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The purpose of this page is to cover the mathematical analysis that I did to try and get a feel as to the performance characteristics of my Toyota MR2 EV Conversion would be. Also provided are some "rules of thumb", useful both for validating calculations and for those of us who hate math. Download hypothetical EV performance analysis spreadsheet: Download EV rolling resistance calculation spreadsheet: Example numbers plugged in to the formulas in the above spreadsheet are estimates and actual measurements from my Toyota MR2 EV. The formulas are (for the most part) in SI units. Conversions to english units are provided where meaningful. Some of the formulas I pulled right out of textbooks and from http://wikipedia.org, and others I derived and/or rearranged, so beware of inaccuracies. As of when I wrote this page, I havent built the car yet to validate these calculations! Page Topics:GoalsWhat is important to know about the potential performance of an EV? In my case I want to know that the following performance aspects would be acceptable to me:
What about top speed? Well, frankly if the car can meets all the above requirements, then top speed should be fine. So, I haven't bothered to explicitly calculate this. However, if one wanted to calculate it, it would be the minimum of either the speed at which all drag forces exactly equal the acceleration force from the motor, or the speed at which the motor reaches its maximum RPM. PreparationThere are many measurements and estimates you need to make in order to have any hope of making a semiaccurate calculation of what the acceleration, hill climb angle, and range for an EV will be. They are listed briefly below. I discuss them in detail on the EV Performance Input Data page.
CalculationsInput DataA summary of all the input data and estimates I recorded for my Toyota MR2 EV (discussed in detail on my EV Performance Input Data page) is listed below.
Download Spreadsheet: Below, is a summary of each calculation (acceleration, hill climb angle and range) on the spreadsheet. I describe the input data, any simplifying assumptions I made, and the results for each of the areas of interest that I identified. The spreadsheet above for the actual calculations. AccelerationGoal: Calculate zeroto55mph acceleration in seconds at maximum power. Input Data: Simplifying assumptions: (and their expected effects on the calculation) Given the above input data, it is fairly straightforward to calculate the acceleration ability of the car. Simply use Sir Isaac Netwon's formula: F=ma, or Force equals mass multiplied by acceleration. This can be rearranged to a=F/m or acceleration equals mass divided by force. My data suggest the car can accelerate at 1.93 m/s^{2}. That is about 1/5th of a G. This isn't quite what I am looking for, which is the time to get from zero to the target speed. But, simple physics says V=At, or velocity equals acceleration multiplied by time. If I know velocity and acceration, I can rearrange the equation to t=V/A and plug in the numbers and solve for the time in seconds. My spreadsheet gives a result of about 13 seconds and change to accelerate to 25 meters per second, or 55mph. in third gear. The spreadsheet gives two calculations, one which factors in deceleration caused by aerodynamic drag, and another which does not. The values are very close, which is logical since the force from aerodynamic drag is much smaller than the force that the motor can produce. 13 seconds to get to 55mph in 3rd gear meets my criteria for performance, but that is a fairly sedate time for a sports car. But, consider that all my simplyfing assumptions resulted in a more conservative acceleration time. For example, I would normally start out in first gear. If I had picked first gear for my acceleration calculation (ignoring RPM limitations of the motor) the spreadsheet gives a subsixsecond time to 55mph. In reality, I could only accelerate to about 25mph (maximum speed per gear ratio is also provided on the spreadsheet) before having to shift. By shifting gears while accelerating as one normally would, the time would be several seconds faster. A backoftheenvelope calculation would average the 1st gear and 3rd gear times and add a bit of time for shifting, but that still gives you 9 or 10 seconds to 55mph which isn't too bad. Also, had I considered constant horsepower instead of constant torque, then the motor torque number would have been vastly higher at low RPMs since horsepower is proportional to RPM multiplied by torque. Figuring that would have meant a much faster acceleration at lower speeds. But performing the calculation that way would be academic anyway as the tires would undoubtedly spin out if I dumped maximum power (64,000 watts with a 128 volt battery and 500 amp controller) into the motor from while in first gear at a standstill. Hill Climb AngleGoal: Calculate hillclimb angle at maximum power in first gear. Input Data: Simplifying assumptions: (and their expected effects on the calculation) For this calculation, I solved for the maximum force at which the motor can push the car forwards in first gear, figuring only rolling resistance of the tires working against it. Then, knowing this, I know that the maximum hill climb angle will be the angle at which gravity pulls the car backward with an equal force. This would mean that the car can maintain its current speed, but not accelerate. By knowing the gravitational force pulling the car backward, and the overall weight of the car, I used trigonometry to come up with the hill climb angle. I also expressed this in a percent grade (more commonly used for describing hills) on the spreadsheet. plugging in all the numbers, my spreadsheet gives a hill climb angle of 28 degrees. This is a 54% grade, which more like a wall than a road. No problem here. None of the simplyfing assumptions in this calculation affect the math, so this should be an accurate number. I would probably lose traction long before I run out of power. RangeOK, first of all, range is the thing that everybody wants to know about first. Its also the weakest aspect of your typical EV. So that means people tend to muddle the numbers, and make up different definitions of range, to suit whatever they are trying to say. But near as I can figure, what really matters is "practical range", which (for lead acid batteries) is defined as 80% of the maximum possible range the car could attain, totally draining its batteries. (that would ruin them if done too often). But maximum possible (theoretical) range is what the math gives you. So I'll start with that. You need to keep in mind that practial range is going to be at most 80% of the theoretical range, and (allowing for real world driving conditions) a factor of 50% is more reasonable. But, in the interest of applestoapples comparison, theoretical range is a useful measurement for comparing the performance characteristics of EVs. Goal: Calculate theoretical range at 25m/s (55mph) Input Data: Simplifying assumptions: (and their expected effects on the calculation) Figuring range is simply a matter of figuring out how many watts of power it takes to keep the car going, and dividing the battery energy capacity by that value. I accounted for the overall efficiency of the car in getting energy from the batteries to the wheels (multipy Eff_{mot}, Eff_{el} and Eff_{dt} together), then divided the total power needed to overcome drag forces on the car at speed by that to get the total watts being drawn from the batteries. Finally, I divided the energy capacity of the battery (in kilowatthours for 75amp draw) by the kilowatts drawn to propel the car, to get its run time in hours. I multiplied the run time by the target speed to get range in miles. I calculated a theoretical range of about 70 miles with the input data above. This is a little under what I am hoping for. However, using slightly more optimistic numbers for Electrical system efficiency, rolling resistance, and aerodynamic drag reduction due to drafting other vehicles will result in a longer range. Note that this means my expected practical range was at minimum about 35 miles. After driving it (and making some efficiency improvements) the car currently has a practical range of about 45 miles. This calculation is probably the most educational of the three. It makes very clear what aspects of the car most heavily influence its range. These are clear: rolling resistance and aerodynamic drag. Anything and everything that can be done to reduce these should be done in an EV. The extra weight from the batteries does affect the rolling resistance, but it is a small component of the overall drag experienced by the car. 