An LTI system is defined by the equation d*y(t) +4 ¢y() + 4y(t) = dx(e) + 0.5x(t) dt dt dt (a) Find the characteristic equation, characteristic roots and characteristic modes of this system (b) Comment on the stability of the system. (c) Find y.(t), the zero-input component of the response fort 2 0, if the initial conditions are y. (0-) = 3 and y,(0 -) = -4
An LTI system is defined by the equation d*y(t) +4 ¢y() + 4y(t) = dx(e) + 0.5x(t) dt dt dt (a) Find the characteristic equation, characteristic roots and characteristic modes of this system (b) Comment on the stability of the system. (c) Find y.(t), the zero-input component of the response fort 2 0, if the initial conditions are y. (0-) = 3 and y,(0 -) = -4
Introductory Circuit Analysis (13th Edition)
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Question
![An LTI system is defined by the equation
d?y(t)
+ 4 dy(t) + 4y(t) = dx(t) + 0.5x(t)
dt
dt
(a) Find the characteristic equation, characteristic roots and characteristic modes of this
system
(b) Comment on the stability of the system.
(c) Find y.(t), the zero-input component of the response fort 2 0, if the initial conditions
are y. (0-) = 3 and yo(0 –) = -4
(d) Mathematically derive an expression for h(t), the impulse response of the system.
(e) Using the convolution tables and the result of part (d), find the zero-state output for the
system given the input x(t) = e-stu(t)
(f) What is the total system response based on the conditions and input specified above?
(g) What is the natural response and what is the forced response of this system?
(h) Using Matlab, plot the impulse and step responses and frequency response (magnitude
and phase) of this system directly from the differential equation.
Hint: Sample code given in Module 6 may be used, but frequency range will have to
be increased.
(1) Compare the impulse response plot from part (h) to the expression from part (d).
Comment on the overall shape and specific points on the plot that confirm that they are
equivalent.
G) Using the frequency response plots from part (h), determine the output of the system
if x(t) = 8+ 10 cos(t – 20°) + 6 cos(30t + 30°)
Hint: You can use the data cursor
feature to read the coordinate values off the
plots.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbbaa86c9-34e2-46ec-92ea-d49a93028650%2F1cf680d6-9625-4935-a290-8985abb04192%2Ffgk986l_processed.png&w=3840&q=75)
Transcribed Image Text:An LTI system is defined by the equation
d?y(t)
+ 4 dy(t) + 4y(t) = dx(t) + 0.5x(t)
dt
dt
(a) Find the characteristic equation, characteristic roots and characteristic modes of this
system
(b) Comment on the stability of the system.
(c) Find y.(t), the zero-input component of the response fort 2 0, if the initial conditions
are y. (0-) = 3 and yo(0 –) = -4
(d) Mathematically derive an expression for h(t), the impulse response of the system.
(e) Using the convolution tables and the result of part (d), find the zero-state output for the
system given the input x(t) = e-stu(t)
(f) What is the total system response based on the conditions and input specified above?
(g) What is the natural response and what is the forced response of this system?
(h) Using Matlab, plot the impulse and step responses and frequency response (magnitude
and phase) of this system directly from the differential equation.
Hint: Sample code given in Module 6 may be used, but frequency range will have to
be increased.
(1) Compare the impulse response plot from part (h) to the expression from part (d).
Comment on the overall shape and specific points on the plot that confirm that they are
equivalent.
G) Using the frequency response plots from part (h), determine the output of the system
if x(t) = 8+ 10 cos(t – 20°) + 6 cos(30t + 30°)
Hint: You can use the data cursor
feature to read the coordinate values off the
plots.
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