# Review of Fourier Series

Nearly any periodic function of time , $f(t)$ such that $f(t)=f(t+T)$, can be written as a sum of sinusoids known as the Fourier Series.

$f(t)= a_0 + \sum\limits_{n=1}^{\infty} a_n cos(n\omega t) + b_n sin(n\omega t)$

where the radian frequency $\omega$ is defined as $\omega = 2\pi/T$. This result applies to signals with jump discontinuities which are often encountered in the world of power electronics. While there are few very special periodic functions without Fourier Series representations, the properties of these exceptions are so bizarre as to be physically unrealistic. Any periodic symbol that can be produced by a real circuit meets the mathematical conditions for a Fourier Series representation. The coefficients in the series expression are given by

$a_0 = \frac{1}{T} \int_\tau^{\tau+T} f(t)dt$

$a_n =\frac{2}{T}\int_\tau^{\tau+T} f(t)cos(n\omega t)dt$

$b_n =\frac{2}{T}\int_\tau^{\tau+T} f(t)sin(n\omega t)dt$

The integrals can be computed beginning at any time $\tau$, as long as the interval of integration includes a full period. We can choose any convenient value for $\tau$, such that $\tau = 0$, or $\tau = -T/2$. Again, $\omega = 2\pi/T$.

In power electronics, it is often helpful to make the change of variables $\theta = \omega t$. The new variable $\theta$ defines an angular time, with units of radians, that helps emphasize the shape of a waveform rather than the explicit time. In the angular time coordinate, with $\tau = 0$, the series coefficients become

$a_0 = \frac{1}{2\pi} \int_0^{2\pi} f(\theta)d\theta$

$a_n = \frac{1}{\pi} \int_0^{2\pi} f(\theta)cos(n\theta)d\theta$

$b_n = \frac{1}{\pi} \int_0^{2\pi} f(\theta)sin(n\theta)d\theta$

The angular scale is very convenient for waveforms that differ in frequency but not in shape. For example, the qualitative behavior of a diode bridge is the same whether the input is 50 Hz, 60 Hz, 400 Hz, or something else. Keep in mind that angular time is formally a change of variables.

An alternative form of the Fourier Series is frequently used by electrical engineers. In this case, the sine and cosine terms are combined into phase shifted cosine functions. The series then has the form

$f(t)= \sum\limits_{n=0}^{\infty} c_n cos(n\omega t + \theta_n)$

Here,

$c_0 = a_0$

$\theta_0 =0$

$c_n = \sqrt{a_n^2+b_n^2}$

$-\theta_n = \tan^{-1}(b_n/a_n)$

In a typical power converter, the function $f(t)$ is a piecewise sinusoid: Many source waveforms are sinusoidal, and jumps are added as switch action selects the connections between inputs and outputs. Integrals for these waveforms for the Fourier coefficients are relatively easy to compute. Modern computer programming languages such as Matlab, Mathematica, Python, and many others can be used to automate the procedure for solving integrals numerically. In solving integrals, the orthagonality relations

are of special importance. It is also helpful to recall some symmetry relationships: If the function $f(x)$ has even symmetry (i.e. symmetry about the y-axis, like $cos(\theta)$), the coefficients $b_n = 0$ for all $n$. If the function has odd symmetry (i.e. symmetry about the origin, like $sin(\theta)$), the coefficients $a_n = 0$ for all $n$.

Each term $c_n cos(n\omega t + \theta_n)$ is called a Fourier component of a harmonic of the function $f(t)$. The Fourier component corresponding to n is called the nth component of the nth harmonic. The coefficient $c_n$ is the component amplitude, and $\theta_n$ is the component phase.

The term $c_0 = a_0$ is called the DC component of $f(t)$. The term $c_1 Cos(\omega t + \theta_1)$ is called the fundamental of $f(t)$. The frequency $1/T$ is the fundamental frequency of $f(t)$.

Each component also has an RMS value, $c_{0(RMS)} = c_0$, $c_{n(RMS)} = c_n/\sqrt{2}$.

It is important to realize that, while a given waveform might have an infinite number of Fourier components, in most practical applications only one is really desired. For example, if $f(t)$ is the output voltage waveform of an AC-DC converter, the value of $c_n$ corresponding to 180 Hz would be of interest. For the purposes of power electronics, it is convenient to define the following terms.

The term wanted component refers to the one Fourier component which is desired by the user or designer in a waveform. All other Fourier components are called unwanted components.

Given that there is only one wanted component in most situations, the others are, in effect, noise.