RLC Series/Parallel Impedance Near Resonance

[raisin_raisin]

Hey,
The impedance of the series and parallel RLC circuit both tend to R near the resonant frequency (one is a min/ one is a max) so if I expand the frequency around this point I should be able to show they are the same for $$\delta \omega$$ small right? For some reason I can't get it to work. Any pointers?
Thanks

 

[berkeman]

Hey,
The impedance of the series and parallel RLC circuit both tend to R near the resonant frequency (one is a min/ one is a max) so if I expand the frequency around this point I should be able to show they are the same for $$\delta \omega$$ small right? For some reason I can't get it to work. Any pointers?
Thanks

Show us your equations, so we can comment...

 

[raisin_raisin]

Show us your equations, so we can comment...

Thanks for your reply, I did not post them initially as I cannot get very far. Sorry for bad texing I can't figure out why it won't work.
Parallel case: Putting over common demoninator
$$\be<br /> Z^{P}=\frac{1}{i\omega C +\frac{1}{i\omega L} + \frac{1}{R}} = \frac{i\omega LR}{-\omega^{2}RLC + R + i \omega L}<br /> \ee$$
But
$$\\ \\<br /> \be<br /> -\omega^{2}RLC =- (\omega_{0}+ \delta \omega)^{2}RLC \approx - \omega_{0}^{2}RLC- 2\omega_{0}\delta \omega RLC=-R - 2\omega_{0}\delta \omega RLC<br /> \ee$$
Last equality follows since resonance frequency $$\omega_{0}=\frac{1}{\sqrt{LC}}$$
So
$$\be<br /> Z^{P} = \frac{i\omega LR}{-2\omega_{0}\delta \omega RLC + i \omega L} \approx \frac{R}{2i\delta \omega RC + 1} <br /> \ee$$
Want to get series case in a similar form but can't even get close.
$$\be<br /> Z^{S} = \frac{1}{i \omega C} + R + i\omega L = \frac{i \omega RC + 1 -\omega^{2} LC}{i \omega C}<br /> \ee$$