Transcript
Introduction
Hey, everyone. Thank you for joining today’s Ask a Pro Webinar. My name is Landon Rhodes. I’m an engineer here at Power Monitors Incorporated. Today, we’ll be covering a white paper on power analysis with distorted waveforms.
This is a really interesting topic. It extends the ideas of the power triangle, where you have real power on the horizontal axis and reactive power on the vertical axis, and apparent power, and it extends those ideas to situations where you have distorted waveforms, where you have harmonics present in the signal.
This paper is based on the work of Francisco Montoya from the University of Almeria in Spain. He developed this theory, and I’m presenting a lot of his ideas.
The Power Triangle Review
So I’m going to start by reviewing what the power triangle is. This is a simple diagram of the power triangle. I’ve got real power on the horizontal axis, I’ve got reactive power on the vertical axis, and then the apparent power is the hypotenuse of the triangle.
This is constructed by representing voltage and current sinusoidal waveforms as phasors. In this section I’ve described how you can go about constructing this phasor. You directly associate a voltage sinusoid and the current sinusoid with a particular complex phasor, and then the power is just the product.
Apparent power is just the product of the two complex phasors, and that tells you this complex number over here. The real power is just the real part of that complex number, and the reactive power is the imaginary part of that complex number. And then you can define the power factor as the cosine of the angle, which is just the real power divided by the apparent power.
This is all pretty standard analysis techniques for power. We directly associate voltage phasor and current phasor with a complex number, and then multiply those complex numbers, and then begin to do some quantities related to that.
Basis Functions for Distorted Signals
In the distorted signal domain, there’s a lot more variation that can happen. What Francisco Montoya proposes is this basis for recording or breaking a signal up into component parts. The first basis function is just a number one. Second basis function is √2 cosine ωT, and then √2 sine ωT, and then fourth basis function is √2 cosine 2ωT, then √2 sine 2ωT, all the way up to the Nth harmonic.
This basis has two N plus one functions. If you want to measure up to the 10th harmonic, this would have 21 elements in it. So depending on how coarse you want your distortion to measure, this is the basis that he proposes.
Each time, we just increase the frequency and we get a cosine signal and we get a sine signal. The reason why we have two at each frequency is to account for variation in the phase of the signal. If we just had one signal at that frequency, then we would only get amplitude variation. We couldn’t parameterize any phase variation. That’s why we need a cosine and a sine.
Orthonormality of the Basis
I’ve described here that this is an orthonormal basis with respect to this inner product here. Orthonormal just means that the length of the vector is one. If you take the product of a vector with itself, you get one. And then if you take a vector multiplied with another vector, you always get zero, if they’re different vectors.
I’ve set up this graphic here. This red line is describing basis function zero, basis function one. This is σ0, σ1, σ2, σ3, σ4, σ5. The black curve is a second function. If they correspond to the same function, then this product ω/2π integral from zero to 2π/ω of F(T) times G(T) dT—pairwise multiply the functions together, integrate from zero to 2π/ω. When they’re the same, then you get one. When they’re different, then you get zero. That’s all that orthonormal means in this context.
This orthonormality condition is really, really nice when you’re trying to evaluate these numbers, because all you have to do to create this KI is take some signal defined from zero to 2π/ω, take the sum from I equals zero to I equals 2N, some constant number times the sigma basis function. And that K number is defined by just taking the product of the function signal with the sigma basis. That is only a condition of the orthonormality. You’re just forming the projection of the signal onto the particular basis function. This is a really nice condition that happens when this orthonormality condition is satisfied. This is really, really nice and easy to evaluate.
Geometric Product of Voltage and Current
What Francisco Montoya then proposes is we do this for my voltage signal and my current signal, and we define a sequence of coefficients on these sigma basis functions. For voltage we have sum from K equals zero up to K equals 2N of VK times the sigma basis function. And we do the exact same thing for current: sum from K equals zero up to K equals 2N of IK times σK.
He defines this multi-vector quantity, this M quantity, as voltage times current. That just means we’re taking the two and multiplying them as vectors, where this product is defined as the geometric product, which is a restriction on the tensor product.
