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Unformatted text preview: Unit 05 Magnetic Fields 7& Magnetic Circuits Electromechanical energy conversion and energy distribution
in power systems is enabled by magnetic ﬁelds created by magnetic
circuits. In such a sense, these ﬁelds and circuits are the main
components of electrical power systems. They are essential
components of generators, transformer and motors. The creation of magnetic ﬁeld by electric current was
discovered by Oersted in 1820 and the creation of the electric ﬁeld
by a change of magnetic ﬂux was discovered by Faraday in 1931.
These two discoveries have created the very physical fundamentals
of electrical power systems. The creation of the magnetic ﬁeld, of intensity I? , is governed
by two laws, BiotSavart Law and Ampere Law.
The Biot—Savart Law. When a point of space has a distance
3 ﬁ'om a conductor element oriented in direction :17 with a current i
then a magnetic ﬁeld intensity
dﬁ = Lian—1' =Lidl sin(dl,d)’
472' d3 472' d2 is created at such a point. Current loop Fig. 1
The vector of the magnetic ﬁeld intensity is perpendicular to the plane of vectors :1? , and the distance ti". Orientation of this vector
can be detemrined by the rightlulu ded screw rule. [if]. I'
 / d
id? _. displacement
d
. 51'
h Rotation
I Fig.2
Ifthe ﬁrst elementefaaeveeterpreaueuﬁhsremeed on tothe second element (a?) of the product then the result of the vector
product is oriented towards the direction of displacement of the
righthanded screw (Fig. 2a). This rule can be applied also in a
different way. To displace a righthanded screw in the direction of
the current ﬂow, it has to be rotated in the direction of the magnetic
ﬁeld intensity, H, (Fig. 2b). Magnetic ﬁeld intensity, H, has dimension [A/m], but its unit
does not have any name. To calculate the magnetic ﬁeld intensity H created by a current
loop at some point of space, integration along the entire current loop
is needed. On the condition that the current loop is on a plane and
the magnetic ﬁeld intensity is calculated at same point of the same plane and is is the plane unit vector (a unit vector perpendicular to
the current loop plane)
a = g m = It; a} Loop Loop d
Since the distance d and the angle between the loop element d1i,=Hi,. d1 and the distance vector d changes along the loop, this integral
could be difﬁcult for calculation. It is easy to calculate it only if
geometry of the loop is simple.
Illustration 1. Calculate the magnetic ﬁeld intensity in the center
of a circular current loop of diameter of D = 2R = 0.5m with the
current i = 500 A. ' m Fig. 3 When the magnetic ﬁeld intensity is calculated in the center of a
circuit as shown in Fig. 3, then the distance d = D/2 and the angle between the loop element 37 and the distance vector d is
constant and equal to 90°. Thus, . ' 0
H: qu=4LSmg° C'Fdlé
Loop 7t R Loop
=_i_; =_i_=_500:_=
47: Ram 2R 2x025 1000M“ MAMA/\AAAAAAAMAMAAMA
The Ampere Law. It says that the integral along a closed path of
the magnetic ﬁeld intensity H is equal to the current ﬂux enclosed
by this path. In this integral the symbolfdenotes the vector of
current density in [Almz] and ds'is the vector of the surface
element. 41%;: may:
Path Area Fig. 4 The direction of integration along the path and the direction of the
current ﬂux across the area satisfy the righthanded screw rule. When the integration path in the Ampere Law encloses N1
conductors with current i1, ﬂawing in direction that satisﬁes the
righthanded screw rule and N2 conductors with current i; ﬂowing in
the opposite direction, then the Ampere Law for any path around
these conductors can be written in the form qﬁdi=NllI1—N2i2 .
Path Illustration 2. Calculate the magnetic ﬁeld intensity H in the
distance d = 0.25m from a conductor with the current i = 500 A. r r?
Fig. 5 Solution. The vector of magnetic ﬁeld intensityH at any point
of the circle with the current in its center has the same
magnitude, H, and it is tangent to the circle. Hence, cjﬁdi=H cfd1=2mH = i.
Path Path
Thus .; ___&. =
H‘ 2727 _27rx0.25 3184M” AAMMAAAAAAAAAAAAAAAAAAAAA The mechanical F exerted on electric charge q in electric and
magnetic ﬁelds are governed by the Lorenz Law. The Lorenz Law. If an electric charge dq moves in the
electromagnetic ﬁeld with velocity ii then a force is exerted on
such a charge. If electric ﬁeld is speciﬁed by the vector of the electric ﬁeld intensity E and the magnetieﬁeld is speciﬁed by the vector of magnetic ﬂux density, B , then the force is equal to
d13=dqa§+r XE).
