Speaker sensitivity, impedance, and calculating amp power


This is an elementary question, but I'm not quite understanding how to match amp power to speakers. When I searched around on this forum, I found many discussions which went deep into the weeds. I am hoping for a way of calculating the level of amplifier power I need for speakers with different sensitivities and impedances.

If you have the patience, here's the basic question. So, I've learned that one must consider a number of factors to calculate the amount of amplifier power to drive the speaker:

Sensitivity of the loudspeaker
Loss of db at the listening position
SPL desired at listening position
Amount of headroom desired

Most discussion of the demands a speaker will make on an amp focus on the speaker sensitivity. But the speakers I'm considering vary also in their impedance. How would I use both of those factors to estimate necessary amplifier power to drive them with a comfortable amount of headroom?


8700e65e 845e 4b1b 91cc df27687f9454hilde45
I have no idea how you would do it.  

How I would do it is not waste one minute of my time on hard to drive inefficient speakers. There are an awful lot of really great easy to drive speakers 86 dB or greater sensitivity. Pick any one of them and a really good 30 watt (tube, or 60 SS) amp is all you need. So why over think it?

Here’s another good reason to KISS: both sound levels and the power needed to achieve them are logarithmic. What this means in layman’s terms is that to go even just a little bit louder- arbitrarily picking 3dB as just a little bit, because it is- calls for not a little more power but TWICE as much.

So you can knock yourself out crunching numbers, for all the good it will do you, because they will always bring you back to what I said in the beginning. Either you get reasonably efficient reasonably easy to drive speakers, or spend the rest of your life hunting for the unicorn 1000 watt into 1 or 100 ohm amp. Which, sad to say, a lot of guys never do figure out and wind up doing exactly that.

Up to you. Choose wisely.
Honestly, as some one who does a lot of measuring and critical listening, this is difficult. It seems amps are more sensitive to speaker impedance than we think, so current is important, but the rated power at 8 Ohms is not really.  Usually 50-100 W is plenty for a modest listening area with your standard 2 or 3 way speaker.

Looking at the speaker impedance, look for dips below 4 Ohms, and that will tell you were it will challenge an amplifier, and what areas of music to listen for while buying.

Use your own ears. :)

Best,
Thanks, Erik. While this question may have a practical upshot, it’s really a theoretical question. I appreciate your efforts to answer it. 
"The Complete Guide to High End Audio" by Robert Harley.




^^^ Correct! Only problem, we now know the OP isn't interested in anything to do with real world results but only theory. Oh well. Still, points for the ref.
It’s not always about educating or helping the OP, as many will read this thread but fewer actually contribute in a positive way other than to ascribe a narrow view , perhaps just personal bias.

we in the business of helping studios, bands, mastering engineers, recording artists and audiophiles start with a discussion of target SPL. Most people, including on this venerable free admission site don’t actually own an accurate SPL meter or have any idea how loud a trumpet is from
say 30’ ?

so OP, let’s start there, how loud ?
The 2nd question, so you can get started on it is: what is the spectral density of the music you listen to ?

fewer still own and pay attention to a RTA ( real time analyzer )

a current “ client “ of mine ( pro bono ) is using three tracks to evaluate potential systems and size his speaker/amplifier...

the difference in spectral density is off the charts different and proof he does not need a subwoofer... just well engineered full range speakers.. 
Question 3 is about duty cycle....

is that “ word salad “ enough ?

its easy but math and science based...
OP:

If you are interested in theory alone, start with understanding power and sensitivity (not efficiency) at 1 meter.

Assume the amplifier is a perfect voltage source, and the speaker an ideal 8 ohms. Then understand that you can use the power in dB to tell you the difference in speaker output.

dB louder = 10 log (power / 1 watt)

So, if you h ave an 89 dB sensitive speaker, and apply 100 watts, the output at 1 meter will be 109 dB.

The truth is that an ideal speaker with flat frequency response speaker has equal voltage sensitivity at all frequencies, but the power consumed at any frequency is proportional to the inverse of the impedance, so the idea of using power to determine output is a cheat. We’re really using the amps equivalent voltage difference.

So long as your speaker is exactly 8 Ohms, then we can use the power calculation (above) interchangeably with the voltage calculation, below:

db louder = 20 log ( voltage / 2.83V)

The real problem we have is that amps are not sold by voltage output, but by power, but in fact, power amps are not power amps at all, they are voltage amps with (ideally) zero output impedance. That is, a power amplifier does not multiply the input power. It multiplies the input voltage and produces whatever current, and therefore power, is needed.

