Electronics in a phy ...

One method of describing electrical signals is the description in the frequency domain. It consists in providing characteristics such as the amplitude, frequency and phase of harmonic waveforms into which the signal \(f(t)\) can be decomposed. The set of these data creates the frequency spectrum of the \(F(\omega)\) signal.

A continuous and periodic time function with period T can be represented as a sum of harmonic functions that form a Fourier series:

\( f(t) = \frac{a_0}{2} + \sum_{k=1}^{\infty}{\left [ a_k cos \left ( \frac{2\pi kt}{T} \right ) + b_k sin \left ( \frac{2\pi kt}{T} \right ) \right ]} \hspace{1cm} (1) \)

where:

\( a_k = \frac{2}{T} \int_{0}^{T}{f(t) cos \left ( \frac{2\pi kt}{T} \right )} \hspace{1cm} (2) \\ b_k = \frac{2}{T} \int_{0}^{T}{f(t) sin \left ( \frac{2\pi kt}{T} \right )} \hspace{1cm} (3) \\ a_o = \frac{2}{T} \int_{0}^{T}{f(t)dt} \hspace{1cm} (4) \)

The circular frequencies of the successive components of the series are multiples of the fundamental frequency \(\omega_1\) and are called harmonic frequencies.

\( \omega_k = k \omega_1 = k \frac{2\pi}{T} \hspace{1cm} (5) \)

The series (1) can be written in the form:

\( f(t) = \frac{a_0}{2} + \sum_{k=1}^{\infty} c_k cos \left ( \frac{2\pi kt}{T} -\phi_k \right )  \hspace{1cm} (6)\)

where:

\( c_k = \sqrt{a_k^2 + b_k^2} \hspace{1cm} (7) \\ \phi_k = arctan \left ( \frac{b_k}{a_k} \right ) \hspace{1cm} (8) \)

Here we may observe ambiguity in phase - e.g. if \(b_k\) and \(a_k\) are negative. So we need to correct the angle. In programming languages this is frequently solved by using the atan2(b,a) function.