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Inductors and AC

The Circuit

An ac voltage applied to an inductor.

As you might expect, the behavior of an inductor with an applied ac voltage is almost exactly the opposite of the behavior of a capacitor. The circuit, as shown to the right, doesn't look much diffent. After all, we have merely substituted one schematic symbol for another. And we can be pretty sure that some current will flow through the inductor.

The circuit shown here would be a real problem for a dc source. Once the current had built up to its maximum value, there would be no change in current and therefore no inductive reaction to the applied voltage. The circuit current would be limited only by the resistance of the wire that makes up the inductor. Usually, that isn't much.

But an ac voltage source is constantly changing, so we can expect the inductor to be constantly reacting to it, but we don't yet know how. Of course, we can speculate about the likely form of the Ohm's Law expression for inductors and ac, the phase relationship between voltage and current, etc. But it would be far better to mathematically derive the appropriate expression, that can then be verified by experiment.



Deriving iL

To determine the actual relationship between voltage and current for any given inductor, we begin with the basic differential expression that describes the behavior of an inductance:

eL = L  diL

dt

As with the capacitor, the applied voltage is a sine wave, described mathematically as vpsin(ωt), where ω = 2πf. For the inductor, however, we must integrate in order to solve for iL. This is another aspect of calculus, and we will simply use that book of math tables again to determine the result:

eL = vpsin(ωt)  =  L  diL

dt
 
vpsin(ωt)dt  =  L  diL
 
vpsin(ωt)dt  =  L  diL
 
vp[-cos(ωt)]  =  L  iL

ω
 
-vpcos(ωt)  =  iL

ωL

This final expression very much resembles the Ohm's Law formula E/R = I. If we define an inductive reactance XL = ωL = 2πfL, we have our inductive counterpart to the capacitive expression we determined previously. Again, it's not really a resistance, but it is a result of the inductor's reaction to the applied ac voltage and changing current, so it can properly be called a reactance.



Voltage and Current

Voltage and current in an inductor.

The final equation above states that the current through the inductor, as a negative cosine wave, will lag the applied voltage by 90°, or ¼ cycle. This is the case, as shown in the graph to the right. This seems intuitively reasonable, since the basic reaction of any inductor is to oppose any change in current through itself.

As we indicated above, we can define inductive reactance according to the equation:

Inductive reactance vs. frequency.
XL = 2πfL = ωL

If we plot this as XL versus ωL using logarithmic scales as we did for capacitance, we get the graph shown to the right. This is very similar to the graph we got with capacitive reactance, but with the opposite slope. This makes sense, because XL is directly proportional to both frequency and the value of L, where XC is inversely proportional to both frequency and the value of C.

The effect of an inductance is in many ways exactly opposite to the effect of a capacitance. This becomes very important when these components are used together in an ac circuit. We'll see how and why on a later page, when we explore such circuits in more detail.


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