# Why three-phase power? Why not a higher number of phases?

Is there a reason, beyond historical reasons, that three phases has become the dominant number of phases?

I am aware of the advantages against one phase and two phase, namely the reduced amount of conductor needed, and that motors can provide torque when stalled (and less pulsation).

Is this solely due to diminishing returns, with only a small increase in smoothness of torque application, at the cost of increased complexity (increased number of wires (albeit of smaller CSA)).

To be clear, the phases are all evenly distributed, that is, five phases separated by 72 degrees.

• Very (!) similar: electronics.stackexchange.com/q/12851/930 – zebonaut Aug 12 '15 at 9:47
• @zebonaut: yes, they both talk about three phase, but here the similaritites kinda stop... – PlasmaHH Aug 12 '15 at 10:21
• @PlasmaHH Agree that the questions are put a different way, but the explanations head into the same direction: Three phases, with the angles spaced equally across 360 deg, is the most basic system possible if one wants to achieve (rotational) symmetry. Don't get me wrong: I didn't want to say "duplicate!", what I wanted to say was "something worth reading over there!". – zebonaut Aug 12 '15 at 11:25
• That is the basis of this question, I said I knew why we use 3 phases over less than 3 phases. I wanted the reasons for not using more. – Hugoagogo Aug 12 '15 at 11:29
• Three phases is the minimum number you can have without having "dead" spots in the cycle. – Hot Licks Aug 16 '15 at 12:59

In addition to PlasmaHH's answer, industry uses almost exclusively three phase power since an induction motor needs at least a three phase supply to start and run in a known direction. Single phase induction motors require lossy, unreliable, and expensive tricks to do the same (extra windings, lossy windings, speed sensitive switch, capacitors, etc).

The supply grid is based on three phase since that is the most efficient in terms of generation and delivery. Using a 9 phase grid for example would require running 9 wires for the entire distribution grid, not cost effective.

The higher order motors mentioned don't use line generated phases. Stepper motors use more phases for finer control. High order polyphase rectifiers are designed often with more 'phases', to reduce ripple, but the phases are generated locally by phase-shifting the line input by some means, either direct LC shifting, or by using a motor-generator set.

• Re rectifiers with lots of phases - for big equipment (2,280 kW hoists) I've mostly seen the phases being derived from a multi-winding transformer, which is very efficient. Using a delta-delta-star (Dd0y5) transformer will turn three phases into six phases. Most of the time when I've seen a motor-generator set it's to turn AC into DC. – Li-aung Yip Aug 12 '15 at 11:51
• Those are also common transformers for feeding a large VFD with regenerative capabilities. For regen capabilities though, one winding is generally provides about a 5% step up to the incoming line to allow for dumping excess power, – R Drast Aug 12 '15 at 13:20
• Your first statement is incorrect. 2 phases 90 degrees apart can also run a motor in a predictable direction and with constant power. Two phase quadrature power is also not inherently less efficient to generate. There are of course other reasons 3 phase power is used, but your answer misses those points. – Olin Lathrop Aug 13 '15 at 22:36
• Thanks I decided on this answer as it described, a good number of possible reasons where more phases could be required. Also @Court Ammons, answer made me realize that mathematically there is no improvement in motor smoothness, 3 is already an optimal case (wolframalpha.com/input/…). – Hugoagogo Aug 14 '15 at 9:58

When you have single phase power distribution, you need one phase and one return, both carrying the same current.

If you now instead use symmetrical three phase power, you use three phases with a third of the current carrying capability, and you can get rid of the neutral. This simply saves some money in copper. If you now add more phases, you can not save any more copper, but only add complexity.

If you have asymmetrical three phase power, you can not get rid of the neutral, but it does not need to be able to handle all the combined current of all three phases in return. Again some copper saved. Adding more phases though will not reduce the copper needed for the neutral that much.

So yes, in the end it is more cost for virtually no gain in the average application. So you will only find more than three phases for very special things.

