LC Oscillations: In electronics, you might have come across a rectifier circuit which is used to convert AC signal to DC signal. But what if we want to go the other way? Is it possible to convert DC signal to AC signal? The answer is yes. So basically, an oscillator circuit is a kind of circuit that is used to convert DC signal to AC signal. Using an oscillator circuit, we can generate waveforms like a sinusoidal wave, square wave, triangular wave, sawtooth wave, etc.

So, in this article, we are going to discuss one such oscillator circuit known as an LC Oscillator, and after going through this article, you will get a complete idea about how to do an LC Oscillator circuit works and some of its important equations.

What is LC Oscillation?

LC Oscillator uses a tank circuit (which includes an inductor and a capacitor) that gives required positive feedback to sustain oscillations in a circuit. As the name suggests, in this circuit, a charged capacitor \((C)\) is connected to an uncharged inductor \((L)\) as shown below;

The circuit shown above is an LC tank circuit. This circuit will contain a capacitor that is fully charged and an inductor that is completely de-energized. The resistance of this inductor must be as low as possible (ideally zero).
Now, think about it, if a charged capacitor is connected to a resistor, the energy from the capacitor will be consumed by the resistor, and the flow of current will eventually come to a halt.

But in this case, this capacitor (which stores electrical energy) is connected to an inductor (which stores magnetic energy) with very low resistance. Hence, as the inductor starts taking energy from the capacitor, it starts getting energized, and its energy increases, which in turn discharges the capacitor. Now, when the inductor gets fully energized, the capacitor loses all its energy, and now, the inductor will start charging the capacitor through the energy stored in it. The transfer of energy from the capacitor to the inductor and from the inductor to the capacitor goes on. This continuous transfer of energy from one device to another is what we call LC Oscillations.

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Equations of Energy Stored in a Capacitor and an Inductor

1. Energy Stored in a Capacitor
   The capacitor is a device that stores electric charge and energy. The amount of charge stored \((q)\) in a capacitor is directly proportional to the potential difference \((V)\) applied across its ends. Hence, the equation of charge stored in a capacitor will be:
   \(q = CV\)
   Here, \(C\) is known as the capacitance of the capacitor. Now, the change in electric potential energy is given by:
   \(dU = q(dV)\)
   \(dU = CVdV\)
   Integrating the equation;
   \(\mathop \smallint \nolimits_0^U dU = \mathop \smallint \nolimits_0^V CVdV\)
   \(\therefore \,U = \frac{1}{2}C{V^2} = \frac{1}{2}qV = \frac{{{q^2}}}{{2C}}\)
   These are the equations for energy stored in a capacitor.
   
2. Energy Stored in an Inductor
   The inductor is a device that on passing a current through it stores energy in a magnetic field. The potential difference \((V)\) across the ends of an inductor is directly proportional to the change in electric current with respect to time \(\left( {\frac{{di}}{{dt}}} \right)\).
   \(V = – L\frac{{di}}{{dt}}\)
   Here, \(L\) is known as the inductance of the inductor. Now, the change in potential energy stored in the inductor will be:
   \(dU = Vi\left( {dt} \right)\)
   \(dU = Li\left( {di} \right)\)
   Integrating the equation;
   \(\mathop \smallint \nolimits_0^U dU = \mathop \smallint \nolimits_0^i Lidi\)
   \(\therefore \,U = \frac{1}{2}L{i^2}\)
   This is the equation for energy stored in an inductor.

Working of an LC Oscillator

When a fully energized capacitor is connected to a de-energized inductor, all the energy of the whole circuit is with the capacitor only, and the energy of the inductor is zero. Let us denote energy stored in the capacitor (electrical energy) as \(U_E\) and energy stored in the inductor (magnetic energy) as \(U_B\). The diagram here depicts the current situation of the circuit.

Current starts flowing from capacitor to inductor, and the inductor starts getting energized and the capacitor discharged. The energy of the inductor starts increasing, and that of the capacitor starts decreasing. The diagram here depicts the current situation of the circuit. The bars below the circuit diagram show that at this point in time, half of the energy stored in an inductor is equal to that of a capacitor, which means the capacitor has transferred half of its energy to the inductor.

Now, as soon as the capacitor gets completely discharged, all the energy of the capacitor will be transferred to the inductor. Hence, the whole lot of electric energy is converted to magnetic energy. The third diagram here depicts the current situation of the circuit.

Since the capacitor is completely discharged and the inductor is completely energized, now the inductor will start charging the capacitor with the same direction of the current. The fourth diagram here depicts the current situation in the circuit. So now, the inductor has transferred half of its energy to the capacitor.

Finally, the capacitor will again get fully charged, and the inductor will get fully energized. But the difference now in the capacitor will be that its polarity is reversed. So, if the current starts flowing in the circuit from the capacitor again, it will flow in the opposite direction. As the current in the circuit now has an opposite flow of current, we can say that it has completed the first half of the AC cycle and started the second half.

Hence, while the whole cycle gets completed, both capacitor and inductor will get fully charged twice.

