The aim of this question is to help you become familiar with phasor diagrams, and in particular to see how the diagram changes when one of the system parameters (in this case resistance) varies. The diagram (right) represents the approximate equivalent circuit of an induction motor (copper and mechanical losses are ignored, and the permeability of magnetic core is considered infinite), supplied from a constant- voltage source V of angular frequency ω. The power in the fictitious resistance R represents the power converted from electrical to mechanical form, and the value of R is a function of the speed of the motor. Sketch phasor diagrams showing the current and the voltages across R and L when a) R = 0; b) R = 0.1ωL; c) R = ωL; d) R = 10ωL. (Use the same scale for each diagram; calculate angles; and use a protractor or drawing package to produce a set of decent diagrams so that they can be compared easily.

1. You may find it useful to recall the geometrical property of a semi-circle shown in the sketch.

2. Find the value of R (in terms of ωL) that yields maximum power, and the corresponding power.