EE422 Sample EXAM I

Problem 1 (25 points)

a) Find the transfer function for the following block diagram:

b) Sketch the Bode plot (magnitude and phase) for the following:
i)

ii)

c) Find the transfer function from the following magnitude plot:

Problem 2 (25 points)
a) Find the transfer function, H(s) = Y(s)/W(s), for the following circuit:

b) Find the output for the following inputs:

i) w(t) = 2u(t)
ii) w(t) = 3e-tu(t)
iii) w(t) = 6u(t-2) - e-(t-1)u(t-1)
d) Show that convolution in the time domain is multiplication in the Laplace domain:


c) Find the Laplace transform of the following signal which is periodic after t=0:

 

Problem 3 (25 points)

a) Find the state variable model, including initial conditions and output equation, for the following circuit:

b) For the state variable model given by:

i) Solve for x(t) if w(t) = 2e-tu(t)
ii) Now solve for x(t) if w(t) = e-(t-3)u(t-3) and

Problem 4 (25 points)

a) Use the two-sided Laplace Transform to show that assuming that x(t) is zero for all t < t0.
b) Use the two-sided Laplace Transform to show that
c) Prove the Final Value Theorem:
d) Specify the Region of Convergence for the following :
i)

ii)

iii)
e) For each H(s) in part d), state if we can use the Final Value Theorem, and if so, find x(¥ ).
f) For each H(s) in part d), state if a Fourier Transform exists, and if it does exist, can we use the substitution s=j2p f to obtain it.
g) For each H(s) in part d), determine if the system represented by H(s) is asymptotically stable, marginally stable, or unstable.