I’m overfilling your bucket here because I want you to reconsider your question. :)

This handy Wikipedia entry may further confuse you:  https://en.wikipedia.org/wiki/Decibel


Best,

E
Avoid any speaker that dips below 4 ohms through the bass section.Above 5-6 ohms is even better.The flatter the impedance curve the better.

For every 3db increase in SPL the amplifier power required doubles. 1 watt, 2 watts, 4 watts, 8 watts, 16 watts, 32 watts, 64 watts, 128 watts ... And a 10db increase in SPL (double the loudness) requires about 10X the power.
SPL isn't the only consideration when choosing an amp. Factors such as headroom for transient spikes and damping factor play key roles. I had an amp from a specific manufacturer that was rated at 200w into 4 ohms. I had a sense (how's that for a scientific analysis) that it just didn't seem to have enough power to control my speakers which are rated at 96db. I sold it and went for the same brand with an output of 370w into 4 ohms. My system improved because the sound was tighter and more detailed than previously. I wasn't listening any louder, that wasn't my goal. The larger amp was able to control my speakers much better and give me more detailed and cleaner performance. Again, power is not only about SPL.
I'll go buy the Harley book. I'll try to consult it before asking questions like this. I assumed there were no prereq's for asking a question.
@millercarbon — I'm interested in both. It starts with a practical question but leads into theoretical ones, which I want to get a handle on. I'm not seeking to ask questions with hidden motives. No need to out me.
@erik I'll reconsider my question. 

Another way to have asked the question would have been:

"What role does impedance have, if any, alongside other factors in the calculation of how much power should be used to drive a speaker comfortably?"

@tomic601 How loud? SPL of about 80 db, which I would assume with classical/jazz would have peaks of about 100 SPL?

I have been using various calculators to try to see how all the factors I mentioned in the OP vary. For example, there's this one from Crown: www.crownaudio.com/en-US/tools/calculators#amp_power_required
And a nice sheet linked to here on the Hans Beekhuyzen Channel www.youtube.com/watch?v=itKqSWH07_Y

The reason I asked the question was because the calculators I found do not mention the impedance of the speakers. Again, I'll reconsider my question based on what Erik said and what the Harley book says. If I could delete the question, I would.

Good explanation of decibel watts in previous posts. OP, you will need to buy a decibel meter to get to some of your answers.
I posted Harley’s book because the answers to the OP’s questions require more detail and explanation than a forum thread can successfully handle. Plus, Harley’s prose is easy to understand, and the book becomes a resource that proves useful again and again. 
At least that’s been my experience.  
+ the Harley book. Should have been mentioned in all of the OP's current threads. I myself should have done so.
Another way to have asked the question would have been:

"What role does impedance have, if any, alongside other factors in the calculation of how much power should be used to drive a speaker comfortably?"

In modern times speaker sensitivities are usually (although not always) based on the SPL produced at 1 meter in response to an input of 2.83 volts, rather than in response to an input of 1 watt. 2.83 volts into 8 ohms corresponds to 1 watt, so in the case of an 8 ohm speaker (that is truly 8 ohms) both numbers will be the same. However 2.83 volts into 4 ohms corresponds to 2 watts, which is 3 db greater than 1 watt. So 3 db should usually be subtracted from the specified sensitivity of a 4 ohm speaker to derive the SPL it will produce in response to 1 watt.

In the case of a 6 ohm speaker the corresponding figure to subtract is about 1.25 db.

Often such specs do not indicate whether they are based on 2.83 volts or 1 watt. In those cases it would be a good bet that they are based on 2.83 volts.

Note though, that since most solid state amplifiers can provide considerably more power into 4 ohms than into 8 ohms (sometimes as much as a factor of 2 more) that increase in amplifier power capability will partially or fully compensate for the 3 db subtraction. That would not be the case with tube amps, though, or with McIntosh solid state amps which have autoformers at their outputs. Tube amps and McIntosh solid state amps having autoformers are usually designed such that maximum power capability when a 4 ohm load is connected to their 4 ohm tap is essentially the same as when an 8 ohm load is connected to their 8 ohm tap.

On another note, the 80 db average and 100 db peak listening levels you mentioned are fairly similar to mine, my listening being mostly to classical music. Although I’ve found that some (relatively few) classical symphonic recordings that have been well engineered with minimal or no dynamic compression can produce 105 db peaks at my 12 foot listening distance while being listened to at average levels in the mid-70s.

