∫∫ Three-Phase Power

We established in a previous post why it is necessary to have AC power – to reduce transmission losses – but why do we have three-phase power? Why not just one phase, like the inside of a house? Wouldn’t it be cheaper to run just one set of power lines instead of three? Wouldn’t there be a third less labor involved in electrical work if there were a third the wires? What is so important about three-phase power that we go through so much cost and effort to utilize it?

It all comes down to power. Let’s look at power in an AC Circuit. If the current and voltage waves are perfectly lined up, power simply equals the current times the voltage, as seen below. (The x-axis is in radians, and shows one complete cycle, or 2 Pi radians. In the US power system, there are 60 of these cycles every second.)

See a problem here? Current and voltage cross zero twice a cycle. If there are 60 of these cycles per second, which there are in the United States, the power from a single phase AC circuit drops to zero 120 times every single second. That means everything essentially turns off 120 times every second. Your toaster briefly stops toasting 120 times a second, your dishwasher briefly stops washing 120 times per second, and your lights turn off 120 times per second.

Thankfully, most of these changes are not visible to the naked eye. The coils of wire in your toaster remain hot even if there is no power in them for a split second and the tungsten burning away in your incandescent light bulb does not cease to be red hot because there is no current flowing through it for a one-thousandth of a second. It does flicker a little bit, which is one of the reasons natural light is more pleasant than artificial. In fact, the choice of 60 hertz for the US power grid was based on the minimally accepted flicker rate of light bulbs. (The English had a high tolerance for flickering lights due to their superior beer, so the UK grid operates at only 50 hertz.)

Okay, so the power stops for a tiny fraction of a second, but none of this matters since we can’t tell. Right? Well, kind of. For most everything in your home, it doesn’t matter, and if you’ve ever noticed the utility pole next to your house, only two electrical wires connect to your home from it. (If there are more you probably have a phone or cable TV.)

Constantly accelerating and decelerating power does matter, however, for large industrial operations and even for most operations in commercial-scale buildings. Imagine for a moment that you are a piece of equipment and a motor is powering you. You could be a compressor or a pump, for instance. It would wear you down badly if that motor drove you and then stopped and then drove you again 120 times per second. If instead the motor just pushed you at a constant, steady rate, you would last much longer and be able to do your job with much higher consistency and quality. This is exactly what three-phase power accomplishes.

Each phase of three-phase power is 120 degrees apart. The 120 degree separation is cleverly chosen to make the sum of all three-phases at any given point constant. This is easier to see then to describe, so below is a graph of three-phase power. You can see the power delivered by each phase, and in the top of the graph the sum of each phase combined.



The black line is the same as the single phase power in the earlier graph, and the red and green line are power in the phase 120 degrees ahead, and 120 degrees behind. Note that the purple line, the sum of the three lines below it, is constant. That is the beauty of three-phase power! Any motor, pump, fan, refrigerator, recycling plant, office building, or manufacturing plant can easily acquire perfectly smooth and constant power from the three-phase grid. With single-phase, or even two-phase power, perfectly constant power is not achievable.

Three-phase power allows us to combine the benefits of AC and DC power: we get low losses in the transmission system due to the use of simple AC transformers, and constant, perfectly smooth power like one gets with a DC circuit! This is why practically every country in the world goes to all the trouble of using three-phase power.




For the ambitious viewer, below is a graph of everything that is going on in three-phase power: each phase of current is represented, each phase of voltage is represented, their resulting power delivery curves, and the sum of those curves is shown below.





18 Responses to ∫∫ Three-Phase Power

  1. This is great for when the power factor is one, but how does it work if voltage and current get offset like is discussed in earlier posts? Is the power factor for all three phases the same or are they independent?

    • This is a fantastic question Emma, which merits an entire post in and of itself. While I unfortunately do not have time to write that post now, I will at least give you a simplified response here.

      The power draw of a motor is based on the load it is serving, and the amps drawn by the motor automatically compensate (to a point) to serve that load. When the amperage and voltage waves get out of phase, which is always the case for an inductive load like a motor, the motor draws more amps to provide the same amount of power. The more out of phase the waves get, the more current the motor draws, but the power drawn remains constant (remember P = I*V*p.f., so as power factor goes down amps go up to compensate to maintain the same power.)

      This works until the voltage and current get so out of phase that constant, or somewhat close to constant, power can no longer be obtained by the motor and it shuts down in a series of jerky false rotations and whimpers. Not a pretty sight.

      Your question has a second part: what happens if the power factor is different in the different phases? A three-phase motor causes the same power factor change in each phase unless something has been done terribly incorrectly with the stator windings, but what if somewhere else in the building one of those phases is siphoned off to power a single phase motor? Then that phase would have a different power factor than the other phases.

      This, in fact, is frequently the case. Given that each phase automatically compensates to provide a constant amount of power based on the load it is serving, however, what we find is that the amps flowing in each phase may be different from each other but the net power draw of the motor is constant. This is why it is important to measure the current and power factor of all three phases, instead of just one.

    • brenden says:

      This is a great point. It’s actually that the other two phases combined act as ground for the first one. If you look (carefully) at the last graph, you can see the sum of all three currents at any given point is zero. This means that current in one phase returns through a combination of the other two.

