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Active and Passive Frequency Filters – Formulas & Equations

Frequency Filters – Active and Passive Filters Equations and Formulas

Active and Passive Frequency Filters - Formulas & Equations

Frequency Filters:

Passive Filters

The type of frequency selecting circuits that are made of only passive components such as resistor, capacitor and inductor.

Low Pass Filter:

It passes low input frequency without any attenuation & blocks high frequency after a fix point known as cutoff frequency.

The output is taken across C & R in RC & RL circuit respectively.

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Cutoff Frequency:

The frequency where the output signal becomes the 70.7% of the input signal is called cutoff, corner or breakpoint frequency, & it is given by;

Cutoff Frequency of Low Pass Filter

Transfer Function:

The transfer function for both series RC & RL circuit is same;

transfer function low pass filter

Time Constant:

Time constant plays an important role in defining the cutoff frequency of the ciruit.

  • τ = 1 / ωc                           ­For Both circuit
  • τ = L / R                      For RL circuit
  • τ = RC                        For RC circuit
High Pass Filter:

This type of filter allows high frequency component from its input signal. The circuit used for HPF is same as LPF but the output is taken across R & L in RC & RL circuit respectively.

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Cutoff Frequency:

Same as the Low pass filter.

Cutoff Frequency of High Pass FilterTransfer Function:

Only transfer function is changed due to changing the output element.

transfer function high pass filter

Time Constant:

It will also remain same.

  • τ = 1 / ωc                           ­For Both circuit
  • τ = L / R                      For RL circuit
  • τ = RC                        For RC circuit
Band-Pass Filter:

it allows a fixed range of frequency & blocks every other frequency component before or after that allowable region.

Center Frequency:

the center of the allowable band of frequency fc is given by;

Center Frequency of band Pass Filter

Cutoff Frequency:

There are two cutoff frequency in band pass filters i.e. Lower cutoff ωc1 & upper cutoff ωc2 , any frequency before ωc1 and after ωc2 is being blocked by the filter.

Cutoff Frequency of band Pass Filter

Bandwidth:

The total range of the allowable frequency is known as bandwidth, from lower cutoff to upper cutoff frequency.

β = ωc2 – ωc1

  • β = R/L                       For Series RLC
  • β = 1/RC                    For Parallel RLC
Quality Factor:

Quality Factor of band pass filter

Transfer Function:

transfer function band pass filter

Band Reject Filter:

Band reject filter has the same circuit to a band pass filter, except the output is taken across both inductor L and Capacitor C. Thus only the transfer function changes.

Transfer Function:

transfer function band reject filter

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Active Filters:

They allow specific frequencies with a gain which can be modified using the resistor network.

First Order Low Pass & High Pass Filter:

The first order filter contains only one reactive component.

Active Low Pass & High Pass Filter

Cutoff Frequency:

The cutoff frequency for both high pass & low pass active filter;

cutoff frequency active filter

Gain:

Total output voltage gain for this filter is given by;

K = R2 / R­1

Transfer Function:

The transfer function for both low pass & high pass active filter with the gain K is given by;

transfer function for both low pass & high pass active filter

Scaling:

Scaling allow us to use more realistic values of resistors, inductors and capacitors while keeping the quality of the filter. It can be used in passive as well as active filters. There are two types of scaling i.e. magnitude scaling & frequency scaling.

Magnitude Scaling

if you only want to scale the magnitude of the filter.

  • R’ = kR
  • L’ = km L
  • C’ = C / km

Frequency Scaling

When you only want to scale the frequency of the filter

  • R’ = R
  • L’ = L / kf
  • C’ = C / kf

Simultaneous Scaling

When you want to scale the both frequency & magnitude of the filter;

  • R’ = kR
  • L’ = (km/kf) L
  • C’ = (1/kmkf) C
  • R’ = scaled resistance
  • L’ = scaled inductance
  • C’ = scaled capacitance
  • km = Magnitude scaling factor
  • kf = frequency scaling factor

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