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Question
Obtain the expression for the applied emf and the effective resistance of the circuit when alternating emf is applied to an LR circuit.
Solution
- Consider an alternating e.m.f. applied to a series combination of the pure inductor of inductance L and resistor of resistance R.
- Let eR and eL be the r.m.s. the voltage across resistor and inductor respectively. As R and L are in series, the current at any instant is given as i = i0 sin ωt.
- Vector diagram,
- From the above figure,
e2 = `"e"_"R"^2 + "e"_"L"^2`
∴ e = `sqrt("e"_"R"^2 + "e"_"L"^2) = sqrt(("iR")^2 + ("iX"_"L")^2) = "i"sqrt("R"^2 + "X"_"L"^2)`
∴ e = iZ
where Z = `sqrt("R"^2 + "X"_"L"^2)` is the impedance of the RL circuit. - a. Expression for applied emf in RL circuit,
e = e0sinωt = `sqrt("e"_"R"^2 + "e"_"L"^2)` = iZ
b. Expression for effective resistance in RL circuit, Z = `sqrt("R"^2 + "X"_"L"^2)`
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