Design rules for opamp bootstrapping
Here is the bootstrapping technique for opamps, as exposed in the Art of Electronics:
This technique is supposed to increase considerably the input impedance of the opamp for an AC input source (usually passed through a cap). The Art of Electronics considers it somewhat outdated, as it is now possible to use extremely low bias input current opamps.
I disagree with him: I see no reason to buy a somewhat expensive femtoamp opamp when you can obtain the same result with a more banal one (as long as we deal with AC sources).
Actually I do have an application where I use the self capacitance of a rotating half cylinder to measure the ambient electric field, which is perfectly tailored to bootstrapping.
Now, to make my question precise:
Assume
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I have an AC sinusoidal source of known frequency w;
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I want the output signal (voltage) be at least n percents of the input signal (e.g. 99%);
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I want the RMS input current be equal to at most I_max.
Question: how should I choose the values of the two resistors and of the capacitor in the schematic above?
Edit: Here is a simulation done with LT spice. I've more or less given random values, following nothing but some intuitive guess:
Simulation with bootstrap:
Result (current through the sine generator):
Result of the same simulation but without cap C1:
Edit 2:
Perhaps the schematic will make more sense if we pass the input signal through a capacitor, as is usually the case. Here is the schematic for the simulation:
And here are the results: the first trace is the current through cap C2, and the second trace is the current at the in+ of the opamp:
So, we see that the amplitude of the current through C2 is about 2nA, and the amplitude of the AC current at in+ is 6nA, more than I_C2.
Without cap C1, the current through the sine generator (and hence through C2) is the same as previously
2 answers
That circuit doesn't do what you think it does. Here is the circuit in question:
The impedance seen at IN is definitely higher at high frequencies, but that is only due to it being artificially low at DC.
At DC, there is a load of 2R on IN, in parallel with the opamp input impedance. At high frequencies, but within the opamp's range, the voltage on the left side of C1 is the same as IN. That means there is no voltage across R1, which means there is no current thru it. At high frequencies, therefore, the load on IN is only the opamp input impedance.
Put another way, the resistors add load on IN, which the circuit removes at high frequencies. However, the load due to the opamp input is always there.
A similar circuit can offset input impedance, but there has to be some positive gain from IN to OUT. Then at high enough frequencies, the left end of C1 is the same signal as IN, but a little larger. This then has the effect of adding a negative resistance load to IN. If that negative resistance exactly offsets the positive resistance of the opamp input, the net effect is apparent infinite input impedance. However, again, that requires a gain above 1 from IN to OUT. Your circuit is a voltage follower, which has a gain just under 1.
The technique above is very difficult to get right, mostly because the opamp input impedance needs to be known accurately. For example, if the opamp input impedance happens to already be 0, then trying subtract a fixed impedance from it will make things worse.
Nowadays, opamps with high input impedance are cheap and available. There is no point trying to play games like this, and the cost is minimal compared to the whole system.
You say you are trying to build a field mill. The cost of the motor and battery alone will swamp any reasonable electronics for detecting the field.
Without cap C1, the current through the sine generator (and hence through C2) is the same as previously
That means that the impedance presented to the sine generator is the same in both cases. Since the voltage is the same, and you say the current is the same, the impedance must also be the same.
However, that doesn't make much sense. At DC, the load should be R1+R2, which is 200 kΩ. There will be some load due to the opamp input, but that should be small relative to 200 kΩ.
The opamp is a voltage follower, so OUT should be driven to the sine generator output. For sufficient frequency, that will also appear on the left end of C1. That results in 0 V across R1, which means 0 current thru it, which means R1 presents infinite impedance to its top end.
This means there should have been a difference in sine generator output current with and without C1.
What is the amplitude of the sine generator output? That divided by 200 kΩ should be the current out of the sine generator with C1 removed.
I noticed that the two voltage sources for powering the opamp are labled AC1 and AC2. Those need to be DC sources.
I answer to myself because I believe this subject is interesting and not very well understood, even to professionals. Also, I want to point out that this answer is the product of my own investigations and simulations, and is not coming from reliable sources or knowledge.
First of all, I want to point out that the bootstrapping technique is mainly used in the context of transistor based amplifiers, mainly BJT.
In the following image, R1, Rc, RE and R2 form the usual topology of a BJT amplifier. The bootstrap is formed by resistor R3 and cap "Cboot":
In this context, the bootstrap technique is alive and well! Excellent related questions and answers can be found here. I will not enter into this matter since it is well known and the original question is related to opamp bootstrapping. I only point out that the main point is that the bootstrapping technique avoids the loading of the AC signal by the biasing resistors.
Now, let me return to the main question: opamp bootstrapping.
Despite what the "Art of electronics" says (and also, to some respect, what Olin said above), this technique, if correctly understood, can be still useful in some circumstances.
Here is the main point of bootstrapping: often, an AC signal needs to be passed through a cap, and then buffered or amplified. There may be several reasons to pass the input signal through a cap, the most obvious one being you want to annihilate some DC bias present in the input signal.
Doing so, it is mandatory to follow immediately the cap with a resistor called "a return path to ground". Here is an illustration of what is meant:
Without a return path to ground, the output of the opamp would be undefined: for example, if there is no input signal at all, you expect that the output of the opamp be 0. But without a return path, the opamp would charge cap C, and its output would stabilize somewhere between its supply voltages, most often near Vcc+ or Vcc-. Usually, the return path R has to be chosen in such a way that the RC time constant is well below the lowest frequency of interest present inside the input signal.
This being said, for very very sensitive input signals, even a return path of 10 Mega ohm may load the signal heavily: it may be necessary to make the return path to ground at least 1Gohm or more, which is not convenient. This is where the bootstrap technique may be useful (see schematic inside the question): it virtually annihilates the the return path at the AC level, and ensures the signal loading is minimal, that is, no more than the input current of the amplifier.
A last question remains: what values should be chosen for the resistors R1=R2=R and C in the schematics of the question. Well, this is again function of the RC time constant. More precisely, the cut off frequency 1/2πRC has to be chosen well below the lowest frequency of interest present in the signal. For example, R = 1 Mega ohm and C = 100 nF fits most signals > 10Hz.
To sum up:
- The opamp bootstrap technique is used to prevent the return path to ground to load the input signal;
- It can be advantageously used for extremely sensitive input signals to ensure the signal is loaded as few a possible, that is, no more than the input current of the amplifier itself;
- Hence, it allows using standard resistor (in the range of 1 M ohm), and in fact opamps with higher input current, since they surely works with return path of 1 M ohm.
- Perhaps most importantly, this technique allows not worrying about the value of the return path to ground: it ensures the circuit works in the optimal way anyway.
- finally, regarding what values to choose for the resistor and the bootstrap cap, it appears that R = 1 Mega ohm and C = 100 nF works in most of the cases, whenever the frequency of the AC signal is > 10Hz.
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