Overview - AC power systems operate at a nominal frequency. In the United States, this is 60 Hertz. In other parts of the world 50 Hertz is also common. Figure 1, below shows what a typical 120V voltage waveform looks like when plotted with a frequency of 60 Hz. Notice that the peak of the waveform is actually higher than 120V. It's common to refer to voltages by their root-mean-square (RMS) values. For sinusoids, the RMS value is just the peak value divided by √2.
Figure 1: A 120V Power Supply in the United States
But why is the waveform a sinusoid? Well, there a several good physical and mathematical reasons for that. First, the physical reason: a sinusoid is a waveform that will naturally come out of a generator when rotated with a constant frequency. This means that getting something other than a sinusoid out of a generator may require a less-than-ideal design. Second is the mathematical reason: Sinusoids are what we call "eigenfunctions" of linear systems. What does that mean? Basically, if we feed a sinusoidal waveform into our electrical power systems (cables, transformers, etc.) we get sinusoids out. This is convenient to design around and easy to engineer.
Other Waveforms - This begs the question, what happens if we put something other than a sinusoid into our power system? What if, for instance, we used a square wave (like in Figure 2) for the basis of our power systems?
Figure 2: A Square Wave Voltage Source
If the electrical world were made out of pure resistances, then the results would be just like our sinusoids (square wave in, square wave out). However, the real world is full of inductances and capacitances that distort waveforms as they pass through. Only a sinusoid will exhibit steady-state behavior that is still sinusoidal after passing through a system with inductances and capacitances. After passing through a transformer ( a component with a high inductance), our output waveform may look very similar to Figure 1!
Figure 3: A Square Wave Input to a Transformer Leads to a Sinusoidal Output
The Fourier Series - Why does this happen though?
It turns out that any periodic waveform can be written as the sum of sinusoids. For something like the square waveform above, it will require an INFINITE number of sinusoids added together with different frequencies and magnitudes to approach the square wave shape. Figure 4 is the exact formula that is required to get a square with with a peak value of 1/-1.
Figure 4: The Fourier Series for a Square Wave
Notice that each term in the sum decreases in amplitude as the frequency increases. This makes sense; otherwise producing a square wave voltage (even with a slow frequency) would be really hard to do and require something to oscillate REALLY quickly! Additionally, notice that the frequency of the sinusoids is always a multiple of some fundamental frequency. This is very important to understand. It means that the dominant frequency, the frequency of the square wave itself, is the lowest frequency in the system.
This explains why the square wave going into a transformer comes out looking sinusoidal. The higher frequency components of the square wave are more drastically attenuated by the transformer's inductance, leading to an effect we call filtering: Higher frequency terms from our Fourier Series disappear while lower frequency terms remain. If we passed through a capacitive system, the reverse would be true: high frequency terms would remain and lower frequency terms would disappear. Combining resistors, capacitors, and inductors in various combinations can create all kinds of elaborate and useful filters.
We could (and many mathematicians do) write elaborate Fourier series equations for all kinds of waveforms. However, that's not that important to understand the big picture. The key takeaways are as follows:
A periodic waveform can always be written as the sum of sinusoids
The fundamental frequency of a waveform is the lowest frequency component in the system. All higher frequency components are multiples of this frequency.
Inductances and capacitances create filtering on periodic waveforms, with certain frequencies experiencing greater attenuation than others.
Practical Applications - So now that we know about Fourier Series, how does this help us in the real world? To understand that, let's return to WHY our power systems use sinusoidal sources for AC power: predictability and quality in design. However, a lot of components in our power systems are not so linear. Things like rectifiers, that convert AC to DC, or inverters, that convert DC to AC, do not work so nicely. They produce undesirable sinusoidal terms of higher frequency, known as harmonics. Harmonics distort our sinusoidal AC waveforms and can lead to undesirable impacts on our power system. Some examples of the problems caused by harmonics:
Induced voltages in otherwise safe equipment
Improper tripping of relays and protective devices
Overheating of equipment
And these are only at the lowest level. These problems can easily cascade into much bigger issues. IEEE 519 sets standards for harmonic content (ratios of harmonic voltage and current levels to fundamental levels) that must be met for designs. Many times, manufacturers will also adhere to IEEE 519 standards for products.
Engineers need to know when harmonic are likely to occur in a power system, the level of harmonics produced by equipment, and ways to mitigate their effects. Generally, this means designing filters and placing them in strategic locations in the power system. A power system without the right harmonic mitigation strategy can be bad news for everyone involved!