# How exactly do inductors work in an LC circuit?

What an interesting thread -- it's so hard to translate from brain to symbols and back again. I'll take a whack at it, mostly for my own edification. I am not a physicist; barely an EE on a good day.

If you hung a mass from a spring on the ceiling, pulled it down until the spring was fully extended, you wouldn't be surprised to see that the mass not only flies up until the spring is completely compressed, but then reverses direction and continues to oscillate. My theory is that you're asking for the same physical intuition about the LC circuit and how it reverses direction, for which we are not born with any natural intuition. Not being that smart, I try to think about things in the simplest possible terms, preferably basic DC if no calculus is really required.

If I hook a battery up to a capacitor, it initially looks like a short circuit. All the electrons go a runnin' towards one plate, but they can't cross the barrier, so a voltage (measured across the capacitor) starts to zoom upwards and the electrons start going slower and slower as the capacitor voltage gets closer and closer to being equal to that of the battery itself. Finally, the capacitor looks like an open circuit to the battery, or like a battery that has the same voltage as the battery that charged it. If I then short the terminals of the capacitor, it will send a current running in the reverse direction, first a whole lot of current, then less and less as the capacitor's voltage decreases because it is using up its electrons.

If I hook a battery up to a coil, it initially looks like a open circuit, 'cause the moving electrons are trying to cause a magnetic field to build up. As the field gets closer to its maximum value (associated with whatever current the battery is capable of supplying), less work is being done to increase it and the current goes faster, until eventually the coil looks like a short circuit, current is flowing as fast as the battery can supply it. Of course, if I then try to unhook the battery, the current that was supporting that magnetic field stops and the field starts collapsing, inducing a current in the reverse direction (and a big ol' spark that made many a child of the 50's think that electronics might be cool to learn).

So both the capacitor and the coil are capable of storing energy via electrons running in one direction, then releasing energy by causing electrons to run in the opposite direction. We've got the basic ingredients for oscillation.

You're proposing to charge a capacitor up to some voltage, then hook a coil to it. So at the beginning, the capacitor looks like a battery, the coil looks like an open circuit. Electrons start to flow slowly, the mag field of the coil starts to build, the voltage across the capacitor starts to drop slowly as its plate loses electrons. Electrons flow faster, because the coil is starting to look more like a short circuit, the mag field is building, and the voltage across the capacitor is dropping faster. Eventually, things reach their peak. The rising current from the capacitor can rise no more, the mag field can build no more, and when the current starts to decrease (as it must because the capacitor voltage keeps dropping), that means the mag field starts to collapse. That's where the first reversal comes in. The collapsing field supplies a reversal, and soon the electrons are flowing in the opposite direction, charging the capacitor with the opposite polarity that you originally charged it with. One half of the oscillation is complete and things are back roughly the starting point except the capacitor has an opposite charge, so it can supply its next reversal. Of course, the process symmetrically reverses itself, supplying the other half of the oscillation, putting things roughly back where it all started.

Just as the spring and mass oscillate by sloshing energy back and forth between two different storage mechanisms (the energy stored by elevating a mass, and the energy stored by extending a spring), the LC circuit also oscillates by sloshing energy back and forth between two different storage mechanisms: the voltage potential created by charging a capacitor, and the current potential created in a magnetic field by running a current through a coil.

Of course, real inductors and capacitors leak and contain resistances, the Second Law pertains, and things peter out over time rather than becoming a perpetual motion machine, but I imagine you were already clear about that.

• • Zack K, Mister T and Dale
Right after I posted this I finally read Ronburk's post. Except for details like battery vs capacitor and a mass vs a flywheel, we're pretty much saying the same thing.

The original question was about how the inductor works in an LC circuit. Consider what happens over the course of a complete cycle of the oscillation. Consider only an ideal capacitor and an ideal inductor, with no losses. Both the capacitor and the inductor are energy storage elements. The energy stored in the capacitor is E=(C*V^2)/2. Notice that it doesn't depend on the instantaneous current "through" (charging or discharging) the capacitor, only on the integral of the current (dv/dt = i/C). The energy stored in the inductor is E=(L*I^2)/2. Notice that it does not depend upon the instantaneous voltage across the inductor. only on the integral of the voltage (di/dt - v/L). The energy stored in the capacitor is in the electric field. The energy stored in the inductor is in the magnetic field. Electrical resonance involves continual cyclic transfer of energy back and forth between the inductor and the capacitor.

Let's say you have the inductor (I'll call it L) and capacitor (I'll call it C) in parallel, and initially you establish a DC current in L by connecting a current source across the L/C pair. To avoid complicating matters by storing charge in C, you start at zero current and slowly turn up the current so the rate of change of current in L (and thus the voltage across it) are negligible.

Now, instantaneously turn off the current source. L, being an ideal inductor, would maintain that constant current indefinitely if the voltage across it were held at zero. The current path is through C, however, so voltage is increasing across it and thus across L. The current direction in L still the same as it was with the current source operating, which means L is now drawing current from C, so the voltage is changing in the negative direction. This voltage across L causes an increasing negative rate of change in the current.

The moment the current crosses zero coincides with voltage being at negative peak, and thus the rate of change of current being at maximum. All the energy from L has been transferred into C. We have peak voltage and zero current. The oscillation has now completed its first quarter-cycle since you turned off the current source.

The negative voltage continues producing a negative rate of change in current through L (and C), di/dt = v/L. As the current (now negative in L) increases, the voltage (negative) is changing towards zero at an increasing rate, dv/dt=i/C. The moment the voltage crosses zero coincides with the current being at negative peak. All the energy that was stored in C has been transferred back into L. The second quarter-cycle has now completed.

The third and fourth quarter-cycles are just like the first two except for voltage and current polarity and direction.

If you observe the flywheel and hair-spring in a mechanical watch, the strain in the hair-spring is analogous to the current, which drives the acceleration of the flywheel. The velocity of the flywheel is analogous to voltage, which is at maximum when acceleration is passing through zero. Position of the flywheel, the integral of velocity, corresponds to strain in the hair-spring which again is analogous to current.

sophiecentaur
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