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LC circuit

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An LC circuit is a resonant circuit or tuned circuit that consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C. When connected together, an electric current can alternate between them at the circuit's resonant frequency.

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. They 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.

Contents

[edit] Operation

[edit] Simplified overview

An LC circuit can store electrical energy vibrating at its resonant frequency. A capacitor stores energy in the electric field between its plates, depending on the voltage across it, and an inductor stores energy in its magnetic field, depending on the current through it. If a charged capacitor is connected across an inductor, charge will start to flow through the inductor, building up a magnetic field around it, and reducing the voltage on the capacitor. Eventually all the charge on the capacitor will be gone. However, the current will continue, because inductors resist changes in current, and energy will be extracted from the magnetic field to keep it flowing. The current will begin to charge the capacitor with a voltage of opposite polarity to its original charge. 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) and the cycle will begin again, with the current in the opposite direction.

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 by power from an external circuit) internal resistance makes the oscillations die out. Its 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. The oscillations can be very fast, typically hundreds to billions of times per second.

[edit] Time domain solution

By Kirchhoff's voltage law, we know that the voltage across the capacitor, V _{C}\, must equal the voltage across the inductor, V _{L}\,:

V _{C} = V_{L}.\,

Likewise, by Kirchhoff's current law, the current through the capacitor plus the current through the inductor must equal zero:

i_{C} + i_{L} = 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

\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 s ^{2} + \omega^ {2} = 0\,, thus

s = +j \omega\,

or

s = -j \omega\,
where j is the imaginary unit.

Thus, the complete solution to the differential equation is

i(t) = Ae ^{+j \omega t} + Be ^{-j \omega 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.

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}}.\,

Thus, the resulting solution becomes:

i(t) = 2 A cos(\omega t).\,

The initial conditions that would satisfy this result are:

i(t=0) = 2 A\,

and

\frac{di}{dt}(t=0) = 0.\,

[edit] Resonance effect

The resonance effect occurs when inductive and capacitive reactances are equal in absolute value. (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 is

\omega = \sqrt{1 \over LC}

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

The equivalent frequency in units of hertz is

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

LC circuits are often used as filters; the L/C ratio determines their "Q" and so 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. Positive feedback around the tuned circuit ("regeneration") can also increase selectivity (see Q multiplier and Regenerative circuit).

Stagger tuning can provide an acceptably wide audio bandwidth, yet good selectivity.

[edit] Series LC circuit

[edit] Resonance

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

Hence, at f_r\, :

X_L = -X_C\,
{\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 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 f_r\,, current is maximum. Circuit impedance is minimum. In this state a circuit is called an acceptor circuit.
  • Below f_r\,, X_L \ll (-X_C)\,. Hence circuit is capacitive.
  • Above f_r\,, X_L \gg (-X_C)\,. Hence circuit is inductive.

[edit] Impedance

First consider the impedance of the series LC circuit. The total impedance is given by the sum of the inductive and capacitive impedances:

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 series, will act as a band-pass filter having zero impedance at the resonant frequency of the LC circuits.

[edit] Parallel LC circuit

[edit] 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.

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

[edit] Impedance

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 and simplification, gives

Z=\frac{-j \omega L}{\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.

[edit] Applications

 This article may require cleanup to meet Wikipedia's quality standards. Please improve this article if you can. (March 2009)

[edit] 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.

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

[edit] See also

[edit] External links

Retrieved from "http://en.wikipedia.org/wiki/LC_circuit"
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