Tutorial Sheet #3
1. The design of a four-pole, three-phase, 230 V, 60 Hz induction motor is to be based on a
stator core of length 21 cm and inner diameter 9.52 cm. The stator winding distribution which
has been selected has a winding factor kw = 0.925. The armature is to be Y-connected, and
thus the rated phase voltage will be 230/√3 V.
a) The designer must pick the number of armature turns so that the flux density in the
machine is large enough to make efficient use of the magnetic material without being
so large as to result in excessive saturation. To achieve this objective, the machine is
to be designed with a peak fundamental air-gap flux density of 1.2 T. Calculate the
required number of series turns per phase.
b) For an air-gap length of 0.3 mm, calculate the self-inductance of an armature phase
based upon the result of part (a). Neglect the reluctance of the rotor and stator iron
and the armature leakage inductance.
2. A two-pole, 60 Hz, three-phase, laboratory-size synchronous generator has a rotor radius
of 5.71 cm, a rotor length of 18 cm, and an air-gap length of 0.25 mm. The rotor field
winding consists of 264 turns with a winding factor of kr = 0.95. The Y-connected armature
winding consists of 45 turns per phase with a winding factor kw = 0.93.
a) Calculate the flux per pole and the peak fundamental air-gap flux density which will
result in an open-circuit, 60 Hz armature voltage of 120 V RMS/phase(line-toneutral).
b) Calculate the DC field current required to achieve the operating condition of part (a).
c) Calculate the peak value of the field-winding to armature-phase-winding mutual
inductance.
3. Figure 1 shows in cross section a machine having a rotor winding f and two identical stator
windings a and b whose axes are in quadrature. The self-inductance of each stator winding is
Laa and of the rotor is Lff. The air gap is uniform. The mutual inductance between a stator
winding depends on the angular position of the rotor and may be assumed to be of the form
Maf = M cos θ0, Mbf = M sin θ0, where M is the maximum value of the mutual inductance.
The resistance of each stator winding is Ra.
a) Derive a general expression for the torque T in terms of the angle θ0, the inductance
parameters, and the instantaneous currents ia, ib, and if. Does this expression apply at
standstill? When the rotor is revolving?
b) Suppose the rotor is stationary and constant direct currents ia = I0, ib = I0, and if = 2I0
are supplied to the windings in the directions indicated by the dots and crosses in Fig.
1. If the rotor is allowed to move, will it rotate continuously or will it tend to come to
rest? If the latter, at what value of θ0?
c) The rotor winding is now excited by a constant direct current If while the stator
windings carry balanced two-phase currents ia = √2 Ia cos ωt, ib = √2 Ia sin ωt.
The rotor is revolving at synchronous speed so that its instantaneous angular position
is given by θ0 = ωt - δ, where δ is a phase angle describing the position of the rotor at t
= 0. The machine is an elementary two-phase synchronous machine. Derive an
expression for the torque under these conditions.
d) Under the conditions of part (c), derive an expression for the instantaneous terminal
voltages of stator phases a and b.
e) Derive an expression for the torque acting on the rotor if the rotor is rotating at
constant angular velocity, such that θ0 = ωt + δ, and the phase currents become
unbalanced such that ia = √2 Ia cos ωt, ib = √2 (Ia+I’) sin ωt.
Fig 1: Cylindrical rotor, two-phase machine for Problem 3.
4. Figure 2 shows in the schematic cross section a salient-pole synchronous machine having
two identical stator windings a and b on a laminated steel core. The salient-pole rotor is made
of steel and carries a field winding f connected to slip rings.
Because of the nonuniform air gap, the self- and mutual inductances are functions of the
angular position θ0 of the rotor. Their variation with θ0 can be approximated as:
Laa = L0 + L2 cos 2 θ0
Lbb = L0 - L2 cos 2 θ0
Mab = L2 sin 2 θ0
where L0 and L2 are positive constants. The mutual inductance between the rotor and the
stator windings are functions of θ0
Maf = M cos θ0
Mbf = M sin θ0
where M is also a positive constant. The self-inductance of the field winding, Lff, is constant,
independent of θ0. Consider the operating condition in which the field winding is excited by
direct current If and the stator windings are connected to a balanced two-phase voltage source
of frequency ω. With the rotor revolving at synchronous speed, its angular position will be
given by θ0 = ωt. Under this operating condition, the stator currents will be of the form
ia = √2 Ia cos (ωt+ δ)
ib = √2 Ia sin (ωt+ δ)
a) Derive an expression for the electromagnetic torque acting on the rotor.
b) Can the machine be operated as a motor and/or a generator? Explain.
c) Will the machine continue to run if the field current If is reduced to zero? Support you
answer with an expression for the torque and an explanation as to why such operation
is or is not possible.
