Re: [Phys-l] Q from amplitude width?

[John Denker <jsd@av8n.com>]

On 04/25/2011 12:03 AM, Bernard Cleyet wrote:

> I varied the drive power to maintain a constant pendulum amplitude. 

That is good practice when dealing with high-Q oscillators.
It's more-or-less mandatory, as far as I can see.
Build a PID controller and:
 ++ measure the drive as a function of frequency (at constant amplitude) 
 -- rather than amplitude as a function of frequency (at constant drive).

> The pendula's magnets were immersed in a water in order to reduce all the Qs for convenience.

That shouldn't be necessary, assuming you're using constant
amplitude.

> This was because my function generator has very poor resolution, and,
> with low Q, equilibrium and decay are obtained quickly.

At constant amplitude, the response is very quick anyway.  So you've
got nothing to gain and much to lose by adding water.

Generator resolution has got nothing to do with response time.

>  function generator has very poor resolution

You can fix that in software.  Synthesize the drive (including the PID
controller) in software.  Double-precision gives you a resolution of
one part in 2^53 i.e. one part in 10^16 ... which should be enough.

Calibrate against an external TCXO to obtain high precision over
the short/medium term.  Phase lock that against WWV using NTP to
obtain unlimited precision over the long term.
  http://www.ntp.org/


> The Qs were found by fitting the free decay to an exponential.   I
> wish to compare the response width to my free decay Q values.  How
> does one do this?

a) When fitting to the free decay, don't just fit the envelope
to an exponential.  Fit the entire wavefunction to a sinusoid
times an exponential ... i.e. a complex exponential.

b) When fitting to the resonant lineshape, the same idea applies.
Look at the phase and amplitude, not just the amplitude.  There is
an imaginary part (not just a real part) to the Lorentzian lineshape.
  http://www.ncsu.edu/chemistry/franzen/public_html/CH433/workshop/lineshape/lineshape.html

Note the contrast:
  A) If you just look at the amplitude A, i.e. the real part of the 
   response, it won't tell you whether you are above or below the 
   resonant frequency;  the symmetry is wrong.  Also, at the peak 
   of the resonance, ∂A/∂f is zero, so that is not very informative.  
  Φ) If you look at the phase Φ, it instantly tells you whether you
   are above or below resonance.  Also, ∂Φ/∂f scales like Q, so it 
   is hugely informative.

> My question was how to extract from the data*** of the response the Q
> defined by the narrowness of the response in order to compare w/ the
> exponential free decay definition.

Again: Don't just look at the narrowness of the line.  Look at the
phase also.  The dual approach gives you much more information.

This is called synchronous detection.  Measuring the phase requires
the detector to be synchronized with the drive.  To eliminate possible
sources of error, use a multi-channel detector and use one channel
to "detect" the drive while another channel detects the response.

To detect the TCXO, the drive, and the response requires more than
two channels, i.e. more than you get with an ordinary stereo audio
input.  This means you need to spend a few bucks to get multi-channel 
audio input.  This will almost certainly take the form of an external 
USB pod (rather than an internal PCI card).

To visualize what the oscillator is doing, it often helps to plot 
the Lissajous pattern.  Put the drive on "X" and the response on 
"Y".  If you want details on this, there is a book on the theory 
of sound by some guy called Rayleigh.

=============

The idea of synchronous detection works over a wide range.  It 
is easy for mechanical oscillators, but that's not the end of 
the story.

In the case of optical spectroscopy of atoms and molecules, back 
in the bad old days it was traditional to use non-synchronous 
detection, i.e. to just look at the real part of the response.  
However, if you work hard enough, you can do synchronous detection 
even at optical frequencies.  You need a well-stabilized laser 
(second-order doppler-free).  Then you can walk the laser frequency 
across the atomic resonance and observe the phase and amplitude.

Being able to do this is a prerequisite for laser trapping and
laser cooling.
  http://nobelprize.org/nobel_prizes/physics/laureates/1997/