#op-amps

Integrator circuit

We can analyze this exactly how we’ve done it previously with op-amps - we’ll sum currents at the inverting input. No current enters the op-amp, so we only have two current paths to worry about, and since the voltages are the same at both inputs and one is grounded, we can say the voltage at the inverting input is zero.

We’ve now got an op-amp circuit that will integrate whatever you put into it over time!

In actual application, you’re much more likely to see this version of the integrator circuit:

Rf will normally be very large so that the current that actually makes it back through this branch of the circuit is negligible and it will behave more or less the same as the other version. There are a few reasons to include Rf in the circuit - it makes the output of the circuit more stable and less prone to drift for reasons that we’ll look at more closely later. It also means that there is a path for feedback if the capacitor becomes fully charged. Recall that a fully charged capacitor looks like an open circuit to a DC voltage - if this occurs without Rf in the circuit, the gain of the op-amp will (theoretically) be infinite. The presence of Rf limits the gain that can occur in this case to Rf/R.

Differentiator circuit

If you can integrate with an op-amp capacitor circuit, you can differentiate, too.

As with the integrator circuit, there is another variation of the differentiator that you may see.

A regular differentiator can be subject to a lot of noise under some circumstances. This variation improves their performance in this respect, again, for reasons we’ll touch on shortly.

So under what circumstances would you actually have a use for a circuit that can integrate or differentiate?

Well, a common one is converting between wave forms. Suppose you have a ramp input that you want to make into a pulse, or vice versa.

But think about what happens here - if your input voltage is higher than your threshold (the voltage difference between the two terminals is positive), it throws the op-amp’s output all the way positive. If the input voltage is lower than the threshold (the voltage difference between the terminals is negative) it throws the output all the way negative. Essentially, the output of this circuit can only have two values - a high and a low.

You could use a regular op-amp to do this, but you probably don’t want to. Instead, you probably want a device designed to do exactly this and only this - a comparator. The output is a little more flexible than a regular op-amp (you aren’t limited to the positive and negative supplies of the chip), and they typically have a much faster response than a regular op-amp.

But anyways. What’s a comparator good for?

Threshold detection - Suppose you need an alarm to sound when the pressure in a tank exceeds its rating, or the fluid in a reservoir gets too low. A comparator will do the job - you compare the voltage you get off a level sensor or pressure sensor to a threshold voltage that you set. You could combine a couple of these to get a window detector that will tell you when your input signal is within a high and low bound.

Zero-crossing detection - Sometimes, you just want to know when a signal goes from positive to negative. You just compare your input voltage to ground. This will also have the effect of converting a sine into a square wave, which can be useful.

The power supply inputs are often ignored in schematics. As long as you provide the op-amp with the power it requires, it will work. The only impact the power supply has on the op-amp’s performance is that the op-amp cannot output a voltage outside the range of its supply rails - that is, if you give it 15 V at the positive supply terminal and -15 V at the negative supply terminal, it cannot output 16 V. If you try to make it do this, it will just clip at 15 V (or a little under).

So we’re left with two inputs and an output.

Here’s the basics of how this works:

The + input is the non-inverting input, and the - input is the inverting input. When the non-inverting input is higher than the inverting input, the op-amp’s output swings towards the positive side of its power supply. When the inverting input is higher than the non-inverting input, it swings to the negative. 

Op-amps are almost always used with negative feedback - that is, the output will be connected to one of the inputs in some way such that it cancels out some of the input. (There are a very few cases where positive feedback is used - the output is connected to the input such that it adds to the input - but this is much less common.) While feedback is an important concept in electronics and controls in general, what it means for us right now is that the output of the op-amp will do its best to make the difference between the inputs zero.

For the purpose of analysis, we will say that the op-amp always succeeds in doing this and the voltages at the inputs are equal.

We will also say, for analysis purposes, that the current into both of the inputs is zero. (In real op-amps, this is not quite true, but the input current is so small it’s frequently measured in picoamps, so it’s a reasonable simplification.)

So let’s try to analyze a basic op-amp circuit and see what happens. The circuit below is called an inverting op-amp.

We’re saying that the voltage at the inputs is equal. Since the non-inverting input is grounded, that means v+ and v- are both zero, here. v- also happens to be the voltage at Node A.

Since we’re also saying no current flows into the op-amp, we can write a node equation for A without worrying about any current flowing from A to v-.

This gets us the relationship between the output and the input for this op-amp configuration. It amplifies the input by the ratio of the resistors used and flips its sign. (Hence the name “inverting amplifier”.)

For contrast, let’s look at the non-inverting amplifier and do the same analysis.

Things come out a little messier here, but we’ve still got the input being multiplied (amplified) by some factor dependent on the resistors. This time, its sign isn’t flipped around, though.