An integrator circuit is a circuit that outputs voltage that is integration of the input voltage. An integrator circuit can be passive or active. A passive integrator circuit is build using passive components like resistor and capacitor. An active integrator circuit is build using op-amps or transistors. Here practical Op-Amp Integrator Circuit design is illustrated.

In the previous op-amp integrator tutorial How to design Basic or Ideal Op-Amp Integrator Circuit we explained how to design basic op-amp integrator and explained its limitation and error. The practical op-amp integrator circuit explained and designed here helps to solve the basic or ideal integrator problems.

Some of the application of integrator circuit includes solving differential equation, are used in analog computers, are used in analog to digital converters, for signal wave shaping circuits, function generator or waveform generator, in frequency modulation circuits etc.

**Practical Op-Amp Integrator Circuit Diagram**

The circuit diagram of a practical op-amp integrator circuit is shown below

The practical op-amp integrator differs from basic or ideal integrator circuit in that a resistor RF is used in parallel with the capacitor CF. Also a bias resistor Rb is used. The resistor RF minimizes the problem due to offset voltage and the resistor Rb minimizes the problem due to bias current. Hence the practical circuit minimizes the errors found in the ideal op-amp integrator.

We can derive the following equation for the output voltage Vout for practical op-amp integrator circuit which is as follows,

Thus for the integrator circuit, the output voltage is \(-\frac{1}{R_{1}C_{F}}\) times the integral of the input voltage. The negative sign indicates that the output voltage is shifted 180 degree out of phase relative to the input voltage.

The value \(R_{1}C_{F}\) is called the time constant of the integrator. For example, if R1=2.4KOhm and CF=0.1uF then time constant is R1*CF = 2.4KOhm*0.1uF=0.1ms.

Also in the above equation t is the time period which is the inverse of the frequency of the input signal. If for example the input signal frequency is 1KHz then the time period t of integration is 1ms.

**Frequency Response of Practical Op-Amp Integrator **

By taking the Laplace transform of the input and output voltage
equation of the integrator we can derive the **gain A as function of frequency** as follows,

$$ A= -\frac{\frac{R_{F}}{R_{1}}}{1+j \frac{f}{f_{a}}}$$

where,

$$f_{a}=\frac{1}{2 \pi C_{F} R_{F}}$$

is the **break frequency** or **corner frequency**.

The **magnitude of gain A** is then,

$$ |A| = \frac{\frac{R_{F}}{R_{1}}} { \sqrt{1+{(\frac{f}{f_{a}})}^2}}$$

At d.c condition, that is when f=0, we get the **DC Gain** as follows,

$$ |A|= \frac{R_{F}}{R_{1}} = 20 log(\frac{R_{F}}{R_{1}} )dB$$

That is unlike in ideal integrator where the dc gain is infinite(very high) for practical integrator the dc gain is limited to \(\frac{R_{F}}{R_{1}}\).

And the bias resistor(or compensation resistor) \(R_{b}\) is calculated by \(R_{1}\) in parallel with \(R_{F}\) ,

$$R_{b} = \frac{R_{1} R_{F}}{R_{1}+R_{F}}$$

Consider the following practical integrator circuit,

For the above circuit, the frequency response and phase response graph is shown below.

The following shows the magnitude response graph with the break frequency fa and the frequency fb where the gain becomes unity(0dB),

Thus for practical op-amp integrator circuit, the d.c gain remains constant for all frequencies below \(f_{a}\) and above \(f_{a}\) the d.c gain decreases with roll-off rate of 20dB/dec.

**Bandwidth of Integrator**

From the above graph, we can see that the practical integrator circuit performs integration when the frequency is between \(f_{a}\) and \(f_{b}\). The bandwidth of the integrator is thus \(f_{a}\).

**Voltage offset and input bias current **

In practical op-amp integrator circuit, the input bias current due to voltage offset flows through the feedback resistor \(R_{F}\) instead of \(C_{F}\). The voltage at the output therefore depends on the \( \frac{R_{F}}{R_{1}} \)

**Proper Integration **

For ensure proper integration of input signal, the d.c gain A= \( \frac{R_{F}}{R_{1}} \) is typical set greater or equal to 10. That is,

$$ \frac{R_{F}}{R_{1}}\geq10 $$

which also means,

$$ f\geq10f_{a} $$

Similarly, for proper integration the time period T of the input signal is made larger than \( R_{F}C_{F} \), that is,

\[T\geq R_{F}C_{F} \]

where, \( R_{F}C_{F}=\frac{1}{2 \pi f_{a}} \)

In this tutorial we have explained the theory of operation of a practical op-amp integrator circuit. We have explained major equation required for the design of a practical op-amp integrator. We have also explained how practical integrator overcomes the limitation and errors in ideal op-amp integrator circuit. Similarly, we have explained the frequency response and bandwidth of practical integrator. Also we have provided some design constraints required to build a practical op-amp integrator circuit.

In the next follow up tutorial How to Design LM358 Op-Amp Practical Integrator a practical design of op-amp integrator is illustrated.