EPDR is Not What I thought
EPDR and Stereophile
In 2007 Keith Howard coined the concept of Equivalent Peak Dissipation Resistance in his article Heavy Load: How Loudpeakers Torture Amplifiers. The idea of EPDR is to help readers understand more about a speaker and how hard it is to drive. Howard writes:
But for a magazine audience, the principal interest in a loudspeaker’s load impedance lies in gaining some indication of its compatibility with a given amplifier.
The rest of the article needs very careful reading, because to many readers it sounds like he’s saying this is a better impedance measurement, but he never actually proves that. He weaves together a lot of charts, and even contradictory data from Otala:
That was left to others to check, and their results suggest that [sigh of relief] this is not a phenomenon with any practical relevance. That is what Dolby Labs’ Eric Benjamin found when he investigated the issue in 1994 (footnote 3). It’s what I found, too, when I unwittingly reprised some of Benjamin’s work in 2005 (footnote 4), albeit using a software-analysis approach rather than an oscilloscope. While in this context you can’t prove a negative—there is always the possibility that some pieces of music will contain just the waveform necessary for a particular speaker to demonstrate the Otala effect—the available evidence suggests that this probably occurs extremely rarely, if at all.
but then Howard turns the ship around, after saying this isn’t important:
No, the problem with conventional impedance measurements lies not in the measurement method itself but the way in which its results are presented.
He then goes onto use Benjamin’s work with a lot of graphs, but this part is key: He labels the charts as showing equivalent dissipation, or rather how much heat will a typical linear amplifier produce. This is not a measure of how a speaker’s output will vary but how hot the amplifier will get. The definition of “hard to drive” speaker gets very muddled if you aren’t very careful. I’m going to argue that for most audiophiles, EPDR is irrelevant. At best, it says to put a fan on your amp.
Hard to Drive Speakers
In the typical parlance of audiophiles, a hard to drive speaker is one with unusually low impedance and / or phase angles. The reason for this has to do with how voltage is divided between the amplifier’s output, and the speaker. Without using complex math, here’s the basic voltage divider formula. We’ll use R (resistance) instead of Z (impedance) to keep things simple. Let \(V_{out}\) be the amplifier’s attempted output and \(V_{spkr}\) be the voltage that actually makes it to the speaker terminals:
\[ V_{spkr} = V_{out} * \frac{R_{spkr}}{R_{amp} + R_{spkr}} \]
Consider with a very good amplifier the amplifier’s output impedance ( \(R_{amp}\) )is very low, so if \(R_{amp} = 0\) therefore \(V_{spkr} = V_{out}\) because the right side becomes equal to 1. This is ideal, and the electrical output remains constant regardless of frequency. However, as \(R_{amp}\) rises and/or \(R_{spkr}\) drops, \(V_{out}\) starts to go along with the impedance curve, which can be a roller coaster. This is why tube amps don’t do so well with electrostatic speakers and their 1 Ohm loads in the treble, or really any speaker with < 4 Ohm areas.
Here’s the fundamental problem with EPDR : None of the formulas involved address the amplifier’s output impedance, or peak current delivery or anything else related to the voltage and current at the speaker terminals. The value and derivation of EPDR is entirely about transistor heat. Howard never crosses the line to claim it isn’t about heat, but he also tries to use this understanding as another, better impedance measurement, and it can’t be. And this is where the confusion has come from.
In addition to the problem of using a measurement in a way it doesn’t seem to be intended for, there’s also the question of exactly which formula is being used. There are at least three different versions of EPDR calculators out in the wild:
Stereophile / Howard
Room EQ Wizard
VituixCAD (Used by Erin’s Audio Corner)
Of all of these I could not find any direct evidence of the actual formula’s used, however we’ll discuss Jack Ocklee-Brown’s derivations later below, which seem to be close to what VituixCAD is using. It may be what Room EQ Wizard uses, we have no idea.
Another thing to note is that an amplifier’s output impedance is the end result of the power supply, output stages and feedback design.
Damping Factor
Quickly, amplifiers rarely publish or measure output impedance, but instead publish “damping factor.” It’s calculation is \(DF = 8 / R_{amp}\)
So an amplifier with a DF of 100 has an output impedance of 0.08. Important to note that DF is usually reported in the bass but is often lower (higher R) in the treble.
