# Why does electricity have a phase?

Anyone who's seen the j in electrical formulas knows that electricity has a phase component as well as a magnitude. But where does the phase come from? Does it only happen in A/C?

Edit: In other words, why is electrical power transmitted as a (sum of phase-shifted sine) wave(s)?

Edit 2: If I were pushing a cart, there would be something like a phase as my legs don't deliver equal power from every position. (But let's assume I push completely the same from a given position.) So there the difference in power delivered from various stances causes phase. But what's the mechanism that causes this in electrical power?

• Because time is a thing. – Ignacio Vazquez-Abrams Mar 5 '15 at 17:43
• @IgnacioVazquez-Abrams What does that mean? – isomorphismes Mar 5 '15 at 17:55
• @isomorphismes: He means that if you sample an AC signal at an instantaneous point in time, it has no phase. Phase only has meaning with respect to time, as shown in the graph in Kynit's answer. – Warren Young Mar 5 '15 at 18:10
• @isomorphismes, in Kynit's picture, he is showing a graph of voltage (at some point on a wire, for example) with respect to time. The axes are very clearly marked. How have you learned about fourier transforms without ever seing a graph of a function with time as independent variable before? – The Photon Mar 5 '15 at 22:13
• It relates to electricity because voltage is an electrical phenomenom. – The Photon Mar 5 '15 at 23:07

Your second edit makes it sound like you're asking about the mechanism behind the phase shift. That is simple: It's designed with three separate windings. This picture is a simplistic representation, and honestly there is more than one way to generate 3 phase power, but these coils are staggered 120 degrees and therefore the current induced by the magnet will be staggered in phase as well.

You could very well generate three phase power (not mains power, but 3 phase power to a motor, for example) with some sort of DAC setup but I think you're talking about what I've linked below.

This website has a neat flash animation that might help clarify the image I've attached at the bottom. http://www.launc.tased.edu.au/online/sciences/physics/3phase/threeph.htm • Oh cool, great answer, thank you! I guess I should go read about DAC / mains / 3-phase / windings … and look up where this picture came from. – isomorphismes Mar 5 '15 at 19:45
• Don't let the DAC comment throw you off, it's more to do with local three phase control (motors in my experience) and not the three phase mains power you're thinking of. I'll edit my answer with a better website. – scld Mar 5 '15 at 20:03
• Ah ok. The flash animation makes it make perfect sense. The driver is delivering different amounts of power when the magnets are closer or farther from the green/red/blue things. – isomorphismes Mar 5 '15 at 22:45
• So is this setup particular to A/C? Or a 3-phase generator could generate any kind of power? – isomorphismes Mar 5 '15 at 22:47
• @isomorphismes: "Mains" is what English speakers outside the US and Canada call the domestic power grid. We'd say "wall power" or "line power" instead. – Warren Young Mar 5 '15 at 23:21

Phase is a difference in time between signals that have the same frequency.

Here is a picture of three sine waves: If you looked at one of A, B, or C at a time without a well-defined starting point, you couldn't tell the difference between them. The only reason why they're different is because they get back to their 'starting point' at different times.

This isn't exclusive to sine waves - you can talk about phase for any periodic signal. Note, however, that every periodic signal is a sum of sine waves, so you're really just talking about the phase of the fundamental.

• Sorry, maybe my question was not very clear. I know what a phase is, but not why electrical power has a phase. – isomorphismes Mar 5 '15 at 18:13
• Oh. Then you're looking for electronics.stackexchange.com/q/64604/49251 - if this answers your question, we can mark this one as a duplicate. – Greg d'Eon Mar 5 '15 at 18:15
• I don't understand that question or the answers to it. – isomorphismes Mar 5 '15 at 18:23
• I really have no idea what you're asking. Your three questions (original + 2 edits) are all asking different questions. You're going to have to be more specific for us to be helpful. – Greg d'Eon Mar 5 '15 at 18:27
• OK, sorry about that. I thought they are all the same question. I understand Fourier domain, sine waves, complex arithmetic, etc. But what I don't understand is how any of that relates to electricity. – isomorphismes Mar 5 '15 at 19:42

Phase is just a way of describing time differences in periodic signals and events. Remember that, from a mathematical standpoint, a true periodic signal is eternal:

$$f(t + T) = f(t), \ \ \ -\infty < t < \infty$$

If you have two (ideal) sinusoidal signals, it doesn't make sense to say that one happens before or after the other. Neither of them really "happens" at all -- they're not distinct events, they're spread out over all time. This is most obvious in a Fourier series, where there's no time variable at all, yet the signal is still completely defined.

