Electronics/Electronics Formulas/Series Circuits/Series RLC

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Circuit Configuration[edit]

RLC series circuit.png

Formula[edit]

Circuit's Impedance[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}

Differential Equation[edit]

The Differential equation of the circuit at equilibrium

L \frac{di}{dt} + \frac{1}{C} \int i dt + iR= 0
\frac{d^2i}{dt^2} + \frac{R}{L} \frac{di}{dt} + \frac{1}{LC} = 0
s^2 + \frac{R}{L} s + \frac{1}{LC} = 0
s = (-\alpha \pm \lambda) t
\lambda = \sqrt{\alpha^2 - \beta^2}
\alpha = \frac{R}{2L}
\beta = \frac{1}{LC}

The Natural Response of the circuit[edit]

  • \lambda = 0 .  \alpha^2 = \beta^2
i = e^(-\alpha t)
  • \lambda = 0 .  \alpha^2 = \beta^2
i = e^(-\alpha t)[e^(\lambda t) + e^(-\lambda t)]
  • \lambda = 0 .  \alpha^2 = \beta^2
i = e^(-\alpha t)[e^(j \lambda t) + e^(-j \lambda t)]

The Resonance Response of the circuit[edit]

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 .
 \omega = 0 .  \omega = 0

Summary[edit]