Output impedance, damping factor and frequency response variability. How good is the 1/8 rule of thumb really?

[JIW]

It is mentioned often in discussions of output impedance and damping factor that the as a rule of thumb the output impedance should be no higher than 1/8 of the speaker or headphone impedance. In other words, damping factor should be 8 or higher.

One source, where such a rule is derived is NwAvGuy in his article Headphone Amp & Impedance in the "Tech section". He claims that 1 dB is the smallest difference audible for most listeners. Using this and the voltage division formula V = VS*Z/(Z + Zout) where V is the voltage across the load, VS is the source voltage, Z is the load impedance and Zout is the output impedance, he shows that a 1 dB reduction in voltage is 10^(-1/20) = 0.89 and if the output impedance is 1/8th of the load V/VS = 8/9 = 0.89 as well. However, using more decimals, 10^(-1/20) = 0.8913 but 8/9 = 0.8889. Thus, the output impedance has to be a bit above 1/8 of the load.

However, if impedance is less variable, a less restrictive requirement can be derived.

Let the desired variability in dB be d, the variability of the load impedance be characterised by the ratio of the maximum and the minimum k and the minimum impedance be Zmin. Thus, the voltage at the minimum impedance is Vmin = VS*Zmin/(Zmin + Zout) and the voltage at the maximum impedance is Vmax = VS*k*Zmin/(k*Zmin + Zout) and the ratio of the two is

Vmax/Vmin = (k*Zmin/(k*Zmin + Zout))/(Zmin/(Zmin + Zout)) = k*(Zmin + Zout)/(k*Zmin + Zout).

Thus,

Vmax/Vmin*(k*Zmin + Zout) = k*(Zmin + Zout)

and thus,

(Vmax/Vmin - k)*Zout = k*(1-Vmax/Vmin)*Zmin

which gives the damping factor

Zmin/Zout = 1/k*(k - Vmax/Vmin)/(Vmax/Vmin - 1) = (1 - 1/k*Vmax/Vmin)/(Vmax/Vmin - 1).

For the desired variability, Vmax/Vmin = 10^(d/20). Thus,

Zmin/Zout = (1 - 1/k*10^(d/20))/(10^(d/20) - 1).

Thus, the higher the impedance variability, the higher the damping factor has to be. The worst case is approximated by the variability approaching infinity. In that case, the impedance ratio approaches

Zmin/Zout = 1/(10^(d/20) - 1) = 10^(-d/20)/(1-10^(-d/20)).

The denominator in the expression on the right is the error in NwAvGuy's simple approach. The exact damping factor at the minimum impedance for 1 dB variability is 1/(10^(1/20)-1) = 8.1955 which is 2.44% higher than 8. For 0.1 dB variability, it is 1/(10^(0.1/20)-1) = 86.36.

If impedance variability is 2 similar to the HD600, the damping factor required for 1 dB variability is Zmin/Zout = (1 - 1/2*10^(1/20))/(10^(1/20) - 1) = 3.5977 but if impedance variability is 10 as is typical for some speakers or IEMs, damping factor must be at least Zmin/Zout = (1 - 1/10*10^(1/20))/(10^(1/20) - 1) = 7.2759 and if impedance variability is 100, damping factor must be at least Zmin/Zout = (1 - 1/100*10^(1/20))/(10^(1/20) - 1) = 8.1035.

In conclusion, the rule of thumb is based on one central claim of audibility which is debatable and some basic math. The former can easily be adjusted for if found lacking while the latter has a rather small error. However, with relatively low impedance variability, a lower damping factor is required.

 

[solderdude]

He claims that 1 dB is the smallest difference audible for most listeners.

That would also depend on the width (Q) of the raised area.

The 1/8th or 1/10th 'rule' is not a rule but a guideline if one wants to be sure the sound signature is not altered too much while not knowing anything else than output resistance of the source (often not even specified) and the (average or at 1kHz) impedance of the load.

Real life issues:
High output R also means limited output power in low impedance loads resulting in low maximum output levels.
Output resistance of most gear is not specified. Things like (AV)receivers, integrated amps with a headphone socket can be anywhere between 50 and 500ohm.
Not all headphones have wildly varying impedance (most planars do not vary at all)

For that reason the 'rule of thumb' at least might ensure that tonality is not affected too much.

 

[DVDdoug]

In conclusion, the rule of thumb is based on one central claim of audibility which is debatable and some basic math.

Right! A rule-of-thumb is NOT a mathematical axiom.

But I'd say it's a good rule-of-thumb. And it's really not hard to meet or exceed that "requirement".

