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We were discussing the Frequency Response
of an Intigrator.
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The basic integrating circuit, integrating
amplifier using operation amplifier, we have
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already discussed and it was this, this is
the input signal v i and resistance R 1, capacitor
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C F and of course, these power supply is there.
This is load resistor across which, we take
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the output, and this is to reduce the offset
effects, about offset effects we will definitely
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talk. So, this is around R 1 and we have seen
that the output of this is minus 1 by R 1
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C F integration v i d t, where v i is the
input signal.
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And if we take that input signal v i is expressed
as, a 0 sin omega t, then the output is then
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the output of the integrator is v 0 minus
1 by R 1 C F and integration a 0 sin omega
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t d t, and this gives v 0 equal to a 0 omega
R 1 C F cos omega t. This is we take as first
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and this is the second.
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Then we can write the ratio of the two for
the peak voltages, taking peak voltages both
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at the input and output, the ratio which is
gain v 0 by v i its magnitude is, a 0 by a
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0 omega R 1 C F and so, it is equal to 1 by
omega R 1 C F and this is gain, gain of the
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intigrator.
So, what we are seeing that, the gain is frequency
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dependent. As the frequency is, this angular
frequency is omega, which is 2 pi F. So, as
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the frequency F increases, the gain falls.
So, gain is proportional proportional to 1
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by omega, the frequency and the plot for this
is, this is the frequency f 1, at which the
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gain falls to 0, this is gain in d B, and
this value this represents v 0 by v i this
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magnitude of this quantity.
Now, the gain is falling and it can be shown
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that, this fall is minus 20 d B per decade
change in frequency. So, if frequency becomes
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10 times, then gain becomes one-tenth that
is the meaning and the gain falls. And here,
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the f 1 this frequency, this is frequency
f 1 is given by 1 by 2 pi R 1 C F, this we
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can get from here actually, this is gain and
when at the frequency at which gain falls
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to 0 is given by this. And so, this is this
is the frequency response of the basic integrator,
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this integrator circuit.
And now we have discussed that, there is a
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problem with this circuit. And the problem
arises, because of the offset, offset currents
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and voltages and the capacitor, gets charged
and it is amplified and there is a output,
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even when there is no input; and these problems
can be tackled by attaching a resistor along
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with the capacitors in shunt.
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So now, we discussed the frequency response
of a practical integrator, frequency response
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of practical integrator. So, the circuit which
we are considering is now this
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this is R F the resistance we have included;
and this is C F and this is connected here
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and this is load across, which we take the
output and here, we connect the input. So,
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this is the practical integrator.
Now, remember the basic that, true integration
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occurs when the when the impedance of this
combination of C F and R F is capacitive,
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and this will be capacitive only when, this
the effect of this capacitance dominates.
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So, and we know that at low frequencies, at
low frequencies the parallel combination of
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the reactance of this capacitance in parallel
with R F at low frequency, this will be very
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high impedance reactive impedance of the capacitance
will be very high, and this parallel combination
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will give the effective value as R F.
So, that means at low frequency is this, inclusion
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of this resistance R F, this limits the lower
frequency for integration; that means, at
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very low frequencies we had this condition
is satisfied, the output will not be the true
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integration of the of the input. And then,
so for true integration, for true integration
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this reactance should be very small as compared
to R F; that means, 1 by 2 pi f C F, this
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should be this should be very small as compared
to R F.
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And we can define a critical frequency beyond
which, integration will be alright. So, we
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define a critical frequency, f c we define
a critical frequency f c such that, X c is
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equal to R F. And substituting the value of
this, 1 by 2 pi f c now into C F, this is
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equal to R F or the frequency f c is equal
to, from here 1 by 2 pi R F C F, this is the
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critical frequency. And this will give; f
c give a break frequency gives break frequency
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in the response
in the response.
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And the response is this, this is frequency,
this frequency is f 1 and this is at, this
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is the 20 d B
and here, this is the 3 d B line and this
gives, f c this cut of frequency which we
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have written this f c, this is this value
and for this f 1 we have written earlier,
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f 1 is equal to were gain falls to 0 d B,
this is gain in d B and this is R F by R 1
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gain. So, and this gain is this frequency
is 1 by 2 pi R 1 C F.
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There is a difference in two break frequencies,
f 1 this is because this resistance will be
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smaller than the resistance, which is used
with the capacitor. So, this contains the
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first resistance here, this is R F and R F
is definitely higher at least 10 times, normally
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10 times of this. So, this is the frequency
response of this is the frequency response
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of practical integrator.
And what we see, that below f c frequency
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which we can calculate from this expression,
below f c true integration will not occur.