Foundation
The underlying principle of EPDR, as Howard notes, is a paper by Eric Benjamin: “Audio Power Amplifiers for Loudspeaker Loads,” JAES, Vol.42 No.9, September 1994. The paper is copyrighted and paywalled, but fortunately formulas cannot be copyrighted, so we can walk through Benjamin’s ideas. Benjamin doesn’t directly publish a formula for EPDR. That’s not actually his goal. Benjamin’s paper is about power dissipation across devices in Class B amplifiers. In other words, he wants to know how much a transistor will heat up by calculating the Watts dissipated based on the load impedance.
I don’t want to make light of Benjamin’s work, but the three Benjamin formulas (4, 6, 7) involved which Howard probably uses as a foundation would be relatively simple for an electrical engineer to derive. The real value of Benjamin’s paper is in forcing EEs to think outside of the box when designing the thermal/power envelope of linear amplifiers. We’ll start with the instantaneous device dissipation formula (4):
\[ P_{d}(\theta) = (V_{s} - V_{0} \sin \theta)\left(\frac{V_{0}}{|Z_{l}|} \sin(\theta + \phi) \right) \]
Where \(V_{s}\) is the supply voltage, \(V_{0}\) is the output voltage amplitude, \(Z_{l}\) is the load impedance, \(\phi\) is the load phase angle, and \(\theta\) is the instantaneous angle of the output waveform.
Benjamin then integrates this over the input signal’s conduction angle to get average power dissipation (Formula 6 and 7). The end results are two formulas for power dissipation based on phase angle of the load, one for phase angles less than 50 degrees (6), and one for phase angles greater than 50 degrees (7).
Formula 6 if \(|\phi| \leq 50^\circ\) then:
\[ P_{d} = \frac{2 V_{ss}^{2}}{\pi^{2} |Z| \cos\phi} \]
Formula 7 if \(|\phi| > 50^\circ\) then:
\[ P_{d} = \frac{V_{ss}^{2}}{2|Z_{mag}|} * \left(\frac{4}{\pi} - \cos(\phi) \right) \]
The point that I want my readers to understand is that Benjamin’s goal, and formulas are to calculate power dissipation in the output devices of an amplifier. He is not trying to create a new impedance measurement for speakers, and is definitely not saying “the speaker will sound different because of this.” If anything, this is about heat sinks, and cooling. Even if you barely understand formulas, the term on the left of all three formulas is \(P_{d}\), power across a device, meaning a transistor. Power across a device is power that must be dissipated or let off as heat.
Deriving EPDR
So how do we get EPDR from Benjamin’s work? Howard doesn’t really explain this, but we can reverse engineer it. The idea is to find an equivalent resistive load that would dissipate the same power as the complex load. In other words, if we had a purely resistive load \(R_{eq}\), what value of \(R_{eq}\) would dissipate the same power as our complex load \(Z_{l}\) with phase angle \(\phi\)
The formula requires a 2 step process. First we calculate a token value for power dissipated given the load, and phase angle, then we calculate what resistance would cause the same amount of power dissipation. I say token because for our case, we don’t actually care about actual power, we care about what would be equivalent resistance. So whether we calculate in the range of 0 to 1 watt, or thousands of watts, it doesn’t matter. If you were making an amplifier and trying to understand heat sink requirements you would care and would not use these formulas as-is.
We first calculate the dissipated power, P, from (6) or (7) based on the phase angle. Then we set this equal to the power dissipated by a resistive load:
\[ \text{EPDR} = \frac{2V^{2}}{\pi^{2} * P} \]
For our use, V doesn’t matter, set it to 10 or any other positive constant for all your calculations. Below is the R code which expresses Benjamin’s two formulas and our own EPDR calculator. Hopefully they will help you translate to your language of choice or even a spreadsheet.
# Benjamin's device dissipation formula.
# This is written for easier use with vectors in R
dev_power <- function (Zmag, phase_deg, V=10) {
phi <- abs(phase_deg) * pi / 180
# Go ahead and calculate both formulas, 6 and 7
P6 <- (2 * V^2) / (pi^2 * Zmag * cos(phi))
P7 <- (V^2 / (2*Zmag)) * ((4/pi) - cos(phi))
# Return either P6 or P7 based on phase angle
ifelse(abs(phase_deg) < 50, P6, P7)
}
# Calculate EPDR
epdr_from_dev_power <- function(Zmag, phase_deg, V=10) {
# Calculate some sample power dissipation
P <- dev_power(Zmag, phase_deg)
# Now we calculate what resitor would give the equivalent as returned, above
EPDR <- 2 * V^2 / (pi^2 * P)
return(EPDR)
}So, how does this calculation compare to VituixCAD, Stereophile or Jack Ocklee-Brown’s? Not very well, we overestimate EPDR by a significant amount, and here’s where things get interesting.
