This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "LC circuit" – news · newspapers · books · scholar · JSTOR (March 2009) (Learn how and when to remove this message)

An LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, is an electric circuit consisting of an inductor, represented by the letter L, and a capacitor, represented by the letter C, connected together. The circuit can act as an electrical resonator, an electrical analogue of a tuning fork, storing energy oscillating at the circuit's resonant frequency.

• 
  LC circuit diagram
• 
  LC circuit (left) consisting of ferrite coil and capacitor used as a tuned circuit in the receiver for a radio clock
• 
  Output tuned circuit of shortwave radio transmitter

LC circuits are used either for generating signals at a particular frequency, or picking out a signal at a particular frequency from a more complex signal; this function is called a bandpass filter. 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. While no practical circuit is without losses, it is nonetheless instructive to study this ideal form of the circuit to gain understanding and physical intuition. For a circuit model incorporating resistance, see RLC circuit.

Terminology

The two-element LC circuit described above is the simplest type of inductor-capacitor network (or LC network). It is also referred to as a second order LC circuit^[1][2] to distinguish it from more complicated (higher order) LC networks with more inductors and capacitors. Such LC networks with more than two reactances may have more than one resonant frequency.

The order of the network is the order of the rational function describing the network in the complex frequency variable s. Generally, the order is equal to the number of L and C elements in the circuit and in any event cannot exceed this number.

Operation

An LC circuit, oscillating at its natural resonant frequency, can store electrical energy. See the animation. A capacitor stores energy in the electric field (E) between its plates, depending on the voltage across it, and an inductor stores energy in its magnetic field (B), depending on the current through it.

If an inductor is connected across a charged capacitor, the voltage across the capacitor will drive a current through the inductor, building up a magnetic field around it. The voltage across the capacitor falls to zero as the charge is used up by the current flow. At this point, the energy stored in the coil's magnetic field induces a voltage across the coil, because inductors oppose changes in current. This induced voltage causes a current to begin to recharge the capacitor with a voltage of opposite polarity to its original charge. 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.

The charge flows back and forth between the plates of the capacitor, through the inductor. The energy oscillates back and forth between the capacitor and the inductor until (if not replenished from an external circuit) internal resistance makes the oscillations die out. The tuned circuit's action, known mathematically as a harmonic oscillator, is similar to a pendulum swinging back and forth, or water sloshing back and forth in a tank; for this reason the circuit is also called a tank circuit.^[3] The natural frequency (that is, the frequency at which it will oscillate when isolated from any other system, as described above) is determined by the capacitance and inductance values. In most applications the tuned circuit is part of a larger circuit which applies alternating current to it, driving continuous oscillations. If the frequency of the applied current is the circuit's natural resonant frequency (natural frequency ${\displaystyle f_{0}\,}$ below), resonance will occur, and a small driving current can excite large amplitude oscillating voltages and currents. In typical tuned circuits in electronic equipment the oscillations are very fast, from thousands to billions of times per second.^{[citation needed]}

Resonance effect

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 frequency at which this equality holds for the particular circuit is called the resonant frequency. The resonant frequency of the LC circuit is

${\displaystyle \omega _{0}={\frac {1}{\sqrt {LC}}},}$

where L is the inductance in henries, and C is the capacitance in farads. The angular frequency ω₀ has units of radians per second.

The equivalent frequency in units of hertz is

${\displaystyle f_{0}={\frac {\omega _{0}}{2\pi }}={\frac {1}{2\pi {\sqrt {LC}}}}.}$

Applications

The resonance effect of the LC circuit has many important applications in signal processing and communications systems.

• The most common application of tank circuits is tuning radio transmitters and receivers. For example, when tuning a radio to a particular station, the LC circuits are set at resonance for that particular carrier frequency.
• A series resonant circuit provides voltage magnification.
• A parallel resonant circuit provides current magnification.
• A parallel resonant circuit can be used as load impedance in output circuits of RF amplifiers. Due to high impedance, the gain of amplifier is maximum at resonant frequency.
• Both parallel and series resonant circuits are used in induction heating.

LC circuits behave as electronic resonators, which are a key component in many applications:

• Amplifiers
• Oscillators
• Filters
• Tuners
• Mixers
• Foster–Seeley discriminator
• Contactless cards
• Graphics tablets
• Electronic article surveillance (security tags)

Time domain solution

Kirchhoff's laws

By Kirchhoff's voltage law, the voltage V_C across the capacitor plus the voltage V_L across the inductor must equal zero:

${\displaystyle V_{C}+V_{L}=0.}$

Likewise, by Kirchhoff's current law, the current through the capacitor equals the current through the inductor:

${\displaystyle I_{C}=I_{L}.}$

From the constitutive relations for the circuit elements, we also know that

${\displaystyle {\begin{aligned}V_{L}(t)&=L{\frac {\mathrm {d} I_{L}}{\mathrm {d} t}},\\I_{C}(t)&=C{\frac {\mathrm {d} V_{C}}{\mathrm {d} t}}.\end{aligned}}}$

Differential equation

Rearranging and substituting gives the second order differential equation

${\displaystyle {\frac {\mathrm {d} ^{2}}{\mathrm {d} t^{2}}}I(t)+{\frac {1}{LC}}I(t)=0.}$

The parameter ω₀, the resonant angular frequency, is defined as

${\displaystyle \omega _{0}={\frac {1}{\sqrt {LC}}}.}$

Using this can simplify the differential equation:

${\displaystyle {\frac {\mathrm {d} ^{2}}{\mathrm {d} t^{2}}}I(t)+\omega _{0}^{2}I(t)=0.}$

The associated Laplace transform is

${\displaystyle s^{2}+\omega _{0}^{2}=0,}$

thus

${\displaystyle s=\pm j\omega _{0},}$

where j is the imaginary unit.

Solution

Thus, the complete solution to the differential equation is

${\displaystyle I(t)=Ae^{+j\omega _{0}t}+Be^{-j\omega _{0}t}}$

and can be solved for A and B by considering the initial conditions. Since the exponential is complex, the solution represents a sinusoidal alternating current. Since the electric current I is a physical quantity, it must be real-valued. As a result, it can be shown that the constants A and B must be complex conjugates:

${\displaystyle A=B^{*}.}$

Now let

${\displaystyle A={\frac {I_{0}}{2}}e^{+j\phi }.}$

Therefore,

${\displaystyle B={\frac {I_{0}}{2}}e^{-j\phi }.}$

Next, we can use Euler's formula to obtain a real sinusoid with amplitude I₀, angular frequency ω₀ = 1/√LC, and phase angle ${\displaystyle \phi }$.

Thus, the resulting solution becomes

${\displaystyle I(t)=I_0}$Zdroj:https://en.wikipedia.org?pojem=Tuned_circuit
Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok. Podrobnejšie informácie nájdete na stránke Podmienky použitia.

---