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Inductive Reactance Help (page 2)

By — McGraw-Hill Professional
Updated on Sep 9, 2011

Reactance And Frequency

Inductive reactance is one of two kinds of reactance. (The other kind will be dealt with in a little while.) In mathematical expressions, reactance is symbolized X . Inductive reactance is denoted X L .

If the frequency of an ac source is f (in hertz) and the inductance of a coil is L (in henrys), then the inductive reactance X L (in ohms) is

X L = 2π fL ≈ 6.2832 fL

This same formula applies if the frequency f is in kilohertz and the inductance L is in millihenrys. It also applies if f is in megahertz and L is in microhenrys. 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 X L versus f is a straight line when graphed. Inductive reactance also increases linearly with inductance. Therefore, the function of X L versus L also appears as a straight line on a graph. The value of X L is directly proportional to f ; X L is also directly proportional to L . These relationships are graphed, in relative form, in Fig. 15-4.

More About Alternating Current Inductive Reactance Points In TheRl Quarter-plane

 

Fig. 15-4 . Inductive reactance is directly proportional to inductance, as well as to frequency.

Reactance And Frequency Practice Problem

Problem

An inductor has a value of 10.0 mH. What is the inductive reactance at a frequency of 100 kHz?

Solution

We are dealing in millihenrys (thousandths of henrys) and kilohertz (thousands of hertz), so we can apply the preceding formula directly. Using 6.2832 to represent an approximation of 2π, we get

 

X L = 6.2832 × 100 × 10.0 = 6283.2 ohms

Because our input data are given to only three significant figures, we must round this to 6,280 ohms or 6.28 kilohms (kΩ).

Points In The Rl Quarter-plane

In a circuit containing both resistance and inductance, the characteristics become two-dimensional. You can orient the resistance and reactance half-lines perpendicular to each other to make a quarter-plane coordinate system, as shown in Fig. 15-5. Resistance is shown on the horizontal axis, and inductive reactance is plotted vertically, going upward.

Each point on the RL quarter-plane corresponds to a unique complex-number impedance . Conversely, each complex-number impedance value corresponds to a unique point on the quarter-plane. Impedances on the RL quarter-plane are written in the form R + jX L , where R is the resistance in ohms, X L is the inductive reactance in ohms, and j is the unit imaginary number, that is, the positive square root of –1. The value j in this application is known as the j operator . (If you’re uncomfortable with imaginary and complex numbers, go back and review that material in Chap. 1.)

Suppose that you have a pure resistance, say, R = 5 ohms. Then the complex-number impedance is 5 + j 0 and is at the point (5, 0) on the RL quarter-plane. If you have a pure inductive reactance, such as X L = 3 ohms, then the complex-number impedance is 0 + j 3 and is at the point (0, 3) on the RL quarter-plane. These points, and a few others, are shown in Fig. 15-5.

 

More About Alternating Current Inductive Reactance Points In TheRl Quarter-plane

Fig. 15-5 . The RL impedance quarter-plane showing five points for specific complex-number impedances.

In the real world, all coil inductors have some resistance because no wire is a perfect conductor. All resistors have at least a tiny bit of inductive reactance because they have wire leads at each end and they have a measurable physical length. There is no such thing as a mathematically perfect pure resistance like 5 + j 0 or a mathematically perfect pure reactance like 0 + j 3. Sometimes you can get close to such ideals, but absolutely pure resistances or reactances never exist (except occasionally in quiz and test problems, of course!).

In electronic circuits, resistance and inductive reactance are sometimes both incorporated deliberately. Then you get impedances values such as 2 + j 3 or 4 + j 1.5.

Remember that the values for X L are reactances (expressed in ohms) and not inductances (which are expressed in henrys). Reactances vary with the frequency in an RL circuit. Changing the frequency has the effect of making the points move in the RL quarter-plane. They move vertically, going upward as the ac frequency increases and downward as the ac frequency decreases. If the ac frequency goes down to zero, thereby resulting in dc, the inductive reactance vanishes. Then X L = 0, and the point is along the resistance axis of the RL quarter-plane.

Practice problems of these concepts can be found at: Alternating Current Practice Test

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