In Problems 4 to 10, the sketches show several practical examples of electrical signals (voltages or currents). In each case we want to know the harmonic content of the signal, that is, what frequencies it contains and in what proportions. To find this, expand each function in an appropriate Fourier series. Assume in each case that the part of the graph shown is repeated sixty times per second

Square wave

A harmonic in an electric signal is the signal content at a particular frequency that is a multiple integral of the main frequency produced by the generators. A complicated signal in the time domain can be observed with an oscilloscope.

The given sketch of an electrical signal is

The value of harmonic content of signal is to be found

$V\left(t\right)=\frac{a_0}{2}+\sum _{n-odd}^{\infty }{a}_n\frac{\cos \left(\frac{2\pi nt}{\tau }\right)}{n}+\sum _{n-odd}^{\infty }{b}_n\frac{\sin \left(\frac{2\pi nt}{\tau }\right)}{n}$

First find the value of b_n

The given sketch is of an odd function which is odd at point $t=\frac{\tau }{2}$where$\tau =\frac{1}{60}$

Calculate b_n as

$$\begin{gathered} b_n=\frac{2}{t}\underset{0}{\overset{t}{\int }}A\sin \frac{2\mathrm{\pi }nt}{\tau }dt \\ =2\times \frac{2}{\tau }\underset{0}{\overset{\tau /2}{\int }}A\sin \frac{2\pi nt}{\tau }dt \\ =\frac{4A}{\tau }\times {{\left.\frac{\tau }{2n}\cos \frac{2\pi nt}{\tau }\right|}^{\tau /2}}_0 \\ =-\frac{2A}{n}((-1{)}^n-1) \end{gathered}$$

Further solve to get value of b_n as

$b_n=\frac{4A}{n\pi }$where n is odd and $$\begin{gathered} a_0=a_n \\ =0 \end{gathered}$$

$V\left(t\right)=\frac{a_0}{2}+\sum _{n-odd}^{\infty }{a}_n\frac{\cos \left(\frac{2\pi nt}{\tau }\right)}{n}+\sum _{n-odd}^{\infty }{b}_n\frac{\sin \left(\frac{2\pi nt}{\tau }\right)}{n}$

$$\begin{gathered} =\frac{4A}{\pi }\sum _{n-odd}^{\infty }\frac{\sin \left(\frac{2\pi nt}{\tau }\right)}{n} \\ =\frac{400}{\pi }\sum _{n-odd}^{\infty }\frac{\sin \left(120\pi nt\right)}{n} \end{gathered}$$

Therefore, the hormonic content of signal of the given sketch

is$$\begin{gathered} \text{V(t) =}\frac{\text{4A}}{\mathrm{\pi }}\sum _{\text{n - odd}}^{\infty }\frac{\text{sin(2πnt/τ)}}{\text{n}} \\ \text{=}\frac{\text{400}}{\pi }\sum _{\text{n - odd}}^{\infty }\frac{\text{sin(120πnt)}}{\text{n}} \end{gathered}$$