LC Circuit

• • • Claimed by Rishab Chawla 11/19/15***

Main Idea

Consider an electrical circuit consisting of an inductor, of inductance L, connected in series with a capacitor, of capacitance C. Such a circuit is known as an LC circuit, for obvious reasons. An LC circuit, oscillating at its natural resonant frequency, can store electrical energy. https://upload.wikimedia.org/wikipedia/commons/8/80/Tuned_circuit_animation_3_300ms.gif

A Mathematical Model

Starting with Kirchoff's Loop Rule, we have V = L * dI/dt + q/C.

Taking the derivative of each term, dV/dt = L(d^2*I/dt^2) + 1/C * dq/dt.

The voltage of the battery is constant, so that derivative vanishes. The derivative of charge is current, so that gives us a second order differential equation: 0 = L(d^2*I/dt^2) + 1/C * I

Rearranging, (d^2/dt^2)I = - I/(LC).

This can be solved by guessing and checking with a generic sine function: I = I0sin(ωt + φ)

(d^2/dt^2)I0sin(ωt + φ) = -I0/(LC) * sin(ωt + φ) − ω^2*I0sin(ωt + φ) = -I0/(LC) * sin(ωt + φ)

The angular frequency is now given by ω = 1/sqrt(LC). This is also known as its resonance frequency. A LC circuit is then an oscillating circuit with frequency ω/(2π) = 1/(2π*sqrt(LC)).

A Computational Model

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Examples

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History

The first evidence that a capacitor and inductor could produce electrical oscillations was discovered in 1826 by French scientist Felix Savary He found that when a Leyden jar was discharged through a wire wound around an iron needle, sometimes the needle was left magnetized in one direction and sometimes in the opposite direction. 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. British 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.[2][4][5] 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. 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. 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.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.

One of the first demonstrations of resonance between tuned circuits was Lodge's "syntonic jars" experiment around 1889.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. When a high voltage from an induction coil was applied to one tuned circuit, creating sparks and thus oscillating currents, sparks were excited in the other tuned circuit only when the circuits were adjusted to resonance. Lodge and some English scientists preferred the term "syntony" for this effect, but the term "resonance" eventually stuck. The first practical use for LC circuits was in the 1890s in spark-gap radio transmitters to allow the receiver and transmitter to be tuned to the same frequency. The first patent for a radio system that allowed tuning was filed by Lodge in 1897, although the first practical systems were invented in 1900 by Italian radio pioneer Guglielmo Marconi.

See also

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Further reading

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References

Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 22 https://archive.org/stream/bstj20-4-415#page/n13/mode/2up