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So, true integration true integration in the
frequency limit, in the frequency between
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the frequency f c and f 1. So, actually f
1 is chosen quite high as compared to f c;
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f c we choose very high as compared to f 1
and in it is a thumb rule that, f c is one-tenth
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of f 1, and and in that case R F will be ten
times of R 1.
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This resistance is ten times of this resistance
in the design. So, that is this, integration
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will occur in this frequency region, the integrator
is not used below the frequencies f c. About
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the applications applications of the integrator,
these are used in analog computation
analog computation and in wave shaping circuits
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and of course, they are similar other uses.
So, this was about the integrator or integrating
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amplifier using.
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Next, we take a differentiator
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that is the, next in which we are considering
the differentiator. In differentiator circuit,
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the output is proportional to the rate of
change of the input at that instant. I repeat
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that, output in the differentiator is proportional
to the rate of change of the input. And rate
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of change is as we will see; that means, what
I am saying is this, is proportional to when
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input signal is v i and output is proportional
to this and this is this circuit is called
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differentiating differentiator or differentiating
amplifier.
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And this can be again realized in inverting
amplifier. Inverting amplifier, the capacitor
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R 1 the the resistor R 1 is replaced by the
capacitance, then the circuit works as a differentiator.
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So, if the resistance R 1 of inverting amplifier
is replaced by the capacitance capacitor,
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then this is the circuit, this is R F, this
is c 1 and here this is v I, the
resistance which is equal to roughly R F and
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here, is the load resistor across which we
take the output, this is the basic differentiator,
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basic.
And of course, this power supplies are there,
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which are not essential to be shown, because
it is implied that, they are they are no electronic
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repeatedly I am saying, no electronic circuit
will function, unless the transistors of this
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in the device are properly biased and for
biasing we need the d c supply. So, this is
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these are two supplies here.
This is the charging current i c, which the
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signal will send v i will send and here is
current i f, this is i f, this is the small
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current i b; and i b this repeatedly is being
said that, i b because, the input impedance
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is very high of the device it is in 1 or 2.
So, i b can be taken as 0, if i b is taken
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0; that means, no current is going in this
arm, in the inverting input then, we can write
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i c equal to i f i c equal to i f.
And how much is i f, i f this current two
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voltages, this is v 2 this is v 0. So, v 2
minus v 0 by R F this is this. But we know
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that, v 2 the inverting input is at virtual
ground potential, so since v 2 is 0 virtual
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ground everywhere we are seeing, the advantages
in inverting amplifier, the advantages of
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virtual ground, because the concept of virtual
ground makes the analysis much simpler. So,
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this is virtual ground. Then i f is equal
to minus v 0 by R F, this is one expression.
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And how much is i c? The charging current
charging current we can write i c, this is
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equal to the capacitance into, from simple
capacitors theory and voltage difference,
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so v i minus v 0 by d t and sorry the two
voltages, this is at v i and this is at v
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2. So, this is 2, v 2 which is 0. But, v 2
is 0 and hence i c equal to c 1 d v i d t
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hence, because the two currents are identical
here, i c is equal to i f.
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So, these two expressions we we have already
got for i f and here this is for i c and so,
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we can write therefore, therefore c 1 d v
i by d t, this is equal to minus v 0 by R
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F or v 0 is equal to minus R F c 1 d v i by
d t. This is what we said in the beginning
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that, this is all constant and this sine inversion
is does not make a difference this also we
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have been talking and so, the output is proportional
to the rate of change of input.
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So, let us take two cases, when input is a
sine wave, input is a sine wave output will
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appear as cosine, at the output we will have
cosine function cosine wave. And for triangular
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input, a triangular triangular input will
produce a square wave output will produce
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square wave output. Now, we talk about again,
because for these integrators and differentiators,
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frequency response is very important. When
the input frequency thus, the frequency of
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the input signal is changing then, how the
output will vary that is of significance.
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Now here, we we should note in this basic
circuit of differentiator. There are two problems,
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one is that gain of the amplifier gain of
the amplifier is R F by X c, where X c is
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the this is you know for the inverting amplifier,
the gain used to be R F by R 1, R 1 was the
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resistance here. Now, here it is the capacitance.
So, the resistance will replaced by the reactance;
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and this reactance will go low very fast with
at higher frequency, because this is X c is
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1 by j omega c.
So, at low frequencies this is at high frequencies,
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this is very low and therefore, gain rises
with frequency with frequency at the rate
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of at the rate of 20 d B per decade of frequency;
that means, when frequency becomes ten times,
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then it changes ten times, so 20 d B. So,
this is the change, and at very high frequencies,
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this will be very low and gain will become
very large. At very high frequencies, gain
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is extremely large and that makes the circuit
that makes the circuit
and that way, the frequencies noise they will
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also may get amplified. Now because of this,
there is a high frequency noise which will
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be heavily amplified, and that will dominate
the differential output
differential output. So, it will it will not
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be a accurate differentiation at very high
frequencies.
