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If $Q=\frac{\omega L}{R}$ , why does the term $-\frac{j}{\omega C}$ become $-\frac{j}{\omega_r}{\omega}$?

The expression $H(j\omega) =
\frac{1}{1+jQ(\omega/\omega_r)[1-(\omega_r/\omega)^2]}$ results from several algebraic steps following $H(j\omega) = \frac{R}{R+j(\omega L
- 1/\omega C)}$. First, pull $R$ out from numerator and denominator. For $Q\sim \frac{\omega_r L}{R}$, we can write $\frac{wL}{R} = \frac{Q\omega}{\omega_r}$. Substitute in and do a bit of manipulation to get the desired form.



Kate Scholberg 2017-02-06