# 13.5: Summary

Power is the product of current and voltage, however, the phase relationship between the two has a major impact on the result. The most simple and straightforward case is when the current and voltage are in phase, meaning the load is a pure resistance. In this case, the true power can be computed directly as the product of RMS current and voltage. At the other extreme, when the voltage and current are 90 degrees out of phase, as in the case of a purely capacitive or inductive load, power is alternately generated and dissipated. That is, in a reactive load, true power dissipation is zero. A mechanical analogy is the storage and release of energy in an ideal spring as it is alternately compressed and then allowed to expand. In between these two extremes, that is, when the load is a complex impedance, the true power dissipation is somewhere between zero and the resistive maximum.

The power triangle is used to make visual sense of this situation. It is a right triangle comprised of three legs. The horizontal leg represents the true, or resistive, power and is denoted by the letter $$P$$. It is measured in watts. The vertical leg represents the so-called “reactive power”. It is denoted by the letter $$Q$$ and has units of VAR (volt-amps-reactive). $$Q$$ can be either inductive or capacitive. The third leg, the hypotenuse, is the apparent power, $$S$$. It is measured in VA (volt-amps). It is called apparent power because that is what the power appears to be if it is naively measured with a voltmeter and an ammeter, ignoring the phase difference between them. The angle between the real and apparent powers, $$\theta$$ (theta), is the phase angle between the voltage and current. In other words, theta is the impedance angle. Knowing theta, real power can be determined using right angle trigonometry, namely, $$P = S \cos \theta$$. The cosine of theta is also know as the power factor, $$PF$$. It ranges from 0 (purely reactive) to 1 (purely resistive). Positive or inductive impedance angles are said to have lagging power factor while negative or capacitive impedance angles produce leading power factor.

Ideally, loads are purely resistive and have a power factor of unity. If this is not the case, then for a given voltage, a higher current is needed to create the same real power as in the purely resistive case. This is not advantageous. Power factor correction is the process of shifting the power factor back to unity for complex loads. This is done by inserting a reactance of the opposite sign to counterbalance the reactive portion of the load, for example, adding capacitive reactance to a system that is inductive. The added reactive power must have the same magnitude as the original reactive power but be of the opposite sign, resulting in cancellation.

Efficiency is the measure of usable output power to applied power. Ideally, electromechanical systems such as motors would be 100% efficient, meaning that there is no power loss, but this is not a practical possibility. For example, there will always be frictional losses and power losses in wires.

## Review Questions

1. Define apparent power, real power and reactive power.

2. Describe the power triangle.

3. What is power factor, $$PF$$?

4. Describe the difference between leading and lagging power factor.

5. What is power factor correction?