Homework # 9 EE517 Fall 2000 Due November 9, 2000

In Woodson & Melcher read chapter 4.

Problem 1

This problem is intended to help you understand why three phase power distribution is used. To do this, calculate the volume of copper wire required to distribute power using two approaches. In the first approach use three single-phase circuits connected in parallel as shown in Fig. 1. This is the same as one circuit with a pair of wires having a cross sectional area equal to three times the cross sectional area of one of the pair of the wires in Fig. 1.

In the second approach use the three-phase circuit shown in Fig. 2.

Assume the distance from the source to the load is 1000 m (meters) in all cases and that the cross sectional area of the wire required to carry 10 A is 2 cm 2. The three-phase voltages are 120 degrees apart.

- For the first approach what are i
_{a}, i_{b}, i_{c}, i_{na}, i_{nb}, and i_{nc}? Plot i_{na}as a function of time. - For the first approach, what is the total power delivered to the loads?
- For the first approach what is the total volume of the copper required to deliver the total power to the loads?
- For the second approach what are i
_{a}, i_{b}, i_{c}? Show they sum to zero so i_{n}is zero. - For the second approach what is the total power delivered to the loads?
- For the second approach what is the total volume of the copper required to deliver the total power to the loads?

Problem 2

The fluxes linked by the rotor and stator windings of a 2-pole 3-phase synchronous machine are

- Write down the expression for conservation of co-energy in terms of l
_{a}, i_{a}, l_{b}, i_{b}, l_{c}, i_{c}, l_{f}, i_{f}, Te, and q. You do not have to substitute in the above expressions for the fluxes. - Write down the values of i
_{a}, i_{b}, i_{c}, i_{f}, Te, and q as well as di_{a}, di_{b}, di_{c}, di_{f}, and dq for each path segment used to compute the co-energy. - Compute the machine's co-energy.
- What is the torque as a function of i
_{a}, i_{b}, i_{c}, i_{f}, and q? - What is the torque produced at the motor's shaft? What is the average output shaft power? What is the advantage of the three-phase machine compared to the single-phase machine. Remember that . Similarly
- What is the voltage at the terminals of phase a and what is the relationship between w
_{e}and w_{m}? Recall that i_{a}+ i_{b}+ i_{c}= 0 so i_{b}+ i_{c}= -i_{a}. - Draw the equivalent circuit for phase a. This is called the machine's single-phase equivalent circuit.
- What is the average power into your equivalent circuit and how does it compare to the average power out of the motor's shaft.

For what follows let i_{a} = I cos(w_{e}t), i_{b} = I cos(w_{e}t-2p/3), i_{c} = I cos(w_{e}t-2p/3), q = w_{m}t + g and i_{rf} = I_{f} = constant. Note the motor is driven by a current source and siusoidal steady state has been achieved.

Problem 3

A single phase synchronous motor with parameters, armature inductance 15 mH and mutual inductance 300mH has a field current of 1.2 A and is connected to standard 110Vrms 60Hz outlet power (driven by a voltage source). It is operating in sinusoidal steady state. It drives a fan with a torque speed curve given by

N-M

This equation gives the average torque the fan needs.

- Plot the fan torque speed curve form zero rpm to the operating speed of the motor. Numerically label both the x and y axis.
- What is the machine's speed in rpm?
- What is the average torque the machine must supply and what is the average mechanical power out of the machine? Remember that the average motor torque equals the average load torque.
- Draw the equivalent circuit for the motor labeling all of the component values.
- What is the angle between the motor's back emf and the applied voltage d (a number)? Recall that the average torque produced by a single-phase synchronous machine is .
- A radio shack true rms Amp meter measures the current into the machine. What does it measure (a number)?
- What is the complex power into the machine in terms of variables? What is the average power into the machine in terms of variables? What is the numerical value of the average power into the machine?
- Using your equivalent circuit in part c) draw the phasor diagram for Kirchoff's voltage law using the input voltage as the reference and using the armature current as the reference. Two phasor diagrams are required. Note E
_{f}has angle d with respect to the input voltage and an angle g with respect to the armature current. - From your result in h) what is the relationship between d and g?