# EV Performance Analysis

This page discusses performance analysis for an EV Conversion, starting from a known chassis and desired performance parameters, e.g. range and horsepower.   This page is heavy on EV Terminology, especially with regard to efficiency and batteries.

# Computing Power

Note:   746W is one horsepower.   So,  one kilowatt is about 1.3 horsepower.

Computing the maximum power for your EV conversion is relatively easy.  You basically need to identify the weakest link,  and it will be the limiting factor in the horsepower the vehicle can deliver to the wheels.   Usually the limiting factor is either the batteries or the inverter/controller.     For example,  if you have a 100KW inverter,  but the 120V battery is only rated to deliver a maximum of 500 amps without damage, the maximum possible power you should expect is 500A*120V or 60KW.   In reality it will be less than this as the battery voltage sags as the current goes up.    The specifications for your battery should give some indication as to voltage sag at a given current  output, and you can use this to refine your estimate of maximum power.

## Sizing Battery for Power

Regardless of battery pack voltage,  if the AH (amp hour) capacity of the battery is too small,  you will not have sufficient range OR power.  Nobody expects to start a car on a string of eight AA cells, even though that is 12V.   Similarly,   you need to ensure the AH capacity of the cells in your EV is large enough to provide adequate power.   For this, you need to look at a couple of factors:

1. maximum amps your inverter/controller can pull.    The batteries you pick should be large enough to be able to supply this much current,  at least for a short time.  If you are building for range and not power,  it is probably OK if the controller can pull a bit more than the optimum C rate for the batteries,  but you should tune your controller and BMS to avoid pulling more than 2 or 3 times the maximum continuous C rate for more than a few seconds.    If you are building a drag car,   the batteries should have a C rate allowing maximum controller amps continuously, or close to it.   That would probably mean something other than prismatic LiFePO4.
2. Average expected power.   This is what you might see cruising on the highway.   Figure the amps at this power (which you can estimate,  or get from intermediate steps of the range calculation in the next section).   You want the continuous amps below the C rate of the battery, to ensure that the battery will not overheat or be damaged by normal driving.   I feel it is best to keep freeway power under 0.5C.
• C Rate:  My GBS cells have a 2C or 3C power density, depending on who you ask.   As these cells are known to have higher internal resistance than some other LiFePO4 I will be conservative and use the 2C rating.   This means that they can theoretically continuously deliver 200A for 30 minutes (equalling 100AH) without damage.
• Maximum Power:  My UMOC445TF inverter can pull a maximum of 250A, or about 125% of the maximum continuous discharge rate for 100AH, 2C battery.  However it can only do this accelerating at full throttle, which I rarely need to do (unless goofing around) and even then it only lasts a few seconds, until the motor RPM comes up enough to start limiting the maximum amps the inverter can pump through it, which owing to my gearing and the torque/voltage curves of the Solectria AC55 motor happens at about 3000rpm or 40mph.  This is well within limits of my batteries.   However if I had a DC motor and a Zilla Z2K controller that can pull 2000A,  a 2C, 100AH battery would be woefully inadequate.
• Average Power:  Going from the intermediate calculations in the range section,    (up to step 11 of the rigorous efficiency calculation)  my xB needs 18KW of power to maintain 60mph.   That works out to 18KW/320V or 56A.   This is 27% of maximum discharge rate from a 100AH, 2C battery, well within safe margins for continuous use.    However if I had chosen the 40AH GBS with the same C rate, this would be 70% of the maximum safe discharge current.  This would not leave much margin for accelerating, hill climbs, and whatnot which dramatically increase the power requirement.

# Computing Range

Even with sufficient power density (especially if using high C rate cells) you need to ensure you have enough AH capacity for your desired range.   A given increase in range requires proportionally more AH capacity,  more or less.   Obviously there are practical limitations that come into play as well:  volume, weight, and cost are the main ones.

