Torque Equation of three phase induction motor

The torque produced in the induction motor depends on the following factors :

1. The part of rotating magnetic field which reacts with rotor and is responsible to produce induced e.m.f. in rotor.

2. The magnitude of rotor current in running condition.

3. The power factor of the rotor circuit in running condition.

       Mathematically the relationship cab be expressed as,

                       T α Φ I_2r cos Φ_2r                                             .........(1)

where              Φ = Flux responsible to produce induced e.m.f.

                        I_2r = Rotor running condition

                        cos Φ_2r  = Running p.f. of motor

       The flux Φ produced by stator is proportional to i.e. stator voltage.

.^..                       Φ α  E₁                                                                                 .........(2)

       while E₁ and E₂ are related to each other through ratio of stator turns to rotor turns i.e. k.

.^..                        E₂/E₁ = K                                                         .............(3)

       Using (3) in (2) we can write,

       Thus in equation (1), Φ can be replaced by E₂.

While                 I_2r = E_2r /Z_2r  = (s E₂)/√(R₂² +(s X₂)²)                   .............(5)

and                     cos Φ2r = R₂/Z_2r  = R₂/√(R₂² +(s X₂)²)                ............(6)

       Using (4), (5), (6) in equation (1),

.^..                       T = (k s E₂² R₂)/(R₂² +(s X₂)²)                              ............(7)

where                  k = Constant of proportionality

       The constant k is provided to be 3/2 for three phase induction motor.

.^..                         k =3/(2 π n_s)                                                        ............(8)

Key Point  :  n_s = synchronous speed in r.p.s. = N_s/60

       Using (8) in (7) we get the torque equation as,

       So torque developed at any load condition can be obtained if slip at that load is known and all standstill rotor parameters are known.

1.1 Starting Torque

        Starting torque is nothing but the torque produced by an induction motor as start. At start, N= 0 and slip s = 1. So putting s = 1 in the torque equation we can write expression for the starting torque T_st as,

Key Point : From the equation (10), it is clear that by changing the starting torque can be controlled.

       The change in R₂ at start is possible in case of slip ring induction motor only. This is the principle used in case of slip induction motor to control the starting torque T_st.

Example 1 : A 3 phase, 400 V, 50 Hz, 4 pole induction motor has star connected stator winding. The rotor resistance and reactance are 0.1 Ω and 1 Ω respectively. The full load speed is 1440 r.p.m. Calculate the torque developed on full load by the motor.

Assume stator to rotor ratio as 2 :1.
Solution : The given values are,
                        P = 4,    f = 50 Hz,    R₂ = 0.1 Ω,      X₂ = 1 Ω,     N = 1440 r.p.m.
           Stator turns/Rotor turns = 2/1
.^..                      K = E₂ /E₁ = Rotor turns/Stator turns = 1/2 = 0.5
                          N_s=120f/P = 120x50 / 4 = 1500 r.p.m.
                          E_1line = 400 V                                        ..............Stator line voltage given
.^..                       E_1ph = E_1line /√3 = 400/√3 = 230.94 V
But                     E_2ph /E_1ph = 0.5 = K
.^..                       E_2ph = 0.5 x 230.94 = 115.47 V
Full load slip,      s = (N_s-N)/N_s = (1500-1400)/1500 = 0.04
                          n_s = Synchronous speed in r.p.s.
                               = N_s/60 = 1500/60 = 25  r.p.s.

                                = 87.81 N-m