Why do we represent A/C currents with sine waves?

Question

Why is it harmonic in nature? I am starting to learn about alternating currents and somehow I can’t get the fact that it’s described by a sine wave and not by any complicated waveform. Is there a reasonable explanation for this? Or do we just rely on it for simplifying things? Also, how are alternating currents different from variable currents?

Answer 1

I am not an engineer, but when one talks about 'alternating current' they mean a specific type of current which is made to vary sinusoidally, for example by rotating a magnet along a wire. Edit: This type of current is useful for certain situations (see this question).

Answer 2

In general, an alternating current need not have a sinusoidal waveform, it can have a different periodic waveform, like for instance a square wave. However, there are two principal reasons to describe AC using sines. The first is mathematical, the other technical:

• using Fourier series, you can describe any periodic function as a (potentially infinite) sum of sines. The square wave for instance can be written as $$f(t)=\sum_{k=1}^{\infty} \frac{\sin(2\pi (2k-1)t)}{2k-1}.$$ This means that if you know what happens for a sine signal, you can often simply apply this to any other periodic signal.
• if you have the simplest AC generator consisting of a conductor loop rotating in a uniform magnetic field, it will generate a sinusoidal signal. In reality, the signal generally won't be perfectly sinusoidal, but for that see point above.

Answer 3

Alternating currents are a peculiar types of general time-varying currents: they are time-periodic harmonic signals. They are used for practical reasons and that can be easily described in frequency domain, since they have (ideally) only one contribution at the "fundamental" frequency of the harmonic signal.

Alternating current ideally have intensity that is a harmonic function of time. This type of regime is characteristic of many applications, since electric voltage and current are usually "produced" ( or better conversion, from other sources) using rotating mechanical components (gas, steam, water turbines, reciprocating motors, ...), and "employed" with rotating components (electric motors, ...). With harmonic signals, it's quite easy and cheap to change voltage and current intensity using transformers.
I say "ideally" because some minor contributions could enter the electric system, produced by minor physical phenomena or interference. If they give minor contributions, you can easily recognize a harmonic signal with some noise/perturbations.

When these mechanical components rotates at constant speed at regime, you get an harmonic behavior of electrical quantities. During transient phase of operation (start/stop) the rotational speed continuously changes and the frequency of the electric variables changes as well, producing a non-harmonic "complex" signal.

Anyway, you can produce ideally any kind of electrical signal you like, both

• non-periodic: think at the charge of a discharge of a capacitor and the exponential trend of the electric quantities;
• periodic: with wave form you like, as an example a square wave, or a "quasi-direct" current from a alternating current through a rectifier.

In these cases, when you deal with linear systems, you can

• use Laplace analysis for transient signals;
• use Fourier analysis for periodic signals, to decomposed it in a sum of its harmonic contributions, to study the harmonic response and the "filtering" behavior of the system itself

Answer 4

You've got serveral questions here... let's try to unpack them (reprahsing them as I understand them)

Why is AC (almost always) a sine wave?

AC is a sine wave function because this is what electromechanic generators produce. It's a function of the physics that goes on...

Why is the sine the basic waveform

If you look at how "sine" is defined you can see that it's derived from a monotnonically rotating point on a circle. As long as its velocity does not change it is the most monotonic movement. So a sine has basically "the least amount of change possible" because of this.

You could use other ways of consturcting arbitary waveforms than sine. But sine have some really nice properties:

• if you add several sine waves of the same frequency (independently of phase shift and amplitude) you get a sine wave as output
• integrating and differenciating of a sine waveform just shifts phase but it's still the same waveform
• Nature seems to "like" sinusodial waveforms. A pendulum swings in a sinusodial pattern for example as do many other things.
• if you put a signal above the cut off frequency trough a single stage low pass filter you'll get a sine wave out (it just removes all the other, higher frequency, parts of the waveform)