Figuring out the very dBSPL coming out of DAPs

Compared to other two devices, iAudio U5 is way cheaper.

I am going to figure out the sound pressure level of these DAPs driving Ultimate Ears Triple.Fi 10 Pro, which I bought from Amazon.com for $99.

@ full volume (0dBFS), their maximum outputs are 717mV, 431mV, and 339mV respectively.
(as constant-voltage sources, and assumed to be free from any distortion components at this level)

And Triple.Fi 10 Pro specs:
•DC resistance: 32Ω
•Sensitivity: 117 dB SPL/mW @ 1 kHz
   
Now a simple math:
I = V / R
P = V * I
10logP = resulting gain

Thus considering the sensivity @ 1 kHz, the SPL of
U5 (at full volume) driving Triple.Fi 10 Pro should be:
120 dB + 12.1 dB = 132.1 dB SPL @ 100 Hz
117 dB + 12.1 dB = 129.1 dB SPL @ 1 kHz
113 dB + 12.1 dB = 125.1 dB SPL @ 10 kHz

HD60GD9 (at full volume) driving Triple.Fi 10 Pro:
120 dB + 7.6 dB = 127.6 dB SPL @ 100 Hz
117 dB + 7.6 dB = 124.6 dB SPL @ 1 kHz
113 dB + 7.6 dB = 120.6 dB SPL @ 100 Hz

MZ-RH1 (at full volume) driving Triple.Fi 10 Pro:
120 dB + 5.6 dB = 125.6 dB SPL @ 100 Hz
117 dB + 5.6 dB = 122.6 dB SPL @ 1 kHz
113 dB + 5.6 dB = 118.6 dB SPL @ 100 Hz

Fair enough, these devices are well capable of driving Triple. Fi 10 Pro, at least in terms of output power. Above results are derived by assuming the devices are of reference-quality, having extremely low output impedance. Of course, conventional DAPs have relatively high output impedance in practice. Then let us see how the output impedance of each DAPs affects Triple.Fi 10 Pro's output.

With the courtesy of my comrades, Salsal and asRay, I was able to obtain the following data.

U5:
@ 100 Hz: 1.9 Ω
@ 1 kHz: 1.4 Ω
@ 10 kHz: 1.3 Ω

HD60GD9:
@ 100 Hz: 11.4 Ω
@ 1 kHz: 10.7 Ω
@ 10 kHz: 12.8 Ω
(note: impedance reaches 42 Ω @ 20kHz!)

MZ-RH1:
@ 100 Hz: 8.8 Ω
@ 1 kHz: 8.5 Ω
@ 10 kHz: 9.6 Ω

This graph, measured by Mr.Fuchinove, shows the following impedance characteristics of Triple Fi. 10 Pro:

@ 100 Hz: 32 Ω (DC resistance)
@ 1 kHz: 60 Ω (resonant frequency? possibly due to the crossover network)
@ 10 kHz: 6 Ω (quite low. due to the multiple drivers coupled in parallel?)

Then with a help of simple formula, acquired from Audioholics.com,

the effects of impedance interaction between the source devices and Triple. Fi 10 Pro can be figured out easily. (Triple. Fi 10 Pro's reference cable has a DC resistance of ~1 Ω)

Considering the reference device(source+cable=1.1 Ω) driving Triple.Fi 10 Pro:
@ 100 Hz: -0.29 dB
@ 1 kHz: -0.16 dB
@ 10 kHz: -1.46 dB

U5 (at full volume) driving Triple.Fi 10 Pro:
@ 100 Hz: 131.6 dB SPL
@ 1 kHz: 129.0 dB SPL
@ 10 kHz: 123.7 dB SPL
Other than -1.36 dB at 10 kHz, no problem with U5.

HD60GD9 (at full volume) driving Triple.Fi 10 Pro:
@ 100 Hz: 125.1 dB SPL
@ 1 kHz:  123.2 dB SPL
@ 10 kHz: 111.7 dB SPL
Impedance interaction seriously skews the upper frequency range, -8.91dB at 10 kHz. Furthermore, at 20kHz, the attenuation is -13.6 dB! (Triple.Fi 10 Pro's output barely reaches 16 kHz though)

MZ-RH1 (at full volume) driving Triple.Fi 10 Pro:
@ 100 Hz: 123.6 dB SPL
@ 1 kHz: 121.5 dB SPL
@ 10 kHz: 111.2 dB SPL
Not as much high frequency deviation as Kenwood, still -7.38 dB at 10 kHz.

Then, we can compare these results to the frequency response(the official data from Ultimate Ears) of Triple.Fi 10 Pro driven with a reference source:

As output impedance rises, the output of TF10 starts to be underdamped, and follows its own input impedance characteristic for maximum energy transfer. We can all conclude that, even with high-end portable devices, you can never avoid Ohm's law. :-D IEMs with multiple transducers are just like speakers- you must drive them with a low impedance voltage-source to prevent the loss of damping.

[From Stereophile: UE-10Pro , UE-11Pro]

(IMO this source dependency could be the reason why none of multi-driver IEMs goes over 15 ~ 16 kHz. As this article suggests, although voltage drive effectively eliminates the damping issue, it has some fundamental problems at higher frequencies. With high output impedance source, speakers can be velocity-driven and may reach higher frequency range. However, as we have seen, this is not possible with multi-driver IEMs. Rather, their HF attenuates greatly, due the reduced impedance possibly resulted by the multiple drivers in parallel.)

How about IEMs with a single transducer? Well, I know my ER-4B is free from such problem and reaches well over 15 ~ 16 kHz limit, because it was originally designed to be equipped with a serial resistance of 100 Ω, raising voltage-to-current ratio(amplifier's output following its own output impedance curve, independent from the load) and underdamping the transducer to achieve the target response.

Now, for the last but not the least, here comes Replaygain.

Originally, the reference model was K-20 when RG was first proposed, but somehow K-14 was implemented instead.

U5 (at full volume) driving Triple.Fi 10 Pro with RG applied:
131.6 dB - *14 dB = 117.6 dB SPL @ 100 Hz
129.0 dB - *14 dB = 115.0 dB SPL @ 1 kHz
123.7 dB - *14 dB = 109.7 dB SPL @ 10 kHz

HD60GD9 (at full volume) driving Triple.Fi 10 Pro with RG applied:
125.1 dB - *14 dB =  111.1 dB SPL @ 100 Hz
123.2 dB - *14 dB =  109.2 dB SPL @ 1 kHz
111.7 dB - *14 dB =  97.7 dB SPL @ 10 kHz

MZ-RH1 (at full volume) driving Triple.Fi 10 Pro with RG applied:
123.6 dB - *14 dB = 109.6 dB SPL @ 100 Hz
121.5 dB - *14 dB = 107.5 dB SPL @ 1 kHz
111.2 dB - *14 dB = 97.2 dB SPL @ 10 Hz

(*this calculation, 89+14, is only a simple representation of how RG works. The actual RG calculates rms value of music, and matches the value to the equal-loudness curve inversely)