Speaker Cable Series Impedance and Loss
Fred E. Davis
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.