Charging time of an RC circuit

This page will describe how to calculate the time it takes for a capacitor to charge from one voltage to another through a resistor.
Let C be the capacitance of a capacitor being charged from a voltage source with voltage V through a resistor of resistance R. Suppose the capacitor starts with 0 volts across it and is allowed to charge. The voltage \(V_c(t)\) can be modelled as a function of time (t): $$ V_c(t) = V - Ve^{-\frac{t}{RC}} $$ This equation can be solved for t to find the time it takes for a capacitor to charge from 0 volts to \(u\) volts $$ u = V - Ve^{-\frac{t}{RC}} $$ $$ \frac{u}{V} = 1 - e^{-\frac{t}{RC}} $$ $$ 1 - \frac{u}{V} = e^{-\frac{t}{RC}} $$ $$ \ln\left(\frac{V-u}{V}\right) = -\frac{t}{RC} $$ $$ RC\ln\left(\frac{V}{V-u}\right) = t $$ Suppose one is only interested in the amount of time T it takes for a capacitor to charge from \(l\) volts to \(u\) volts. One would calculate the time for it to go from 0 to \(u\) and subtract the time for it to go from 0 to \(l\). $$ T = RC\ln\left(\frac{V}{V-u}\right) - RC\ln\left(\frac{V}{V-l}\right) $$ $$ T = RC\left(\ln\left(\frac{V}{V-u}\right) - \ln\left(\frac{V}{V-l}\right)\right) $$ $$ T = RC\ln\left(\frac{\frac{V}{V-u}}{\frac{V}{V-l}}\right) $$ $$ T = RC\ln\left(\frac{V-l}{V-u}\right) $$

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