What the heck are Power Factor and Three Phase Power?

I’ve read several articles on both these topics, and I’m more confused now than ever.

Apparently power factor is a measure of inefficiency in AC systems where the current lags behind the voltage or something? Why are they connected at all, besides being multipliable to deliver watts?

And is three phase power at all related to this “lag”, or is it something different? Something about overlapping inverse AC wave forms something something… what?



Say you have a magnet spinning near a big wire. That’s a generator, and it makes alternating current, because the spinning magnet pushes and then pulls the electrons in the wire as it spins around. If you hook that wire up to an oscilloscope, you’ll see a sine wave.

Now suppose you add a second wire opposite the first, so the magnet is now inducing currents in both wires. Because the north side of the magnet is near the top wire while the south side of the magnet is near the bottom wire, if you graph both wires on the oscilloscope, you’ll see two sine waves that are exactly opposite of each other. You could call this two-phase power.

As it happens, if you add these two sine waves together, you get zero. Because when one is at +10v, the other will be at -10v. And when one is at zero, so is the other one.

If you connect a load to each of these phases, and the loads draw the exact same amount of current, the return current would add up to zero. Which means you could omit the return wire completely, saving some money. In real life, the loads are never perfectly balanced, so you need a return wire, but it can be a lot smaller than the supply wires.

Now to add more efficiency, instead of two wires 180 degrees apart on the magnet, make it three wires 120 degrees apart, forming a triangle. Again, assuming you have three perfectly balanced loads, they add up to zero and you can omit the (or in real life, have a very small) return wire.

It turns out that this is a very efficient way to distribute electricity. You get to have three supply wires going to your customers, and a single small return wire for all of them. You could of course do four or five or a dozen phases, but then you’d need more wires. It turns out that three is a pretty good number.

Most residential electrical services in North America take one of these three phases, put it through a transformer to get it down to 240v, and then give you a grounded “center-tap” for your neutral. The two hot wires coming into your house are connected to the two ends of the transformer, so between either hot and the neutral is 120v, and between both hots is 240v. This is actually a two-phase system from the transformer to your house.

For industrial and commercial buildings, it’s more common to have all three phases of power come into the building to supply power to large equipment. By connecting all three phases to a big electric motor, for instance, you can deliver a lot of current. To deliver the same power with a single phase would require a much higher voltage which would require much thicker wires.

I mentioned that in the residential split-phase system, if you connect something between the two hot wires, you get 240v, because they are 180 degrees out of phase with each other. In the three-phase system, if you connect something between any two of the three 120v phases, you get 208v. In a previous thread I made this diagram to explain why.

Now, power factor.

When you connect something like a motor to a power source, a rotating magnetic field is created inside the motor which causes it to spin. (In fact, it’s basically the opposite of a generator.) It takes energy to maintain that magnetic field, and the energy that goes into that magnetic field is not used directly by the motor. (This is kind of an oversimplification.) What this means, basically, is that less power is being returned to the grid while the motor is in use. When the motor shuts off, it dumps all that field energy back into the grid.

Power companies will therefore charge industrial clients with lots of big motors and stuff more money, due to the temporary losses from power factor.

You know that volts times amps equals watts. With AC power it isn’t quite that simple. With AC, both voltage and current exist as sine waves, but they aren’t necessarily aligned with each other. That is, the peaks and troughs of voltage don’t necessarily happen at the same time as the peaks and troughs of current. The way they line up is called “phase” - a phase difference of zero means they line up exactly with each other, while a phase difference of 90 degrees means that voltage is ahead of (or behind) current by a quarter of a wave. (Electrical engineers usually express phase in radians, but I assume you’re more familiar with degrees).

AC power is volts times amps times the cosine of the phase difference. If voltage and current are exactly in phase with each other, the phase difference is zero, the cosine is one, so the power is volts times amps. If current lags voltage by 60 degrees, the cosine is 0.5, so the power is volts times amps times a half. This multiplier - the cosine of the phase difference - is the power factor. AC power is most efficient when the power factor is one - that is, when the phase difference between voltage and current is zero.

