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Encyclopedia > LC circuit

An LC circuit consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C. When connected together, an electrical current can alternate between them at an angular frequency of Image File history File links This is a lossless scalable vector image. ... An inductor is a passive electrical device employed in electrical circuits for its property of inductance. ... Capacitors: SMD ceramic at top left; SMD tantalum at bottom left; through-hole tantalum at top right; through-hole electrolytic at bottom right. ... In electricity, current is the rate of flow of charges, usually through a metal wire or some other electrical conductor. ... It has been suggested that this article or section be merged into Angular velocity. ...

omega = sqrt{1 over LC}
where L is the inductance in henries, and C is the capacitance in farads. The angular frequency has units of radians per second.

LC circuits are key components in many applications 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. For a model incorporating resistance see RLC circuit. Inductance (or electric inductance) is a measure of the amount of magnetic flux produced for a given electric current. ... An inductor. ... Capacitance is a measure of the amount of electric charge stored (or separated) for a given electric potential. ... Examples of various types of capacitors. ... Some common angles, measured in radians. ... Oscillation is the periodic variation, typically in time, of some measure as seen, for example, in a swinging pendulum. ... Electrical resistance is a measure of the degree to which an electrical component opposes the passage of current. ... An RLC circuit (also known as a resonant circuit or a tuned circuit) is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. ...

Contents

Resonance effect

The resonance effect occurs when inductive and capacitive reactances are equal. See: Reactance. [Notice that the LC circuit does not, by itself, resonate. The word resonance refers to a class of phenomena in which a small driving perturbation gives rise to a large effect in the system. The LC circuit must be driven, for example by an AC power supply, for resonance to occur (below).] The frequency at which this equality holds for the particular circuit is called the resonant frequency. The resonant frequency of the LC circuit (in radians per second) is It has been suggested that Electric reactance be merged into this article or section. ... This article is about resonance in physics. ... In an electrical circuit, resonance occurs at a particular frequency when the inductive reactance and the capacitive reactance are of equal magnitude, causing electrical energy to oscillate between the magnetic field of the inductor and the electric field of the capacitor. ... Some common angles, measured in radians. ...

omega = sqrt{1 over LC}

The equivalent frequency in units of hertz is The hertz (symbol: Hz) is the SI unit of frequency. ...

f = { omega over 2 pi } = {1 over {2 pi sqrt{LC}}}

Series resonance

Here R, L, and C are in series in an ac circuit. Inductive reactance (XL) increases as frequency increases while capacitive reactance (XC) decreases with increase in frequency. At a particular frequency these two reactances are equal in magnitude but opposite in phase. The frequency at which this happens is the resonant frequency (fr) for the given circuit.


Hence, at fr :


XL = XC


{omega {L}} = {{1} over {omega} {C}}


Converting angular frequency into hertz we get


{2 pi fL} = {1 over {2 pi fC}}


Here f is the resonant frequency. Then rearranging,


f = {1 over {2 pi sqrt{LC}}}


In a series ac circuit, XC leads by 90 degrees while XL lags by 90. Therefore, they both cancel each other out. The only opposition to a current is coil resistance. Hence in series resonance the current is maximum at resonant frequency.

  • At fr, current is maximum. Circuit impedance is minimum. In this state a circuit is called an acceptor circuit.
  • Below fr, XL < XC. Hence cct is capacitive.
  • Above fr, XL > XC. Hence cct is inductive.

Parallel resonance

Here a coil (L) and capacitor (C) are connected in parallel with an ac power supply. Let R be the internal resistance of the coil. When XL equals XC, the reactive branch currents are equal and opposite. Hence they cancel out each other to give minimum current in the main line. Since total current is minimum, in this state the total impedance is maximum.


Resonant frequency given by: f = {1 over {2 pi sqrt{LC}}}.


Note that any reactive branch current is not minimum at resonance, but each is given separately by dividing source voltage (V) by reactance (Z). Hence I=V/Z, as per Ohm's law. poo A voltage source, V, drives an electric current, I , through resistor, R, the three quantities obeying Ohms law: V = IR Ohms law states that, in an electrical circuit, the current passing through a conductor from one terminal point on the conductor to another terminal point on the...

  • At fr,line current is minimum. Total impedance is maximum. In this state cct is called rejector circuit.
  • Below fr, cct is inductive.
  • Above fr,cct is capacitive.

Applications of resonance effect

  1. Most common application is tuning. For example, when we tune a radio to a particular station, the LC circuits are set at resonance for that particular carrier frequency.
  2. A series resonant circuit provides voltage magnification.
  3. A parallel resonant circuit provides current magnification.
  4. 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.
  5. A parallel resonant circuit can be used in induction heating.

Carrier frequency is the fundamental frequency used in both amplitude modulation and frequency modulation i. ...

Circuit analysis

By Kirchhoff's voltage law, we know that the voltage across the capacitor, VC must equal the voltage across the inductor, VL: Kirchhoffs circuit laws are a pair of laws that deal with the conservation of charge and energy in electrical circuits, and were first described in 1845 by Gustav Kirchhoff. ...

