# Fourier Series of Full-Wave Rectifier Circuit

In an earlier post, we established that there is typically only a single component of the Fourier Series desired in most cases. All other components are effectively noise. An example of a typical full-wave rectifier circuit will help to illustrate this idea.

Let’s compute the Fourier Series of a full-wave rectified signal $|V_0 cos(\omega_{in}t)|$. What is the fundamental frequency? Which is the wanted component, and what is the RMS value? what are the amplitude and frequency of the largest unwanted component?

Full-wave rectified cosine function.

The waveform in question is plotted to the right. This waveform has period $T= \pi/\omega_{in}$. Choose $\tau = -T/2$ to take advantage of symmetry for the integrals. The radian frequency associated with the Fourier Series in this case is, by definition, $\omega = 2\pi/T = 2\omega_{in}$. The Fourier coefficients are then,

$a_0 = \frac{\omega_{in}}{\pi} \int_{-\pi/(2\omega_{in})}^{\pi/(2\omega_{in})} V_0 cos(\omega_{in} t) dt$

$= \frac{2\omega_{in}}{\pi} \int_{0}^{pi/(2\omega_{in})} V_0 cos(\omega_{in} t) dt$

$= \frac{2\omega_{in}}{\pi}[\frac{V_0}{\omega_{in}} sin(\omega_{in}t)]_0^{\pi/(2\omega_{in})}$

$a_0 = \frac{2V_0}{\pi}$

$a_n = \frac{\omega_{in}}{\pi} \int_{-\pi/(2\omega_{in})}^{\pi/(2\omega_{in})} V_0 cos(\omega_{in} t) cos(2\omega_{in} tn) dt$

$b_n = \frac{\omega_{in}}{\pi} \int_{-\pi/(2\omega_{in})}^{\pi/(2\omega_{in})} V_0 cos(\omega_{in} t) sin(2\omega_{in} tn) dt$

The coefficients $a_n$ and $b_n$ can benefit from the change of variables $\theta = \omega_{in} t$. Then integral tables or computer tools can readily be applied. The results are

$a_n = \frac{4}{\pi} \int_{0}^{\pi/2} V_0 cos(\theta) cos(2n\theta) dt$

$= \frac{4V_0 cos(n\pi)}{\pi(1-4n^2)}$

$b_n = \frac{2}{\pi} \int_{-\pi/2}^{\pi/2} V_0 cos(\theta) sin(2n\theta) dt$

$=0$ for all $n \geq 1$

In summary, the Fourier Series for a full wave rectified cosine can be written as

$f(t) = \frac{2V_0}{\pi} + \frac{4V_0}{\pi} \sum\limits_{n=1}^{\infty}\frac{cos(n\pi)}{1-4n^2} cos(2\omega_{in} tn)$

The fundamental frequency is twice that of the AC input. The wanted component is the DC term, with an RMS value of $2V_0/\pi$. The amplitudes $c_n$ decrease rapidly as n increases, and the largest unwanted component is the $n=1$ term.

$\frac{rV_0}{3\pi} cos(2\omega_{in} t)$

An example of building a square wave with the summation of the first four sinusoidal Fourier components.

Look at the components of the full wave rectified signal to see the reconstruction of the original waveform. Since the coefficients fall so rapidly with increasing $n$, only a few terms are needed to give an excellent approximation of the original signal.

An inverter might switch a DC input to approximate an AC output. The simplest version would produce a square wave of a given radian frequency $\omega_{out}$, the series for a square wave will be useful in several types of converters.