(By the way, I’m playing a little fast and loose with the terms “volts” and “amps.” To figure out true AC power, you need to use a kind of average voltage and current. The averaging technique is called “root mean square” or “RMS.”)

Three-phase power and power factor are different things, although they both involve phase. To understand three-phase, you should know that the power you have in your home is single-phase: each power outlet has a “hot” side that delivers power, and a “neutral” side that absorbs it. The hot side delivers one sine wave of AC voltage.

In three-phase power, there are three “hot” sides. They all deliver sine waves of AC voltage, but they are out of phase with each other by 120 degrees. That is, their peaks are evenly spaced out: if the frequency is 60 cycles per second, the peaks of the different “hot” wires will be separated by 1/180 of a second. This type of power is useful because certain types of electric motors run efficiently on it (because it’s possible to arrange the wiring to create a rotating electric field). Three-phase power is used a lot in industry. I don’t know of any home that uses three-phase power.

Industry uses three phase motors because the rotating three phase field has a defined direction, so three phase (or other polyphase) motors will always start and run in the same direction depending on how they are wired.

Single phase motors have no directional preference unless they are designed to be inefficient by shading a pole for example. Single phase motors that can deliver any kind of significant torque on startup need to use capacitors, start/run switches, or other complex electro-mechanical solutions to get them running in the same direction every start.

Not all, but many microwave ovens with turntables have an extremely inexpensive single phase AC motor driving the turntable. If you watch one, you will see that it sometimes starts clockwise, and sometimes counter-clockwise.

Let’s say you have 3 big circuits that you want to run. Each circuit needs a wire going out and a return wire coming back, so you’ve got a total of 6 wires.

Now let’s do the same thing with 3 phase. We take each individual circuit and we delay its sine wave by 120 degrees. Why? Because 3 sine waves that are 120 degrees apart add up to zero when you add them together. This means that if you tie all of the return wires from our three circuits together, it adds up to zero and there’s no current flowing through the wire. If there’s no current, then the wire isn’t doing anything and you don’t need it at all. In reality, the three individual circuits will never be perfectly balanced, and so the return won’t add up to zero, but it will be small. So instead of needing 6 wires like you do above, now you only need 3 big wires and one little one. You save a huge amount in wiring costs.

The power company will run 3 phase from the generators to the transmission lines and to the substations. From there, it varies. If you have a lot of houses that are relatively close together, they may run all 3 phases through that neighborhood. Maybe the first street gets phase A, the second street gets phase B, the third street gets phase C, and so on, splitting the 3 phases down each individual street. Less common, but still in use in a few areas, is that they’ll run all 3 phases through the entire area and each house gets 2 wires from the 3 phase. It’s easy to tell if you have this since instead of 120 and 240 volts in your house, you’ll have 120 and 208 instead.

If the houses are much less densely packed, the power company may split the 3 phases at the substation and run phase A off in one direction, phase B off in another, etc. You lose the benefit of being able to share the neutral return at that point, but sometimes they don’t have enough of a load to justify running all three phases to each area.

Large business buildings are often fed from 3 phase.

Now for power factor.

If you take a coil of wire and run current through it, the coil of wire forms a magnetic field and stores energy. Remove the current and the magnetic field collapses and the energy that was stored in the magnetic field gets converted back into electricity. So a coil of wire is a simple energy storage device. We call this an inductor.

Take two metal plates and put them close together, and they will also store energy, except they will store it in an electric field instead of a magnetic field. This is a simple capacitor.

These energy storage devices as a group are called “reactors”.

Because AC is alternating current and is a sine wave, it cycles constantly. This means that inductors and capacitors are constantly charging up and discharging. They don’t use energy like a light bulb does. All they do is temporarily store the energy and release it later. But, it takes current to charge them up, so it’s a greater load on the generator.

The energy that gets used for things like light bulbs is called watts. The energy that is used just to charge up reactors is called vars, for “volt-amp reactive”.

