Range speculation

[WarpedOne]

EDIT: This numbers are wrong. Take a look at my next post down below!

It is known fact that required power raises aproximately with CUBE of speed. Doubling the speed requires eight times the power.
Tesla has 185kW electric motor that is capable of accelerating it to maximum speed of 135mph. It also has 50kWh of charge.
I did some calculations based on these numbers and came up with this table:


SpeedPowerTimeRange
(mph)(kW)(h)miles
50.01532026599
100.16656650
150.31972955
302.024.6739
404.810.4416
456.97.30328
509.45.32266
5512.54.00220
6016.23.18185
6520.62.4157
7531.71.6118
9054.80.982
105870.5760
12013090.3846
1351850.2736


Range in this table means how far will you get driving at given constant speed and how long it will take.
Regenerative braking is obviously not counted in.
These numbers are not absolute but they do ilustrate the situation to some degree.
Some observations:

• Driving it to the max will drain the batteries in less then half an hour, in that time you've driven about 40 miles
• Driving it mainly at low city speeds will give quite good range - way above advertised 250 miles
• Ataining good range at higher speeds is still quite far away. Baterries will have to increase their capacity several times.
• People will be inclined to drive more slowly - in this way their range will increase considerably

EPA Highway standard demands about 48.3mph average speed and Tesla is quoted to have 250+ miles range acording to this standard.
My calculations predict 266miles range at constant 50mph, so these numbers are not way off.

 

[tonybelding]

I'm sure your numbers at very low speeds can't be trusted, because at those speeds you still have the "overhead" of rolling resistance, mechanical friction in the drivetrain, cooling system, computers and controller circuitry, etc. As you move toward higher speeds the aerodynamic drag becomes by far the largest factor, and your numbers are probably in the ballpark.

If you follow the typical highway cruising speed that most people drive here in Texas, which is about 80 MPH, then you are looking at. . . a good 100 miles of driving, roughly? That fits my needs pretty well, but some people might find it restrictive.

On the other hand. . . Tesla Motors said they thought the Roadster was capable of 200 miles of "typical sports car driving". Maybe their idea of typical sports car driving is different from mine, I dunno.

 

[WarpedOne]

>> I'm sure your numbers at very low speeds can't be trusted,

I agree.

Those numbers are all derived from that cube/speed rule which holds generaly at higher speeds. At lower speeds it transforms into square power/speed relation and at even lower speeds into linear power/speed relation. In below estimation I included square and linear relations at lower speeds. I choose 55mph as a point where cube relation is replaced with sqare one and 30mph where square relation is replaced by linear relation. Unfortunately I don't have any references so I choose those speeds quite arbitrarely. Any input would be apretiated.

Another thing crossed my mind. I started off with assumption that maximum speed is achived at precise maximum engine power. According to chart at http://www.teslamotors.com/performance/performance.php that maximum power is achived at 8000rpm. Tesla Roadster has fixed gearbox ratios and second gear is stated to have 7.4:1 final ratio. This translates 8000 engine rpm into 1081 wheel rpm. Rear wheel dimensions are 225/45 R17, those have 225x45% mm + 17inch = 634mm diameter. Lets round that to 63cm or 24,8inch. 1081rpm of such a wheel gives us PI * 1081rpm * 0,63m (!!!) * 60 / 1000 = 128km/h or 80mph. Maximum speed of 135mph is thus not achived at maximum power at 8000rpm but at redline in second gear at 13500rpm where motor is capable of outputing mere 105kW! This kinda goes in line with my standard familly sedan with 88kW engine which is capable of 125mph max speed.