 

[terryforsythe]

It is mentioned often in discussions of output impedance and damping factor that the as a rule of thumb the output impedance should be no higher than 1/8 of the speaker or headphone impedance. In other words, damping factor should be 8 or higher.

To properly determine the damping factor, it is calculated starting from the the terminal of the speaker's drivers and going back through the amplifier's output stage. Thus, the calculation should include the impedances of the crossover components if a passive crossover is used in the speaker. Speaker cables also should be included if their resistance is not negligible in comparison to the crossover's resistance.

If the speaker's crossovers have high impedance at bass frequencies, typically driven by the DC resistance of the series inductors, the output impedance of the amplfier may be negligable unless you are using a tube amplifier with output transformers.

If using passive speakers, I highly doubt you will hear a difference from one amplifier to another if they all are modern solid state amplifiers not being pushed to their limits. If someone claims to hear a difference, ask them to again try in a blind comparison.

On the other hand, if using active speakers or headphones with short speaker cables, there may be subtle differences, but I have not tested that myself.

I have compared a passive speaker to one with the crossovers removed and setup to be all active, and with that I heard a significant difference. But, it was not a blind test, so take that as you will. Also, in that speaker (Elac UBR62), the DCR of the woofer is 4.15 ohm and the woofer portion of the passive crossover had a total of 1.6 ohms series resistance, which is pretty high. The damping factor at low frequencies with the passive crossovers was around 3.6. All active it was around 28.7. In both cases, the amplfier's output impedance was so low (< 50 uOhm) that it was negligible in the damping factor calculations.

Generally, in a speaker, damping factor is mostly a concern for the woofer, and typically is not an issue for midranges and tweeters.

 

[DonH56]

I think high variability is part of it, but if the impedance is variable but remains fairly high, then output impedance is less a factor. That is, varying from 1 ohm to 10 ohms I would expect more frequency deviation from a low damping factor (high output impedance) than say 6 ohms to 20 ohms.

A brief look for various amplifiers and speakers: https://www.audiosciencereview.com/...amping-factor-and-speakers.23968/#post-807327

 

[JIW]

I think high variability is part of it, but if the impedance is variable but remains fairly high, then output impedance is less a factor. That is, varying from 1 ohm to 10 ohms I would expect more frequency deviation from a low damping factor (high output impedance) than say 6 ohms to 20 ohms.

A brief look for various amplifiers and speakers: https://www.audiosciencereview.com/...amping-factor-and-speakers.23968/#post-807327

For the same damping factor based on the minimum load impedance, only the ratio of maximum and minimum impedance matters for the amount of deviation. For the same output impedance, two loads with the same maximum-minimum-ratio but different minimum impedance, the one with lower minimum impedance will have greater deviations due to the lower damping factor. This can all be derived straightforwardly from

Vmax/Vmin = k*(Zmin + Zout)/(k*Zmin + Zout).

so no need to speculate. It is increasing in k which is evident when dividing each term by k and it is then also apparent that it is decreasing in Zmin if k > 1 since the part added to Zmin in the denominator is smaller than the part added to it in the numerator. Further, the log-derivative with respect to Zmin is 1/(Zmin + Zout) - k/(k*Zmin + Zout) = (k*Zmin + Zout - k*(Zmin + Zout))/((Zmin + Zout)*(k*Zmin + Zout)) = (1-k)*Zout/((Zmin + Zout)*(k*Zmin + Zout)) < 0 for k > 1.

 

[JIW]

To properly determine the damping factor, it is calculated starting from the the terminal of the speaker's drivers and going back through the amplifier's output stage. Thus, the calculation should include the impedances of the crossover components if a passive crossover is used in the speaker. Speaker cables also should be included if their resistance is not negligible in comparison to the crossover's resistance.

If the speaker's crossovers have high impedance at bass frequencies, typically driven by the DC resistance of the series inductors, the output impedance of the amplfier may be negligable unless you are using a tube amplifier with output transformers.

If using passive speakers, I highly doubt you will hear a difference from one amplifier to another if they all are modern solid state amplifiers not being pushed to their limits. If someone claims to hear a difference, ask them to again try in a blind comparison.

On the other hand, if using active speakers or headphones with short speaker cables, there may be subtle differences, but I have not tested that myself.

I have compared a passive speaker to one with the crossovers removed and setup to be all active, and with that I heard a significant difference. But, it was not a blind test, so take that as you will. Also, in that speaker (Elac UBR62), the DCR of the woofer is 4.15 ohm and the woofer portion of the passive crossover had a total of 1.6 ohms series resistance, which is pretty high. The damping factor at low frequencies with the passive crossovers was around 3.6. All active it was around 28.7. In both cases, the amplfier's output impedance was so low (< 50 uOhm) that it was negligible in the damping factor calculations.

