I am working on a problem involving an RC circuit and I just cannot figure it out. The circuit consists of a voltage source, a 1k resistor and a 400nF capacitor connected in series. The source a 5V square wave source with on/off time of .5ms. Since the time constant of the circuit is .4ms, I know that the capacitor cannot fully charge/discharge before the input changes again.
I have done a PSPICE simulation of the circuit, but I cannot figure out a mathematical equation to describe what is happening to the capacitor voltage. The graph of the capacitor voltage obtained from PSPICE is attached to this post.
I have tried using the equations for vc charging(Vs*(1-e^(-t/RC))) and discharging (Vs*e^(-t/RC)), but they don't work because the relatively large time constant of the circuit does not allow the capacitor to charge up to 5V or discharge to 0V.
How can I find the equation for capacitor voltage, both charging and discharging, when the RC time constant is greater than five times the on/off time of the source?

 

You assume the circuit response has reached steady state.

For the charging condition

Vmax=Vmin+(5-Vmin)*(1-exp(-t/RC))

For the discharging condition

Vmin=Vmax*exp(-t/RC)

You then simultaneously solve for either Vmax or Vmin to satisfy the conditions at t=0.5ms and RC=0.4ms

As a check you require that (Vmax+Vmin)/2=2.5V [the steady state average capacitor voltage]

 

I got it. Thanks for your help. Your solution makes complete sense and I'm not sure how I missed that before.
Thank you!