Can LC Oscillations Be Generated Using a DC Source?

[saubhik]

Homework Statement

Suppose we have a circuit with the inductor, uncharged capacitor, ideal battery with emf E all in series. At t=0, the circuit is switched on. The following takes place sequentially:
1. Current at t=0 is max. Battery charges capacitor, current decreases.
2. inductor senses change in current, sets up back emf, decreasing the rate of current decay from exponential decay to a sinusoidal decay.
3. capacitor gets charged to max i.e. C*E. Current falls to 0.
4. inductor's back emf vanishes as current vanishes.
5. electric field energy tends to transform to magnetic field energy. Thus, capacitor discharges to the inductor.
6. the back emf set by inductor again describes the rate of change of current growth in opposite direction.
7. charge on capacitor falls to 0 and current reaches its max value, this time in opposite.
8. capacitor gets charged again and the whole process gets repeated in opposite direction for the next half cycle and so on.

Homework Equations

from above analysis, current and charge have phase diff of 90. Both sinusoidal functions.
max charge is C*E and max current is (resonant angular frequency)*(max charge).

We can get these by solving the differential equation obtained using KVL.

The Attempt at a Solution

we have an alternating current from a d.c. voltage.
Please see if my analysis is correct! also please point out if i am lacking in finer details...
I didn't know we could get AC from a DC supply.
Also i didnt find any transient characteristics of this circuit? which circuits have transient solutions?

 

[gneill]

The inductor in the series circuit is not going to allow "Current at t=0 is max". So you've got a problem at step 1.

Have a look http://en.wikipedia.org/wiki/RLC_circuit" .

 

[saubhik]

yah...
that's right.
So the equations don't hold for t=0?
is that the DC transcience?

what then happens exactly at t=0; how to express this mathematically?

 

[gneill]

yah...
that's right.
So the equations don't hold for t=0?
is that the DC transcience?

what then happens exactly at t=0; how to express this mathematically?

You just need the right equations. Write out the differential equation for the circuit and solve for the "step response". It's a second order system, characterized by containing two different passive energy storage components (the capacitor and the inductor), so you get a second order differential equation, which means it'll have characteristics like damping factors and resonance. The web page I linked to in my post above has the details.

 

[rwisz]

Your analysis is correct. In this circuit, the inductor and capacitor act as a resonant circuit, with the inductor providing the back emf to counter the decrease in current and the capacitor storing the energy transferred between the electric and magnetic fields. This results in a sinusoidal oscillation of current and charge.

Regarding your question about getting AC from a DC supply, it is possible through the use of an inductor as it allows for the creation of a varying magnetic field, which in turn induces a varying voltage. This is known as an AC-DC converter.

As for the transient characteristics, this circuit does have a transient solution as it takes time for the current to reach its maximum and for the capacitor to fully charge. The transient solution can be found by solving the differential equation obtained using KVL.

Circuits with transient solutions are those that exhibit a time-dependent behavior, such as in this case where the current and charge vary over time. Other examples include RC circuits, RL circuits, and RLC circuits. These circuits have a transient solution as they take time to reach their steady-state behavior.