When converting a gas car,   In order to get to desired range you need some idea of the efficiency of the chassis.   Once you get that,  you can do some math to get to the required battery pack size.   There are a lot of ways of going about this.   Here are three ways:   A rather drawn out Rigorous method that to mathematically model major aspects of the car to get to a calculation of needed battery pack size for given circumstances;  a Easier method for doing the same that takes a short cut by observing that gas mileage is a proxy for the overall efficiency of the car, and finally a One Liner to go from gas mileage and desired range to battery pack size that sums up my experience pretty well.    Finally,  If you are totally math phobic (you should have that looked at) I include some Rules of Thumb as well.

## The Hard (Rigorous) Way

This calculation attempts to include all forces acting on the car,  and account for all efficiencies (or lack of) in the system to come up with a range, given known properties of the car body and components chosen.

• Note that these numbers are from my scion xB; and apologies for switching between SI and english units willy nilly; these numbers come from my notes;  I did not recalculate them here.
• Note also several of my assumptions are intended to be pretty conservative.   It is better to error on the side of caution here.  For example,  It might be safe to figure better drive train efficiency or better electrical efficiency,  and a few percent in a couple of places can make a significant difference in the final number.
• Note that speed makes a big difference owing to the fact that wind drag increases as the square of speed.   This means (roughly speaking) at 10% reduction in speed will work out to a 10% increase in range.   This is because 10% less speed will use about 20% less power, allowing you to go about 10% farther on the same amount of power)
1. Define driving conditons, say 60mph straight and level.
2. Define desired range, e.g. 100 miles.
3. Compute Wheel RPM at that speed from tire diameter.    Tire diameter for the  Dunlop Enasave Tires LRR on my xB is 23.3″ according to Tire Rack.  This works out to 866 RPM at 60mph.    Since I do not have better numbers,   I will assume 0.01 Rolling Resistance coefficient for these tires,  though they are probably considerably better than that.
4. Compute the aerodynamic drag at that speed.  Start with the CdA of the car (drag coefficient multiplied by frontal surface area) and you can calculate force of wind resistance at some given speed.   CdA for my xB is 0.32(Cd) * 2.48(A, m^2) or 0.8.  Running this through the drag force calculation behind the link gives 80 pounds of drag at sea level.
5. Compute the Rolling Resistance. Calculated Rolling Resistance.  This is roughly constant regardless of speed for a given car;  but is proportional to the weight of the car and is greatly affected by tire choice, alignment, and road surface.   If using Low Rolling Resistance Tires,  a good starting point is 0.01 multiplied by the weight of the car.   You will of course have to estimate the weight of the car since that includes the battery pack whose size you are figuring, but you can get pretty close.   My xB weighs 3000lbs, so this works out to about 0.01*3000 or 30lbs of rolling resistance drag force.
6. Compute drive wheel torque knowing the raw drag forces acting on the car (wind and rolling resistance) required to overcome those forces.     80 + 30 or 110lbs of drag force.  Torque (ft/lbs) the drive wheels then must produce is this force divided by wheel radius:  110/(23.3/24) or  113 ft/lbs.
7. Compute drive wheel power from RPM and Torque.   If using Horsepower,  It is simply (RPM * Torque) / 5252.   To get from Horsepower to Kilowatts (KW),  multiply horsepower by 0.746.   In other words,  100 horsepower is about 75KW.    For my xB at 60mph,    Horsepower requirement is (866RPM*113ft/lb)/ 5252 or 18.68 horsepower.   18.68HP*0.746 = 14KW.
8. Estimate drivetrain Efficiency.   All bearings, brakes, transmissions, differentials, universal joints, splines, and CV joints, and rubber seals will cause a small amount of friction.   I typically use:
• 99% for wheel bearings
• 99% CV/universal joint shafts (non extreme drive angle)
• 97% for disk brakes and 99% for drum brakes (to allow for brake drag).  Use 98% if the car has disks in front and drums in back.
• 95% for a manual transmission, transfer case, or transaxle (includes differential),
• 98% for a belt or chain drive
• 96% for a hypoid differential (with pinion perpendicular to axle, as in most rear drive vehicles).
• These are my best conservative estimates based on the the complexity of the components and what I have learned.  Simply multiply all the efficiencies of the components you have together to get an estimate of your drivetrain efficiency.   It will probably be around 90% for most two wheel drive cars.    For my Xb,   I figure 99%(wheel bearings) * 98%(disks and drums) * 99%(CV driveshafts) * 96%(Hypoid Diff) * 98%(Belt Drive) or 90.3% drivetrain efficiency.
9. Compute Motor shaft power by simply dividing the Wheel power in KW by the efficiency of the drivetrain.    For my xB, 14KW/.903 or 15.5KW shaft power.
10. Estimate electrical efficiency.   You can usually find specifications for the motor and inverter you are using that will give you an idea of the efficiency of the motor at the RPM and Torque output it will be running at given your gear ratio and shaft power.   But failing that,   90% motor efficiency is a good number to start with.   Figure 98% efficiency for the inverter/controller and 99% efficiency for the high voltage wiring and 98% for the batteries themselves.   Like with the drivetrain,  Multiply all these efficiencies together to get an overall electrical efficiency.   For my Solectria AC55 Motor and UMOC 445TF Inverter These numbers are good to use.  The AC55 motor efficiency peaks at 93% but I use 90% to account for non-ideal conditions,  I get 90%(motor) * 98%(inverter) * 99%(wiring) * 98%(batteries) or 85.5% electrical efficiency.
11. Compute Battery output power.   To do this, divide motor shaft power by electrical efficiency.   For my xB:  15.5KW/0.855 or 18.1KW from the batteries.  Note from here you can get amps and volts.  E.G. for a 320V battery,   It will need to put out approximately 18100W/320V = 56A.   In reality,  Volts will be a little lower, and amps a little higher, owing to voltage sag of the batteries.
12. Figure time at desired speed to achieve desired range.   For 100 miles at 60mph,  you need 1.66 hours.
13. Compute Usable KWh.   Multiply hours drive time by battery output power to get usable KWh or kilowatt hours the battery needs to deliver.  1.66*18.1 or about 30KWh.
14. Compute Battery Nominal Size.  Divide needed KWh by maximum DOD (Depth of Discharge) of the battery.   This is usually 80%, or 0.8.    30/0.8 = 37.5KWh
15. (Bonus) Compute WH/Mile.   You can do this by taking the usable KWh and divide by the distance.   This works out to 30/100 or .3KWH, or 300WH/Mile.

