When study the performance of three-phase induction motors, understanding slip, speed, and rotor resistance and current and other parameters are more essential. These concepts form the foundation of how induction motors operate under different load conditions . The interaction between the stator’s rotating magnetic field and the rotor determines the motor’s efficiency and behavior. Key terms such as synchronous speed, slip, and rotor impedance , rotor resistance, rotor current, help explain the motor’s dynamic response and power transfer characteristics.
1. Synchronous Speed
When Three phase electrical supply applied to induction motor stator, a magnetic field is formed. This magnetic field is looking likes rotating in nature. The speed , which rotating magnetic field rotates called as synchronous speed. This synchronous speed depends on supply frequency (f) and number of stator poles (P)
The general formula of synchronous speed written as
\[ N_s = \frac{120 f}{P}\]
This formula indicates that, synchronous speed increases as supply frequency (f) increases. As same as it decreases when number of poles decreases. But number of poles in stator is physically constructed. Unless tapped speed control system it is very hard to change it. Assuming number of poles constant , synchronous speed of an induction motor can be varied by changing supply frequency (f)
2. Slip (s)
The magnetic flux lines generated by rotating magnetic field generated in stator are cuts by rotor conductor. So emf induces in it. This EMF causes current circulation in it as those conductors are short circuited. Hence a repulsion force (As per Lenz’s law) generated between stator and rotor. Hence rotor starts in direction of synchronous speed. When load connected in rotor shaft , it tends to overcome to it and maintain its nominal speed. Under load condition actual speed of rotor is less than synchronous speed.
The difference between synchronous speed (Ns) and actual rotor speed (N) called as slip (s)
The slip (s) is always represented in percentage.
Formula of slip
\[ \% s = \frac{\omega_s – \omega}{\omega_s}\]
(or)
\[ \%s = \frac{N_s – N}{N_s} x 100\]
To derive actual rotor speed (N) from expression of slip (s)
N = Ns– s . Ns
N = Ns (1-s)
From above expression ,The following observation made for various operation in an induction motor
| Value of slip (s) | Speed of Rotor (N) | Nature of Operation |
|---|---|---|
| 0 | Maximum Speed | Motoring |
| 1 | Stand Still, Zero Speed | Motoring |
| >1 (More Than 1) | Less than rated speed , But in reverse direction | Plugging |
| <1 (Less than ) | More than rated speed, In same direction | Generator |
Slip Frequency (fr)
Emf induced in rotor winding due to relative velocity between rotor speed and synchronous speed. But Stator Emf causes to induces Emf in rotor , this will be AC in nature. So EMF has certain amount of frequency.
Rotor frequency fr – s f
This equation indicated than rotor frequency in equal to stator frequency when slip (s) = 1. When slip is maximum (Near s =1) rotor frequency will be minimum
Hence frequency of rotor induced voltage is proportional to slip, it is called as slip frequency (fr)
Rotor Induced Voltage (E2)
Induction motor is popularly known as rotating transformer. Its stator winding acts as primary winding and rotor winding act as secondary winding
So it can be compared with transformation ratio formula
\[ \frac{E_2}{E_1} = \frac{N_2}{N_1}\]
Where,
E1 = Voltage applied in stator winding
E2 = Voltage induced in rotor winding
N1 = No of turns in stator winding
N2 = Number of turns in rotor winding
i. At Stand Still condition (When Rotor Speed N = 0)
\[E_2 = \frac{N_2}{N_1} x E_1\]
If induction motor , turns ration considered that K then
E2 = KE1
ii. Under running condition (E2r)
Under running conditions, relative velocity between rotating magnetic field and rotor is reduces. So EMF induced in rotor is also reduces.
E2r = rotor induced voltage per phase under running condition
E2r α Ns – N
E2 = Ns
\[\frac{E_{2r}}{E_2} = \frac{N_s – N}{N_s}\]
\[s = \frac{E_{2r}}{E_2}\]
\[E_{2r} = s E_2\]
Rotor resistance (R2)
Rotor resistance is independent to rotor frequency (fr). Either stand still N = 0 (or) approximately equal to synchronous speed N = Ns, rotor resistance is always same. Rotor frequency does not affect rotor frequency.
Rotor Reactance (X2)
The rotor frequency at standstill condition fr = f
Then rotor reactance
X2 = 2πfL Ohm/Phase
Under running condition rotor frequency fr = sf
X2r = 2π sf L
= s2π f L
X2r = sX2
This shows that rotor reactance is maximum at stand still condition and reduces when rotor speed decreases.
Rotor impedance (Z2)
i. At stand still condition
The rotor impedance at stand still condition is expressed as
\[Z_2 = \sqrt{R^2 _2 + X^2 _2}\]
ii. Under rotor running condition
The rotor impedance under running condition is given by
\[Z_{2r} = \sqrt{R^2 _2 + s^2 X^2 _2} \]
This shows frequency of rotor induced is dependent on slip s the leakage resistance is also a function of slip. Hence rotor impedance also changes when speed changes
Rotor Power factor
i. At stand still condition
The rotor power factor at stand still condition expressed as
\[cos\phi = \frac{R_2}{Z_2}\]
\[ = \frac{R_2}{\sqrt{R^2 _2 + X^2 _2}}\]
ii. Under running condition
\[cos\phi = \frac{R_2}{Z_{2r}}\]
\[ = \frac{R_2}{\sqrt{R^2 _2 + sX^2 _2}}\]
Rotor current
The rotor current is derived from ratio of induced motor voltage per phase and the rotor impedance
i. Under stand still condition
Rotor current under stand still condition derived as
\[I_2 = \frac {E_2}{Z_2}\]
\[\frac{E_2}{\sqrt{R^2 _2 + X^2 _2}}\]
ii. Under running condition
The rotor current under running ,condition given by
\[I_{2r} = \frac{E_{2r}}{Z_{2r}}\]
\[\frac{sE_2}{\sqrt{R^2 _2 + s^2 X^2 _2}}\]
\[I_2r = \frac{E_2}{\frac{R^2 _2}{s^2} + X^2 _2}\]
Power Transferred to rotor
The power transferred to rotor can be expressed as
\[P_{2r} = I^2 _{2r} x \frac{R_2}{s}\]
