Impedance at high frequency in audio cables

[pwnell]

I am trying to understand at a technical level why impedance is never mentioned as a major problem at the upper end of the audio frequency spectrum in speaker cables.

I have reviewed http://www.audioholics.com/audio-video-cables/calculating-cable-inductance-of-zip-cord and used http://chemandy.com/calculators/round-wire-impedance-calculator.htm to calculate impedance at say 22kHz.

If I use a very long run of AWG18 wire (yes, I know - one should use thicker wire but this is not about resistance as much as inductive reactance as the thickness of the cable does not adversely affect the inductive reactance), with the following parameters:

Frequency: 22kHz (0.022MHz)
Length: 40000mm (a 20m surround speaker run, thus 40m in total length including return path)
Diameter: 1.024mm
Relative permeability of insulator picked as 1 (for air, PVC should be similar)

I get Z = 0.85 +j12.4 Ohms.

The magnitude of this vector is 12.43 Ohms. Therefore, the inductive reactance contributes 93% of the total impedance. All the articles mention that the most important part is minimising R, i.e. picking a thicker cable but in these calculations it would make no real difference. Picking a thick cable such as AWG10 would result in:

Z = 0.22 + j11.28 Ohms which is 11.3 Ohms, and inductive reactance still contribute to 98% of the losses.

Can someone explain to me what I am missing - why this high impedance caused by inductance at high frequencies is not an issue?

 

[Speedskater]

You may have used a calculation for a single wire rather than a closely spaced pair.

 

[Bizarro_Stormy]

I get Z = 0.85 +j12.4 Ohms.

I get 1.21 gigawatts...

1.21 gigawatts?!?
Great Scott..!

 

[pwnell]

Sure. I did consider that but it appears to only increase inductance - unless the signs somehow cancels out.

Using https://www.eeweb.com/toolbox/parallel-wire-inductance/

D = 1.024mm
S = 4mm
L = 2000cm
µ = 1

I get an inductance of 16.3µH, which is considerably less than the 80µH caused by external inductance. So regardless whether it adds or subtracts, if this is the right calculation, the inductance is still high.

 

[j_garcia]

http://www.roger-russell.com/wire/wire.htm

 

[Speedskater]

I would guess about 2uH.

http://www.audioholics.com/audio-video-cables/calculating-cable-inductance-of-zip-cord

 

[pwnell]

http://www.roger-russell.com/wire/wire.htm

Thanks, I did review that earlier but it does not speak directly to my question.

 

[pwnell]

I would guess about 2uH.

http://www.audioholics.com/audio-video-cables/calculating-cable-inductance-of-zip-cord

It differs from the calculations using http://www.g3ynh.info/zdocs/refs/NBS/Rosa1908.pdf for external inductance...

 

[mtrycrafts]

... less than the 80µH caused by external inductance. ....

And, your next step is to use that 80uH to calculate the signal loss at 22kHz with real output and input impedances and see if it matter audibly. Maybe use 16kHz that most can still hear than, not 22.

 

[Speedskater]

Looking back at an old cable test:
http://www.audioholics.com/gadget-reviews/speaker-cable-face-off-1/cable_table_lg.gif/image_view_fullscreen

13 to 16 uH for 20 meters seems to be correct.

 

[Swerd]

I am trying to understand at a technical level why impedance is never mentioned as a major problem at the upper end of the audio frequency spectrum in speaker cables.

I'm responding to this post despite my belief that the OP understands the answer to his own question.

If I use a very long run of AWG18 wire (yes, I know - one should use thicker wire but this is not about resistance as much as inductive reactance as the thickness of the cable does not adversely affect the inductive reactance), with the following parameters:

Frequency: 22kHz (0.022MHz)
Length: 40000mm (a 20m surround speaker run, thus 40m in total length including return path)
Diameter: 1.024mm
Relative permeability of insulator picked as 1 (for air, PVC should be similar)

I get Z = 0.85 +j12.4 Ohms.

Now do those calculations again, changing the frequency and wire length.

Why did you choose 22 kHz? That frequency is barely heard by most listeners; few tweeters or recording microphones work well at 22 kHz; and there is little or no musical content at that frequency. What would the inductive reactance be one octave or several octaves lower – at 11 kHz, 5.5 kHz, 2.75 kHz, 1.38 kHz, etc.?

A speaker wire 40 meters (131 feet) is rather long. How about 15 meters?

Can someone explain to me what I am missing - why this high impedance caused by inductance at high frequencies is not an issue?

Here is what you're overlooking. The inductive reactance may be 93% of the total impedance at 22 kHz, but what is the magnitude of the total impedance at that frequency? Is it significant? And what is that impedance at lower audio frequencies?

And finally, is any of that audible to human listeners while listening through audio speakers? No calculations can answer that question, only listening tests can.

 

[Speedskater]

The impedance of most (but not all) tweeters at 22kHz is rather high. Maybe 2 or 3 times the loudspeaker system's nominal impedance.

 

[mtrycrafts]

...
Why did you choose 22 kHz? That frequency is barely heard by most listeners; ....

