Transcript
Introduction
Welcome to this white paper webinar. My name is Landon Rhodes. I’m a hardware engineer, and I’m going to be walking you through this white paper that I wrote about phasor decomposition via the method of symmetrical components in a three-phase system.
A set of perfectly balanced phasors where you have the A phase sine wave, and then you have the B phase sine wave delayed by one hundred and twenty degrees, and the C phase, which is exactly like B and C, delayed by two hundred and forty degrees. That’s a very easy, nice mathematical system to deal with. There’s not a lot of problems. If you understand basic circuit theory, then a balanced system is not too difficult to deal with.
Unfortunately, in the real world, three-phase systems are not perfectly balanced. Sometimes there’s distortion on one, sometimes there’s capacitive coupling on one leg more than the others, sometimes you get more phase shift on one, sometimes you get voltage sags on one leg. All kinds of things can happen to make what should be a three-phase system of perfectly balanced sinusoids unbalanced. It makes it a lot harder to deal with. So this paper describes a mathematical tool called symmetrical components that simplifies and clarifies the analysis of this unbalanced case.
Review of Phasors
A phasor is a complex number representation of a sine wave. You should see a graph I did of a circle and a point traveling around the circle. If I hit play, then it’s just traveling nicely around the circle. So a spinning point in the complex plane can be directly associated with a sine wave.
The way we do that is we just do a time series plot of where this point is along its path. We do a time series plot, and we get a sine wave. So there’s a very close relationship between a spinning complex number like this and a sine wave. This section walks through how that happens.
You can represent a spinning complex point by this expression here, in the section on review of phasors in my paper. A spinning complex number can be represented by a magnitude and a phase offset and a complex number that continuously spins and increases. It goes around the circle as T increases in a counterclockwise direction.
Then this paragraph talks about the algebraic justification. It turns out that the pointwise addition of sine waves corresponds exactly to two complex numbers being added as complex numbers. And this algebraic fact justifies the existence of phasors.
Derivation of Symmetrical Components
The idea is we’re going to take an unbalanced system of phases. In this graph, I have phase A, phase B, and phase C. These are not in a balanced setup. B and C are closer together. A is a little bit longer. I have in orange this point here, this point here, and this point here are the one hundred and twenty volt balanced phases. This is what a balanced one hundred and twenty volt system would look like. And we’re trying to model this unbalanced system of phase A, phase B, and phase C.
We’re going to do that by breaking it up into three different pieces. We’re going to break this up into a sum of three different balanced phases. There are three different ways we can construct a balanced set of phases based on this K value, which describes an integer number of rotations.
Zero Sequence
The first way we can construct a balanced sequence of phasors is when K equals zero, and that happens when all three phasors point in the same direction. This is called the zero sequence. We’re constructing the same complex number all pointing in the same direction.
Positive Sequence
The second way we can construct a balanced sequence is when K equals one. The K value is an integer multiple of hundred and twenty degrees from each other. So all the angles are either zero times hundred and twenty degrees, or K equals one where they’re all hundred and twenty degrees apart, or they’re all two hundred and forty degrees apart in the K equals minus one case.
When K equals one, they’re all hundred and twenty degrees apart. This happens when you have phase A, and then you have phase B that lags phase A by hundred and twenty degrees, and then phase C lags phase B by hundred and twenty degrees. If you look at a graph of all of them together, you’re going to see phase A hit the peak, and then phase B hit the peak, and then phase C hit the next peak. So that is a positive sequence.
Negative Sequence
K equals minus one corresponds to the negative sequence, which is the reverse pattern of the positive sequence. You have phase A, and then you have phase C, which lags phase A by hundred and twenty degrees, and then you have phase B that lags phase C. So you have the ACB pattern, where you see the A peak, and then you see the C peak, and then you see the B peak.
