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1. Consider a C.T. system below. (Assume all initial conditions are zero). (a) Find transfer function, H(s), for the system. (include all detailed steps, no points for just the answer). dt2 + 4 dt (b) State Response, y2s(1) when the system when input just signal the answer). is x(t) Find = Zero e -5tu(t). (include all detailed steps, no points for x(1) = 8(1) X(s) =1 1 x(1) =1 X(s) = s dx(t) 1 sX(s)~x(0") x(t) = e-d X(s)= dt s+a 5 d²x(t) 52x(s). - sx(07-107 x(1) = cos(bt) x(s)==167 = dt² b 1 x(t) = sin(bt) X(s)= 3+6 x(1)dr -X(s) s Laplace Transform Properties Basic Laplace Transform Pair 1 2. Consider a C.T. system implemented by using op-amps. Complete steps below. (a) Find Transfer Function, A(s) = Y(s) / X(s), for a C.T. system with op-amps below. Assume all initial conditions are zero. (Partial credits will be given, be sure to include ALL detailed steps) (b) Find Transfer Function, B(s) = Y(s) / X(s), for a C.T. system with op-amps below. Assume all initial conditions are zero. (Partial credits will be given, be sure to include ALL detailed steps) M Cn Rn Ro W If M Cp R11 Rn - + + + N(s) Ca Y(s) X(s) Y(s) - - A Figure 2(a) For A(s) = Y(s)/X(s), Figure 2(b) For B(s) = Y(s) / X(s), an mostom with below + - -0 + X(s) Cn + Y(s) T(t) - - Figure 2(a) For A(s) = Y(s) / X(s), Figure 2(b) For B(s) = Y(s)/X(s), (c) Use results from (a) and (b) to realize (build) a C.T. system with below description. Find values for R/1, Ril, R/2, Ri2, and Cp when C=0.01 [uF] and Cp = 0.01 [uF]. (Assume all initial conditions are zero.). Use an additional inverting gain op-amp if needed. (Partial credits will be given, be sure to include ALL detailed steps) d²y(t) dt2 + 35 dy(t) dt + 5 + 50x(t) 3. Consider a periodic signal x(t) shown below, where Amplitude = 4 [V], To=2 (ms) (period of signal), wo (fundamental frequency), and n fo (harmonic frequencies). 5 4 3 2 1 -1 -3 -2 -1 1 2 3 4 -4 -10°3 (a) Find Co, C1, C3, and C5 using the definition of Compact Trigonometric Fourier Series Coefficients. (Partial credits will be given, be sure to include ALL detailed steps) (Note) (b) Draw frequency-domain representation of signal (C6/v5(n)6), n: integer). = 0). 2-D graph, similar to spectrum analyzer traces. (note, G=G=C= mol Co=ao Cn= an+b2 b, x(1)=af+) an If n= nol On = tan"¹ -bey) an 1 af= - To S x(t)dt T3 2 dt (m = 0) Toir X.=7 1 2 dt (m # 0) To is a) Find Fourier Co, C1, C3, and C5 using the definition of Compact steps) Series Coefficients. (Partial credits will be given, be sure Trigonometric to include ALL detailed (Note) an=2n f==2x/To (b) Draw frequency-domain representation of signal n: integer). 2-D graph, similar to spectrum analyzer traces. (note, C=G=C6 =0). x x(1)=C C, Il n=1 Co =ao Cr= an+b2 00 80 11 nost n=1 br 1 On =tan~¹ = To x 2 x(I) = an-T!! (m # 0)

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