Problem 101-1    Source:   Adapted from example 1-10 - page 43 - CHAPMAN, Stephen J. -    Book: Fundamentals of Electric Machines - 5th edition - Ed. McGrawHill - 2013.

The linear machine shown in Figure 101-01.1 is supplied with a voltage V = 120 V and a resistance internal equal to 0.3 Ω. The magnetic flux density is 0.1 T with orientation entering the page. The length of the bar is L = 10 m. Find:

a) The maximum starting current of the machine.

b) The steady state speed without load.

c) The steady state speed if a force of 30 N pointing to the right were applied to the bar.

d) The power that the bar would be producing or consuming.

e) The power that the voltage source would be producing or consuming.

f) Explain the difference between these last two numerical values. Does this machine work as a motor or as a generator?

Solution of the Problem 101-1   

Item a

Before the machine starts the bar is at rest, therefore v = 0 m/s. In this case, from eq. 101-02, we conclude that ε_ind = 0 volts. So, using the eq. 101-05, we get:

Item b

We know that when the bar reaches steady state the net force on it is null. When this occurs, the voltage induced in the bus will be equal to the source voltage, reducing the current flowing through the circuit to zero. So, as ε_ind = V = 120 volts and using eq. 101-02, we get:

Item c

Applying a force of 30 N pointing to the right on the bar, the steady state will be reached when the force (F_ind) is equal and opposite to the applied force. Thus, the total force acting on the bar is null. So using the eq. 101-01, we get:

In this case, the current is flowing up the bar because the induced force is pointing to the left. So, using the right-hand rule, we conclude that the current flows upward. Now we need to calculate the voltage induced in the bar. She is given by:

With these data we can calculate the displacement speed of the bar, or:

Item d

As the current flows up the bar, i.e. the current leaves through the positive of the induced voltage, then the bar is producing power of the order of:

Note the negative sign of the power, indicating that the bar is supplying power to the circuit, according to the adopted convention.

Item e

In the voltage source that feeds the circuit, the current enters the positive pole of the source. In this way, the source is consuming power supplied by the bus. Its value is:

Item f

Note that there is a difference between the power produced by the bus and the power consumed by the source. This is due to the presence of the R resistor that dissipates a power of:

We know from the energy conservation theorem that the algebraic sum of the powers must be equal to zero. So:

We must point out that as the bar is producing power, then this machine is operating as a generator.