Problem 49 - University Physics with Modern Physics

Question

In a shunt-wound dc motor with the field coils and rotor connected in parallel (Fig. \(27.56 ),\) the resistance \(R_{t}\) of the field coils is \(106 \Omega,\) and the resistance \(R_{r}\) of the rotor is 5.9\(\Omega\) . When a potential difference of 120 \(\mathrm{V}\) is applied to the brushes and the motor is running at full speed delivering mechanical power, the current supplied to it is 4.82 \(\mathrm{A}\) (a) What is the current in the field coils? (b) What is the current in the rotor? (c) What is the induced emf developed by the motor? (d) How much mechanical power is developed by this motor?

Step-by-Step Solution

Answer

(a) 1.13 A, (b) 3.69 A, (c) 98.2 V, (d) 362.3 W

Step 1: Calculate Current in Field Coils

Use Ohm's law to calculate the current in the field coils, \( I_f \). The voltage across the field coils, \( V_f \), is the same as the applied voltage: \( V_f = 120\, \text{V} \). Using the formula \( I_f = \frac{V_f}{R_t} \) where \( R_t = 106\, \Omega \), compute \( I_f = \frac{120}{106} \approx 1.13\, \text{A} \).

Step 2: Determine Current in Rotor

The total current supplied by the source is 4.82 A. The current in the rotor, \( I_r \), can be calculated using the principle of parallel circuits. From \( I = I_f + I_r \), deduce \( I_r = 4.82 - 1.13 = 3.69\, \text{A} \).

Step 3: Calculate Induced EMF

For the rotor, use the relationship \( V = I_r R_r + \text{EMF} \). Solve for \( \text{EMF} = V - I_r R_r = 120 - (3.69 \times 5.9) = 98.2\, \text{V} \).

Step 4: Compute Mechanical Power Developed

The mechanical power developed by the motor is given by \( P = \text{EMF} \times I_r \). Thus, substituting the values, \( P = 98.2 \times 3.69 \approx 362.3\, \text{W} \).

Key Concepts

Shunt-Wound Motor

A shunt-wound motor is a type of DC motor where the field windings are connected in parallel to the armature or rotor windings. This is a common configuration used in many applications like fans, blowers, and conveyors. In a shunt-wound motor, the field coils and rotor woundings share the same voltage supply but carry different currents.

The parallel arrangement allows for better speed regulation, which means whether the motor is under light or heavy load, the speed remains relatively constant. This happens because the field current remains mostly unaffected by changes in load, as it is determined by the supply voltage and the field resistance. This type of motor provides:

• Good speed regulation
• Ability to start under load
• Simplicity of control

Shunt-wound motors are widely used in applications where a steady, continuous motion is desired.

Induced EMF

Induced EMF (Electromotive Force) is a critical concept in the functioning of DC motors. In a motor, as the rotor (also known as the armature) spins, it moves through a magnetic field, which induces an EMF. This induced EMF opposes the applied voltage due to Lenz's Law, helping regulate the motor's speed.

The magnitude of the induced EMF can be calculated using the formula:\[\text{EMF} = V - I_r \times R_r\]where \( V \) is the supply voltage, \( I_r \) is the current in the rotor, and \( R_r \) is the resistance of the rotor.

This induced EMF is crucial as it determines how efficiently the motor converts electrical energy into mechanical energy.

Mechanical Power

Mechanical Power is the power that our motor produces when it's running at full speed. This is the energy converted from electrical to mechanical form, used to perform work, such as turning a fan or moving a conveyor.

In a DC motor, mechanical power can be calculated using the formula:\[P = \text{EMF} \times I_r\]where EMF is the induced electromotive force in the rotor, and \( I_r \) is the current through the rotor. This calculation helps us understand how much useful work the motor can do.

Efficient utilization of mechanical power is essential for minimizing energy losses and ensuring the motor operates cost-effectively.

Ohm's Law

Ohm's Law is a fundamental principle in electrical engineering that relates the voltage, current, and resistance in an electrical circuit. It states that the current (I) flowing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) in the path:\[I = \frac{V}{R}\]In our exercise, Ohm's Law was used to determine the current flowing through the field coils of the motor, calculated by dividing the applied voltage (120 V) by the resistance of the field coils (106 Ω).

Ohm's Law is a vital tool for analyzing electric circuits, especially for understanding the distribution of current in shunt-wound motors.

Parallel Circuits

Parallel circuits are electrical circuit configurations in which components are connected across common points, creating multiple paths for the current to travel. In a shunt-wound motor, this means that both the field windings and the rotor windings are connected directly across the voltage source.

Parallel circuits have several advantages:

• Constant voltage across all components.
• If one path fails, the other paths can continue to function.
• Individual components can be operated independently.

This configuration is beneficial in electric motors because it ensures that despite the changes in load, the speed remains consistent, making the motor highly reliable and efficient in handling various tasks.