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.