Line voltage in 3 phase as single sine wave

[Elquery]

Howdy all.

The typical image of a three phase electrical system involves 3 sine waves, phase shifted 120 degrees. These sine waves each, individually, represent the 'phase voltage,' which is to a common neutral in a wye configuration. In this wye configuration the line to line voltage is determined by multiplying phase voltage by the square root of three. (Correct me if I've gone wrong anywhere so far.)

In other words, there is a distinct voltage between the lines (between the original sine waves).
My question is: can this line to line voltage be displayed as a single sine wave of its own? (Either practically or theoretically)
Is there any accuracy to such a notion that for any given voltage, it can only be a single sine wave (if original phases are sine waves at least)? What would be the best representation of what the load is experiencing?

Maybe another way to ask this is to ask if the load experiences one peak voltage per half cycle (i.e. one sine wave) or does it experience 2 distinct peaks in the same time.

If anything is unclear or imprecise in the wording, let me know.

 

[phyzguy]

Well, we can look at it analytically. The voltage on line 1 is V0*sin(ωt), and the voltage on line 2 is V0*sin(ωt+2π/3). The voltage between them is just the difference. We can simplify this using the sum and difference rules:
$$V = V_0(\sin(\omega t) - \sin(\omega t + \frac{2 \pi}{3} ))= 2 V_0 \sin(\frac{\pi}{3}) \cos(\omega t + \frac{\pi}{3}) = \sqrt{3} V_0 sin(\omega t - \frac{\pi}{6})$$

So the voltage between the two phases is a sine wave with a higher amplitude (1.7 times higher), the same frequency, and an intermediate phase.

 

[Elquery]

Thank you! I can make some sense of this when considering a linear load.

I am tripping myself up when I try to think of a 3 phase motor.
First, since each phase oscillates 60 times a second (in the U.S.) a three phase motor will experience peak voltage 3 times as often as a singe phase motor, correct?

Assuming yes, then are the peaks it experiences the peak of the original sine waves or of the new shifted and amplified waves?
(I want to say the original, since the magnetic field is referencing only a single phase at a given time, yes?)
Can a 3 phase motor be wired to utilize line to line voltage?

 

[Elquery]

Related: I've read several times now that phase is 'a relationship between voltages' and that you need 'three points to measure phase.'
I don't get it.
I can understand that in order to measure a relationship between phases, you need that third point, but a voltage in itself can be depicted as a single sine wave (phase?) using just the two points of reference for that voltage, no?

 

[NascentOxygen]

Can a 3 phase motor be wired to utilize line to line voltage?

If things are balanced (meaning the load on each phase is identical), then for a Y- connected machine there will be no current in the neutral line—so you can leave it off! The common point where the 3 loads connect will assume a voltage of around 0 because of the balanced phases, and it's called the star point. The 3 currents feeding into that point sum at every instance to ZERO.

 

[Elquery]

then for a Y- connected machine there will be no current in the neutral line—so you can leave it off!

Yes it is precisely because of this that I am getting confused about the phase /voltage relationships! If no current flows on the neutral, is that evidence that the relevant voltage potential is between the lines (i.e. is the line to line voltage, a vector of the phase voltage)?

I can't seem to reason this. Perhaps where I'm going wrong is that I am trying to assign a 'voltage experienced by the motor' and maybe that's the wrong question to ask. I understand that since its AC, the actual voltage is always varying, and the voltages we usually refer to are RMS. But whether the wingdings of the motor experience a peak voltage correlated to the phase RMS or to the Line to Line RMS (the root three vector of the phase) is where I'm tripping up. This may all stem from a lack of understanding the details of the motor wingdings and its connections.

 

[NascentOxygen]

The line voltage is a sinusoid ahead of the phase voltage by 30 degrees, and of magnitude larger by a factor of √3.

 

[anorlunda]

Yes it is precisely because of this that I am getting confused about the phase /voltage relationships! If no current flows on the neutral, is that evidence that the relevant voltage potential is between the lines (i.e. is the line to line voltage, a vector of the phase voltage)?

It sounds like you are confusing yourself because of the words used and because things can be measured more than one way.

The simplest way to say it has been given already by @phyzguy .

You can express it as 3 phase-to-neutral voltages, or as 3 line-to-line voltages. That applies to both Wye and Delta connections. But for Delta connections the neutral point is in our heads, not physical.

All those quantities are simple sin waves They differ only in magnitude and phase, but are all the same in form and frequency.

The secret to 3-phase systems is that the sum:

sin(0)+sin(120 degrees)+sin(240 degrees)=0

That is an identity. It is true at every point in time.
You can add a constant to the phase and it doesn't change.

sin(1)+sin(121)+sin(241)=0 when all angles are in degrees.

Also

sin(ωt)+sin(ωt+2$\pi$/3)+ sin(ωt+4$\pi$/3)=0.I you really want to visualize it, such as for a motor, I recommend animations on YouTube rather than words. If you want, I can find some to recommend. But you can do it yourself. Just search for "AC motor animation" on YouTube.

 

[Elquery]

It sounds like you are confusing yourself because of the words used and because things can be measured more than one way.

I'm thinking this is the case. I've watched quite a few animations and they 'make sense' to me. It also makes sense to me that the line to line voltage is a sine wave shifted and amplified. I have to admit that while the math is probably the most precise, I'm not quite up to snuff on it, and it isn't typically where I will gather an intuitive understanding of something.

I understand that the voltages can be measured more than one way, and the voltage (and sine wave corresponding) depends on where we measure.
Where I lose it is when I ask myself what voltage is being applied to the rotor. For sure, it is changing, but what RMS voltage is being applied? Is this question a nonsensical question?

