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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
1- 2
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.
1- 6
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.
1- 17