The formulas presented here are only useful if we are talking about DC, or direct current. Since our sound is not a constant stream of electrons, but rather a sophisticated waveform, we need to upgrade our equation. So, if the formulas we have work for DC, and our sound is a complex, sophisticated waveform, the first piece of information that we are missing is the details about the waveform. And here is the part where things get complicated. It's not practical to work with these complex waveforms when trying to get basic information about a system—like the power output of an amp. So, we need to be smart about it. Instead of working with real-world waveforms, we will use something very similar. We're going to use a sine wave. Why? Well, in most cases, it is the most similar waveform to the real-world one when we play our instrument. And not only that—it's also very simple to represent and work with mathematically. So, let's ask ChatGPT about some physical elements of a sine wave:
A sine wave is a smooth, periodic oscillation that describes a repetitive variation, often used in physics, engineering, and audio. It is one of the simplest and most fundamental waveforms, mathematically defined by the sine function.
Great. Since I don't want to go super-detailed on the sine wave here (you can do it by yourself if you feel curious), I will just highlight some additional things that we need to know about sine waves.
Sine waves are periodical, and to describe them, we need to know the period at which they oscillate (or the frequency they oscillate at, or the length of the waveform), and we need to know the amplitude. We are not really interested in phase at the moment. Plus, we need to know how these values correlate to the DC values we were using up until now. Let's take a look at a picture of a sine wave:
As can be seen from the image, what is labeled as A corresponds to our amplitude. It is measured in volts, and it is similar to our DC voltage value, but not quite the same. Unlike the DC voltage, which is constant in time at a certain amplitude, the amplitude of a sine wave goes up and down over time.
Now, why is this important? Well, if we want to measure the power of a certain system, we need to understand that a constant amplitude will do "more" work than an oscillating one. So we need to add a few more values here.
To simplify things a little bit, we need to accept that if we want to do the same amount of work, therefore produce the same amount of power as DC, we need to use an RMS (root mean square) value of our sine wave. Which, as you can determine from the picture, is a bit lower than the peak value of the amplitude. Basically, this accounts for the fact that, because of the periodic nature, the sine wave doesn't generate the same amount of heat as DC does. Therefore, we need a bit higher amplitude with sine waves so that the power generated would be equal to the power generated by DC. This is very important for what's coming next.
Oh, and by the way, did you know that usually, when we talk about AC values, we talk in RMS values? The 230 volts or 120 volts are actually RMS values, which means that the actual swing of the waveform (if you were to look at our line voltage on a scope) is quite a bit higher—around 325 volts in Europe and around 170 volts in the States.
Ok, so back to our talk about power. In our basic formula, where P = I * V, we now understand what the V value would be—it would be the RMS value of the amplitude of our sine wave. But things do get even more complicated.
Remember that to describe a sine wave, we also need to know the frequency of the wave? So, what is the frequency?
Well, the practical, real-world waveform that is actually present while we play an instrument is a complex waveform. And this waveform, being complex in nature, is comprised of many waveforms of different frequencies. For audio, we usually only think about the frequency range between 20Hz and 20kHz, and for guitar amplifiers, this bandwidth is even narrower (in most cases). Usually, when we work with audio amplifiers, we use a 1kHz sine wave for testing purposes. Practically, we measure the RMS amplitude at 1kHz and we use that value in our power formula. And yes, this is how it's done. This is how the whole industry does it. And these are the values that you see in the specifications of your amp.
For those who were paying close attention, it is clear now that, in practice, the power output of an amplifier at 1kHz tells us very little about the amplifier. Because the amplifier can have a different power output at 100Hz or at 10kHz. So the power output that the industry serves us is only relevant to us as a relative value when compared to other units of a similar type. Realistically, it tells us very little about how the amp sounds, or how we will perceive it.
Alright, before I move into some real-world numbers, hi-fi, Marshalls, and all the fun stuff, we need to talk about one more piece of the puzzle. It's something called THD—total harmonic distortion. The mathematical procedures for producing this value are very complex; therefore, we will not discuss them here. Rather, we will oversimplify this and say that THD is a number that tells us the percent of distortion present in our amplifier. Rather simplistic, but let's go along with it.
Now, why is this important? Well, we were talking about some industry standards and specifications. Let's take an example. Let's say that we are looking at an amplifier, and the specification says that it is rated at 100W at 1kHz with 0.01% THD. That basically means that the amplifier was measured reproducing a 1kHz sine wave with very little to no distortion and delivering 100W of power to the speaker. Amazing. In guitar player world, this means 100W of clean mids. So, yet again, we know quite little about the amplifier.
Ok, now that we understand all these numbers which we can see in amplifier specifications, let's talk about some practical values that we run into in the real world. Let's say we have an 8-ohm speaker, and we plug the speaker into a 100W amplifier. The amplifier needs to be capable of producing a certain amplitude (voltage), and it needs to be capable of delivering a certain amount of current to the speaker so we achieve this 100W that the amplifier is rated at. We can calculate these values:
\[P = V \cdot I\]
and
\[V = I \cdot R\]
and if we combine these two formulas, we get:
\[P = \frac{V^2}{R} \]
From this formula, we can derive that the voltage (or amplitude) needed to achieve 100 Watts of Power into an 8 Ohm speaker is
\[V \approx 28.3 V\]
Keep in mind that this is actually the RMS amplitude of a 1khz sine wave!
