Series LC circuit

Resonance

Here L and C are affiliated in alternation to an AC ability supply. Anterior reactance consequence (X_L\,) increases as abundance increases while capacitive reactance consequence (X_C\,) decreases with the access in frequency. At a accurate abundance these two reactances are according in consequence but adverse in sign. The abundance at which this happens is the beating abundance (f_r\,) for the accustomed circuit.

Hence, at f_r\, :

X_L = -X_C\,

{\omega {L}} = {{1} \over {\omega} {C}}\,

Converting angular abundance into hertz we get

{2 \pi fL} = {1 \over {2 \pi fC}}

Here f is the beating frequency. Then rearranging,

f = {1 \over {2 \pi \sqrt{LC}}}

In a alternation AC circuit, XC and XL abolish anniversary added out. The alone action to a accepted is braid resistance. Hence in alternation resonance the accepted is best at beating frequency.

At fr, accepted is maximum. Ambit impedance is minimum. In this accompaniment a ambit is alleged an acceptor circuit.

Below fr, X_L \ll (-X_C)\,. Hence ambit is capacitive.

Above fr, X_L \gg (-X_C)\,. Hence ambit is inductive.

edit Impedance

First accede the impedance of the alternation LC circuit. The absolute impedance is accustomed by the sum of the anterior and capacitive impedances:

Z = ZL + ZC

By autograph the anterior impedance as ZL = jωL and capacitive impedance as ZC = (jωC)−1 and substituting we have

Z = j \omega L + \frac{1}{j{\omega C}} .

Writing this announcement beneath a accepted denominator gives

Z = \frac{(\omega^{2} L C - 1)j}{\omega C} .

The numerator implies that if ω2LC = 1 the absolute impedance Z will be aught and contrarily non-zero. Therefore the alternation LC circuit, if affiliated in alternation with a load, will act as a band-pass clarify accepting aught impedance at the beating abundance of the LC circuit.