This leads me two two conclusions:

• Tesla Roadster could be fitted with a third gear in which it could go way faster then 135mph
• Range calculations in my first post are way too conservative

1. Maximum speed

Using cubic speed/power relation and knowing that Tesla needs 105kW to go 135mph, we can calculate how fast it can go at 185kW:

max speed at 185kW = (1353 * 185 / 105)1/3 = 163mph

This maximum speed would be achived at maximum power of 185kW at 8000 engine rpm and 2209 wheel rpm. To achive this speed Tesla Roadster needs another longer gear - third gear - that wouldn't hit into redline to quickly. Needed third gear would have ratio of maximum 1,06 : 1 or 3,62 : 1 overall ratio. Accelerating in this third gear wouln't be mindblowing, but it would give additional 30mph of speed for maximum of 163mph. That is some serios speed.

2. Range

Range/speed numbers in my first post are based on assumption that maximum speed is attained at maximum power. As we've seen this isn't so.
Here is new estimation based on the fact that Tesla Roadster only needs about 105kW to go 135mph:


Speed Power Time Range
(mph) (kW) (hours) (miles)
10 0.7 71 714
20 1.4 36 714
30 2.1 24 714
35 2.9 17 609
40 3.8 13 533
45 4.8 11 473
50 5.9 8.5 426
55 7.1 7.0 387
60 9.2 5.4 325
65 11.7 4.3 277
70 14.6 3.4 239
75 18.0 2.8 208
80 21.9 2.3 183
90 31.1 1.6 145
105 49.4 1.0 106
120 73.7 0.68 81
135 105 0.48 64
150 144 0.35 52
160 175 0.29 46

I've also factored in the fact that at lower speeds that cube relation turns into square relation (blue numbers) and at even lower speeds into linear relation (green numbers). At the end I added potential (red) numbers for higher speeds attainable with longer third gear. Those speeds would be possible if Tesla Roadster was fitted with third longer gear.

At constant speed of 48.3 mph (average speed of EPA Highway standard test) Tesla Roadster has around 440 miles range but only around 250+ miles when driving at EPA Highway standard demands - lots of accelerating and braking. This means 250 mile range is quite conservative estimation and many people could quite easily achieve better results.

Some observations are thus:

• Driving it to the max will drain the batteries in less then an hour, in that time you've driven about 60 miles
• Driving it mainly at low city speeds will give quite good range - maybe even double the advertised 250 miles
• Ataining good range at higher speeds is still quite far away. Baterries will have to increase their capacity
• People will be inclined to drive more slowly - in this way their range will increase considerably
• As Tesla probably wont add another longer gear there is clear room for 'racing' version of Roadster

(!!!)* I found the error in my previous calculations: wheel diameter is 0,63m and not 0,53m. When I corrected this error it became obvious to me that Tesla Roadster is gerabox limited in view of maximum speed. I redone the range calculations and maximum top speed with third gear and replaced that old post.

 

[Tesla2Go]

Perhaps maxi speed is calculated using first gear?

 

[vfx]

I thought I had read (or heard) that Martin said that the roadster far exceeded the 250 miles EPA stated. (somehing like 400 miles?). Tesla just refused to use those numbers because they ae trying to be conseravitive with thier claims.
I also wonder if they used calculated projections to get the numbers or was it emperical?

Milege test breakdown to do the math.

http://www.fueleconomy.gov/feg/how_tested.shtml

and found it intersting that they read the exhaust for the totals not the gas gauge...

http://www.fueleconomy.gov/feg/fe_test_schedules.shtml

 

[tonybelding]

Maximum speed of 135mph is thus not achived at maximum power at 8000rpm but at redline in second gear at 13500rpm where motor is capable of outputing mere 105kW!

I figured something like that. My Lotus Esprit V8 is supposed to have a top speed of about 175 MPH, and the way you get there is by putting it 5th gear and running the engine up to its redline. I didn't know if that would also hold true for an electric car, which made me hesitant to say anything. . . .

Incidentally, a GM EV1 was able to hit 183 MPH on a test track. But, they had to modify the gear ratios to get those results. The standard EV1 available to consumers was electronically limited to 80 MPH anyhow.

