Re: [Phys-l] Q, mass (or L), and the driven response of an oscillator

[John Denker <jsd@av8n.com>]

On 03/23/2012 01:56 PM, Bernard Cleyet wrote:
> ... the response is greater and narrower w/ increasing Q.  I have no
> problem w/ this if the Q increase is a result of decreased damping,
> but it's not intuitive why.

Executive summary: The reactive parts of the impedance add to
zero at the natural frequency.

Do the electrical version.  Use phasors and complex impedances.

1) Draw the circuit diagram for a series RLC circuit.  We
 drive it with constant voltage and observe the current, so 
 the "response" i.e. the output-vs-input relationship is I/V.

2) Convince yourself that the complex impedance of the inductor
 Z_L is always 180 degrees out of phase with the complex impedance
 of the capacitor Z_C

3) Define the natural frequency (ω_0) of the system to be the 
 point where Z_L and Z_C are exactly equal and opposite.  They 
 add to zero.

4) Use Ohm's law to show that at ω_0, the impedance of the circuit 
 as a whole (Z) is equal to R.  The reactive components drop out.

5) Write down the full expression for Z for all frequencies, as 
 a function of frequency.

6) Convert Z from phasor representation to (amplitude, phase) 
 representation.  Show that |Z| is at a minimum exactly at ω_0. 
 This corresponds to the maximum of I/V, which is necessarily 1/R.

7) Convince yourself that changing L and/or C (at constant R) 
 changes ω_0, but leaves the maximum response the same, namely 1/R.

It doesn't get any more intuitive than that.  The reactive parts
of the impedance add to zero at ω_0.  (For a /parallel/ RLC circuit, 
the reactive parts add to zero conductance i.e. infinite impedance 
at ω_0.)

Note that I call ω_0 the /natural/ frequency of the system rather 
than the "resonant" frequency of the "oscillator", because all of 
the above remains true even when the system is grossly overdamped, 
so that there is no resonance and no oscillation.

Note that for any finite Q, you can drive the system off-resonance
just fine, so there is no such thing as "the" frequency or "the"
period of the system.