
A 208-V, 60-Hz, 4-pole, three-phase induction motor has a full-load speed of 1755 rpm.

Calculate: a.
b.
c.
its synchronous speed, the slip the rotor frequency.
51


The rated speed is given by:

The slip is given by:

The rotor frequency at full load:
52


For f =50Hz
 p n s
2 4 6 8 12 24
3000 1500 1000 750 500 375
53


At standstill ω m
= 0 so that the slip rotor currents have a frequency f r
= f s
.
=1 and the that is, the machine then acts as a simple transformer with short circuited secondary
 torque is produced, hence the poly-phase induction motor is self-starting.
54


If the rotor attains the synchronous speed, ω m so that the slip s = 0.



= ω s
,
No induction takes place because there is no relative motion between the flux and rotor conductors.
The frequency of the rotor EMF & currents : f r
= 0
Thus, at synchronous speed, the value of the secondary
MMF will be zero, and no torque will be produced.
 The induction motor cannot run at synchronous speed
.
55


The no-load speed of the induction motor is of the order of 99.5% of the synchronous speed.
 the full-load per-unit slip is of the order of 0.05.
 a stator with any number of phases will develop torque in a rotor of the same or any other number of phases, single-winding stators only excepted.
 the number of poles in the stator and rotor must be the same for torque production.
56

Sen, Chapter 5, section 5.3
57


The rotating magnetic field can be represented by a pair of magnets, for a 2 pole machine, rotating in the air gap.

Consider that the phase coils are full-pitch coils of N turns.
 as the rotating field moves the flux linkage of a coil will vary.

The flux linkage for coil aa' will be maximum (= N ϕ p
) at ω t = 0° and zero at ω t = 90°.
58


The flux linkage λ a
( ω t) will vary as the cosine of the angle ω t. Hence:
 the voltage induced in phase coil aa' is obtained from Faraday's law as:
59


The rms value of the induced voltage is:

The above equation shows the rms voltage per phase with the turns forming a concentrated fullpitch winding.

For an actual ac machine each phase winding is distributed in a number of slots. We must introduce the distribution factor k d
={0.85 to 0.95}
60


The induction machine can be operated in three modes:


 motoring generating plugging.
61


If the stator terminals are connected to a threephase supply, the rotor will rotate in the direction of the stator rotating magnetic field.

The steady-state speed n is less than the synchronous speed n s as shown in Fig
62


IF the speed of the system is higher than the synchronous speed and the system rotates in the same direction as the stator rotating field, The induction machine will produce a generating torque, that is, a torque acting opposite to the rotation of the rotor.
 the power flow is reversed and system energy will be fed back to the supply
63


Suppose an induction motor is running at a steady-state speed.

If its terminal phase sequence is changed suddenly, the stator rotating field will rotate opposite to the rotation of the rotor, producing the plugging operation.

The motor will come to zero speed rapidly and will accelerate in the opposite direction, unless the supply is disconnected at zero speed.
64

65

Part (V)
66


Since the stator and the rotor windings are coupled inductively, an induction motor resembles a 3 phase transformer with a rotating secondary winding.

The 3 phase induction motor can be represented by an equivalent circuit at any slip s as:
67






R
1
= per phase resistance of the stator winding
L
1
=per-phase stator winding leakage inductance
X
1
= 2 π f L
1
=per-phase stator winding leakage reactance
R r
L r
= per-phase rotor winding resistance
= per-phase rotor winding leakage inductance
68





X b
= 2 π f L r
= per-phase rotor winding leakage reactance under blocked-rotor condition
X r
= 2 π s f L r
= sX b
= per-phase rotor winding leakage reactance at slip s .
X m
R c
= per-phase magnetization reactance
= per-phase equivalent core-loss resistance
69

70


The per phase induced EMF in the rotor winding under blocked rotor condition = E b

The rotor winding is distributed around the rotor periphery, the magnetic axes of the winding coils are different and therefore the total E b is less than the sum of the voltages induced in the coils
71


Therefore, we have to introduce a factor k d2 to count for the distribution of the winding:

 k d2
= 0.85 to 0.95
Similarly:
72



 k d2
N
2
= winding factor for the rotor winding
= actual turns per phase of the rotor winding
The per phase induced EMF in the rotor winding at slip s is given by:
73


X b
=leakage reactance of the rotor under blocked rotor condition.

Based on the above equation, we develop the equivalent circuit as shown below:
74

 the hypothetical resistance R r
/s in the rotor circuit is called the effective resistance
.


At standstill, s = 1 → effective resistance is very low rotor current very high → starting torque
→
At synchronous speed , s = 0 → effective resistance is infinite → rotor current = 0 → No torque
75

 the transformation ratio, a :

The equivalent circuit referred to the stator is as shown:
76

77

78

Actual resistance of the rotor load resistance (or dynamic resistance)
The load resistance:
- Depends on the speed of the motor
- Represents the load on the motor
- Is the electrical equivalent of the mechanical load on the motor.
79

80

The net power that is crossing the air-gap and is transported to the rotor by electromagnetic induction is called the airgap power
81

82

83

84

85


A 6-pole, 230V, 60Hz,Y-connected, three-phase induction motor has the following parameters on a per-phase basis:




R
1
X
1
= 0.5
Ω , R
2
= 0.25
Ω
= 0.75
Ω , X
2
= 0.5
Ω
X m
= 100 Ω , R c
= 500 Ω .
The friction and windage loss is 150 W.

Determine the efficiency of the motor at its rated slip of
2.5%.
86

87

88

89

90

91

92


If the voltage drop across R
1 the terminal voltage branch (i.e., R c and
V
X m
1 does not appreciably differ from the induced voltage E
1
, and X
1 is small and the magnetizing
) can be moved to the machine terminals as shown
93


In Induction machine, the exciting current I ϕ
, is high-of the order of 30 to 50 percent of the full-load current.

The resistance R c is, however, omitted, and the core loss is lumped with the windage and friction losses

The leakage reactance X m is also high.
94