Circuit Theory/LC Tuned Circuits

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[edit] CL in Series

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

There is one frequency at which ZL = ZC

j\omega L = \frac{1}{j\omega C}
(j\omega)^2 = \frac{1}{LC}
\omega = \frac{1}{\sqrt{LC}}

The frequency \omega = \frac{1}{\sqrt{LC}} is called The Resonant Frequency is denoted as ωo

At resonance frequency VL = VC or VL - VC = 0 or VC = - VL or

The total voltage of the circuit is equal to zero . The voltage in the circuit flactuates between Inductor's voltage and Capacitor's Voltage between two phase angle 0 and 180 . The voltage completes one cycle at 180 then inverted and travel back to origin at angle 0 . This process is called Resonance and it creates voltage's oscillation between phase angle 0 - 180 . The frequency when the oscillation occurs is called Resonance Frequency, Oscillation Frequency or Tuned frequency

[edit] CL in Parallel

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

There is one frequency at which ZL = ZC

j\omega L = \frac{1}{j\omega C}
(j\omega)^2 = \frac{1}{LC}
\omega = \frac{1}{\sqrt{LC}}

The frequency \omega = \frac{1}{\sqrt{LC}} is called The Resonant Frequency is denoted as ωo

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