Question

1. For the function (t) below, T 2 and Vm-100 V. vt) 3 2 012 3 (a) Sketch v'(t) and derive the Fourier coefficients for '(t). (b) Use your results from part (a) to determine for Fourier coefficients for v(t). Express your solution in the complex form of the Fourier series, nugt and verify your solution by plotting your results using Matlab. 2. Assume that the signal above is the input to the bandpass filter shown below. y (t (a) Choose L=1H and the appropriate value for C if you wish to filter out the third harmonic from this signal (i.e. n- 3). Choose your resistance R assuming Q - 5, 10 and 100. (b) Derive an expression for the output voltage vo(t) in terms of the complex Fourier series en and the transfer function H(jnwo). Plot your results separately for the three values of Q.

Answer 1

V(t) = 50(t + 1), - 1 lessthanorequalto t lessthanorequalto 1 t = 2 w_0 = 2 pi/t = pi rad/sec c_n^1 = 1/t integral_- t/2^t/2 v^1 (t) e^- jnw_0 t dt = 1/2 integral_- t^t [50 - 100 (t -))] e^- jn pi t dt = 1/2 [50 e^- jn pi t/ - jn pi Vertical-bar _- 1^1 - 100 middot e^- jn pi (- 1)] = 1/2 [50 e^- jn pi - 50 - j n pi - 100 e^j n pi] = 50/2 [e^j n pi - e - j n pi/ j n pi - 2 e^j n pi] but e^j n pi = (- n)^n = e-^j n pi = 25 [0 - 2(- 1)^n] = -50(- 1)^n therefore c_n^1 = + 50 (- 1)^n + 1 v^1 (t) = d/dt x(t) doubleheaderarrow F. S

MATLAB CODE:

syms Ck ker t
prompt='what is the value of N?';
N=input(prompt);
for k=1:N
    w0=pi;T=2;
    ker=exp(-j*k*w0*t);
    Ck=(1/T)*(int(50*(t+1)*ker,-T/2,T/2));
    simplify(Ck)
end
M=abs(Ck);
P=angle(Ck)*180/pi;
subplot(1,2,1);stem(M);
xlabel('k');ylabel('magnitude of Ck');
subplot(1,2,2);stem(P);title('phase plot');

Result:

The above result is C1,C2,C3,C4 and C5 values.This verifies our answer is correct.

Please solve for Q=10 and Q=100 cases,the programme given above is general ,you can verify for output fourier series coefficients just by changing integrand value.

Thank you.