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So, to take care of this fact, that gain becomes
exceedingly high, when the frequencies are
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very high. We attach a small value resister
and we get a practical differentiator
practical differentiator is this, this is
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R F, this is the resistance R 1 which we have
attached, and R 1 is quite a smaller as compared
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to R F; and this is c 1 and here, we attach
the input signal, this resistance is equal
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to R F and these are the supplies. This is
the practical differentiator
practical differentiator.
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And here, now at very low frequencies frequency
of the signal tending towards infinity at
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very high frequencies at very high frequencies.
Here, the gain was rising very fast with the
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fall in X c at in this case, now when this
is almost is a short, then gain will be restricted
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to this. And this normally may be 10, 12,
15, so gain will be this at very high frequencies;
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and the frequency response of this practical
differentiator is this and here, this
was the gain going up very fast with frequency,
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this is frequency, and this is in d B.
Now here, for the practical one, the gain
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will state very high frequencies when this
capacitor is almost acts as a short at those
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frequencies, the gain will be this here; this
is R F by R 1. So, this is the response of
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a practical differentiator, this is
practical differentiator, this was for the
ideal one, this is for the the basic which
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we have showed the integrator, basic integrator
and this is the practical integrator.
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And here, this frequency is f 2 or f c we
can say and this is f 1, f 1 is the frequency,
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where the gain will fall to 0. And f 1 will
be given by, f 1 is 1 by 2 pi R F c 1 at the
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frequency at which this becomes a the gain
becomes 0. So, this is the frequency.
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And the inclusion of R 1 that has restricted
that limits the high frequency range of the
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differentiator, and this f c will be given
by, f c is given by 2 pi R 1 C 1. So, again
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as in differentiator we have two frequencies,
f 2 which is same as f c which I have written
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here, this we can write f 2 or f c, this is
this frequency, and this is the the difference
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is of R 1 and R F. So, beyond this frequency
f 2, no no good differentiation, no true differentiation
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of the input will appear.
And so, differentiation is there in the frequency,
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true differentiation occurs in the frequency
range f 1 to f c or f 2 whatever you write.
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And this frequency is set f c is set quite
high in comparison to f 1, so that there is
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a good range of frequencies over which true
differentiation will be possible about applications,
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this is we have discussed.
So, the basic circuit is not used for differentiator,
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but this circuit is used by inclusion of resistance
R 1 and this is the frequency response of
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the practical differentiator, the inclusion
of R 1 puts a limit on the high frequency;
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that means, a safes the circuit from of very
high gain; and so, a true differentiation
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occurs between the range f 1 and f c.
And about the applications of differentiator,
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they are the same as the integrator that,
they are used in analog analog computation
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and wave shaping. So, this is all about the
differentiator. So, we have taken several
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applications, we continue with few more and
that will show the versatility of the operation
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amplifier. So, after these summing circuits
and differentiator, integrators, sine changer
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and we now go for another class of of applications,
and these are active filters.
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Active filters first thing is, what is a filter?
Filter is a frequency selective circuit. Frequency
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selective; that means, you have a spectrum
of frequencies and you want to select for
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your system that your system should respond
to certain frequencies. So, how to get them?
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Frequency selection can be done in by using
filters, filter circuits.
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Now, there are two types of filters, passive
filters and active filters. Filters, these
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are frequency
selective circuits, and they can be two types,
passive filters and active filters
passive filters and active filters. Now, passive
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filters can be realized frequency selection
circuits you can make by using capacitors,
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resistors and inductors, you must have done,
you have done a resonance circuit.
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What is a resonance circuit? Resonance circuit
is a frequency selective circuit, and resonance
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can be obtained by the combination of capacitor,
resistance and inductors. But, these filters
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the passive, these are called passive filters,
they are inefficient; inefficient in the sense,
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there are two points, one is that these components
will dissipate certain signal. So, signal
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is attenuated in passive filters number 1.
Second thing is that frequency response of
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passive filters is not as good as in the case
of active filters.
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So, active filters are the ones, which make
use of the active device, transistors or to
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make filters, operational amplifiers are very
widely used in fact, they have replaced B
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J T and all other circuits, for making active
filters; because, handling of operation amplifiers
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is not at all difficult plus, they are not
expensive. And hence, active filters make
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use of operation amplifiers and these circuits
are very widely used.
