Express the voltage pulse vt in figure mathematically using

Question

Question 1:

A voltage pulse is shown in Figure 1.

Figure 1. A Voltage Pulse

(a) Express the voltage pulse, v(t), in Figure 1 mathematically using unit step functions.

Find the Laplace transform, V(s), of the voltage pulse in Figure 1

Question 2:

A signal function, g(t): has the Laplace transform G(s). where

G(s) = 8s³ + 40s² + 56s + 12/s.(s +3)(s + 2)²

(a) Use G(s). directly to End the intial value of g(t). ie. g(0).

(b) use G(s) directly to find the final value of g(t). i.e. g(∞).

(c) G(s) can be expressed in partial fraction form as:

G(s) = (8s³ + 40s² + 56s +12)/(s (s +3)(s +2)²) =  A/s + B/ (s + 3) + C /(s + 2)² + D/(s + 2)

where A. B. C and D are constants (and, is this particular case, integers).

(1) Use the residue method to find A.

Use the residue method to find B.

(iii) Use the residue method to find C.

(iv) Use the residue method to find D.

(d) G(s) can also be expressed as:

G(s) = ((A + B+ D)s³ + (7A + 4B + C + 5D)s² + (16A + 4B + 3C+6D)s +12A)/(s(s +3)(s +2)²)
There A B. C and D are the same integer co:nstants as in part (c).

Use the expression above to validate your answers to pad (c) of this question.

(e) Use the partial fraction expansion of G(s) to find an expression for go.

(1) Use the expression for go in part (e) to find the initial value 0}. Compare your answer to that determined in part (a).

Use the expression for g(t) in part (e) to find the Eno] value g(x). Compare your answer to that determined in part (b).

Hint: You should make use of l'Hopitals rule.