I must say the other posts on this topic are bang on. I am going to expand on this subject a bit, hopefully without getting too lengthy or complicated. If we start by just looking at this relationship in regards to its effect on the speakers it may help to simplify the equation. There are two ways to damage a speaker, one is pure overpowering and the other is clipping. Fortunately both of these will manifest as audible distortion prior to damaging the speakers. The key is to turn it down right away if distortion is detected. Overpowering of a speaker is a combination of frequency, time, and power.
Essentially the higher the frequency the speaker is asked to reproduce the better it will be at sustaining high amounts of power. This statement can be skewed if the tweeter x-over point is too low or too linear, as the lower frequencies the tweeter can sustain are much higher than the woofer. Therefore in designing a speaker it is important to monitor this, especially since there can be (not always) off axis advantages to keeping the tweeter x-over point low.
By there design conventional speaker motor assemblies are excellent at reproducing very high amounts of power for very short periods of time. This allows for the reproduction of music, or movies, with all of their varying levels without being limited to the short bursts of high power that are required, and all this assuming the designer has been diligent about the mechanical capabilities of the speaker.
Sustained high power will at some point, depending on the speaker, over heat and burn the voice coil or damage some mechanical part of the speaker component. Sustained high power tends to be where most speaker companies get their power ratings. The basic test is modified pink noise for 8 hours at a given power. Though it is nice to have a standard the reality is that unless the requirement for the short burst of power is also considered the rating can be somewhat meaningless in the real world. At Axiom we have two separate tests to encompass both of these considerations in order to come up with our ratings.
As for the amplifier the situation is somewhat the same. Amplifiers are rated at their ability to produce a stable input signal without clipping. Since the real world involves a myriad of various input signal levels this rating can be a little bit meaningless, though at least a standard benchmark. The amplifiers ability to produce high power for short periods of time without clipping can really boost the power available in the real world.
So to tie all this together and get back to the original question, unless you have reached the mechanical or heat maximum capability of the speaker, the only and most common way to damage a speaker is through amp clipping. This is when the amplifier is asked to produce a level of power beyond its capabilities. This capability can be reached quite quickly in amplifiers that have limited ability to produce bursts of high power for even short amounts of time. So there is no amplifier power which is too low for the speaker, but you may be quite limited in the overall volume your system can achieve without clipping. Relating this back to impedance does not change any of the above equation. The lower the impedance of the speaker the more it will demand from the amplifier. Some amplifiers can produce quite a bit more power when presented with a lower impedance. By continuously lowering the impedance though at some point it will get so low as to overheat the amplifier. Obvious 0 ohms would be a dead short to the amplifier. As to how low an impedance a given amp can sustain or how much extra power you will achieve by lowering the impedance of the speaker is a question that has to be answered by the amplifier manufacturer or by measuring this data from the amplifier in question.
I hate to say it but this is the short answer!
Ian Colquhoun
President & Chief Engineer