• Would you have any comment on what these special applications may be – Hugoagogo Aug 12 '15 at 9:22
• @Hugoagogo: I have seen 5 phase for stepper motors and 12 phase for high power DC rectification, and then there are historical experiments that might still run on other varieties... – PlasmaHH Aug 12 '15 at 9:26
• Yeah, but 3 phases will have 400 V voltage between the wires, not 230. A single 100 A wire will deliver 40kW with that voltage. – Dmitry Grigoryev Aug 12 '15 at 15:38
• @PlasmaHH Why would the voltage to the ground matter if it doesn't carry any current? You can see a one-phase line as two phases with half phase-to-ground voltage if you prefer. In that case a 230V * 100 A line will transfer 46kW with two lines. – Dmitry Grigoryev Aug 13 '15 at 8:33
• @DmitryGrigoryev: because GND is a convenient and (almost) arbitrary point in every circuit that we use as a reference for easy calculation. In the symmetrical case you can as well call any of the phases GND and then calculate it, but since then voltage and current are not in phase with the delivering lines over the resistors in delta configuration, this makes calculation much harder, so we rather look at the rms values of the delivering lines, all referenced to a common point. – PlasmaHH Aug 13 '15 at 9:25

Three is the lowest number of phases which are equally spaced around the circle, and which can be used to create a rotating magnetic field in a given direction.

Any more phases just require more wires, and more windings in an induction motor.

Two phases can set up a rotating magnetic field if they are 90 degrees apart ("quadrature"). Quadrature-generating tricks like run capacitors are used with induction motors that run off single phase power.

Two phase power turns out not to have advantages. Motors run more smoothly on three phases, and balanced two phase requires four conductors whereas three phase requires only three. That is to say, we can link a three phase generator with a three phase induction motor using exactly three wires. Three-wire two-phase is possible, but it won't be balanced. Two of the conductors will carry the phases, and the third conductor acts as the neutral. This means that one wire has to handle more current since it is acting as a return for the other two. The three conductors under three phase all carry the same current: they are balanced.

For all these reasons, three phases represents an optimum. If it is a given that electricity is used for induction motors, more than three phases is wasteful, and so is fewer than three.

However, two-phases systems have been used, as well as higher order phase systems, like six and twelve phase, continue to be because they have some special advantages.

• By two-phase system, do you mean quadrature, or are you referring to U.S. split-phase wiring with two anti-phase hot wires and a neutral between them? – supercat Aug 13 '15 at 17:51
• @supercat Quadrature. I had the split-phase/two-phase distinction in there at some point; guess I didn't save that edit! – Kaz Aug 13 '15 at 18:42

The main purpose is that having at least three phases allows your motor to start in expected direction. For one-phase induction motors some workarounds are necessary (like putting additional wiring with a capacitor used during the start-up). It was correctly explained in previous answers.

Why not more? Simply - it is not necessary and it generates costs. It's not only the problem of wires (so use of copper, insulation) but also construction problem. Can you imagine a tower for overhead lines having nine phases? Well, probably you can - sometimes one can meet towers that hold two 3-phase lines, or even more:

(pic from Wikipedia)

The main problem here is to secure proper insulating distance between conductors and conductors and ground (or tower structure), which requires large use of materials.

Also, if you have more phases, the chance of failure is higher. Of course, in this case (say - a broken conductor) the total asymmetry will be lower, but a risk of necessity of switching off the entire line will be higher.

Building a generator for more phases is also complicated. Typically, hydrogenerators, with small speed, do have many pole pairs, so it would be ok not to give 24 pole pairs, but one or two (for example, for 12 phases), but it is complicated for thermal generator-turbine units. There is usually one pole pair, sometimes two. This leads to speed 3000 rpm (for 50 Hz network). It is necessary for the stator to receive power from such a machine with the lowest risk possible, so less phases means less in-turn short-circuits chance. Introducing more phases would require much more expensive stator construction.

Please also note, that even if today it is no problem to have a power electronics frequency converter, also multiplying phases, rectifying etc., it was a problem only 30 years ago, and more of course. Then people decided to use three phases, and now it is impossible to switch.