The magnetic ﬂux density is measured in Teslas [T]. It is a vector,
tangent to the magnetic ﬂux, ¢, line, with the magnitude
M
B = g: . The magnetic ﬂux, ¢, is measured in Webers [Wb] and cones
quently, the magnetic ﬂux density, [T] = [Wb/mz]. =l‘he magnetic component ofthe force exerted on ameving
charge di ~ dq dﬁm=dq17x3=quXB=Edlx§=idlx§. The magnetic ﬂux density li’ in vacuum, and approximately in
air and in a great majority of materials, is proportional to the
magnetic ﬁeld intensity, H B=ﬂoH. The symbol ,uo denotes magnetic permeability of free space and it is
equal to yo = 47r><10'7 [H/m]. The magnetic ﬂux density B in iron, cobalt and nickel and in
alloys of these metals with each other, depends on the magnetic ﬁeld
intensity H thousands time stronger than in vacuum. Such materials
are called ferromagnetics. For such materials B=y,uoH, where y is referred to as a relative permeability.
Its approximate value for some ferromagnetics:  annealed iron: ,u = 5 500
 ironsilicon laminations (96% Fe, 4% Si) ,u = 7 000
 permalloy(55%Fe, 45% Ni) ii = 25 000
 suppermalloy y = 100 000 Due to electron spin and their rotation around the nucleus,
individual atoms form elementary magnetic dipoles. In a great
majority of materials, these dipoles are oriented randomly and
consequently, their net effect is zero. Ferromagnetism is caused be
an alignment of elementary magnet dipoles in material
microstructine. They form domains. These dipoles in a domain have
the same orientation, different from the orientation in neighboring I \\
Mas Fig. 6 External magnetic ﬁeld enlarges and rotates individual magnetic
domains in the direction of the external ﬁeld intensity, causing a
magnetic ﬂux to occur. This process is strongly nonlinear, speciﬁed
in terms of hysteresis loop. The reorientation of magnetic domains
remains to some degree permanent. Even if the external magnetic
ﬁeld disappears, the ferromagnetic remains magnetized. It is
speciﬁed by the residual magnetic ﬂux density, Br Negative
magnetic ﬁeld is needed for material remagnetization. It is speciﬁed
by the magnetic ﬁeld intensity coersion, Hc. When all domains are
oriented towards the external magnetic ﬁeld, the ferromagnetic
material becomes saturated and the relative permeability p declines to unity.
B 5' Fig. 7 Mechanical work is needed for changes of magnetic domains
structure. This work at one eyele of ehange of ﬁeld
intensity is proportional to the area of the hysteresis loop. When the
ﬁeld intensity changes periodically, then energy loss in the
ferromagnetic material increases with the area of the loop and with
ﬁequency. This energy loss is observed as an increase of the
material temperature. Therefore, magnetic circuits used for creation
of variable magnetic ﬂux (for example, in transformers) should be
built of ferromagnetic materials with the hysteresis loop area as
small as possible. Such ferromagnetics are referred to as soft
ferromagnetics. The hysteresis loop of a soﬁ ferromagnetic material
is shown in Fig. 7. Ferromagnetic material for a permanent magnet should fulﬁll
different requirements. It should have the residual magnetic ﬂux
density, 3,, as high as possible. The higher E, the stronger magnet
could be formed. A permanent magnet in an external magnetic ﬁeld
can loose its magnetization, however. This unfavorable feature
declines with the increase of the magnetic ﬁeld intensity coersion,
Hc of the ferromagnetic material. The higher coersion intensity the
better is the material for a permanent magnet. Ferromagnetics with
high residual ﬂux density and high coersion intensity are referred to
as hard ferromagnetics. Table l. The values of Br and Hc of ferromagnetic materials used for
permanent magnets 04 Neod ’ 'umironboron allo 1.25 Tesla
1.25 50 ~o~.— E
UI\IUI : The hysteresis loop of a hard ferromagnetic material is shown in
Fig. 8. Fig 8 Magnetic circuits. At the same value of the magnetic ﬁeld inten—
sity H, the ﬂux density B in ferromagnetic materials is p—time higher
than in air. Consequently, magnetic ﬂux is mainly conﬁned to
ferromagnetic material. If this material is formed to create a loop or
loops for the magnetic ﬂux, a magnetic circuit is created. Winding
with a current is usually the main source of the magnetic field and
the ﬂux in such a magnetic circuit. A permanent magnet or external
magnetic ﬁeld could serve as such a source as well. Fig. 9
The magnetic ﬂux a) in the circuit shown in Fig. 9a has the same
value along the entire magnetic loop. Consequently, the ﬂux density _ A ,
does not change along the loop. Since the magnetic permeability
changes along the ﬂux loop, changes also the magnetic ﬁeld
intensity. Its value in the ferromagnetic core is Hc=——B ,
##o
andintheairgap
B s
#0
The Ampere Law for the average path along the core deI = Hcl+HgA =Ni.
path When the magnetic field intensity is expressed in terms of magnetic
permeability y no, ﬂux (D and the core crosssection areaA, then 4’ [+3441 =Ni. A ##o A .110
The term
I
R = —— ,
c ##o A
is referred to as a reluctance of the magnetic core, while
A
R 3—!!014’ is the reluctance of the airgap. Consequently, the Ampere Law for
the magnetic circuit shown in Fig. 10a can be written in the form
d>(Rc +Rg) = Ni. Consequently, the magnetic ﬂux in such a circuit can be calculated
from formula Ni
(D = —.
