1 - principles of electrical machines and power electronics p_c_sen
634 pág.

1 - principles of electrical machines and power electronics p_c_sen

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lagging current and the exciting winding can be represented by a pure inductance, as shown in Fig. l.l8b. The phasor diagram for fundamen-
tal current and applied voltage is shown in Fig. 1.18c. 
With Hysteresis 
We shall now consider the hysteresis loop of the core, as shown in Fig. l.l9a. The waveform of the exciting current iq, is obtained from the sinusoidal flux waveform and the multivalued <P-i characteristic of the core. The excit-ing current is nonsinusoidal as well as nonsymmetrical with respect to the 
voltage waveform. The exciting current can be split into two components, 
one (ic) in phase with voltage e accounting for the core loss and the other Cim) in phase with <P and symmetrical with respect toe, accounting for the 
magnetization of the core. This magnetizing component im is the same as 
the exciting current if the hysteresis loop is neglected. The phasor diagram is shown in Fig. l.l9b. The exciting coil can therefore be represented by a 
resistance Rc, to represent core loss, and a magnetizing inductance Lm, to 
represent the magnetization of the core, as shown in Fig. 1.19c. In the phasor diagram only the fundamental component of the magnetizing current is 
A permanent magnet is capable of maintaining a magnetic field without any 
excitation mmf provided to it. Permanent magnets are normally alloys of 
Permanent Magnet 2 7 
(a) (c) 
FIGURE 1.19 Exciting current with hysteresis loop. (a) <I>-i loop and exciting 
current. (b) Phasor diagram. (c) Equivalent circuit. 
iron, nickel, and cobalt. They are characterized by a large B-H loop, high 
retentivity (high value of Br), and high coercive force (high value of He)· These 
alloys are subjected to heat treatment, resulting in mechanical hardness of 
the material. Permanent magnets are often referred to as hard iron and 
other magnetic materials as soft iron. 
Consider the magnetic circuit shown in Fig. 1.20a. Assume that the magnet 
material is initially unmagnetized. A large mmf is applied, and on its removal 
the flux density will remain at the residual value Br on the magnetization 
curve, point a in Fig. 1.20b. If a reversed magnetic field intensity of magni-
tude H 1 is now applied to the hard iron, the operating point moves to point 
b. If H 1 is removed and reapplied, the B-H locus follows a minor loop as 
shown in Fig. 1.20b. The minor loop is narrow and for all practical purposes 
can be represented by the straight line be, known as the recoil line. This line 
is almost parallel to the tangent xay to the demagnetizing curve at point a. 
The slope of the recoil line is called the recoil penneability JLrec. For alnico 
magnets it is in the range of 3-Sp.,0 , whereas for ferrite magnets it may be 
as low as 1.2p.,0 . 
As long as the reversed magnetic field intensity does not exceed H 1 , the 
magnet may be considered reasonably permanent. If a negative magnetic 
field intensity greater than H 1 is applied, such as H 2 , the flux density of the 
permanent magnet will decrease to the value B 2 \u2022 If H 2 is removed, the 
operation will move along a new recoil line de. 
28 chapter 1 Magnetic Circuits 
Soft iron keeper 
Soft iron 
(a) (b) 
FIGURE 1.20 Permanent magnet system and its B-H locus. 
Let the permanent magnet in Fig. l.20a be magnetized to the residual flux density denoted by point a in Fig. l.20b. If the small soft iron keeper is removed, the air gap will become the active region for most applications as shown in Fig. l.2la. 
In order to determine the resultant flux density in the magnet and in the air gap, let us make the following assumptions. 
1. There is no leakage or fringing flux. 
2. No mmf is required for the soft iron. 
a Soft iron B, 
r c Hard _l 
lm iron lg l f 
-He 0 
FIGURE 1.21 Permanent magnet with keeper removed and its B-H locus. 
