Experiment 9: Low Pass Filter

CTNA

Aim

To design RC low pass filter and study its frequency response.

Apparatus Required

  • Bread Board
  • CRO (Cathode Ray Oscilloscope)
  • Resistance
  • Capacitor
  • Connecting Wires and CRO Probes
  • Function Generator

Theory

A filter is a circuit that passes a specific range of frequencies while rejecting other frequencies. A passive filter consists of passive circuit elements such as capacitor, inductor, and resistor. A typical low pass filter is formed when the output of an RC circuit is taken off the capacitor as shown in the circuit diagram.

The transfer function is:

$$ H (\omega) = \frac{V_O}{V_I} = \frac{1/j\omega C}{R + 1/j\omega C} = \frac{1}{1 + j \omega RC} $$

This figure shows the plot of $|H (\omega)|$, along with ideal characteristics.

The half-power frequency, which is equivalent to the corner frequency on the Bode plots but in the context of filters is usually known as the cutoff frequency, is obtained by setting the magnitude $H(\omega)$ equal to $1/\sqrt{2}$:

$$ H (\omega_C) = \frac{1}{\sqrt{1 + (\omega^2 R^2 C^2)}} = \frac{1}{\sqrt{2}} $$

$$ \omega_C = \frac{1}{RC} $$

The cutoff frequency is also called rolloff frequency. A low-pass filter is designed to pass only frequencies from DC up to the cutoff frequency.

Formula Used

$$ \omega_C = \frac{1}{RC} $$

$$ \omega_C = 2 \pi f_C $$

Thus, the cutoff frequency is:

$$ f_C = \frac{1}{2 \pi RC} $$

In terms of the capacitor when the resistor is constant:

$$ C = \frac{1}{2 \pi R f_C} $$

Circuit Diagram

Procedure

  1. Set up the circuit as shown in the circuit diagram taking the output on the capacitor.
  2. The input for the filter is taken from the output of the function generator.
  3. The input of the filter is also connected to channel 1 and the output is connected to channel 2 of the CRO.
  4. Vary the frequency of the input signal over a wide range. Note the $V_{OUT}$ for each frequency and calculate the corresponding gain.
  5. Plot the values of gain versus frequency graph.

Observation Table

Sr. No. Frequency $V_{IN}$ $V_{OUT}$ $V_{OUT}/V_{IN}$

Result

A low pass filter for a given cut-off frequency is designed successfully.

Precautions

  • Check connections twice.
  • Make the circuit neat and clean.