Fig 2: Salient pole, two-phase machine for Problem 4 and 5.
5. Figure 2 shows a two-pole rotor revolving inside a smooth stator which carries two coils of
110 turns each. The rotor produces a sinusoidal space distribution of flux at the stator surface;
the peak value of the flux density wave being 0.85 T, when the rotor current is 15 A. The
magnetic circuit is linear. The inner diameter of the stator is 11 cm, and its axial length is
0.17 m. The rotor is driven at a speed of 50 rev/sec.
a) The rotor is excited by a current of 15 A. Taking zero time as the instant when the
axis of the rotor is vertical, find the expression for the instantaneous voltage generated
in the open-circuited stator coil.
b) The rotor is now excited by a 50 Hz sinusoidal alternating current whose peak
value is 15 A. Consequently, the rotor current reverses every half revolution; it is
timed to be at its maximum just as the axis of the rotor is vertical. Taking zero time as
the instant when the axis of the rotor is vertical, find the expression for the
instantaneous voltage generated in the open circuited stator coil.
6.
a) A three phase, 16 pole synchronous generator has a star connected winding with
144 slots and 10 conductors per slot. The coils are full pitched. The flux per pole is
0.03 Wb, sine distributed. The rotational speed is 375 RPM. Find RMS induce phase
and line voltages and also the frequency of the induced voltage.
b) A three phase 10 pole star connected alternator runs at 600 RPM. It has 120 stator
slots with 8 conductors per slot. The conductors of each phase are series connected
forming full pitch coils. The flux per pole is 56 mWb. Determine the phase and line
induced voltages.
7. The air gap flux density waveform of a 2-pole salient pole synchronous generator can be
assumed to be rectangular over the central two third part of the pole pitch. The peak value of
the flux density is 1.2T. Draw one complete cycle of the emf induced in a single turn coil
which is short corded by 30o and moves with a uniform velocity of 20 m/sec through the air
gap. The active conductor length is 2m. What is the RMS value of the induced voltage?
8. The flux distribution in the air gap of a 50 Hz salient pole synchronous generator may be
taken as rectangular, the base being two third of the pole pitch. Calculate the RMS value of
the induced voltage per phase with the following particulars: No. of full pitch turns per phase
= 120, phase spread = 60°, pole pitch = 50 cm, stator length = 75 cm, peak flux density in the
air gap = 0.7 T. Also find the RMS induced phase voltage if the same total flux had a
sinusoidal distribution.
9. A 2 pole, 3 phase ac winding is housed in 18 slots, each slot having 12 conductors and
forms full pitched coils. Consider the time instant at which the current in phase a has its
maximum value of 10 A. Sketch all the 18 slots on a horizontal axis. Mark the direction of
currents in the conductors occupying the slots relevant to phase a. Make a proportional sketch
of the MMF wave of phase a only. Calculate the peak value of the fundamental component of
the MMF wave. What are the other harmonic components present in this MMF wave? Find
out the relative amplitudes (as % of the fundamental) of the first three dominant harmonics.
10. The two-pole stator winding distribution of Figure 3a is found on an induction motor with
an air-gap length of 0.381 mm, an average rotor radius of 6.35 cm, and an axial length of 20.3
cm. Each stator coil has 15 turns, and the coil phase connections are as shown in Fig 3b (only
for a-phase). Calculate the a-phase self inductance Laa and the a-phase to b-phase mutual
inductance Lab.
(a)
(b)
Fig 3: (a) Distributed two-pole, three-phase, fractional-pitch armature winding for Problem
10. (b) Coil connection for a-phase.
11. The flux distribution curve of a 2 pole cylindrical rotor 50Hz star connected synchronous
generator is given by B(θ) = sin θ + 0.2 sin3θ + 0.2 sin 5θ + 0.2 sin 7θ T where the angle ‘θ’
is measured from the magnetic neutral (q-axis) of the rotor. Find the RMS values of the first
three frequency components in the induced line voltage of the generator with the following
particulars. No. of full pitch turns per phase = 120; phase spread = 60o; No. of
slots/pole/phase = 5; pole pitch = 50 cm; stator length = 75 cm.
12. A 3 phase 8 pole 50 Hz synchronous generator is designed such that the ratio of the
fundamental RMS phase induced voltage to the fundamental flux per pole (in mWb) is 48.5.
The third harmonic induced phase voltage is 25% of the fundamental induced phase voltage
while the third harmonic flux per pole is 19.93% of the fundamental flux per pole. The stator
phase winding coils are short corded to eliminate the fifth harmonic induced phase voltage.
The phase spread of the stator coils is 60o (electrical). Find out
(a) The coil span of the stator coils in degrees (electrical).
(b) Total number of slots in the machines
(c) The number of conductors per slot