An electric filter is often a frequency selective circuit that passes a specified band of frequencies and blocks or alternates signal and frequencies outside this band.

**Active
filters:**

An
electric filter is often a frequency selective circuit that passes a specified
band of frequencies and blocks or alternates signal and frequencies outside
this band. Filters may be classified as

1.
Analog
or digital.

2.
Active
or passive

3.
Audio
(AF) or Radio Frequency (RF)

Analog
filters are designed to process analog signals, while digital filters process
analog signals using digital technique.

Depending
on the type of elements used in their construction, filter may be classified as
passive or Active elements used in passive filters are Resistors, capacitors,
inductors. Elements used in active filters are transistor, or op-amp.

1.
Gain and Frequency adjustment flexibility:

Since
the op-amp is capable of providing gain, the i/p signal is not attenuated as it
is in a passive filter. [Active filter is easier to tune or adjust].

2.
No loading problem:

Because
of the high input resistance and low o/p resistance of the op-amp, the active
filter does not cause loading of the source or load.

3.
Cost:

Active
filters are more economical than passive filter. This is because of the variety
of cheaper op-amps and the absence of inductors.

The
most commonly used filters are these:

1.
Low
pass Filters

2.
High
pass Filters

3.
Band
pass filters

4.
Band
–reject filters

5.
All
pass filters.

1. It has a constant gain from 0 Hz to
a high cutoff frequency f_{1}.

2. At fH the gain in down by 3db.

3.
The frequency between 0 Hz and fH are known as the pass band frequencies where
as the range of frequencies those beyond fH, that are attenuated includes the
stop band frequencies.

4. Butterworth, Chebyshev and Cauer
filter are some of the most commonly used practical filters.

5. The key characteristics of the
butter worth filter are that it has a flat pass band as well as stop band. For
this reason, it is sometimes called flat- flat filters.

6. Chebyshev filter -> has a ripple
pass band & flat stop band.

7. Causer Filter -> has a ripple
pass band & ripple stop band. It gives best stop band response among the
three.

High
pass filter with a stop band 0 <f< f _{L and} a pass band f> f
_{L}

f_{L}
-> low cut off frequency

f
-> operating frequency.

It
has a pass band between 2 cut off frequencies f_{H} and f_{L}
where f_{H} > f_{L} and two, stop bands: 0<f< f_{L}
and f > f_{H} between the band pass filter (equal to f_{H} -
f_{L}).

Band
–reject filter: (Band stop or Band elimination)

It
performs exactly opposite to the band pass.

It
has a band stop between 2 cut-off frequency fL and fH and 2 pass bands:
0<f< fL and f> fH fC -> center frequency.

**Note:**

The
actual response curves of the filters in the stop band either R or S
or both with R_{in} frequencies.

The
rate at which the gain of the filter changes in the stop band is determined by
the order of the filter.

Ex: 1^{st} order low pass filter the gain
rolls off at the rate of 20dB/decade in the stop band.

(i.e)
for f > fH.

2^{nd}
order LPF -> the gain roll off rate is 40dB/decade.

1^{st}
order HPF -> the gain rolls off at the rate of 20dB (i.e.) until f:fL

2nd
order HPF -> the gain rolls off at the rate of 40dB/decade

First
order LPF that uses an RC for filtering op-amp is used in the non inverting
configuration. Resistor R1 & Rf determine the gain of the filter. According
to the voltage –divider rule, the voltage at the non-inverting terminal (across
capacitor) C is,

Gain
A= (1+R_{f}/R_{1})

Voltage
across capacitor V_{1}= V_{i}
/ (1+j2πfRC)

Output
voltage V0 for non inverting amplifier =A V_{1}

=
(1+R_{f}/R_{1}) Vi/(1+j2πfRC)

Overall
gain V_{0}/V_{i} = (1+R_{f}/R_{1}) Vi/(1+j2πfRC)

Transfer
function H(s) =A/(jf/f_{h}+1) if f_{h} =1/2πRC

H
(jω) = A/( jωRC+1) = A/( jωRC+1).

The
gain magnitude and phase angle of the equation of the LPF can be obtained by
converting eqn. (1) b into its equivalent polar form as follows.

1.
At very lowω)|frequency, f < fH

|H
(jω) =A

2. At f =fH

|H (jω)| =A/√2=0.707A

3. At f> fH

|H
(jω)| <<A ≅ 0

When
the frequency increases by tenfold (one decade), the volt gain is divided by
10. The gain falls by 20 dB (=20log10) each time the frequency is reduces by
10. Hence the rate at which the gain rolls off fH = 20 dB or 6dB/octave
(twofold Rin frequency). The frequency f = fH is called the cut off frequency
because the gain of the filter at this frequency is down by 3 dB (=20 log
0.707).

A
LPF can be designed by implementing the following steps.

1. Choose a value of high cut off
frequency f_{H}**.**

2. Select a value of C less than or
equal to 1μf.

3. Choose the value of R using
fh=1/2πRC

4. Finally select values of R1 and RF
dependent on the desired pass band gain AF Using A=(1+Rf/R1)

A
second order LPF having a gain 40dB/decade in stop band. A First order LPF can
be converted into a II order type simply by using an additional RC network.

The
gain of the II order filter is set by R_{1} and R_{F}, while
the high cut off frequency f_{H} is determined by R_{2}, C_{2},
R_{3} and C_{3}.

1.
Choose a value for a high cut off freq. (fH ).

2.
To simplify the design calculations, set R_{2} = R_{3} = R and
C_{2} = C_{3} = C then choose a value of C<=1μf.

3. Calculate the value of R *R *=1/2πf_{h}C

4.
Finally, because of the equal resistor (R_{2} = R_{3}) and capacitor (C_{2} = C_{3}
) values, the pass band volt gain A_{F} = 1 + R_{F} / R_{1}
of the second order had to be = to 1.586. R_{F} = 0.586 R_{1}.
Hence choose a value of R_{1} <=100kΩ.

5. Calculate the value of R_{F}.

High
pass filters are often formed simply by interchanging frequency-determining
resistors and capacitors in low-pass filters.

(i.e)
I order HPF is formed from a I order LPF by interchanging components R & C.
Similarly II order HPF is formed from a II order LPF by interchanging R &
C.

Here
I order HPF with a low cut off frequency of fL. This is the frequency at which
the magnitude of the gain is 0.707 times its passband value.

Here
all the frequencies higher than f_{L} are passband frequencies.

The
output voltage V_{0} of the first order active high pass filter is

·
At
high frequencies f>fL gain = A.

·
At
f= fL gain = 0.707 A.

·
At
f < fL the gain decreases at a rate of -20 db /decade. The frequency below
cutoff frequency is stop band.

I
order Filter, II order HPF can be formed from a II order LPF by interchanging
the frequency

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