Understanding Induction Motor Fields

Single-phase induction motors are characterized by two rotating magnetic fields: a forward field and a backward field. Both fields rotate at synchronous speed, but in opposite directions. The rotor current frequency depends on the relative speed between the rotor and these fields. The question asks for the frequency of rotor currents specifically due to the backward field.

Calculating Synchronous Speed

First, we need to determine the synchronous speed ($N_s$) of the motor. The formula for synchronous speed is:

$$ N_s = \frac{120 \times f}{P} $$

Where:

• $f$ = Supply frequency = 50 Hz
• $P$ = Number of poles = 10

Substituting the values:

$$ N_s = \frac{120 \times 50 \text{ Hz}}{10} = \frac{6000}{10} = 600 \text{ RPM} $$

The synchronous speed is 600 RPM.

Determining Rotor Speed and Slip

The motor's actual running speed ($N_r$) under load is given as 540 RPM.

Slip ($s$) is the difference between synchronous speed and rotor speed, expressed as a fraction of synchronous speed:

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

Calculating the slip:

$$ s = \frac{600 \text{ RPM} - 540 \text{ RPM}}{600 \text{ RPM}} = \frac{60}{600} = 0.1 $$

The slip of the motor is 0.1.

Rotor Current Frequency Derivation

The frequency of induced currents in the rotor ($f_r$) is directly proportional to the slip and the supply frequency ($f$):

$$ f_r = s \times f $$

For the forward field, the slip is $s = 0.1$, so the frequency is $0.1 \times 50 \text{ Hz} = 5 \text{ Hz}$.

However, the question asks for the frequency due to the backward field. The backward field rotates at $-N_s$ (synchronous speed in the opposite direction). The slip ($s'$) relative to the backward field is calculated as:

$$ s' = \frac{\text{Speed of backward field} - \text{Rotor speed}}{\text{Speed of backward field}} = \frac{(-N_s) - N_r}{(-N_s)} $$

$$ s' = \frac{-N_s - N_r}{-N_s} = \frac{N_s + N_r}{N_s} $$

Substitute the values:

$$ s' = \frac{600 \text{ RPM} + 540 \text{ RPM}}{600 \text{ RPM}} = \frac{1140}{600} = 1.9 $$

Now, calculate the frequency of the rotor currents due to the backward field ($f_{rb}$) using this slip ($s'$):

$$ f_{rb} = s' \times f $$

$$ f_{rb} = 1.9 \times 50 \text{ Hz} = 95 \text{ Hz} $$

Therefore, the frequency of the induced rotor currents due to the backward field is 95 Hz.