A low pass filter is a filter that passes frequencies from 0Hz to critical frequency, f_{c}, and significantly attenuates all other frequencies.

At low frequencies, X_{C} (Impedance given by the capacitor) is very high, and the capacitor circuit can be considered an open circuit. So under this condition, V_{out}= V_{in} or Gain = 1.

At very high frequencies, X_{C} is very low, and the V_{out} is small as compared with V_{in}. Hence the gain falls and drops off gradually as the frequency is increased.

Active low pass filters

Calculation of transfer function of an active low-pass filter

The voltage gain of an active low pass filter can be calculated as follows,

To construct this filter, we are using 741 op-amp. Here, noninverting input (V+) input has been grounded. Therefore, the voltage at V+ is zero. There is no voltage difference between the noninverting input (V+) and the inverting input (V-). So, the voltage at inverting input V- is also zero. According to Ohm’s low, V_{in} can be given by,

To calculate V_{o}, the total impedance (Z) given by resistor (R_{2})and capacitor (C) should be calculated.

Where,

Z = Total impedance given by resistor and capacitor

R_{2} = Resistance of the R_{2} resistor

ω = Angular frequency of the signal

X_{c} = Impedance given by the capacitor

C = Capacitance of the capacitor

According to Ohm’s law, V_{o} is given by,

Let’s take

Where,

G_{0} = DC gain (Low-frequency gain)

So, the voltage gain is given by,

The above equation is the transfer function of an active low-pass filter. Using this equation, the voltage gain at each ω (Angular frequency of the signal) value can be calculated.

Calculation of voltage gain at different ω (frequency) values using the transfer function of an active low-pass filter

When ω <<< ω_{c}

The voltage gain is expressed in decibels (dB).

When ω = ω_{c} where the frequency is equal to the critical frequency (f = f_{c}).

When ω >>> ω_{c}

Passive low pass filters

Calculation of transfer function of a passive low-pass filter

The voltage gain of a passive low pass filter can be calculated as follows,

Let’s take

The above equation is the transfer function of a passive low-pass filter. Using this equation, the voltage gain at each ω (Angular frequency of the signal) value can be calculated.

Calculation of voltage gain at different ω (frequency) values using the transfer function of a passive low-pass filter