We're going to be talking about generic terms on power and harmonics in non-sinusoidal systems. So this is a very, very generic representation of the issue of transmitting power from source to load, and it applies equally well to DC systems and AC systems. We are talking about the interface between a source and a low that we call a surface. And at that interface port, if you look at these guys being black boss boxes in their own right, you can just focus on the voltage and current at that surface. And in general, if the system is periodic, where both voltage and current can be expressed in terms of DC terms and harmonics, from the fundamental n is equal to 1, to higher harmonics potentially. So no restrictions on whether these voltages and currents are distorted or not, but there is an assumption that the system is periodic in steady state operation. These are just fourier series representing voltages, voltage and current at the interface port or the surface. In general, the energy transmitted from source to load, through that surface, by definition, that's just going to be instantaneous power, product the voltage and current, integrated over the period of the system, period assumed to be equal to t. Don't be confused here, T is not switching period. It really is the period of this AC system. Now, if you take that energy over a line cycle, then you have an average power that is transmitted through that surface. This is the voltage, this is the current, the product of the two, you integrate over a period, and you divide by the line period, you get a result that is pretty basic, but really important. The products of unequal harmonics result the integration being equal to zero. So we have zero power transmitted by interaction of harmonics of different order, you have to have harmonics of equal order in order to be able to transmit net power through the surface. Product of the amplitudes of that harmonic divided by two and times the cosine of the angle between the harmonics of order n. Applying this to all terms in that fourier series gives us an impression for the average power, and that average powers contains DC part, and contains the products of the harmonics. A couple of examples on that. Let's suppose the voltage has sinusoidal waveform, it's fundamental only, and the current is only a third harmonic. What is the average power transmitted from source to load in this case? Well, since there are no equal harmonics in the two waveforms, the average power is going to be equal to zero. Just to emphasize, again, the key difference between what we call instantaneous power, there's going to be just a product of voltage and current at any given point in time. And that instantaneous power, of course, is not zero everywhere, right? It goes up here, is positive here, and it goes down, it's negative here, and so on. In terms of the energy transfer through the surface from an AC system to the load, it is the average power that matters. All right, second example is when we have third harmonic of both voltage and current. There is, of course, net average power transfer. The third example is the example that looks more complicated, voltage has 1st, 3rd, and 5th harmonic. Current has 1st, 5th, and 7th harmonic. And now, we have the capability of producing average power, because both do include first harmonic, and both do into the 5th harmonic. And those are two harmonics where we will have some average power transfer. Cosine of the angle between the harmonic voltage, and harmonic current is what affects the amount of average power transfer at that particular harmonic. An important factor to keep in mind with respect to AC systems in general, is the notional root mean square value of a waveform. General definition for a periodic waveform is given right here, you can find the RMS value as being a sum of the square of the component is correspond to the harmonics. Here, V sub n is the amplitude of the harmonic, right? So keep that in mind, that's why we have V n squared over 2 as contributing really to the rms value. Rms value with sine wave is the peak value over square root of 2. Rms square is the peak value square over 2. The same expression, of course, goals for current, and you can note here that the presence of harmonics will always increase the rms value. The more harmonics you have, the more of the terms you have in this part right here, you have a larger value of the rms current or voltage. But on the other hand, we understand also that the net power transfer is not necessarily increased by the presence of harmonics, increased rms value generally imply increase losses. Which brings us to the definition of what is called power factor. Here is a general definition of the power factor. It's an important quantity in AC systems to understand, it's the ratio of the average power, the numerator over the product of rms voltage and rms current in the denominator. So in the best possible case, you have power factor equal to one, and it can be certainly less than one, and we'll see example of that coming up. Power Factor really tells you how effectively is this AC system utilized. An important example to look at carefully is the example of a linear resistive load, but the voltage is non-sinusoidal, all right. So you have a case where the load is just resistor, so voltage and current are proportional to each other, and the voltage itself is not necessarily purely sinusoidal. So voltage has harmonics, because the load is resistive, we have current harmonic, each is equal to voltage harmonic over resistance. And so, the rms current can be found in terms of the harmonics of the voltage. For each one of these, we have just the scale factor due to the resistance, so rms current is equal to one over r times the rms voltage. Which is great, because when you look at the expression for the average power right here, you realize that in the case when the load behaves as a resistor, even though the voltage itself is non-sinusoidal, the power factor is going to come out to be equal to one. The power factor for the other interesting case, when the voltage is sinusoidal, but the load is nonlinear and dynamic, it's quite important, because that really goes into the heart of how do we actually design our power electronics components tied to the grid. Let's suppose for a second degrade is perfect, and we have just a sinusoidal voltage coming from the grid. On the other hand, as we construct our power electronic system, there is going to be plugged into the the AC grid. Invariably, it is going to end up being nonlinear and dynamic in nature. We want to make it as perfect as possible, right, but it's never completely perfect, and what is the effect of having that imperfection in two aspects, one is the nonlinearity of the load, it's not just a resistor. And the second one is that's a dynamic load, it may actually have phase shifts between the current fundamental with respect to the voltage fundamental. So if you do the power factor calculation for this important case, you find