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Unformatted text preview: et cutting across a conducting ladder. 4.4 The Rotating Field and Induced Voltages
Consider a simple stator having 6 salient poles, each of which
carries a coil having 5 turns (Fig.4.4). Coils that are diametrically
opposite are connected in series by means of three jumpers that
respectively connect terminals a-a, b-b, and c-c. This creates three
identical sets of windings AN, BN, CN, that are mechanically
spaced at 120 degrees to each other. The two coils in each winding
produce magnetomotive forces that act in the same direction. 222 Chapter Four Fig.4.4 Elementary stator having terminals A, B, C connected to
a 3-phase source (not shown). Currents flowing from line to
neutral are considered to be positive.
The three sets of windings are connected in wye, thus forming a
common neutral N. Owing to the perfectly symmetrical
arrangement, the line to neutral impedances are identical. In other
words, as regards terminals A, B, C, the windings constitute a
balanced 3-phase system.
For a two-pole machine, rotating in the air gap, the magnetic
field (i.e., flux density) being sinusoidaly distributed with the peak
along the center of the magnetic poles. The result is illustrated in
Fig.4.5. The rotating field will induce voltages in the phase coils
aa', bb', and cc'. Expressions for the induced voltages can be
obtained by using Faraday laws of induction. 223 Three-Phase Induction Machine Fig.4.5 Air gap flux density distribution.
The flux density distribution in the air gap can be expressed as: B(θ ) = Bmax cos θ (4.1) The air gap flux per pole, φ P , is:
π /2 φ p = ∫−π / 2 B(θ )lrdθ = 2 Bmax lr (4.2) Where,
l is the axial length of the stator.
r is the radius of the stator at the air gap.
Let us consider that the phase coils are full-pitch coils of N
turns (the coil sides of each phase are 180 electrical degrees apart
as shown in Fig.4.5). It is obvious that as the rotating field moves
(or the magnetic poles rotate) the flux linkage of a coil will vary.
The flux ( = N φ P at
linkage linkage for coil aa' will be maximum ωt = 0o ) (Fig.4.5a) and zero at ωt = 90o . The flux λa (ω t ) will vary as the cosine of the angle ω t . Hence; 224 Chapter Four λa (ωt ) = Nφ p cos ωt (4.3) Therefore, the voltage induced in phase coil aa' is obtained
from Faraday law as: ea = − dλa (ωt )
= ω Nφ p sin ωt = Emax sin ωt
dt (4.4) The voltages induced in the other phase coils are also
sinusoidal, but phase-shifted from each other by 120 electrical
degrees. Thus, eb = E max sin (ωt − 120) (4.5) ec = E max sin (ωt + 120) . (4.6) From Equation (4.4), the rms value of the induced voltage is: E rms = ωNφ p
2 = 2πf
Nφ p = 4.44 fNφ p
2 (4.7) Where f is the frequency in hertz. Equation (4.7) has the same
form as that for the induced voltage in transformers. However, φP in Equation (4.7) represents the flux per pole of the machine.
Equation (4.7) shows the rms voltage per phase. The N is the
total number of series turns per phase with the turns forming a
concentrated full-pitch winding. In an actual AC machine each
phase winding is distributed in a number of slots for better use of 225 Three-Phase Induction Machine the iron and copper and to improve the waveform. For such a
distributed winding, the EMF induced in various coils placed in
different slots are not in time phase, and therefore the phasor sum
of the EMF is less than their numerical sum when they are
connected in series for the phase winding. A reduction factor KW ,
called the winding factor, must therefore be applied. For most
three-phase machine windings KW is about 0.85 to 0.95. Therefore, for a distributed phase winding, the rms voltage per
phase is Erms = 4.44 fN phφ p KW (4.8) Where N ph is the number of turns in series per phase.
4.5 Running Operation
If the stator windings are connected to a three-phase supply and
the rotor circuit is closed, the induced voltages in the rotor
windings produce rotor currents that interact with the air gap field
to produce torque. The rotor, if free to do so, will then start
rotating. According to Lens law, the rotor rotates in the direction of
the rotating field such that the relative speed between the rotating
field and the rotor winding decreases. The rotor will eventually
reach a steady-state speed n that is less than the synchronous speed
ns at which the stator rotating field rotates in the air gap. It is 226 Chapter Four
obvious that at n = ns there will be no induced voltage and current
in the rotor circuit and hence no torque.
In a P-pole machine, one cycle of variation of the current will
make the mmf wave rotate by 2/P revolutions. The revolutions per
minute n (rpm) of the traveling wave in a P-pole machine for a
frequency f cycles per second for the currents are: n= 2
P f * 60 = 120 f (4.9) p The difference between the rotor speed n and the synchronous
speed ns of the rotating field is called the slip s and is defined as s= ns − n (4.10) ns If you were sitting on the rotor, you would find that the rotor
was slipping behind the rotating field by the slip rpm = ns − n = sns . The...
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This document was uploaded on 03/12/2014 for the course ENGINEERIN electrical at University of Manchester.
- Spring '14