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Inductors and Capacitors Together

As we learned earlier, a capacitor stores energy in an internal electric field, while an inductor stores energy in a surrounding magnetic field. When combined with a resistance in an electrical circuit, either component will lose energy as it forces current to flow through the resistor.

But what happens when we combine an inductor and a capacitor? We'll start with no resistor at all, and explore the theoretical behavior of the circuit. We'll insert resistance in the next page to see how that affects the behavior of the circuit. The results are very interesting, as you're about to see.

An ideal LC resonant circuit.

Consider the circuit shown to the right. The components are assumed to be ideal, so there is no resistance anywhere in the circuit. Also, we have charged the capacitor to some initial voltage. Since these are theoretically ideal components, this charge does not leak off, and will remain in place as long as switch S remains open.

When S is closed, the voltage across C is suddenly applied to L. At this moment, C begins to discharge itself through L. However, the current through L cannot change instantaneously, and the voltage across C cannot change instantaneously. These are the basic properties of these two components. Therefore, the current through L (and C) increases only gradually as the voltage across C (and L) decreases gradually.

The curve is not exponential as you might expect. Remember that there is no resistance present in this circuit, so there is no loss of energy. Instead, the rate at which C discharges itself through L actually increases, thus more rapidly increasing the current through L.

The voltage and current in C and L.

When C has been fully discharged, the current through L and C is at its maximum value. The magnetic field around L has been built up, and cannot be dissipated at once. Instead, it begins to collapse, as L takes on whatever voltage is necessary to continue the current flow at its current rate. As it does so, it recharges C, but with the opposite polarity. This continues eternally (as long as S remains closed). The green curve in the graph to the right shows the voltage across L and C, assuming S was closed at time t = 0.

When C is fully charged, the magnetic field is exhausted and the circuit current is once again zero. Now, C begins to discharge once again, driving current through L in the opposite direction. The red curve to the right shows the current flowing through L and C, beginning at the instant S is closed. The end result is a continuous oscillation, as shown in the graph.

Note that the voltage and current curves look the same, but peak at different times. This is the normal behavior of this type of circuit.

In fact, the waveforms for both voltage and current in this circuit are sine waves. Technically, the (green) curve for circuit voltage is a cosine wave. Both are true sine waves, but are 90° out of phase with each other.

This circuit operates by transferring energy back and forth between C and L. A circuit that does this is said to resonate, and the particular frequency at which this phenomenon occurs is known as the resonant frequency of the circuit. This resonant frequency is determined by the values of L and C, in accordance with the equation:

When studying this circuit and its behavior, keep in mind that all components here are theoretically ideal, having no resistance to absorb and dissipate energy. In the real world, this is impossible. (Although we have made strides towards room-temperature superconductors, we don't have perfect conductors at room temperature as of this writing.) The inductor especially, being made of many turns of relatively fine wire, has an inherent internal resistance. We can reduce that resistance by using heavier-gauge wire, but that increases the cost rapidly. It is much easier to allow for that resistance, and to deal with it as it is. That is the subject of the next page, on circuits containing resistance, inductance, and capacitance.

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