LC circuit
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Image:Lc circuit.png LC circuits behave as electronic resonators, which are a key component in many applications such as oscillators, filters, tuners and frequency mixers.
An LC circuit consists of an inductor and a capacitor. The electrical current will alternate between them at an angular frequency of
- <math>\omega = \sqrt{1 \over LC}</math>
- where L is the inductance in henries, and C is the capacitance in farads. The angular frequency has units of radians per second.
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.
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Resonant frequency
The resonant frequency of the LC circuit (in radians per second) is
- <math>\omega = \sqrt{1 \over LC}</math>
The equivalent frequency in the more familiar unit of hertz is
- <math>f = { \omega \over 2 \pi } = {1 \over {2 \pi \sqrt{LC}}} </math>
Circuit analysis
By Kirchhoff's voltage law, we know that the voltage across the capacitor, <math>V _{C}</math> must equal the voltage across the inductor, <math>V _{L}</math>:
- <math>V _{C} = V_{L}</math>
Likewise, by Kirchhoff's current law, the current through the capacitor plus the current through the inductor must equal zero:
- <math>i_{C} + i_{L}</math> = 0
From the constitutive relations for the circuit elements, we also know that
- <math>V _{L}(t) = L \frac{di_{L}}{dt}</math>
and
- <math>i_{C}(t) = C \frac{dV_{C}}{dt}</math>
After rearranging and substituting, we obtain the second order differential equation
- <math>\frac{d ^{2}i(t)}{dt^{2}} + \frac{1}{LC} i(t) = 0</math>
We now define the parameter ω as follows:
- <math>\omega = \sqrt{\frac{1}{LC}}</math>
With this definition, we can simplify the differential equation:
- <math>\frac{d ^{2}i(t)}{dt^{2}} + \omega^ {2} i(t) = 0</math>
The associated polynomial is <math>s ^{2} + \omega^ {2} = 0</math>, thus
- <math>s = +j \omega</math>
or
- <math>s = -j \omega</math>
- where j is the imaginary unit.
Thus, the complete solution to the differential equation is
- <math>i(t) = Ae ^{+j \omega t} + Be ^{-j \omega t}</math>
and can be solved for <math>A</math> and <math>B</math> by considering the initial conditions.
Since the exponential is complex, the solution represents a sinusoidal alternating current.
If the initial conditions are such that <math>A = B</math>, then we can use Euler's formula to obtain a real sinusoid with amplitude <math>2A</math> and angular frequency <math>\omega = \sqrt{\frac{1}{LC}}</math>.
Thus, the resulting solution becomes:
- <math>i(t) = 2 A cos(\omega t) </math>
The initial conditions that would satisfy this result are:
- <math>i(t=0) = 2 A</math>
and
- <math>\frac{di}{dt}(t=0) = 0</math>
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:
- <math>Z = Z_{L} + Z_{C}</math>
By writing the inductive impedance as <math>Z_{L} = i \omega L</math> and capacitive impedance as <math>Z_{C} = \frac{-i}{\omega C}</math> and substituting we have:
- <math>Z = i \omega L + \frac{-i}{\omega C}</math>
Writing this expression under a common denominator gives:
- <math>Z = \frac{(\omega^{2} L C - 1)i}{\omega C}</math>
Note that the numerator implies if <math>\omega^{2} L C =1</math> the total impedance Z will be zero and otherwise non-zero. Therefore the series connected circuit will act as a band-pass filter having zero impedance at the resonant frequency of the LC circuit.
Parallel LC
The same analysis may be applied to the parallel LC circuit. The total impedance is then given by:
- <math>Z=\frac{Z_{L}Z_{C}}{Z_{L}+Z_{C}}</math>
and after substitution of <math>Z_{L}</math> and <math>Z_{C}</math> we have:
- <math>Z=\frac{\frac{L}{C}}{\frac{(\omega^{2}LC-1)i}{\omega C}}</math>
which simplifies to:
- <math>Z=\frac{-L\omega i}{\omega^{2}LC-1}</math>
Note that <math> \lim_{\omega^{2}LC \to 1}Z = \infty </math> but for all other values of <math>\omega^{2} L C</math> 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.
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 capacity, the narrower the filter bandwidth. For a parallel resonant circuit the opposite applies.
Applications
LC circuits behave as electronic resonators, which are a key component in many applications:
See also
pl:Obwód rezonansowy pt:Circuito LC ru:Колебательный контур sl:Nihajni krog fi:Värähtelypiiri