Path: newsfeed.pitt.edu!scramble.lm.com!news.math.psu.edu!news.ems.psu.edu!news.cse.psu.edu!uwm.edu!vixen.cso.uiuc.edu!newsfeed.internetmci.com!tank.news.pipex.net!pipex!dispatch.news.demon.net!demon!peer.news.xara.net!xara.net!netcom.net.uk!netcom.com!ix news1.ix.netcom.com!ix.netcom.com!news From: inspctec@ix.netcom.com(Fred E. Davis) Newsgroups: rec.audio.tech Subject: Speaker Cable Series Impedance and Loss Date: 2 May 1996 19:41:50 GMT Organization: Netcom Lines: 205 Message-ID: <4mb35u$sk2@dfw-ixnews6.ix.netcom.com> NNTP-Posting-Host: nhv-ct4-05.ix.netcom.com X-NETCOM-Date: Thu May 02 2:41:50 PM CDT 1996 Speaker Cables: Series Impedance and Loss Fred E. Davis 2 May 1996 What I present here are data about wire sizes and some of the loss effects encountered in speaker cables. This data examines some of the properties of cables that influence the series impedance of the cable in a frequency-dependent manner. The two most major of these are AC resistance (skin effect) and inductive reactance. Unless it is a special cable design, the capacitance of a speaker cable is too small to have much of an effect. In the skin effect, current density falls off exponentially from the surface of the conductor toward the center. The 'critical depth', or depth of penetration, is where the current density has fallen to 1/e, or 0.368. Viewed as a cross section of the wire, the current density distribution is plotted as: Current density I | 1.0 ---|* |* | * delta | * | * | * | * 0.368--| . . . . . * * | | -------------> depth towards center of wire Fig. 1: Current density vs. depth An expression for critical depth is: delta = sqrt( 1 / (2*pi*f * sigma * mu)) where: delta = critical depth (meter) f = frequency (Hz) sigma = conductivity (mhos/meter) mu = permeability (henrys/meter) {Source: "Secrets of RF Circuit Design" by Joseph Carr, pp. 7-8, (ISBN 0-8306-8710-6)} The effect of the skin effect on resistance in copper wire can be expressed as: Rac = k * sqrt( f ) * Rdc where: Rac = resistance with AC (ohms) Rdc = resistance at DC (ohms) f = frequency (MHz) k = constant related to conductor diameter Some k values for different wire sizes: Wire Size (AWG) k 18 10.9 14 17.6 10 27.6 8 34.8 6 47.9 4 55.5 2 69.8 {Source: "Grounding and Shielding Techniques in Instrumentation" by Ralph Morrison, p. 126 (ISBN 0-471-02992-0)} There is a *very* nice wire calculator, the 'EdTb Wire Selector,' from dTb Software (with a free sampler to download at www.dtbsware.com/wire.html). Among *many* other things, it will display Rdc, current capacity, and Rac at your choice of frequency. Using this data, let's take 20 feet of solid-core wire of differing gauges. Using DC and AC resistance, and computed inductances, let's compare the effective series impedance of the cable using just AC resistance (ie, skin effect, ignoring inductive reactance), and just DC resistance and inductive reactance (ignoring skin effect). Then let's compare the inductive properties against the skin effect on the cable's series impedance. 20 ft ----------- @ 20kHz ----------- Wire(AWG) Rdc Rac XL Rdc&XL Rdc&XL/Rac 18 0.255 0.261 j0.717 0.761 292% 14 0.101 0.114 j0.582 0.591 519% 10 0.040 0.065 j0.452 0.454 696% 8 0.025 0.054 j0.389 0.390 719% 6 0.016 0.044 j0.329 0.329 742% 4 0.010 0.035 j0.270 0.271 768% Fig. 2: Cable series impedance, Rac vs. Rdc and XL The series impedance of the cable is increased far more by inductive reactance compared to skin effect alone. These examples are for a cable with no load, so what happens when there is a load? To keep things simple, the loads used will be simple resistive values. (If you think 2 ohms is low, and for an interesting look at speaker impedances and models of speaker systems, see "Peak Current Requirement of Commercial Loudspeaker Systems" by M. Otala and P. Huttenen, JAES, Vol. 35, pp. 455-462) Once again, let's compare skin effect to inductive reactance