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),
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; 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
, 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 , 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".
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.
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,
First published in Proc IEEE, vol 71, No. 6, June 1983, p772.