Ok, here's the thread. So what makes you think I have my fundamentals mixed up a bit?
se
One example is the comment that "damping" is meaningless outside of resonance. ironically, you said that immediately after your link to wikipedia on damping.
I think its important to flesh out the details and get the whole story of "damping factor" organized into a coherent storyline.
Any speaker design is an electromechanical system. Let's first start by looking at the mechanical system before introducing the electro- bit....
I've argued that the zeroth order model for a speaker is a mass-spring-damper system. Fortunately, this system is extremely well studied and understood.
The classic 1 degree of freedom dynamical system is the mass-spring-damper system. If the position of the oscillator is
x and letting ' denote the derivative with respect to time, then the system can be written as
M*x'' + C*x' + K*x = F. [1]
M is the mass,
C is the the viscous damping, and
K is the spring stiffness. The system may be subjected to an external forcing
F.
The point here is that the dynamics of the system are influenced by the damping,
C, whether the system is subject to a sinusoidal excitation, an impulse, a step response, etc. The damping in the system affects the response whether resonance is of interest or not.
Of interest in the above single DOF system are the quantity sqrt(
K /
M ), which is called the natural frequency and is usually denoted by omega, and the quantity
C / (2 * sqrt(
M * K ) ), which is called the damping ratio and is usually denoted by zeta. These will look familiar to anybody who's peaked at the wiki article on damping to which SE linked.
The natural frequency is important because it gives the frequency that the system would oscillate at in the absence of damping.
The damping ratio is important because it tells us strong the damping in the system is. When the damping ratio is < 1, then the system will be oscillate about equilibrium with decaying amplitude in time. When the damping ratio is > 1, then the damping is strong such that the system will slowly decay toward equilibrium without ever over-shooting. Finally, there is the critical case where the damping ratio is exactly 1. In this case, the perturbed system decays toward equilibrium as fast as possible without ever overshooting.
Now, let's add in the forcing, which is where the "electro" in "electromechanical" comes into play... Here, the field equations required to analyze a the electromotive forces on a speaker are a bit more complicated that the single degree of freedom oscillator; however, a good starting point for
electromagnetic induction can be found on wikipedia. basically, the forcing term,
F, will be proportional to the current in the headphone coil/etched traces (in a dynamic/orthodynamic headphone) times the strength of the magnetic field. Namely, one integrates
i x
B over the contour of the circuit to find
F, where
i is the current and
B is the magnetic field.
When you plug a speaker into an amplifier, the speaker is (to some approximation) a mass-spring-damper system and the amplifier's voltage drives current through the speaker coils/etched traces that generates a force that excites the system. This would be the force
F. The force on the transducer is due to the magnetic field through with the speaker coils/etched traces run through. Likewise, as the coils/etched traces move through this magnetic field, current is induced in the coils/traces which forms an magnetic field that tends to resist further motion of the coils/traces (given that the coils/traces experience little resistance). That force is proportional to the rate which the speaker travels through the magnetic field and opposes the motion (
Lenz's law)---it is essentially a viscous damping force!
This electrical damping force could be moved from F and included on the left hand side of the dynamics equation [1], augmenting the physical damping. Hence, the electrical damping is an effect which effects the mechanics of the transducer.
A neat visual for the mechanical forces generated by the relative motion of magnetic field lines and closed circuits can be seen in any of the demos where magnets are dropped through electrically conducting, yet nonmagnetic tubes. The eddy currents generated by the moving magnetic field lines relative to the conductor generate their own magnetic field which opposes the motion. Because the tubes are very low resistance conductors forming closed circuits, large induced currents are allowed to flow, generating significant mechanical resistance to motion. In the absolute absence of resistive losses
(i.e., superconduction), the magnets may even be completely suspended (levitated).
Moving the opposite direction of superconduction.... if you increase the resistance in the circuit through which the induced current flows, then the increased resistive losses reduces eddy current and consequently reduces the counterelectromotive force.
One nice visualization of this is to compare the speed of the magnets falling through a solid copper tube vs the magnet falling through a copper tube with slots cut into the side (thus decreasing the conductivity of the eddy current circuit.) This is what is happening with the "damping factor" in amplifier outputs. The "eddy current circuit" in headphones consists of the circuit from the amplifier, through the cable, through the headphone coil/traces, and back through the cable and into the amp. The amplifier output impedance is the resistance contributed by the amp to this circuit. The larger the output impedance, the less induced current can flow, and the lower the counterelectromotive force which damps the speaker.
This is what the damping factor is here to tell us---how the output impedance of the amp compares to the rest of the headphone circuit. The higher the damping factor, the less the resistance in the amplifier output reduces the effectiveness of electrical damping in the speaker. It doesn't matter whether the speaker design is classic or orthodynamic. If the speaker design is such that it is driven by current flowing through wires/traces in a magnetic field, then the motion of that speaker through the magnetic field will induce a back emf which can drive a (damping) current if the circuit impedance allows it.
So, that's my overview of speakers and damping from first principals. I can try to elaborate on specific points. I'm not arguing whether electrical damping is or isn't critical in the performance of headphones/speakers/orthos/etc. I just want to bring transparency to the physical processes at play and hopefully learn more about when, where, and why they make an impact on sound (when they do) and when, where, and why they don't matter (when they don't).
Cheers