Rc + Rg Illustration 3. A magnetic core made of siliconiron laminations
with magnetic permeability ,u = 7000 has the length l = 0.5m and the
crosssection area A = 100 cm2. The air gap A = 1 mm. The core
saturates at the ﬂux density BM = 1.2 T. Calculate the maximum
current in the winding with N = 100 turns that do not saturate the
core. Solution. To avoid saturation the magnetic ﬂux should not be higher
than em = ABsat = 0.01 [m2] x 1.2[Wb/m2] = 12 x10'3 Wb.
The reluctance of the core is c=——=——————_7——=5700A/Wb.
##oA 7000x47rx10 x0.01
The reluctance of the airgap is
—3
g=__4 =—__—1X1_9 =79500A/Wb.
ﬂoA 471x10 x0.01 Observe, that the reluctance of 1 mm long airgap is almost 14 times
higher than the core reluctance. From the Ampere Law 3
ism = ‘13:;ch +le) = Lag—(5.7 +79.5)x1o3 = 10.2A. AMMAAAAAAAAAMAAMAAAM The Faraday Law. Change of the magnetic ﬂux (I) in a loop induces
voltage e in such a loop. When the induced voltage e and the ﬂux
CD are mutually oriented according to righthand screw rule as shown
in Fig 1 l, then the induced voltage is
_ dd! 8 — "' '2‘— .
thus, if the ﬂux in the loop increases, then the voltage induced in the
loop is oriented towards terminal b, i.e., opposite to the direction
marked by the arrow. Orientation of the voitage induced by a change
of the magnetic ﬂux could be determined by the Lenz Rule. It says,
that the voltage inducedby a change of the magnetic ﬂux has such
orientation, that the current caused by this voltage creates a magnetic ﬂux ((1),) that counteract this change. Fig. 12 It means, that orientation of the magnetic ﬂax and the induced
voltage satisfy a left—hand screw rule. At such orientation e=— dt '
When a loop with magnetic ﬂux is composed of N turns, as shown
in Fig. 13, Fig. 13. then the induced voltage is proportional to the rate of magnetic ﬂux
change and the number of turns,
d (D
e — N dt— .
Observe, that if the current in the coil increases, i.e., di/dt > 0, then
the arrow e speciﬁes true orientation of the voltage induced in the
coil.
Since the magnetic ﬂux could be expressed as Ni
¢—ﬁ.
Thus,
_ R dt ' dt’
where
N2
=?’ is the inductance of the coil. It increases with the square of turn
number N and declines with the core reluctance. For a core without
air gap
A
= ill/:0 N2,
thus, it increases with the core magnetic permeability ,u and cross
section area, A. Hysteresis power loss. When a coil with a magnetic core as shown inFig. 14 Fig. 14. is connected in AC circuit then the instantaneous power at the coil terminals is equal to
._ d H114 _ __
p(t)rul—(NE¢)(7) dt(AB)Hl—VH dt, where V=Al is the magnetic core volume. Thus, the instantaneous
power is proportional to the change in the magnetic ﬂux density, dB,
at magnetic intensity H, i.e., to the area HdB as shown in Fig. 10.
Due to hysteresis, magnetic ﬁeld intensityH at the same ﬂux density
B is higher when the core is magnetized than when it is de
magnetized. Censequently, the area of HdB is larger at dB magnetization than at demagnetization. The difference is released
as a heat of the core increase. Fig. 15 The active power at the coil terminals for one cycle T of the current
variability, i.e., when B and H change in the whole range shown in Fig. 15, is equal to T T 1 1 1 P=T Jp(t)dt =V? [HdB=?VABH = ABHVf= APh,
0 0 thus it is proportional to the area ABH of the hysteresis loop, the core
volume V and the supply voltage frequency, f. This power is referred to hwteresispawer loss.
Since soft ferromagnetics have much smaller hysteresis loop area, ABH, than hard ferromagnetics, coils with soft ferromagnetics
have lower power loss than coils with hard ferromagnetics.
Therefore, magnetics with a hysteresis loop as narrow as possible
are used for AC coils. When sinusoidal voltage u=\/E’Ucoswt is applied to the coil with a magnetic core, then the ﬂux density B in
the core has to satisfy F araday’s Law dd) dB
u—Nw—NA—dr
hence, ,
_ 1 _J§U _ . _ .
—m udt— NA coswtdt—NAwsmwt—Bmxsmmt, then the ﬂux density B in the core has to change as a sinusoidal
function, with
J3 U NAa) Thus, the RMS value U of the voltage applied to the coil cannot be
higher than <Bsa. Bmax = a) J5 N A 3m = 4.44NAme < 4.44NAstat Fig. 16 \ ...
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 Fall '08
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