DT 770 250Ω are quite hard to drive, substantially harder than something like the Sennheiser HD600, and I would think you probably should be using something more powerful than the single ended out of a KA3, yes.

You could mod them to balanced, or get a more powerful desktop amp. The Douk U3 at $30 might be a good option, it does 1.3W and you can use the KA3 as a DAC into it.

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Thank you everyone for your advices! The existance of helpul ppl in this community make me belive in humanity!

Now that iv'e understood that the KA3 isn't enough, and modding the headphones is out of the table, is there any portable amp-dac combo/ amp that could drive the DT770??

It depends what a person means when they say headroom. There is a big difference between having enough power for your average db vs having enough power for transients. There is no need to have “headroom” more than you need for transients but in the OPs case it clearly does not do that.

If OP listens at 80 db average at 90-97% volume then the peaks in the music will be cut off. Depending on the type of music you listen to you could have anywhere for say 15-30 db transients. The rub is that the power needed is not linear with the increase in db but instead exponentially. So where you may only need 50mW of power at 80db you will need 150mW at 85 db (I made up those numbers but a headphone power calculator can tell you exactly what they are depending on your headphone).

My suggestion would be to use a headphone power calculator and input your resistance (ohms) and sensitivity (db/mW) and then see how much power it needs at 115db. I typically listen at around 80-85 so this covers me for an additional 30-35 db of “headroom” for transients. That way even if you typically only listen at 70 you know you are covered if you want to really go crazy for a bit (but be sure to limit your time doing this if you do).

It is also worth noting that if you plan to EQ (and you should if you can because why not) that you will need to take into account your negative preamp into the equation. So it you listen at 80 but have a negative preamp of -5db you will need to include that in your power calculation by adding another 5db to your listening level.

So if people mean they want headroom so that they can listen at 95db with full transients for 1 song well then maybe? But otherwise there is no headroom needed past what your headphone requires to fully show transients. Just another warning that while listening at this kind of level for a short time is “ok” but definitely don’t do it for long periods of time and you can always see what OSHA says as far as exposure at certain levels that is a good reference.

It's scientific that you can push an amp into clipping if it doesn't have the current, although it more typically happens with lower impedance headphones than higher ones.

Volume controls voltage and with low impedance headphones you usually can turn the volume up higher than the amp's capability to supply current, if you do this you do get clipping and a huge spike in distortion.

You can see this in the 16Ω measurements for the KA3 unbalanced, and the 32Ω as well with the balanced. It's not apparent in the 68Ω or 300Ω though, that goes to max voltage without clipping, although distortion does start to rise again above 20mW (not to the level it would likely be audible).

https://www.l7audiolab.com/f/fiio-ka3/

Headroom is simply having more power than you will need. Powering a 25 watt rms speaker with a 100 watt rms amp leaves 75 watts of headroom. If that amp is putting that 100 watts rms out with .005% THD. Imagine what the THD is if the amp is only putting out 15 watts. Probably not measurable at all. Definitely is not audible. So there is plenty of headroom for distortion also. Which is the important one really.

Have you paired the 250 ohm DTT70 with the btr5? I'd imagine it won't be enough so I might as well get the 80 ohm one