Last week you learned about the need for a balancing a battery pack and some balancing design criteria and strategies that are involved in that. You also learned about some different passive-balancing circuit designs that you could use when you're balancing a battery pack. This week you will also learn about active balancing and circuits that can be used for active balancing, and active balancing circuits break down into three general categories. The first category moves charge from one cell to another using capacitors in switching mechanisms. The second category moves energy from one cell to another using the principle of inductance, basically using transformers. The final category uses DC-DC converter techniques to discharge high state of charge cells and to charge a low state of charged cells. We're going to discuss all of these designs this week, but in this lesson our focus is on the switched capacitor based designs. I have drawn a circuit on this slide for you that is one type of switched capacitor based balancing scheme. In this circuit, you will notice that there is one fewer capacitor than there are battery cells in the battery pack. In the design, we have single-pole double-throw transistor switches that repeatedly cycle back and forth without any kind of intelligence. So, that as all of the switches are moved to there right position for a period of time and then all of their switches are moved to their left position for a period of time and then this just repeats forever. So, how does this really work? To understand this consider two cells that are right next to each other in the battery pack. Suppose that the capacitor is connected by the switches to a battery cell that has a higher voltage than the capacitor. The battery cell will discharge some of its charge into that capacitor and it will raise the capacitor's voltage and this will continue either until the capacitor has reached the same voltage as this cell or until the switches change position. So, let's consider that the capacitor voltage has reached the same voltage as the cell, then the switch has changed position, and the capacitor is now placed in parallel with the cell that has a lower voltage than itself. So, in that case the capacitor is going to discharge some of its energy into that battery cell until the battery cell reaches the same voltage level as the capacitor either by the battery cells voltage increasing or the capacitor's voltage decreasing to that point. So, over the course of time, the entire battery pack is equalized by equalizing the voltages of all of the cells in the battery pack. But you might imagine this could take a long time. Suppose that the low-voltage cell is at one end of the battery pack and high-voltage cell is at the opposite end of the battery pack, in order to equalize the battery pack the high-voltage cell must first transfer to the cell next to it and then it needs to transfer into the cell next to that and next to it and it propagates throughout the pack from one end to the other, but this propagation can take a significant amount of time. So, we can slightly modify the design on the previous slide to use only a single capacitor and some more intelligence in the switching logic. In this design, the software intelligently connects the capacitor in parallel with the highest voltage cell. That cell charges the capacitor up to its voltage and then the intelligent control switches the capacitor to be in parallel with the lowest voltage cell, and the capacitor discharges directly into that cell and raises its voltage. So, this allows more direct movement of charge from a high-voltage cell to a low-voltage cell without having to spend time changing the voltage of all of the intermediate cells in the process. Both of these designs have a drawback and in fact all capacitor-based designs have a drawback and that is that they depend on there being a significant voltage difference between these unbalanced cells for the method to work. Most lithium ion battery chemistrie's have very little voltage difference variation between cells even if the states of charge of the cells are quite different. So, let's see what this means with regard to how quickly we can balance a battery pack. We're going to approach the analysis from an energy perspective. The energy stored in a capacitor is equal to 1.5 multiplied by the capacitance, multiplied by voltage squared. So, the maximum energy that could be transferred is equal to 1.5 multiply in the capacitance times the difference in squared voltages of the two cells. We can also relate energy to a difference in state of charge. Remember from the first course in this specialization that the energy stored in a battery cell is approximately equal to the state of charge ranges by that battery cell multiplied by the total capacity and multiplied by the nominal voltage of the cell. To simplify our analysis in this case, we're going to assume that the nominal voltage of a cell is approximately equal to the average of the highest and lowest voltages of the cell. That's not going to be perfectly true, but it's going to be close enough for us to get an idea of how quickly a capacitor based design is going to balance a battery pack. So, if we set the two equations for energy equal to each other then we get the following relationship; on the left side of the equation we have the energy related to the change in state of charge and on the right side of the equation we have energy-related to a difference in squared voltages. Now, notice that I've factor the difference of squares into a product that shows how we can cancel one term from both sides of the equation. We really are dividing both sides of the equation by the nominal voltage and also by the total capacity and when we do that we get the final result that one balancing operation changes the state of charge by the capacitance divided by the total capacity multiplying the change in voltage. Let's use the equation from the previous slide with some example numbers to get a sense for how quickly a capacitor based design is able to balance a battery pack. So, consider one cell that has 10 amp hour total capacity and a different cell having the same total capacity. The two cells and the battery pack have a voltage difference though that we'll say is 0.1 volts and we'd like to compute the change in state of charge due to one switching operation. In our design we are free to select any value of capacitance that we want. But notice that real high valued capacitors tend to have high series resistances, and that means that they will charge and discharge relatively slowly. In my simple energy analysis on the previous slide, there was no element of time and the equations it was really a steady-state analysis. So, if we wanted to really understand what happened with high resistance capacitors, we would need to do a much more in-depth circuit analysis. But for this example, I'm going to still select a very large value of capacitance of one farad. This is much larger than we would reasonably use because one farad capacitors are super-capacitors and they tend to have much higher resistance than standard electrolytic or ceramic capacitors and so forth. But even using this really very large capacitance of one farad, we will get a change in the state of charge as you can see by plugging the numbers into the equation that's approximately equal to three times 10 to the negative six or in percent we can multiply by 100 to get a percent change in state of charge of 3.3 times 10 to the minus four percent in one switching operation. So, we're going to need on the order of 10,000 switching operations to change the state of charge by one percent. At this rate it's going to take forever to equalize the cells in the battery pack. So, I find it difficult to think of applications where a capacitor based design would work very well. Now it's possible that these designs might work for some electric vehicle applications because they tend to use a very broad range of state of charge and if the battery cells are really quite out of balance, then there might be an appreciable voltage difference that would enable the balancing or continue to work well enough. But the design was certainly not work well for hybrid electric vehicles where we need to keep the states of charge all in a very narrow band and in fact a very narrow band of state of charge is used. So, the differences in voltage is also very small. To summarize this lesson, you've learned that capacitor circuits can be used to move charge from high-voltage cells to low-voltage cells in a battery pack. This design is less wasteful than passive balancing since most of the energy is conserved. Now, there will be some energy loss dissipated through parasitic resistances in the circuit, but there's certainly no resistance that we add intentionally to the design that dissipates this energy as heat. But you've also learned a significant disadvantage to capacitor based designs is that they balance very slowly. Capacitor based balancing depends on there being a large voltage difference between the battery cells and in lithium ion battery packs voltage differences among cells tend to be very small even if the battery pack is far away from being balanced. There are different capacitor based designs in the literature that charge a single capacitor using the entire battery stack voltage and so that they can charge the capacitor up to a much higher value and then they discharge the capacitor through a resistance to a low voltage cell. The resistance is required so that we don't dump a huge voltage across a cell and cause a safety concern. So, these designs can be faster since the voltage difference between the overall batteries stack and one individual cell can be large, but it does require very careful computer monitoring and control so that the capacitor voltage rating is not exceeded in the first place and also the cell voltage ratings are not exceeded in real time. That brings us to the end of this lesson and in the next lesson you will learn about methods that are based on inductance using transformers and these designs can be faster.