with three different loads: dB loss @ 20kHz driving: 20 ft 8 ohms 4 ohms 2 ohms Wire(AWG) Rac Rdc&XL Rac Rdc&XL Rac Rdc&XL 18 -0.279 -0.306 -0.549 -0.659 -1.064 -1.462 14 -0.123 -0.131 -0.244 -0.303 -0.481 -0.749 10 -0.071 -0.057 -0.140 -0.140 -0.279 -0.380 8 -0.059 -0.037 -0.117 -0.095 -0.232 -0.266 6 -0.048 -0.024 -0.096 -0.063 -0.191 -0.182 4 -0.038 -0.016 -0.076 -0.041 -0.152 -0.121 Fig. 3: Cable loss driving loads at 20kHz, Rac vs. Rdc and XL Now the combination of cable and load impedance make the losses from skin effects appear greater, especially with heavier cables. The crossover point where skin effect losses catch up with inductive reactance losses is around 12 AWG. But we're still dealing with a 'fantasy' cable that has skin effect and no inductance, or inductance but no skin effect. Next, let's combine AC resistance with inductive reactance (which more closely models a real cable at 20kHz) driving the three loads: dB loss @ 20kHz driving: 20 ft 8 ohms 4 ohms 2 ohms Wire(AWG) Rac&XL Rac&XL Rac&XL 18 -0.311 -0.670 -1.480 14 -0.145 -0.330 -0.798 10 -0.084 -0.194 -0.482 8 -0.069 -0.157 -0.386 6 -0.055 -0.124 -0.301 4 -0.043 -0.096 -0.228 Fig. 4: Cable loss driving loads at 20kHz using Rac and XL No real surprises here. The heavier cables appear to be doing a pretty good job. If we now subtract the loss using Rdc from the loss at 20kHz using Rac and XL, we get a picture of how flat the cable responses will be across the audio band, which is what your ear would be listening to (unless you were switching between cables). Graphically speaking, this is like taking the plots for each of the cables and overlaying one upon the other in order to get some idea of how similar their shapes are. Again, driving the three load values: dB loss at 20kHz re:DC driving: 20 ft 8 ohms 4 ohms 2 ohms Wire(AWG) Rac&XL Rac&XL Rac&XL 18 -0.038 -0.132 -0.436 14 -0.036 -0.113 -0.370 10 -0.041 -0.107 -0.310 8 -0.042 -0.102 -0.277 6 -0.038 -0.090 -0.233 4 -0.032 -0.074 -0.184 Fig. 5: Cable loss difference from DC to 20kHz Now this is interesting! This shows that the relative losses are strikingly similar until you start driving very low impedance loads, well below 4 ohms. This similarity can also be seen in complete plots of cable impedance vs. frequency. Except for a fixed offset from the DC resistance, the plots all have the same overall shape. The exception, of course, are the flat-impedance cables, which typically have sufficient capacitive reactance to balance the inductive reactance for audio frequencies. The responses may be similar, but what this figure does not show you are the current carrying capacities of the different gauges. Just for fun, here are approximate current capacities for the cables and the equivalent power into the three loads. This may help you match the properly sized cable to your power amplifier. Peak Power (Watts) driving: Wire(AWG) Amps 8 ohms 4 ohms 2 ohms 18 5.18 215 107 54 14 10.39 864 432 216 10 20.83 3,471 1,736 868 8 29.49 6,957 3,479 1,739 6 41.75 13,945 6,972 3,486 4 59.12 27,961 13,981 6,990 Fig. 6: Cable current capacity and peak power (:D Perhaps the similarity in losses among cables illustrates why speaker cables sound so similar in blind tests, and why speakers with very low impedance dips might sound different with different cables. On the other hand, these differences only span 0.252 dB at 2 ohms from 18 AWG to 4 AWG! And at 20kHz, to boot. The losses at 10kHz will be much less. For JNDs (Just Noticeable Differences), I refer you to "Speaker Cables: Measurements vs. Psychoacoustic Data" by Edgar Villchur, Audio, July 1994. The combination of skin effect and inductive reactance *will* cause higher frequencies to roll off, but the numbers are just too small to be significant, even when including load impedance variations. I'm making no claims as to 'sound' or audibility here. What I show are just numbers.