Exercise:
Using alternating exponentially increasing and decreasing signals we can model a wave that looks like a series of shark fins: vt cases V_ left - e^-t/tau right & textfor quad leq t T/ V_ lefte^-t-T//tau - e^-T/tau right & textfor quad T/ leq t T cases with periodical continuation. The figure below shows an example with tauT/. center includegraphicswidthtextwidth#image_path:shark-fin-no-fill-# center abcliste abc Derive a formal expression for the mean value. What is the value for the time constant tau where the mean value corresponds to V_mV_/? abc Derive a formal expression for the root mean square value. Calculate the ratio sscVrms/V_ for the time constant tauT/. abcliste

Solution:
abcliste abc The mean value is V_m fracT_^T vt textdt fracV_T left _^T/ left-e^-t/tauright textdt + _T/^T lefte^-t-T//tau - e^-T/tau right textdt right fracV_T left fracT - _^T e^-t/tau textdt + _^T/ e^-t'/tau textdt' - e^-T/tau fracT right where we have used the substitution t't-T/ in the last step. The two remaining egrals are identical so this can be simplified to V_m fracV_T fracT left - e^-T/tau right fracV_ left - e^-T/tau right The mean corresponds to V_/ if V_m fracV_ fracV_ left - e^-T/tau right Longrightarrow frac - e^-T/tau Longrightarrow e^-T/tau frac Longrightarrow -fracT tau lnfrac -ln Longrightarrow tau fracTln abc The root mean square is given by sscV^rms fracT _^T v^t textdt fracV_^T left _^T/ left-e^-t/tauright^ textdt + _T/^T lefte^-t-T//tau - e^-T/tau right^ textdt right fracV_^T left _^T/ left-e^-t/tauright^ textdt + _^T/ lefte^-t'/tau - e^-T/tau right^ textdt' right fracV_^T _^T/ left - e^-t/tau + e^- t/tau + e^- t/tau - e^-t/tau e^-T/tau + e^-T/tau right textdt fracV_^T left fracT left+e^-T/tauright - left+e^-T/tauright _^T/ e^-t/tau textdt + _^T/ e^-t/tau textdt right fracV_^T left fracT left+e^-T/tauright - left+e^-T/tauright left-tau e^-t/tauright_^T/ + left-fractau e^-t/tau right_^T/ right fracV_^T left fracT left+e^-T/tauright + tau left+e^-T/tauright lefte^-T/tau-right - tau lefte^-T/tau - right right fracV_^T left fracT left+e^-T/tauright + tau lefte^-T/tau-right - tau lefte^-T/tau - right right fracV_^T left fracT left+e^-T/tauright + tau lefte^-T/tau-right right fracV_^ left + e^-T/tau -fractauTleft-e^-t/tauright right The root mean square is thus sscVrms fracV_sqrt sqrtleft + e^-T/tau -fractauTleft-e^-t/tauright right For tauT/ we find fracsscVrmsV_ sqrtfracleft + e^- -fracleft-e^-right right sqrtfracleftfrace^ +frac right sqrtfrac+e^ e^ approx resP The figures below show the signal and the squared signal respectively. center includegraphicswidthtextwidth#image_path:shark-fin-fill-# includegraphicswidthtextwidth#image_path:shark-fin-squared-# center abcliste