Resonance -- why do we calculate the frequencies w1 & w2 at half power frequencies?

1. Mar 10, 2016

Hassan Raafat

at electric circuits , at RLC circuits , in case resonance :why do we calculate the frequencies w1 & w2 at half power frequencies ?

2. Mar 10, 2016

Twigg

The linewidth at half power (full-width half-max, FWHM) is a worthwhile number because it's directly related to your damping ratio. The FWHM is just the difference between w1 and w2: the full linewidth at half maximum power. Think back to your solution of the RLC differential equation. At one point you had to have gotten a quadratic formula for the decay rate/frequency of the circuit: $s = (-R/2L) \pm \sqrt{ (R/2L)^{2} - \omega_{0} ^{2}}$, whose real part gives you a decay rate and whose imaginary part gives you a frequency. You can see from there why your decay rate $\gamma = -R/2L$ is equal to half the FWHM, $\Gamma /2$ (aka the half-width half-max HWHM, equal to $\gamma$). In the high quality limit, the quantity $\gamma = R/2L$ tells you both how fast energy is being dissipated AND how much the damped resonance frequency is shifted from $\omega_{0}$.

3. Mar 10, 2016

Hassan Raafat

thanks Twigg , but I can't understand that ⇒
Γ/2 (aka the half-width half-max HWHM, equal to γ). In the high quality limit

4. Mar 10, 2016

Twigg

My bad. The big gamma $\Gamma$ is the full-width half-max. The little gamma $\gamma$ is half the big gamma $\Gamma/2$ and is equal to BOTH the half-width half-max AND the damping ratio R/2L.

"in the high quality limit" meaning for underdamped systems with high Q factor

5. Mar 10, 2016

Hassan Raafat

so If I got it ... you mean that big gamma equals band width ?
and little gamma equals R/2L = Γ/2 ?
Doctor tells us that if Q ≥ 10 , we can use the approximation .. you mean that ?

6. Mar 10, 2016

Twigg

Yes, that's correct. Apologies for any confusion.

7. Mar 11, 2016

Hassan Raafat

OK , Twigg , no need to apologise
thanks for your time . :)

8. Mar 11, 2016

LvW

Hassan, have you ever looked at the phase shift at both frequencies?
For very low and very high frequencies the phase shift approaches +90 and -90 deg, respectively (and 0 deg at resonance).
And at the mentioned "corner frequencies" (w1, w2) the phase shift is +45 and -45 deg., respectively.
Another nice reason to use these two characteristic frequencies for defining the bandwidth.

Are you interested in another reason?
If we define the bandwidth BW based on these two 3dB frequencies f1=w1/2π and f2=w2/2π the filter quality factor Q is defined as
Q=fo/BW (both in Hz).
And this definition gives a Q factor which is identical to the "pole Q" which is defined based on the pole location in the complex s-plane.

But don`t forget: It is a DEFINITION only.
For some specific applications we are free to use another bandwidth definition.

Last edited: Mar 11, 2016
9. Mar 13, 2016

Hassan Raafat

Thanks a lot LvW , you have declared it clearly , now I understand it really , but I have a small question .. Please can you give me an example to those applications where we use another difinitions for bandwidth and what is this definition ?

10. Mar 13, 2016

LvW

Hassan - in principle, you are free to use your own definition.
For example, you can require that the magnitude of the transfer function does not deviate from the value at the center frequency by more than 0.1 db or any other value.
Here is an example: For a higher-order bandpass (n=4,6,8..) it is common practice to start with a corresponding lowpass design and using the lowpass-bandpass transformation for finding the bandpass function and the corresponding parts values.
If you start, for example, with a Chebyshev lowpass function you will arrive at a bandpass with two "corner frequencies" other than "half-power" values.
This is because the passband of Chebyshev low pass functions is specified according to the allowed passband ripple which may be less than 3dB (0.1dB or 0.5dB or 1dB). Looking at Chebyshev design tables you will see that the given values are grouped for different ripple values within the pass band.
As a result, the "Chebyshev-bandpass" also has a ripple within the passband - and the advantage of higher out-of-band damping if compared with Butterworth design of the same order.

By the way: Here is another relation between the bandwidth B of a second-order bandpass and the phase response φ(ω):
It is possible to show that the slope d(φ)/dω of the phase function at the center frequency fo follows the following relation: 2/B=d(φ)/dω .
This relationship is valid only in case the bandwidth B is defined as "half-power bandwidth".