Generally, in a speaker, damping factor is mostly a concern for the woofer, and typically is not an issue for midranges and tweeters.

Fair point. I did not consider that. I was thinking of headphones primarily and thought it should apply to speakers just as well.

As to your crossover removal, how well did you reproduce it before the amplifiers?

Edit: While the term damping factor may not be appropriate in that case, the voltage deviations at the speaker input and hence its frequency response deviations are still calculated the same way.

 

[JIW]

That would also depend on the width (Q) of the raised area.

The 1/8th or 1/10th 'rule' is not a rule but a guideline if one wants to be sure the sound signature is not altered too much while not knowing anything else than output resistance of the source (often not even specified) and the (average or at 1kHz) impedance of the load.

Real life issues:
High output R also means limited output power in low impedance loads resulting in low maximum output levels.
Output resistance of most gear is not specified. Things like (AV)receivers, integrated amps with a headphone socket can be anywhere between 50 and 500ohm.
Not all headphones have wildly varying impedance (most planars do not vary at all)

For that reason the 'rule of thumb' at least might ensure that tonality is not affected too much.

Agreed and possibly also the affected frequency range. 1 dB around 4 kHz may be different than 1 dB at 40 Hz or 16 kHz.

This raises the question how much larger the specified impedance is than the minimum impedance. The required damping factor is simply increased by the proportionally by the ratio of nominal to minimum impedance. For single driver dynamic headphones, it should be less than 2-to1 and for planar magnetics it should be exactly 1 as you say. Since dynamic single driver headphones typically also do not vary in impedance more than the maximum being twice the minimum, a damping factor of 8 is likely still enough. For multi-driver IEMs or speakers, this might be quite different though.

 

[DonH56]

For the same damping factor based on the minimum load impedance, only the ratio of maximum and minimum impedance matters for the amount of deviation. For the same output impedance, two loads with the same maximum-minimum-ratio but different minimum impedance, the one with lower minimum impedance will have greater deviations due to the lower damping factor. This can all be derived straightforwardly from

so no need to speculate. It is increasing in k which is evident when dividing each term by k and it is then also apparent that it is decreasing in Zmin if k > 1 since the part added to Zmin in the denominator is smaller than the part added to it in the numerator. Further, the log-derivative with respect to Zmin is 1/(Zmin + Zout) - k/(k*Zmin + Zout) = (k*Zmin + Zout - k*(Zmin + Zout))/((Zmin + Zout)*(k*Zmin + Zout)) = (1-k)*Zout/((Zmin + Zout)*(k*Zmin + Zout)) < 0 for k > 1.

Look at two cases, both having amplifier Zout = 0.1 ohms.

1) Zmin = 1 ohm, Zmax = 10 ohms, thus Zmax/Zmin = k = 10: Vmax/Vmin = 1.089 (0.74 dB)
2) Zmin = 4 ohms, Zmax = 40 ohms, thus Zmax/Zmin = k = 10: Vmax/Vmin = 1.022 (0.19 dB)

That agrees with my post and your derivation. If I increase Zout, the deviation goes up, but will still be greater for the lower-impedance load even though the impedance max/min ratio is the same. I am not sure what your disagreement is with my statement (or were just expanding on it?) The amount of response deviation depends upon the load (speaker) and the output impedance of the amplifier.

 

[JIW]

Look at two cases, both having amplifier Zout = 0.1 ohms.

1) Zmin = 1 ohm, Zmax = 10 ohms, thus Zmax/Zmin = k = 10: Vmax/Vmin = 1.089 (0.74 dB)
2) Zmin = 4 ohms, Zmax = 40 ohms, thus Zmax/Zmin = k = 10: Vmax/Vmin = 1.022 (0.19 dB)

That agrees with my post and your derivation. If I increase Zout, the deviation goes up, but will still be greater for the lower-impedance load even though the impedance max/min ratio is the same. I am not sure what your disagreement is with my statement (or were just expanding on it?) The amount of response deviation depends upon the load (speaker) and the output impedance of the amplifier.

My response was based on you saying that you would expect it. To me that sounds like guessing and I wanted to show that it can be known. I did not mean to say your example is incorrect.

 

[DonH56]

My response was based on you saying that you would expect it. To me that sounds like guessing and I wanted to show that it can be known. I did not mean to say your example is incorrect.

Language barrier. Expect implies some knowledge of the anticipated result of the analysis, which I have, and I sometimes word things "softer" than always speaking in absolutes. I try not to guess and usually clearly say I do not know in those circumstances.