So Given these calculations and assumptions;  I would want a 37kwh battery to go 100 miles, achieving 300WH/mile.    37KWh is a pretty big battery.  I ended up using a 32KWh battery, or One Hundred 3.2V, 100AH cells.

Using the above 300wh/mile and 80% DOD assumptions,   the actual range of my scion xB using a 32KWh battery instead of the 37KWH one is (32*0.8)/0.3 or 85 miles.

I know this calculation is conservative,  as I have been measuring driving efficiency more like 250WH/mile now that the car is running.   Most likely this difference is due to better than anticipated rolling resistance and drivetrain drag.    Only a 20% improvement in Rolling resistance (Using 0.008 instead of 0.01) once compounded through the other efficiencies in the system,  would add about 5 miles of range; adding back 5% of efficiency to the electrical and drivetrain each would add another 9 miles or so).   This would give 100 miles of range, and that would work out to about 250wh/mile which is what I am seeing.

## The Easier Way

You can short-circuit an awful lot of the calculation by simply observing that the vast majority of the external forces on a given car at a given speed are the same whether it is gas or electric.   Wind drag is exactly the same,  and rolling resistance (if going from normal tires to LRR) is likely to be the same or a bit less, even accounting for extra battery weight.   If you then know the approximate gas mileage the car gets,  You can work backwards from that to an estimate of the range of the car as an electric.