Why only 3 phases? Well if we need more phases we can convert 3 phase easily into 6 phase/12 phase etc using a transformer wired to do so. The main application of more phases is for less ripple voltage into a full bridge rectified capacitor bank. I've never seen one but learned about them from an ancient lecturer at university while doing electrical engineering.

Also lets say we had a delta configuration of 3 matched resistors connected to a 3 phase connection. The power used over time will be identical to a DC powered resistor because when one phase is at 0% the other two phases will be at 66.66% & 33.33% if I remember correctly. This relationship also means that the power from one phase will return down the other phases. Isn't 3 phase awesome!

So to summarize, there's no need for additional phases because you can very easily convert it to more phases at your end. It's typically not done though as 3 phase is already awesome.

Hope this helps.

• Only person to mention that if you want more phases you can do it (in a balanced manner) with at least 3-phases, making more phases a bit redundant and expensive. – user1512321 Aug 12 '15 at 20:02
• Indeed, 6 phase transmission lines are common, but tend to be referred to as 3 phase, dual circuit. Conversion between 3 and 6 phases is trivial. The 90 degree phase shift required to make 12 phases is not much more difficult, requiring only a transformer with one star winding and one delta winding. My brother works on distribution networks and this actually caused a problem once: If legacy equipment introduces the 90 degree shift in one supply, it cannot be used as a backup for another without the shift, due to incompatible phases. – Level River St Aug 12 '15 at 21:30

Three phase has a very important property: if you look at power (V^2/R) across all three phases and sum them, that power is CONSTANT across the entire cycle. This means 3 phase motors can drive at a constant power and the generators see a constant load. 2 phase is insufficient to get this relationship.

One could use higher phase counts, but it costs more to wire, and would not really offer any additional advantage in most situations. 3 phase is chosen because it is a minimum number of wires with good properties.

• Two-phase quadrature could achieve such a relationship. The biggest problem with two-phase quadrature in many applications is that it requires a return wire to carry more current than the "hot" wires, while three-phase feeds the same amount of current through all three wires. – supercat Aug 13 '15 at 22:16
• Nifty! I never knew it worked with two-phase as well! Thanks! – Cort Ammon Aug 13 '15 at 22:44
• @supercat (Or you could do two-phase quadrature with separate return wires, which is identical to four-phase, and therefore wastes a wire compared to three-phase.) – immibis Aug 14 '15 at 13:09
• It's easy to see that quadrature modes could give you constant power -- $\cos^2\theta+\sin^2\theta=1$. – Landak Mar 26 '16 at 18:21

Many of the other answers erroneously state that you need 3 phases for a motor to start reliably or turn in a specific direction, and to use constant power. Actually, this could be done with two phases, 90° off from each other. You still get defined direction and constant power draw over a cycle.

However, such a two phase system would require a minimum of three wires, but the current thru the three wires would not be symmetric for a constant-power load. So if you need three wires anyway, what's the best way to use these three wires as efficiently and flexibly as possible? The answer is the three-phase system we actually use. Instead of one common and two "hot" lines 90° out of phase, you have three symmetric hot lines, each 120° out of phase from the other two. Note that the average voltage (and current for a balanced load) is always 0 for a symmetric 3 phase system. This is not true of a 2 phase system.

More phases doesn't give you any additional desirable properties, so would just add complexity and cost.

A voltage is, definitionally, between two conductors. If you have one conductor, you have no voltages. No voltage, no power, nothing happens. Not terribly useful.

If you have two conductors, you have one pair (2C2), which allows for one voltage. We call this single-phase. Now we can actually make things happen, which is a substantial advantage over having only one conductor. But you can only make one thing happen; there is no possible variation in how the load can be connected. Put another way, there's only one dimension to the voltage: it's positive, or it's negative. One common problem is that if you hook up a single-phase motor directly to an AC line, you have no guarantee about which way it will spin, or if it will at all.

