Analize the circuit response to applied voltage asociated with the function :(s + 1)/(s + 2)(s^2+1)(s^2+4)

1 Answer | Add Yours

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

According to the Laplace transform, the circuit responses are associated with rational functions.

We'll decompose the given rational function in the elementary quotients, which are arising from standard circuits responses.

(s + 1)/(s + 2)(s^2+1)(s^2+4) = A/(s+2) + (Bs + C)/(s^2 + 1) + (Ds + 2E)/(s^2 + 4)

We'll calculate A,B,C,D,E.

We'll determine LCD for the ratios from the right side:

s+1 = A(s^2+1)(s^2+4) + (Bs + C)(s+2)(s^2 + 4) + (Ds + 2E)(s+2)(s^2 + 1)

We'll remove the brackets:

s+1 = As^4 + 5As^2 + 4A + (Bs^2 + 2Bs + Cs + 2C)(s^2 + 4) + (Ds^2 + 2Ds + 2Es + 4E)(s^2 + 1)

s+1 = As^4 + 5As^2 + 4A + Bs^4 + 4Bs^2 + 2Bs^3 + 8Bs + Cs^3 + 4Cs + 2Cs^2 + 8C + Ds^4 + Ds^2 + 2Ds^3 + 2Ds + 2Es^3 + 2Es + 4Es^2 + 4E

We'll combine like terms:

s+1 = s^4(A + B + D) + s^3(2B + C + 2D + 2E) + s^2(5A + 4B + 2C + D + 4E) + s(8B + 4C + 2D + 2E) + 4A + 8C + 4E

The correspondent coefficients must be equal:

A + B + D = 0

2B + C + 2D + 2E = 0

5A + 4B + 2C + D + 4E = 0

8B + 4C + 2D + 2E = 1

4A + 8C + 4E = 1

The response is:

Ae^-2t + Bcos t + Csin t + Dcos 2t + Esin 2t

We’ve answered 318,911 questions. We can answer yours, too.

Ask a question