Once the equation is boiled down, the final two steps to do is make the numerator a polynomial (no s's with negative exponents--including #/s). And second, have the denominator's highest s-term have no constant out in front of it.
After that when finding the roots (zeros or poles, which all mean the same thing), and you're left with a complex form of something like s = α + jβ, take the (+) & (-) expression. The other expression isn't needed; in the end it equals 2||A1||cos(βt + θ)e^(-αt). And if you're lucky, you can sometimes check that time constant with regular AC analysis methods.
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