In this lecture, we look again at the boost PFC rectifier where we employ average current mode control to have the input current follow the input voltage wave shape. So remember, our goal was to design a rectifier that can operate from universal input anywhere in the world from any frequency 50 or 60 Hz or any input voltage between 85 and 260 volt RMS, and produce a DC output voltage across the load, a DC output voltage being selected to be higher than the worst case peak of the AC input voltage. A particular issue that we want to address in this lecture is the need for energy storage in single-phase, low-harmonic or PFC rectifiers. After viewing this lecture, you will understand how we need to size the output filter capacitor C, and how that need for energy storage results in requirements with respect to the energy storage capacitor C and the output of the boost rectifier. As a reminder, the input voltage is assumed to be sinusoidal with a peak value of V sub M, and the input current, given that our average current mode controller is operating properly, is also a sinusoidal waveform with a scale factor equal to the emulated resistance R sub e. We constructed an average circuit model to verify operation of our average current mode controller for the boost PFC rectifier. And as a reminder, we obtain these waveforms that show the input voltage, the input current, and also the output current and the output power for a given numerical example of 170 volt, peak AC live voltage which corresponds to the 120 volt RMS AC line at one kilowatt. And the point that we made is that the output power that is feeding into the output node of the rectifier is not a constant value, but is instead moving up and down from zero to a maximum value with an average that represents the average power taken from the AC line. Let's look into that in a little more detail. Here's what we can say about these wave forms. The output current is equal to the input power divided by the output DC voltage. Now, if the input voltage is sinusoidal, the output current turns out to have a sinusoidal square waveform. A sine square can be written as 1- cosine 2 omega t. And so the output power will in fact follow the same waveshape. The output power is going to follow average value times 1- cosine 2 omega t, where omega represents the line frequency. The waveform shown in the previous slide showed that the output current, and thereby the output power, have an AC component at twice the line frequency. If the input frequency is 50 Hz, the frequency of the output current feeding the output filter capacitor and the load is going to be 100 Hz. For a 60 Hertz case, we will have 120 Hertz component in the output current. Now notice that that does not depend at all on what switching frequency we select in the realization of a DC-DC converter. Switching frequency can be 100 kilohertz or 200 kilohertz, or megahertz, it doesn't really matter. In all those cases, the output current is going to have an AC component at twice the line frequency because of the AC nature of the single-phase input. The AC input voltage and the AC input current in phase with the input voltage result in the time-varying power output that follows this twice line frequency AC component. And that is the fundamental reason why we need the substantial energy storage capacitor at the output of the rectifier to produce a DC voltage across the load, or close to DC voltage across the load, in the presence of this time varying output current. The waveforms that illustrate the operation on the output side in a PFC rectified are shown right here. Power has an average component that's equal to Pload, the Pload that's delivered to the DC load is essentially DC. Around that, there is the AC component of the output power that follows this twice line frequency waveform. Now for a given value of the output filter capacitor, the energy storage capacitor, this time varying component in the power is going to result in a time varying component of the voltage, inevitably. How large is the ripple in the output voltage will be determined by the size of the output filter capacitor. The output capacitor voltage will vary between the maximum value and the minimum value with twice the ripple delta V that is going to be a design set point. So we are going to say we will allow the ripple of so many volts and that in turn, will tell us how large the filter capacitor needs to be. Let's do that. So, the energy storage on the output filter capacitor at a maximum voltage is given right here 1/2 C Vcmax squared minus the energy storage at a minimum value of the output voltage. That has to be equal to the integral of power, the AC component of the power over the half line cycle. When you compute this integral here, you get Pload over omega, where omega is 2 pi times the line frequency. Now sorting here the difference and representing that difference in the form of 1/2 C times 2 times the DC output voltage, the output voltage being equal to the midpoint value between the maximum and the minimum, and twice the ripple voltage delta V, equate to that to Pload over omega gives us the final expression for the required value of the capacitance for a given ripple. As an example, let's suppose we have a power of 1 kilowatt, the output voltage of 400 volts, with a desired ripple of 12 volts and at a frequency of 50 hertz, the required value of the capacitance comes out to be 330 microfarads. Again notice that this value of the energy storage capacitance does not depend at all on the switching ripple considerations or the switching frequency. Presumably, such