# How does an op amp integrator work?

I know there are at least two questions related to this on stackoverflow but neither really answer my question, and in any case, both questions got downvoted. What I am after is an operational understanding of how an op-amp integrator works. I know how a simple RC circuit can integrate, what I don't understand is how the feedback loop in an op-amp configuration helps. I understand how feedback works in a noninverting amplifier. I took the figure below from www.electronics-tutorials.ws. This web site has an explanation but I don't follow it. My understanding so far is this:

1. Apply a positive voltage to input vin. Current flows through Rin resulting initially in a non-zero voltage at X (Correct?).

2. Due to the high impedance of the op-amp at X, we can assume that all the current then flows to the capacitor (initial discharged).

3. The capacitor starts to charge resulting in a voltage across the capacitor.

4. The difference in voltage at the two op-amp inputs (the positive input is at zero, hence the difference is negative) resulting in the output, vout, going negative (we assume that vout was zero initially).

My question is what happens next? How does the feedback act to bring the difference between the two inputs back to zero? Or have I got this wrong?

I am very familiar with the proofs for showing that the configuration will integrate but they don't give any real intuition and many videos, wikpedia, and books but almost all regurgitate the proof without giving much insight. I'm after an intuitive understanding, not a mathematical proof.

Out of interest I also redrew the op-amp circuit next to the RC integrator shown below which gives the suggestion that the op-amp is amplifying the small vollage across C (assuming high R1) while having high impedance from the resistor/capacitor node. Not sure if that is a legitimate way to look at it.

• Put in a square wave, the opamp strives to hold the (-) pin very close to zero volts, which imposes a constant voltage across the input resistor. This constant current has to go somewhere, and the only path is into the capacitor. Give constant current into a capacitor, you get a perfect RAMP output, which is the integral. – analogsystemsrf Jan 18 at 3:04
• To me - this small comment (if compared with all the answers) is the best explanation of the circuits behaviour in the time domain. – LvW Jan 18 at 14:57
• But it doesn't explain why the op-amp strives to hold the (-) pin close to zero. An op-amp on its own won't do that, it's the combination with the feedback that makes that happen. This is the bit I wasn't sure about. – rhody Jan 18 at 18:47

This may help:

• Remember that when current flows into the RC junction of your op-amp that the voltage at that point will tend to rise.
• If the inverting input voltage rises the slightest bit above the non-inverting input voltage then the op-amp output will start to swing negative.
• The output swinging negative will, through the capacitor1, tend to pull the inverting input down towards zero again where it stabilise (for the moment).

The result is that feeding current into the RC node causes the op-amp output to go negative.

Out of interest I also redrew the op-amp circuit next to the RC integrator shown below which gives the suggestion that the op-amp is amplifying the small voltage across C (assuming high R1) while having high impedance from the resistor/capacitor node. Not sure if that is a legitimate way to look at it.

That's correct. It might be better than you think. The simple RC circuit has the advantage that it's non-inverting but the disadvantage that it's non-linear. With a constant input voltage the output will be an exponential charge curve.

Putting the op-amp in as you have shown still allows the capacitor to charge up but maintains the top terminal at virtual ground. The advantage is a linear change in output. The disadvantage is that there is a minus sign on the integral obtained.

1 You can think of a capacitor as holding the voltage across it as a constant in the short term. That means that if the voltage on one side is changed the voltage on the other side will try to change by the same amount.

One question. what is the orientation of the capacitor in terms of conventional current? i.e. if vin goes positive the capacitor is I assume negative on its right-hand side (nearest vout). Now vout goes negative and therefore reduces the voltage across the capacitor until the potential at X is zero?

I think your understanding is correct.

If Vin goes positive then current flows into the X node charging up C. (Remember the op-amp's voltage hasn't changed yet.) This tends to increase the voltage on the inverting input and that causes the output voltage to decrease. This draws some charge from the right hand side of C. Now the inverting input is pulled back down to zero volts but there is charge on C so there is a voltage across it. Since the conventional current flowed to the right there is a negative voltage remaining on the capacitor.

• I think this the kind of answer I was looking for. One question. what is the orientation of the capacitor in terms of conventional current? ie if vin goes positive the capacitor is I assume negative on its right-hand side (nearest vout). Now vout goes negative and therefore reduces the voltage across the capacitor until the potential at X is zero? – rhody Jan 17 at 22:44
• See the update. – Transistor Jan 17 at 23:15

The op-amp is going to try its best to keep the voltage between it's plus and minus input the same. In an ideal op-amp, no current flows into the inputs, so the only way that it can do that is by changing its output voltage.

