Derive an expression for the impedance of an a.c. circuit consisting of an inductor and a resistor.

Solution

Let, the circuit contain a resistor of resistance R and inductor of inductance L in series.
Voltage across R is V_R
Voltage across L is V_LCurrent flowing through the circuit is I.

Applied voltage, V = V_o sin $ωt$
V_R and I are in phase whereas, V_Lleads the current by an angle by $π$ /2.

Applied voltage is the resultant of V_R and V_L
i.e., $V = \sqrt{{V_{R}}^{2} + {V_{L}}^{2}}$

But, V_R = Ri and,
V_L = $ωL i$ $ωL i$
where, $χ_{L} = ω L$ is the inductive reactance.

V = $\sqrt{(Ri)^{2} + (χ_{L} i)^{2}}$
and
Impedence, Z = $\frac{V}{i} = \sqrt{R^{2} + {χ_{L}}^{2}} = \sqrt{R^{2} + (ω L)^{2}}$
which is the required expression for impedence.

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Electrostatic Potential And Capacitance

Derive an expression for the impedance of an a.c. circuit consisting of an inductor and a resistor.

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