# 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 90^{o} 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 90^{o}, 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 \(\sqrt{R^2+L^2\omega^2}\).