LC Oscillation (specially its waveform) is understood when we perform some mathematical operations on circuit like Laplace. However, it can be easily understood conceptually by applying basic Inductor and Capacitor equation which you might have read in high school.
Inductor:
It is an element which induces a reverse voltage in itself governed by the equation given below:
V = L(di/dt).
Thus, the voltage across it governs the rate at which current rises in the inductor and vice-verse. This is the most important equation and it can help you to solve any circuit involving the inductor.
Capacitor:
By virtue, it is an element which stores charge and hence develops the voltage across its terminal. Here, we have
Q = C.V or
V = (1/C)∫I.dt
Now lets discuss the LC Oscillations in detail.
Initially, assume all the switches are ideal and there is no charge in Capacitor and no magnetic field energy in Inductor.
When we close switch S1(S2 is still open), then the current in the inductor starts charging at the rate of Vdc/L (from eqn di/dt = V/L). Now, the time for which you keep the switch close, determines the maximum current and thereby helps in determining the voltage of the sine wave which you will see. You can design all this and compare the results practically in your lab and its fun.
Now, let say time is t1 for which S1 is closed. So,
Im = (Vdc/L)*t1
And also since there is conservation of energy, we can write
(1/2)L*Im^2 = (1/2)C*Vm^2, where Vm is the maximum voltage across the capacitor.
From these two equation you can determine the voltage and current peak value and its shape is generated as follows.
When S2 is closed and S1 is open, then since there is no charge across the capacitor, the voltage induced in the inductor is zero(0). Thus, the (di/dt) is also zero. This means that current remains at Im and this constant current(since there is no di/dt) flows through the capacitor, which results in its charging.
As the capacitor gets charged, a voltage is developed across it governed by the capacitor equation. Now this voltage acts across inductor and determines its (di/dt), since voltage is in opposite direction to that of current, it causes the current waveform to go down(or current to decrease at that rate).
As voltage gradually builds up, the rate increases higher and higher and at Vm, it becomes maximum. By that time current would have drop to zero and will not further charge up the capacitor as it was doing earlier. Since, there is nothing to charge capacitor, now capacitor will try to discharge via inductor.
Now, there is voltage across inductor, current will now flow as if we have connected a battery across it with the correct polarity. Thus, current will rise, but in negative direction and the rate will be governed by the voltage across the capacitor. As the capacitor discharges more and more, the rate of current rise in the negative direction will decrease, but it will keep rising. By the time current would have reached Im, capacitor would have discharged completely and completes half a cycle. Now, there is nothing which can supply energy to the inductor, so here again inductor will try to find a path to discharge its energy and the same loop will come again but in opposite direction.
Here is the actual current and voltage waveform. The yellow line indicates that S1 is open and S2 is closed. Although they might be shifted in space(as in pic), but they mark the same beginning, which means that there is a phase difference of 90 degree between current and voltage waveform.
LC Oscillator are widely used ranging from filters to the Converters, there exact working can give a good insight, how they work and at particular time, how they will behave, how much current will flow through them, how much voltage will they give. All these information might help you to design optimally.
Regards.
Amit Mishra