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Now, there are four types of filters, four
types, four types of filters. What are these
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types? First I name them and then, I will
elaborate them. One is low pass filter
low pass filter then, we have high pass filter,
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we have band pass filter
and then finally, we have band stop
band stop or band reject filter or
band reject filter. These are the four different
kinds of filters, depending on the selection
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of frequencies for the working of the circuit
at which frequencies we want to choose. And
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in signal processing, this filtering of frequencies
is a very important process and so, we now
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first talk about these filters.
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The low pass filter, first we take low pass
filter. Low pass filter is the one, which
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is supposed to pass, by pass we mean that
they will go in the circuit for further processing.
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So, certain frequencies from very low to a
certain high frequency, they will be allowed
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to propagate beyond that frequency signals
will attenuate heavily very heavily and they
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will be prohibited from going for further
processing, so that will be low pass filter.
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The ideal first I take ideal low pass filter,
then the actual low pass filter practical.
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So, this is the case, this is gain and this
is frequency, this is the pass band, this
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is frequency, so this is pass band, and all
these is stop band and let us, call this frequency
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as f H and this is ideal.
So, what is the low pass filter, from 0 frequency
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to upto a maximum of say f H frequency, it
will pass; that means, signal will pass through
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the circuit without attenuation and it will
be it has been selected in this frequency
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range it will be selected and it will be further
processed. And in ideal, this is the case
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of ideal filter, in which gain the attenuation
in the pass band attenuation in the pass band
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in the ideal filter is 0 d B that means no
attenuation; while attenuation in the stop
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band, stop band is infinity infinitely high
attenuation. So, this is for ideal filter.
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Now, electronically we cannot have this ideal
gain frequency response, because there is
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no process by which this break can be that
sharp. So, the practically what we have is
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this
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here
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we have this frequency is f H. And so here,
this gain this is 3 d B fall 3 d B fall and
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this is frequency.
So, this is the case that gain will start
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falling here and then, it will go and fall
very fast here and upto this frequency 3 d
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B fall will be corresponding very close to
H f here. The the fall will start and 3 d
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B has falling by this frequency, cut off frequency
and beyond that, again in the ideal case,
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it was infinite attenuation here, that at
loss the attenuation will be high, but it
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continues for certain frequency region.
So, this is the response and we will drive
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an expression for this cut off. What is the
highest cut off, for the low pass filter?
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In this, we can design we can choose it can
be 5 kilo hertz; it can be 50 kilo hertz depending
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on our requirement. So, this is the case of
low pass filter, this is the practical filter,
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this is practical case
practical filter.
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Then we go for high pass filter high pass
high pass filter the ideal case is here, this
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is pass band, and this is stop band, this
is gain and here, this is of course, frequency.
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So, this is the ideal ideal filter, where
the attenuation in the stop band is infinity;
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and from this frequency, which is the lowest
frequency from where the pass band start,
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the attenuation is 0. But, this 0 and infinity
again, these are ideal and this sharp boundary
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between stop band and pass band, this is also
only ideal.
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In the practical case, this circuit this response
for the the actual filter will be like this,
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this is gain and this gain is 3 d B, and this
frequency will be f L say, this is frequency.
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So, this is the response of the high pass
filter, this is the practical. And we will
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arrive at an expression for this f L that
is lowest frequency in this we can choose
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as I said, depending on the circuit components.
Now, the active filters as I said, will make
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use of capacitors and resistors, just few
capacitors and resistors and filter design
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will be complete, the inductors are not used;
there are reasons, and these reasons are inductors
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are bulky and heavy, and they are not and
hence, inductors are not used with active
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filters. So, this is the case of high pass
filter.
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And then, we can have the band pass filter,
c band pass filter. The band of frequencies
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is allowed before that, they it is not permitted,
after that it is not permitted; and for the
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ideal, this is the case, this is gain and
this is pass band, this is stop band, this
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is also stop band. But practically, these
are the frequencies f H 1 and f H 2; that
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means, for this these frequencies line between
the pass band, the attenuation will be very
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low; very heavy in the smaller frequency region
and very heavy in the higher frequency region.
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And for the practical one, this plot is like
that, and here this is 3 d B change again,
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3 d B and this is f H 1, f H 2 gain. So, this
is the the band pass, band of frequencies
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will pass, rest will be stopped.
And finally, we have the last one, the band
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reject or band stop. This is ideally it is
this, this is pass band, this is pass band,
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this is the stop band, this is gain, this
is frequency and this is that. And practically
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this will be like this, where this is gain,
this is frequency, and these are those two
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frequencies, L 1, f L 2. So, this band will
not be permitted to go in the circuit for
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further processing. So now, we will go one
by one about these in details, the circuits
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and their analysis.