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:

• evPerformance.ods (open office)
• evPerformance.xls (excel)

Download EV rolling resistance calculation spreadsheet:

• evCoastDown.ods (open office)
• evCoastDown.xls (excel)

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!

Goals

What 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:

• It must have a suitable theoretical range (arbitrarily defined as about 80 miles under "ideal" conditions at 55mph)
• It must have acceptable acceleration to 55mph (arbitrarily defined as an unremarkable, but safe 15 seconds)
• It must be able to climb all the local hills. (arbitrarily defiined as a 25% grade, which is steeper than any road around here)

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.

Preparation

There are many measurements and estimates you need to make in order to have any hope of making a semi-accurate 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.

• Weight: Weight factors into all calculations for the performance of an EV. See my EV Weight Change Page for more details on how a car's weight will be affected by the conversion.
• Motor Characteristics: Motor efficiency, horsepower (peak and continuous), voltage and RPM ratings all need to be known for calculating performance of an EV.
• Drivetrain Characteristics: The drivetrain gear ratios are needed to figure acceleration and hill climb angle. Its efficiency will affect range. See my MR2 EV Transmission Modification page for more information.
• Electrical System: The efficiency of the electrical system is needed for the EV range calculation. For acceleration and hill climb ability, the maximum current and voltage ratings of the motor controller is needed.
• Battery Characteristics: Battery energy capacity will directly affect the range of the car. Battery weight affects acceleration and hill climbing ability. See the EV Battery Considerations page.
• Aerodynamic Drag: Aerodynamic drag is the largest external force acting against a car at freeway speeds. It must be factored in when calculating range. It also affects acceleration. Look at some of the links on the EV Reference Material page for more information.
• Rolling Resistance: Rolling resistance is the second largest force external acting on a moving car. It factors into the same calculations as aerodynamic drag, and also hill climb angle. Look at some of the links on the EV Reference Material page for more information.

  Update! After getting the car running, I performed some coast-down tests to deterministically measure rolling resistance (given known curb weight and aerodynamic properties) You can download a spreadsheet above for doing the calculation. My rolling resistance (prior to getting an alignment job and low-rolling-resistance tires) is 0.014 which is not bad, but not as good as what I planned for. (Using that number in my performance analysis spreadsheet makes the calculated range almost exactly match what I have been seeing in actual driving which was encouraging.)

  After the alignment job and installation of my Bridgestone B381 tires, the measured rolling resistance went down to 0.0124. This is still not where I want it (0.01) but a step in the right direction. Probably the reason is that the old (nearly bald) tires rolled a bit better than I gave them credit for. It is well known that old tires roll better than new ones. So, as the new tires break in the rolling resistance should improve further. Also, I can still attempt to reduce brake drag in the rear calipers.

  Note that a reduction in the rolling resistance ratio of 0.0015 is enough to add about 5 miles to my practical range!

Calculations

Input Data

A 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.

Property	Variable Name	Value	Units
Weights used for my calculations
Total Weight (Mass)	Wtot	1450kg (3200lbs)	Kilograms (1 kilogram equals 2.2 pounds)
Motor Characteristics used for my calculations
ADC 9" Rated Voltage	Mvmax	120 V	Volts
ADC 9" Rated RPM	Mrpmmax	5000 rpm	Revolutions per Minute
ADC 9" Rated Power	Mpcon	22000 W (30hp)	Watts (746 watts equals 1 horsepower)
ADC 9" Peak Power (500A controller limited)	Mppk	64000 W (85hp)	Watts (746 watts equals 1 horsepower)
ADC 9" Efficiency (at 20lb/ft of torque)	Effmot	0.88	None
ADC 9" Torque Constant	Mtc	0.312 Nm/A (0.23LbFt/A)	Newton-Meters per Ampere (1 newton meter equals 0.74 foot pounds)
Drivetrain Characteristics used for my calculations
Overall Efficiency	Effdt	0.92	None
Highest gear ratio (Low gear)	Dgrhi	13	None
Lowest gear ratio (High gear)	Dgrlo	4	None
Electrical System Characteristics used for my calculations
Overall Efficiency	Effel	0.95	None
Controller maximum current	Ccmax	500A	Amperes
Controller maximum voltage	Cvmax	144V	Volts
Battery Characteristics used for my calculations
Battery Weight	Wbat	500kg (1100lbs)	Kilograms
Battery volts	Bv	128V	Volts
Battery Energy Capacity (20 hour rating)	Be	24kwh	kilowatt hours
Battery Energy Capacity (75 amp load rating)	Be75	14kwh	kilowatt hours
Aerodynamic drag characteristics used for my calculations
Drag coefficient times frontal surface area	CdA	0.535m2 (5.76sqft)	square meters (1m2 equals 10.76sqft)
Rolling resistance characteristics used for my calculations
Total Rolling Resistance (as a proportion of vehicle weight)	Rr	0.01	None

