Analyzing Integrator Circuits Using Operational Amplifiers

The transfer function of an operational amplifier (op amp) at DC is established by using resistors as gain adjustment setting elements. In the general case, these elements are impedances, which may include some reactive elements. Let's look at this general case shown in Figure 1.

Figure 1

The General Case of Op Amp Feedback Rewriting the results from the first article in this series using these terms, the transfer function is: Gain = V(out)/V(in) = - Zf/Zi In the steady state of the circuit of Figure 2, this reduces to: V(out) = -V(in)/2πfRiCf which is valid for a sine wave signal in the steady state.

Figure 2

Op Amp Configured as Integrator As we did in the initial analysis, the current flowing into the summing node must equal the current flowing out of the node. In other words, the current flowing through Ri must equal the current flowing through Cf. This situation can be expressed as the following transfer function: Using this transfer function, we have a general integrator. Because the DC error term of the op amp is included in the integration, this circuit is not usually used in a direct signal chain. However, it is a powerful circuit that is widely used in control loops. Please review the instrumentation amplifier described in Part 5 of this series, "Introduction to Instrumentation Amplifiers" (link below). In many high-gain applications, the voltage offset of the INA reduces the effective dynamic range, although the DC value has nothing to do with it.

Figure 3

Figure 3 shows an ideal application of an integrator. The input DC offset voltage from both the INA and the signal source appears at the input and is multiplied by the INA gain. This voltage appears at the integrator input. The op amp integrator drives the inverting input equal to the non-inverting input (which in this case is ground (GND)), so that the voltage offset of the INA is cancelled. This application makes the circuit look like a single-pole high-pass filter. The cutoff frequency is as follows: When Ri = 1 MΩ and Cf = 0.1 μF, the cutoff frequency is 1.59 Hz. The DC offset of the circuit is reduced to the Vos of the op amp. In some single-supply applications, it is necessary to bias the non-inverting input of the op amp above GND. The integrator is an inverting circuit, so a positive input signal will try to drive the output below the negative rail, GND. The bias voltage that appears at the non-inverting input of the op amp is the voltage that will maintain the zero input when the INA output is turned on.