Understanding Rotor Current Frequency in Induction Motors

The question asks us to determine the frequency of the current flowing in the rotor of an induction motor, given the slip 's' and the supply frequency 'f'. This is a fundamental concept in the operation of three-phase induction motors.

Defining Key Terms

• Supply Frequency (f): This is the frequency of the AC power supply connected to the stator windings of the motor. It is typically measured in Hertz (Hz).
• Slip (s): Slip is a measure of the difference between the synchronous speed (the speed of the rotating magnetic field in the stator) and the actual rotor speed. It is usually expressed as a fraction or a percentage of the synchronous speed. Mathematically, slip is defined as:

$$ \text{s} = \frac{\text{N}_{\text{s}} - \text{N}_{\text{r}}}{\text{N}_{\text{s}}} $$

• Where:
• $ \text{N}_{\text{s}} $ is the synchronous speed (speed of the rotating magnetic field).
• $ \text{N}_{\text{r}} $ is the rotor speed (actual speed of the rotor).

Relating Slip to Rotor Frequency

The current in the rotor is induced because the rotor conductors cut the magnetic field created by the stator. The frequency of the induced current in the rotor depends on the *relative* speed between the rotor conductors and the rotating magnetic field. This relative speed is directly proportional to the slip.

When the rotor is stationary (s = 1, starting condition), the relative speed is maximum and equal to the synchronous speed. In this case, the frequency of the induced rotor current is equal to the stator supply frequency, 'f'.

When the rotor rotates at synchronous speed (s = 0, ideal condition, no load), there is no relative speed between the rotor conductors and the rotating magnetic field. Therefore, no voltage is induced, and the rotor current frequency is zero.

For any speed between standstill and synchronous speed (0 < s < 1), the relative speed is proportional to the slip. Consequently, the frequency of the induced voltage and current in the rotor is proportional to the slip 's' and the supply frequency 'f'.

The formula for the rotor current frequency ($ \text{f}_{\text{r}} $) is given by:

$$ \text{f}_{\text{r}} = \text{s} \times \text{f} $$

Where:

• $ \text{f}_{\text{r}} $ is the rotor current frequency.
• $ \text{s} $ is the slip.
• $ \text{f} $ is the supply frequency.

Analyzing the Options

Let's look at the given options:

Based on the analysis, the frequency of the current in the rotor is indeed given by the product of the slip (s) and the supply frequency (f).

Conclusion on Rotor Frequency

For a slip 's' and supply frequency 'f', the frequency of the current in the rotor of an induction motor is sf. This relationship is crucial for understanding various aspects of induction motor operation, including rotor induced voltage and impedance.

Revision Table: Induction Motor Frequencies and Speeds

Additional Information on Induction Motor Concepts

Induction motors work based on the principle of electromagnetic induction. The rotating magnetic field produced by the stator induces voltage and current in the rotor conductors.

• Rotating Magnetic Field: When a three-phase AC supply is connected to the stator windings, it creates a magnetic field that rotates at the synchronous speed ($ \text{N}_{\text{s}} $).
• Rotor Induction: As the rotating magnetic field sweeps past the rotor conductors, it induces an electromotive force (EMF) in them. Because the rotor circuit is closed (either short-circuited in a squirrel cage motor or connected to external resistance via slip rings), this induced EMF causes current to flow in the rotor conductors.
• Torque Production: The interaction between the rotor currents and the rotating magnetic field produces a torque, which causes the rotor to rotate in the same direction as the field.
• Necessity of Slip: For voltage and current to be induced in the rotor, there must be a relative speed between the rotor and the rotating magnetic field. This relative speed exists only when the rotor speed ($ \text{N}_{\text{r}} $) is less than the synchronous speed ($ \text{N}_{\text{s}} $). This difference in speed is what we define as slip (s > 0). If the rotor were to catch up to synchronous speed (s = 0), the relative motion would cease, induction would stop, rotor current would become zero, and consequently, torque would become zero. This is why an induction motor always operates with some amount of slip.

Understanding the relationship between supply frequency, slip, synchronous speed, rotor speed, and rotor frequency is fundamental to analyzing the performance characteristics of induction motors, such as torque-speed curves, efficiency, and power factor.