Calculating Induction Motor Synchronous Speed

The synchronous speed ($N_s$) is the speed at which the stator's magnetic field rotates. It's determined by the supply frequency and the number of poles in the motor. The formula is:

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

Given the frequency ($f$) is 50 Hz and the number of poles ($P$) is 6:

$$ N_s = \frac{120 \times 50}{6} = \frac{6000}{6} = 1000 \text{ rpm} $$

Therefore, the speed of the stator magnetic field is 1000 rpm.

Stator Magnetic Field Relative to Rotor Magnetic Field

In a squirrel cage induction motor, the stator's rotating magnetic field induces currents in the rotor bars. These induced currents create their own magnetic field (the rotor magnetic field). Due to the principle of electromagnetic induction, the rotor field essentially 'locks' onto the stator field, rotating together at the synchronous speed ($N_s$) relative to the stationary stator frame. This means that, from the perspective of the rotor's magnetic field, the stator's magnetic field appears stationary. Hence, their relative speed is zero.

Speed of stator magnetic field to rotor magnetic field = 0 rpm

Rotor Speed Relative to Stator Magnetic Field

This refers to the speed difference between the rotor and the stator's magnetic field, often called the slip speed. The slip ($s$) quantifies how much slower the rotor runs compared to the synchronous speed ($N_s$).

The relationship is defined as:

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

Where $N_r$ is the actual rotor speed.

The speed of the rotor relative to the stator magnetic field is $N_r - N_s$. Rearranging the slip formula gives us $N_s - N_r = s \times N_s$. Therefore, the speed of the rotor relative to the stator field is:

$$ N_r - N_s = - (N_s - N_r) = - (s \times N_s) $$

Given the slip ($s$) is 5% or 0.05, and the synchronous speed ($N_s$) is 1000 rpm:

$$ N_r - N_s = - (0.05 \times 1000 \text{ rpm}) = -50 \text{ rpm} $$

The negative sign indicates that the rotor is rotating slower than, or 'slipping back' relative to, the stator magnetic field.

Summary of Results

The analysis shows:

• The speed of the stator magnetic field relative to the rotor magnetic field is 0 rpm.
• The speed of the rotor with respect to the stator magnetic field is -50 rpm.