Also, here is another SPL calculator you may find useful:

https://myhometheater.homestead.com/splcalculator.html

In contrast to the Crown calculator you cited this one adds 3 db if two speakers are specified, which is a reasonable approximation, and it attempts to address the effects of room reflections, i.e., what is sometimes referred to as "room gain."

Also, be aware that these calculators will significantly underestimate the resulting SPL in the case of planar speakers, such as electrostatics and Magnepans, for which SPL falls off much less rapidly as distance increases, compared to dynamic (box-type) speakers.

Finally, be aware that speaker sensitivity specs are frequently optimistic by a few db. Before using these calculators it would be prudent to Google the specific make and model of the speaker together with the word "measurements." If Stereophile, SoundStage, or some other publication has reviewed the speaker the measurements that may be presented in conjunction with the review will usually be much better to rely on than the manufacturer’s spec. And their measurements and the associated text will also often provide useful insight into the impedance of the speaker, how it varies as a function of frequency, and how accurate or inaccurate the specified number may be.

Regards,
-- Al
^^^^^^Very good advice from almarg and erik.
I was about to type Almarg's basical post.  That is solid advice.  Even was going to recommend the same calculator.

Good luck to you!


Excellent. Thank you, all.
"What role does impedance have, if any, alongside other factors in the calculation of how much power should be used to drive a speaker comfortably?"



Zero, because in my experience, the math is not complete. We don’t get damping factor for the amp, most of the time, and it too, is frequency dependent. In addition, my experience says that amps are more sensitive to impedance dips than the math would lead us to believe if we did know this.

The way to approach this is, for a given speaker, note where the impedance is worse, then listen to amps with music in that range and see if you hear or feel the deficiency in amplifier output.

One speaker I feel is notorious like this is Focal. They often have these narrow dips around 100 Hz and anything but the most robust amp will noticeably sag.

One decent indicator of an amp’s current drive (ability drive low impedance speakers) is how well it doubles power as impedance drops in half.

8 Ohms -> 100 W
4 Ohms -> 200 W
2 Ohms -> 400 W

But again, I feel noting the impedance dips and listening is a better indicator than this.

Best,

E
Speaker impedance, and particularly uneven impedance curves, has a great deal to do with selecting a proper amplifier to drive a specific speaker. I refer you once again to Harley's book.

The aforementioned pure Class A Pass Labs XA series amplifiers are exceptional at handling speakers with uneven impedance curves. 
as Al referenced, manufacturer sensitivity specs are often seriously overstated...
Impedence and phase angle, there is always that nasty and...

I always liked the late great Roger Modjeski advice on the subject ( which you find w search this site ), measure SPL and use a DMM at speaker terminals.

OP there was nothing wrong with your post, as you can see there are some awesome contributors this site
@tomic601 There sure are!
Erik (or anyone),

You stated:
So, if you have an 89 dB sensitive speaker, and apply 100 watts, the output at 1 meter will be 109 dB. 
What is the formula for calculating what the SPL would be at 3 meters (typical listening position) rather than 1 meter?

Thanks,
J.Chip
What is the formula for calculating what the SPL would be at 3 meters (typical listening position) rather than 1 meter?

For a single speaker, and assuming it is a conventional box-type (i.e., it is non-planar and non-line source), and neglecting room reflections, the attenuation of SPL resulting from the increase in distance is:

20 x log(distance from listener/1 meter)

where "log" is the base-10 logarithm.

So in this case, and under those assumptions:

20 x log(3 meters/1 meter) = 9.5 db of attenuation relative to the SPL produced at 1 meter (rounding off slightly).

For Erik’s example that you quoted:

109 db - 9.5 = 99.5 db SPL at 3 meters.

The presence of a second speaker and the effects of room reflections will typically add several db to that.

The 20 x log formula is built into the online calculators which have been linked to in some of the posts above.

Regards,
-- Al


^Or...you simply could position a $40 decibel meter 3 meters from the speakers and measure the decibel level. ;)
Al, Erik, Miller C, Tomic601 and others, thank you for the discussion/knowledge above.  Tried both calculators, they are within a db or so.  The above shared knowledge is priceless, much better than a school.  Thank you folks for helping me in understanding a very complex aural subject.  Enjoy the music.
How can an amp have different current output with them same "wattage" as a different brand?  I would think current output capabilities would go a long way into driving a given speaker.   Also it seems hard to get impedance vs frequency curves for a speaker system.  Or do I just not know where to look.  It also seems that in an effort to boast low frequency response in a single cabinet, we lose efficiency.  Am I wrong with any of this?  Just curious.
Glad you didn't delete the thread. I wanted to know this too
@daledeee1 This conversation with Jim Salk has a graph, perhaps of the kind you're asking about? Of course, you're asking about getting graphs for *any* speaker we want to buy. Salk's graph is not in this excerpt but you can find it here:

https://www.innerfidelity.com/content/beauty-sight-and-sound-salk-wow1-mini-monitor-page-2

