In this lecture we will introduce low-harmonic or power factor correction rectifiers and show how what we have learned about the design of average current mode control can be applied directly to this important application case. Let's look at what a low harmonic rectifier actually does. A rectifier is an interface between an AC line, or a grid, and a DC load. So the input side of the rectifier is a voltage that comes from the grid. In the single phase case that voltage can be anywhere between 85 and 260 volt RMS. And so when we talk about universal input rectifiers we will talk about the capabilities of operating that unit from anywhere in the world, from any of these input voltages. Similarly the line frequency or the grid frequency can be 50 Hz or 60 Hz depending on the part of the world. And so for universal input operation, we have a wide range of input voltages, and we have one or the other input line frequency. The line voltage, or Vac, can be assumed to be close to sinusoidal. Now of course, in practice that line voltage is not exactly sinusoidal. For our purposes, it will be sufficient to assume that is close to sinusoidal. And so an AC line voltage can be written as an amplitude V sub M, times sine of omega t, where omega is the line frequency in radians per second, or two pi times f line frequency in Hz, where f is 50 hertz, or 60 hertz. Now, to perform rectification, a very common way of doing that is to bring, on the input side, a full-wave diode bridge rectifier. So that the output voltage of that full wave bridge rectifier is full-wave rectified, so vg(t) is going to be a full-wave rectified sinusoid. That full wave rectified sinusoid then becomes an input voltage for a DC-DC converter that is going to be operated with a duty cycle that varies in time, so that the output voltage can be equal to a DC value and so that the power can be delivered to a DC load in that form. To perform that function, the converter will be controlled so that the input current, the iac, the line current or the grid current taken from the power grid, follows the same wave shape as the grid voltage. Ideally, if the line voltage vac is sinusoidal, iac is similarly a sinusoidal waveform with the scale factor equal to what is called emulated resistance. So in the ideal case of a low-harmonic or power factor correction rectifier, the input current follows the same wave-shape as the input voltage and ideally it is sinusoidal, with a scale factor being represented by the emulated resistance. The input port of an ideal low harmonic rectifier looks like a resistor. There are numerous international regulations that in fact dictate that the rectifiers with respect to the AC line do behave in this ideal manner as opposed to taking current with harmonics which is undesirable with respect to the disturbances on the AC line. So to accomplish this goal of having an input current that is sinusoidal, given an input voltage that is sinusoidal, the controller around this DC/DC converter in the low harmonic rectifier will have to modulate the duty cycle so as to control the input current to follow a waveform proportional to the input voltage. And you see the main control function in the low-harmonic or power factor correction rectifier is the current control, and so this is why this presents an ideal case to apply what we know about average current mode control ideas. The Power Factor Correction refers to the fact that when a sinusoidal current is taken from a sinusoidal voltage in phase with the sinusoidal voltage, the power factor at that interface with the grid is equal to one, which is the best utilization of the power from the AC grid line. The coverage in this lecture, and the lecture that follows, will be self-contained. The topic is covered in much more detail in our reference textbook in Chapter 18. But, do not worry, we will cover all details necessary, in particular, to show how average control can be applied to have the input current follow this sinusoidal reference that is obtained by sensing the input voltage. Let's examine the realization of that ideal rectifier in a little more detail. So we will employ a DC-DC converter exposed on the input side to a voltage that is a full-wave rectified AC line voltage. To perform the control function, the controller will typically sense the input voltage v sub g, which is a full-wave rectified AC input voltage and will have to sense the input current, and, based on sensing these two quantities, we'll make decisions about how to modulate the duty cycle so as to control the input current to follow the input voltage wave shape, with a scale factor equal to the emulated resistance, Re. Waveforms in the ideal rectifier are illustrated on this page. The AC voltage is ideally sinusoidal with an amplitude V sub M. The AC current is ideally sinusoidal with an amplitude of V sub M over Re. The converter input voltage is a full-wave rectified sine wave with the same amplitude V sub M. The converter input current, is a full wave rectified sine wave, with an amplitude V sub M over Re. The output voltage is ideally DC voltage, that's the objective, the output should be rectified DC voltage. And when you compare that DC voltage now to the input voltage v sub g you realize that the conversion ratio of the converter must go up and down shooting up to, ideally, an infinity at the zero crossings of the input voltage. In the ideal case, the conversion ratio of the converter is given by V over V sub M, sinusoidal function of time. And so that is what really represents the shoot ups to infinity at the zero crossings of the input voltage right here. The conversion ratio will have