The L-C Oscillator Circuit.

When a charged capacitor is connected to an inductor, the conventional analysis is to equate the voltage across the capacitor with the voltage across the inductor

.

Differentiating, we get

This is then recognised as having as a solution simple harmonic motion (SHM),

,

where

.

The traditional analysis assumes that when current is switched into the inductor, it appears instantaneously at all points in the inductor; the use of the single, lumped quantity L implies this. Similarly, it is assumed that the electric charge density at all points in the capacitor is the same[1]; that there are no transient effects such that the charge density is greater in certain regions of the capacitor plates.

Work on high-speed logic systems led to a reappraisal of the conventional analysis, particularly insofar as it bears on the choice of type and value of decoupling capacitor for logic power supplies.

In Figure 54, consider a capacitor (or open-circuit transmission line) which is connected to a single-turn inductor (or short-circuited transmission line). The initial state is that the capacitor was charged to a voltage v and then connected to the inductor by closing the switches.

Figure 55 shows the coefficients which apply when signals reflect or pass through discontinuities in the circuit.

If at a certain time the signal in the capacitor PQ has an amplitude  and the signal in the inductor QR is y, then the sequence in Figure 56 will occur.

, coming from the left to Q, breaks up into a reflected signal

and a forward signal

because the two relevant coefficients are . At the same time, the signal y, coming from the right towards Q, breaks up into a forward going  and a reflecting , because the relevant coefficients are   , travelling to the left from Q, combines with the leftwards travelling  .
When they reach the open circuit at P, where the reflection coefficient is +1, they reflect back towards Q.
The value of this signal is now , the next in the sequence, and we have calculated it to equal .
Similar arguments explain all other amplitudes in the sequence.

The bottom line in the sequence gives us the value of   in terms of y and .
If we add  to this value of , we get

.

.

But the middle of the sequence tells us that

.

Therefore,

.

, etc., is a sequence of amplitudes seen in the capacitor, and they obey the above formula. Now since

we can see that one possibility is that the sequence in  represents a series of steps which approximate to a sine wave.

The conclusion is that one waveform which can be supported by an L - C circuit is a sine wave, where the C is an open-circuit transmission line and the L is a short-circuited transmission line. The larger the value of  [2], the smaller is the forward flow of current each time across the central node at Q between the C and the L. This means that there is more time between maxima in the voltage level in C, and a lower "resonant frequency"[3].

[1]
Bleaney B.I. and Bleaney, Electricity and magnetism, 2nd Edn., pub. Oxford, Clarendon, 1965, p258.
Fewkes J.H. and Yarwood, Electricity and Magnetism vol.1, pub. University Tutorial Press, London, 1956, p505.

[2]
i.e., the bigger the discrepancy between  and , or to put it another way, the more capacitive the capacitor and/or the more inductive the inductor,

[3]
First published in Proc IEEE, vol 71, No. 6, June 1983, p772.