Effect of Slip on Rotor Parameters

Effect of Slip on Rotor Parameters

In case of a transformer, frequency of the induced e.m.f. in the secondary is same as the voltage applied to primary. Now in case of induction motor at start N = 0 and slip s = 1. Under this condition as long as s = 1, the frequency of induced e.m.f. in rotor is same as the voltage applied to the stator. But as motor gathers speed, induction motor has some slip corresponding to speed N. In such case, the frequency of induced e.m.f. in rotor is no longer same as that of stator voltage. Slip affects the frequency of rotor induced e.m.f. Due to this some other rotor parameters also get affected. Let us study the effect of slip on the following rotor parameters.

1. Rotor frequency

2. Magnitude of rotor induced e.m.f.

3. Rotor reactance

4. Rotor power factor

5. Rotor current 

 

1. Effect on Rotor Frequency

In case of induction motor, the speed of rotating magnetic field is,

N_s = 120 f / P      … (5.7.1)

where             f = Frequency of supply in Hz

At start when N = 0, s = 1 and stationary rotor has maximum relative motion with respect to R.M.F. Hence maximum e.m.f. gets induced in the rotor at start. The frequency of this induced e.m.f. at start is same as that of supply frequency.

As motor actually rotates with speed N, the relative speed of rotor with respect R.M.F. decreases and becomes equal to slip speed of Ns - N. The induced e.m.f. in rotor depends on rate of cutting flux i.e. relative speed Ns - N. Hence in running condition magnitude of induced e.m.f. decreases so as its frequency. The rotor is wound for same number of poles as that of stator i.e. P. If fr is the frequency of rotor induced e.m.f. in running condition at slip speed Ns - N then there exists a fixed relation between (Ns - N), fr and P similar to equation (5.7.1). So we can write for rotor in running condition,

Thus frequency of rotor induced e.m.f. in running condition (fr) is slip times the supply frequency (f).

At start we have s = 1 hence rotor frequency is same as supply frequency. As slip of the induction motor is in the range 0.01 to 0.05, rotor frequency is very small in the running condition.

Example 5.7.1 A 4 pole, 3 phase, 50 Hz induction motor runs at a speed of 1470 r.p.m. speed. Find the frequency of the induced e.m.f in the rotor under this condition.

Solution : The given values are,

It can be seen that in running condition, frequency of rotor induced e.m.f. is very small.

 

2. Effect on Magnitude of Rotor Induced E.M.F.

We have seen that when rotor is standstill, s = 1, relative speed is maximum and maximum e.m.f. gets induced in the rotor. Let this e.m.f. be,

E₂ = Rotor induced e.m.f. per phase on standstill condition

As rotor gains speed, the relative speed between rotor and rotating magnetic field decreases and hence induced e.m.f. in rotor also decreases as it is proportional to the relative speed Ns - N. Let this e.m.f. be,

E_2r = Rotor induced e.m.f. per phase in running condition

Now  E₂ ∝ Ns       while E_2r ∝Ns - N

Dividing the two proportionality equations,

E2r = s E2

The magnitude of the induced e.m.f. in the rotor also reduces by slip times the magnitude of induced e.m.f. at standstill condition.

 

3. Effect on Rotor Resistance and Reactance

The rotor winding has its own resistance and the inductance. In a case of squirrel cage rotor, the rotor resistance is very very small and generally neglected but slip ring rotor has its own resistance which can be controlled by adding external resistance through slip rings. In general let,

R₂ = Rotor resistance per phase on standstill

X₂ = Rotor reactance per phase on standstill 

Now at standstill, fr = f hence if L₂ is the inductance of rotor per phase,

X₂ = 2πf_r L₂ = 2πfL₂ Ω/ph

while R2 = Rotor resistance in Ω/ph

Now in running condition,      f_r = sf  hence,

X_2r = 2πf_r L₂ = 2πfsL₂ = s . (2πf L₂)

X_2r = sX₂

where   X_2r = Rotor reactance in running condition

Thus resistance as independent of frequency remains same at standstill and in running condition. While the rotor reactance decreases by slip times the rotor reactance at standstill.

Hence we can write rotor impedance per phase as :

Z₂ = Rotor impedance on standstill (N = 0) condition

= R₂ + j X₂  Ω/ph

Z₂ =   R²₂ + X²₂ + X² Ω/ph

while Z_2r = Rotor impedance in running condition

= R₂ + j X_2r = R₂ + j (s X₂) Ω /ph

Z_2r = R²₂ + (sX₂ )² Ω/ph   … magnitude

 

4. Effect on Rotor Power Factor

From rotor impedance, we can write the expression for the power factor of rotor at standstill and also in running condition.

The impedance triangle on standstill rotor condition is shown in the Fig. 5.7.1. From it we can write,

The impedance in running condition becomes Z2r and the corresponding impedance triangle is shown in the Fig. 5.7.2. From Fig. 5.7.2 we can write,

cos ϕ_2r = Rotor power factor in running condition

Key Point As rotor winding is inductive, the rotor p.f. is always lagging in nature.

 

5. Effect on Rotor Current

Let     I₂ = Rotor current per phase on standstill condition

The magnitude of I2 depends on magnitude of E₂ and impedance Z₂ per phase.

I₂ = (E₂ per phase / Z₂ per phase) A

Substituting expression of Z₂ we get,

I₂ = E₂ / √R₂² + X₂² A

The equivalent rotor circuit on standstill is shown in the Fig. 5.7.3.

The ϕ₂ is the angle between E₂ and I₂ which determines rotor p.f. on standstill.

In the running condition, Z₂ changes to Z_2r while the induced e.m.f. changes to E_2r. Hence the magnitude of current in the running condition is also different than I₂ on standstill. The equivalent rotor circuit on running condition is shown in the Fig. 5.7.4.

I_2r = Rotor current per phase in running condition .

The value of slip depends on speed which intum depends on load on motor hence X_2r is shown variable in the equivalent circuit. From the equivalent circuit we can write,

ϕ_2r is the angle between E_2r and I_2r which decides p.f. in running condition.

Key Point Putting s = 1 in the expressions obtained in running condition, the values at standstill can be obtained.

 

Examples for Practice

Example 5.7.2 A 3-phase induction motor runs at a speed of 1485 r.p.m. at no-load and at 1350 r.p.m. at full-load when supplied from a 50 Hz, 3-phase line.

i) How many poles does the motor have ?

ii) What is the % slip at no-load and at full-load ?

iii) What is the frequency of rotor voltages at no-load and at full-load ?

iv) What is the speed at both no-load and full-load of, the rotor field with respect to rotor conductors, the rotor field with respect to the stator and the rotor field with respect to the stator field.

[Ans.: i) 4, ii) 1 %, 10 %, iii) 0.5 Hz, 5 Hz, iv) on no load 15 r.p.m. 1500 r.p.m. 0 r.p.m., on full load 150 r.p.m. 1500 r.p.m. 0 r.p.m.]

Example 5.7.3 A 50 Hz, 4 pole induction motor has an induced e.m.f. in the rotor with a frequency of 2 Hz. Calculate i) Synchronous speed ii) Slip iii) Speed of the motor.        

[Ans.: i) 1500 r.p.m. ii) 0.04 or 4 % iii) 1440 r.p.m.]

Review Question

1. Explain the effect of slip on the following rotor parameters :

i) Frequency ii) Induced e.m.f. iii) Current iv) Power factor v) Reactance vi) Impedance.