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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.

Actual response and the ideal response of a low pass filter
Figure 01: Actual response and the ideal response of a low-pass filter

Active low pass filters

Circuit diagram of an active low pass filter
Figure 02: Circuit diagram of an active low-pass filter

Calculation of transfer function of an active low-pass filter

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

Introduction to Digital Filters eq 01

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,

Low pass filter eq 01

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

Low pass filter eq 02

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,

Low pass filter eq 03

Let’s take

Low pass filter eq 04

Where,

  • G0 = DC gain (Low-frequency gain)

So, the voltage gain is given by,

Low pass filter eq 05

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
Low pass filter eq 06

The voltage gain is expressed in decibels (dB).

Low pass filter eq 07

  • When ω = ωc where the frequency is equal to the critical frequency (f = fc).
Low pass filter eq 08
  • When ω >>> ωc
Low pass filter eq 09

Voltage gain at different ω values (Active low pass filter)
Table 01: Voltage gain at different ω values (Active low-pass filter)

Voltage Gain (dB) vs frequency response Graph (Active low pass filter)
Figure 03: Voltage Gain (dB) vs frequency response Graph (Active low pass filter)

Passive low pass filters

Circuit diagram of a passive low pass filter
Figure 04: Circuit diagram of a passive low pass filter

Calculation of transfer function of a passive low-pass filter

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

Low pass filter eq 10

Let’s take

Low pass filter eq 11

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
Low pass filter eq 12
  • When ω = ωc
Low pass filter eq 13
  • When ω >>> ωc
Low pass filter eq 14

Voltage gain at different ω values (Passive low pass filter)
Table 02: Voltage gain at different ω values (Passive low-pass filter)

Figure 05: Voltage Gain (dB) vs frequency response Graph (Passive low pass filter)
Figure 05: Voltage Gain (dB) vs frequency response Graph (Passive low pass filter)

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References and Attributes

Figures:

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


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