From Ampere's circuit law, 
For continuity of flux, 
Hm[m + Hg[g = 0 
Hm = -~Hg 
From Eqs. 1.43, 1.45, and 1.46, 
Permanent Magnet 2 9 
( 1.44) 
( 1.46) 
( 1.4 7) 
Equation 1.47 represents a straight line through the origin, called the 
shear line (Fig. 1.21b ). The intersection of the shear line with the demagneti-
zation curve at point b determines the operating values of B and H of the 
hard iron material with the keeper removed. If the keeper is now reinserted, 
the operating point moves up the recoil line be. This analysis indicates that 
the operating point of a permanent magnet with an air gap is determined 
by the demagnetizing portion of the B-H loop and the dimensions of the 
magnet and air gap. 
From Eqs. 1.43, 1.45, and 1.46, the volume of the permanent magnet 
material is 
Vm = Amlm 
BgAg Hglg 
Bm Hm 
where Vg = Aglg is the volume of the air gap. 
Thus, to establish a flux density Bg in the air gap of volume Vg, a minimum 
volume of the hard iron is required if the final operating point is located 
such that the BmHm product is a maximum. This quantity BmHm is known 
as the energy product of the hard iron. 
A family of alloys called alnico (aluminum-nickel-cobalt) has been used 
for permanent magnets since the 1930s. Alnico has a high residual flux 
density, as shown in Fig. 1.22. 
30 chapter 1 Magnetic Circuits 
B (tesla) 
FIGURE 1.22 Demagnetization curve 
for alnico 5. 
Ferrite permanent magnet materials have been used since the 1950s. These 
have lower residual flux density but very high coercive force. Figure 1.23 
shows the demagnetization curve for ferrite D, which is a strontium ferrite. 
Since 1960 a new class of permanent magnets known as rare-earth perma-
nent magnets has been developed. The rare-earth permanent magnet materi-
als combine the relatively high residual flux density of alnico-type materials 
with greater coercivity than the ferrites. These materials are compounds of 
iron, nickel, and cobalt with one or more of the rare-earth elements. A 
commonly used combination is samarium-cobalt. The demagnetization 
curve for this material is shown in Fig. 1.24. Another rare-earth magnet 
material that has come into use recently is neodymium-iron-boron. The 
demagnetization curve for this alloy is also shown in Fig. 1.24. The residual 
flux density and coercivity are both greater than those for samarium-cobalt. 
It is expected that this neodymium-iron-boron will be used extensively 
in permanent magnet applications. For further discussion of the use of 
permanent magnets in machines see Sections 4.6, 4.7, and 6.13. 
B (tesla) 
FIGURE 1.23 Demagnetization curve for fer-
rite D magnet. 
Permanent Magnet 31 
FIGURE 1.24 Demagnetization curve 
for samarium-cobalt magnet and neo-
dymium-iron-boron magnet. 
The permanent magnet in Fig. 1.21 is made of alnico 5, whose demagnetiza-
tion curve is given in Fig. 1.22. A flux density of 0.8 T is to be established 
in the air gap when the keeper is removed. The air gap has the dimensions 
Ag = 2.5 cm2 and lg = 0.4 em. The operating point on the demagnetization 
curve corresponds to the point at which the product HmBm is maximum, 
and this operating point is Em = 0.95 T, Hm = -42 kA/m. 
Determine the dimensions (lm and Am) of the permanent magnet. 
From Eqs. 1.43 and 1.46, 
lg lgBg 
lm = Hm Hg HmJ.Lo 
From Eq. 1.45, 
0.4 X 10-2 X 0.8 
42 X 103 X 477 X 10-7 
= 0.0606 m = 6.06 em 
0.8 X 2.5 X 10-4 
= 2.105 cm2 \u2022 
32 chapter 1 Magnetic Circuits 
1.1 The long sole~·10id coil shown in Fig. P1.1 has 250 turns. As its length is much greater than its diameter, the field inside the coil may be considered uniform. Neglect the field outside. 
(a) Determine the field intensity (H) and flux density (B) inside the solenoid (i = 100 A). 
(b) Determine the inductance
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