# Calculating output in Frequency domain, RC Circuit.

Cuthbert Nyack
In this page a Square wave is assumed to be passed through an RC circuit and the output is to be calculated in the frequency domain. To do this each component in the spectrum of the square wave is multiplied by the transfer function of the RC at that frequency. A new Fourier series is then generated by multiplying each component of the spectrum by the appropriate value of the transfer function. Summing this new series gives the output from the RC circuit. The square wave and its spectrum is shown below. The spectrum converges as 1/n If the time constant of the RC is much longer than the period of the square wave then the output approximates that of a triangular wave. The triangular wave and its spectrum is shown below. The spectrum converges as 1/(n**2). The Fourier series representation of the square wave input is given by:- The Fourier series representation of the output is given by:- This can also be represented by the usual Fourier Series. The proceedure for finding the output is shown below. First the input function Fi(t) is represented by the fourier series coefficients. Each coefficient is then multiplied by the transfer function of the RC circuit at that frequency to obtain the output coefficient and phase. The resulting output coefficients and phase are then used in a Fourier Series to obtain the output function Fo(t). The applet below reproduces the output from the RC circuit by summing the contributions from the different frequency components. The magenta line shows the input square wave with amplitude 0.5, the red line shows the reconstructed output. The spectra of the input is shown in green(even function) while the even and odd parts of the output spectra is shown in red and orange respectively. The phase change introduced by the RC circuit causes an even function to become part even and part odd. For small time constants, the output is similar to the input while for large time constants, it approximates a triangular wave with spectrum converging as 1/(n**2). This is a result of the 1/n dependence of the square wave series and the 1/n dependence of the RC transfer function for large n. The first scrollbar changes the number of terms summed. Ratio of RC time constant to square wave period can be varied by the second scrollbar. The gain scrollbar can be used to make the higher components in the spectra visible.

When activated the following gif image show how the applet should appear. 