 

[JIW]

Language barrier. Expect implies some knowledge of the anticipated result of the analysis, which I have, and I sometimes word things "softer" than always speaking in absolutes. I try not to guess and usually clearly say I do not know in those circumstances.

Yeah. Maybe guessing was not the right term to use on my part but the way I read it there seemed to be some uncertainty on your part. Also, I know expectation more from statistics rather than engineering and so that is maybe why my first thought went there.

 

[DonH56]

Yeah. Maybe guessing was not the right term to use on my part but the way I read it there seemed to be some uncertainty on your part. Also, I know expectation more from statistics rather than engineering and so that is maybe why my first thought went there.

"If I drop a hammer on a high-gravity planet, I expect it to hit the ground." There is always room for uncertainty so I tend to avoid absolutes like "it will hit the ground". The planet could explode, the hammer could get knocked aside by something, etc. Decades of experience has made me wary of absolutes.

I need to remember to stay out of your threads, sorry, as my manner of speaking/writing is inconsistent with your approach.

 

[terryforsythe]

As to your crossover removal, how well did you reproduce it before the amplifiers?

I'm not sure what you are asking. The passive speaker had 1 amplifier channel. The active one had 3. Comparing the two, the bass in the passive speaker was not as tight. The bass in the all active setup was tighter and had more impact.

 

[JIW]

I'm not sure what you are asking. The passive speaker had 1 amplifier channel. The active one had 3.

In terms of filters.

 

[terryforsythe]

In terms of filters.

In the project I first made one speaker to be all active and left the other passive. I did a quick comparison of the sound between the two speakers (not a blind listening test) just to verify whether I was heading in the right direction for what I was tring to achieve. After that comparison I went full steam ahead.

For the active speaker I setup active crossovers in a miniDSP HTx. I output the woofer channel to one channel of my Hypex Nilai amplifier. I output the midrange and tweeter channels to respective channels of a Toppling LA90D amplifier. For the passive speaker, I used one channel in the HTx full bandwidth, without any crossovers, and sent that to the other channel of the Hypex Nilai connected to the passive speaker.

I determined the damping factors by measuring the DC resistance of the woofer, the inductors in the passive woofer filter, and the resistance of the speaker cables. The output impedance of the amplfier I determined from its specification.

 

[JIW]

In the project I first made one speaker to be all active and left the other passive. I did a quick comparison of the sound between the two speakers (not a blind listening test) just to verify whether I was heading in the right direction for what I was tring to achieve. After that comparison I went full steam ahead.

For the active speaker I setup active crossovers in a miniDSP HTx. I output the woofer channel to one channel of my Hypex Nilai amplifier. I output the midrange and tweeter channels to respective channels of a Toppling LA90D amplifier. For the passive speaker, I used one channel in the HTx full bandwidth, without any crossovers, and sent that to the other channel of the Hypex Nilai connected to the passive speaker.

I determined the damping factors by measuring the DC resistance of the woofer, the inductors in the passive woofer filter, and the resistance of the speaker cables. The output impedance of the amplfier I determined from its specification.

I see. So you did not recreate the exact transfer function of the cross-over in the DSP it seems. That may explain at least in part you hearing a difference.

 

[terryforsythe]

I see. So you did not recreate the exact transfer function of the cross-over in the DSP it seems. That may explain at least in part you hearing a difference.

No. That definitely explains part of the difference. Ineed, my original goal was to improve the crossover. The improvement, subjectively speaking, was more than I anticipated. I was pleasantly surprised to say the least. I knew inductor core losses would have some impact, but at the 200Hz crossover frequency of the original woofer filter they should not have had high enough impact to account for what I subjectively heard. In previous tests, negative effects from inductor core losses were most audible to me in the upper midrange. I had an interesting mystery I wanted to solve. That is when I started investigating the damping factor. To keep things simple, I only used DCR of the passive filter since it is the predominant factor for the woofer at low frequencies, but a full analysis would require determining the actual impedances vs frequency. Nonetheless, looking at DCR alone, the damping factor improved by a factor of 8 by going all active. That solved the mystery for me.

 

[terryforsythe]

Out of curiosity, I modeled my Elac's original passive crossover, connected to my amplifier, in Quc-S and generated source impedance data. I used that data along with my woofer's measured impedance data (port plugged) to compute the damping factor as seen by my woofer. I used the magnitudes for the impedance values in the spreadsheet, so it is not 100% accurate, but it is close enough to provide insight. (EDIT: I updated the spreadsheet with the phase angles for the calculations (not much change), and created a chart.) Here are the results: