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A fundamental aspect of electric motors is Back Electromotive Force, often abbreviated as back EMF. This occurs in motors when they are running. Back EMF is a phenomenon where the motor, while operating, generates its own voltage opposing the applied supply voltage. This acts much like a brake on the current flow within the motor circuit.

This resistance to current is crucial for the motor's efficiency and longevity. It can be calculated using the formula:

• \( \text{Back Emf} = V - I \times R \)

Here,

• \( V \) is the constant supply voltage applied to the motor.
• \( I \) is the current through the motor while it is at its standard operational speed.
• \( R \) denotes the resistance offered by the motor’s armature windings.

The back EMF grows as the motor speeds up, naturally regulating the motor’s current to prevent overheating. It helps maintain the motor at a desired speed and power under various load conditions. In the given problem, substituting the known values results in a back EMF of \( 114.96 \text{ V} \). Understanding back EMF is crucial in ensuring motors are neither overworked nor underpowered.

When calculating the current at the motor's start, things change significantly from its normal operational state. At the moment a motor is switched on, it is at rest without any motion. This means back EMF is not yet generated.

Without back EMF to counteract the supply voltage, the entire potential difference is exerted across the armature’s resistance. Using Ohm’s law, this initial current spike can be calculated as:

• \( I = \frac{V}{R} \)

Since the resistance of the motor is still present and all the supply voltage acts purely on it, the motor sees a generally significant burst of current. In our example, substituting the appropriate values gives a starting current of approximately \( 166.67 \text{ A} \).

It's important to note that this high startup current can damage components if not regulated, which is why proper control mechanisms are often installed in motor systems to limit this excessive current during starting periods.

Given that the starting current can be significantly high, it is often necessary to add resistance in series with the motor to limit this current. This is especially important for preventing potential damage and ensuring the motor’s prolonged operational efficiency.

To restrain the starting current, an additional series resistance \( R_s \) can be calculated using the equation:

• \( I = \frac{V}{R + R_s} \)

Rearranging the equation helps isolate \( R_s \), resulting in:

• \( R_s = \frac{V}{I} - R \)

By plugging in the given values, the calculations yield a series resistance of \( 7.28 \Omega \) needed to cap the initial current to \( 15.0 \text{ A} \).

This additional resistance ensures the motor starts smoothly, safeguarding the circuit and components from potential current surges. Properly accounting for and controlling the starting current not only prolongs the life of the motor but also ensures safety and reliability in its operation.