Download Spreadsheet:

• evPerformance.ods (open office)
• evPerformance.xls (excel)

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.

Acceleration

Goal: Calculate zero-to-55mph acceleration in seconds at maximum power.

Input Data:

• Target speed: V_target = 25m/s (56mph)
• Car weight: W_tot = 1450kg (3200lbs)
• Car aerodynamics: C_dA = 0.535m² (5.76sqft)
• Rolling resistance factor: R_r = 0.01
• Motor torque constant: M_tc = 0.312nm/A
• Drivetrain gear ratio (4th): Dgr_lo = 4
• Maximum controller current: C_c = 500A

Simplifying assumptions: (and their expected effects on the calculation)

1. Motor output torque is constant. This assumption allows me to use a constant value for acceleration force. This assumption, however means that effective horsepower output by the motor will increase as the car speeds up. Horsepower is proportional to RPM multiplied by torque. This assumption will result in a more conservative acceleration time calculation.
2. All acceleration in 3rd gear. This is the lowest gear that will not exceed the motor's maximum RPM at the target speed. This simplification will make for a longer overall acceleration time than if I figured in shift points.
3. Perfect tire traction. There is a good chance that under hard acceleration (especially in 1st gear with low rolling resistance tires) I would spin out, but for this calculation, that possibility is ignored.

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/s². 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 sub-six-second 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 back-of-the-envelope 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 Angle

Goal: Calculate hill-climb angle at maximum power in first gear.

Input Data:

• Car weight: W_tot = 1450kg (3200lbs)
• Rolling resistance factor: R_r = 0.01
• Motor torque constant: M_tc = 0.312nm/A
• Maximum controller current: C_c = 500A
• Drivetrain gear ratio (1st): Dgr_hi = 13

Simplifying assumptions: (and their expected effects on the calculation)

1. Perfect tire traction. There is a good chance that on an extremely steep hill, the tires could slip. (especially in 1st gear with low rolling resistance tires) For this calculation, that possibility is ignored.

2. Aerodynamic drag is ignored. Extremely steep hill climbing is not done at speeds sufficient to consider this.

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.

Range

OK, 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 apples-to-apples comparison, theoretical range is a useful measurement for comparing the performance characteristics of EVs.

Goal: Calculate theoretical range at 25m/s (55mph)

Input Data:

• Target speed: V_target = 25m/s (56mph)
• Car weight: W_tot = 1450kg (3200lbs)
• Car aerodynamics: C_dA = 0.535m² (5.76sqft)
• Rolling resistance factor: R_r = 0.01
• Drivetrain gear ratio (4th): Dgr_lo = 4
• Battery capacity (75amp): Be₇₅ = 14KWh
• Electrical system Efficiency: Eff_el = 0.95
• Motor Efficiency: Eff_mot = 0.88
• Drivetrain Efficiency: Eff_dt = 0.92

Simplifying assumptions: (and their expected effects on the calculation)

• Ideal conditions assumed: Constant speed. Neutral wind. Perfectly level. Sea level. These conditions won't be achieved normally, but the assumption must be made since their effects are difficult to quantify. The good news is that under moderate traffic conditions, there should be a net tailwind due to drafting other vehicles.

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 kilowatt-hours for 75-amp 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.