Q Factor of Circuits: Definitions & Applications

[tomz]

On my textbook, there are several definition given for this Q factor.

1) 2 pi * maximum energy stored in reactive element / energy dissipated in a period

2)resonance frequency (in terms of ω) / band width

3)Q = 1/(2 * damping factor)I have tried a couple of random circuits, and its seems not all of them are correct for arbitrary circuit. (Some o them may only true for simple RLC series of parallel).

May I ask which statement is always true?

My textbook also says resonant frequency = natural frequency *(1-2*zeta^2) where zeta is damping factor

and undamped natural frequency is = natural frequency *(1-*zeta^2) where zeta is damping factor

Are these true for RLC series only??

Thank you!

 

[rbj]

first of all, Q essentially has meaning only for 2^nd-order circuits. if it's 4^th or higher order, it will have more than one Q to talk about. now, for a 2^nd-order circuit, you will get a transfer function that will look like:$$\begin{align} H(s) & = \frac{b_0 + b_1 s^{-1} + b_2 s^{-2}}{a_0 + a_1 s^{-1} + a_2 s^{-2}} \\ \\ & = \frac{b_0 s^2 + b_1 s + b_2}{a_0 s^2 + a_1 s + a_2} \\ \\ & = \frac{(b_0/a_2) s^2 + (b_1/a_2) s + b_2/a_2}{(s/\omega_0)^2 + (1/Q) (s/\omega_0) + 1 } \\ \end{align}$$

put your 2^nd-order transfer function in the form shown and then $\omega_0$ is your resonant frequency and the thing that multiplies your $s/\omega_0$ term is 1/Q. that is the definition from the POV of a transfer function.

 

[NascentOxygen]

On my textbook, there are several definition given for this Q factor.

1) 2 pi * maximum energy stored in reactive element / energy dissipated in a period

2)resonance frequency (in terms of ω) / band width

3)Q = 1/(2 * damping factor)


I have tried a couple of random circuits, and its seems not all of them are correct for arbitrary circuit. (Some o them may only true for simple RLC series of parallel).

May I ask which statement is always true?

For me, they all ring a bell.

I'd say they are all correct (for, as rbj explains, a second-order underdamped system).

 

[Bassalisk]

Quality factor is all 3.
Well I am familiar with first 2, but never used(yet) 3rd one.

I can connect physically first 2.

Quality of your RLC series combination is a measure how "good" your circuit filters one specific frequency.

If the circuit is ideal, with no resistance, it will have infinite quality factor meaning it will oscillate. The oscillation will be a sinusoid. Which means you will have a delta dirac function at your resonant frequency f0 given by $f0=\frac{1}{2\pi \sqrt{LC}}$. And the bandwidth would there for be 0.
(from $B=\frac{f0}{Q}\text{ Q ->}\infty \text{ B ->0}$

But if you have some resistance there, your delta will broaden and you will have a non-zero bandwidth.

So yes both of them are correct. Third one is beyond me, never used it.

 

[alphysicist]

I can provide some clarification on the definition and applications of Q factor in circuits. The Q factor, also known as quality factor, is a measure of the efficiency of a resonant circuit. It is defined as the ratio of the energy stored in the circuit to the energy dissipated per cycle.

The first definition given in your textbook is the most general and can be applied to any circuit. It calculates the Q factor by considering the maximum energy stored in the reactive element (inductor or capacitor) and the energy dissipated in a period. This definition is applicable to both series and parallel resonant circuits.

The second definition is specific to series resonant circuits and calculates the Q factor as the ratio of the resonant frequency (in terms of angular frequency, ω) to the bandwidth of the circuit. This definition is not applicable to parallel resonant circuits.

The third definition is also specific to series resonant circuits and calculates the Q factor as the inverse of two times the damping factor. This definition assumes that the damping factor is small, and therefore, is not applicable to circuits with significant damping.

To answer your question, the first definition is always true for any circuit, while the second and third definitions are only applicable to series resonant circuits.

The equations for resonant frequency and undamped natural frequency that you mentioned are also specific to series RLC circuits. They are used to calculate the resonant frequency and undamped natural frequency in terms of the natural frequency and damping factor.

In summary, the Q factor is a useful measure in characterizing the efficiency of resonant circuits. It is important to understand the different definitions and their applications in order to accurately calculate the Q factor for different types of circuits.