  2. Thanks, Brenden, this is the first time that I’ve seen an explanation of why we use three-phase power. I have two questions:

    1. How are the three phases combined in a simple socket like the one that I plug my toaster into (as opposed to a three-phase motor)? Are two just wired together (for ground as per your response to Dev)?

    2. One can’t help but see the symmetrical beauty in the three phases, but you could have two phases adding up to a constant if they were a quarter cycle apart (sin^2 + cos^2 = 1). Other than the ugly lack of symmetry, is there a more practical reason not to do it that way?

    • brenden says:

      Hi Susan, great questions here!

      1. We actually have a whole blog post planned on this exact issue, but the simple version of it is that your house only gets two phases from the utility pole, 180 degrees apart. A normal socket in your house just has one of those phases and a ground, giving you 120 volts. Your washing machine socket has both phases, allowing you to get 240 volts. No three-phase power in a house though.

      2. It’s a very clever realization that a sine wave shifted by a quarter cycle becomes a cosine wave, and it then makes sense to wonder why we can’t get constant power from the two waves since sin^2 + cos^2 = 1. Power, however, is simply the sum of the waves, not the sum of their squares which is what is constant in the equation above. Sin(x) + Sin(x + some.phase.change) will never be constant, no matter how cleverly we attempt to line up the phases.

      • I thought power was current * voltage and those are in-phase sine waves (as per your first illustration on that page), so that would make power a square of a sine. Am I missing something here?

        • Susan, great question again. Let me see if I can paraphrase your question: the sum of two sine waves a quarter cycle apart is equal too sine plus cosine. Since each sine or cosine is made up of its individual in-phase sine and cosine waves of current and voltage, two phase power where each phase is a quarter cycle apart equals sin^2(x) + cos^2(x), which we know is one. Why don’t we use two-phase power since we can make constant power from only two phases, and we don’t need three?

          Your assumption above requires the voltage and current coming through each phase to be identical for two-phase power to deliver constant power. Unfortunately, this is never the case. Phases are frequently split off to power different single-phase pumps, fans, and entire buildings, and thus the current and voltage available to each phase is never the same. With three-phase power, you can provide constant power without requiring the current and voltage to be identical in all three-phases. What a great question!

          • Robert says:

            To what extent does the construction of the motor itself also have a bearing on the use of three-phase power?

            With two-phase power, you would have an even number of pole-pairs in the motor windings, which could easily lead to “dead spots” in the magnetic field as the rotor turns within the windings. Three-phase power would be more likely to eliminate these dead spots and keep the rotor turning more smoothly.

            Single phase motors have to use tricks with the windings to keep the rotor moving, which I believe is a big reason for their relative inefficiency. Perhaps you could write a future post on the different types of motors…

  3. Awesome article Brenden, I’d long wondered why the UK and other parts of the world sometimes use a 50hz system.

    I think what you’ve provided is very helpful to understand the implications of three phase power at the load level and from an equipment operation standpoint.

    However, overall my impression has been that the greatest advantage of three phase is that from a total power delivery perspective it is actually cheaper to run 3 wires vs. 2. Is that not the case?

    • brenden says:

      Hi Harris, thanks for the positive feedback. I’ve never heard running three wires would be cheaper than two. Could you direct me to a source on this? I’d love to learn a little more about it.

      • Harris Schaer says:

        Hey Brenden, the best source I found states that since power for a three-phase system is P = I*V*sqrt(3)*pf, with that sqrt(3) term you’re essentially able to support 73% more power, but over only 50% additional wire.

        In concert with that, each individual leg has to support less current than a single-phase system which means the conductors themselves can be higher gauge and thus smaller, meaning less copper, less space, less cost.

        • brenden says:

          Hi Harris, thanks for the clarification, I understand the confusion now. The idea of getting 73% more power from 50% more lines is obviously appealing, but that’s unfortunately not how it works. P = I*V*sqrt(3)*pf in a three-phase system when you use line to line voltage. A better way of thinking power in any system is that it is just the sum of the power running through each phase, where the power in an individual phase is given by P = I*V*pf, where V is the line to ground voltage. Thus, the total power in a three-phase system is given by P = I*V*3*pf, where voltage is line to ground, NOT line to line. The voltage difference in a three phase system between line to line and line to ground is sqrt(3), so the two equations are equivalent. P = I*V(line to ground)*3*pf = I*V(line to ground)*sqrt(3)*pf, but when you look at it based on the line to ground voltage, you see that you only get 50% more power for 50% more lines.

          It’s a very clever argument they’re making. Too bad it doesn’t work out that way!

  4. Hey brenden,

    Great info,

    But I have a question why do they go only for 3 phase system? why don’t they go for 5 phase system for example? won’t that allow us to transfer more power?

    • A 5-phase system would allow for more power, but it would also be 66% more expensive to build out. Utility poles in a lot of places where more power can be carried on the existing three-phase lines now have a second set of three-phase lines on them, however, so in some places we have effectively switched to 6-phase power to enable the transfer of more power. The engineering of it is really two parallel sets of 3-phase power, which is important because most big motors are designed 3-phase power, but it’s not super different than your idea!


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