Circuit Theory/LC Tuned Circuits

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Series LC[edit]

A circuit of one Capacitor and one inductor connected in series

Circuit Impedance[edit]

Z = Z_L + Z_C
Z = j\omega L + \frac{1}{j\omega C}
Z = \frac{1}{j\omega C} (j\omega^2 + 1)
Z = LC

Natural Response[edit]

At equilibrium , the total volatge of the two components are equal to zero

L\frac{dI}{dt} + IR = 0
\frac{dI}{dt} = - I \frac{R}{L}
\int \frac{dI}{I} = - \frac{R}{L} \int dt
ln I =  - \frac{t}{T} + C
I =  e^(- \frac{t}{T} + C)
I =  A e^(- \frac{t}{T})

The Natural Response of the circuit is a Exponential Decrease in time

Resonance Response[edit]

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

In Resonance, Impedance of Inductor and Capacitance is equal and the sum of the Capacitor and Inductor's voltage are equal result in Standing Wave Oscillation . Therefore, Lossless LC series can generate Standing Wave Oscillation

LC in Parallel[edit]

A circuit of one Capacitor and one inductor connected in parallel

Circuit Impedance[edit]

\frac{1}{Z} = \frac{1}{Z_L} + \frac{1}{Z_C}
Y = \frac{1}{j\omega L} + j \omega C
Y = \frac{1}{j\omega L} (j \omega^2 + 1)