the minimum value when the input voltage reaches the maximum value of V sub M. So how to do we actually realize an ideal rectifier? One popular approach is to employ average current mode control, and the diagram that is shown right here around a boost converter example is exactly the same diagram we have seen earlier in the realization of the average current mode control loop. We typically place a sense resistor of small value so that it doesn't dissipate a significant amount of power. We have a gain block that brings that signal up to a sensed value that is convenient for comparison with a current control reference signal. The error between these two is feeding a current loop compensator that we have already learned how to design. The output of that compensator is the input to a pulse width modulator, and pulse width modulator in turn controls the boost switch. So by controlling the input current to follow the input voltage we will obtain the function of an ideal low harmonic rectifier. But how do we actually generate this reference current right here? Well, remember, the goal is to have the input current follow the input voltage wave shape, and so the most convenient way of generating that current reference is to have the signal proportional to the input voltage, v sub g. The voltage that will be full wave rectified AC line voltage. Here is is that implementation. So we sense the input voltage, that sensed input voltage is then used to generate a control input for the current control loop. The input current is sensed and compared to the voltage proportional to the input voltage. The outer signal is passed through a current compensator, the pulse modulator and controlling the switch Q1. Now, the scale factor between the input voltage and the control input of the average current mode controller is determined by this multiplier block right here. Why do we do that? Because we want to be able to adjust the amplitude of the input current to control the power delivered to the output load. We want to be able to scale the input voltage to a controlled level by an adjustable quantity. That adjustability is achieved by the multiplier block right here. So if our control input to the multiplier has a higher value, that means that the control input for the current control loop will have a higher value, a higher amplitude but still the same shape as the voltage v sub g. And that means that you will have a higher current amplitude as a result, and we will take a larger amount of power from the AC line. Conversely, if we reduce v control down, we obtain a lower value of the control input for the current control loop. The amplitude of the input current goes down, while the wave shape still remains sinusoidal, and a smaller amount of power is taken from the AC line and delivered to the output node. We will focus next on the design of the current loop compensator. How do we design this compensator right here so that we accomplish the goal of having the input current follow a wave shape proportional to the input voltage. To do that we need a model. Well in this case here, the model is really a repeat of what we had done already. In a large signal average sense for the inductor in the boost converter, on the left hand side we have L d per dt of the average inductor current. On the right hand side we have a difference between vg, the average value of the input voltage, minus d prime times the DC output voltage. In a rectifier, the DC output voltage is not going to change much at all. The DC output voltage can be assumed to be constant. And so when you linearize this equation right here, you obtain an equivalent circuit model that, in the small-signal sense, shows that the response from duty-cycle perturbation to the input current perturbation is simply constant over s where that constant is equal to V/L, V being the DC value of the output voltage, and L being the inductance in the boost converter. This is exactly the same result we have used earlier in the design of the current control loop for a boost dc-dc example. In fact we realize here that the same approach we have developed and applied to design average current mode control loop in a boost dc-dc example applies in exactly the same manner to average current mode control design in the boost low-harmonic rectifier. Let's do that. The example that is numerically shown right here follows the values that we have used earlier in the dc-dc case. In fact, one motivation behind that dc-dc example was to enable us to design a low harmonic PFC rectifier. The same value for the inductance, the same value for the switching frequency, V sub M of the pulse width modulator, the same equivalent current sensing resistance, and the same desired DC output voltage of 400 volts. That, by the way, is around the typical value that you would have in practical PFC rectifiers based on the boost converter because that voltage is going to need to be higher than the peak value of the full wave rectified AC line voltage for universal input operation. Worst case being 260 volt RMS times square root of 2. That's going to give us the highest value of around 370 volts which is less than the DC voltage that we assumed in this example. You may, at this point, want to go back and review the lecture on the design example for the average current mode controlled boost converter, the dc-dc case. We found in that case that our uncompensated loop gain follows this constant over s function. We design the compensator as a simple PI or leg compensator with an additional high-frequency pole, and these are the numerical values we have used in that example, and exactly the same design