Active Band Pass Filter

Active Band Pass Filter

As we saw previously in the Passive Band Pass Filter tutorial, the principal characteristic of a Band Pass Filter or any filter for that matter, is its ability to pass frequencies relatively unattenuated over a specified band or spread of frequencies called the "Passband". For a low pass filter this passband starts from 0Hz or DC and continues up to the specified cut-off point at -3dB. Equally, for a high pass filter the passband starts from this -3dB cut-off frequency and continues up to infinity or the maximum open loop gain for an active filter.

However, the Active Band Pass Filter is slightly different in that it will only pass frequencies or signals within a certain "Band" or range of frequencies that are set between two cut-off or corner points labelled "lower frequency" (ƒL) and "higher frequency" (ƒH) while attenuating any signals outside of these two points.

A simple Active Band Pass Filter can be easily made by cascading together a single Low Pass Filter with a single High Pass Filter. The cut-off or corner frequency of the low pass filter (LPF) is higher than the cut-off frequency of the high pass filter (HPF) and the difference between the frequencies at the -3dB point will determine the "Bandwidth" of the filter while attenuating any signals outside of these points. One way of making a simple Active Bandpass Filter is to connect the basic RC high and low pass filters to an op-amp circuit as shown.

Active Band Pass Filter Circuit

The high or upper corner point (ƒH) as well as the lower corner frequency cut-off point (ƒL) are calculated the same as before in the standard 1st-Order low and high pass filter circuits. The amplifier defines the overall voltage gain of the circuit. The Bandwidth of the circuit is the difference between the upper and lower -3dB points. For example, if the -3dB cut-off points are at 200Hz and 600Hz then the bandwidth of the filter would be given as: Bandwidth (BW) = 600 - 200 = 400Hz.

Band Pass Frequency Response Curve

An Active band pass filter can also be made using an Inverting Operational Amplifier by rearranging the resistors and capacitors as shown. The lower cut-off -3dB point is given by ƒc2 while the upper cut-off -3dB point is given by ƒc1.

Inverting Band Pass Filter Circuit

Resonant Frequency

The actual shape of the frequency response curve for any passive or active band pass filter will depend upon the characteristics of the filter circuit with the curve above being defined as an "ideal" band pass response. An active band pass filter is a 2nd Order type filter because it has "two" reactive components (two capacitors) within its circuit design and will have a peak response or Resonant Frequency (ƒr) at its "centre frequency", ƒc. The centre frequency is generally calculated as being the geometric mean of the two -3dB frequencies between the upper and the lower cut-off points with the resonant frequency (point of oscillation) being given as:

• Where:
• ƒ_r is the resonant or Centre Frequency
• ƒ_L is the lower -3dB cut-off frequency point
• ƒ_H is the upper -3db cut-off frequency point

and in our simple example above the resonant centre frequency is given as:

The "Q" or Quality Factor

In a Band Pass Filter circuit, the overall width of the actual passband between the upper and lower -3dB corner points of the filter determines the Quality Factor or Q-point of the circuit. This Q Factor is a measure of how "Selective" or "Un-selective" the band pass filter is towards a given spread of frequencies. The lower the value of the Q factor the wider is the bandwidth of the filter and consequently the higher the Q factor the narrower and more "selective" is the filter.

The Quality Factor, Q of the filter is sometimes given the Greek symbol of Alpha, (α) and is known as the alpha-peak frequency where:

As the quality factor of a band pass filter (Second-order System) relates to the "sharpness" of the filters response around its centre resonant frequency (ƒr) it can also be thought of as the Damping Factor or Damping Coefficient because the more damping the filter has the flatter is its response and likewise, the less damping the filter has the sharper is its response. The damping ratio is given the Greek symbol of Xi, (ξ) where:

The "Q" of a band pass filter is the ratio of the Resonant Frequency, (ƒr) to the Bandwidth, (BW) between the upper and lower -3dB frequencies and is given as:

Then for our simple example above the quality factor "Q" of the band pass filter is given as:

346Hz / 400Hz = 0.865.     Note that Q is a ratio and has no units.