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# Low-Pass Filters (LPF) - Active Low Pass Filter and Passive Low Pass Filter  B.Sc (Hons, USJ) (Polymer Science and Technology, Chemistry, Physics)
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Last Updated On : January 20, 2023
Published Date : July 25, 2022

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

At low frequencies, XC (Impedance given by the capacitor) is very high, and the capacitor circuit can be considered an open circuit. So under this condition, Vout= Vin or Gain = 1.

At very high frequencies, XC is very low, and the Vout is small as compared with Vin. 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, Vin can be given by,

To calculate Vo, the total impedance (Z) given by resistor (R2)and capacitor (C) should be calculated.

Where,

• Z = Total impedance given by resistor and capacitor
• R2 = Resistance of the R2 resistor
• ω = Angular frequency of the signal
• Xc = Impedance given by the capacitor
• C = Capacitance of the capacitor

According to Ohm’s law, Vo is given by,

Let’s take

Where,

• G0 = 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 <<< ωc

The voltage gain is expressed in decibels (dB).

• When ω = ωc where the frequency is equal to the critical frequency (f = fc).
• When ω >>> ωc

## 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

• When ω <<< ωc
• When ω = ωc
• When ω >>> ωc ## References and Attributes

### Figures:

The cover image was created using an image by Marijn Hubert from Pixabay