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# Need help with A-C HWP 06.03: A "parabolically oscillating voltage of half-period T" is defined by:...

## Question

###### Need help with A-C HWP 06.03: A "parabolically oscillating voltage of half-period T" is defined by:...

Need help with A-C

HWP 06.03: A "parabolically oscillating voltage of half-period T" is defined by: o)-A(2) V(t) = A if t falls into the so-called "nth period interval", defined by: where n is an integer Note that for any given time t, there is always exactly only one integer n so that t falls into the nth period interval. The nth period intervals cover the entire real t-axis contiguously and without overlap, for n - 0,1, 2, +3 (a) Make a careful full-page plot of V (t) vs. t, either on graph paper or using any graphics- capable software (Excel, gnuplot, Matlab, Python, ...etc.), for the following parameter values A = 2.0V , T 10ms with V(t) plotted over the plotting time interval 70ms t S +70ms Show V(t) as a full line, Do not use any plot symbols (no "markers" !). Label the coordinate axes with "t [ms]" and with "V [V", respectively On the t-axis, mark the mid-point of each nth period interval shown and label it "0T", "+2T, "-27, ,'t4r", ,,-4r", etc. Also mark all the end points of these intervals and label them " +17, "-lr,''t3r, ,,-3r" , etc. On the V-axis, mark the value of A and label it ,,A". Include this plot in your HW solution paper Do not assume the foregoing specific values of r and A in the following parts of this problem Instead, solve all the remaining parts for arbitrary real, positive τ and A, unless stated otherwise (b) Let Prove that V(t) is indeed a periodic function, with a period of T, ie., prove that V(t + mT) = V(t) for all real t and all integer m Hint: If lies in the nth period interval, which period interval does the shifted time,- t +mT, fall into? In other words, assuming that (2n 1)T< tS (2n1)T, find the integer n, for which (2n,-1)T < t' < (2n' + 1)T. By above definition of V(t) you then know that:

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