Is the resonant frequency as defined for a second order transfer function same as the frequency at resonance in a second order electrical circuit?

When I studied second order system and thus natural, damped and resonant frequency I wondered if the resonant frequency I find using the formula $$\ w _ { r } = w _ { n } \sqrt { 1 - 2 D ^ { 2 } } \$$ where D is damping factor, can be applied in any RLC circuit to find the resonant frequency. So I considered a series RLC circuit. Then its impedance is $$\ z = R + s L + \frac { 1 } { C s } \$$, and I used its reciprocal to form a second order transfer function. By general definition of electrical resonance if I put s = jw in the expression of z and put imaginary part = 0, I get resonant frequency as $$\ w = \frac { 1 } { \sqrt { L C } } \$$, which is, disappointingly, not equal to the resonant frequency derived from considering 1/Z as a second order transfer function, which is, $$\ w = \sqrt { \frac { 1 } { L C } - \frac { R ^ { 2 } } { 2 L ^ { 2 } } } \$$ and I don't understand why. EDIT - I am including derivation for the above formula with relevant diagram:

simulate this circuit – Schematic created using CircuitLab

Then, the transfer function = Output / Input = $$\ \frac { 1 } { z } = \frac { 1 } { R + s L + \frac { 1 } { C s } } = \frac { C S } { L C S ^ { 2 } + R C S + 1 } \$$ Now using the standard form of second order transfer function, natural frequency can be derived as $$\w _ { n } = \frac { 1 } { \sqrt { L C } }\$$ and then, resonant frequency can be given as $$\w _ { r } = w _ { n } \sqrt { 1 - 2 D ^ { 2 } } \$$, thus $$\w _ { r } = \frac { 1 } { \sqrt { L C } } \sqrt { 1 - 2 ( \frac { R } { 2 L } ) ^ { 2 } }\$$.

• Just add \ before \$ ... May 17 at 8:31

I don't understand why

You appear to be confusing two things; the natural undamped resonant frequency of an RLC circuit and, the damped resonant frequency of the transfer function. There are several notable points in the spectrum of an RLC low-pass filter (for example): -

• $$\\omega_n\$$ is the natural undamped resonant frequency
• The damped resonant frequency is $$\\omega_n\sqrt{1-\zeta^2}\$$
• The frequency at which the response amplitude peaks is this: $$\\omega_n\sqrt{1-2\zeta^2}\$$

Without seeing your derivation or your exact circuit and where you place the input and output nodes, it's a bit of a guess but, I suspect, that your more complicated formula is the latter (above).

Regarding the pole-zero diagram you have this: -

Images from my basic website.

• Images from your own website {claps}. May 17 at 12:51
• Thank for your reply. Should the damped resonant frequency not be defined as the frequency at which the peak of the magnitude occurs, because I have always understood resonance as a phenomenon at which a system reaches its peak? Also, I actually wanted to use the transfer function for deriving the expression for resonant frequency in any series or parallel combination of RLC circuit (i,e RL-C, RC-L etc.), in which cases I got different results on putting imaginary part of impedance 0, and using second order transfer function approach. P.S. - I have edited my question for further clarification. May 17 at 17:40
• I think what I have found since using this site is that different people call them different things. Some use the term pole frequency when I call it natural undamped frequency and some call the pole frequency the frequency where the pole projects at 90 degrees to the jw axis. I've given up on arguing about names. May 17 at 18:56