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(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?