Constructing the Unbalanced System
What we’re going to do is build an unbalanced set of phases. We’re going to construct it based on a sum of some positive sequence set of phasors, some zero sequence set of phasors, and some negative sequence set of phasors. These are all going to be balanced. The reason why that’s really helpful is because balanced phasors are a lot easier to deal with mathematically. We can split this problem up into three simpler problems where each balanced case is simpler to deal with.
We’re going to assign the zero sequence to the complex number V zero. We’re going to assign the positive sequence to the complex number V one. And we’re going to assign the negative sequence to the complex number V two. This coefficient will describe the magnitude and the phase shift of the A component corresponding to this set of phasors.
Visualizing the Sequence Components
On this screen, you see I’ve got the setup where the first arrow is positive arrow, the second arrow is the zero arrow, and then the third arrow is the negative sequence arrow. In the positive sequence case, I’ve got arrow one here, and then arrow two kind of pointing in a similar direction as phase B, and then the third arrow pointing here in the similar direction to phase C.
For the next case, I’ve got the zero sequence, and the corresponding zero sequence is pointing in the same direction as this arrow. The zero sequence arrow corresponding to phase C is also pointing in the same direction as B and A.
The negative sequence is the last arrow that goes from zero sequence to the end. The negative sequence component points about thirty degrees off from phase A. The last arrow is increasing counterclockwise, not clockwise. It’s going a different direction. The negative sequence component pointing towards phase B goes in a different opposite orientation.
So this arrow, this arrow, and this arrow are positive sequence balance phasors. This arrow, this arrow, and this arrow here are zero sequence balance phasors. And then this arrow, this arrow, and this arrow here are negative sequence balance phasors.
Computing the Symmetrical Components
It’s helpful to define a complex number alpha, which represents a one-third of a turn rotation, a rotation by one hundred and twenty degrees. That’s going to be a complex number sitting on the unit circle, about right here on the complex plane, and it represents a one hundred and twenty-degree rotation. That’s how I’m defining alpha, is e to the two pi over three, which is just one-third of a rotation of a circle. Two pi on three.
Setting Up the Equations
We need to write down equations for each phase to set up the system. For phase A, we have that the zero sequence component, plus the positive sequence component, plus the negative sequence component, all of those have to sum to phase A.
This first equation says that the zero sequence component, plus the positive sequence component, plus the negative sequence component, all of those sum together as a vector to get to phase A.
The phase B equation says that we take the zero sequence component, and we take the positive sequence component rotated by two hundred and forty degrees. So we take that, rotate it by two hundred and forty degrees. And then we take the negative sequence component and rotate by one hundred and twenty degrees. We’re taking this component, rotate it all the way around to here. We’re taking this component as it is, and we’re taking this component and only rotating it by a hundred and twenty degrees. So that corresponds to phase B.
Phase C is defined very similarly. We need the zero sequence component, the same as for A and B. Then we need the positive sequence component rotated by alpha, rotated by one-third of a rotation. And then we need the negative sequence component rotated by two rotations. Rotate once to get up to this direction, and then rotate again to get to this direction.
Matrix Formulation
These are the three equations that define this relation. We can pack this into a nice matrix equation by saying that phase A, phase B, phase C is this matrix here: one, one, one; one, alpha squared, alpha; one, alpha, alpha squared; times this matrix of zero sequence component, positive sequence component, and negative sequence component.
Using some linear algebra tools, we can find the inverse of this matrix as one-third times one, one, one; one, alpha, alpha squared; and one, alpha squared, alpha. We need to take a one-third out front, and we need to switch all the powers. If it’s an alpha squared, then it becomes alpha. If it wasn’t alpha, it becomes alpha squared. So we’re just mirroring this matrix. This gives the solution for how we’re going to find what the zero sequence, positive sequence, and negative sequence components are.
Example in Desmos
I have this example here in the paper. This is a Desmos graph that I built to simulate. I can just plug in any A, B, and C component, and it calculates the zero component, positive component, negative component, and then I can also do it in polar representation as well.
These points here talk about what I was describing before of how the first and biggest arrow is the positive sequence, and then the next arrow is the zero sequence arrow, and then the next arrow is the negative sequence arrow. The first and biggest set of arrows emanating from the origin are the positive sequence arrows.
