#
OP AMP INTEGRATOR CIRCUIT DESIGN AND WORKING

The **OP AMP Integrator**,
is a circuit which performs the integration of an input signal. The output of
an integrator will be the mathematical integral value of continuous input
signal. Such a circuit is obtained by using a basic inverting amplifier
configuration if the feedback resistor Rf is replaced by a capacitor Cf.

**V**_{o} = -1/R_{1}C_{F
} ∫_{0}^{T}(Vin dt +
C)

The above equation shows that the output voltage is directly
proportional to the negative integral of input voltage and inversely
proportional to the time constant R1Cf.

**If the input is a
sinewave, the output will be a cosine wave**.

**If the input
to the integrator is square wave, the output will be a triangular wave.**

When Vin=0, the integrator shown in figure will act as an **open loop amplifier**. This is because
the capacitor Cf acts as an open circuit (The impedance of feedback capacitor
approximates to infinity) to the input offset voltage Vio. In other words, the
input offset voltage Vio and the part of the input current charging capacitor
Cf produces the error voltage at the output of the integrator.

In **PRACTICAL
INTEGRATOR**, to reduce the error voltage at the output, a resistor Rf is
connected across the feedback capacitor Cf. Thus, Rf limits the low frequency
gain and hence minimizes the variations in the output voltage. Both the
stability and Roll-off problems can be eliminated by the addition of a resistor
Rf in the *practical Integrator*.

__STABILTY__:
The term stability refers to the constant gain as frequency of an input signal
is varied over a certain range.

__Roll-off__:
The term Roll-off refers to the rate of decrease in gain at lower frequencies.

By applying Kirchoff’s current equation at node V2,

The input bias current I_{B} is negligibly small so
we can ignore it,

Current through the feedback capacitor,

**V**_{in}-V_{2}/R_{1}
= C_{F} (d/dt) (V_{2}-V_{0})

However **V**_{1 }= V_{2} = 0 because A (Gain) is very large.

**V**_{in}/R_{1}
= C_{F} d/dt (-V_{0})

** =C**_{F }(-V_{0}) + V_{0} at t=0

**V**_{o} = -1/R_{1}C_{F
} ∫_{0}^{T}(Vin dt +
C)

Where C is the integration constant.

### Tittu

#### Author & Editor

I'm Tittu Thomas, An Electronics and Communication Engineer from India, Kerala. I love doing hobby electronics circuits, blogging, programming, etc. Started this blog while doing my B-Tech degree under CUSAT university.

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