I originally posted this on another forum (iFans) here: http://www.ifans.com/forums/showthread.php?t=364522
I do think that it is an important concept to get down for a headphone forum, so I am going to repost here as well. Some disclaimers before I begin, I have double checked the units and they do add up to dB when multiplied out (stage by stage), so that part is accurate. However, I still have to warn that these results may not be 100% accurate and will only give you a ballpark estimate at your actual listening levels. The best way to get the actual number would to get a dummy head and an SPL meter (kind of expensive). This method requires a calculator, and a little understanding of high school physics.
On the other forum I did include a quick and dirty command line program (Windows only, sorry Mac and Linux users) that would calculate it for you granted you inputted correct information. I don't know what the rules are like here concerning executables, so I won't upload it here. It is uploaded on the other forum if you need it (it comes with source code to be compiled for other machines; use a C-based compiler).
Again, no guarantees of 100% accuracy, but it will get decent numbers to give you a good estimate. Let me know if you have any questions. This guide also assumes that your volume bar increases voltage at a parabolic rate (what I found most devices to do: phones, iPods, laptops, etc).
Ideal listening levels should be around 60-80 dB. I generally listen myself around 70 dB (calculated). So how exactly do we calculate this (estimated) listening level? Well it's simple, first we need some information though:
- Maximum voltage output of your device (search google for this), the iPod Touch 4G puts out a maximum of 1.110 v (60 mW max or 30 mW per channel).
- Headphone specs (specifically the impedance and the sensitivity ratings) - Find it on the box of your headphones, or online through the manufacturer's website (or even the reseller's site sometimes)
So let's start by calculating the actual voltage that comes out of your headphone jack. When you push up on your volume, you actually don't increase in a linear, straight manner. Instead, it's squared. To get the voltage your increasing use this formula below:
The next thing we need is actually the current going through your headphones. The current is generated using the formula below for current through a parallel circuit.
Actual Voltage (volts) = Max Output Voltage (volts) * (volume level fraction)^2*
The volume level fraction is actually the amount of the total volume you are outputting. So if you listen at half the max volume of the iPod, your volume level fraction is 1/2 = .5. If you listen to it at 30%, then the fraction would be 30/100 = 3/10 = .3. If you listen at 3/8, then your fraction would be 3/8 = .375. It's important that you keep this a fraction under 1.
With the voltage and current we can calculate power generated by your device at a given volume level. This is why we calculated the current and voltage. So use the equation below:
Current = Voltage * (1/R1 + 1/R2 + 1/R2 + 1/R4...); where R1, R2, R3, etc are resistors. Your headphones have two of these both rated the same. So we can simplify that whole statement to:
Current (amps) = Voltage (volts) * (2/R); Where R = Impedance rating (ohms) of the headphones. Thusly, we get the equation:
Current (amps) = Voltage (volts) * 2 / Impedance (ohms)*
Now to get the actual sound pressure level (SPL) coming out of your headphones (note this is in Pa), we can use the formula below:
Power = Current (amps) * Voltage (volts)*
Or if you want the long one:
Power = (Voltage (volts) * 2 / Impedance (ohms)) * (Max Output Voltage (volts) * (volume level fraction)^2)
Now, this number is going to range anywhere from 0 to 15 (maybe higher if you have special headphones). We need to convert this number into one we can use. To convert the SPL (Pa) to SPL (dB) to get the decibel rating we want you can do one of two things: Use this converter*
SPL (Pa) = Sensitivity * Power*
Or the long one:
SPL (Pa) = Sensitivity * (Voltage (volts) * 2 / Impedance (ohms)) * (Max Output Voltage (volts) * (volume level fraction)^2)
Or you can use this formula:
SPL (dB) = 20 log(SPL (Pa) / (2*10^-5)) / log(10) (dB)
*We are stating that the log has a base of ten when we divide by log(10). On any given calculator, the log button has a default base of 10 (check this for your self), that is, log(10) = 1. Ln is the natural log which has a base of e. In the equation above, log can be replaced with ln, but you'll have to divide by ln(10) still since ln(10) is not 1.
So we get the final equation as:
SPL (dB) = 20log(SPL (Pa) / (2*10^-5)) dB*
Plugging in what SPL (Pa) is, you get the long equation:
SPL (dB) = 20 log(Sensitivity * (Voltage (volts) * 2 / Impedance (ohms)) * (Max Output Voltage (volts) * (volume level fraction)^2) / (2*10^-5))
^^If you do this equation, don't type it in wrong And double check your work).
Please note you'll get more accurate results (since you'll make less human errors) if you just calculate each thing and right down the number it produces and plug those into the small equations (which are marked with a star, *, at the end of them.. The big equations scare me (especially the last one)... If you are using the bigger equations, just skip to the final equation; plug and chug it all down and hope you didn't type anything in wrong
And there you have it, the actual listening level you are listening at (approximated). Now let's see how scary your numbers are... Happy (safe) listening everybody.
Sources used: http://www.innerfidelity.com/content/loud-music-sucks