Great, now we can get the amount of current that is going to flowing through this circuit:
\[I = \frac{V}{R} \]
\[I \approx 3.53 A \]
This is all good stuff. Now we know that our amplifier needs to generate a nice and clean 1kHz sine wave, with an RMS amplitude of approximately 28V, so it would transfer 100W of power to the speaker. And this is the actual measurement that is listed in the specs. This is the measurement used as an industry standard when measuring amplifier power. Unfortunately, there are many special cases that require special attention. First of all, there are many different amplifier architectures. Many of us know that most tube amps have output transformers, and different outputs (that magically need to be matched to the speaker), and many of us also know that transistor amps don't have that, but they have a "minimum" rating. I'd say this topic alone requires a separate paper, and it will get one.
What I would like to do here is actually stay focused on measuring the power in guitar amplifiers. The main reason for that is the fact that guitar amplifiers are not really used in a "normal" way. I don't really think anyone expected the early guitar players cranking those amps and distorting the crap out of them. But it happened nonetheless, and it kind of dictated how amplifiers were designed in the future.
So I want to get right to the point. Let's take a Marshall Plexi-style amplifier, my favorite example. That amplifier is loaded with 4 EL34s, it's rated at 100W, but somehow this particular amplifier seems to be so much more powerful than many other 100W amplifiers, especially compared to their 100W transistor-based counterparts, and I want to explain why.
Firstly, I need to mention that I will not be getting into speaker efficiency—this is a topic for another time. We will assume that we are comparing amplifiers on the same speaker cab. Older amplifier designs, much like the Marshall Plexi-style circuit, didn't really have volume controls that came "at the end" of the preamplifier. That came later, with the first Master Volume 2204/2203 designs. The volume control on the Plexi is more like a gain control, and the amplifier is wide open. Whoever played a good Superlead or something knows that the amp is already extremely loud when the volume pot is at 2. It really depends on the amp, but most early Marshalls would be nice and clean around 2, but if you go past 2 on the volume pot, the sound gets progressively louder and more distorted, and obviously more glorious. But what's the deal with the power measurement here? Well, basically...
Early Marshalls are 100W amps, and they deliver 100W of "clean" - when the volume pot is at ~2.
If we go full naughty and go past 2 on the lead channel, the volume increases, as well as our excitement and our neighbor's rage. But what actually happens is that our signal starts to compress, and eventually distorts. This scenario completely wipes out all the measurements I was doing earlier. It simply doesn't work like that anymore. Now, we absolutely love Marshalls for the way they compress and distort, and that is how we play them. But if we were to compare that type of usage to a hi-fi amplifier scenario, it is completely not normal. We would never use a hi-fi amplifier as distorted as we use guitar amplifiers. With guitar amps, we are basically constantly punishing them, and they are designed to be punished.
Going back to our example with the sine wave. If we ran a 1kHz sine wave into a Marshall and went past 2, the wave immediately starts compressing, and eventually distorting and transforming into a full square wave. But the interesting thing is, that the amplitude does not grow. It cannot. Because of the limitations of the amplifier design. The peaks simply hit a ceiling and can't go anywhere from there, other than to change and distort. But the question now is: if the amplitude doesn't grow, and the speaker impedance is the same, why is the amplifier getting louder as if it has more power?
The answer is actually very simple. We discussed the RMS value, which we need to do a proper power calculation. The RMS value we discussed is only for a sine wave. The problem is that when distorted, we are not working with a sine wave anymore. There are different formulas according to which we derive the RMS value for different waveforms. Since in our case we are moving towards a square wave, let's explore the RMS amplitude value of a perfect square wave:
The RMS (Root Mean Square) value of a perfect square wave is equal to its peak value. This is because, for a square wave, the magnitude of the waveform is constant (either at the positive or negative peak) over each half-cycle.
\[V_{\text{RMS}} = V_{\text{Peak}}\]
This basically means that the RMS value of a perfect square wave is actually higher than the RMS value of a perfect sine wave. So, to make the new calculation for the power output, we need to determine what the Peak value of the sine wave was that we were working with:
\[V_{\text{Peak}} = V_{\text{RMS}} \cdot 1.41\]
\[V_{\text{Peak}} = 40 V\]
And now, to calculate the power output of the amplifier when it's playing a sine wave, we use this new value:
\[P = \frac{V^2}{R} \]
\[P = \frac{40^2}{8} \]
\[P = 200W \]
Now, obviously, this is an extreme. A lot of amplifiers' power sections will sag before the amp reaches this value, meaning there are some real-world limitations. Plus, our real-world signal is never a perfect square wave, but the more distorted and compressed it gets, the closer it gets to a perfect one. However, it remains a complex wave, making this calculation approximate. So the ultimate answer to our question about the power output of the Marshall Plexi in question really depends on how much we distort it. It never quite gets to 200W, but I measured 170W on my 1976 Marshall Superlead.
One more thing to mention is that even though we use these amps in such a way and they sound great, not all amplifiers are designed to sound good when distorted. Many transistor amplifiers don't sound good when distorted, and their power sections are actually designed to stay super-clean at all times. That's why tube amps appear to be louder—it’s because we are used to playing them pushed over the limit, and we’ve accepted that as normal, and it makes us happy.