Capacitor Equations

[mikec]

Hi,
I have a simulation/scenario where I can't use the standard capacitor equations to find the capacitor voltage or current while charging. I have run the simulation in LTSpice and the basic scenario is where the risetime of the pulse signal is much slower the time constant (RC) of the circuit. I have attached the circuit and the plot. I had never noticed this before as any graphs show an ideal very fast risetime for a pulse and basically I do the same when I am normally simulating. Must literature explaining capacitors deal with 2 scenarios where the pulse length is either 1. Shorter than the time constant or 2. Longer than the time constant but I have never seen the risetime been considered.

The first thing I note from the plot is the maximum charge current that is reached which is only 1mA, now I know that I=C x dv/dt so because the rate of change of voltage is "slow" the circuit never reaches the max current in the circuit which as a rule of thumb I use I=V/R = 1V/100 = 10mA in this case. I know when I decrease the risetime of the pulse to be much faster than the time constant I get about 10mA.

The other thing I note is that the equation Vcap = Vsource(1-e(-t/RC)) doesn't hold for the first portion of the graph (in the orange box), again this clear because Vsource has not reached its max or final value.

The third thing I note is that at the very start after applying Vsource the Vcap is exponential which gives rise to an exponential Icap, after this the rate of change of Vcap seems to follow Vsource. I can't explain this part of the graph?


What I'd like to know is are there equations for this scenario that can be used to calculate say Vcap when the rise time of the pulse is longer than the RC?

What a practical consequences of this when designing a circuit? Has or does anyone frequently consider this when designing a circuit?

Cheers!