LC circuit

They are key components in many electronic devices, particularly radio equipment, used in circuits such as oscillators, filters, tuners and frequency mixers.

An LC circuit is an idealized model since it assumes there is no dissipation of energy due to resistance.

Any practical implementation of an LC circuit will always include loss resulting from small but non-zero resistance within the components and connecting wires.

The purpose of an LC circuit is usually to oscillate with minimal damping, so the resistance is made as low as possible.

An LC circuit, oscillating at its natural resonant frequency, can store electrical energy.

Due to Faraday's law, the EMF which drives the current is caused by a decrease in the magnetic field, thus the energy required to charge the capacitor is extracted from the magnetic field.

When the magnetic field is completely dissipated the current will stop and the charge will again be stored in the capacitor, with the opposite polarity as before.

Then the cycle will begin again, with the current flowing in the opposite direction through the inductor.

In typical tuned circuits in electronic equipment the oscillations are very fast, from thousands to billions of times per second.

[citation needed] Resonance occurs when an LC circuit is driven from an external source at an angular frequency ω0 at which the inductive and capacitive reactances are equal in magnitude.

The resonant frequency of the LC circuit is where L is the inductance in henries, and C is the capacitance in farads.

The equivalent frequency in units of hertz is The resonance effect of the LC circuit has many important applications in signal processing and communications systems.

LC circuits behave as electronic resonators, which are a key component in many applications: By Kirchhoff's voltage law, the voltage VC across the capacitor plus the voltage VL across the inductor must equal zero: Likewise, by Kirchhoff's current law, the current through the capacitor equals the current through the inductor: From the constitutive relations for the circuit elements, we also know that Rearranging and substituting gives the second order differential equation The parameter ω0, the resonant angular frequency, is defined as Using this can simplify the differential equation: The associated Laplace transform is thus where j is the imaginary unit.

Since the exponential is complex, the solution represents a sinusoidal alternating current.

As a result, it can be shown that the constants A and B must be complex conjugates: Now let Therefore, Next, we can use Euler's formula to obtain a real sinusoid with amplitude I0, angular frequency ω0 = ⁠1/√LC⁠, and phase angle

In real, rather than idealised, components, the current is opposed, mostly by the resistance of the coil windings.

In the series configuration, resonance occurs when the complex electrical impedance of the circuit approaches zero.

We begin by defining the relation between current and voltage across the capacitor and inductor in the usual way: Then by application of Kirchhoff's laws, we may arrive at the system's governing differential equations With initial conditions

is needed: The final term is dependent on the exact form of the input voltage.

Using the partial fraction method: Simplifiying on both sides We solve the equation for A, B and C: Substitute the values of A, B and C: Isolating the constant and using equivalent fractions to adjust for lack of numerator: Performing the reverse Laplace transform on each summands: Using initial conditions in the Laplace solution: The first evidence that a capacitor and inductor could produce electrical oscillations was discovered in 1826 by French scientist Felix Savary.

He correctly deduced that this was caused by a damped oscillating discharge current in the wire, which reversed the magnetization of the needle back and forth until it was too small to have an effect, leaving the needle magnetized in a random direction.

American physicist Joseph Henry repeated Savary's experiment in 1842 and came to the same conclusion, apparently independently.

[8][9] Irish scientist William Thomson (Lord Kelvin) in 1853 showed mathematically that the discharge of a Leyden jar through an inductance should be oscillatory, and derived its resonant frequency.

[6][8][9] British radio researcher Oliver Lodge, by discharging a large battery of Leyden jars through a long wire, created a tuned circuit with its resonant frequency in the audio range, which produced a musical tone from the spark when it was discharged.

[8] In 1857, German physicist Berend Wilhelm Feddersen photographed the spark produced by a resonant Leyden jar circuit in a rotating mirror, providing visible evidence of the oscillations.

[6][8][9] In 1868, Scottish physicist James Clerk Maxwell calculated the effect of applying an alternating current to a circuit with inductance and capacitance, showing that the response is maximum at the resonant frequency.

[6] The first example of an electrical resonance curve was published in 1887 by German physicist Heinrich Hertz in his pioneering paper on the discovery of radio waves, showing the length of spark obtainable from his spark-gap LC resonator detectors as a function of frequency.

[6] One of the first demonstrations of resonance between tuned circuits was Lodge's "syntonic jars" experiment around 1889.

[6][8] He placed two resonant circuits next to each other, each consisting of a Leyden jar connected to an adjustable one-turn coil with a spark gap.