Because this is a bilinear operation, we can write this multiplication as a long list of sums. Sum number times σ0 plus sum number σ1 plus sum number times σ2, and so on up to σ2N. And then same for current. And then we can just distribute that as a giant product result into a double sum of just the products: sum from I equals zero up to I equals 2N, sum from J equals zero up to J equals 2N of my voltage number times my current number, times the product of the two sigmas.
That just derives from the fact that this is a restriction on the tensor product. We can distribute like normal multiplication. It works like normal multiplication. However, it is not commutative. You can’t flip σI and σJ. Flipping them is not commutative like multiplication is.
Defining the Geometric Product
For this particular restriction, the geometric product is defined by this condition: if you have any given vector V in the vector space, then the vector times itself, V times V, is just defined to be the dot product between a vector and itself. In this case the inner product of any sigma with itself is just one. So whenever we have σI σI together, whenever the indices match, then that just collapses down to one in this product.
We’re going to apply this restriction to this expansion. We have vector V plus W times vector V plus W. Because of this restriction, we’re just going to say this is equal to the dot product. When we expand this, we pairwise multiply each term: VV, VW, WV, and WW. We do the same thing with the dot product: V·V, V·W, W·V, and W·W. We expand all this, and eventually collapse down to this condition: vector V times vector W plus vector W times vector V is just twice the dot product between the two.
What that tells us is that if these two are perpendicular, then V·W is zero, based on the definition of perpendicularity. So this becomes zero. And so VW equals negative WV. What this means is that we can separate our sum from up here, which is just a pairwise multiplication of all the terms.
This sum here comes from the condition when we have the same basis functions. And this condition here is when they’re different. When they’re different, you can flip the I and J and get the negative sign. So you get this cross product term. You get a sum of numbers and a sum of cross products times a particular σI σJ.
The Multi-Vector and Bi-Vectors
This is a multi-vector. We’ve got a number plus a two-dimensional term. σI σJ is a two-dimensional entity. This is the equivalent of the complex imaginary unit i. σI σJ is equivalent to the complex imaginary unit i because if I take σI σJ and I square it, and then run through all the computations based on the rules I’ve defined above, you get negative one. Squaring this thing gives negative one. So it behaves exactly like the imaginary unit i.
However, this gives a lot more specific information about what this rotation is talking about. In normal complex numbers, in two dimensions, there’s only one plane of rotation that is available, so you don’t really need to be careful about specifying that. Whereas if you’re talking about up to the 10th harmonic, then this is a 21-dimensional vector space. So you need to clarify exactly which rotations you’re talking about. And so this specifies the plane of rotation.
If you have vector σI and vector σJ, this is a representation of the bi-vector σI σJ. It’s oriented area based on the oriented arrows σI, σJ. This bi-vector encodes just the plane parallelogram segment spanned by σI, σJ. And if this is in 3D space or 4D space or any higher dimensional space, it still works. The bi-vector term is just the oriented parallelogram spanned by these two vectors.
So we’ve got M as just a dot product, just a number, plus a bunch of these planar terms, these imaginary unit bi-vector terms. If you had σ0 through σ4, five dimensions, then you’re going to get essentially ten of these σI σJ terms. Every different pair of combinations you can get. This term is going to grow by n squared. And the first term is just a number.
Obviously, this is a lot more complicated than the power triangle story because in the power triangle, you’re just assuming one plane of rotation, one way that you can get reactive power. Whereas if we go up to σ0 to σ4, you could get up to 10 different degrees of rotation. You would get a lot of different terms.
Apparent Power, Power Factor, and Reactive Power
We can define the apparent power as just the size of M or just the norm of M. I’m just going to take each term—remember, M is just this thing up here—and take each of these inner coefficients and square it, sum them all together, and then take the square root. Sum from I equals 0 up to I equals 2N, sum from J equals 0 up to J equals 2N of VI times IJ. Take each coefficient, square it, sum them, and then take the square root. This gives me the size of this multi-vector quantity.
We can also split up the products of the voltage and the current, and we just get RMS of voltage quantities times RMS of current quantities. Take each quantity, square it, take the square root, and then multiply the two for voltage and current. So it’s |V| times |I|.
