Calculating Rotor Frequency in an Induction Motor

In a 3-phase induction motor, the rotor does not rotate at exactly the same speed as the rotating magnetic field produced by the stator. This difference in speed is called slip. The frequency of the voltage and current induced in the rotor depends on this slip and the supply frequency.

Understanding Slip and Frequency

The stator of a 3-phase induction motor is connected to an AC supply, which creates a rotating magnetic field. This field rotates at the synchronous speed, which is determined by the supply frequency and the number of poles in the motor.

As the rotating magnetic field cuts the rotor conductors, it induces an electromotive force (EMF) and causes current to flow in the rotor. This induced current creates its own magnetic field, which interacts with the stator field, causing the rotor to rotate. The rotor speed is always slightly less than the synchronous speed under normal operation. This difference is the slip.

The frequency of the induced EMF and current in the rotor is not the same as the supply frequency. It is proportional to the slip. At standstill (when the rotor is not moving), the slip is 1 (or 100%), and the rotor frequency is equal to the supply frequency. As the rotor speeds up, the relative speed between the rotor and the rotating field decreases, reducing the slip and thus reducing the frequency of the induced EMF.

Formula for Rotor Induced EMF Frequency

The relationship between the rotor frequency ($f_r$), the supply frequency ($f$), and the slip ($s$) is given by the formula:

$\qquad f_r = s \times f$

where:

• $f_r$ is the frequency of the induced EMF in the rotor.
• $s$ is the slip, expressed as a decimal.
• $f$ is the supply frequency.

Applying the Formula to the Given Problem

We are given the following information for a 3-phase induction motor:

• Supply frequency ($f$) = 100 Hz
• Slip ($s$) = 4%

First, we need to convert the slip percentage into a decimal value:

$\qquad s = 4\% = \frac{4}{100} = 0.04$

Now, we can use the formula to find the frequency of the motor induced EMF ($f_r$):

$\qquad f_r = s \times f$

Substitute the given values into the formula:

$\qquad f_r = 0.04 \times 100 \text{ Hz}$

Performing the multiplication:

$\qquad f_r = 4 \text{ Hz}$

Therefore, the frequency of the motor induced EMF is 4 Hz.

Summary of Calculation

Understanding Slip and Speed Relationship

Slip can also be expressed in terms of synchronous speed ($N_s$) and rotor speed ($N_r$) as:

$\qquad s = \frac{N_s - N_r}{N_s}$

The difference $(N_s - N_r)$ is often called the slip speed.

The synchronous speed is determined by the supply frequency ($f$) and the number of poles ($P$) of the motor:

$\qquad N_s = \frac{120 \times f}{P}$ (in RPM)

Combining these relationships, you can see how the rotor frequency ($f_r = s \times f$) is directly related to the difference in speed between the rotating field and the rotor.

Revision Table - Induction Motor Frequency

Additional Information - Induction Motor Operation

The frequency of the induced EMF in the rotor is crucial for the operation of the induction motor. When the rotor is at standstill (slip = 1), the rotor frequency is equal to the supply frequency. As the rotor starts rotating, the slip decreases, and so does the rotor frequency. Under normal running conditions, the slip is small (typically 2-5%), resulting in a low rotor frequency. This low frequency is important because it affects the rotor reactance and the power factor of the rotor circuit.

A lower rotor frequency means lower inductive reactance ($X_r = 2 \pi f_r L_r$), allowing rotor current to flow more easily and produce torque. If the rotor frequency were high (like the supply frequency at low slip), the high inductive reactance would limit rotor current and reduce torque production.