If you have three conductors, you have three pairs (3C2), which allows for three voltages. We call this three-phase. Now we can make three things happen, at different times. For example, you could have three electromagnets arranged in a circle and turn them all on in a sequence. Now we can guarantee that a motor will rotate, and in which direction. This is a substantial advantage over single-phase. Put another way, we now have two dimensions to the voltage; it's represented by a vector in a two-dimensional space. There are only two possible distinct arrangements of conductors ((3-1)!), which corresponds to the two possible directions of rotation.

If you extend this to four conductors, you have six pairs (4C2), so the next step is six-phase voltage. What advantages would six-phase have over three-phase? Well, now there are (4-1)! = 6 possible distinct arrangements of conductors, which means that if you're trying to cause something to rotate in a plane, you could hook things up in a manner that is inconsistent with that. So if you had a six-winding induction motor, it would be possible to hook it up in a manner that would vibrate horribly and rotate at half the normal speed, rather than just pick one direction or the other. That's not a plus.

But suppose that your rotor had three degrees of rotational freedom instead of one. With six-phase, and an appropriate mechanical arrangement of magnetic poles, you could induce rotation (roll, pitch, and yaw) in a floating spherical rotor of fixed position. Since such a thing does not exist to my knowledge, this does not really qualify as a useful application. (Maybe in a null gravity environment, where the magnetic poles are orbiting some body? But then, how are they all hooked up to the same six-phase AC line?) Of course, in a four-dimensional space, where we could have such a system and still translate all three directions of rotation to some other load outside our spherical stator/rotor arrangement, this arrangement could be hella useful.

Meanwhile, back in 3+1-space, I work in the world of industrial power electronics, and I've seen systems that use the kind of phase-shift transformers other answers have mentioned. As a matter of nomenclature, nobody I've talked to would describe using a phase-shift transformer to generate three more out-of-phase AC legs to be creating "six-phase". (By my math, you'd have fifteen-phase, but that's still not the language used.) When running three-phase through a rectifier into a cap, you get six pulses of current per cycle. For this kind of system, you'd get twelve pulses, so that kind of system would be called twelve-pulse.

(In general, the twelve-pulse rectifier is two six-pulse rectifiers. If you have two motor drives, you can connect their DC busses directly together and feed each with a different three-phase set. Or you can get a stand-alone rectifier for one set and feed its DC input into the remaining drive.)

If you're comparing a six-pulse rectifier to a twelve-pulse rectifier, with identical loads, each current pulse must be smaller to compensate for there being more of them driving the same load. This makes the overall current out of the line look somewhat more like a sine wave, meaning the harmonics are reduced. Ripple on the caps is also lower, but I've never known anyone to be terribly concerned about that.

Greater harmonics improvements can be had with an eighteen-pulse system and three rectifiers. (36-phase!) At higher voltages and powers, even higher numbers of paralleled rectifiers may exist. This document on a medium-voltage VFD line references a 54-pulse rectifier at 11 kV!

TL;DR

Three-phase power gives us one rotational degree of freedom, which is the limit of what is useful in a three-dimensional space.

Another simple reason: Additional phases would be "two similar" to the existing ones. Put differently: Any additional phase would simply be a linear combination of the voltages among the existent three wires - the vector space spanned by sine and cosine is just two-dimensional.

Another aspect of the problem is the matter of conductor geometries for high-tension transmission lines. With three lines, the problems of inductance and induced crosstalk currents are minimized and more easily filtered, than if there were an additional multiple of conductors. The costs keep going up faster than the benefits with more conductors.

• Its been known for over 100 years that transformer magnetisation make mainly 3rd harmonic and same with AC motors 3 phase is best for supressing 3rd harmonic which would be more sensible than say 5 or 7 phase – Autistic Aug 13 '15 at 2:17

Lionel Barthold, founder of Power Technologies, Inc., explained this well:

He says that although he has designed higher phase systems, they are not practical due to, as you say, diminishing returns, especially with regards to all the more transformers needed at the substations. When you double the number of phases, you also have to double the amount of equipment at the substations.