Question

I ONLY NEED B AND 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 ? 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:

Answer 1

C00 4 0 Cb 10 d) Phra he voly 10

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2-0 IO I0

80 70 60 V(t)=A(t-2n τ) 2/t 2 50 Σ40 30 20 10 .4τ t[ms]

parabolic.voltage.gp × 1 set terminal png enhanced 2 set output "parabolic-voltage.png" 3 set xrange [-70:70 4 unset key 5 set xlabel "t [ms]" font ",12" 6 set ylabel "V [V]" font ", 12" 7 set label. "V(t)-A(t-2n/Symbol t])*2/{/Symbol ty2" at 20,60 8 set label "A" at-74,2 font",14" 9 set xtics ("-7/Symbol t" 70, -61/Symbol tj" -60, "-41/Symbol t]"4,-21/Symbol tj" -20, /Symbol t" 0, "21/Symbol t" 20, "41/Symbol t]" 40, "61/Symbol t" 60, "71/Symbol t]".70) 12 14 plot-v(x)