This part of the Inner Fidelity conversation bears a bit on this topic:
"Q: Tying in with the above - desktop users don't typically have space for a massive amp. They tend to use compact units with relatively low power ratings, with Class D implementations being more and more common. Any comment on amplifier matching with these speakers?

A: Today, the trend in speaker design is to develop smaller and smaller speakers that play deeper and deeper. The problem is, the laws of physics dictate that the resulting speakers will be somewhat insensitive and will require more power to drive. But since that is what speaker manufacturers want to produce, most of the R&D for cutting-edge drivers these days is focused on insensitive drivers that play deeper in smaller cabinets. So if you want to use the cutting edge drivers (which we do), the models you develop will tend to be lower sensitivity. They will require more power to drive, but will be more accurate and exhibit higher sound quality.

At the same time, amps are becoming more powerful to address this need. This is especially true of Class D amplifiers. These provide a lot of power at a very reasonable price and are very efficient as well. While some people feel that Class D amps are not quite ready for prime time and can be a bit analytical, they are certainly accurate and provide a lot of performance for the money. Plus, they run cool which is another advantage.

As for the small class D amps, these are quite attractive cost-wise, but don't work all that well with low sensitivity speakers. Years ago, some friends did some interesting experiments at the Rocky Mountain Audiofest. They set up both average power meters (RMS) and peak-reading meters and monitored the power levels while playing music. While the RMS meter registered 5 - 8 watts, the peak-reading meters hit peaks of 200 - 250 watts during instantaneous transients (drum hits and the like). So while, on average, 5 - 8 watts is sufficient, if you want the cleanest possible performance, the more watts you have, the better. So while these small "T-amps" are quite inexpensive, many of the lower power amps won't have the muscle required to control the woofer in the bass region. So I'd pass on trying to marry them up." (Jim Salk)
There are two kinds of audiophiles when it comes to amps and this subject; the good enough type, and the overkill type. The good enough type look at the rating and usually stay within that range, i.e. 50-100wpc as good enough. The overkill type, such as myself will use amps that are seemingly far more powerful than necessary.

I have seen no preference emerge as to any given simple formula that will assure satisfaction. I disagree that you must stay with very efficient speakers; the sonic characteristics vary so widely in terms of frequency extension, tonality, etc. That simple formulas are not helpful either in regard to speakers or amps.

Actual comparisons reveal one’s priorities quickly. :)
Thanks, Doug. I was listening to Paul McGowan on why overkill buys you the benefits of "linearity" in the speakers' response on a video yesterday, but he was careful to point out that *how* you listen makes a huge difference as to whether it will matter. For critical listening, it can make a genuine difference and it's best to know that before purchasing. That's why I'm researching these questions.

And of course the character of the amp's sound is critical. But to make a careful appraisal in the listening, I want to ensure that power levels  are sufficient to do the speakers justice, with perhaps a little to spare. Of course, if I cannot tell, I cannot tell, and then saving money would be my priority, at least to shepard it toward elsewhere in the system.

The larger questions I've raised in a couple threads on this forum — on sensitivity, on impedance, on crossover controls — are seeking a better theoretical understanding so that I can make some good guesses as to what to audition.

If, for example, I know that a 4 ohm/88db sensitivity speaker would benefit from an amp of 200 watts more than 100 watts, I should try to arrange *that* as my auditioning lineup. After all, dealers are patient — they'll give me a chance to try out different combinations — but only to a point. So, in order to try out what might be something I want to actually buy, I need to do as much homework, first. And I want to try my hardest to buy from a dealer, if they carry the gear I actually prefer. I'm convinced that brick and mortar listening opportunities are a value, and if they treat me reasonably and offer reasonable prices, I want to buy with them.

daledeee1 2-20-2020
How can an amp have different current output with the same "wattage" as a different brand?

