Mechanical oscillations, such as the pendulum, involve a constant exchange of energy between kinetic and potential forms. In the absence of energy dissipation from friction, a pendulum can continue to oscillate indefinitely. Similarly, in an electrical circuit where a capacitor and an inductor are connected in parallel, the initial energy of the system is transferred back and forth between electric and magnetic energy forms, resulting in electrical oscillations. When the parallel LC circuit reaches a state of resonance, the phenomenon is crucial for wireless radio communications technology, which relies on this principle. Here we explores the behavior and key parameters of electrical resonant circuits.

How ideal LC Circuit Oscillator Works

An oscillatory electrical circuit can be created by connecting an inductor L and a capacitor C in parallel, as shown in the figure below.

Assuming the capacitor initially contains a charge of q, the voltage V across the parallel network can be expressed in terms of the charge as,

\(q = C V_C = C v_{(max)}\)

The voltage across the charged capacitor reaches its maximum, v(max), and its associated electric field and stored energy are also at their maximum at time t = 0. At this point, the network current is still at zero, and the inductor is still perceived as an ideal wire by the capacitor charge. Despite this, the charges stored in the capacitor are compelled to move through the inductive wire due to the electric field. Once the charges(electrons) leaves the capacitor plate, this charge flow constitutes a change in current, causing the previously "ideal wire" to exhibit significant inductive properties and generate a magnetic field, as per the following Lenz's law.

\(v(t)=-L\frac{di(t)}{dt}\)

Therefore, the current rising through the inductor must adhere to Lenz's law and create a magnetic field that opposes the change that initiated it. When the capacitor is fully discharged, the current reaches its peak, \(i_{max}\), at t = T/4, and the entire energy of the LC system is now stored in the magnetic field of the inductor. From here, the inductor serves as the energy source in the circuit, pushing charges through the wire while gradually transferring the magnetic energy into the capacitive electrostatic energy. The constant flow of current continues to cause charges to accumulate at the opposite capacitor plate, generating an electric field in the opposite direction relative to the initial state. This process persists until the capacitor is fully charged again at t = T/2. This time, the voltage across the capacitor is at its minimum, v(min) = -v(max), and since the system is considered ideal, no thermal dissipation occurs in the wires, capacitor, or inductor. Therefore, energy conservation law must be upheld for a sustained, repetitive exchange of energy between the inductor and capacitor to occur.

#### Mathematics of ideal LC oscillator

The ideal LC circuit depicted in above circuit diagram generates an electrical current with a sinusoidal waveform when analyzed in the time domain. By applying the Kirchhoff's Voltage Law (KVL) equation around the loop, we can write this phenomenon in mathematics as follows,

\(v_C-v_L=0\)

or, \(\frac{q(t)}{C}-L\frac{di(t)}{dt}=0\)

and since, \(i(t)=\frac{q(t)}{dt}\), by differentiation, we have,

\(\frac{d^2i(t)}{dt^2}+\frac{1}{LC}i(t)=0\)

The solution of the above second order differential equation is,

\(i(t) = I_o sin(w_ot+\theta)\)

and the oscillating frequency is,

\(w_o = \frac{1}{\sqrt{LC}}\)

that is, \(f_o = \frac{1}{2\pi \sqrt{LC}}\)