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. At steady-state, it is customary to express the waveforms of all the currents and potentials in the circuit (including the current of current sources and voltge 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 methods that we used to study DC circuits can also be used to study AC circuits (Ohm's law, current division, voltage division, nodal analysis, mesh analysis, superposition, Thévenin's' theorem, Norton's' theorem, and source transformation) provided that we perform the analysis in frequency (or complex) domain. When we transform a circuit from time-domain to the frequency domain, the resistors, capactors and inductors become impedances measured in ohms, $Ω$. The voltage and current sources are are becoming complex sources, in which the values of the voltages and currents are complex numbers. Table 1 provides a list of conversion formulas from time-domain to frequency domain.

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;

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.

• 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 writting the signal of each source as a superposition of cosine waveforms (Fourier transforms).
• Complex numbers can be represented mathematically in rectangular from, 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 $\arctan(\frac{-1}{-1})=45^\circ$.