Wednesday, May 30, 2012

Impedance and Loss: An Ohm Story

I got into this a bit previously in a post about splitters, here. Ever run some speaker wire a bit too far and get temped to splice in a length of mismatched cable to reach a terminal.  Theoretically each splice creates a change in resistance the circuit, and if measuring cumulatively, it adds to the total resistance. Incidentally, since there is no difference between impedance and resistance in DC circuit you can measure this change in either unit. FYI: The International unit for electrical resistance is the ohm Ω, and whaddyaknow, so is impedance. Though, with impedance it represents a ratio. Generally speaking, the total loss should be trivial unless you screw up.  So read how NASA splices here, and get out your soldering iron.

So my example today is not on the space shuttle, it is right in your home stereo. (With thanks to Richard L. Hess for the math. You can see his fine audio restoration work here.) Today's example is speaker wire. The signal loss you experience between the amplifier and the speaker cone is reliant on three primary factors
1. The gauge of the cable run
2. The length of the cable
GAUGE
Typically speakers are just a few feet from the speakers. But maybe you ran speakers all through out your house and yard like Ken Kesey. In those extreme, psychedelic cases higher gauges you  may use up to 14 or 12 gauge running a high power, tube-driven amplifier. But typically, even running cables around a living room for a surround-sound system needs a16 gauge. The term "gauge" comes from the Brown & Sharpe wire gauge, a standardized wire gauge system used since 1857. The smaller the number the larger the wire.You can see a chart here.

The measurements for stranded wires are a bit different than solid wire. Stranded wires are identified by three numbers, the AWG size as "gauge"  the number of strands, and the AWG size of each strand. For example, a 16 gauge wire with 7 strands of 24 AWG would be: 16 AWG 7/24. Not too complicated. Thank you mister Henry Sharpe. (Stranded AWG PDF chart here. )I recommend using stranded over solid wire for numerous reasons. It's more flexible, easier to splice, and solid wire gets bent, and each bend adds its own changes in resistance.Over a short length this doesn't add up to much, but the price difference is trivial.

LENGTH

DC current loss occurs at a calculable rate over the length of the wire.  Here is a calculator. I'm going to use Mr Hess' example and say that we'll use 70 and 90 feet (note calculator is in cm) and for 16 gauge and 12 gauge wire at 1 kHz and 10 kHz.  The loss calculations can't be exact without actually knowing the reactance of the speaker at a  given frequency, so  1 kHz and 10 kHz are just examples. The reality is a wiggly curve across the whole EQ expressible by the speaker.  Reactance, in this case is the opposition of a circuit  (the wire) to the change in current due to that element's inductance. Inductance is measured here in microhenrys as µH.

16 Gauge:
@70 feet  0.281 ohms resistance, 44 µH inductance
1 kHz  0.278 ohms reactance
10 kHz  2.778 ohms reactance

@90 feet  0.361 ohms resistance, 58 µH inductance
1 kHz  0.365 ohms reactance
10 kHz  2.778 ohms reactance

12 gauge:
@70 feet  0.111 ohms resistance, 42 µH inductance
1 kHz  0.265 ohms reactance
10 kHz  3.655 ohms reactance

@90 feet  0.143 ohms resistance, 56 µH inductance
1 kHz  0.349 ohms reactance
10 kHz  3.495 ohms reactance

Loss Calculation:
Z = resistance + reactance = impedance
Z(s) = impedance of the speaker
Z(w) = impedance of the wire
Z(t) = total impedance of wire and speaker = Z(s) + Z(w)

Loss (in decibels) = 20 log ( Z(s) / Z(t) )

Let's proceed with rounded numbers. At 1 kHz with a nominal 8-ohm speaker,  the 16 gauge cable at produces a loss of -0.59 dB at 1 kHz. Through the longer 16 gauge cable produces a loss of  -0.75 dB, at 1 kHz . That's a difference of -0.16 dB more loss.Bumping up that cable run to 12 gauge wire produces losses of -0.40 dB and -0.52 dB or about -0.12 dB more loss on the 90 foot run. It's not zero, but it's what I'd call trivial. Your average audiophile would surely disagree.