the average current is in this form right here. Notice that we have, by assumption, just the first harmonic right here. So average value of the power is going to include only the first harmonics of voltage and current, and it will include the cosine of the angle between the two. Why do we have a cosine of the angle between the two? Because the load could in general, be dynamic, right? It could have a phase shift, where it's not necessarily a pure, purely resistive load, it could have a phase shift in the fundamental with respect to the voltage. Now, rms current, of course, we have a nonlinear loads, we may have harmonics in the current, you plug those into the expression for the power factor. And you get this important expression right here that tells you the power factor is actually the product of two terms. Both of which can be only up to 1, cannot exceed 1. So the first term is called distortion factor. And this is a result of nonlinearity, nonlinearity of the load is going to produce harmonics in response to a sinusoidal input, and the second term is cosine angle between the fundamental voltage and fundamental current. That's called displacement factor, and that's due to the fact that the load may be dynamic. That distortion factor has an expression that depends on how large is the rms value of the fundamental current, over the total rms current, ideally, it's equal to one. And that's the case when there are no harmonics. Distortion factor is maybe not as well known as another metric that is typically used to characterize nonlinear system responses, it's called total harmonic distortion. Total harmonic distortion by definition is equal to the RMS value of the harmonics in the current, excluding fundamental, divided by the RMS value of the fundamental, and so, distortion factor and total harmonic distortion are related in this manner right here. Here's a graph that shows distortion factor versus THD. THD, total harmonic distortion can be larger than 100%. It doesn't end here. You'd have a total harmonic distortion, arbitrarily large, if you have no fundamental, and only harmonics, total harmonic distortion goes to infinity. On the other hand, the distortion factor can only be between zero and one. If you have no harmonic distortion, no harmonics, distortion factor is 100%, and total harmonic distortion is 0%. And that's really what we will be trying to do, right, when you construct these systems. You want to have no harmonics, and you want to have a zero total harmonic distortion, and distortion factor equal to 2,1, or 100%. All right, there is an example here that we always bring up as an example of how not to do rectification. This has been done many, many times in through history of just doing a diode bridge rectifier, followed by a large capacitor, that's it, right? How more complicated do we need to go? So on the input side, you have an AC line voltage. On the output side, you have really a large capacitor that's supposed to just filter out all these large, line frequency harmonics, and keep the output voltage DC value across the load. So the advantage of this approach, that's called the peak detection rectifier, is that it's extremely simple, and in fact, it's actually very efficient. Just kept diodes, there is no switching, no theory needed to design this, throwing diodes, Diode bridge, and put the big capacitor, and you're done. The problem with this approach is either added on with respect to the AC line, it generates really, really large harmonics, okay? Why is that? Well, because the conduction of the diodes here is only happening. Add a time when peak value of the voltage is larger than the output, the DC voltage, so this is your AC voltage. And you say right now, what is my DC voltage going to be at the output, the DC voltage is going to be very close to this peak value right here, it's going to charge up the output capacitor up to close to the peak value. The only time when you can have conduction of the diode is around these peaks. And when you look at the current waveform at this point right here, this current waveform here, you will have large spikes of current at these peaks of the AC line voltage, which is why we call that the peak detection rectifier, because the output voltage is approximately equal to the peak of AC line, but the effect of this type of rectifier is huge harmonic content. So what you have here, shown on the bottom diagram here, is the harmonic numbers, you have odd harmonic numbers to high values. The distortion factor is 59%, so it's far below 100%. Total of harmonic distortion is 136% for a typical P detection rectifier, there are standards that don't allow you to actually produce such large harmonics. From the practical point of view, it's also important to realize the importance of the power factor being very simply explained here in terms of how much power can you actually get from the AC line. If you have a wall outlet that is rated at 50 names, so rms current has to be less than 50 names. If you have a peak detection rectifier, let's see how much power can we actually get out of that wall socket. So we have 120 volts rms voltage. We want to derate this, right? You don't want to touch 15 amps, you want to go, maybe less than 20% with respect to that. The power factor is just 0.55, mainly because the distortion factor is so low. And the rectifier efficiency can be high, but it doesn't really help that much. You end up having a maximum power of about 770 watts. At unity power factor, if you do this with a properly designed rectifier, you have the same voltage, same derating, the power factor can be easily very close to 1. Unfortunately, we have a more complicated rectifier, it's going to pay a little bit inefficiency. So it may go down here, but the overall power that you can take is certainly much higher. And so, people have come up with different ways of regulating how much harmonics can be present with respect to the AC line. And the one that really, really matters for commercial systems these days is what is now called European norm 61000-3-2, because it's actually enforced in European Union. And because when you make products, you want to be able to sell them anywhere across the world. It's really a worldwide standard these days. What is really regulated in that norm is the value of harmonic currents you can have in a system that's plugged into the wall. The standard is complicated. There are many different sub classes of devices, and there are a lot of details involved in it. But this is an example, this class A, which would, for example, apply to a computer plugged into a wall. They tell you how large can harmonic content be in terms of the absolute value of the current. And you have these numbers shown right here for harmonics from 3 to 39, and even harmonics from 9,8,2 to 40, we realized that, ideally, a grid tied power electronic should take or deliver power at unity power factor, and with the low current harmonics. And so, what we're going to do now is actually learn how to do that.