Impedance in headphones vs. speakers - confused

Yeah, I can understand your confusion!! I believe that it exists part because of your admitted lack of understanding of the subject & part because people do not always clearly explain what they mean when they write "easier to drive".
In a nutshell, the issue is the same whether it is a pair of speakers or if it's a pair of headphones. It has to be since it's an amp drive effectively a speaker load.

An amplifier (whether it's a power amp driving a loudspeaker or a head-amp or an inbuilt head-amp inside the iPhone) always delivers power into the (speaker) load. And, power = voltage times current. You an associated voltage swing going to some positive peak, crossing thru zero volts & going to some negative peak (most amplifiers have positive & negative supply rails) & you have associated current flow into (push) the speaker load & out of (pull) the speaker load.
When you have a speaker with an average impedance of, say, 16 Ohms, the amplifier does not need to deliver much current into 16 Ohms to create a fairly large voltage swing because, from Ohms law, voltage = current times resistance. if resistance = 16 Ohms, you need less current compared to when resistance = 8 Ohms to create the same voltage swing. It's the voltage swing that gets the loudspeaker drivers to act in a pistonic fashion to create sound. So, we say that a 16 Ohms speaker is an easy load to the attached power amplifier. Hence the oft written phrase "this is an easy to driver speaker". HOWEVER, there's a flip side: when the current is on the lower side, the voltage is on the higher side!! This happens because power = voltage times current. For comparison purposes, if you are going to feed into a 16 Ohms speaker a fixed amount of power, the amplifier will have to deliver less current (compared to a, say, 4 Ohms speaker) but since the speaker impedance is higher, the corresponding voltage generated is much larger than the, say, 4 Ohms speaker. What can happen is that with very little increase in volume (on your preamp), you can very easily clip the power (or head) amp because the peak-peak voltage swing of the 16 Ohms speaker can escalate very quickly to a large peak-peak voltage swing. Therefore, you often need a power amplifier with a lot of headroom to drive a higher impedance speaker. Power amplifiers with large wattages usually have very large positive & negative power supply rails that can accomodate large peak-peak voltage swings. For example, if you had a 50VAC power transformer secondary, it would generate a ~71VDC supply rail. A 50VAC power transformer secondary into an 8 Ohm load creates a 300W/ch power amplifier capable of accomoating ~70V peak voltages. If you dropped that power transformer to, say, 35VAC then you would have a ~50VDC power supply & a 150W/ch into 8 Ohms power amplifier capable of accomodating only a 50V peak voltage. Normally, a 150W/ch power amp would sufficient to drive most speakers but if you attach a 16 Ohm speaker to a 150W/ch amp & crank up the volume you could easily reach 50Vpk during cresendos in the music thereby clipping the amplifier momentarily. In this case, a 300W/ch power would be better as it could accomodate a 70Vpk voltage swing.

The reverse is true for a loudspeaker that is ,say, 4 Ohms. Here the power amplifier has to deliver 4X more current into 4 Ohms (as compared to a 16 Ohms loudspeaker) to generate the same voltage swing as we tried to generate with the 16 Ohms loudspeaker (remember voltage = current times resistance). So, if the power amplifier does not have a robust power supply, the amplifier will struggle or fail to meet the challenge. Hence, the oft written & mentioned phrase "this speaker is hard to drive". Looking at it from the power delivered angle, since the current drawn from the amplifier is higher (than the 16 Ohms case), the voltage swing is 4X lower (compared to 16 Ohms case) for the same amount of power delivered by the amp to the loudspeaker because if the current goes up 4X, then the voltage has to go down 4X so that the product voltage times current (which is = power) remains constant. In the 4 Ohms loudspeaker case, one can easily do well with the 150W/ch amplifer because the 50VDC power rails will suffice as the voltage swing is less likely to hit 50V during a cresendo BUT the power amplifier needs to have a very robust power supply otherwise the music playback is going to sound very anemic.

