Electronics/Electronics Formulas/Series Circuits/Series LC

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Circuit Configuration[edit]

RL Series Open-Closed.svg

Formula[edit]

The total Impedance of the circuit

Z = Z_R + Z_L
Z = R + j\omega L
Z = \frac{1}{R} (1 + j\omega T)
T = \frac{L}{R}

The Differential equation of the circuit at equilibrium

L \frac{di}{dt} + \frac{1}{C} \int i dt = 0
\frac{d^2i}{dt^2} + \frac{1}{LC} = 0
s^2 + \frac{1}{LC} = 0
s = \pm j \sqrt{\frac{1}{LC}} t
s = \pm j \omega t

The Natural Response of the circuit

i = A Sin \omega t

The Resonance Response of the circuit

Z_L - Z_C = 0 . Z_L = Z_C . \omega L = \frac{1}{\omega C} . \omega  = \sqrt{\frac{1}{LC}}
V_L + V_C = 0 . V_C = -V_L

Summary[edit]

Series LC can be characterised by

2nd order Differential Equation

\frac{d^2i}{dt^2} + \frac{1}{T} = 0
T = LC

With Natural Response of a Wave function

i = A Sin \omega t

With Resonance Response of a Standing Wave function

i = A Sin \omega t