Integrator and Integrator Circuit

Integrator

The Integrator, also known as an Integrator op amp, is a circuit in which the output voltage Waveform is the Integral of the input voltage Waveform. If the Feedback Resistor is swapped out for a Capacitor, a basic Inverting Amplifier Configuration can be used to create the desired circuit.

So The expression for the output voltage V₀ can be Obtained by KVL eqn at node V₂.

where C Integration constant A

So eqn (3) Indicates that the output is directly Proportional to the negative Integral of the input volts and Inversely Proportional to the time constant R₁ 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 R₁C_f = 1, V_out =0V (i.e) C =0.

Because the capacitor CF acts as an open circuit to the input offset voltage Vio when Vin = 0, the integrator functions as an open loop amplifier.

Or

The error voltage at the integrator’s output is created by the input offset voltage Vio and the portion of the input that charges the capacitor CF.

Integrator Circuit

Practical Integrator

A resistor RF is connected across the feedback capacitor CF of the practical integrator in order to lower the error voltage at the output. Thus, RF restricts the low frequency gain and lessens output voltage variations. A resistor RF can be added to the practical integrator to address the stability and low frequency roll-off issues. So The frequency response of the basic integrator is shown from this fb and is the frequency at which the gain is dB.

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.

So From the fig of practical Integrators, f is some relative operating frequency. And for frequencies f to fa to gain R_F / R₁ 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 R₁ 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.