#### tomizett

##### Active Member

I'm wondering if someone can help me tidy up my algebra? My problem concerns the time taken for a capacitor to charge from one voltage to another (I'm sure this must be a classic electrical engineering exam question).

So...

The capacitor C is charged from 0v through a resistor R towards a supply voltage Vmax. At time t the voltage is:

V(t) = Vmax( 1 - e^(-t/RC) ) [1]

...and so the capacitor reaches an arbirary voltage V1 at time t1, where:

t1 = -RC ln( 1 - V1/Vmax ) [2]

...and likewise sometime later it reaches a higher voltage V2 at time t2, where:

t2 = -RC ln( 1 - V2/Vmax ) [3]

the time taken to charge from V1 to V2 is (t2 - t1), or:

RC ln( 1 - V1/Vmax ) - RC ln( 1 - V2/Vmax ) [4]

and removing the common factor of the time constant RC gives:

**t1,2 = RC ( ln( 1 - V1/Vmax ) - ln( 1 - V2/Vmax ) )**[5]

Now, I'm pretty confident that this is correct, but is there a more elegent way to express it? Is there a cleaverer way to aproach this problem? I know that at some point, when Vmax >> (V1 and V2) the charge current becomes nearly constant and the time tends towards:

t1,2 = (V2-V1)RC / Vmax [6]

is there a way to see this limit from [5] above?

This isn't really important, but - as often happens - although I can come up with a solution I believe to be correct, my mathematics is not really good enough to give me a neat and tidy answer.

Thanks!