Q.Derive an expression for the impedance of an LCR circuit connected to an AC power supply. Draw phasor diagram.

Impedance of a series LCR circuit derived from the phasor diagram.

In a series LCR circuit carrying current $I$ (common to all elements, taken as reference phasor along the x-axis):

• Voltage across R, $V_R=IR$, is in phase with $I$.
• Voltage across L, $V_L=IX_L$, leads $I$ by $90°$.
• Voltage across C, $V_C=IX_C$, lags $I$ by $90°$.

Since $V_L$ and $V_C$ are exactly opposite in phase (both perpendicular to $I$ but in opposite senses), their resultant is $(V_L-V_C)$, still perpendicular to $V_R$. By the phasor (vector) diagram, the applied voltage $V$ is the resultant of the perpendicular phasors $V_R$ and $(V_L-V_C)$:

$V = \sqrt{V_R^2+(V_L-V_C)^2} = I\sqrt{R^2+(X_L-X_C)^2}$ …