A Basic Introduction to Band-pass Filter

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Electronic 2

Band-pass

active filter

Done by:

Mohammed Ahmed AL_Aqil

Talal alsindi

Instructor:

Eng.

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Introduction:

An active filter is a type of analog electronic filter , distinguished by the use of one or more active components i.e. voltage amplifiers or buffer amplifiers. Typically this will be a vacuum tube, or solid-state (transistor or operational amplifier ).

Active filters have three main advantages over passive filters:

• Inductors can be avoided. Passive filters without inductors cannot obtain a high Q (low damping), but with them are often large and expensive (at low frequencies), may have significant internal resistance, and may pick up surrounding electromagnetic signals.

• The shape of the response, the Q (Quality factor ), and the tuned frequency can often be set easily by varying resistors, in some filters one parameter can be adjusted without affecting the others. Variable inductances for low frequency filters are not practical.

• The amplifier powering the filter can be used to buffer the filter from the electronic components it drives or is fed from, variations in which could otherwise significantly affect the shape of the frequency response.

All the varieties of passive filters can also be found in active filters. Some of them are:

• High-pass filters – attenuation of frequencies below their cut-off points.

• Low-pass filters – attenuation of frequencies above their cut-off points.

• Band-pass filters – attenuation of frequencies both above and below those they allow to pass.

• Notch filters – attenuation of certain frequencies while allowing all others to pass.

Combinations are possible, such as notch and high-pass (for example, in a rumble filter where most of the offending rumble comes from a particular frequency), e.g.Elliptic filters.

Active Filters

Low-Pass filters - the integrator reconsidered.

In the first lab with op-amps we considered the time response of the integrator circuit, but its frequency response can also be studied.

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If you derive the transfer function for the circuit above you will find that it is of the form:

Which is the general form for first-order (one reactive element) low-pass filters. At high frequencies (

ω

>>

ω

o) the capacitor acts as a short, so the gain of the amplifier goes to

zero. At very low frequencies (

ω

<<

ω

o) the capacitor is an open and the gain of the

circuit is Ho. But what do we mean by low (or high) frequency?

We can consider the frequency to be high when the large majority of current goes through the capacitor; i.e., when the magnitude of the capacitor impedance is much less than that of R1. In other words, we have high frequency when 1/

ω

C<<R1, or

ω

>>1/R1C=

ω

o. Since R1 now has little effect on the circuit, it should act as an

integrator. Likewise low frequency occurs when

ω

<<1/R1C, and the circuit will act as an

amplifier with gain -R1 /R2= Ho.

High-Pass filters - the differentiator reconsidered.

The circuit below is a modified differentiator, and acts as a high pass filter. First Order High Pass Filter with Op Amp

Using analysis techniques similar to those used for the low pass filter, it can be shown that

Which is the general form for first-order (one reactive element) low-pass filters? At high frequencies (

ω

>>

ω

o) the capacitor acts as a short, so the gain of the amplifier goes to

H0= -R1 /R2. At very low frequencies (

ω

<<

ω

o) the capacitor is an open and the gain of

the circuit is Ho. For this circuit

ω

₀=1/R₂C. Therefore this circuit is a high-pass filter (it passes high frequency signals, and blocks low frequency signals.

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Band-Pass circuits

Besides low-pass filters, other common types are high-pass (passes only high frequency signals), band-reject (blocks certain signals) and band-pass (rejects high and low frequencies, passing only signal around some intermediate frequency).

Combining the first order low pass and high pass filters that we just looked at can make the simplest band-pass filter.

Simple Band Pass Filter with Op Amp

This circuit will attenuate low frequencies (

ω

<<1/R2C2) and high frequencies

(

ω

>>1/R1C1), but will pass intermediate frequencies with a gain of -R1 /R2. However,

this circuit cannot be used to make a filter with a very narrow band. To do that requires a more complex filter as discussed below.

High Q (Low Bandwidth) Bandpass Filters.

For a second-order band-pass filter the transfer function is given by

where

ω

o is the center frequency,

β

is the bandwidth and Ho is the maximum amplitude

of the filter. These quantities are shown on the diagram below. The quantities in parentheses are in radian frequencies, the other quantities are in Hertz (i.e. f o=

ω

o /2

π

,

B=

β

/2

π

). Looking at the equation above, or the figure, you can see that as

ω−>

0 and

ω

->infinity that |H(s=j

ω

)|

−>

0. You can also easily show that at

ω

=

ω

o that |H(s=j

ω

o)|

=H0. Often you will see the equation above written in terms of the quality factor, Q,

which can be defined in terms of the bandwidth,

β

, and center frequency,

ω

o, as

Q=

ω

o /

β

. Thus the Q, or quality, of a filter goes up as it becomes narrower and its

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If you derive the transfer function of the circuit shown below: High-Q Bandpass Filter with Op Amp

You will find that it acts as a band-pass filter with:

and the center frequency and bandwidth given by:

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The notation R1||R2 denotes the parallel combination of R1 and R2,

.

