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 either a low pass or a high pass filter is obtained.
One simple use for these types of filters is in audio amplifier 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 but are within a certain frequency band,
either narrow or wide.
By connecting or "cascading" together a single Low Pass Filter circuit with a
High Pass Filter circuit, 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 passive filter arrangement produces a frequency selective filter known commonly as a Band Pass Filter
or BPF for short.
Band Pass Filter Circuit
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 passes signals within a certain "band" or "spread" of
frequencies without distorting the input signal or introducing extra noise. This band of frequencies can be any width
and is commonly known as the filters Bandwidth. Bandwidth 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 for widely spread frequencies, we can simply define the term "bandwidth", BW
as being the difference between the lower cut-off frequency ( ƒcLOWER ) and
the higher cut-off frequency ( ƒcHIGHER ) points. In other words,
BW = ƒH - ƒL. Clearly for a pass band 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.
The "ideal" Band Pass Filter can also be used to isolate or filter out certain frequencies
that lie within a particular band of frequencies, for example, noise cancellation. Band pass filters are known generally as
second-order filters, (two-pole) because they have "two" reactive component, the capacitors, within their 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 above shows the characteristics of the
band pass filter. Here the signal is attenuated at low frequencies with the output increasing at a slope of
+20dB/Decade (6dB/Octave) until the frequency reaches the "lower cut-off" point ƒL.
At this frequency the output voltage is again 1/√2 = 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
ƒH 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 ƒr 2 =
ƒ(UPPER) x ƒ(LOWER).
A band pass filter is regarded as a second-order (two-pole) type filter because it has "two" reactive
components within its circuit structure, then the phase angle will be twice that of the previously seen first-order filters,
ie 180o. The phase angle of the output signal LEADS that of the input by +90o up
to the centre or resonant frequency, ƒr point were it becomes "zero" degrees (0o)
or "in-phase" and then changes to LAG the input by -90o 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 for both the low and high pass filters, For example.
Then clearly, the width of the pass band of the filter can be controlled by the positioning
of the two cut-off frequency points of the two filters.
A second-order band pass filter is to be constructed using RC components that
will only allow a range of frequencies to pass above 1kHz (1,000Hz) and below 30kHz (30,000Hz). Assuming that both
the resistors have values of 10kΩīs, calculate the values of the two capacitors required.
The High Pass Filter Stage.
The value of the capacitor C1 required to give a cut-off frequency
ƒL of 1kHz with a resistor value of 10kΩ
is calculated as:
Then, the values of R1 and C1 required for the
high pass stage to give a cut-off frequency of 1.0kHz are,
R1 = 10kΩīs and
C1 = 15nF.
The Low Pass Filter Stage.
The value of the capacitor C2 required to give a cut-off frequency
ƒH of 30kHz with a resistor value of 10kΩ
is calculated as:
Then, the values of R2 and C2 required for the
low pass stage to give a cut-off frequency of 30kHz are,
R = 10kΩīs and
C = 510pF. However, the nearest preferred value of the calculated capacitor value of
510pF is 560pF so this is used instead.
With the values of both the resistances R1 and R2
given as 10kΩ, and the two values of the capacitors C1 and
C2 found for both the high pass and low pass filters as 15nF and
560pF respectively, then the circuit for our simple passive Band Pass Filter
is given as.
Completed Band Pass Filter Circuit
We can also calculate the "Resonant" or "Centre Frequency" (ƒr) point
of the band pass 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 ƒr 2 =
ƒc(UPPER) x ƒc(LOWER) for example:
Centre Frequency Equation
- Where, ƒr is the resonant or centre frequency
- ƒL is the lower -3dB cut-off frequency point
- ƒH is the upper -3db cut-off frequency point
and in our simple example above, the calculated cut-off frequencies were found to be
ƒL = 1,060 Hz and ƒH = 28,420 Hz
using the filter values.
Then by substituting these values into the above equation gives a central resonant frequency of:
Band Pass Filter Summary
A Band pass 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
The amplitude of the output signal from a band pass filter or any passive RC filter for that matter, will
always be less than that of the input signal. In other words a passive filter is also an attenuator giving a voltage gain of
less than 1 (Unity). To provide an output signal with a voltage gain greater than unity, some form of amplification is
required within the design of the circuit.
A Passive Band Pass Filter is classed as a second-order type filter because it
has two reactive components within its design, the capacitors. It is made up from two single RC filter circuits that
are each first-order filters themselves.
If more filters are cascaded together the resulting circuit will be known as an "nth-order"
filter where the "n" stands for the number of individual reactive components and therefore poles within the filter circuit.
For example, filters can be a 2nd-order, 4th-order, 10th-order, etc. The higher the filters
order the steeper will be the slope at n times -20dB/decade. However, a single capacitor value 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 pass band and zero gain in the stop bands. 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.
In the next tutorial we will look at filter circuits which use an operational amplifier within their design to not only
to introduce gain but provide isolation between stages. These types of filter arrangements are generally known as