The power factor, just like how in the power triangle case we define the power factor as the real power divided by apparent power, in this more complicated distorted case we’re going to do real power, which is just the dot product between the two voltage and current, and then divide that by the size of V times the size of I. This is the equivalent power factor as the power triangle case.
Visualizing Rotation Vectors
One thing I wanted to show you is what these rotation vectors look like. This is the normal power triangle case. This quantity here in green represents rotating in the plane between the red sinusoid and the black sinusoid. That’s what the green vector represents. I’m going to move it along, and it just corresponds to a phase shift of the quantities.
But now watch what happens if I increase TA2B. This is σ4 and σ1, so now I’m just rotating between these two. And so you obviously get some waveform distortion. You also get some weird harmonic effects. Now this is rotating between σ4 and σ9. This is between σ4 and σ19.
If they represent the same frequency, for example σ5 and σ6, then it just corresponds to a phase shift of the elements. It’s very normal to parameterize that as a phase shift. But if you’re talking about two different frequencies, then it’s really kind of strange and weird.
Compensation Methods
Montoya mentions that the bivector terms where the σI and σJ are sine waves of the same frequency can be compensated by passive networks, like with capacitors and inductors. However, bivector terms where the σI and σJ terms are different frequencies—you get weird harmonic effects there. What that means is you have to compensate it with active filter networks. You can’t use capacitor rings like you can do with the normal case.
Comparing to the Power Triangle
I’ve gone through a calculation of comparing this calculation to the power triangle results and showing that they are exactly the same. We have σ1 is √2 cosine ωT, and σ2 is √2 sine ωT. I’m going to write the voltage as VRMS times √2 times sine ωT, or VRMS times σ2. I’m going to write the current I as IRMS times √2 times sine(ωT minus θ). θ is the phase shift. That’s going to correspond to this θ right here in the power triangle.
I’m going to write that as some number times cosine ωT and some number times sine ωT, or α times σ1 plus β times σ2. And then these are the conditions that make that work: α² plus β² equals IRMS², and θ is the arctan of minus α over β plus π times (1 minus sign of β) over 2. This just guarantees that we get all the right properties, and that α cosine ωT plus β sine ωT adds up correctly.
Computing the Multi-Vector M
Using this geometric power theory framework, we’re going to compute this multi-vector M. Multi-vector M is voltage times current, which is just VRMS times σ2 times (α σ1 plus β σ2). Expand this out and we get a number, β VRMS, and then a number times this planar quantity: minus α VRMS σ1 σ2.
The power factor is just the real part divided by the sum of the squares. This is β VRMS over VRMS IRMS, which is just the product of the voltage RMS and the current RMS. The VRMS cancel and we just get the power factor is β over IRMS. That’s a very simple calculation. We just multiply and rearrange, and then we can easily read off the power factor from that.
The reactive power is just the bi-vector component of M. This is minus α times VRMS times σ1 σ2. That’s just the reactive power component.
Traditional Power Triangle Comparison
Using the traditional power triangle method, we can associate the voltage with the phasor VRMS angle zero, and the current can be associated with the phasor IRMS angle θ, and now set the complex number S to be voltage phasor times current phasor. The power factor is just the cosine of this angle, cosine θ. Take this θ and plug that in, and then simplified, we get the exact same results. We get β over IRMS as we got up here.
It doesn’t really make sense for this β to ever be negative because that corresponds to applying a voltage to some load, and then that becomes the current source of that load. You would basically get your power triangle in this domain here. It doesn’t really make sense. Usually, the power triangle is always in the first quadrant or in the fourth quadrant down here. So we can just say that this is β over IRMS, which matches what we got up here.
The reactive power is the same. We just do magnitude of S times sine θ, and we do the algebra and we get down to that, which exactly matches what I have up here.
Conclusion
This is a cool new method that Francisco Montoya developed to characterize and talk about power and power analysis within the context of distorted signals when you have harmonics of different frequencies. This is a way to talk about power factor and talk about apparent power and talk about things like real power. This is a mathematical framework that allows you to talk about those quantities in a distorted state.
Thank you everyone for joining today’s webinar. If you have any questions, you can reach us anytime by calling 800-296-4120 or emailing support@powermonitors.com. Thank you and have a good afternoon.
