Band Pass Filter

Band Pass Filters

The cut-off frequency or ƒc point in a simple RC passive filter can be accurately controlled using just a single resistor in series with a non-polarized capacitor, and depending upon which way around they are connected a Low Pass or a High Pass filter is obtained. One simple use for these types of filters is in audio type applications or circuits such as in loudspeaker crossover filters or pre-amplifier tone controls. Sometimes it is necessary to only pass a certain range of frequencies that do not begin at 0Hz, (DC) or end at some high frequency point. By connecting or "Cascading" together a single Low Pass Filter circuit with a High Pass Filter circuit in series we can produce another type of passive RC filter that passes a selected range or band of frequencies that can be either narrow or wide while attenuating all those outside of this range. This new type of filter arrangement is commonly called a Band Pass Filter or BPF for short.

Unlike a Low Pass Filter that only pass signals of a low frequency range or a High Pass Filter which pass signals of a higher frequency range, a Band Pass Filters only pass signals within a certain "band" or "spread" of frequencies without distorting the input signal or introducing extra noise. This band of frequencies is commonly known as the Bandwidth and is defined as the frequency range between two specified frequency cut-off points (ƒc), that are 3dB below the maximum centre or resonant peak while attenuating or weakening the others outside of these two points.

Then we can simply define the term "bandwidth" as being the difference between the lower cut-off frequency ( ƒc_LOWER ) and the upper cut-off frequency ( ƒc_UPPER ) points. In order for a passband filter to function correctly, the cut-off frequency of the low pass filter must be higher than the cut-off frequency for the high pass filter.

This type of Band Pass Filter is known generally as a Second Order Filter, (2nd-Order) because it has "two" reactive component within its circuit design. One capacitor in the low pass circuit and another capacitor in the high pass circuit.

Frequency Response of a 2nd Order Band Pass Filter.

The Bode Plot or Frequency Response Curve shows the characteristics of the band pass filter. Here the signal is attenuated at low frequencies and the output increases at +20dB/Decade (6dB/Octave) until the frequency reaches the "lower cut-off" point ƒc(HP). At this frequency the output voltage is again 70.7% of the input signal value or -3dB (20 log (Vout/Vin)) of the input. The output continues at maximum gain until it reaches the "upper cut-off" point ƒc(LP) where the output decreases at a rate of -20dB/Decade (6dB/Octave) attenuating any high frequency signals. The point of maximum output gain is generally the geometric mean of the two -3dB value between the lower and upper cut-off points and is called the "Centre Frequency" or "Resonant Peak" value ƒr.This geometric mean value is calculated as being ƒr² = ƒc-upper x ƒc-lower.

A band pass filter is a 2nd Order type filter because it has "two" reactive components within its circuit structure, then the phase angle will be twice that of the previously seen 1st order filters, ie 180^o. The phase angle of the output signal LEADS that of the input by +90^o up to the centre or resonant frequency point ƒr were it becomes "zero" degrees (0^o) or "in-phase" and then changes to LAG the input by -90^o as the output frequency increases.

The upper and lower cut-off frequency points for a band pass filter can be found using the same formula as that of the low and high pass filters, For example.

Example No1.

A 2nd order band pass filter is to be constructed using RC components that will only allow a band of frequencies to pass above 1kHz and below 30kHz. Assuming that both resistor values are 15.9kΩ´s, calculate the values of the two capacitors required.

The High Pass Filter Stage.

The value of the capacitor C1 need to give a cut-off frequency fc(HP) of 1kHz with a resistor value of 15.9kΩ is:

Then, the values of R1 and C1 required for the high pass stage to give a cut-off frequency of 1.0kHz is,
R1 = 15.9kΩ´s and C1 = 10nF.

The Low Pass Filter Stage.

The value of the capacitor C2 need to give a cut-off frequency fc(LP) of 30kHz with a resistor value of 15.9kΩ is:

Then, the values of R2 and C2 required for the low pass stage to give a cut-off frequency of 30kHz is,
R = 15.9kΩ´s and C = 334pF.

With the values of both the resistances R1 and R2 given as 15.9kΩ, and the two values of the capacitors C1 and C2 found for both the high pass and low pass filters as 10nF and 334pF respectively, then the circuit for our simple passive Band Pass Filter is shown below.

Completed Band Pass Filter Circuit

Resonant Frequency.

We can also calculate the "Resonant" or "Centre Frequency" (ƒr) point of the bandpass filter were the output gain is at its maximum or peak value. This peak value is not the arithmetic average of the upper and lower -3dB cut-off points as you might expect but is in fact the "geometric" or mean value. This geometric mean value is calculated as being ƒr² = ƒc_UPPER x ƒc_LOWER for example:

Centre Frequency Equation

• 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 calculated cut-off frequencies were found as ƒ_L = 1.0kHz and ƒ_H = 30kHz.

Then by substituting these values into the above equation gives a central resonant frequency of:

Band Pass Summary

A Bandpass Filter can be made by cascading together a Low Pass Filter and a High Pass Filter with the frequency range between the lower and upper -3dB cut-off points being know as the filters "Bandwidth". The centre or resonant frequency point is the geometric mean of the lower and upper cut-off points. At this centre frequency the output signal is at its maximum and the phase shift of the output signal is the same as the input signal. The output signal from a bandpass filter or any passive RC filter will always be less than that of the input signal giving a voltage gain of less than 1 (Unity). To provide an output signal with a voltage gain greater than 1, some form of amplification is required within the design of the circuit.

A Band pass Filter is a 2nd order type filter because it has two reactive components within its design. It is made up from two single RC filter circuits that are each 1st order filters. If more filters are cascaded together the resulting circuit is known as an "N^th-order" filter where the "N" stands for the number of individual reactive components within the circuit, ie, 4^th-order , 10^th-order, etc and the higher the filters order the steeper will be the slope. However, a single capacitor made by combining together two or more individual capacitors is still one capacitor.

Our example above shows the output frequency response curve for an "ideal" band pass filter with constant gain in the passband and zero gain in the stopbands. In practice the frequency response of this Band Pass Filter circuit would not be the same as the input reactance of the high pass circuit would affect the frequency response of the low pass circuit (components connected in series or parallel) and vice versa. One way of overcoming this would be to provide some form of electrical isolation between the two filter circuits as shown below.

Buffering Individual Filter Stages

One way of combining amplification and filtering into the same circuit would be to use an Operational Amplifier or Op-amp, and examples of these are given in the Operational Amplifier section. Filter circuits which use an operational amplifier within their design are generally known as Active Filters.