Editing Ohm's law (section)

===Reactive circuits with time-varying signals===

When reactive elements such as capacitors, inductors, or transmission lines are involved in a circuit to which AC or time-varying voltage or current is applied, the relationship between voltage and current becomes the solution to a [[differential equation]], so Ohm's law (as defined above) does not directly apply since that form contains only resistances having value ''R'', not complex impedances which may contain capacitance (''C'') or inductance (''L'').

Equations for [[time-invariant]] [[alternating current|AC]] circuits take the same form as Ohm's law. However, the variables are generalized to [[complex number]]s and the current and voltage waveforms are [[complex exponential]]s.<ref>{{cite book | title = Fundamentals of Electrical Engineering | first = Rajendra|last=Prasad | publisher = Prentice-Hall of India | year = 2006 | url = https://books.google.com/books?id=nsmcbzOJU3kC&q=ohm%27s-law+complex+exponentials&pg=PA140 | isbn = 978-81-203-2729-0 }}</ref>

In this approach, a voltage or current waveform takes the form ''Ae''{{sup|''st''}}, where ''t'' is time, ''s'' is a complex parameter, and ''A'' is a complex scalar. In any [[LTI system theory|linear time-invariant system]], all of the currents and voltages can be expressed with the same ''s'' parameter as the input to the system, allowing the time-varying complex exponential term to be canceled out and the system described algebraically in terms of the complex scalars in the current and voltage waveforms.

The complex generalization of resistance is [[electrical impedance|impedance]], usually denoted ''Z''; it can be shown that for an inductor,
<math display="block">Z = sL</math>
and for a capacitor,
<math display="block">Z = \frac{1}{sC}.</math>

We can now write,
<math display="block">V = Z\,I</math>
where ''V'' and ''I'' are the complex scalars in the voltage and current respectively and ''Z'' is the complex impedance.

This form of Ohm's law, with ''Z'' taking the place of ''R'', generalizes the simpler form. When ''Z'' is complex, only the real part is responsible for dissipating heat.

In a general AC circuit, ''Z'' varies strongly with the frequency parameter ''s'', and so also will the relationship between voltage and current.

For the common case of a steady [[Sine wave|sinusoid]], the ''s'' parameter is taken to be <math>j\omega</math>, corresponding to a complex sinusoid <math>Ae^{\mbox{ } j \omega t}</math>. The real parts of such complex current and voltage waveforms describe the actual sinusoidal currents and voltages in a circuit, which can be in different phases due to the different complex scalars.