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You can surmise by now that leading phase and lagging phase are different ways of looking at similar animals. In practice, ac sine waves are oscillating rapidly, sometimes thousands, millions, or even billions of times per second. If two waves have the same frequency and different phase, how do you know that one wave is really leading the other by some small part of a cycle, instead of lagging by a cycle and a fraction, or by a few hundred, thousand, million, or billion cycles and a fraction The answer lies in the real-life effects of the waves. Engineers and technicians think of phase differences, for sine waves having the same frequency, as always being between 0 and 180 degrees, either leading or lagging. It rarely matters, in practice, whether one wave started a few seconds earlier or later than the other. So, while you might think that the diagram of Fig. 12-9 shows wave X lagging wave Y by 270 degrees, or that the diagram of Fig. 12-10 shows wave X leading wave Y by 270 degrees, you would get an odd look from an engineer if you said so aloud. And if you said something like This wave is leading that one by 630 degrees, you might actually be laughed at. Note that if wave X (the dotted line in Fig. 12-10) is lagging wave Y (the solid line), then wave X is somewhat to the right of wave Y.

Inductive reactance produces a change in the way the current flows with respect to the voltage..

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The circular renditions of sine waves, such as are shown in the four drawings of Fig. 12-5, are well suited to showing phase relationships. If a sine wave X is leading a sine wave Y by some number of degrees, then the two waves can be drawn as vectors, with vector X being that number of degrees counterclockwise from vector Y. If wave X lags Y by some number of degrees, then X will be clockwise from Y by that amount. If two waves are in phase, their vectors overlap (line up). If they are in phase opposition, they point in exactly opposite directions. The drawings of Fig. 12-11 show four phase relationships between waves X and Y. At A, X is in phase with Y. At B, X leads Y by 90 degrees. At C, X and Y are 180 degrees opposite in phase; at D, X lags Y by 90 degrees. In all cases, you can think of the vectors rotating counterclockwise at the rate of f revolutions per second, if their frequency is f Hz.

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Inductive reactance is one of two form of reactance. (The other form, called capacitive reactance, will be discussed in the next chapter.) Reactance in general is symbolized by the italic uppercase letter X. Inductive reactance is symbolized XL. If the frequency of an ac source is given, in hertz, as f, and the inductance of a coil in henrys is given as L, then the inductive reactance in ohms, XL, is calculated as follows: XL = 2 f L In this formula, the symbol stands for the mathematical constant pi, which is the number of diameters around the circumference of a circle. It is equal to approximately 3.14. We can consider the value of 2 to be equal to 6.28 in most practical situations. Therefore, the preceding formula can be written a little more simply as: XL = 6.28f L This same formula applies if the frequency, f, is in kilohertz and the inductance, L, is in millihenrys. And it also applies if f is in megahertz and L is in microhenrys. Just remember that if frequency is in thousands, inductance must be in thousandths, and if frequency is in millions, inductance must be in millionths. Inductive reactance increases linearly with increasing ac frequency. This means that the function of XL versus f is a straight line when graphed. Inductive reactance also increases linearly with inductance. Therefore, the function of XL versus L also appears as a straight line on a graph. The value of XL is directly proportional to f, and is also directly proportional to L. These relationships are graphed, in relative form, in Fig. 13-4.

12-11 Vector representation of phase. At A, waves X and Y are in phase; at B, X leads Y by 90 degrees; at C, X and Y are 180 degrees out of phase; at D, X lags Y by 90 degrees.

Problem 13-1 Suppose a coil has an inductance of 0.500 H, and the frequency of the ac passing through it is 60.0 Hz. What is the inductive reactance

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