[g] + Hm[lm]=0 With the iron assumed to have infinite permeability, Hi = 0 and

1- 14

( ) 01.1

00

0

0

=⎟⎟⎠

⎞⎜⎜⎝

⎛−+

=⎥⎦

⎤⎢⎣

⎡−+==+

mr

cmg

mcr

mcgmmg

lBHBgB

lHB

BHgB

lHgH

μ

μ

or B·795.77 + (B-1.1)·6818=0

B=0.985 (T)

Figure 14. Magnetization curves for common permanent-magnet materials [1].

1- 15

H (A·t/m)

B (Wb/m2)

O-Hc

Br

(Hm, Bm)

Figure 15. Remnant flux and coercive field for a piece of iron.

2.6 Torque and Force In Figure 12 we found the energy density in the field for the positive half-cycle is the total area or

(2.32) ∫=a

e

B

Bf dBHw

If this is simply approximated as a triangular area, then wf = ½ BH (J/m3). The total energy, Wf, would be found by multiplying this by the volume of the core

( )( ) ( )

R

F

221

21

21

21

φ

φ

=

=== HlBAAlBHW f (2.33)

Mechanical energy is done by reducing the reluctance (as the armatures of a relay are brought

together, for example), thus

RddWm2

21φ−= (2.34)

Since force is given as , the magnetic force is given by mdWFdx =

1- 16

dxdF R2

21φ−= (2.35)

and has units of newtons.

In a mechanical system with a force F acting on a body and moving it at velocity v, the power Pmech is vFPmech ·= (2.36) For a rotating system with torque T, rotating a body with angular velocity ωmech: mechmech TP ω·= (2.37) On the other hand, an electrical source e, supplying current, i, to a load provides electrical power Pelec

iePelec ·= (2.38) Since power has to balance, if there is no change in the field energy, mechmechelec TiePP ω·· === (2.39) Notes • It is more reasonable to solve magnetic circuits starting from the integral form of Maxwell’s

equations than finding equivalent resistance, voltage and current. This also makes it easier to use saturation curves and permanent magnets.

• Permanent magnets do not have flux density equal to BBr. Equation ( ) defines the relation between the variables, flux density Bm

2.31B

and field intensity Hm in a permanent magnet. • There are two types of iron losses: eddy current losses that are proportional to the square of the

frequency and the square of the flux density, and hysteresis losses that are proportional to the frequency and to some power n of the flux density.

1 A. E. Fitzgerald, C. Kingsley, Jr., S. D. Umans, Electric Machinery, 6th edition, McGraw-Hill, New York, 2003.

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