Step 1: Understanding the Square Wave Input

The input voltage  is a square wave defined as:

This pattern repeats every T seconds.

Step 2: Charging Phase (  )

At t=0, the source switches to +1V, and the capacitor begins charging from its initial voltage  (where  ).

The charging equation for the capacitor voltage is:

Here:

•  = +1V (target voltage).
•   (starting voltage).

Substituting these values:

At , the capacitor voltage reaches +V (due to steady-state symmetry):

Step 3: Solving for V

Rearranging the equation:

For high RC(  ) , the exponential term can be approximated using the Taylor series:

Substituting this approximation:

Simplifying:

Neglecting the small term  :

The exact solution is:

For , so .

Step 4: Discharging Phase

At , the source switches to , and the capacitor begins discharging toward  from .

The discharging equation is:

At , the capacitor voltage reaches , completing the cycle.

Final Expression for Capacitor Voltage

For  :

For  :

Conclusion

The capacitor voltage  oscillates between  and , where:

This is because the capacitor cannot fully charge/discharge due to the high RC time constant. The waveform resembles a clipped exponential curve, oscillating below the input square wave amplitude.