# Voltage and Power Equations of DC Motor | Condition of Maximum Power

- Link: https://studyelectrical.com/2018/01/voltage-and-power-equations-of-dc-motor-maximum-power-condition.html
- Published: 2018-01-19T09:49:00+00:00

![dc%2Bmotor%2Bvoltage%2Band%2Bpower%2Bequations](https://1.bp.blogspot.com/-6AEBbpaDMko/
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2Bvoltage%2Band%2Bpower%2Bequations.jpg)

In this post, we will look at the **voltage and power equation of a dc motor** and
the condition for maximum power in a dc motor.

## Voltage Equation of a DC Motor

The voltage V applied across the motor armature has to

 1. overcome the [back emf E_(b)](http://www.studyelectrical.com/2015/02/back-emf-and-significance-in-dc-motor.html)
    and
 2. supply the armature ohmic drop I_(a)R_(a) .

![voltage%2Bequation%2Bdc%2Bmotor](https://2.bp.blogspot.com/-zBvibL9JEso/WohfWEoBhhI/
AAAAAAAACas/AjfyWvMalPYN4wW7VPGWJtpDgm9ovYPrACLcBGAs/s1600/voltage%2Bequation%2Bdc%
2Bmotor.png)

**This is known as voltage equation of a motor.**

> [Back EMF and Its Significance in DC Motor](http://www.studyelectrical.com/2015/02/back-emf-and-significance-in-dc-motor.html)
> 
> [Working Principle of DC Motor](http://www.studyelectrical.com/2014/12/working-principle-of-dc-motor.html)

## Power Equation of DC Motor

Now, multiplying both sides of voltage equation by I_(a) , we get
 ⌊power%2Bequation%
2Bdc%2Bgenerator%2Bdc%2Bmotor⌉ Hence, out of the armature input, some is wasted 
in I^(2)R loss and the rest is converted into mechanical power within the armature.

The gross mechanical power developed by a motor is
 ⌊power%2Bequation%2Bdc%2Bgenerator%
2Bdc%2Bmotor%2B1saf⌉ It may also be noted that motor efficiency is given by the 
ratio of power developed by the armature to its input ⌊power%2Bequation%2Bdc%2Bgenerator%
2Bdc%2Bmotor%2B2⌉ Obviously, the higher the value of E_(b) as compared to V, higher
the motor efficiency.

### Condition for Maximum Power

The gross mechanical power developed by a motor is
 ⌊gross%2Bmechanical%2Bpower%
2Bdc%2Bmotor%2Bdc%2Bgenerator⌉ Differentiating both sides with respect to I_(a) 
and equating the result to zero, we get  Thus maximum efficiency of a dc motor occurs
when [back EMF](http://www.studyelectrical.com/2015/02/back-emf-and-significance-in-dc-motor.html)
is equal to half the applied voltage.

#### Important Points

Thus gross mechanical power developed by a motor is maximum when back EMF is equal
to half the applied voltage.
  This condition is, however, not realized in practice,
because in that case current would be much beyond the normal current of the motor.
Moreover, half the input would be wasted in the form of heat and taking other losses(
mechanical and magnetic) into consideration, the motor efficiency will be well below
50 percent.

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