connection between the conductors, resulting in very large current values with correspondingly high arc temperatures.

Transformer impedance is also useful for calculating the degree to which the output voltage of a power transformer will “sag” below its ideal value when powering a load. Suppose we had a power transformer with a 5:1 turns ratio, designed to receive 120 VAC at its primary winding and output 24 VAC. Under no-load conditions the transformer’s internal impedance will be of no e ect, and the transformer will output 24 VAC exactly. However, when a load is connected to the secondary terminals and current begins to flow to power this load, the transformer’s internal impedance will result in the secondary voltage decreasing by a small amount. For example, if this transformer happens to have an impedance of 5.5%, it means the secondary (output) voltage will sag 5.5% below 24 VAC at full load, assuming the primary voltage is maintained at the standard 120 VAC level. 5.5% of 24 volts is 1.32 volts, and so this transformer’s secondary voltage will “sag” from 24 volts down to 22.68 volts (i.e. 1.32 volts less than 24 volts) as load current increases from zero to its full rated value.

5.5Phasors

Phasors are to AC circuit quantities as polarity is to DC circuit quantities: a way to express the “directions” of voltage and current waveforms. As such, it is di cult to analyze AC circuits in depth without using this form of mathematical expression. Phasors are based on the concept of complex numbers: combinations of “real” and “imaginary” quantities. The purpose of this section is to explore how complex numbers relate to sinusoidal waveforms, and show some of the mathematical symmetry and beauty of this approach.

Since waveforms are not limited to alternating current electrical circuits, phasors have applications reaching far beyond the scope of this chapter.

5.5.1Circles, sine waves, and cosine waves

Something every beginning trigonometry student learns (or should learn) is how sine and cosine waves may be derived from a circle. First, sketch a circle, then sketch a set of radius vectors from the circle’s center to the circle’s perimeter at regular angle intervals. Mark each point of intersection between a vector and the circle’s perimeter with a dot and label each with the vector angle. Sketch rectangular graphs to the right and below the circle, with regularly-spaced divisions. Label those divisions with the same angles as the vectors and then sketch dashed “projection” lines from each vector tip to the respective divisions on the rectangular graphs, marking each intersection with a dot. Connect the dots with curves to reveal sinusoidal waveshapes.

The following illustration shows the dots and projection lines for the first five vectors (angles 0 through π2 radians10) only. As you can see, the circle’s vertical projection forms a sine wave, while the circle’s horizontal projection forms a cosine wave:

π2 /2π3

10A full circle contains 360 degrees, which is equal to 2π radians. One “radian” is defined as the angle encompassing an arc-segment of a circle’s circumference equal in length to its radius, hence the name “radian”. Since the circumference of a circle is 2π times as long as its radius, there are 2π radians’ worth of rotation in a circle. Thus, while the “degree” is an arbitrary unit of angle measurement, the “radian” is a more natural unit of measurement because it is defined by the circle’s own radius.