The motor was invented by Michael Faraday who discovered the laws of electromagnetic induction. Actually, the invention began with the discovery of Direct current. This invention led to conduct detailed experiments for the invention of the motor. Earlier direct current was developed in 1800 and further investigations have been made for better utilization of DC. This DC helped in investigating the current supplied to the machine to operate. At the same time, investigations have done on the usage of alternating current. However, for the operation of any machine torque plays a key role. It is the basic twisting phenomenon to rotate the shaft of any machine. So, in this article, we shall discuss the basics of DC motor, how torque is developed, and the derivation of torque and power equation of DC motor.

DC Motor is a type of motor that uses input dc supply and transforms this input electrical energy to the required output mechanical power. The basic DC motor is shown in the figure below.

### Working

It works on the principle of electromagnetic induction. It also works consistently with the statement made by Lorentz. In accordance with Lorentz, the current-carrying conductor positioned beneath the magnetic field experiences a force. This force enables a unidirectional torque on the conductors which in turn rotates the rotor. The working of a DC motor is shown in the figure below.

## Torque Equation of DC Motor

The equivalent circuit diagram of this type of motor is shown in the figure below.

The voltage equation of the motor is given by

**V _{ }= E_{b} + I_{a }R_{a}**

The current equation is expressed as

**I _{a }= V – E_{b }/ R_{a}**

### Torque in DC Motor

The torque is defined as the turning moment created at a particular point. The torque at a particular point is shown in the figure below.

**T = F × R**

where,

F is the force acting on the body

R is the radius of the rotor

The speed N at which the force enables to rotate is expressed in rps.

The work done per revolution is given by F × distance covered **F × 2πr Joule**

Work done per second is given as **F × 2πr × N expressed in J/s.**

### Torque Equation of Permanent Magnet DC Motor Derivation

#### Torque and Power Equation of DC Motor

The energy generated by the motor is** E _{b }I_{a}**

The mechanical power developed = The electrical power developed

**T _{e} 2π N = E_{b }I_{a}**

**T _{e }= E_{b }I_{a} / 2π N**

The back emf is expressed as below

**E _{b} = N φ Z P / 60 A**

Substitute E_{b} expression in the torque equation

**T _{e }= E_{b }I_{a} / 2π N**

**T _{e }= N φ Z P / 60 A / 2π N**

**T _{e } = (1/ 2π) . (φ Z P / 60 A).**

This is the required torque equation that enables the rotor to rotate.

The torque is expressed in N-m. K = (1/ 2π), where K is constant

**T _{e } = K. (φ Z P / 60 A).**

From the above equation, we can say T_{e }α φ. I_{a}

### Torque Equation for Shunt Motor

The shunt type diagram is shown in the figure below.

The torque equation of a normal motor is given as **T _{e }α φ. I_{a}**

From the diagram, we can observe that φ is constant for a shunt motor as the winding is connected in parallel with the armature.

### Torque Equation for Series Motor

The series type diagram is shown in the figure below.

The torque equation of a normal motor is given as **T _{e }α φ. I_{a}**

φ α I_{a} for a series motor as the winding is connected in series with the armature. The torque equation can be written as **T _{e }α I_{a}. I_{a }**

**T _{e }α (I_{a}**

**)2**for a series motor. As the flux is directly proportional to armature current ( φ α I

_{a }) at a certain point the flux remains constant after its saturation point. In this situation, the Dc series motor acts normal to that of a shunt motor

**T**.

_{e }α I_{a}Thus, in this article, we had discussed the basics of DC motor, how torque is developed. It is basically developed due to the interaction of two forces that creates a unidirectional torque which enables the rotor to rotate. Apart from this, we had also discussed the derivation of torque and power equations. Here is a question for the readers, how torque is developed in a DC motor?

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