Question about time constant of RC circuit

The formula is:

$$V_{avg}=duty\_cycle \cdot (V_{high}-V_{low})$$

You can find the exact mathematical derivation on the web. I'll try to provide some intuition.

So, if the average output voltage of the RC filter driven from a PWM signal is not dependent on RC values, then what is? Mainly two factors:

1. High RC time constant leads to a longer "charging time" - the time it takes the output voltage to reach its average value. You can think of this time as a "turn-on" time. The "discharging" or "turn-off" time is also increases with the RC time constant.
2. High RC time constant leads to a lower ripple of the output voltage around its average value.

#1 is easy to explain - just recall that high time constant means longer raise/fall time of the voltage in a simple RC circuit. The same applies to this RC filter.

#2 follows from the same argument - the higher the raise/fall time of the RC filter, the lower the change in voltage that will be seen on the output during the high/low phases of the input PWM signal.

Please note that the above two factors are conflicting - you will (usually) want a fast turn-on/turn-off time and a low output ripple. The former requires low RC while the latter requires high RC. You will choose the RC values according to the constraints of your application, and if they can't be satisfied with a simple RC network - you'll search for a more sophisticated filters.

A transient simulation simulates the circuit's conditions over time. You seem to have no component in your circuit which introduces a time varying voltage.

Use a vpulse source instead of the vdc (V17), fill in it's properties so that it will step the supply voltage. Simulate that, then check that the supply voltage has the transient step that you want to simulate. If that is OK, look at the voltages inside the circuit.