Differential Equation of LC Oscillations and its Solution

Applying Kirchhoff’s Voltage Law in the tank circuit, we can say that the summation of potential difference across capacitor and inductor will be equal to zero.
\({V_L} + {V_C} = 0\)
\( – L\frac{{di}}{{dt}} + \frac{q}{C} = 0\)
But, \(i=- \frac{dq}{dt}\) because the charge on the capacitor decreases with respect to time:
\(L\frac{{{d^2}q}}{{d{t^2}}} + \frac{q}{C} = 0\)
\(\therefore \,\frac{{{d^2}q}}{{d{t^2}}} = – \frac{q}{{LC}}\)
This equation is known as the differential equation for LC Oscillations. From this equation, we can write the angular frequency of LC oscillations as;
\(\omega = \frac{1}{{\sqrt {LC} }}\)
Therefore, its frequency will be:
\(f = \frac{1}{{2\pi \sqrt {LC} }}\)

The solution to the differential equation of LC Oscillation will be;
\(q = {q_m}\sin \;\left( {\omega t + \phi } \right)\)
Here, \(q_m\) is the maximum charge on the capacitor. Differentiating this equation with respect to time, we get the equation of current;
\(i = \frac{{dq}}{{dt}}\)
\(i = \frac{d}{{dt}}\left( {{q_m}\sin \;\left( {\omega t + \phi } \right)} \right)\)
\(\therefore \,i = {q_m}\omega \cos \;\left( {\omega t + \phi } \right)\)
But, at time \(t = 0\), the current flowing through the circuit will be zero. Hence,
\(\cos \;\left( \phi \right) = 0\)
\(\therefore \,\phi = \frac{\pi }{2}\)
Thus, the equation of charge will be;
\(q = {q_m}\sin \;\left( {\omega t + \frac{\pi }{2}} \right)\)
\(\therefore \,q = {q_m}\cos \;\left( {\omega t} \right)\)
The graph of charge vs. time on the capacitor will be;

Here, \(T\) is the time period of LC Oscillations.

Total Energy of LC Oscillations

The equation of charge in LC Oscillations is given by:
\(q = {q_m}\cos \;\left( {\omega t} \right)\)
Differentiating this equation, we get the equation of current:
\(i = \frac{{dq}}{{dt}}\)
\(\therefore \,i = – {q_m}\omega \sin \;\left( {\omega t} \right)\)
The energy stored in a capacitor is given by the formula:
\({U_E} = \frac{{{q^2}}}{{2C}}\)
Substituting the equation for a certain time \(t\);
\({U_E} = \frac{{{q_m}^2}}{{2C}}\left( {\omega t} \right)\)
Energy stored in an inductor is given by the formula:
\({U_B} = \frac{1}{2}L{i^2}\)

Substituting the equation for the same time as that of the capacitor;
\({U_B} = \frac{1}{2}L{q_m}^2{\omega ^2}\left( {\omega t} \right)\)
Since the angular frequency, \(\omega = \frac{1}{{\sqrt {LC} }}\)
\({U_B} = \frac{{{q_m}^2}}{{2C}}\left( {\omega t} \right)\)
Therefore, the total energy of the LC Oscillations will be;
\(U = {U_E} + {U_C}\)
\(U = \frac{{{q_m}^2}}{{2C}}\left( {\omega t} \right)\; + \frac{{{q_m}^2}}{{2C}}\left( {\omega t} \right)\)
\(\therefore \,U = \frac{{{q_m}^2}}{{2C}}\)

Here, the value of the maximum charge on the capacitor and its capacitance is constant. Hence, from this, we can say that the total energy of LC Oscillations always remains constant.

Applications of LC Oscillations

LC Oscillations are used in many electronic devices such as transmitters, radio devices, filters, frequency mixers, television, RF generators, etc.

1. Used to convert DC signal to AC signal
2. This resonance circuit can help in voltage amplification
3. It is used to generate a signal of some specific frequency by selecting the values of inductance and capacitance accordingly
4. They can be used in induction heating

There are many more uses of LC Oscillators in our daily life appliances, and these are just a few examples of them.

Some Other Types of Oscillators

Apart from the LC oscillator circuit, there are other oscillator circuits too. Some of them are as mentioned below;

1. Armstrong Oscillator
2. Clapp Oscillator
3. Crystal Oscillator
4. Hartley Oscillator
5. Colpitts Oscillator
6. RC Phase shift Oscillator, etc.

Summary

Summarizing the whole article, LC Oscillations are caused by LC Oscillator circuits also known as tank circuit which includes a capacitor and inductor. Energy transfer keeps on taking place between capacitor and inductor, but still, the total energy of LC oscillations remains constant (ideally if the inductor has zero resistance). Due to the continuous transfer of energy from the capacitor to the inductor and from the inductor to the capacitor, the direction of current eventually changes according to the polarity of charging and discharging and hence generating an alternating signal.

Frequently Asked Questions (FAQs) on LC Oscillations

Q.1. What happens in LC Oscillations?
Ans: Continuous transfer of energy from capacitor \((C)\) to the inductor \((L)\) generates LC Oscillations. The capacitor is initially fully charged, then discharged, then charged again, and the process goes on.

Q.2. Why do LC circuits oscillate?
Ans: When a charged capacitor is connected to a de-energized inductor, the capacitor will start charging the inductor while discharging itself, and the same goes with the inductor when it is fully energized, and the capacitor is completely discharged. Hence, the short oscillation of current takes place here.

Q.3. Why are LC oscillations non-realistic?
Ans: In real life, the inductor and capacitor will have some amount of resistance in them because it is obvious that they are non-ideal. Due to this, in every cycle of LC oscillation, some amount of energy is lost in the resistance, and it can’t continue to go on forever.

Q.4. What is the frequency of LC oscillation?
Ans: The frequency of LC oscillation is given by \(f = \frac{1}{{2\pi \sqrt {LC} }}\). Hence, it depends on the values of capacitance \((C)\) and inductance \((L)\).

Q.5. What is the use of an LC oscillator?
Ans: LC oscillators are mostly used to generate a particular frequency, and apart from this, it is used in many other ways too.

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