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# q factor

## q factor

(OP)
Hello,

I am having some difficulty understanding the meaning of the "Q factor."  One definition is that it is the ratio of the reactance to the resistance.  This perplexes me, because for an RLC circuit, this is zero at resonance!

Let's take the example of a non-ideal inductor (in my application I am actually attaching this to a MEMS pressure sensor to create a wireless pressure sensor).  It is an inductor in series with a resistor, all in parallel with a cap.  What is the "quality factor" of this circuit?  Is this even meaningfull without a "load capacitance" also added in?

Thanks in advance and I apologize for the open-endedness of my question.  Really been banging my head over this one for a while.

### RE: q factor

There are some reasonably good explanations on Wiki.

Look for "Q-factor". Also "Inductor" and so on.

Short answer is you don't compare the resistance to the cancelled-out reactances. The Wiki article provides the formulas for both series and parallel RLC circuits.

But these appear to assume ideal components. For extreme cases, you might have to include more complex models for some or all of the components. In other words, the L and the C will have a finite Q before you even add the R into the circuit.

Not to mention that the R may have some L and C.

But if the R value swamps out the non-ideal aspects of the L and C, then those details might be ignored.

### RE: q factor

http://en.wikipedia.org/wiki/Q_factor#RLC_circuits

Although I don't think it's listed here, one of the definitions for Q involves characterization of the natural response (decaying sinusoid at resonance): (Maximum Energy Stored) / (Energy Dissipated per cycle).   This may be limited to lightly-damped (hi-Q) circuits, I'm not sure

That definition has a resemblance to the ratio you mentioned.  At resonance, Xc = Xl.  I'll bet Q = Xc/R = Xl/R

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### RE: q factor

#### Quote:

I'll bet Q = Xc/R = Xl/R
This would apply with Xc and Xl calculated at the resonant frequency

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### RE: q factor

Q-factor is short for 'quality factor', so basically you can consider it the definition of how good your component is.  A high quality inductor will have low equivalent series resistance (ESR), and a poor quality inductor will have a high ESR.  Same goes for capacitors.  So theoretical inductors and capacitors have infinite Q.

A control circuit with infinite Q would ring forever; lower Q defines the damping factor of how the oscillation decays.

Another definition of Q for a RLC is 'center frequency' / bandwidth.  The higher your Q the narrower the filter, the lower the Q the wider the filter.

Unloaded-Q is your RLC circuit without the load.  Typically you can use a loop of wire to 'sniff' this with a scope to measure this (i.e. radiated coupling).

This is not to be confused with Q from Star Trek!

John D

### RE: q factor

(OP)
Thank you all so much for the responses!  John D.. you mentioned "it will be less than the unloaded Q due to the added load capacitance."  Can you elaborate a bit on this?  The reactance/resistance ratio is at its peak somewhere before the intrinsic, unloaded resonance frequency.  Achieving this freqency would require adding a cap in parallel, no?

Note by "reactance/resistance" above I am referring to the reactance of the entire "equivalent circuit" for the non-ideal inductor, and not just the inductor portion.  Is this an incorrect assumption?

Maybe it would help to step back a bit.  I am trying to optimize a spiral coil inductor which will be loaded with about 4 puffs.  I am running some EM simulations to optimize this inductor.  To compute Q, I am looking at the impedance over frequency (impedance magnitude) and doing the resonant frequency divided by the FWHM.  Should I be optimizing the Q of _unloaded_ spiral coil inductor, or that loaded with my four puffs?  Does it even make a difference?  If one is improved, is the other improved?  Thanks so much for the help!

### RE: q factor

Unloaded Q is your tank (RLC) in parallel with nothing.  Nothing has infinite Q!

As soon as you put a load on with parasitics it has a Q less than infinity, and thus your loaded Q is lower.  Think of your Q-value like resistance; two in parallel have a lower value.  Note that I'm talking about parallel RLC circuits; if you have a series RLC then everything is inverted.  Your unloaded Q is in series with 0+j0 ohms.

You'll have maximum Q at your resonant frequency of your RLC circuit; that is where the power is 100% real and 0% imaginary.  The denominator doesn't go to zero, it goes to the parasitic resistance.

RF Circuit Design by Bowick is a great reference for this.  It has a whole chapter dedicated to Q.

http://www.amazon.com/RF-Circuit-Design-Christopher-Bowick/dp/0750699469

John D

### RE: q factor

Q is a factor on how fast things (Impedance, gain, etc) changes when you deviate from the resonance point.

Hi Q  = fast change
Lo Q  = slow change

Q is not based on resonance frequency, only the rate of change when you deviate away from your resonant frequency.

k

### RE: q factor

(OP)
Thank you again for your replies!  John D... what is the Q of a parallel component? X/R or R/X?  If it's R/X.. then it makes sense that it is "infinite" at resonance...  thanks again!

### RE: q factor

Once again when computing Q, you don't use the total impedance of capacitance and inductance (which is 0), you use either one (see my posts above).

Q is a characteristic of the circuit and does not depend upon the frequency.   One definition of Q is in terms of X which is itself a function of frequency, but that definition of Q relies on X computed at the resonant frequency.

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