# Rewrite (Laplace) transfer function

For a sinusoidal oscillator I've constructed the following transfer function: $$H(s)=\frac{sC_2R_1}{s^2C_2C_1R_2R_1 + s(C_2R_2+C_1R_1)+1-sC_2R_1\frac{R_f}{R_b}}$$ I expect I should be able to rewrite this to a form like: $$\frac{s+a}{(s+a)^2+b^2}$$ as that would convert to: $f(t)=e^{-at}cos(bt)$, which is the time domain behavior of the oscillator.

Then $b$ will probably contain the R's and C's as they determine the frequency and $a$ will probably contain the factor $\frac{R_f}{R_b}$ as they determine the gain and when their ratio is not exactly 2 the oscillator's amplitude changes exponentially.

Unfortunately I've no clue how to do this rewriting. Are there general steps one can take?

• @Jean-ClaudeArbaut It took me a while to realize that I've to add a $e^{-at}\frac{a}{b}sin(bt)$ term in the time domain to get the $+a$ in the numerator, but then I get a solution using your hint. If you put your hint in an answer, I can flag it. Many thanks. – Remco Poelstra Aug 17 '18 at 9:25

$$as^2+bs+c=a\left[\left(s+\frac{b}{2a}\right)^2+\frac{4ac-b^2}{4a^2}\right]$$