Question

A shunt-wound DC motor with the field coils and rotor connected in parallel (see the figure) operates from a 135 V DC power line. The resistance of the field windings, $R_f$ , is 248Ω. The resistance of the rotor, $R_r$ , is 4.60Ω. When the motor is running, the rotor develops an emf ε. The motor draws a current of 4.94 A from the line. Friction losses amount to 47.0 W.

A) Compute the field current $I_f$ .

$I_f$ = ? A

B) Compute the rotor current $I_r$ .

$I_r$ = ? A

C) Compute the emf .

= ? V

D) Compute the rate of development of thermal energy in the field windings.

$P_th,windings$ = ? W

E) Compute the rate $P_th,rotor$ of development of thermal energy in the rotor.

$P_th,rotor$ = ? W

F) Compute the power input to the motor $P_in$ .

$P_in$ = ? W

G) Compute the efficiency of the motor.

Answer 1

Concepts and reason

The concepts used to solve this problem are Ohm’s law, current, potential difference, emf, power, and efficiency of the motor.

Initially, use Ohm’s law to find the field current.

Use the relationship between the total current in the circuit and the field current to find the rotor current.

Use the relationship between the field potential difference, rotor current, and resistance of the rotor to find the emf developed by the rotor.

Use the relationship between the field resistance and field current to find the rate of development of thermal energy in the field windings.

Use the relationship between the rotor resistance and rotor current to find the rate of development of thermal energy in the rotor.

Use the relationship between the field potential difference and total current in the circuit to find the power input of the motor.

Use the relationship between the power input of the motor, the rate of development of thermal energy in the fields, the rate of development of thermal energy in the rotors, and friction loss to find the power output of the motor.

Finally, use the relationship between the power input of the motor and power output of the motor to find the efficiency of the motor.

Fundamentals

Ohm’s law states, “at constant temperature, the steady current flowing through the conductor is directly proportional to the potential difference developed across the conductor”.

The expression for Ohm’s law is as follows:

Here, the current is , the potential difference is , and the resistance is .

The expression for the potential difference is as follows:

Here, the emf is .

The expression for the power in terms of current and resistance is as follows:

Here, the power is .

The expression for the input power is as follows:

Here, the total current in the circuit is .

The expression for the efficiency is as follows:

Here, the output power is , the input power is , and the efficiency is .

(A).

The expression for Ohm’s law is as follows:

Here, the field current is , the field potential difference is , and the field resistance is .

Substitute for and for .

(B).

The expression for the rotor current is as follows:

Here, the rotor current is and the total current in the system is .

Substitute for and for .

(C).

The expression for the potential difference is as follows:

Here, the resistance of rotor resistance is .

Substitute for , for , and for .

Rearrange the above equation to get .

(D).

The expression for the rate of development of thermal energy in the field windings is as follows:

Here, the rate of development of thermal energy in the field windings is .

Substitute for and for .

(E).

The expression for the rate of development of thermal energy in the field rotor is as follows:

Here, the rate of development of thermal energy in the field rotor is .

Substitute for and for .

(F)

The expression for the power input of the motor is as follows:

Here, the power input of the motor is .

Substitute for and for .

(G).

The expression for the power output of the motor is as follows:

Here, the power output of the motor is and the friction loss is .

Substitute for , for , for , and for .

The expression for the efficiency of the motor is as follows:

Substitute for and for .

Ans: Part A

The field current is .

Part B

The rotor current is .

Part C

The emf developed by the rotor is .

Part D

The rate of development of thermal energy in the field windings is .

Part E

The rate of development of thermal energy in the field rotor is .

Part F

The power input of the motor is .

Part G

The efficiency of the motor is .