With a very few exceptions, nearly all solid state amps are designed such that for a given input voltage their output voltage remains essentially constant regardless of the load impedance, as long as the amp is operated within the limits of its voltage, current, power, and thermal capabilities. And for a given output voltage from the amp, per Ohm’s Law the current drawn by a given load impedance will increase as that impedance decreases. And correspondingly the power delivered into that load will increase as that impedance decreases, since the power delivered into a resistive load corresponds to the voltage applied to it multiplied by the current it draws when that voltage is applied to it. (Speaker impedances are not purely resistive, of course, but I won’t get into that for purposes of this explanation).

So as an example if we consider two amps rated at say 100 watts into an 8 ohm resistive load, one amp may be able to deliver 200 watts into a 4 ohm resistive load, and 400 watts into a 2 ohm resistive load, if it is able to provide the correspondingly increased amounts of current. The other amp might be able to deliver only 150 watts into 4 ohms, and 200 watts into 2 ohms. So the second amp has less current capability than the first amp, even though they are both rated at 100 watts into 8 ohms. (This assumes, of course, that both ratings are accurate, and it also assumes that the 100 watt/8 ohm rating of the first amp has not been understated to create a false impression that the amp can double the maximum amount of power it can deliver into halved load impedances).

Tube amps are a different story altogether, though. While most solid state amps have output impedances that are near zero, which enables them to maintain essentially constant voltage into varying load impedances (within some limits), tube amps have output impedances which are significant relative to speaker impedances, and which vary widely among different tube amps. As a result, depending mainly on their output impedance, and assuming they are operated within their capabilities, different tube amps tend to fall at various points along a spectrum whose end points are maintaining constant voltage and maintaining constant power into varying load impedances. Usually not at either of the end points of that spectrum, but somewhere in between.

Finally, as you probably realize the majority of speakers these days are designed to be driven by solid state amps having near zero output impedances, while others are designed such that they are best driven by tube amps, while others are happy with either type.

Regards,
-- Al

P.S: @dwmaggie, thank you for the kind words.

I know that a 4 ohm/88db sensitivity speaker...
This is what I would suggest to cut through the noise and get you to a result in the fastest and most valuable manner.

Call Mark at Reno HiFi and tell him you’re on a budget, but would like his recommendation for a Pass Labs amplifier or integrated amplifier to drive your speakers. Give him the specifications...if it’s 4 ohm/88db, then tell him that. Also tell him the dimensions of your listening room/environment.

See what he says. His recommendation might be too expensive. Or, it might be within the realm of possibility, but regardless you can count on him to give you a suggestion that will absolutely work and sound exceptional.

If he makes a recommendation, and if it’s anywhere in the realm of possibility, then you should avail yourself of the opportunity to audition the amp in your home.

He offers home trial with a return. He is a brick and mortar shop.

You have nothing to lose by calling him.

I am a past Reni HiFi customer, but I now own Audio Note tube amplification, so I receive no benefit by my suggestion.
Hilde45 quoting the great Jim Salk: " While the RMS meter registered 5 - 8 watts, the peak-reading meters hit peaks of 200 - 250 watts during instantaneous transients (drum hits and the like). " 

This is very interesting information. The implication is that we would benefit from 14 dB to 17 dB of headroom above our average SPL, which corresponds with having from 25 to 50 times more power available.  

Obviously it’s easier to get there with high efficiency speakers, though elsewhere in the passage hilde45 quoted Jim makes some insightful arguments for the lastest drivers having modest efficiency. 

Duke
@audiokinesis Right -- and, FWIW, I think the speakers mentioned here are that way because they're small and yet can do bass pretty well. Other speakers Salk makes are not as insensitive, e.g. the  Song3 Beats. 

@tvad Good suggestion. That may be the way to go. I'm going to hear some Atolls tomorrow and some Quicksilver. Just to hear them -- and they'll play through Dyn Evoke 10s, so we'll see how they perform. Later, when I have the Salks, I'll try out some amplification in my home, and maybe via Reno. 
@hilde45, my apologies. I did not read all your posts carefully, and missed that you are assembling a new system including loudspeakers. I thought you were only looking for a tube preamp.

By the way, @audiokinesis (Duke), is a great resource and builder of excellent loudspeakers. A fine gentleman as well.
Thanks -- I am so glad to know that, and will go take a look at his speakers. 
Thank you very much, Tvad.

Hilde45, at the moment the best place to see my home audio designs is actually the James Romeyn Music and Audio website. My own website is under revision.

But my feelings would not be hurt if you ended up with something by Jim Salk. He offers a wide variety of extremely well-thought-out designs in heirloom-quality enclosures.

Duke
My speakers are so sensitive I have to watch what I say around them or  they can get upset....nobody needs a sulking speaker.