VC = VL

Likewise, by Kirchhoff's current law, the current through the capacitor plus the current through the inductor must equal zero: Kirchhoffs circuit laws are a pair of laws that deal with the conservation of charge and energy in electrical circuits, and were first described in 1845 by Gustav Kirchhoff. ...

iC + iL = 0

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

V _{L}(t) = L frac{di_{L}}{dt}

and

i_{C}(t) = C frac{dV_{C}}{dt}

After rearranging and substituting, we obtain the second order differential equation A simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ...

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

We now define the parameter ω as follows:

omega = sqrt{frac{1}{LC}}

With this definition, we can simplify the differential equation:

frac{d ^{2}i(t)}{dt^{2}} + omega^ {2} i(t) = 0

The associated polynomial is s2 + ω2 = 0, thus

s = + jω

or

s = − jω
where j is the imaginary unit.

Thus, the complete solution to the differential equation is In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ...

i(t) = Ae + jωt + Be jωt

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. City lights viewed in a motion blurred exposure. ...


If the initial conditions are such that A = B, then we can use Euler's formula to obtain a real sinusoid with amplitude 2A and angular frequency omega = sqrt{frac{1}{LC}}. Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ... A sine wave or sinusoid is a waveform whose graph is identical to the generalized sine function y = Asin[ω(x − α)] + C where A is the amplitude, ω is the angular frequency (2π/P where P is the wavelength), α is the phase shift, and C is the vertical offset. ... Amplitude is a nonnegative scalar measure of a waves magnitude of oscillation, that is, magnitude of the maximum disturbance in the medium during one wave cycle. ... It has been suggested that this article or section be merged into Angular velocity. ...


Thus, the resulting solution becomes:

i(t) = 2Acost)

The initial conditions that would satisfy this result are:

i(t = 0) = 2A

and

frac{di}{dt}(t=0) = 0

Impedance of LC circuits

Series LC

First consider the impedance of the series LC circuit. The total impedance is given by the sum of the inductive and capacitive impedances: Electrical impedance, or simply impedance, is a measure of opposition to a sinusoidal alternating electric current. ...

Z = ZL + ZC

By writing the inductive impedance as ZL = jωL and capacitive impedance as Z_{C} = frac{1}{j{omega C}} and substituting we have

Z = j omega L + frac{1}{j{omega C}} .

Writing this expression under a common denominator gives

Z = frac{(omega^{2} L C - 1)j}{omega C} .

Note that the numerator implies if ω2LC = 1 the total impedance Z will be zero and otherwise non-zero. Therefore the series connected circuit, when connected to a circuit in parallel, will act as a band-pass filter having zero impedance at the resonant frequency of the LC circuit. The frequency axis of this symbolic diagram would be logarithmically scaled. ...


Parallel LC

The same analysis may be applied to the parallel LC circuit. The total impedance is then given by:

Z=frac{Z_{L}Z_{C}}{Z_{L}+Z_{C}}

and after substitution of ZL and ZC we have

Z=frac{frac{L}{C}}{frac{(omega^{2}LC-1)j}{omega C}}

which simplifies to

Z=frac{-Lomega j}{omega^{2}LC-1} .

Note that lim_{omega^{2}LC to 1}Z = infty but for all other values of ω2LC the impedance is finite (and therefore less than infinity). Hence the parallel connected circuit will act as band-stop filter having infinite impedance at the resonant frequency of the LC circuit. A generic band-stop filter, showing both positive and negative angular frequencies In signal processing, a band-stop filter or band-rejection filter is a filter that passes most frequencies unaltered, but attenuates those in a range to very low levels. ...


Selectivity

LC circuits are often used as filters; the L/C ratio determines their selectivity. For a series resonant circuit, the higher the inductance and the lower the capacitance, the narrower the filter bandwidth. For a parallel resonant circuit the opposite applies. Selectivity is a measure of the performance of a radio receiver to respond only to the tuned transmission (such as a radio station) and reject other signals nearby, such as another broadcast on an adjacent channel. ...


Applications

LC circuits behave as electronic resonators, which are a key component in many applications: Most instruments includes parts which vibrate with and amplify the sound of the instrument. ...

Television signal splitter consisting of a hi-pass and a low-pass filter. ... A tuner is a device to adjust the resonant frequency of an antenna or transmission line to work most efficiently at one frequency or band of frequencies. ... In telecommunication, a mixer is a nonlinear circuit or device that accepts as its input two different frequencies and presents at its output (a) a signal equal in frequency to the sum of the frequencies of the input signals, (b) a signal equal in frequency to the difference between the... The Foster-Seeley discriminator is an FM detector circuit that works on the same principle as most commonly used FM detectors, which is through variations in frequency. ... Proximity card is a generic name for contactless integrated circuit device used for security access or payment systems. ...

See also

This article is about resonance in physics. ... An RLC circuit (also known as a resonant circuit or a tuned circuit) is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. ...

External links

  • Lessons In Electric Circuits

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