Reactance and Frequency

Inductive reactance is one of two form of reactance. (The other form, called capacitive reactance, will be discussed in the next section.) 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 = 2πf L

In this formula, the symbol π 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 2π 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 following figure.

Problem 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?
Using the preceding formula, calculate XL = 6.28 × 60.0 × 0.500 = 188 Ω. This is rounded to three significant figures.

Problem 2

What will be the inductive reactance of the preceding coil if the supply is a battery that supplies pure dc?
Because dc has a frequency of zero, XL = 6.28 × 0 × 0.500 = 0 Ω. That is, there will be no inductive reactance. Inductance doesn’t have any practical effect with pure dc.

Problem 3

If a coil has an inductive reactance of 100 Ω at a frequency of 5.00 MHz, what is its inductance?
In this case, you need to plug numbers into the formula and solve for the unknown L. Start out with the equation 100 = 6.28 × 5.00 × L = 31.4 × L. Because the frequency is given in megahertz, the inductance will come out in microhenrys. You can divide both sides of the equation by 31.4, getting L = 100/31.4 = 3.18 μH.