Since absolute time is meaningless for periodic signals, the only thing that matters is relative time between two signals, which we call phase. But a relative time longer than the period doesn't really make sense, since $\Delta t \pm T = \Delta t$. So the relative time has to be a number between zero and the period, regardless of what the period is. Instead of messing around with physical units of time, it's more convenient to measure time in fractions of a period. For various historical and mathematical reasons, we've settled on two common fractions:

$$\frac{T}{360} \to 1^\circ$$ $$\frac{T}{2 \pi} \to 1\ \mathrm{radian}$$

So why is this useful? In electricity, it often happens that we have a power source that produces a sinusoidal voltage. Sinusoids are related to rotation, so you can get them from a generator rotating at a constant speed, for instance. Phase comes in in two places:

1. In capacitive and inductive devices, the voltage and current have the same frequency but different phases. In a circuit with several such devices, the voltages at different nodes can be out of phase with each other.

2. You can make a more powerful motor by using several voltages that are out of phase. The principle behind this is called polyphase power.

In communications, sinusoidal signals are used as carriers to allow multiple signals to be transmitted over the same medium at the same time. When working with such signals, it's often necessary to talk about phase, as in #1 above.

Mathematically, a lot of things can be described using the Fourier transform, which represents a signal as a sum of sinusoids at different frequencies, each with its own amplitude and phase. This is helpful because things like electrical circuits respond differently to different frequencies.

(Of course, no signal or process is really perfectly, eternally periodic. But ideal periodicity is a very useful mathematical approximation.)

The complex notation is a compact way to represent sinusoidal signals, i.e. voltages and currents, changing in time in a way that can be described by sine waves of the form $V(t)=A\cdot \sin(\omega t+\phi)$, where $A$ is an amplitude, $\omega$ is the angular frequency and $\phi$ is the phase. So, as you can see two different signals which are even the same frequency can differ by phase, so it is needed to have the representation of the signal to be complete.

• Sorry, maybe my question was not clear enough about where I'm coming from. I understand complex arithmetic and sine waves, but not why electricity is described with them. – isomorphismes Mar 5 '15 at 18:14
• Because the rotational generators are generating this type of signals. Because the resonance circuits are generating this type of signals. Because the radio waves are inducing (and induced by) this type of signals. Sine wave is very fundamental and can be found almost everywhere in the nature. – Eugene Sh. Mar 5 '15 at 18:16
• Ah ok. What do the rotational generators and resonance circuits look like? maybe should have been my question. – isomorphismes Mar 5 '15 at 18:25
• I don't know about "almost everywhere in nature". I've heard this said about groups, linear dynamical systems, n-categories, probability distributions, and any mathematical object you can think of. But it's just as easy to think of things that don't involve [that object]. – isomorphismes Mar 5 '15 at 18:26
• Ok then Is the term "simple harmonic motion" familiar to you? This is how a state of a system that has some internal energy is changing around some equilibrium point. This model is describing a vast number of physical phenomena, like spring-mass systems, pendula, and almost any other system where potential and kinetic energies are involved. This is for mechanics. For waves.. Ok a wave is a sine propagating in space. A wave can be a sum of different sines, but still. In quantum mechanics everything is described as a wave function. So when I am saying everywhere I mean it. – Eugene Sh. Mar 5 '15 at 18:35

I think single electrical signals don't have a phase. But - if we have more signals, or reference (such as time from, for example, time counter), phase is a gauge of relativity. It's used to describe offset between signals and what effects this offset will cause. It's like a car - if you know position - it's not enough. You need for example to know the speed to describe how this car is behaving. But this is my personal consideration.

• A complex impedance will cause a phase shift in a single electrical signal. – Ignacio Vazquez-Abrams Mar 5 '15 at 17:57
• Maybe he meant if you only had knowledge of current OR voltage, as to say of one quantity - you couldn't define phase. So just measuring one thing, not knowing the previous values(for different circuit parameters) and not looking at the circuit you can't know the phase. – WalyKu Mar 6 '15 at 8:53

Electric power is generated by a magnet spinning past a coil over and over. The output oscillates because the magnet generates more power when it's closer to the coil and less when it's farther away.

• That would describe the oscillation itself, but not phase. – dext0rb Mar 5 '15 at 22:57
• @dext0rb What's the difference? – isomorphismes Mar 5 '15 at 23:09
• Phase has nothing to do with the shape of the signal. It is only the time difference between two signals. – dext0rb Mar 5 '15 at 23:41
• @dext0rb ah ok. Right, so it's the motion in a circle that creates the sine wave. – isomorphismes Mar 6 '15 at 0:25
• That is not that right, the speed of change of the magnetic field is related to generated power. So the movement creates a change of field, this in turn induces voltage according to Faradays law. If you have a voltage and a closed circuit, current will flow - through what you get power. – WalyKu Mar 6 '15 at 8:49