You didn't mean to say this, did you? Most people can barely hear 22kHz?

ps. he should really calculate the HF roll off with that inductance.
Fred Davis's paper shows it might be capacitance that would do it.

 

[Swerd]

You didn't mean to say this, did you? Most people can barely hear 22kHz?

Yes I meant what I said. At 22 kHz, most listeners barely hear anything for a variety of reasons, including:

• Few dome tweeters or recording microphones perform well at 22 kHz.
• There is little or no musical content at that high a frequency.
• And human perception of frequencies that high is at best variable, and is often lacking, especially among middle age or older men.

ps. he should really calculate the HF roll off with that inductance. Fred Davis's paper shows it might be capacitance that would do it.

Yes, that is a good suggestion.

The OP focused on inductance due to 40 meters of 18 g wire while only looking at the upper limit of the audio frequency range. My main point (which I didn't say directly) was he ended up ignoring the forest for the trees.

 

[Swerd]

After re-reading this thread, the OP doesn't seem to understand the difference between inductive reactance and capacitive reactance as it might apply to long runs of speaker cable. Straight speaker wire lacks an inductor coil, therefore the inductive reactance is insignificantly low, and the OP's original premise is wrong. However capacitive reactance could apply to very long runs of parallel conductors, such as zipcord.

See the definitions which I paraphrased from wikipedia:

Inductive Reactance – a property exhibited by an inductor. Exists because an electric current produces a magnetic field around it. In the context of an AC circuit, this magnetic field changes constantly as a result of current that oscillates back and forth. It is this change in magnetic field that induces another electric current to flow in the same wire, in a direction such as to oppose the flow of the current originally responsible for producing the magnetic field. Hence, inductive reactance is an opposition to the change of current through an element.

Capacitive Reactance – A capacitor consists of two conductors separated by an insulator or dielectric. Capacitive reactance is an opposition to the change of voltage across an element.

In my opinion, the OP is trolling and should be ignored. I'm sorry I didn't realize this sooner.

 

[Speedskater]

I that that the OP has good math & engineering skills, just an unusual prospective on cables.

 

[3db]

The first link is probabally the most important link to show why the transmission line model does not work in audio frequencies. The bolded part should answer your question.

http://www.allaboutcircuits.com/text...mission-lines/

Basically, it has to do with cable lengths or what the article describes as electrically long or electrically short. If our interconnects and speaker cables approached the quarter wave length of the highest audio frequency, then that cable would be considered electrically long. To get an idea how long this, I punched in 20 KHz into the calculator in this link; http://www.1728.org/freqwave.htm and I got an electrical wavelength (not an acoustic wavelength) to be 1.4999 e^4 meters or 14.999 kilometers. To meet the minimum requirement for "electrically long", divide this by 4 to get our quarter wavelength defintion which will be 3.75 Km. Based on these numbers, it would be safe to say that most of us have speaker cables that are electrically short.

In an electrically short cable, the voltage/current is considered constant through out the length of the cable at a given instant in time. The voltage/current does vary with time but if one could freeze time and take out a voltage meter and measure along the cable length, one would get the same reading along any point on the cable. This means that the amplifier doesn't see the affects of the capacitance or inductance of the cable but it does see the resistance of the cable and the load attached to the end of the cable. This resistance is relatively small compared to the speaker and can be ignored if appropriately sized for the cable run length and the power its expected to carry.

To illustrate another point... remember traveling down a road beside telephone lines and every once in a while, you would see a big lump in the line? That is a transformer along with capacitors and inductors to shape up the signal before it continues to travel along the lines and match impedances. We would require to insert this in our speaker cables if they were many times longer than the 3.75km calculated from above.


Its stands to reason that if the speaker cable is not seen by the amplifier (capacitance and inductance values of the cable does not affect the amplifier), then swapping in and out different speaker cables will not affect the sound provided once again that the cables have sufficient cross sectional area for run length and intended power delivery.

I hope this helped in your understanding of signal propagation in the audio world.

 

[everettT]

Guess the OP was making a statement not starting a discussion, it's a shame because I always love the EEs posts, my chances to learn even if it's not relevant audibley.

 

[Speedskater]

For sure, transmission line theory and cable Radio Frequency Characteristic Impedance have nothing to do with analog interconnects or loudspeaker cables. But for telephone companies 75 years ago, it was a major concern.

But high total cable capacitance can make some power amplifiers unhappy.
Also high cable self inductance can have a measurable change in treble response. Only in rare cases will it be large enough to be audible.

 

[3db]

For sure, transmission line theory and cable Radio Frequency Characteristic Impedance have nothing to do with analog interconnects or loudspeaker cables. But for telephone companies 75 years ago, it was a major concern.

But high total cable capacitance can make some power amplifiers unhappy.
Also high cable self inductance can have a measurable change in treble response. Only in rare cases will it be large enough to be audible.

I agree with your statements. I just wanted to emphasize that telephone companies were concerned because the cables were /are electrically long compared to the wave length of the signal being propagated.