Question

Determine the Fourier series for the square voltage pulse shown in Fig.  and plot its line spectrum.

Answer 1

The wave represents a periodic function of θ or ωt or (2πt/T) having a period extending over 2π radians or T seconds. The general expression for this wave can be written as

f(θ) = a₀ + a₁cosθ + a₂cos2θ + a₃cos3θ + .....+ b₁sinθ + b₂sin2θ + b₃sin3θ + .....

Hence, substituting the values of a₀, a₁, a₂ etc. and b₁, b₂, etc. in the above given Fourier series, we get

f(θ) = V/2 + 2V/πsinθ + 2V/3π sin3θ + 2V/5πsin5θ + ..... = E/2 + 2V/π(sinω₀t + 1/3sin3ω₀t + 1/5sinω₀t + ....)

It is seen that the Fourier series contains a constant term V/2 and odd harmonic components whose amplitudes are as under:

Amplitude of fundamental or first harmonic = 2V/π

Amplitude of second harmonic = 2V/2π

Amplitude of third harmonic = 2V/5 and so. on

The plot of harmonic amplitude versus the harmonic frequencies (called line spectrum) is shown in Fig..