Torque equation of single phase induction motor?

Question

Torque equation and generated emf in rotor of single phase induction motor.

Answer 1

The analysis of a single-phase induction motor is fascinating because, unlike its three-phase counterpart, it is not self-starting. The key to understanding its torque and EMF lies in the Double Revolving Field Theory.

Let's break it down.

The Core Concept: Double Revolving Field Theory

A single-phase AC supply to the stator winding produces a magnetic field that is pulsating, not rotating. It simply grows stronger and weaker in one direction.

The Double Revolving Field Theory states that this single pulsating magnetic field can be mathematically resolved into two magnetic fields:

1. Forward Revolving Field ($Φ_f$): Rotates at synchronous speed ($N_s$) in the clockwise (or forward) direction.
2. Backward Revolving Field ($Φ_b$): Rotates at synchronous speed ($N_s$) in the counter-clockwise (or backward) direction.

The magnitude of each of these revolving fields is half the magnitude of the peak pulsating field ($Φ_{max}/2$).

Now, we can analyze the single-phase motor as if it were two separate three-phase motors with their rotors mechanically coupled, being acted upon by these two opposing fields.

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1. Generated EMF in the Rotor

The EMF induced in the rotor bars depends on the relative speed between the rotor conductors and the magnetic fields. Let's consider two conditions:

A. At Standstill (Rotor Speed, N = 0)

• Slip relative to the Forward Field ($s_f$):
  $s_f = \frac{N_s - N}{N_s} = \frac{N_s - 0}{N_s} = 1$
• Slip relative to the Backward Field ($s_b$): The backward field rotates in the opposite direction, so its relative speed is additive.
  $s_b = \frac{N_s - (-N)}{N_s} = \frac{N_s + N}{N_s} = \frac{N_s + 0}{N_s} = 1$

Since both slips are equal to 1, the forward and backward fields cut the rotor conductors at the same speed ($N_s$). Therefore, they induce EMFs of equal magnitude but at the same frequency ($f_r = s \cdot f = 1 \cdot f$).

• EMF due to Forward Field ($E_f$): Proportional to slip $s_f$.
• EMF due to Backward Field ($E_b$): Proportional to slip $s_b$.

At standstill, $|E_f| = |E_b|$.

B. During Rotation (Rotor Speed, N > 0)

Let's assume the motor is rotating in the forward direction.

• Slip relative to the Forward Field ($s_f$): The rotor is "catching up" to the forward field, so the relative speed is low.
  $s_f = \frac{N_s - N}{N_s}$ (This is a small value, e.g., 0.04 or 4%)
• Slip relative to the Backward Field ($s_b$): The rotor is running opposite to the backward field, so the relative speed is very high.
  $s_b = \frac{N_s + N}{N_s} = \frac{2N_s - (N_s - N)}{N_s} = 2 - \frac{N_s - N}{N_s} = \mathbf{2 - s_f}$

Since $s_f$ is small, $s_b$ is a large value (e.g., if $s_f = 0.04$, then $s_b = 1.96$).

This has two major consequences for the induced EMF:

1. Magnitude: The magnitude of the EMF induced by the backward field ($E_b = sb E{2o}$) is much larger than the EMF induced by the forward field ($E_f = sf E{2o}$), where $E_{2o}$ is the standstill rotor EMF.
2. Frequency: The frequency of the rotor currents is different for each field.
   • Frequency of forward currents: $f_f = s_f \cdot f$ (very low frequency)
   • Frequency of backward currents: $f_b = s_b \cdot f$ (very high frequency, almost $2f$)

This difference in frequency is critical for understanding the torque.

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2. Torque Equation

Torque is produced by the interaction of each magnetic field with the rotor currents it induces.

A. At Standstill (N = 0, s = 1)

• The forward field induces currents and produces a forward torque ($T_f$).
• The backward field induces currents and produces a backward torque ($T_b$).

Since at standstill everything is symmetrical ($s_f = s_b = 1$), the conditions for torque production are identical for both fields. They produce torques of equal magnitude but in opposite directions.

Net Torque ($T_{net}$) = $T_f - T_b = 0$

This is the fundamental reason why a single-phase induction motor is not self-starting. It has zero net torque at standstill.

B. During Rotation (N > 0)

Once the motor is given an initial push (by an auxiliary winding or by hand), the situation changes dramatically.

• Forward Torque ($T_f$):
  The slip $s_f$ is small.
  The rotor reactance for this field, $X_f = sf X{2o}$, is very low.
  * The rotor circuit is therefore highly resistive ($R_2' > X_f$). This results in a high power factor for the rotor circuit, producing a strong forward torque, similar to a normal induction motor.

• Backward Torque ($T_b$):
  The slip $s_b$ is very large (close to 2).
  The rotor reactance for this field, $X_b = sb X{2o}$, is very high.
  * The rotor circuit is therefore highly inductive ($X_b >> R_2'$). This results in a very low power factor (current lags voltage by almost 90°), producing a very weak backward (braking) torque.

The Net Torque is the difference between the two:

$T_{net} = T_f - T_b$

Since $T_f$ is strong and $T_b$ is weak, there is a significant net torque in the forward direction, which keeps the motor running.

The Torque Equation

The torque produced by each field is analogous to the torque in a 3-phase motor. The approximate torque equation for the net torque ($T_{net}$) is the sum of the individual torque expressions:

$T_{net} = T_f - T_b$

$T_{net} = \frac{1}{2\pi Ns} \left[ \frac{I{2f}^2 R_2'}{sf} - \frac{I{2b}^2 R_2'}{s_b} \right]$

Where:
$N_s$ is the synchronous speed in rps.
$I{2f}$ and $I{2b}$ are the rotor currents due to the forward and backward fields, respectively.
$R_2'$ is the rotor resistance referred to the stator.
$s_f$ is the forward slip, and $s_b$ is the backward slip ($s_b = 2 - s_f$).

This can be visualized with a torque-slip curve.

Explanation of the Graph:
1. Tf Curve: The torque-slip curve for the forward-rotating field. It's a standard induction motor curve.
2. Tb Curve: The torque-slip curve for the backward-rotating field. It's a mirror image.
3. Tnet (Resultant Torque): The algebraic sum of Tf and Tb. Notice that at slip = 1 (standstill), the net torque is zero. Once the motor starts rotating (slip decreases from 1), a positive net torque develops.