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LCR Circuit
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LCR circuit

RLC series circuit.png

For the case where the source is an unchanging voltage, the second order differential equation governing the current in the LCR circuit is :

{{d^2 i(t)} over {dt^2}} +{R over L} {{di(t)} over {dt}} + {1 over {LC}} i(t) = 0

 

For the RLC circuit, there are two important parameters,

alpha = {R over 2L}   and  omega_0 = { 1 over sqrt{LC}}

 

is the natural frequency of the circuit.

A useful parameter is the damping factor, ΞΆ which is defined as the ratio of these two,

 zeta = frac {alpha}{omega_0}

The general solution of the differential equation is an exponential in either root or a linear superposition of both,

 i(t) = A_1 e^{s_1 t} + A_2 e^{s_2 t}

where,

 s_1 = -alpha +sqrt {alpha^2 - {omega_0}^2}

 s_2 = -alpha -sqrt {alpha^2 - {omega_0}^2}

Based on above one gets different condition of oscillatory response, via., underdamped, overdamped and critically damped.

 

For more details see

http://en.wikipedia.org/wiki/RLC_circuit

 

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