Introduction
In dc circuits, resistance is a simple thing. It can be expressed as a number ranging from zero (a perfect conductor) to extremely large values, increasing without limit through thousands, millions, and billions of ohms. Physicists call resistance a scalar quantity because it can be expressed on a one-dimensional scale. In fact, dc resistance can be represented along a half-line (also called a ray ).
Given a certain dc voltage, the current decreases as the resistance increases in accordance with Ohm’s law. The same law holds for ac through a resistance if the ac voltage and current are both specified as peak, pk-pk, or rms values.
Inductors And Dc
Suppose that you have some wire that conducts electricity very well. If you wind a length of the wire into a coil and connect it to a source of dc, the wire draws a small amount of current at first, but the current quickly becomes large, possibly blowing a fuse or overstressing a battery. It does not matter whether the wire is a single-turn loop, lying haphazardly on the floor, or wrapped around a stick. The current is large. In amperes, it is equal to I = E/R , where I is the current, E is the dc voltage, and R is the resistance of the wire (a low resistance).
You can make an electromagnet by passing dc through a coil wound around an iron rod. However, there is still a large, constant current in the coil. In a practical electromagnet, the coil heats up as energy is dissipated in the resistance of the wire; not all the electrical energy goes into the magnetic field. If the voltage of the battery or power supply is increased, the wire in the coil, iron core or not, gets hotter. Ultimately, if the supply can deliver the necessary current, the wire will melt.
Inductors And AC
Suppose that you change the voltage source connected across a coil from dc to ac. Imagine that you can vary the frequency of the ac from a few hertz to hundreds of hertz, then kilohertz, and then megahertz.
At first, the ac will be high, just as is the case with dc. However, the coil has a certain amount of inductance, and it takes a little time for current to establish itself in the coil. Depending on how many turns there are and on whether the core is air or a ferromagnetic material, you’ll reach a point, as the ac frequency increases, when the coil starts to get sluggish. That is, the current won’t have time to get established in the coil before the polarity reverses. At high ac frequencies, the current through the coil has difficulty following the voltage placed across the coil. Just as the coil starts to “think” that it’s making a good short circuit, the ac voltage wave passes its peak, goes back to zero, and then tries to pull the electrons the other way. This sluggishness in a coil for ac is, in effect, similar to dc resistance. As the frequency is raised, the effect gets more pronounced. Eventually, if you keep on increasing the frequency of the ac source, the coil does not even come near establishing a current with each cycle. It then acts like a large resistance. Hardly any ac current flows through it.
The opposition that the coil offers to ac is called inductive reactance . It, like resistance, is measured in ohms (Ω). It can vary just as resistance does, from near zero (a short piece of wire) to a few ohms (a small coil) to kilohms or megohms (bigger and bigger coils or coils with ferromagnetic cores at high frequencies). Inductive reactance can be depicted on a ray, just like resistance, as shown in Fig. 15-3.
Fig. 15-3 . Inductive reactance can be represented on half-line or ray. There is no limit to how large it can get, but it can never be negative.
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.
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.
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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From Physics Demystified: A Self-Teaching Guide. Copyright © 2002 by The McGraw-Hill Companies, Inc. All Rights Reserved.