Show All Your Work

1 - Find the Laplace transform of the following functions (Use Table):

a) f(t) = 6

b) f(t) = 4t

c) v(t)= 8 sin 4t

d) v(t) = 6 cos 3t

e) i(t) = 7e^-2t cos3t

f) i(t) = 5e^-3t sin 6t

g) f(t)= 8e^-2t

2- Find the inverse Laplace transform of the following functions:

a) F(s) = (16s + 16) / (s³ + 6s² + 8s)

b) F(s)= (10/s) + (15 / s²) + (20 / s+3)

c) F(s) = (8s +30) / (s² + 25)

d) F(s) = (6s + 42) / (s² + 7s +10)

e) F(s)= (10s² + 42s + 24) / (s³ +4s² +3s)

3- The initial current in an RL circuit shown in Figure 1 is 1 A in the direction as indicated.

A) Find the Current in Laplace domain(s domain).(show all your work).

B) Find the Current in time domain and calculate its value at t=5ms

4- The transfer function of a RC circuit, when subjected to an input voltage, is given by the s-domain expression

TF = V_o(s) / V_i(s) = [R₂C_s +1] / [ (R₁+ R₂)C_s+ I]

Determine the output voltage at time t = 1ms when a voltage of 10 V dc is applied. The component values are R₁ = 3.2kΩ, R₂ = 800 Ω and C =1000 μF. (show all your work).

5- Develop an expression Z(t) for motor position when subjected to a unit step input is applied to a servomotor control system. The transfer function for position Z(s) and input voltage E(s) is given as

Z(s) / E(s) = 15 /  (s + 4)(s + 5)

6- Determine the output displacement X(s) if a signal of 10 V dc is applied to it. Also, find the final steady-state displacement of the system by applying the final value theorem.

X(s) / E(s) = 0.1 / (0.1s² + 0.25s + 1)

7- Determine the steady-state value of the following systems when subjected to a unit impulse

a) 1 / (s+10)

b) (s +1) / (s² +3s +5)

c) (5s +1 ) / s (s +2) (s² + 0.5s +1)

8. Find the Laplace transform of the periodic triangular wave shown in Figure 2 by integration. Where

                           Figure 2

L[f (t)] = ₀∫³2te^-2st dt + ₃∫^∞ -1e^-st dt