Skin Effect Relevance in Speaker Cables
Some so called "exotic" Cable Companies enjoy spreading the fallacy that Skin Effect can cause deleterious effects on your audio performance. While Skin Effect is a real world problem in high frequency applications such as RF Power and Transmission, it is negligible at audio frequencies as I will demonstrate in this article based on fundamental engineering and scientific principles.
It is a good idea to brush up on speaker cable resistance before diving deeper into the topic of Skin Effect and AC resistance.
Skin Effect Defined
Skin Effect has been known and equations derived for skin depth vs frequency as far back as 1915. It is the tendency of alternating current to flow near the surface of a conductor, thereby restricting the current to a small part of the total cross-sectional area and increasing the resistance to the flow of current.
Note: The skin effect is caused by the self-inductance of the conductor, which causes an increase in the inductive reactance at high frequencies, thus forcing the carriers, i.e., electrons, toward the surface of the conductor. At high frequencies, the circumference is the preferred criterion for predicting resistance than is the cross-sectional area. The depth of penetration of current can be very small compared to the diameter.
Skin Depth is defined as the distance below the surface where the current density has fallen to 1/e or 37% of its value at the surface. Because of Skin Effect, the AC to DC resistance of round wire is dependent on the ratio of the wire diameter to skin depth as can be seen in the equation below:
Rratio = Rac/Rdc = (pi*r^2) / (pi*r^2 - pi(r-S)^2)
where S < r ;
S = 2837 / sqrt(f) (Skin Depth)
(This equation has been derived by many sources and for copper wire at 70 degrees C, the above relationship holds true)
Note: This is a worst case calculation as cable resistance increases with temperature and most people will not be listening to music in a 70 degree C environment as they would probably be more occupied by trying to cool down or get out of such a heated room!
Further, since skin depth is inversely proportional to the square root of frequency, different size wires will have different AC to DC ratios, and these ratios will increase with frequency.
First lets make some basic assumptions for a high fidelity two channel audio set-up:
- Speaker Cables Length: 10ft / ea
- Speaker Cable Gauge: 12AWG round multi stranded OFC wire
- Diameter of 12AWG wire = 80.8 mils
- Maximum Audio Frequency (F) of Concern 20kHz (limit of human hearing)
Illustrated below is the cross sectional area of 12 AWG wire showing the radius of the wire (r) and skin depth (S).
Note: Multi stranded 12 AWG wire cross sectional area is slightly larger than equivalent gauge solid core wire, thus we shall model our wire as solid core for a more conservative calculation as depicted in the figure below. As you may recall my elementry Algebra, the Area of a circle = pi*r^2.
Update: This is a very conservative calculation for increase in AC resistance due to skin effect because it assumes the current density profile is uniform within the first skin depth. This equation also assumes solid core wire which will exhibit higher skin effect compared to a similar gauge stranded wire. Further study shall be reserved on this topic for a later article.
Actual measured increase in AC Resistance due to Skin Effect at 20 kHz is less than 3%. See the results in our Cable Face Off Article for more details.
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