Ideally, most of your information in your system is contained in the positive sequence case. That’s usually the case, that the positive sequence is significantly larger than the zero sequence and the negative sequence. Negative sequence usually indicates some type of problem.
The second set of arrows is the zero sequence, and the zero sequence describes a neutral offset. If you have a neutral conductor in your system and the neutral is bouncing around, then you will see a zero sequence offset. If you’re working in a delta system, there is no neutral conductor, and so the zero sequence vectors will all be zero in that case. The third set going from the zero sequence to the end is the negative sequence.
Unbalance
This section talks about unbalance, which is the ratio of the negative sequence component. In this example, it would be this vector here, 12.5 angle minus 4.9 degrees, divided by the positive sequence, which is 119.3 angle minus 1.9 degrees. I have that calculated here as 12.5 over 119.3, which is 10.5%, which is quite high usually.
The reason why we care about balance is this is a good metric for characterizing how different a system is from balanced. If your unbalance is like .1%, your system is basically balanced. There’s very little distinguishing it from a balanced case. Whereas if it’s significantly higher, then you run into a lot more problems.
Effects on Electric Machines
If you’re having an induction motor, for example, a negative sequence current will drive the shaft of the motor in the opposite direction, which will obviously go against the positive sequence orientation. So you have one set of vectors that will drive the motor in one direction, and you have another set of vectors that will drive the motor in the other direction. The more negative sequence current you have, the worse your efficiency is in your motor. And it will turn into a lot of heating loss in the windings of the motor.
In a transformer, you’re going to get a similar effect happening in a three-phase transformer, where a negative sequence voltage is going to make the magnetic field inside the transformer go the opposite direction as the positive sequence vectors are trying to turn it. So high negative sequence voltage can be a problem in electric machines like transformers and motors. They harm efficiency. They cause core losses and conduction losses, which hurts efficiency.
Voltage Unbalance vs. Current Unbalance
This section talks about how voltage imbalance is a good metric, but current balance is not really a good metric because a much better metric for current is just negative sequence magnitude because there’s no baseline for current.
I have an example of a factory that is regularly working with less than 5% current unbalance, which I think is pretty reasonable. But then everything shuts down in the factory and someone leaves a desk lamp on. Obviously, the current in the factory drops to almost nothing. But the current unbalance jumps up to 100%.
Watch what happens in my graph if I turn phase B to zero and phase C to zero. Now I just have phase A, and phase B and C are zero. And then my negative sequence component is 40, and my positive sequence component is 40. If you take a ratio of that, you get 100%, which sounds like it might be a problem, but it’s not really a problem. A high current imbalance doesn’t really indicate anything substantial or anything problematic if it doesn’t correlate to a high negative sequence current. So a better metric for current is just going to be magnitude of the negative sequence component itself. And that would be much higher when the factory is in full operation rather than at night.
Correlating Voltage Unbalance and Negative Sequence Current
It can be beneficial to plot voltage imbalance and negative sequence current. When those two correlate well, that means the load that’s downstream of that measurement is causing that unbalance, or at least a good portion of that unbalance. But if they don’t correlate well, then that tells you that the load is not causing that voltage imbalance. So if you’re trying to find what load is causing a voltage unbalance problem, it’s good to correlate the voltage imbalance graph with the negative sequence current graph.
Summary
This method of symmetrical components is a useful mathematical tool to analyze three-phase power systems, and it works by breaking down an unbalanced set of phases into a sum of three balanced phases by a positive ABC sequence, a negative ACB sequence, and a zero sequence of balanced phases. Each of these sequences describe useful information for power quality engineers. It’s useful to look at what each of these sequences are telling you, and it’s also useful in defining more advanced metrics like unbalance. This is crucial for diagnosing faults and improving equipment performance.
If you have any questions, you can reach out to us at support@powermonitors.com, or you can call us at 1-800-296-4120. Thank you, and have a good day.
