Arrive at the expression for the impedance of a series $L C R$ circuit using phasor diagram method and hence write the expression for the current through the circuit.

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Solution

Consider a series LCR circuit connected to an AC source
AC voltage applied is $v = v_{m} sin ω t$
Let the current through the series be $i = i_{m} sin (ω t + ϕ)$
where $ϕ$ is phase difference between voltage and current. (Ref. image 1)
Let $\to I$ be the phasor representing current and ${\to V}_{R}, {\to V}_{L}, {\to V}_{C}$ and $\to V$ be the phasors representing voltage across $R, L, C$ and source respectively and we assume that ${\to V}_{C} > {\to V}_{L}$ . (Ref. image 2)
The length of the phasors ${\to V}_{R}, {\to V}_{L}, {\to V}_{C}$ are
$V_{R m} = i_{m} R$
$V_{c m} = i_{m} X_{C}$ and
$V_{L m} = i_{m} X_{L}$
From Kirchhoff's loop rule $V_{L} + V_{R} + V_{C} = V$
and the phasor relation is represented as,
${\to V}_{L} + {\to V}_{R} + {\to V}_{C} = \to V$
Expression for Impedence:
Applying Pythagorean Theorem to the right angle triangle,
$V_{m}^{2} = V_{R m}^{2} + {(v_{c m} - v_{L m})}^{2}$
$V_{m}^{2} = {(i_{m} R)}^{2} + {(i_{m} X_{c} - i_{m} X_{L})}^{2}$
$V_{m}^{2} = i_{m}^{2} [R^{2} + {(X_{c} - X_{L})}^{2}]$
$V_{m} = i_{m} \sqrt{R^{2} + {(X_{c} - X_{L})}^{2}}$
$i_{m} = \frac{V_{m}}{\sqrt{R^{2} + {(X_{c} - X_{L})}^{2}}}$
$i_{m} = \frac{V_{m}}{Z}$
Where $Z$ is called impedance of the coil
$Z = \sqrt{R^{2} + {(X_{c} - X_{L})}^{2}}$

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