Question

Derive the torque equation of a transformer.

Answer 1

The torque equation of a transformer can be derived by analyzing the mechanical power developed in the transformer’s rotating parts, specifically in the case of a synchronous generator or motor-type transformer.

However, a traditional transformer (like the one used in electrical systems) doesn’t have any moving parts. But, when transformers are used in applications like induction motors (where a rotor rotates), we can derive an equation for the torque on the rotor.

Steps to Derive Torque Equation for a Transformer:

1. Basic Transformer Principles:
   

   - A transformer consists of two windings: the primary winding and the secondary winding, coupled through a magnetic flux.
   - In an ideal transformer, the flux generated in the core by the primary winding induces an electromotive force (EMF) in the secondary winding.

1. Electromagnetic Force in the Transformer:
   

   - The transformer operates based on the principle of electromagnetic induction, where the primary current creates a magnetic field that induces a secondary current.
   - The induced EMF in the secondary winding is proportional to the rate of change of the magnetic flux.

1. Magnetic Flux and Electromagnetic Induction:
   

   - If the primary winding current \( I_p \) generates a magnetic flux \( \phi \), this magnetic flux induces a voltage \( V_s \) in the secondary winding.
   - The power transferred from the primary to the secondary side is given by:
     \[
     P = V_s \cdot I_s
     \]
   where:
   - \( P \) is the power transferred to the secondary winding.
   - \( V_s \) is the voltage induced in the secondary winding.
   - \( I_s \) is the current flowing in the secondary winding.

1. Mechanical Power and Torque:
   

   - When the transformer is used in a rotating machine, like an induction motor, the rotating magnetic field created by the transformer’s primary and secondary windings causes a mechanical force on the rotor. This force generates torque.

1. Torque Equation:
   

   The mechanical power \( P_{\text{mech}} \) developed in a rotating machine is related to the torque \( T \) and the angular velocity \( \omega \) of the rotor by the following equation:
   \[
   P_{\text{mech}} = T \cdot \omega
   \]
   Here, \( P_{\text{mech}} \) is the mechanical power, \( T \) is the torque, and \( \omega \) is the angular velocity of the rotor.

   The total mechanical power can be written as:
   \[
   P_{\text{mech}} = \frac{V_s \cdot I_s}{\sqrt{3}} \cdot \cos \theta
   \]
   where:
   - \( V_s \) is the secondary voltage.
   - \( I_s \) is the secondary current.
   - \( \theta \) is the angle between the voltage and current waveforms.

   Substituting the mechanical power into the torque equation:
   \[
   T = \frac{P_{\text{mech}}}{\omega}
   \]
   By substituting the mechanical power expression:
   \[
   T = \frac{V_s \cdot I_s \cdot \cos \theta}{\sqrt{3} \cdot \omega}
   \]

1. Final Torque Equation:
   

   The equation for the torque developed in the transformer’s rotating part (e.g., rotor) can be expressed as:
   \[
   T = \frac{V_s \cdot I_s \cdot \cos \theta}{\sqrt{3} \cdot \omega}
   \]
   Where:
   - \( T \) is the torque developed on the rotor.
   - \( V_s \) is the secondary voltage.
   - \( I_s \) is the secondary current.
   - \( \cos \theta \) is the power factor.
   - \( \omega \) is the angular velocity of the rotor.

This equation represents the torque generated by the interaction of the magnetic field created by the transformer’s windings and the rotor in an application where the transformer induces motion, such as in an induction motor.