## Inductance

Inductors seem to be very difficult components for many people to understand. Appendix B is devoted to explaining why inductors do what they do (induct). Here, I try to give an intuitive understanding of what happens when there is an inductor in a circuit.

The hydraulic analog for an inductor (Figure 3-7) is a paddle wheel with significant mass. When the pump applies pressure through the pipe, the paddle wheel's inertia resists the initial pressure, and pressure builds up behind it in the pipe. The pressure begins to overcome the inertia, and the paddle begins to turn. As long as there is pressure behind it, the paddle wheel will continue to speed up until it reaches an equilibrium with the flow.

**Figure 3-7. Hydraulic analog for an inductor.**

Most of us probably don't see many paddle wheels, so let's consider a perhaps more familiar illustration of inertia first. When your car is sitting in the driveway and won't start, think of what happens when you try to push it. It is very hard to get started, and you have to push it very hard to get it started. But once it starts to roll, it can be equally hard to stop. Some of us have had the unfortunate experience of having our car roll into another car or into the side of our garage before we could stop it again.

I had an opportunity once to open and close a huge (5,000 lb.) bank vault door. It was beautifully balanced on its hinges, and swung easily after I got it started. It took considerable effort to stop it again before it swung so far that it closed. It was a beautiful example of almost pure inertia.

So now think what happens when we turn on the pump (voltage). Pressure builds instantly against the vanes of the paddle wheel (inductor) but no water (current) flows. The pressure must "work" against the inertia for a moment before the wheel begins to turn. After a period of time, the wheel begins to turn and allow water (current) to flow. How long this takes is a function of how much force is applied and how much inertia there is in the paddle wheel. The force will continue to speed up the wheel until the wheel reaches an equilibrium speed such that all the water will flow unimpeded around the loop and there will be no pressure at all across the wheel.

Now, suppose we turn off the pump. The paddle wheel's inertia will cause it to continue to turn, forcing water to continue to flow around the loop. It is as though the paddle wheel takes over the pump's function. Reverse pressure will build up across the pump as water is forced to its "back" side and drawn away from its front. This will continue until the force that is built up is finally able to counteract the inertia of the paddle wheel, and the paddle wheel slows to a stop.

So it is with electrons. When we place a battery across an inductor, there is force (voltage) across the inductor, but in the first instant there is no current flow. Current does begin to flow, however, and after a moment the current will reach an equilibrium and flow unimpeded around the loop. At that point the voltage across the inductor has dropped to zero. How quickly the current begins to flow and how quickly it reaches equilibrium depend on how large the inductor is. It takes longer for this to happen with a larger inductor than it does with a smaller inductor.

The impedance to the initial flow of current has its source in the buildup of the magnetic field around the conductor. For a more extended discussion of this characteristic see Appendix B, "Why Inductors Induct." After the current stabilizes there is a stable magnetic field around the wire. Now when we remove the source of current, the magnetic field around the conductor begins to collapse, generating the continuing force to keep current flowing in the same direction. Electrons will continue to flow and build up on the far side of the inductor. As they build up, the charge (voltage) increases. When the voltage increases enough, it will repel the further flow of electrons. The force will finally balance against the "inertia" of the inductor. At that point, the flow will stop.

So, as with a capacitor, what happens to the relationship between voltage and current with an inductor is time dependent.
At the instant when voltage is first applied, voltage appears across the inductor but no current flows. Thus, voltage *leads* current, or current *lags* voltage through an inductor. In this sense, an inductor behaves exactly the opposite as a capacitor.

Also, in contrast to a capacitor, consider what happens when we pump water in one direction, then reverse the pump and pump it in the other direction. If we reverse the direction of the pump quickly, before the "inertia" can be overcome, very little water flows at all. On the other hand, if we pump for a longer time in one direction, then reverse the pump and pump in the other direction for a while, a reasonable flow will exist. A large inertia (inductor) will resist the flow more and for a longer time, and a smaller inertia will let the flow start more quickly.

Thus, the effect of the inductor is a function of both the size of the inductor and the frequency. Flow is small if the inductor is large or if the frequency is high; flow is larger if the inductor is smaller or the frequency is lower (see Table 3-2).

**Table 3-2. Inductor size and frequency both affect the flow of current.**

Suppose two inductors are connected in series (Figure 3-8). If L1 is much larger than L2 (i.e., the first paddle wheel is much larger than the other), the current (flow) cannot be larger than that allowed by the larger one. The second, smaller inductor will not have much effect, unless it is about the same size as the larger one. By the time the force can overcome the inertia of the larger one, the smaller one's inertia has already been overcome.

**Figure 3-8. Hydraulic analog of two inductors connected in series.**

As is the case with resistors, inductors in series add together. So the equivalent inductance, Leq, if two inductors are connected in series is given by Equation 3-11:

Note that if L2 is very much larger than L1, then the equation for Leq reduces to Equation 3-12:

If there are two inductors in parallel, however (Figure 3-9), the current will not be less than what the smaller one will allow. The smaller inductor's inertia is quickly overcome by the force, so current flows through it whether or not the inertia of the larger one has been overcome. If the other inductor is small enough, additional current can flow through it. Thus, parallel inductors combine into a single equivalent, Leq, in the same manner that parallel resistors do (see Equation 3-13):

**Figure 3-9. Hydraulic analog of two inductors connected in parallel.**

Note also that if one inductor, say L2, is very large, the equivalent parallel inductance simply reduces to Equation 3-14:

Note that inductors combine exactly as resistors do. Inductors and capacitors are opposites in this characteristic also.

The measure of inductance is the henry (Joseph Henry). Remember the measure of capacitance is the farad, and the voltage across
a capacitor will rise one volt when one amp flows onto the plates. A current flowing through an inductor may generate a voltage
across it, also. But from the preceding it is clear that, in the case of an inductor, we are not talking about a DC current.
A DC current does *not* generate voltage across an inductor just as a steady flow past the paddle wheel does *not* need a force to keep it going (once the paddle wheel is up to speed). Force builds up across an inductance when the current
flow through it is changing, working against the inertia of the inductance. One henry of inductance will cause one volt to
appear across it when the current through it is *changing* at the rate of one amp per second, or, more generally: