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# (Solved): (4 marks) Question 4. Consider the circuit shown below, where i(0)=i_(0) and v_(c)(0)=v_(0). a) (4 ...

(4 marks) Question 4. Consider the circuit shown below, where

i(0)=i_(0)

and

v_(c)(0)=v_(0)

. a) (4 marks ) Write the KVL equation for the system. Hint: recall that

v_(c)=(1)/(C)\int_0^t id\tau +v_(c)(0)

. b) (4 marks ) Define the initial conditions problem. c) (4 marks) Write the problem in Laplace domain. Hint: if you take the Laplace transform of a constant,

K

, the result is the same as the Laplace transform of

Kh(t).L{K}=(K)/(s)

. d) (4 marks) Solve for

I(s)

. Carefully simplify your solution to the form

I(s)=U(s)(N_(1)(s))/(P_(1)(s))+

(N_(2)(i_(0),v_(0),s))/(P_(2)(s))

, where

N_(1),N_(2),P_(1),P_(2)

are polynomial functions of

s,N_(2)

may also depend on

i_(0)

and

v_(0)

, and

U(s)

is the Laplace Transformation of

u(t)

. e) (2 marks) What is the relationship between

P_(1)

and

P_(2)

? f) (20 marks) Use Simulink or any other simulation software you are familiar with to construct a simulation model of the system above. Consider the following cases: \table[[Parameter,Case 1,Case 2,Case 3,Case 4],[L,1,1,1,1],[C,1,1,1,1],[R,4,4,1,1],[i_(0),0,5,0,5],[v_(0),0,0,0,0],[u,h(t),0,h(t),0]] For each case answer the following: i. What are the roots of the polynomial

P_(1)(s)

for

sinC

, where

C

is the set of complex numbers? ii. Find an expression for

i(t)

that is valid for

t>0

. iii. Plot your solution and the output of the simulations. Hint: They should be exactly the same. Consider using dashed lines to make your result visible. g) (2 marks) What happens when the roots of

P_(1)(s)

have non-zero imaginary part?

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