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# 13.4: Resistance and Inductance in Series

The impedance is just the sum of the resistance of the resistor and the impedance of the inductor:

$\label{13.4.1}Z=R+jl\omega .$

Thus the impedance is a complex number, whose real part $$R$$is the resistance and whose imaginary part $$L\omega$$ is the reactance. For a pure resistance, the impedance is real, and $$V$$ and $$I$$ are in phase. For a pure inductance, the impedance is imaginary (reactive), and there is a 90o phase difference between $$V$$ and $$I$$.

The voltage and current are related by

$\label{13.4.2}V=IZ = (R+jL\omega )I.$

Those who are familiar with complex numbers will see that this means that $$V$$ leads on $$I$$, not by 90o, but by the argument of the complex impedance, namely $$\tan^{-1}(L\omega /R)$$. Further the ratio of the peak (or RMS) voltage to the peak (or RMS) current is equal to the modulus of the impedance, namely $$\sqrt{R^2+L^2\omega^2}$$.