RLC Series/Parallel Impedance Near Resonance

  • #1

Main Question or Discussion Point

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 [tex]\delta \omega[/tex] small right? For some reason I can't get it to work. Any pointers?
Thanks
 

Answers and Replies

  • #2
berkeman
Mentor
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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 [tex]\delta \omega[/tex] small right? For some reason I can't get it to work. Any pointers?
Thanks
Show us your equations, so we can comment...
 
  • #3
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
[tex]
\be
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}
\ee
[/tex]
But
[tex]
\\ \\
\be
-\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
\ee
[/tex]
Last equality follows since resonance frequency [tex] $ \omega_{0}=\frac{1}{\sqrt{LC}} $ [/tex]
So
[tex]
\be
Z^{P} = \frac{i\omega LR}{-2\omega_{0}\delta \omega RLC + i \omega L} \approx \frac{R}{2i\delta \omega RC + 1}
\ee
[/tex]
Want to get series case in a similar form but can't even get close.
[tex]
\be
Z^{S} = \frac{1}{i \omega C} + R + i\omega L = \frac{i \omega RC + 1 -\omega^{2} LC}{i \omega C}
\ee
[/tex]
 

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