-And their consequences on system Q
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When it comes to designing woofer enclosures, the necessity to increase the volume of enclosures
due to the intrusions of drivers, vents, and bracing are well known, and generally compensated for. A
less considered concern may be the effects of additional electrical impedance to the system. This
increased impedance is due to several factors, including the source and cable impedance, impedance
of the crossover, and thermal effects, and is the focus of this article.
The Math: Any change in series impedance affects the Qes of the driver, and consequently Qts, which
of course is one of the factors used to calculate the optimum enclosure size. The Qes + additional
impedance, Qes' is calculated using the formula:
Qes' = [(Rg + Re) / Re] * Qes
Re is the DC resistance of the driver
Rg is the additional impedance
Qes is the Electrical Q of the driver
Qts' the new total Q of the driver can then be found by:
Qts' = (Qes' * Qms) / (Qes' + Qms)
Where Qms is the Mechanical Q of the driver. For the purposes of this article, the additional
impedance will be expressed as scalar quantities rather than their complex impedance.
The output impedance of the source amplifier is the first factor affecting Qe. For a tube amplifier,
this can be as much as several ohms. Solid state amps on the other hand will are typically a small
fraction of an ohm. A source impedance of 0.1 ohms will be assumed for the purposes of this
discussion, and was included in all of the following plots.
Speaker cable and interconnect impedance can also be of some minor concern. The resistance
of the cable itself may range from 0.016 to 0.1 ohm typically for a 10 foot length of 12 to 20 gauge
wire respectively. While this is pretty insignificant, the connections at the terminations, speaker,
crossover and driver will all add some small resistance. For this exercise, 0.1 ohm will be assumed as
a nominal value.
Passive crossover insertion losses: The DCR of the inductors can significantly increase the
series impedance. This of course will vary depending on the gauge, length of wire in the inductor, and
core material. I'll suggest 0.4 ohms as an average value for a woofer in a 3 way system.
The increase in voice coil resistance with temperature. I have to thank Keith Howard for his
article in the November 2006 issue of Stereophile as the stimulus to write this treatise. While he tested
only one speaker, I suspect his results are reasonably representative of a well-designed driver. I
found several things interesting in his study. One is that the bulk of the increase in voice coil
temperature occurred rather quickly, within 30 seconds or so. Another was the tweeter was relatively
immune to thermal effects. Most significant was with the woofer tested, the voice coil only increased 36
degrees, which resulted in an 8% increase in Re. The author dismissed this small increase in Re as
insignificant, but I suggest that if it is not considered, along with the other factors I've noted previously,
it can make a discernible difference in the optimum calculated enclosure volume.