A circuit in which the output voltage waveform is the integral of the input voltage waveform is the integrator or Integration Amplifier.

**Integrator:**

A circuit
in which the output voltage waveform is the integral of the input voltage
waveform is the integrator or Integration Amplifier. Such a circuit is obtained
by using a basic inverting amplifier configuration if the feedback resistor RF
is replaced by a capacitor C_{F}.

The
expression for the output voltage V_{0} can be obtained by KVL eqn at
node V_{2}.

*where C *__integration c__

eqn (3)
indicates that the output is directly proportional to the negative integral of
the input volts and inversely proportional to the time constant R_{1} C_{F}
.

Ex: If
the input is sine wave -> output is cosine wave.

If the
input is square wave -> output is triangular wave.

These
waveform with assumption of R_{1}C_{f} = 1, V_{out} =0V
(i.e) C =0.

When V_{in}
= 0 the integrator works as an open loop amplifier because the capacitor C_{F}
acts an open circuit to the input offset voltage V_{io}.

Or

The Input
offset voltage V_{io} and the part of the input are charging capacitor
C_{F} produce the error voltage at the output of the integrator.

**Practical Integrator:**

Practical
Integrator to reduce the error voltage at the output, a resistor R_{F}
is connected across the feedback capacitor C_{F}.

Thus RF
limits the low frequency gain and hence minimizes the variations in the output
voltages. The frequency response of the basic integrator, shown from this fb is
the frequency at which the gain is dB and is given by,

Both the
stability and low frequency roll-off problems can be corrected by the addition
of a resistor R_{F} in the practical integrator.

Stability
-> refers to a constant gain as frequency of an input signal is varied over
a certain range.

Low
frequency -> refers to the rate of decrease in gain roll off at lower
frequencies.

From the
fig of practical Integrators,

f is some
relative operating frequency and for frequencies f to fa to gain R_{F}
/ R_{1} is constant. After f_{a} the gain decreases at a rate
of 20dB/decade or between f_{a} and f_{b} the circuit act as an
integrator.

Generally
the value of f_{a} and in turn R_{1} C_{F} and R_{F}
C_{F} values should be selected such that f_{a}<f_{b}.
In fact, the input signal will be integrated properly if the time period T of
the signal is larger than

or

equal to
R_{F} C_{F}, (i.e) T >= R_{F} C_{F} @@@@ 6

Uses:

Most
commonly used in analog computers.

ADC

Signal
wave shaping circuits.

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