a large value of the energy storage capacitor will also be sufficient to attenuate sufficiently the switching ripple at the output voltage. As a verification, let's re-simulate our circuit with a working average current mode control loop. But now, with the output, which used to be an ideal voltage sink, replaced by a combination of an actual resistive load, and the actual energy storage capacitor Cout, with 330 microfarads. So, we simulate this circuit in time domain, in the transient, and waveform shown right here, show the approach to steady state operation, with the output voltage shown, with the AC ripple at twice the line frequency, with the amplitude that is close to the desired value. We planned for 12 Volts of delta V. We have 2 times delta V of about 24 Volts. So the choice of the output filter capacitor meets the output ripple requirements. Finally, we also now discussed how the output voltage is in fact regulated. So far we have simply adjusted the amplitude of the input current to obtain the desired 400 volts across a given load resistance. But in a practical case the load varies in time and we would like to maintain the output voltage regulator and the desire value of let's say 400 volts regardless of what the load resistance we have, or, to be more precise, regardless of how much average power is taken from the AC line. A typical way of obtaining that output voltage control, which is really what should be viewed as a control of the average power, balancing the power taken from the AC line and the power that is delivered to the load, is shown on this diagram. The output voltage is sensed with a senseing gain H, it's compared to a reference, The error signal is processed by a voltage loop compensator to produce a control signal which then scales the input voltage waveform. This control voltage here is really a factor that determines the scale between the input voltage sensing and the control signal for the current control. So a larger control input is, the larger the amplitude of the signal at the input or the control input of the current control loop is, the larger the amplitude of the current is going to result, the larger the power is drawn from the AC line. Notice that it is important for this control signal here to remain essentially constant over the entire line cycle so that the input current remains undistorted. If the control input were to move up and down quickly with respect to the AC line waveform, the wave shape of the control current for the current mode control loop would no longer reproduce the sinusoidal waveform on the input side of the rectifier and we would not meet the low harmonic rectification requirement. The voltage loop compensator in conclusion needs to be relatively slow. Control the output voltage, but at a relatively slow rate so that the control input does not change much in time over a line cycle period. More details about how to design a voltage loop compensator can be found in the reference textbook in section 18.4. In this lecture we will only look at one particular example without going into further details of how exactly that voltage loop compensator is designed. So the example is shown right here with a scale factor to bring the reference voltage down to a practical value of 3 volts, and the simple PI compensator in the voltage loop that provides that control input which, in turn, scales value of the input voltage to the value that shows up as the control input for the current control loop. So again, the input current needs to follow the input voltage waveshape, but the amplitude of that input current is determined really by the control signal that comes out of voltage loop compensator. A transient simulation is shown right here, an approach to steady state operation, with the output voltage now showing this ripple we are now familiar with. The input current, inductive current showing a rectified sinusoidal waveshape and a control voltage showing that it doesn't change much over an AC line period. Finally, this diagram here shows a complete power system tied to AC line voltage. Typically, we have a PFC rectifier, that is the front end interface to the grid voltage. That rectifier provides a DC voltage across the energy storage capacitor, and then, in most cases, the DC voltage is taken as the input voltage for follow-up downstream DC-DC converters that provide tightly regulated voltages to the loads in the system. This is the type of diagram that you have present in computer power supplies or really any power supplies that are providing electronics with DC power from the AC line. Details of the realization can be different. In many cases different types of converters are used, in many cases different control schemes are used, but the principles that we have explained in this lecture are the same. So, in conclusion, low harmonic rectifiers are the way to interface AC grid to DC loads. Limits in current harmonics are mandated by various international standards, which really require this effort in low harmonic rectification on the front end. This is a standard system for powering electronics from the AC power line. A low harmonic rectifier emulates resistive load with respect to power AC line which means that it has a resulting input current with low harmonics and close to unity power factor. We found that energy storage is required in single-phase systems. There are multiple control loops involved. An average current mode controller is typically used for the input current, and a slow voltage control loop is used to regulate the DC voltage across this energy storage capacitor.