applies to the PFC rectifier case as well. Here's the average circuit model of the PFC rectifier based on that design example. You will recognize exactly the same circuit as we had in the dc-dc case, except now the input voltage is sinusoidal and it is the interface to the converter through a full wave diode bridge rectifier. We employ the average library, and we perform transient simulation with parameters that correspond to the amplitude of the AC line voltage, the desired value of the power, which then implies the desired value of the amplitude of the input current. That value of the amplitude of the input current is what is used as a scale factor in setting up the control input for average current mode control compensator. In this example here, the output voltage is set to a DC value. A little bit later on we will see how that is actually accomplished. But right now, for the verification of operation of the current control loop, we will simply assume that the output voltage is set to that desired value of 400 volts. And the converter is delivering power to that voltage source as a voltage sink. So the voltage source will be absorbing the output power delivered by the converter. Here are the results. The results are in transient simulation and so the top waveform shows the AC line voltage, which is sinusoidal, with the RMS voltage value of 120V, the peak value of 170V, and the power level of 1 kW. The frequency's assumed to be 50 Hertz. You can try to reproduce the same results for 60 Hertz as an example. The inductor current indeed follows the wave shape of the rectified line voltage, full wave rectified line voltage. And you see that that rectified line voltage produces a waveform that is nicely reproduced by the inductor current. As a result, the AC current is essentially sinusoidal in phase with the AC line voltage as required from a low-harmonic PFC rectifier. An interesting waveform is the waveform of the duty cycle. The converter in this case operates in continuous conduction mode. And so the duty cycle follows the wave shape of approaching 1 at the zero crossings of the AC line voltage and going down to a minimum value corresponding to the peak value of the AC line voltage. So the wave shape of a duty cycle follows our intuition of how the converters should operate to convert full-wave rectified AC line voltage into a DC value at the output and, in the process, consume sinusoidal current from the AC line voltage. If you operate at a voltage RMS of 240V, you obtain similar results. But now, of course, at a same power level, the peak of the AC current drawn from the AC line is lower. For the same power, with a higher RMS value of the input voltage, we are drawing less RMS current from the AC line. The wave shape of the inductor current is shown right here, and the value of the duty cycle, varying over time, is shown at the bottom diagram. We can look at the same input voltage, but at a much lower power level. In this case here, discontinuous conduction mode happened over a much wider fraction of the AC line period, and you still see that we retain an essentially sinusoidal waveform, with some distortion in the response at a much smaller amplitude compared to the earlier simulations. The fact that the converter now operates in discontinuous conduction mode is represented well also by the wave-shape that shows how the duty cycle is changing in time. So we have an average circuit model that is powerful enough to examine operation of this universal input rectifier over a variety of power levels and over a variety of input voltage levels. Let's examine, in addition to the input voltage and input current, we also look at the output current and the output power. Those are important waveforms to understand. When you look at this output current waveform right here, you see that it actually varies from 0 up to a maximum value close to 5 amps, then back down to 0, and again up to about 5 amps, and so on. The output voltage is DC, and so the output power, which is the product of the output current and the output voltage, follows the same pattern. p_out(t) is not constant. p_out(t) varies between 0, to a maximum value, and back down to 0. The average value of that p_out is the desired value of 1 kW. But, in the process of operation from the AC line, we are drawing the time varying power from the AC line and delivering that time varying power to the output DC voltage in the form that is shown right here. We will examine this behavior in more detail in the next lecture. To summarize this introduction to low-harmonic power factor correction rectifiers, we know that this is an important application case because the limits in current harmonics are mandated or required, or at least recommended, by various international standards, including this standard that is European but is essentially considered worldwide. Low-harmonic rectifier is a standard front-end converter in systems powered from the AC power line so the application range is very, very wide. A low-harmonic rectifier emulates resistive load Re with respect to the AC power line, which means that it has to have low harmonics in the input current and close to unity power factor since the input AC line current follows the input voltage waveshape. Average current mode control is not the only approach that can be applied to achieve the goals of a low harmonic rectifier. But it is a very practical and very popular approach to control of the input current waveshape. In a single-phase PFC rectifier we know that power has a time-varying component around an average value and that's the fact that we will explore some more in the next lecture.