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I'm trying to derive the first order response of a sensor with step input. However, I'm stuck at step just before conducting the Laplace transform. In the following image, how is the left side equal to the right side? I know the right side equation is necessary to obtain to perform Laplace transform. enter image description here

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  • $\begingroup$ I'm not sure I understand you question: are you asking what the "." should be or why A*K/((\tau*s + 1)*s) = A*K * (1/s - 1/(s-(-1/\tau))? The latter is simple maths, the former requires more information on what the "." is. $\endgroup$
    – am304
    Dec 15, 2017 at 13:55
  • $\begingroup$ Yes I mean how is AK/((\taus + 1)*s) = A*K * (1/s - 1/(s-(-1/\tau))! $\endgroup$
    – Ace
    Dec 15, 2017 at 14:44
  • $\begingroup$ Partial fraction decomposition. $\endgroup$ Dec 15, 2017 at 14:57

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It's simple maths. Starting from the RHS to get to the LHS:

$$ AK \left[ \frac{1}{s} - \frac{1}{s - \frac{-1}{\tau}} \right] = AK \left[ \frac{1}{s} - \frac{1}{s + \frac{1}{\tau}} \right] = AK \left[ \frac{1}{s} - \frac{\tau}{\tau s +1} \right] = AK \left[ \frac{\tau s + 1 - \tau s}{s(\tau s +1)} \right] = \frac{AK}{s(\tau s + 1)} $$

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