Advanced Circuits, Laplace Transform, Find Initial Conditions

[Color_of_Cyan]

Homework Statement

For a circuit, applying Laplace transform, the output is related to input by:

V_o(S)=(N(s) V_i(s) - s^2+s-2)/D(s),

where D(S) = (s^3 + s^2 + 1).

Find ALL initial conditions (there is no need to know N(s) ).

Relevant Equations

Partial Fraction Expansion, Laplace Transforms:
d^n x(t) / dt^n <---> s^n X(S) - s^n - 1x(0-) – s^(n-2)x'(0-) – s^(n-3)]x''(0-) ….
dx(t)/dt <---> sX(s) – x(0-)

d^2x(t)/dt^2 <---> s^2X(s) – sx(0-) – x/(0-)

Vo(S) = [ N(s)V_i(s) + (- s² + s - 2) ] / s³ + s² + 1 ;

can ignore (-s^2 + s - 2).

From relevant equations:

Vo(S) = [N(s)*Vi(s)]/(s^3 + s^2 + 1); -> (d³V_o(t)/dt³) + (d²V_o(t)/dt²) + Vo(t) = N(t)(dvi)/dt

L[vi(t)] = t to s domain: [s³V_o(s) - s²V_o(0-) - SV'_o(0-) - V_o''(0-)]V_o(s) + s² - SV_o - V'_o(0)] V_o(s) + V_o(s) = N(V_i)(s)

= [s³ V(s) - s² + s + 2] V₀(s) + [s² - s - 1]V₀(s) + V₀(s) = N(V_i(s))

I'm sort of lost simplifying this now. I know I'm supposed to set up for partial fraction expansion eventually but not sure how.

 

[rude man]

Vo(S) = [ N(s)V_i(s) + (- s² + s - 2) ] / s³ + s² + 1 ;

can ignore (-s^2 + s - 2).

From relevant equations:

Vo(S) = [N(s)*Vi(s)]/(s^3 + s^2 + 1); -> (d³V_o(t)/dt³) + (d²V_o(t)/dt²) + Vo(t) = N(t)(dvi)/dt

This is a good start but where did you get the right-hand side? It's just L^-1{Vi(s)N(s)} so leave it at that. You will not be concerned with N(s) in any way anyway.

NOTE: You should use v(t) instead of V(t). Don't confuse the two. Also, always use lower-case s fot the Laplace variable, not S.

Also, the initial condition is v(0+), not v(0-) etc.

You have the right idea equating v(0+), v'(0+) etc. with the given terms not functions of N(s). But now I've kind of lost you. Don't use partial fraction expansions.

Equate like powers of s to your given terms & solve for v(0+) and the necessary additional derivatives of v(0+).