Keep It Short
The good news for most live sound applications is that we don’t have to tolerate extremely long wire runs. By locating power amplifiers near the loudspeakers we can keep wire runs reasonably short. At these shorter distances we can easily afford heavier gauge wire.
While power losses are now manageable, it is worthwhile investigating the next dominant consideration in sizing loudspeaker wire.
Frequency response errors will be caused by the voltage divider created between the wire’s fixed resistance and the loudspeakers changing impedance versus frequency.
Figure 2 and Figure 3 show two representative loudspeaker impedance plots, pulled from the Internet.
These are not offered as either worst case or typical.
From the impedance plot in Figure 2, if we ignore the extreme low frequency, this loudspeaker exhibits a maximum impedance greater than 17 ohms, with a significant region of the upper bass down around five ohms.
Figure 2: This loudspeaker exhibits a maximum impedance greater than 17 ohms. Click to enlarge.
Meanwhile, Figure 3, while more complex, covers a similar impedance range, with a maximum around 16 ohms and a minimum around six ohms.
To derive a frequency response error we need to compare the drop at maximum impedance to the drop at minimum impedance. The equations below calculate that drop for a given wire resistance.
Note: To simplify this analysis we will assume all loudspeaker impedances to be resistive. While not strictly accurate, loudspeaker impedances will typically be resistive at impedance minimums and any errors caused by load phase angle at the impedance maximums will not be significant for the sake of this analysis.
Minimum Voltage drop= V max = Z max /(Z max +Z wire)
Maximum Voltage drop= V min = Z min /(Z min + Z wire)
Frequency Response deviation= FR max = -20 Log10 (V min/ V max)
Solving for 1-, 0.5-, and 0.1-ohm wire resistance we get:
Loudspeaker….......1 ohm…...... 0.5 ohm….. 0.1 ohm
Spkr 1 (17/5)........ -1.09 dB…... -.57 dB…... -.12 dB
Spkr 2 (16/6)....... -.81 dB…..... -.42 dB…... -.09 dB
Figure 3: While more complex than the loudspeaker in Figure 1, this covers a similar impedance range, with a maximum around 16 ohms. Click to enlarge.
Another related consequence is how wire resistance degrades effective damping factor.
While damping factor is usually though of as a power amplifier characteristic, in reality the wire selection can easily dominate actual damping available at the loudspeaker.
In the above examples the 1-ohm wire would by itself cause a rather weak damping factor of 5 or 6 (regardless of the amplifier’s rated damping factor).
Using the 0.1-ohm wire predicts a more respectable 50-60 damping factor, with some small additional degradation due to the amplifier’s output impedance.
Damping factor deserves a more extensive discussion, but for this exercise we will assume that the amplifier’s output impedance is small with respect to our wire’s resistance.