Question

64) A Sinusoidal Volage E Sin that clips the negative portion of the wave. is passed through a half-coave tech fler oldal Voltage E Sincest), Lobere tis time hind the Fourier Series of the resulting pertodie function given by : fet) - -/octco 2 E Sincut) 0 <t21%. Your answer should be Fet) + Sin Cut)- DELFs cos cowt) + ss.com (4.c)+ ] where an - - -_ = 2€ Can- Con+1 (402-1) bi = bn = o for all n #1 and where fet) = E + E Sin Cut) - Cos (Incot) Cun2-1) n=1 05) Expand fct) - Sinct) Cos?ct in a finite fourier Series . Hint: Use Simple trigonometrie Identiftes to corite fct A+ B cos Ct) + C Coscat) + D Coscat) + E Cos Cut) + Flos (5t). Find A, B, C, D, E and F. You Should get the following: f(t) = te Cocos (t)- cos (3+) - Cos (5)) therefore, A=0 c=0 E=0 B = 1/2 D = 0/6 F= -1/16.

please answer question 4 and 5

Answer 1

fto awtLO Esint out to i time period T = 2T By fowier sewes f(t); ao ta an cos new t-tbn sin neot ao = I l feat w/w0 0 T E skn not at lor wt - - 0. (1-0) - the flt at cos neat an - I Eston ut dr- lod next E I in Rot E by (sin whHilto sin wordt] at ( los tom-ut Won - cos w nilt ww winti . -

les (ntin I cos (n-1) no 7 ) - nt ut ntl nd W It los nit -l nal Cosnt ntl for an even for an even odd. an ao G+ - 2 , n even even ntln -n-lan-1 in one, no con 2 / T E sin eut sen nat at In - (nti) et + los (nar) ut ] at es over one Integration of half period cos is function you. = bn to na

Th be - - l E sin ut sin ut at It cos z wt at 5 (4) = sin(t) carlt) SP 2 = 1 / [esint cast)? 14 (sin att = 1/2 [ 1- cos at = A= / E = //

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CHECK:

%MATLAB

t=0:0.01:2;
E=1;
T=1;
w=2*pi./T;
x=(1/pi).*ones(size(t));
for n=1:100
an=(-2./(pi.*((n^2)-1))).*(n~=1).*(rem(n,2)==0);
if isnan(an)==1
an=0;
end
bn=0.5.*(n==1);
x=x+an.*cos(n.*w.*t)+bn.*sin(n.*w.*t);
end
plot(t,x)

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02- 0 02 04 06 08 1 12 14 16 18 2

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%matlab
t=0:0.01:2*pi;
x1=(sin(t).^2).*(cos(t).^2);
x2=(1/8).*(1-cos(4.*t));
plot(t,x1);
title('x(t)=sin^2(t)cos^2(t)');
figure
plot(t,x2);
title('x(t)=(1/8)(1-cos(4t)');

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x(t)=sin?(t)cos? (t) 1 2 3 4 5 6

x(t)=(1/8)(1-сos (4) 1 2 3 4 5 6

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The answers obtained by you are wrong, therefore, I included a matlab script to check the validity of the answers I got,

the first one is a fourier summation of first 100 terms which approximated to a half wave rectified signal

the second one is individual graph if sin^2(t)*cos^2(t) and (1/8)*(1-cos(4*t)) , which comes out to be same