Remaining ripple on a filtered PWM output

Dear PWM enthusiasts,

I calculated the remaining ripple voltage on a filtered PWM output. The 
filter is a mere 1.order RC lowpass. I found an analytic expression for 
the stationary upper and lower ripple voltage and verified the results 
with LTSpice simulations.

Thus it is possible to calculate the corner frequency of the lowpass or 
the PWM frequency for a given ripple voltage.

The calculation was done in the time domain, I enclose a scan of the 
derivation for fun and recreation. It was a bit tricky but great fun.

math rulez!
Cheers

Detlef

$$U_{max} \mbox{ output voltage PWM, } T \mbox{ PWM periode, }\delta \mbox{ PWM duty cycle, }\tau \mbox{ lowpass time constant, }$$

$$U_{pp} \mbox{ upper ripple voltage, }U_{pn} \mbox{ lower ripple voltage}$$

$$U_{pp} = U_{max} \cdot {\frac {1-e^{-\delta T/ \tau}}{1-e^{- T/ \tau}}}$$

$$U_{pn} =U_{pp} \cdot e^{- (1-\delta) T/ \tau}$$