All of the above holds exactly true for a headphone - 32 Ohms low impedance headphones require a headamp (internal to iPhone or external) that can deliver lots of current into 32 Ohms so that the headphone boogie when the music plays. This headamp needs to have only a modest power supply rail such as 12V. OTOH, for 600 Ohms higher impedance headphones you need a headamp that has very large voltage supply rails (such as 20V) to accomodate the large peak-peak voltage swings when the music is playing otherwise the headamp will clip. This particular headamp needs to supply only modest amounts of current into 600 Ohms to get the same voltage peak-peak swing as in the 32 Ohms case. In fact, the headamp driving a 600 Ohms headphone needs to provide 600Ohms/32Ohms = 18.75 times less current.

Most of the time 99% of the public is using a headamp that does not have large voltage supply rails to drive a 600 Ohms headphone & the obviously the results are disastrous! A 600 Ohms headphone is not hard to drive - you just have the wrong sort of headamp for the job! The headamp on-hand is very likely the better candidate to drive a 32 Ohm headphone.....

Hope that this explanation helps to clarify. Thanks.

My head is still swimming!!!!!!

Thank you for your kind words Elizabeth. That was indeed gracious of you....

Bombaywall, thanks so much. I really appreciate it. I admit I had to read your post a couple of times to finally understand everything but I think I got it. Math was never my strong point.

So my new integrated amplifier states in the manual that only headphones with a minimum of 50 ohms should plugged into the headphone jack. So I take it this means that there is not a lot of current available through the amp powering the headphones? But an acceptable power supply?

The one thing that I find confusing though is that most home amplifiers that can handle lots of current are usually expensive. My guess would be because of dealing with the heat generated. But in terms of headphones, an iPhone can easily muster enough current to power a lower impedance headphone such as 16 or 32 ohms. Is it because the amount of current is so small at that point thats not really a big deal? But voltage is because its battery powered?

Hope I got that all right. :) I'm still learning.

So my new integrated amplifier states in the manual that only headphones with a minimum of 50 ohms should plugged into the headphone jack. So I take it this means that there is not a lot of current available through the amp powering the headphones? But an acceptable power supply?

Thanks, glad that I could help some. It's not really too hard to understand once you realize that you have to think in terms of voltage/current & power into the load.

what the int. amp manuf. is telling you is that do not apply a heavy (current) load than a 50 Ohms headphone or else the headphone will ask for too much current that will stress the internal headamp. Nothing's going to happen if you drive a less-than-50 Ohms headphone occasionally but over time, the electronics in the headamp will fatigue. You can go higher in the headphone impedance - 100 Ohms, 120 Ohms, 220 Ohms & keep checking how the headphones sound. As the headphone impedance increases, the current pulled from the int. amp headamp will go down & down but the voltage swing will go up & up. At some headphone impedance, your ears will let you know that it's not longer sounding dynamic enough. Once you find that, you've found the upper limit of your headphone impedance.

The one thing that I find confusing though is that most home amplifiers that can handle lots of current are usually expensive. My guess would be because of dealing with the heat generated.

yes, this is one reason - larger heatsinks get very expensive. There are many other reasons: the transformers get very heavy & physically very large in diameter to accomodate very large amounts of current. Thus, they become very expensive. To accomodate a larger xformer, the chassis becomes bigger. More machined metal, more expense. To handle large current, you need higher amperage recitifier diodes - these are also expensive. Then, you have to heatsink these rectifier diodes really well otherwise their performance derates very rapidly. Again, internal heatsinking takes space & money. This is passed onto the customer.
It's a snowball effect when you make a high current amp.... ;-)

But in terms of headphones, an iPhone can easily muster enough current to power a lower impedance headphone such as 16 or 32 ohms. Is it because the amount of current is so small at that point thats not really a big deal? But voltage is because its battery powered?

