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An RC network is configured as a low-pass filter, and a voltage step input is provided to the indicated circuit. How could one quantify the duration it takes for the output voltage waveform to travel from a specified initial value to a final one? How do we know that the output waveform visits any voltage value in the indicated history only once?

A low pass filter is generally analyzed assuming that the circuit which follows is of an infinite resistance and so, in parallel to the capacitor, that infinite resistance may be neglected. We are then left with a loop made of a voltage source, V, a resistance R and a capacitance C.

If V0 is the initial internal voltage related to the energy initially stored in the capacitor, and if the voltage of the source, V, is constant through time (for the time duration under consideration), then the voltage drops across the capacitor, through time, is

Vc = V0 + (V-V0) exp(-t/RC)

So, for Vc to reach a given value (between V0 and V) the required time is logarithmic.

A logarithmic function is not only monotonic but also continuous. (Discutable though, accordingly to mathematicians)

Note though that as in a river flowing in one direction, it is possible for a time that a small eddy creates a small local flow of water running in the reverse direction, the same is possible for an electrical circuit, as we zoom increasingly into the details. That is called "noise" and so, at the noise level, voltages values can be revisited. So, the proposed (exponential) equation is said to be exact far from the noise level in addition to consider ideal elements (even a simple resistor can create noise, capacitance, etc.), thermal currents and others possible considerations not taken into account.