In the schematic below, $$\v_+ = 0\mathrm{V}\$$. That means that the op-amp will try to hold $$\v_-\$$ at zero, also.

Whatever voltage is generated by V2 gets turned into a current by R1. Because $$\v_-\$$ is being held at $$\0\mathrm{V}\$$, that same current has to flow in C1. And because $$\v_-\$$ is being held at $$\0\mathrm{V}\$$, the op-amp has to drive the output voltage such that the current in C1 matches the current in R1.

So if $$\v_2\$$ is constant, then the current into the node around the negative input is constant, which means that the current out of that node from the cap must be constant -- and that can only happen if the output voltage is falling at a constant rate. The end result is that the op-amp integrates the input voltage into the output voltage.

More complicated voltages at $$\v_2\$$ cause more complicated behavior, but the op-amp is always going to be trying to drive $$\v_-\$$ to $$\0\mathrm{V}\$$. It can only do that by satisfying $$\ \frac{d}{dt} C_1 v_{out} + \frac{v_2}{R_1} = 0 \$$. If you solve that differential equation, it says that $$v_{out} = -\frac{1}{R_1 C_1} \int v_2 dt$$

HTH

simulate this circuit – Schematic created using CircuitLab

• In your comment "The op-amp is going to try its best to keep the voltage between it's plus and minus input the same.", strictly speaking, it's the op-amp combined with the next feedback that does this. This is the bit I was stuck on, how the feedback coupled with the op-amp manages to maintain the difference close to zero. – rhody Jan 18 at 18:50
• Well, I'm glad you worked through it! These are difficult concepts and hard to simplify. – TimWescott Jan 18 at 18:57

Rhody - have you heard about the MILLER effect? Well - the shown circuit is called "MILLER integrator" because the MILLER effect is exploited. Remember: This effect reduces the feedback impedance between an amplifier output (for example: collector) and the inverting input (example: base node of the transistor). And the factor of increase is the gain.

Here, we have the same principle. Hence, there will be a very small capacitive impedance (that means: A very large capacitor) between input and output of the opamp. And the factor of increase is the open-loop gain Aol of the opamp.

Hence, you can make a comparison with a simple RC circuit. However, because of the very large capacitor the cut-off frequency is very low (nearly DC).

Frequency domain: The transfer function between the opamps inverting node and the signal input is

Ho(s)=1/(1+sCo * R) with Co=Aol * C (MILLER effect).

Because of the very large value Aol, we can neglect the "1" in the denominator and arrive at

Ho(s)=1/(sC * Aol *R)

We are lucky and can use the low resistive opamp output (and multiply the function Ho(s) with the gain -Aol) and arrive at the final result (opamp output-to-signal input):

H(s)=Ho(s) * (-Aol) = - 1/sR*C (Transfer function of an ideal integrator)

The inputs of the opamp don't take input current and the opamp will keep its input voltage equal as long as it is wired for negative feedback.

So effectively, the current that the input sinks against the 0V it sees goes straight into charging the capacitor. Usually when you charge a capacitor through a resistor, the charge building up on the capacitor reduces the voltage across the resistor and thus also the charging current, leading to an exponential decay of the charge current.

Here, however, the opamp output actively adjusts the voltage at the other side of the capacitor so that the resistor never gets to see the difference it makes, thus keeping the current through resistor (and into capacitor) independent from the charge across the capacitor.

It's like Útgarða-Loki handing Þorr the mouth of the ocean as a drinking horn and Þorr does not actually find himself able to drain it, not noticing that he causes the tides with his attempt.

The Passive cap. integrator current decays with voltage as it approaches the input.

The active cap. integrator saturates after some time if Vin ≠ 0 because the output voltage drives current into Vin- to maintain 0V diff. until the output saturates at the supply rail.

So input offset is critical and you need an analog switch to discharge and initialize to 0V output.

## anecdotal

I remember knowing nothing about this in 1st yr Eng and my once famous brother-in-law medical Dr of Anesthesia, Intensive Care and open heart surgery, gave me a tour around the hospital and said he needed an integrator to measure O2 content in blood for the brain, after a heart attack victim stops, to know the best treatment ( such as hypothermia )to give when no response to defib. and meds with the likelihood of success. I had no idea ! and was embarrassed to not know! (circa '75) Don't be. Just research it.