Assuming an ideal driver or one that that has conjugate/Zobel filters such that the driver’s effective impedance is “Re”. We know the efficiency is cut because of “Rg” (resistance of everything between the amp and the woofer including your amplifier’s output impedance (I.E.:“Zo”), cable impedance, L-pads, & RDC of inductor coils) by:
SPL = 20 × Log[Re/(Rg+Re)]
Speaker “Qes” (I.E.: electrical component of “Qts”) is proportional to Re/BL². But, in reality the “Re” includes the “Rg”. Thereby:
Qes = ß ×( Re + Rg )/ BL²
where: ß = Sqrt[Mms/Cms] ... I provide for purposes of completion. However, since they’re constant thereby irrelevant in this example, we’ll just through it into a single arbitrary constant for clarity.
Now, if we plug “Qes” into “Qts” (I.E.: total speaker Q which is analogous to acoustic “Qh”) which also includes “Qms” (I.E.: mechanical component).
Qts = Qms × Qes /( Qes + Qms ) = Qms /( 1 + Qms/Qes )
Given a generic woofer’s Theil-Small parameters:
BL 10 Qes ½ Re 5W Qms 10
Here’s a classic example of comparative RDCs of 2 different 1.5mH inductor coils.
RDC Qts increase efficiency cut 0.114W 2% -0.2dB 0.75W 13% -1.2dB
This enough os to significantly change the required box size & other sonic characteristics. We know about low frequency behavior, but hat others you ask? If you have read Basic Circuits page, you know different Q’s have drastically different transients.
However, it is more ominous than that. In a vented box the size goes with Qts² so if the Qts is 0.314 for 0.114DCR it will be 0.35 for 0.75DCR in this example squaring the Qts gives us an increase of 25% in box size to keep the same bass response. Also consider what happens when you go from an Zo=0.05W (E.G.:solid state) to Zo=0.6W (E.G.:pentode), or Zo=1.2W (E.G.:triode). You may have to double the box size (& people wonder why tubes sound different).
An audio power amplifier’s damping factor is defined as the ratio of the load impedance to the output impedance of the amplifier. Plus, it assumes a 8W speaker. Thus, a damping factor of 100 translates into Zo=0.08W. Now if we insert our generic 5W speaker, the real damping factor of the amplifier on our speakers is reduced to 62½ & we’re assuming ideal conditions. Add speaker cables of 0.12W & a series inductor of 0.4RDC, we get a Rg=0.6W. That means your amp’s published 100 damping factor is actually 8.333...
Many audio engineers are of the opinion that an amplifier damping factor of 10 or greater is adequate. We OTOH failed with a purchased damping factor of 100. At this point a common misconception is buying an astronomical damping factor. Let’s expose this malady by assuming a perfect amplifier with an infinite damping factor. Wow!!! That ought to fix ... well nothing, because we still have Rg=0.52. You just spent the price of a house on an amp & you got a damping factor of 9.615. You ought to attack the largest component of Rg, not the least. Upgrade the inductor & reduce that RDC. If you manage to reduce the RDC by 50% to 0.2W, it would still be the largest single component of Rg, but the 8.333 damping factor will be raised to 12.5 without ever changing the amp.