Science and Math for Audio Humans – Integration and Power
by Danny Maland
It's disclaimer-time again: Everything that I set before you should be read with the idea that “this is how I've come to understand it.” If somebody catches something that's flat-out wrong, or if you just think that an idea is debatable, please take the time to start a discussion via the comments.
Before we move away from talking about power delivered to loudspeakers, there's one more topic I want to cover. The compact form of the topic is that “power is effectively the area under the voltage curve.” What this naturally brings up is integral calculus, because mathematical integration is the finding of the area under a curve.
Now, don't worry. We're not going to get into actually doing integrals on various waves, mostly because I myself can't do them “by hand” in a meaningful way. I don't have the background. However, we can talk about the implications of the concept, and how it works in a general sense.
The first thing to define is what “under” a curve means. For our purposes, “under the curve” means “between the curve and the x-axis.” This is important to understand, because one's first instinct is to define “under” as the area from the graph, all the way down to the bottom of the page. However, this wouldn't give you much in the way of meaningful answers – you'd get “negative infinity” all the time, if you think about it. No, what we're concerned about is how far away our graph/ function/ voltage/ whatever is from “0,” or any other reference point that we might choose. Figure 1 shows the area under the curve for the function y=x. We're only looking at x from -1 to 1, for the sake of simplicity.
(By the way, for generating graphs like these, the tools at wolframalpha.com
Now lets look at Figure 2. This is a sine wave that peaks at 2 volts. We're looking at it from 0 to 0.001 seconds (or 1 millisecond), and we can see that one full cycle of the wave has occurred. In other words, it's a 1 kHz sine wave. I've highlighted the area under the curve.
Figure 2 helps us to visualize, say, the power being delivered to a loudspeaker by an amplifier. You can see that, if the voltage delivered is time variant, so to is the power derived. At each “zero crossing” (three being visible in Figure 2), the voltage delivered to the loudspeaker is zero, and so there is no power being delivered to the loudspeaker. However, that is only for those instantaneous points in time. At every other point in the wave cycle, some amount of voltage is generating some amount of power. In the same vein, maximum power is generated at the peaks of the wave – but only for an instant. The majority of the time, the loudspeaker is subjected to a voltage that is both non-zero and non-peak. This is why pro audio types are so keen on “continuous power,” or the not-correctly-named but accepted-anyway “RMS power.” Continuous/ RMS power ratings give us a meaningful way to predict what we might observe from a power amplifier delivering voltage to a loudspeaker.
Now, let's examine the curious case of a power amplifier being driven into VERY hard clipping or limiting. In the case of clipping, we get a whole bunch of harmonic distortion products being added to our desired signal. In the case of limiting, especially if the limiter is well designed, large amounts of harmonic distortion are avoided while the audibility of the limiter's activity is also minimized. In either case, the output signal becomes more and more like a square wave. Figure 3 shows our Figure 2 graph with a square wave added on.
What you should immediately notice is just how much more area the square wave has under its curve. As the power amplifier is driven harder and harder, the continuous power delivered to the loudspeaker more and more closely approaches the peak power. As opposed to a condition where most of the voltage is non-zero and non-peak, the “mathematically ideal” square wave spends all of its time at peak voltage, instantaneously switching from positive to negative-going amplitude.
Side note: Mathematically ideal square waves aren't possible with power amplifiers and speakers. We'd need infinitely fast and powerful power supplies, as well as a way to make the loudspeaker cone move instantly from positive to negative displacement. We'd also have to remove the modern protection systems from our power amplifiers that keep us out of really hard clipping. Still, we can get “plenty close enough” to be dangerous...
The major point in all of this is to be aware of the assumptions being made between the rated continuous power output of an amplifier, and the rated continuous power of a loudspeaker. From the standpoint of preventing thermal failure – that is, not overheating or “cooking” your loudspeaker components, the thing to realize is that the continuous power rating of an amplifier is not actually “maximum power deliverable.” Instead, it is “the DC (direct current) equivalent power deliverable with distortion characteristics deemed to be tolerable.” The amplifier is actually capable of delivering a good deal more than its rated continuous power, if one is willing to accept distortion and limiting as a side effect.
This is one of the roots (there are other important ones, of course) of the myth of “underpowered amps cause people to blow up loudspeakers.” People want more SPL than a system can safely deliver, so they drive their amplifiers into clipping, and then cause thermal failure of their loudspeaker components. They then wrongly conclude that clipping is “dangerous” without respect to the power being delivered, and advocate for larger power amplifiers into the same loudspeakers. The reality is that the amplifier connected to their loudspeakers became too powerful as the RMS voltage approached the peak voltage. They kept asking for more area under the curve, and the amplifier obliged – to a point, of course.