# A practical integrator that does not behave like a low-pass filter?

Op-amp integrators and some common integrators actually are low-pass filters in practice. So if a signal is DC, then it does not integrate but only multiplies by some amount plus some transient response.

Now of course integration over DC cannot be done for a long period of time, as this would definitely break the circuit. But for a limited time, is there any integrator that does proper integration for DC and some range of frequencies?

• Sure, an op-amp integrator will do what you want. Commented Aug 25, 2015 at 23:20
• stage 1: convert your voltage into a current... stage 2: put a capacitor after stage 1. Commented Aug 25, 2015 at 23:26
• I prefer to think of low-pass filters the other way around -- a low pass filter is a leaky integrator. Commented Aug 26, 2015 at 15:44

This circuit:

simulate this circuit – Schematic created using CircuitLab

will do exactly what you want. Many times a designer will place a resistor in parallel with C1 to limit the gain at DC. If you don't want this behavior you can simply leave it off. Just be prepared to deal with your output hitting the rails in some circumstances.

• Another very useful variation is a switch (FET or relay) across the cap, which is used to zero the integrator when necessary. Commented Aug 26, 2015 at 3:59
• Yet another useful variation would be a switched capacitor or switched current source which could be used to inject or drain measured amounts of charge. Commented Aug 26, 2015 at 16:18
• I have some doubts if this circuit can "do exactly what you want" - unless you have an IDEAL opamp. In reality and without a parallel resistor RF the output will continuously rise - integrating the offset voltage (even without any input). On the other hand, with a parallel resistor, this integration will stop (depending on the ratio RF/R1) but the circuit will not be able to integrate low-frequency signals. As mentioned in my detailed answer an ideal integrator is not realizable.
– LvW
Commented Aug 27, 2015 at 6:26

I suppose I know what you mean - and you are right. For an in ideal integrator we require a phase shift of 90 deg between input and output for ALL frequencies (including, for example 0.1 Hz). Because this phase shift requires a 20dB gain roll-off also for low frequencies (down to DC) the corner frequency would be in the vicinity of 0Hz (resp. infinite gain).

This is a non-realizable requirement for a realistic opamp. As a consequence, integrating a DC voltage does not result in a linear output voltage increase but in a function known from the step respoonse of any RC lowpass (1-exp). However, the first part of this function is very close to a linear increase.

As another option you could use an OTA ("current" output) to charge a load capacitor. Hower, in principle, there is the same limitation: There is no ideal OTA. Each real device has a finite output resistance forming an RC lowpass in conjunction with the load capacitor.

Summary: It is not possible to build an ideal integrator. But that`s no surprise because there are no ideal electronic circuits.

UPDATE: Of course, as another problem - a capcitor in the feedback path cannot provide the necessary dc feedback for stabilization of a suitable operating point. Therefore, any practical opamp based integrator circuit needs a feedback resistor in parallel to the capacitor. This results in a larger corner frequency of the lowpass function.

• A practical integrator can be realized without a resistor in parallel with the capacitor. You are correct that it is not "stable" at DC but for an integrator that is the desired behavior. I often use such a circuit as part of a larger global feedback circuit (for example an analog control loop). Commented Aug 26, 2015 at 10:38
• Yes - you are right, in case the integrator is used within an overall dc stabilizing negative feedback circuit (active filters, oscillators,...). However, it was my impression that the question was related to an integrator as a "stand-alone" unit.
– LvW
Commented Aug 26, 2015 at 10:58

Short answer - no. There is no such thing as an amplifier with absolutely zero bias that can produce a perfect integrator.

Longer answer. Yes, but you have to be very specific about what your requirements are. You determine what rate of drift is acceptable, you determine what your minimum cutoff frequency for a high pass filter, and you design from there, probably using some sort of self-zeroing amp like a chopper to minimize drift.

As a thought experiment, build a digital system (A/D -> microcontroller -> D/A) and try to program the ideal behavior. It's true that execution speed represents a delay so we could substitute an FPGA. Let's have the delay approach zero, since this is a thought experiment.

My guess is there would be some code added to handle some desired behavior (like being able to instantly reset the integration), and then that code will reveal what it is you really are trying to get this circuit to do.