Problem 50 - University Physics with Modern Physics

Question

A shunt-wound de motor with the field coils and rotor connected in parallel (Fig. 27.56 ) operates from a \(120-\mathrm{V}\) dc power linc. The resistance of the ficld windings, \(R_{f},\) is 218\(\Omega\) . The resistance of the rotor, \(R_{r}\) is 5.9 . When the motor is running, the rotor develops an emf \(\mathcal{E}\) . The motor draws a current of 4.82 A from the line. Friction losses amount to 45.0 W. Compute (a) the field current; \((b)\) the rotor current; \((c)\) the emf \(\mathcal{E} ;\) (d) the rate of development of thermal energy in the field windings; (e) the rate of development of thermal energy in the rotor; \((f)\) the power input to the motor; (g) the efficiency of the motor.

Step-by-Step Solution

Answer

(a) 0.5505 A, (b) 4.2695 A, (c) 94.85 V, (d) 66.1 W, (e) 107.5 W, (f) 578.4 W, (g) 76%.

Step 1: Calculate Field Current

The field current, \( I_f \), can be calculated using Ohm's Law for the field windings: \( I_f = \frac{V}{R_f} \). Substituting the given values: \( I_f = \frac{120}{218} \approx 0.5505 \text{ A} \).

Step 2: Calculate Rotor Current

The rotor current, \( I_r \), is the total current drawn from the line minus the field current: \( I_r = I - I_f \). With \( I = 4.82 \space A \) and \( I_f = 0.5505 \space A \), we find \( I_r = 4.82 - 0.5505 \approx 4.2695 \text{ A} \).

Step 3: Calculate Back Emf (\(\mathcal{E}\))

The back emf \( \mathcal{E} \) developed by the rotor is given by \( \mathcal{E} = V - I_r \cdot R_r \). Substituting, \( \mathcal{E} = 120 - 4.2695 \times 5.9 \approx 94.85 \text{ V} \).

Step 4: Calculate Thermal Energy Rate in Field Windings

The rate of thermal energy development in the field windings is given by \( P_f = I_f^2 \cdot R_f \). Substituting, \( P_f = (0.5505)^2 \times 218 \approx 66.1 \text{ W} \).

Step 5: Calculate Thermal Energy Rate in Rotor

The rate of thermal energy development in the rotor is \( P_r = I_r^2 \cdot R_r \). Substituting, \( P_r = (4.2695)^2 \times 5.9 \approx 107.5 \text{ W} \).

Step 6: Calculate Power Input to Motor

The power input to the motor, \( P_{in} \), is the product of the line voltage and current: \( P_{in} = V \cdot I \). Substituting, \( P_{in} = 120 \times 4.82 \approx 578.4 \text{ W} \).

Step 7: Calculate Motor Efficiency

Motor efficiency is the useful power output (back emf times rotor current minus friction losses) divided by the power input: \( \text{Efficiency} = \frac{\mathcal{E} \cdot I_r - \text{Friction Losses}}{P_{in}} \). Substituting, \( \text{Efficiency} = \frac{94.85 \times 4.2695 - 45}{578.4} \approx 0.76 \text{ or } 76\% \).

Key Concepts

Shunt-Wound DC Motor

A shunt-wound DC motor is a type of electric motor where the field coils and the rotor (or armature) are connected in parallel. This design allows for better control of the motor's speed and torque. One advantage of this setup is that the field current remains constant, which provides stable performance.
Since the field windings are connected across the supply voltage, the magnetic field remains relatively constant despite variations in the armature current. This feature makes shunt-wound motors suitable for applications requiring consistent speed under varying loads, such as in conveyor systems or fans.
Understanding the connection in parallel is crucial because it affects how the currents and voltages are distributed across the motor's components. Each part can thus operate optimally without interfering with the other.

Efficiency Calculation

Efficiency calculation in a motor is about gauging how well the motor converts electrical energy into mechanical energy. Knowing the efficiency helps us determine the motor's performance and operating costs.
To calculate efficiency, find out how much useful work the motor does compared to the energy put into it from the electrical supply. The formula is:

\[ \text{Efficiency} = \frac{\text{Output Power}}{\text{Input Power}} \times 100 \]

Output power is usually less than input power due to losses like friction and thermal energy. These losses must be accounted for when calculating efficiency.
In our solution, the motor's efficiency was determined by considering the power developed inside the motor after subtracting losses, like friction, from the electrical input.

Ohm's Law

Ohm's Law is one of the fundamental principles for understanding electrical circuits. It provides a relationship between voltage (V), current (I), and resistance (R) through the formula:

\[ V = I \times R \]

This law allows us to calculate one of these three variables if we know the other two.
In the context of our DC motor exercise, Ohm's Law helps us determine the current through the motor's components, like the field and rotor windings. By knowing the resistance and the voltage supplied to each parallel section, we computed the respective currents.
These currents are crucial for subsequent calculations, like finding the thermal energy loss or back electromotive force ( \(\mathcal{E}\) ) generated in the rotor.

Thermal Energy in Electrical Circuits

Thermal energy in electrical circuits often manifests as heat when electrical energy encounters resistance, according to Joule's Law. This can lead to energy losses in electric motors and needs management to ensure efficiency and longevity of the device.
The formula for calculating the thermal energy generated is:

\[ P = I^2 \times R \]

where \( P \) is the power (thermal energy per unit time) in watts, \( I \) is the current in amperes, and \( R \) is the resistance in ohms.
In motors, heat develops in both the field and rotor windings due to the resistance they offer to the flow of electric current. Properly calculating this thermal energy helps in designing cooling measures and assessing the total energy losses, which is vital for efficiency calculations.