As it turns out, inductors charge up during the part of the AC cycle where capacitors discharge, and capacitors charge during the part where inductors discharge. This means that you can use one to balance out the other. Most homes are slightly inductive due to things like motors in clothes dryers, refrigerator compressors, etc. The power company switches capacitors on and off of the line at the substation to balance out the inductive vars from the homes. If you get it perfectly balanced, then the vars all add up to zero, and what happens is that the capacitors discharge their energy into the inductors and then later in the AC cycle the inductors discharge their energy back into the capacitors, and the vars basically just go back and forth and back and forth between the inductors and capacitors. Meanwhile, the generator only has to supply the watts. Since the generators don’t have to supply the vars, this makes the power system much more efficient.

How well balanced the vars are is called the power factor, and it ranges from 0 to 1. A power factor of 1 means that the vars are perfectly balanced. A power factor of 0 would be all vars and no watts.

Most people have heard of watts since that’s what the power company charges you for, and it’s what light bulbs are rated by (or used to be). The power company doesn’t charge you for vars, so most folks have never heard of those.

If you are a business or industrial customer, the power company will charge you for vars, and they charge out the wazoo for them, too. For this reason, industrial customers with lots of motors and such will usually add their own power factor correction capacitors on site to balance out their vars. For business and industrial customers, the “bend over and squeal like a piggy” charges don’t usually kick in until the power factor drops below some particular number, usually around 0.7 or so. Basically, the power company charges you a huge amount for vars just to force you to add your own power factor correction.

You sometimes see devices designed for home use that claim they will save you money with power factor correction. These devices are just capacitors. Since they are constantly in the circuit, they don’t properly balance out the vars, so they don’t actually work. But even if they did, the power company doesn’t charge home users for vars, so these devices can’t possibly save you money.

Oh, one more thing.

Lead and lag refers to the voltage and current in a reactor. This unfortunately requires a bit of hairy math to explain.

Voltage and current in an inductor are related by this formula:

v = L di/dt

Voltage and current in a capacitor are related by this formula:

i = C dv/dt

(you could also express those as integrals, but that would require me looking up the code for the integral symbol).

For those not familiar with calculus, the derivative of a sine wave is a cosine, and a cosine is just a sine wave shifted by 90 degrees.

This phase shifting means that, in an AC sine wave system, the current leads the voltage by 90 degrees in a capacitor and the current lags behind the voltage by 90 degrees in an inductor.

The basic mechanism behind this integral/derivative relationship is the energy storage, magnetic or electric field, depending on whether it’s an inductor or capacitor.

Is that true? I understand that it takes current to charge them up during one part of the cycle, but then AFAIK they just throw that current back at the generator during a different part of the cycle. Since the energy is just boucing back and forth, I don’t think it’s a higher time-averaged/net load on the generator simply because of the increased current.

The problem, as I understood it, is one of increased dissipative/ohmic losses in the whole power transmission network as a result of the larger current associated with crappy power factor. If your power factor is 0.75, then the current between the power plant and the customer is 1/0.75 = 33% larger than it really needs to be. Ohmic losses in the wiring are I[sup]2[/sup]R, so increasing the current by 33% means the ohmic losses in the system increase by 77%. Basically, low power factor means energy is being wasted as it bounces back and forth between the customer and the power plant (and power transmission system capacity is being wasted) , and the power company will make you pay for that waste.

Also, motors aren’t the only cause for poor power factor; large installatons of fluorescent lighting can also cause problems.

Let’s say you have 200 amps of real current and 50 amps of reactive current. The total current load on the generator will be 206 amps and will be 14 degrees out of phase (if I did my math right).

So yeah, it’s not like it adds up to 250 amps, but it does increase the load on the generators. And of course also the I[sup]2[/sup]R losses increase, etc. as you noted.

Another reason utilities like 3 phase power is that they can supply 73% more power with only 50% more wire, pole capacity, and insulators…so they gain about a 15% improvement in capacity for a given cost of installing a transmission line.

In much of the world, three phase power is standard even in residential areas. This can make motorized appliances a little cheaper and more reliable. In the US it is usually difficult to get a 3 phase drop except in industrial or agricultural areas. Heck, my 1970’s neighborhood is supplied by a single-line ground return system.

Single phase power jumps up and down. Three phase power runs around in a circle. So it is much more efficient and simpler to make a motor run using three phase power.

In three phase power, the three wires take turns being at the most positive voltage, and take turns in the same order being the most negative.