Charging of capacitor in RC circuit

My answer to SE EE question Charging of capacitor in RC circuit

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There is a great idea on which all op-amp inverting circuits are built. I can not stop discussing it when I come across a circuit solution in which it can be seen. Maybe the reason for this is that no one wants to see it... but it is so great that we can see its manifestations all around us. This is what happened today when I saw this question in StackExchange EE.

Answer

My answer will be a little unexpected for you because I will answer not "why not linear" but I will show how it can be made linear. In fact, this is the goal in most cases of practice; exponential relation, with few exceptions, is undesirable.

The trick is extremely simple if only you can guess. The voltage VC across the capacitor does not linearly change because it is subtracted from the input voltage (it is a loss) and the current decreases - I = (VIN - VC)/R. So we have to compensate this voltage drop.

For this purpose, we connect a variable voltage source in series to the capacitor and with the same polarity as the input voltage source (travelling the loop)... and adjust its voltage equal to the voltage drop across the capacitor. As a result, the voltage drop will be removed and the current will be as in the beginning - I = (VIN - VC +VC)/R = VIN/R.

If we feel bored doing this tedious job, we assign it to an op-amp. This is the idea of the op-amp inverting integrator - Fig. 1.

Fig. 1. Op-amp RC integrator (a geometrical interpretation visualized by voltage bars and current loops).

I created this Corel Draw picture in the 90's (the element designations do not correspond to the generally accepted ones). Here are some explanations for the inscriptions inside the figure. The circled op-amp (including the bipolar power supply) is a "helping voltage source"; UOUT (VOUT) is a "copy" and UC (VC) is the "original" voltage. So, the op-amp "copies" the capacitor voltage and adds it in series to the input voltage EIN (VIN).

And here is the hydraulic analogy (a little unusual communicating vessels) of the inverting integrator - Fig. 2. This conceptual picture corresponds to the op-amp circuit above.

Fig. 2. Hydraulic analogy of the inverting integrator.

Some explanations about the text inside the figure: The little man on the left is a "helper" and the capacitor on the right is a "thief":)

EDIT: This is a great idea (removing disturbance by anti-disturbance) that we can see everywhere... even in SE EE. There are contributors here for whom the powerful explanations of others are disturbance for his ego; so they try to compensate it with an anti-disturbance (-4)...

My comments

1. Thanks for the interpretation. There is no problem; we all know each other very well and understand what we are talking about. A little humor is never superfluous. I just saw the downvoter's reaction and made the connection with the great principle. BTW it can be implemented without negative feedback, e.g., by a "negative capacitor". Really, this is not the exact answer but it is closely related to it; this is the answer to the next question that logically follows, "How do we make the capacitor charge linearly?" In your place, I would first admire the idea and then criticize...
2. My answer is closely related to this question; so it is useful. If you take the trouble to follow the link above, you will see a 5-step scenario; the third step is dedicated to this question. Showing how something nonlinear can become linear is an indirect (and more original) way to explain what causes this nonlinearity over time...
3. I think it has become clear that this cannot be done through intuition but through mathematics. Unfortunately, intuition only tells us that the current is progressively decreasing but it can't tell us exactly how ... But nevertheless, I keep thinking about this phenomenon of "communicating vessels". But now the question arose how to make a -1 ohm resistor. Interesting topics have no end.
4.  I have a rule: qualitative things should be explained by qualitative means and quantitative things should be calculated by quantitative tools. I have noticed that processes in nature where the difference between two pressure-like quantities determines through a flow-like quantity the second pressure-like quantity, lead to an exponent.
5. This is all true but for "normal" people :) You know that there are some slightly obsessed people who want to know these things:) I think this is some kind of mental illness:)
6. Aren't you impressed by my geometric interpretation? It derives directly from the hydraulic analogy of "communicating vessels". If necessary, should I translate the captions inside the illustration? Have you come across many such illustrations on the web? If so, tell me to contact the author and express my admiration. I have browsed about 100 million pages and finally managed to find something similar. He "opened my eyes" what a transimpedance amplifier is.