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Derive an expression for the impedance of an a.c. circuit consisting of an inductor and a resistor. |
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Answer» Solution :Consider a circuit having an inductance L and a RESISTANCE R, joined in series, to an a.c. supply. Let voltage provided by a.c. supply be `V= V_(m) sin omega t`. Let an instantaneous CURRENT I flows through the coil. Then instantaneous values of potential drops across inductance and resistance are given by:  `vecV_(L) =I X_(L)` and `vecV_(R) = IR`, where `X_(L) = omega L` is the reactance due to the inductance. Moreover phasor `vecV_(R)` and `vecV_(L)` be represented by OA and OB in a phasor diagram.  Then RESULTANT voltage `vecV` will be given by the phasor OC. Hence, `V = OC = sqrt(OA^(2) + OB^(2))` `=sqrt(V_(R)^(2) + V_(L)^(2))= Isqrt(R^(2) + X_(L)^(2))` `therefore` Impedance of the coil `Z = V/I = sqrt(R^(2) + X_(L)^(2)) = sqrt(R^(2) + L^(2) omega^(2))` Moreover, the circuit voltage V is ahead in phase as compared to circuit current I (or current I is lagging behind the source voltage V) by a phase angle `phi` , where `tan phi = (AC)/(OA) = (OB)/(OA) = (IX_(L))/(IR) =X_(L)/R = (Lomega)/R` |
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