2.003 Fall 2002 Final - Sample problems Problem 1 Match each of the following pole-zero diagrams to the corresponding Bode plot and step response from the following two pages. For example is you think that pole-zero diagram (1) corresponds to step response (q) and Bode plot (r), write ”1,q,r” on your exam paper. 5 4 3 2 1 0 −1 −2 −3 −4 −5 −5 −4 −3 −2 −1 0 1 2 3 4 5 5 5 44 33 2 2 11 00 −1−1 −2−2 −3−3 −4−4 −5 −5 −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 5 5 4 4 3 3 2 2 1 1 0 0 −1 −1 −2 −2 −3 −3 −4 −4 −5 −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 5 5 4 4 3 3 2 2 1 1 0 0 −1 −1 −2 −2 −3 −3 −4 −4 −5 −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −5 −4 −3 −2 −1 0 1 2 3 4 5 5 4 3 2 1 0 −1 −2 −3 −4 −5 −5 −4 −3 −2 −1 0 1 2 3 4 5 12.003 Fall 2002 Final - Sample problems Step Response Step Response 0.252.5 0.2 2 0.15 1.5 0.1 4 3.5 3 Amplitude Amplitude Amplitude Amplitude Amplitude 1 0 1 2 3 4 5 6 7 8 9 10 Amplitude Amplitude Amplitude Time (sec) 0.05 0 −0.05 Step Response 1 −0.1 0 1 2 3 4 5 6 7 8 9 10 Time (sec) 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0 1 2 3 4 5 6 7 8 9 10 Time (sec) Step Response Step Response 0.04 0.8 0.03 0.6 0.02 0.4 0.01 0.2 0 0 Time (sec) Time (sec) 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0.1 0.09 0.08 0.07 0.06 2 1.8 1.6 1.4 1.2 0.05 1 Step Response 10 8 6 4 2 0 −2 −4 Step Response 0.25−6 0 1 2 3 4 5 6 7 8 9 10 Time (sec) 0.2 0.15 0.1 0.05 Step Response 1 0 −0.05 0.9 −0.1 0.8 −0.15 −0.2 0.7 −0.25 0 1 2 3 4 5 6 7 8 9 10 Time (sec)0.6 0.5 0.4 0 1 2 3 4 5 6 7 8 9 10 Time (sec) 22.003 Fall 2002 Final - Sample problems Bode Diagram 0 −5 Bode Diagram 10−10 −15 −20 5 0 −1 0 1 2 10 10 10 10 Frequency (rad/sec) Bode Diagram 0 −10 Bode Diagram −20 12 −30 10 −40 Phase (deg) Magnitude (dB) Phase (deg) Magnitude (dB) Phase (deg) Magnitude (dB) Phase (deg) Magnitude (dB) Phase (deg) Magnitude (dB) Phase (deg) Magnitude (dB)−25 0−30 −35 −5−40 90 −10 9045 0 45−45 −90 0 1 2 10 10 10 Frequency (rad/sec) 8−50 −60 6 −70 4 −80 2 0 0 −45 0 −90 −10 −135 −20 −180 0 1 2 −30 10 10 10 Frequency (rad/sec) −40 −1 0 110 10 10Frequency (rad/sec) Bode Diagram 0 −10 Bode Diagram 0 −20 −30 −1 −40 −2 −50 −3 −60 −4 −70 −5 −80 −6 −7 180 −8 135 20 90 15 45 10 0 0 1 2 10 10 10 5 Frequency (rad/sec) 0 −2 −1 0 110 10 10 10Frequency (rad/sec) Bode Diagram Phase (deg) Magnitude (dB) −50 0 50 100 150 −90 −45 0 45 90 Phase (deg) Magnitude (dB) 0 1 10 10 Frequency (rad/sec) Bode Diagram 10 5 0 −5 −10 −15 −20 −25 −30 0 −45 −90 −1 0 110 10 10Frequency (rad/sec) 32.003 Fall 2002 Final - Sample problems Problem 2 x Mkv2 v1b2 v(t)=1-u(t) b1 The figure above shows a system which consist of a mass (M), spring (k), and two dampers (b1, b2). The spring is attached at one end to the mass and to the damper (b2), which is massless at the other. The mass is pulled by a velocity source (v(t)). x denotes the extension of the spring. v1 is the velocity of the mass and v2 is the velocity of the second damper. M = 1 kg, b1 = 1 Ns/m, b2 = 9 Ns/m, k = 9 N/m. a) find v1(0−), v2(0−), and x(0−) b) find x(0+) and ˙x(0+) c) write the differential equation for the system in terms of x. d) find x(t) given the initial conditions in c. 42.003 Fall 2002 Final - Sample problems Problem 3 R2 C + -R1 � o Vi Va(s) a) If a(s) = 106, find the transfer function of Op-Amp circuit above. b) If this circuit were used as a controller, what type of controller would it be? c) If a(s) = 106 what is the transfer function of the circuit? s d) C = 1 µF , R1 = 1e4 �, R2 = 5e4 �. Plot M (�) and �(�)