# Series LCR Circuit

The applied voltage V divides into three parts, VL (across L), VC (across C) and VR (across R) such that

We knowBy KVL, V = VR + j (VL – VC)

V = √ (V²R + (VL – VC)²)

VL  Voltage across Inductor

VR Voltage across Resistor

VC  Voltage across Capacitor

The impedance of the circuit is

VR = IR; VC =$$\frac{1}{\omega C}$$; VL = IωL;

V = IZ

$$IZ=\sqrt{{{\left( IR \right)}^{2}}+{{I}^{2}}{{\left( \omega L-\frac{1}{\omega C} \right)}^{2}}}$$.

$$Z=\sqrt{{{R}^{2}}+{{\left( \omega L-\frac{1}{\omega C} \right)}^{2}}}$$.

But at some particular frequency both Inductive effect and Capacitor effect cancels each other and the circuit start oscillating. This is called resonance and the resonating frequency is

$$\omega L=\frac{1}{\omega C}$$.

$${{\omega }^{2}}=\frac{1}{LC}~$$.

$$\omega =\frac{1}{\sqrt{LC}}\Rightarrow f=\frac{i}{2\pi \sqrt{LC}}~$$.

At this frequency the current in the circuit is maximum as impedance (Z) is minimum.