AC (linear) circuits are circuits containing resistors, capacitors, inductors and other linear components (e.g. transformers, linear sensors, actuators, etc.) driven by sources with sinusoidal (or cosinusoidal) waveforms. Since all the sources in the circuit are AC, it can be shown that all the signals in the circuit— including branch currents and nodal potentials—are also AC and have the same frequency as frequency of the sources. Therefore, under steady-state conditions, it is customary to represent the waveforms of all voltages and currents in the circuit—including the current of current sources and the voltage of voltage sources—as cosinusoidal functions of time $$\begin{equation}X(t)=X_0 \cos(\omega t+\phi)\end{equation}$$ where $X_0$ is the magnitude (expressed in volts or amperes), $\phi$ is the phase (expressed in degrees or radians), and $\omega$ is the angular frequency of the signal.

The angular frequency of the signal can be related to the frequency, $f$, or to the period, $T$ of the signal using the following relationships $$\begin{equation}\omega = 2\pi f=\frac{2\pi}{T}\end{equation}$$

It turns out that most of the techniques used to analyze DC circuits can also be applied to AC circuits—including Ohm's law, current division, voltage division, nodal analysis, mesh analysis, superposition, Thévenin's theorem, Norton's theorem, and source transformation— provided that the analysis is performed in the frequency (complex) domain. To do so, we first transform the circuit from the time domain into the frequency domain. Next, we compute the desired variables (currents, voltages, powers, etc.) in the frequency domain, and finally, we transform the results back into the time domain.

Table 1 summarizes the corresponding relationships between the time-domain and frequency-domain representations.

Table 1. Conversion formulas from time-domain to frequency domain ($j=\sqrt{-1}$).

		Value in time-domain	Value in frequency-domain (i.e. complex form, impedance)
Resistor		$R$	$R$
Capacitor		$C$	$\frac{1}{j \omega C}=-\frac{j}{\omega C}$
Inductor		$L$	$j \omega L$
			Phasor form	Rectangular form
Voltage and current sources, potentials, voltages, currents		$X_0 \cos(\omega t +\phi)$ $X_0 \sin(\omega t +\phi)$	$X_0 \angle{\phi}$ $X_0 \angle{(\phi-90^\circ)}$	$X_0 (\cos \phi+j \sin \phi)$ $X_0 [\cos(\phi-90^\circ)+j \sin(\phi-90^\circ)]$

To compute the values of the voltages and currents in an AC circuit we:

Step 1. Convert the circuit to frequency domain using Table 1. When a circuit is expressed in the frequency domain, resistors, capacitors, and inductors are represented by their complex impedances (measured in ohms, ${\class{mjunit}{Ω}}$), and all voltages and currents become complex quantities.

Step 2. Solve the circuit to compute the complex values of the sought variables (usually currents and voltages);

Step 3. Express the sought variables in time-domain using again Table 1.

Because all circuit voltages and currents oscillate at the same frequency as the driving sources, we only need to determine the magnitude and phase of each sought variable.

• The following formulas might be useful when converting from time-domain to frequency domain$$\begin{equation}\sin x=\cos(x-90^\circ)=\cos(x-\frac{\pi}{2})\end{equation}$$
• Linear circuits driven by sources that provide other periodic signals (e.g. rectangular or triangular waveforms) can be analyzed using the techniques of AC analysis by writing the signal of each source as a superposition of cosine waveforms (Fourier transforms).
• Complex numbers can be represented mathematically in rectangular form, in which the real and imaginary parts are shown explicitly (such as $3+4j$) or in polar form in which the magnitude and angle are shown explicitly ($5 \angle 53.1^\circ$). To convert between the two formats you can use the transformations shown in Table 2. Function $\arctan _2(b,a)$ in this table takes two arguments and is not the same as the $\arctan(\frac{b}{a})$ (inverse tangent) function. Function $\arctan _2(b,a)$ looks at the signs of both values to determine the correct quadrant and gives the angle for which the real part of the complex number is $a$ and the imaginary part is $b$. Most programming languages define function $\arctan _2(b,a)$: in C/C++ it is atan2(b,a), in C# is Math.Atan2(b,a), in Java/JavaScript it is Math.Atan2(b,a), in Python is numpy.arctan2(b,a). To understand the difference between the two functions, you can check that $\arctan _2(-1,-1)=225^\circ$, which is different from $\arctan(\frac{-1}{-1})=45^\circ$.

Table 2. Conversion of complex numbers from rectangular form to polar form and from polar form to rectangular form.

Rectangular form $a+b j$	Polar form $X_0 \angle \phi$
$a=X_0 \cos \phi$ $b=X_0 \sin \phi$	$X_0=\sqrt{a^2+b^2}$ $\phi=\text{atan2}(b,a)$