Jack Ocklee-Brown’s EPDR Derivation
We believe, without being certain, that the formula being used by VituixCAD is derived from work done by Jack Ocklee-Brown which he published in a post at Audio Science Review or at least a close cousin. We think this because his formula values match some spot testing we’ve done with VituixCAD’s EPDR calculator.
We understand that Jack Ocklee-Brown was a pricipal engineer at KEF. Looking at Jack Ocklee-Brown’s (JOB) PDF, he independently derives a formula for EPDR that is based on peak device dissipation. In other words, he goes all the way back to Benjamin’s formula 4, and not the average formulas (6 and 7). Well, OK, nothing wrong with that in a sense, except that again, the point of Benjamin’s work is to try to understand the thermal load on an output device. JOB’s formula (25), simplified, is as follows:
\[ \text{EPDR} = \frac{|Z|}{4\, \bigl( 1 - \sin\!\left( \tfrac{5\pi}{6} + \tfrac{2}{3}|\phi| \right) \bigr) \sin\!\left( \tfrac{5\pi}{6} - \tfrac{1}{3}|\phi| \right) } \]
We think the results are similar to what VituixCAD is offering, but again, JOB’s derivation, accurate or not, doesn’t tell us a damn thing about how well an amplifier’s voltage output behaves when confronted with a difficult speaker load. If this is derived from Benjamin’s formula 4, then it’s about how hot the output devices would get, not how well the speaker sounds or how well the amplifier can drive it.
JOB’s numbers seem close to Stereophile’s numbers as well but we can’t be sure because Stereophile has never published their formule nor do they explicitly mention the amplitude and phase angles from their measurement testing. At best, estimating from impedance curves and phase angle charts, JOB’s numbers are a close estimate of Stereophile’s. Certainly closer than our derivation, above. For instance, for the Magico S5, Steroephile measures a low in the treble of 1.05 Ohms. We traced the impedance and phase angles. JOB and VituixCAD settled around 1.25 Ohms, and our own EPDR calculator (above) resulted in about 2.5 Ohms. So, ours was the most optimistic, Stereophile most pessimistic, and JOB and VituixCAD more or less agreed.
To be clear, neither Benjamin nor Howard published a definitive EPDR formula, so JOB’s decision to start from peak device dissipation is defensible — but it produces a different numeric behavior than the average-based method implied by Benjamin’s formulas 6 and 7. JOB’s numbers however are closer to what Stereophile seems to be publishing in the speaker reviews
Stereophile and Transparency
In almost all other cases, Stereophile is very transparent about their measurement equipment and methods. For instance, when they use a simulated speaker load you can find out exactly what that simulation looks like, and reproduce their findings using your own components.
By contrast, Stereophile has never published their EPDR formula. Keith Howard’s original article doesn’t have a formula, and none of the speaker reviews that include EPDR mention the formula used. This is unfortunate, because it makes it impossible to verify or reproduce their results. At the end of the day we must take them at their word that the EPDR calculator they use is accurate and was accurately used in each speaker review.
It is also unfortunate that John Atkinson does not publish a full EPDR curve for a speaker but cherry picks a couple of worst case numbers. This also is unfortunate because for a reader to understand amplifier heating, which is all we can do with EPDR, seeing the complete EPDR vs. Impedance curve would be more useful.
Conclusion
We’ve walked through the publicly available sources (and sometimes not so public) for how Equivalent Peak Dissipation Resistance came into existence. We’ve shown the original formulas were intended to help amplifier designers understand how much heat their output devices would generate when driving complex speaker loads.
While we have shown our own example formulas for EPDR, Vituix and JOB’s formulas differ, we have no evidence that any version of EPDR is a better way to examine amplifier/speaker matching than the old impedance and phase angle charts. None of the derivations or formulas have anything to do with how a speaker sounds, or how well an amplifier can drive a speaker. As far as we can tell EDPR and the formulas from which it is derived have nothing to do with voltage or current across a speaker input.
In addition, if we were designing amplifiers I would want to know the power output (Benjamin formulas 4, 6 and 7) directly instead of an equivalent resistance.
We note that we are working in the realm of publicly available data we can find. We look forward to hearing from Stereophile or others if they have more information about the formulas they are using, and will undertake a swift online update and correction if pointed to better data than we have so far.
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