That's right. The diaphragm of the ear-bud headphone takes that kind of current that can be supported by a micro-electronics circuit inside the iPhone. As you correctly stated, the voltage peak-peak swing is limited by the battery which can be 4.5V when fully charged & then decays to 3.5V when discharged & everything stops working. The electronics inside an iPhone or iPod Classic is just about good enough to drive the Apple-provided ear-buds but try to drive, say, an integrated amp line-in input (I did that before I got my iPod dock) & it sucked big time! No umph at all & a very flat/flacid playback. If I went a little above 75% of the volume, I'd get distortion thru my speakers (I was clipping that internal headamp) as it was struggling to drive the line-in of my int. amp.

Ok. This is all starting to make sense to me now and I really appreciate your very clear and helpful answers. This has been bothering me for quite a while so it feels good to start to understand it :) Thanks!

Theres's only one thing that I still don't understand. Why is it that an iPod or iPhone won't get stressed out delivering the current required for 32 ohm headphones (apple earbud's or a pair of Grado's for example) but my $3000 integrated amplifier which requires a minimum of 50 ohms has a problem with it? Yet the integrated amp can apparently supply enough current to power 4 ohm home speakers with no problem.

Excellent explanations by Bombaywalla.

A couple of additional points:

1)Another possible reason for the recommendation by your integrated amp manufacturer to use headphones having impedances of 50 ohms or more is that while its headphone amplifier circuit might have no trouble providing the voltage, current, or power required by low impedance headphones, its output impedance may be too high, and vary too much as a function of frequency, to provide good sonics when driving low impedances. That same problem sometimes arises in interfacing preamp outputs to power amp inputs.

If the headphone amp within the integrated amplifier has an output impedance that is a substantial fraction of the impedance of the headphones, and if that output impedance varies significantly over the audible frequency range (as may often be the case), the resulting frequency response will be significantly non-flat, as a result of the voltage divider effect. The voltage that the headphone amp "tries" to put out being divided between the headphone's impedance and its own output impedance. For a given output impedance, the higher the impedance of the headphones the less significant that effect will become.

2)Keep in mind that to produce a given volume level at the listener's ears, the power required by a set of headphones is VASTLY less than the amount of power required by a pair of speakers. Perhaps 100,000 times less, or thereabouts. Headphones commonly produce sound pressure levels of around 100 db in response to an input of 1 milliwatt (0.001 watts). Speakers having typical efficiencies that are listened to at typical distances will require, as a rough order of magnitude, something like 100 watts to produce that same spl.

So a hypothetical 32 ohm speaker, were such a thing to exist, would require vastly more current, voltage, and power to produce a given volume at the listener's ears than a 32 ohm headphone.

Regards,
-- Al

Theres's only one thing that I still don't understand. Why is it that an iPod or iPhone won't get stressed out delivering the current required for 32 ohm headphones (apple earbud's or a pair of Grado's for example) but my $3000 integrated amplifier which requires a minimum of 50 ohms has a problem with it? Yet the integrated amp can apparently supply enough current to power 4 ohm home speakers with no problem.

The electronics for the integrated amp to drive 4 Ohm speakers should be totally different & separate from the electronics that is the headamp. There should be 2 separate circuit boards for these 2 separate functions. The electronics that is driving your 4 Ohm loudspeaker is NOT the circuit driving your headphone. Once you have switched off your int. amp & let is sit for several hours to let the power supply caps discharge, pop open the top lid & take a look inside. You should see a separate circuit board for the headamp.
All the money was spent in making a much better int amp to drive a loudspeaker than a headamp to drive headphones. The headamp was thrown in there as a feature with minimal cost & effort. People who are serious about headphone listening more often than not have a dedicated/separate headphone amp.

Nadman12

Power delivered to headphones P=U^2/R, therefore in order to keep the same power/loudness, with higher impedance headphones, output voltage has to be also higher. Loudness becomes limited either by gain that is too low (tailored for common lower impedance earphones) or supply voltage (that is low in battery operated devices) - whatever occurs first.

Same exists for the speakers but in order to deliver more power (especially with SS amps) speakers are low impedance hence amp has to be able to deliver more current (since power is also P=I^2*R) and sometimes reaches amp's current limit.

I really appreciate all the help! Thanks everyone!