Explanation:

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 "Pass Band". For a low pass filter this pass band starts from 0Hz or DC and continues up to the specified cut-off frequency point at -3dB down from the maximum pass band gain. Equally, for a high pass filter the pass band 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 is a frequency selective filter circuit used in electronic systems to separate a signal at one particular frequency, or a range of signals that lie within a certain "band" of frequencies from signals at all other frequencies. This band or range of frequencies is set between two cut-off or corner frequency points labeled the "lower frequency" (ƒL) and the "higher

frequency" (ƒH) while attenuating any signals outside of these two points.

Simple Active Band Pass Filter can be easily made by cascading together a single Low Pass Filter with a single High Pass Filter as shown.

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 band pass filter while attenuating any signals outside of these points. One way of making a very simple Active Band Pass Filter is to connect the basic passive high and low pass filters we look at previously to an amplifying op-amp circuit as shown.

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This cascading together of the individual low and high pass passive filters produces a low "Q-factor" type filter circuit that has a wide pass band. The first stage of the filter will be the high pass stage that uses the capacitor to block any DC biasing from the source. This design has the advantage of producing a relatively flat asymmetrical pass band frequency response with one half representing the low pass response and the other half representing high pass response as shown.

The higher corner point (ƒH) as well as the lower corner frequency cut-off point (ƒL) are

calculated the same as before in the standard first-order low and high pass filter circuits. Obviously, a reasonable separation is required between the two cut-off points to prevent any interaction between the low pass and high pass stages. The amplifier provides isolation between the two stages and defines the overall voltage gain of the circuit. The bandwidth of the filter is therefore the difference between these 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. The normalised frequency response and phase shift for an active band pass filter will be as follows.

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While the above passive tuned filter circuit will work as a band pass filter, the pass band (bandwidth) can be quite wide and this may be a problem if we want to isolate a small band of frequencies. Active band pass filter can also be made using inverting operational amplifiers, and by rearranging the positions of the resistors and capacitors within the circuit, we can produce a much better filter circuit as shown below. The lower cut-off -3dB point is given by ƒC2 while the upper cut-off -3dB point is given by ƒC1.

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This type of band pass filter is designed to have a much narrower pass band. The centre frequency and bandwidth of the filter is related to the values of R1, R2, C1 and C2. The output of the filter is again taken from the output of the op-amp.

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

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The "Q" or Quality Factor

In a Band Pass Filter circuit, the overall width of the actual pass band 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, ( ) where:ξ

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:

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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.

When analysing active filters, generally a normalised circuit is considered which produces an "ideal" frequency response having a rectangular shape, and a transition between the pass band and the stop band that has an abrupt or very steep roll-off slope. However, these ideal responses are not possible in the real world so we use approximations to give us the best frequency response possible for the type of filter we are trying to design. Probably the best known filter approximation for doing this is the Butterworth or maximally-flat response. In the next tutorial we will look at higher order filters and use Butterworth approximations to produce filters that have a frequency response which is as flat as mathematically possible in the pass band and a smooth transition or roll-off rate.

Also you can see another design of bandpass filter which

consist of two operational amplifier

The usage of Band-pass filters:

Personal Radios

Personal radios use active bandpass filters to select only the signal

coming from the frequency you have dialed. The active bandpass filter

will sit in the circuit board between the antenna and the frequency

modulator. The raw signal will come in over the antenna, the active

bandpass filter will check what frequency range it needs to monitor

and the filter will then only send the correct signal to the speaker

circuits.

Wireless Routers

Wireless routers use active bandpass filters to keep out unwanted

interference. Depending on your router, it will use the active band pass

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filter to only listen on the A, B, G or N Wi-Fi bands. If the active

bandpass filter was not present, then your router would not only pick

up all of these bands and the data broadcast over them but also

interference from cellular broadcasts.

Cell Phones

Cell phones use active bandpass filters to keep out other cell phone

transmission as well other radio signals. Cell phones are essentially

regular walkie-talkie radios, except they have additional circuitry that

allows for signals to be sent and received at the same time.

Televisions

TVs that get their images from over the air radio waves also use active

bandpass filters. The filter selects the frequency of the channel that

you want and then only displays the images broadcast over that

frequency band. In the beginning the TV spectrum of radio waves were

only for TV broadcast; however, they are now being used for cell phone

data transmission. This means that active bandpass filters are even

more important as they must filter out cell phone broadcasts as well.

DSL Modems

DSL modems also use active bandpass filters for picking out the digital

signals from the analog signal on a phone line. In a DSL setup, the

digital signal that goes to your computer "piggybacks" on top of the

normal analog signal that goes to your phone. By using a filter, you can

use both an analog and a digital signal on a single wire.

How to design a band-pass filter?

A bandpass filter can be designed by using on operational amplifier a shown below:

Or it can designed by connecting output of a high-pass filter to the input of a low-pass filter or vice versa as shown below.

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Note that the fact that the high-pass section comes “first” in this design instead of the low-pass section makes no difference in its overall operation. It will still filter out all frequencies too high or too low.

An example of designing a band-pass filters circuit:

To design this band-pass filter circuit, first we should specify the range of frequencies and and apply these two rules: and , then we choose the value of either the resistors ( ) or the capacitors ( ) according to availability of them.

For instance:

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 We fixed the value of the capacitor at because resistors are more available and can be modeled easily.