# Signal Chain Basics (Part 18): The op amp as integrator

(Editor's Note: There are links to the previous parts of this series at the end, below the author's biography.)

In the first article of this series, “Operational Amplifier–The Basic Building Block” (link below), the transfer function for the operational amplifier (op amp) was developed for the DC case with resistors as the gain-setting elements. In the general case these elements are impedances that may contain reactive elements. Consider the general case shown in Figure 1.

Figure 1: Op amp feedback general case
(Click on image to enlarge)

Rewriting the results of the first article of this series directly in these terms, the transfer function becomes:

Gain = V(out)/V(in)= – Zf/Zi

This result, in the steady state for the circuit in Figure 2, reduces to:

V(out) = -V(in)/2p fRi Cf

This is valid for steady-state sine wave signals.

Figure 2: An op amp configured as an integrator.
(Click on image to enlarge)

Just as in the initial analysis, the current into the summing node must equal the current out of that node. In other words, the current though Ri must equal the current through Cf. This is expressed as the transfer function:

With this transfer function, a general case integrator is realized. Because the DC error terms of the op amp are included inside the integral, this circuit does not usually see service in the direct signal chain. It is, however, used as a powerful function inside control loops.

Consider the instrumentation amplifier developed Part 5 of this series, “Introduction to the Instrumentation Amplifier” (link below). In many high-gain applications, the DC value is not of any interest, but the voltage offset of the INA reduces the dynamic range available.

Figure 3: Zeroing the offset with an integrator
(Click on image to enlarge)

Figure 3 demonstrates an ideal application for an integrator. The input DC offset voltage, both from the INA and the signal source, appears at the input and is multiplied by the gain of the INA. This voltage appears at the input of the integrator. The op amp integrator drives to make the inverting input equal to the non-inverting input, which is ground (GND) in this case. In so doing, the voltage offset of the INA has been canceled. This application makes the circuit appear as a single-pole high-pass filter. The corner frequency is at the point where:

When Ri = 1 MO and Cf = 0.1 ÂµF, the corner frequency is 1.59 Hz. The DC offset of the circuit is reduced to the Vos of the op amp.

In single-supply applications, it may be necessary to bias the non-inverting input of the op amp above GND. The integrator is an inverting circuit, so positive input signals would try to drive the output below GND, which is the negative supply rail. The bias voltage present on the non-inverting input of the op amp is the voltage that the INA output will maintain with zero input.