1. Examples: Write the transfer function of the electrical circuit including Op-Amp (Operational Amplifier). Write the Matlab program to calculate the eigenvalues of the system. + - V 2 (t)V 1 (t) L R1R1 R2R2 C clc;clear a=[120 412 7500]; roots(a)

2. Examples: The equation of motion of the system whose mass can be neglected is given as x(t) kc f(t) k=4000 N/m, c=50 Ns/m a) Find the transfer function. b) Find the response to the input in the form of f(t)=40e -4t. c) Find the response to the input in the form of f(t)=40cos(3t-1). d) Find the response to the input in the form of f(t)=100e -2t cos(5t+1). e) Find the response to the input in the form of f(t)=10δ(t) (Impulse input with magnitude 10 Ns) f) Find the response to the input in the form of f(t)=50u(t) (Step input with magnitude 50 N) g) Write the form of the exponential input, for which the system works at resonance. a) b)

3. Examples: c) f(t)=40cos(3t-1)  s=3i d) f(t)=100e -2t cos(5t+1)  s=-2+5i >>s=3i; >>hs=1/(50*s+4000) >>gen=abs(hs) >>fi=angle(hs) >>s=-2+5i; >>hs=1/(50*s+4000) >>gen=abs(hs) >>fi=angle(hs) Re Img Re Img φHφH φHφH

4. Examples: e) f(t)=10δ(t) f) f(t)=50u(t) clc;clear num=[1]; den=[1 80 0]; [r,p,k]=residue(num,den) Residue Theorem

5. Examples: x ss =0.0125 Steady State Value g) Exponential input for resonance; The s value, which makes the denominator of the transfer function zero causes the system resonance. The s value for resonance is s=-80. So, the form of the exponential function for resonance condition is The resonance condition does not depend on the input amplitude A.

6. Examples: Consider the electrical circuit including Op-Amp. Where v 1 is the input and v 2 is the output a) Find the Laplace transform of the output V 2 (s) to a step input with magnitude 12 V. Calculate the steady-state value of the output v 2ss. b) Write the matlab code to calculate the |H(s)| and φ Hs for an exponential-harmonic input in the form of v 1 (t)=24e -0.2t cos(5t-0.2). clc;clear s=-0.2+5i; hs=-2500/(120*s^2+412*s+7500); hg=abs(hs) hfi=angle(hs)

7. Examples: In the calculation procedure of a system response, the result of the Matlab command [r,p,k]=residue(pay,payda) gives the r and p vectors as r=-2+3i, -2-3i, 6, -2 ve p=-4+6i,-4-6i, -3, 0. Find the system response x(t) using the given r and p. Find the steady-state response. x ss =-2 Steady-state value -2 -3 Re Img or