What are the pros/cons of adding more coils or getting a stronger magnet for a speaker?
Are there other methods to make them louder keeping the power supplied to the speaker the same?
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Buy a new one, that has a better efficiency (in amateurish: higher dB value). This is not a bigger is better scenario. Speakers are delicate devices, needing precious balance to provide the quality, not just the power...
What specifies how loud the speaker is going to be? The main point is: louder means more air moved
Also volume is one thing. Balanced acoustical properties is a completely different one. You can make a super loud speaker at a given frequency - that is practically not able to produce any other frequency than that.
Actually, the mechanical limitations of the speakers lead to designing the 2, 3, and other multiple way systems. That is not just putting together speakers and hoping for the best: it is a science in its own.
Taking these into count, there is one thing left out to make a speaker louder, but still useful for something: designing a box, that emphasizes the right frequency band. By using the Thiele-Small parameters, it is possible to do so, and there are quite some design programs helping designing a box for certain applications - mostly enhanced bass. However, this is not an "enter values - magic happens - build box - instant goodnes" - sometimes boxes that are good on paper are horrible in life...
That's an interesting limitation; to keep the power supplied to the speaker the same. It leaves only:
If you remove coil turns, the force is reduced, and so the speaker is quieter. But the mass is reduced, so the speaker is louder. The balance will be lost, and so the frequency response will be changed beyond normal.
If you add coil turns, the force is increased, and so the speaker is louder. But the mass is increased, so the speaker is quieter. Again, the balance will be lost.
If you increase the magnetic field strength, in order to increase the force, the frequency response will be shifted.
If you change the cone at all, changing the mass or volume, the frequency response will be shifted.
All these changes mean retuning of the whole system. You will need many test instruments for the task, and a lot of learning. A new speaker may be cheaper.
If you are just interested in a small band of frequencies at (say) 1kHz, you can parallel tune the speaker to be electrically resonant at 1kHz. Say for instance it is an 8 ohm speaker, it might have somewhere in the region of 1mH leakage inductance. To tune this at 1kHz would require a capacitance of 25uF. Use a 22uF ceramic and feed the output amp to the parallel circuit via a 3.3uF.
You'll probably have to experiment with the tuning components but you are bound to find a resonance somewhere but, don't drive it too hard because you'll probably break the cone or melt the windings. You might even get the magnets so warm they lose their magnetism.
You get better power transfer out of a speaker, and hence better power output and better power efficiency, by fitting a Horn to it. This gives you a horn speaker, used where better efficiency and power output is required.
It doesn't normally use exactly the same power -- when you improve the power transfer characteristic, you change the imput impedence characteristic -- but you get more power output connect to the same amplifier.
You are still limited by the cone stiffness. Now that we have electronic amplifiers, horn speakers normally use button drivers, so they can be very stiff and drive a lot of power. And you are limited by the non-linear nature of air: when you put too much power through it, it starts to distort. And you are limited by wind noises through the horn: the air rushing past the surfaces gets loud if you have too much power.
A horn is a flared tube. The flare can be exponential, or you can approximate it with a simpler shape. At the input end, the air loading is very stiff and heavy, which loads the speaker driver more efficiently. Along the length of the tube, the input impedance is transformed into one which better matches the impedence of open air.
Like everyone else, I can't see what you can do to the speaker it's self with out causing damage but you could change the signal going to it.
As no speaker creates a perfect recreation of the signal so you could process the signal before sending it to the speaker without it changing the overall sound too much.
As you can guess if the signal is louder the speaker will be louder. However you can't just increase the signals size with out damage the speaker so you need to increase the size that is perceived when the speaker creates its representation of the signal.
This can be done by inserting a audio compressor (or limiter) to reduce the size of the signal's peaks followed by a amp to boost the signal. Reducing the peaks will cause odd harmonics to be added (or in other words squaring of the wave form). When the signal is recreated the odd harmonics add to the frequencies of the original signal and create a sound that is perceived as being louder.
If the compressor/limiter reduces the signal by 3db and the amp then boosts by 3db the speaker output should seem as though it is 3db louder