Frequency Response Plot of a Low-Pass Filter

The circuit shown in Figure 5.3-4 consists of a resistor and a capacitor and is thus called .an RC circuit. If we apply a sinusoidal voltage Vi, called the input voltage, to the circuit as shown, then eventually the output voltage V o will be sinusoidal also, with the me frequency but with a different amplitude and shifted in time relative to the input voltage. Specifically, if Vi = A i sin cot, then V o = A o sin (wt +<p). The frequency-response plot is a plot of A o/ A i versus frequency w. It is usually plotted on logarithmic axes. Upper-level engineering courses explain that for the RC circuit shown, this ratio depends on wand RC as follows:

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where s = w i. For RC = 0.1 second, obtain the log-log plot of A o/ A d versus wand use it to find the range of frequencies for which the output amplitude A o is less than 70 percent of the input amplitude A i .
Solution
As with many graphical procedures, you must guess a range for the parameters in question. Here we must guess a range to use for the frequency ca. If we use 1 :5 w :5 100 rad/s, we

An RC circuit

An RC circuit

Frequency-response plot of a low-passRC circuit.

Frequency-response plot of a low-passRC circuit.

will see the part of the curve that is of interest. The MATLAB script file is as follows:
RC = 0.1; s = [l lDD] *i M = abs(l./(RC*s+l»; log 1og (imag(s),M),grid,x label (‘Frequency(rad/s) ‘),
y label (‘Output/Input Ratio”) title(‘Frequency Response of a Low-Pass RC Circuit (RC = 0.1 s) ‘) Figure 5.3-5 shows the plot. We can see that the output/input ratio A o / A i decreases as the frequency w increases. The ratio is approximately 0.7 at w = 10 rad/s. The amplitude of any input signal having a frequency greater than this frequency will decrease by at least 30 percent. Thus this circuit is called a low-pass filter because it passes low-frequency signals better than it passes high-frequency signals. Such a circuit is often used to filter out noise from nearby electrical machinery.

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