Question

When a current Iflows through a resistance, the heat energydissipated per secondis the average value of$\mathit{R}{\mathit{I}}^{\mathbf{2}}$. Let a periodic (not sinusoidal) current I(t) be expanded in a Fourier series$\mathbf{I}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{\sum }}_{\mathbf{-}\mathbf{\infty }}^{\mathbf{\infty }}{\mathbf{c}}_{\mathbf{n}}{\mathbf{e}}^{\mathbf{120}\mathbf{i}\mathbf{n}\mathbf{\pi }\mathbf{t}}$.Give a physical meaning to Parseval’s theorem for this problem.

Short answer

By Parseval theorem, the power of the signal is equal to the sum of the powers output by the individual harmonic components:

$P=R\sum _{n=-\infty }^{\infty }{\left|I_n\right|}^2=R\sum _{n=-\infty }^{\infty }{\left|c_n\right|}^2$

Step-by-step solution

Given Information.

The givenFourier series $I\left(t\right)={\sum }_{-\infty }^{\infty }{c}_n{e}^{120in\pi t}$.

Definition of Parseval’s theorem

Parseval’s theorem is a theorem stating that the energy of a signal can be expressed as its frequency components’ average energy.

Physical meaning of Parseval’s theorem

It is know that the power for one harmonic component of the current is

$P_n=<R{\left|I_n\right|}^2>=R<{\left|I_n\right|}^2>=R{\left|c_n\right|}^2$

Now,$P_n=R\left(c_n{e}^{120in\pi t}\right)(c_n{e}^{120in\pi t}{)}^{*}=R{c}_n{c_n}^{*}=R{\left|c_n\right|}^2$

If resistance is constant the power due to the entire signal is given by

$P=<R{\left|I\right|}^2>=R<{\left|I\right|}^2>=R\sum _{n=-\infty }^{\infty }{\left|c_n\right|}^2$

Thus the power output by the entire signal is equal to the sum of the powers of the individual harmonic components,