When we consider a one phase motor, often-times we can choose to use 120 volts or 240 volts, and this voltage is measured from one side of the coil to the other in both cases, correct? In a 3 phase motor, measuring across one coil would give us only phase voltage, correct? So if the rotor is indeed experiencing 1.7 times this, it would be experiencing the peak somewhere between the coils (30 degrees, as Nascent says)?

 

[anorlunda]

I'm having as much trouble with your words describing your question as you are with words and math in the answers.

Try this video. The key concept is that the 3-phase voltage applied to the stator appears as a field that rotates in time. So it is not a question of measuring a voltage value at one point in time, you must think of the time variations.

 

[Baluncore]

First, since each phase oscillates 60 times a second (in the U.S.) a three phase motor will experience peak voltage 3 times as often as a singe phase motor, correct?

Consider two phases, sine and cosine, each with a neutral supplying field coils in the motor that are mutually perpendicular, so the vector sum of the field windings describe a magnetic field, rotating in a circle, doing 60 rotations per second. That rotating field drags the conductive rotor around with a small amount of slip, at close to 60 times per second.

Then consider the vector sum of three phase windings separated by 120° in position and phase. That also generates a field rotating 60 times per second.

 

[Elquery]

Ok. I think I'm putting it together.

so the vector sum of the field windings describe a magnetic field, rotating in a circle, doing 60 rotations per second

I think this pretty much gets at what I was wondering. The magnetic field is described by the vector sum discussed above in terms of voltage.

I think I wasn't considering the 'poles' of a motor separately from the phases. I always pictured a single phase motor as just 2 poles opposite each other. But even a 6 pole motor, if powered by 1 phase, will not generate a rotating (but rather an oscillating) magnetic field. Clearly my understanding of motors is wanting.

I was picturing the magnetic field jumping from one pole to the next (every 1/3 of a cycle) and was trying to figure out where the maximum strength polarity of the field was 'taking place' at each of these distinct points in time.

So perhaps one last point of clarity: Is this rotating magnetic field actually rotating with 'smooth continuity', or is it a bit choppy? Or said differently, at any given point in time, will the magnetic field have the same strength, simply a different position of polarity? Or is there some waxing and waning of the field strength based on where in the cycle we are for that time?

 

[anorlunda]

So perhaps one last point of clarity: Is this rotating magnetic field actually rotating with 'smooth continuity', or is it a bit choppy?

Not choppy.

r said differently, at any given point in time, will the magnetic field have the same strength, simply a different position of polarity?

That sounds like the rotating field, so yes.

Or is there some waxing and waning of the field strength based on where in the cycle we are for that time?

That also sounds like the rotating field, so yes.

But if you watched the video of the rotating field, those things should be clear, so I don't understand why you asked those questions. Please watch it again starting at 2:52, and pay attention to those red lines that depict the rotating field.

 

[Asymptotic]

But even a 6 pole motor, if powered by 1 phase, will not generate a rotating (but rather an oscillating) magnetic field.

That's why single phase motors must use one of several different methods (which one depends on required start-up torque, and other factors) to get them to rotating.

https://en.wikipedia.org/wiki/AC_motor#Split-phase_motor

A classic symptom of a split-phase, single phase motor with one of the windings burned out is remain motionless and hum, unless the shaft is (very carefully) nudged either CW or CCW. If the motor is lightly loaded, this may be enough to start rotation.

Is this rotating magnetic field actually rotating with 'smooth continuity', or is it a bit choppy?

3 phase is a good trade-off between simplicity, performance, and cost, and "good enough" for most applications. Adding more phases reduces torque ripple and other effects, but is seldom warranted.

Before PWM became practical, AC variable speed drives were often based around "6 step" inverters which generated the lower waveform (image courtesy Researchgate) by converting DC into a very blocky version of a sine wave by sequencing the six transistor bridge on and off at the correct times and in the right order. When dialed to low speed settings (say, to 1 Hz) the motor shaft would "cog" in observably discrete steps.

This doesn't quite address your question, but this paper describes a 6 phase induction machine, and reasons why one can be preferable.

Torque Density Improvement in a Six-Phase Induction Motor With Third Harmonic Current Injection
http://lipo.ece.wisc.edu/2002pubs/2002_12.pdf

 

[Elquery]

"'Or is there some waxing and waning of the field strength based on where in the cycle we are for that time?'
That also sounds like the rotating field, so yes."

To clarify, I meant waxing and waning of the maximum field strength, not at a given location. My understanding at this point is that it is constant. "The result of adding three 120-degrees phased sine waves on the axis of the motor is a single rotating vector which remains always constant in magnitude. " - Wiki. I'm sure this is exactly what the math was telling me, but I wasn't sure that it described the magnetic flux itself, or some other variable.

Concerning the video Anorlunda: at 2:56 the lines are actually shown stepping in a choppy manner, so there is some conflicting imagery, but by and large it is depicted as quite smooth, you are right. Regardless, I should not have used the word choppy anyways, as I more meant 'fluctuating'.
My reasoning was that the magnetic field on a single phase motor is oscillating, but doing so as depicted by a sine wave (typically). So there is distinct maximum and minimum flux magnitude (I am thinking). I realize that with 3 phase there is a vector sum, but I guess it didn't sink in that this vector sum would mean constant flux magnitude. I was thinking there was still fluctuation in magnitude, only with 1/3 less time in between the flux peaks. I see now that its constant.
Thanks for the patience.

 

[Elquery]

The fact that the magnetic field (should I refer to the Magneto Motive Force (MMF) magnitude?) is constant, but rotating, is specific to 120degrees for 3 phases, and also 90degrees for 2 phases correct?

In trying to understand what the MMF would be with a phase angle shift of some other number (say 2 phases 15 degrees shifted, for example) would it be accurate to picture a rotating magnetic field, but with varying magnitude?