1. According to edmunds.com,  the 2004 scion xB gets 31MPG highway.
2. One gallon of regular gasoline contains 33.6KWh of energy.
3. This means, measured as Wh/Mile,  a gas powered xB gets about 33.6/31 or 1083Wh/Mile.
4. A typical gas drivetrain is 20% efficient; most of this horrible efficiency is in the ICE, or internal combustion engine.   A typical EV is about 80% efficient Battery-To-Wheels, or in the case of my xB from the above calculation,  The overall efficiency battery to wheels is estimated at the electrical efficiency multiplied by the driveline efficiency,  or 0.855*0.903 or  0.77 or 77% efficient.
5. This means that the electric xB should be 0.77/0.20 or 3.86 times more efficient.
6. Taking the 1083Wh/mile gas efficiency and dividing by 3.86 should yield estimated electric Wh/mile:  1083/3.86=280WH/Mile
7. From this point,  You can calculate needed battery size.  For a 100 mile range,  you will need 28KWh of usable battery, and 35KWH nominal battery size to stay under 80% DOD.

So only looking at the relative efficiency of gas vs. electric and thus needing fewer assumptions,  you get a similar answer.   If I use the 280Wh/Mile and my 32KWh nominal battery pack,  I can work backwards to get estimated range:  32KWh*0.8/280Wh/Mile = 91 Miles.

## The One Liner

if you are impatient:

((1/MPG) * 8 * Range) / (Max DOD) == KWh battery size

The “8” term is the catch-all conversion factor.  Based on my derivation and observations below the correct term could range from 7 to 10.  I use 8 as a moderately optimistic value.

An example from my xB (I go into detail later) is:

((1/31MPG) * 8 * 100mi) /  0.8DOD == 32.5KWh

Meaning I should have a 32.5KWH battery in order to expect 100 miles of range.

#### Derivation of one liner

Observing that Gas Mileage is a proxy for many of the attributes of the chassis that affect efficiency,  and assuming relatively normal driving conditions (60mph freeway driving) , The “easy” calculation above can all be boiled way down to a really quick estimate to convert from Gas mileage to WH/Mile:

(1/MPG) *  K.

(1/MPG) is simply converting from Distance Per Energy to Energy Per Distance.  K is the issue.  For any given car, K is a constant that factors in all of the efficiencies of the car, the expected range, and driving conditions to convert from 1/MPG to Wh/Mile.  If I calibrate K according to the first (rigorous) calculation, which yielded 300Wh/Mile,  then K should be 9,000.    If I calibrate K according to the second calculation,  which gave a bit more optimistic of an answer, K should be 8400.    If I calibrate K to match the actual measured 250Wh/Mile I am seeing driving at 60mph,  K is 7,500.

I will use a “K” value of 8000 since it is somewhat more conservative that the value computed from what I measured actually driving.

So the, a formula to go From gas mileage (MPG) to battery capacity (KWh) for a given desired range (Miles) then would be:

((1/MPG) * (8000) * Range) / 1000 * (Max DOD) == KWh battery size

The 1000 term converts from Watts to Kilowatts., so factor it out:

((1/MPG) * 8 * Range) / (Max DOD) == KWh battery size

Plugging in numbers from my xB:

((1/31MPG) * 8 * 100mi) /  0.8DOD == 32.5KWh

This says I should need a 32.5KWh battery to get 100 miles of range in a car like my xB that originally got 31MPG highway.    This is almost exactly what I am actually seeing, in a fairly simple formula.    This formula should be a relatively good estimate for most passenger cars,  though I would be less trusting of it for a full size pickup, van, or other large vehicle.   Still, it does use the gas mileage as a proxy for the result of most of the complex calculations (Rolling Resistance and Wind Drag) so it should be pretty close if the gas mileage measurement is accurate.  Again, apologies for the SI to English mix.   I used units that were most convenient for the given calculation.

