LC circuits, comprising of inductors (L) and capacitors (C), are key components in electrical and electronic systems. These circuits are characterized by their ability to oscillate and resonate, storing and exchanging energy between the inductor’s magnetic field and the capacitor’s electric field. LC circuits are used in various applications, including filters, oscillators, and tuned circuits.

LC circuits can be classified into two types:

- Series LC Circuit: The inductor and capacitor are connected in series, and the total impedance of the circuit is the sum of the individual impedances.
- Parallel LC Circuit: The inductor and capacitor are connected in parallel, and the total admittance of the circuit is the sum of the individual admittances.

Resonance:

At a specific frequency called the resonant frequency (f_r), the reactive components of an LC circuit cancel each other out, resulting in a purely resistive impedance (in a series LC circuit) or a purely conductive admittance (in a parallel LC circuit). The resonant frequency is determined by the values of the inductor and capacitor:

f_r = 1 / (2 * π * √(L * C))

Applications:

- Filters: LC circuits can be used as band-pass or band-stop filters, allowing specific frequencies to pass through while attenuating others. In a band-pass filter configuration, the output is taken across the LC circuit, while in a band-stop filter configuration, the output is taken in series or parallel with the LC circuit.
- Oscillators: LC circuits can be combined with active components, such as transistors or operational amplifiers, to create oscillators that generate continuous, periodic waveforms. These oscillators can be used in signal generation, frequency synthesis, and clock circuits.
- Tuned Circuits: LC circuits can be employed as tuned circuits in radio frequency (RF) applications, such as tuning and impedance matching in antenna systems, frequency selective circuits in receivers and transmitters, and in RF filters.
- Energy Storage and Transfer: LC circuits can be used to store and transfer energy between the magnetic field of the inductor and the electric field of the capacitor. This property is exploited in various applications, including energy harvesting, wireless power transfer, and energy storage systems.

Understanding the behavior, equations, and applications of LC circuits is crucial for designing and analyzing various electrical and electronic systems. These fundamental circuits are widely used in signal processing, communication, and power systems, making them an essential topic for engineers and technicians.

## Example of Calculation

An LC (Inductor-Capacitor) circuit is a simple electrical circuit composed of an inductor and a capacitor connected either in series or parallel. The LC circuit, also known as a resonant or tank circuit, can store electrical energy and oscillate between the inductor and capacitor when excited by an external voltage. Here, we will discuss a series LC circuit.

Let’s consider an example of an LC circuit calculation involving the natural frequency and energy stored in the circuit:

Given values:

- Inductor (L): 100 mH (0.1 H)
- Capacitor (C): 10 µF (10 × 10^(-6) F)
- Initial voltage across the capacitor (V_C0): 5 V

We will calculate the natural frequency (f) of the LC circuit and the energy stored in the circuit (E) at the initial time (t=0).

- Calculate the natural frequency (f) of the LC circuit: The natural frequency (also called the resonant frequency) is the frequency at which the circuit oscillates when there is no external source connected to it.

f = 1 / (2 * π * √(L * C)) f = 1 / (2 * π * √(0.1 H * 10 × 10^(-6) F)) f ≈ 1 / (2 * π * √(1 × 10^(-6) H*F)) ≈ 1 / (2 * π * 1 × 10^(-3) Hz) ≈ 1 / (6.283 × 10^(-3) Hz) ≈ 159.15 Hz

The natural frequency of the LC circuit is approximately 159.15 Hz.

- Calculate the energy stored in the circuit (E) at the initial time (t=0): At the initial time (t=0), the energy is stored entirely in the capacitor as electric potential energy. The energy stored in the capacitor can be calculated using the following equation:

E_C = 0.5 * C * (V_C0)^2 E_C = 0.5 * 10 × 10^(-6) F * (5 V)^2 E_C = 0.5 * 10 × 10^(-6) F * 25 V^2 ≈ 1.25 × 10^(-4) J

The energy stored in the LC circuit at t=0 is approximately 1.25 × 10^(-4) J (joules).

During the oscillation, the energy stored in the circuit will transfer back and forth between the capacitor and the inductor without loss. However, in a practical circuit, there would be some resistance due to the non-ideal components, leading to energy dissipation in the form of heat, and the oscillation would eventually decay.

This example demonstrates how to calculate the natural frequency of an LC circuit and the energy stored in the circuit at the initial time.