# Why are AC quantities represented by sine waves always?

Usually we use a sinusoidal wave form to represent a alternating quantity. Why not a cosinusoidal wave or a ramp wave form?

In sine wave forms we can indicate the maximum and minimum amplitude and its variation with respect to time. Ramp waves have the same properties; then why not a ramp wave form?

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Sine and cosine waves are, physically, the most common. They are definitely the best description to what comes out of a wall socket, not because we like them mathematically, but because it's what comes out; electromotive force is generated in the power plant as a sinusoidal pattern with frequency 50/60 Hz. In the usual kind of generator, this is because in the generator, the rotating motion of the magnetic rotor leads to sinusoidal variation of EMF in the winding of the stator and consequently in any circuit connected to the socket. (Note that sine and cosine waves are equivalent, and choice between them is merely convention; the neutral word that can be used to describe the shape is harmonic.)

Even better, if we have some more complicated waveform - from slightly deformed all the way to ramp and step waveforms - then we can use Fourier series to decompose them into a sum of sine waves. We can then study the response of the circuit to each sine wave independently, and add up the responses at the end; this is usually much simpler than using the waveform directly.

The reason this works is because most circuits are linear. That is, if we input some voltage $v(t)$ and we measure some property $p(t)$ of the circuit, then adding a voltage $v'(t)$ will result in the property being $p(t)+p'(t)$. Thus if our complicated waveform $v(t)$ can be expressed as a Fourier series, say, as $$v(t)=\sum_n v_n \sin(n\omega t),$$ and we know (because it's easier to study) that a sinusoidal stimulus $v_n\sin(n\omega t)$ will result in a property $p_n(t)$, then the full waveform $v(t)$ will induce a response $$p(t)=\sum_n p_n(t).$$

This can, of course, break down if you have nonlinear elements in your circuit, such as diodes or overdriven vacuum tubes or transformers. These will behave differently and will induce distortion of your waveform, which may or may not be a good thing depending on what the application is. This distortion is, of course, the same distortion as you get with an electric guitar amp.

A final word on square and ramp waveforms. Fourier treatments of these waveforms are sometimes a bit difficult, and convergence near the sharp edges can be very slow (see e.g. the Gibbs phenomenon). This represents some very important physics of square waves: the instantaneous change in voltage is not actually possible in any physical circuit. This is because if the source circuit has any inductance $L$, the instantaneous change in current $i$ will mean an infinite current change $\frac{di}{dt}$ and therefore an infinite inductance back-voltage. Typical sources have very small inductances, but they can never be zero. The comparison of this inductance and the source internal resistance will give the timescale in which the voltage can suddenly change sign.

If you want a good description of an actual physical square wave, then, you have two choices: you can account for a finite "ramp" time, or equivalently you can simply take a finite number of terms in the Fourier series. The latter is, of course, a lot easier!

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Firstly, a cosinusoid is just a sine wave which has been shifted by a phase of $\pi / 2$. So the two are equivalent ways of describing a wave.

One can describe a wave as a triangular wave, or a square wave or any other wave form. In fact, square waves are used as the input to all sorts of logical circuits and digital electronics.

But a square/rectangular/triangular/ramp/funny shaped wave can always be described as a sum of sines and cosines. The mathematics behind doing that is called a fourier series. So even if one does have a square wave, one can describe it in terms of sines and cosines (which can be easier to work with mathematically) but that may not always be the best way to describe a signal in the circuit.

But in basic circuit analysis and AC signal analysis, we tend to use sine waves because they are the easiest to generate (as compared to square/ramp signals) as well as the easiest to analyze mathematically (all our calculus tools work wonderfully for curves like the sine, but tend to be more inconvenient around things like a triangular wave).

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The crucial mathematical property of sinusoidal waves is that if you add two sinusiodal waves with the same frequency, you get another sinusoidal wave. Square waves, etc., don't have this property. –  Ben Crowell Jul 28 '14 at 16:21

Actually it is the human interpretation that it is a sin wave.

Here the angle θ is the angle between the field lines and a plane perpendicular to armature coil. So by being considering the angle such that, the flux linkage becomes a cosine function and the induced emf becomes a sin function. (Emf is the rate of change of flux and hence it becomes a sine function as the derivative of a cosine function)

The reason why it is represented as a sin wave I think it represents function which changes both magnitude and direction continuously and easy to understand.

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By using sinusoidal waveform we can convert the waveform in to any type of waveform like triangular, square , etc so in my idea sinusoidal waveform is the correct one to express ac quanatities

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Because when you take the sine of 0º to 360º and plot the graph of these values, AC current behaves the same way. It can also be represented by a cosine function, but in this case we assume that the initial value of the AC current shouldn't be zero.

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## protected by Qmechanic♦Jul 28 '14 at 13:29

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