Use the gas mileage YOU get, not the mileage on the dealer sticker, if at all possible.

## Rules of Thumb

As promised.  Some scenarios. Range numbers assume 60mph freeway cruise are of course estimations based on some combination of my observations, experience, and calculations. Battery pack parameters all assume 3C LiFePO4 prismatic cells. All scenarios assume motor/inverter/controller sized to match the battery, chassis is in good condition and has LRR tires, or at least properly inflated 44psi or higher stock size tires. In the freeway column, if I indicate No or Marginal “V” indicates for voltage too low, and “C” for AH capacity too low.

Chassis Type GVW Example Battery Volts Battery AH Battery KWH Range Freeway?
Subcompact Hatch under 2000lbs Geo Metro 3 Door 72V 100AH 7.2KWH ~30mi No(V)
Subcompact Hatch under 2000lbs Geo Metro 3 Door 156V 60AH 10KWH ~50mi Yes
Subcompact Hatch under 2000lbs Geo Metro 3 Door 156V 100AH 15.6KWH ~80mi Yes
Compact Hardtop Sports Car 2000-2500lbs MR2, Paseo, Del Sol, etc 120V 60AH 7.2KWH ~25mi Marginal(VC)
Compact Hardtop Sports Car 2000-2500lbs MR2, Paseo, Del Sol, etc 156V 100AH 15.6KWH ~70mi Yes
Compact Hardtop Sports Car 2000-2500lbs MR2, Paseo, Del Sol, etc 250V 60AH 15KWH ~70mi Yes
Compact Hardtop Sports Car 2000-2500lbs MR2, Paseo, Del Sol 320V 60AH 18KWH ~80mi Yes
Compact Hardtop Sports Car 2000-2500lbs MR2, Paseo, Del Sol 250V 100AH 25KWH ~100mi Yes
Compact Hardtop Sports Car 2000-2500lbs MR2, Paseo, Del Sol 320V 100AH 32KWH ~130mi Yes
Compact Wagon or Midsize Sedan 2500-3000lbs xB, xA, Datsun 120V 100Ah 12KWH ~35mi Marginal(V)
Compact Wagon or Midsize Sedan 2500-3000lbs xB, xA, Datsun 156V 60Ah 10KWH ~30mi Marginal(C)
Compact Wagon or Midsize Sedan 2500-3000lbs xB, xA, Datsun 156V 100Ah 15.6KWH ~50mi Yes
Compact Wagon or Midsize Sedan 2500-3000lbs xB, xA, Datsun 250V 60AH 15KWH ~50mi Marginal(C)
Compact Wagon or Midsize Sedan 2500-3000lbs xB, xA, Datsun 250V 100AH 25KWH ~75mi Yes
Compact Wagon or Midsize Sedan 2500-3000lbs xB, xA, Datsun 320V 60AH 18KWH ~60mi Yes
Compact Wagon or Midsize Sedan 2500-3000lbs xB, xA, Datsun 320V 100AH 32KWH ~100mi Yes
Large Sedan/Wagon, Small Van/SUV 3000-4000lbs Integra, Taurus, VW Bus 156V 100AH 15.6KWH ~35mi Marginal (VC)
Large Sedan/Wagon, Small Van/SUV 3000-4000lbs Integra, Taurus, VW Bus 156V 130AH 21KWH ~50mi Marginal (V)
Large Sedan/Wagon, Small Van/SUV 3000-4000lbs Integra, Taurus, VW Bus 250V 60AH 15KWH 40mi Marginal(C)
Large Sedan/Wagon, Small Van/SUV 3000-4000lbs Integra, Taurus, VW Bus 250V 100AH 25KWH 60mi Yes
Large Sedan/Wagon, Small Van/SUV 3000-4000lbs Integra, Taurus, VW Bus 320V 100AH 32KWH 80mi Yes