Understanding the Back EMF in DC Motors: Why It Is Always Less Than the Applied Voltage

The back electromotive force (EMF) generated in a DC motor is always less than the applied voltage due to the presence of electrical resistance in the motor's windings. This fundamental principle is crucial for understanding the operation and efficiency of DC motors. Let's delve into the details to clarify this concept.

The Role of Back EMF in DC Motors

When a DC motor is in operation, the rotation of the armature induces a voltage within the armature windings, known as the back EMF. This back EMF opposes the applied voltage from the power supply, creating a difference between the two.

Electrical Resistance and Back EMF

The reason the back EMF is less than the applied voltage is because some of the applied voltage is dropped across the electrical resistance of the motor's windings. This resistance is inherent and is determined by the materials used and the length of the wire. The relationship between the applied voltage (V), the back EMF (E_b), and the winding resistance (R) can be described by the equation:

V E_b IR

where (I) is the current flowing through the motor's windings.

Implications of Increasing Speed

As the motor's speed increases, the back EMF also increases. However, it will always be less than the applied voltage due to the voltage drop across the winding resistance. This voltage drop is essential to drive the current required to produce the motor's torque and rotation. In practical applications, this relationship ensures that the motor can operate efficiently and smoothly.

The Application of Lenz's Law

Another way to understand why the back EMF is less than the applied voltage is through the application of Lenz's law. Lenz's law states that the direction of the induced EMF is such that it opposes the change that produced it. In the case of a DC motor, the induced back EMF opposes the applied voltage, hence the distinction.

Coasting and Eddy Currents

If the back EMF were to be equal to the applied voltage, no current would flow to the armature, and no torque would be produced. This situation can occur, for example, in an electric vehicle coasting downhill where the shaft is externally driven. Such scenarios highlight the importance of the back EMF in maintaining the motor's functionality.

Mathematical Representation

The mathematical representation of the back EMF in the above circuit can be expressed as:

V_t E_a I_a R_a

Where:

(V_t) is the applied voltage (E_a) is the back EMF (I_a) is the current flowing through the armature windings (R_a) is the armature winding resistance

From this equation, it can be seen that if the current (I_a) is negative, the back EMF can be higher than the applied voltage. This condition is typical of generators where the mechanical energy input causes a higher back EMF than the applied voltage.

Conclusion

Understanding why the back EMF generated in a DC motor is always less than the applied voltage is essential for optimizing the performance and efficiency of these motors. The presence of electrical resistance in the windings, the application of Lenz's law, and the implications of varying speed all play critical roles in this fundamental principle.