As for your revised speed-versus-range chart, it looks a lot closer to my impression of how the Roadster should perform based on Tesla's public statements about it. If we get real world results similar to these, I'll be happy.

 

[iisjsmith]

I am a bit confused about some of these range calculations. The Roadster uses a PWM variable frequency drive system, not a potentiometer system.

In a potentiometer system, you control speed by controlling input voltage. So the faster you drive the more power you use. So in that case these range numbers would make sense.

But using a variable frequency drive you are only adjusting the frequency of the AC current to change the motor's rpm according to the equation RPM = 120 x f / p where f is the frequency in Hertz and p is the number of poles.

Since the Roadster uses a 4-pole motor you can calculate that to reach the redline RPM of 13500 you would adjust the frequency to 450 Hz. To reach the max power level of 8000rpm you would adjust the frequency to 267Hz. All of this frequency adjustment happens in the inverter.

So range shouldn't change with regards to the speed. It would change if you use more power in the car, such as running the lights and heater and other accessories.

Or am I missing something?

 

[WarpedOne]

???

faster you go, more drag there is. to overcome it you need more power. tehnical details will not save you from that.

 

[iisjsmith]

I didn't realize your range calculations were based on the drag equation...I thought they were based on a power-speed ratio.

What is the reference area and drag coefficient of the Roadster?

 

[WarpedOne]

no no no

I don't have those coeficients. Also, calculating the range based on such detailed information would mean taking into account a million of little details. Friction losses in gearbox, rolling resistance, electrical losses, suspension losses etc etc. Never ending story that would produce a result you couldn't realy trust until you tested it on road. You never know what you've overlooked.

I just took the numbers that TM put on their web site:
- maximum power
- maximum speed
- power and torque curve
- gear ratios
- tire specifications

As I said above, Roadsters max speed is attained in second gear at 13500rev where it runs into electronic blocade. From power curve we see that at this rpm motor is capable of outputing 105kW. Because I don't know how much drag the car has, I presumed that all this power is used for overcoming the drag. This probably ain't so, the car would keep accelerating into redline, but this only mean that my numbers are conservative. The car uses even less energy then I've calculated and it has even better range than my numbers suggest.

 

[iisjsmith]

I thought that the Roadster hit maximum speed at 8000rpm? That's where it puts out maximum power.

But again, I don't know what speed has to do with range. When you accelerate from 50mph to 60mph you are just increasing the rpm of the motor, which increases the rpm of the axle, which increases the rpm of the wheels.

In an ICE vehicle you need to burn more fuel to increase the engine rpm, which will drain the fuel tank quicker and decrease your range.

In an electric vehicle that uses a potentiometer throttle system, you are increasing the voltage supplied to the motor which will drain the battery quicker which will decrease your range.

In a Tesla Roadster, or any other EV that uses a variable frequency drive, you are simply increasing the frequency of the current. You aren't using more power or draining the battery faster, so your range shouldn't change.

Of course, I'll give you that drag will increase by a factor of 4 as your speed increases. But we can't know how that will affect the range unless we know about the drag factors of the vehicle. For all we know the Roadster has incredible drag characteristics and going from 50mph to 60mph will have a small drag increase.

 

[WarpedOne]

>> I thought that the Roadster hit maximum speed at 8000rpm?

Me too, until I've taken a look at gear ratios and tire dimensions. Then I found out the maximum speed is reached ad 13500 rpm. At 8000 rpm in II. gear it goes only 80mph.

>> But again, I don't know what speed has to do with range.

When you have limited amout of enerergy, everything. Because drag forces are not NOT in linear relation with speed.

To go twice as fast, you need more then twice the power. With 500hp you can go about 200mph, with 1000hp and same aerodynamics only about 250mph. So, you doubled the consumption of your energy supply (500hp -> 1000hp), but only increased the speed for 25% (200mph -> 250mph). In the same time you now travel for 25% more, but total travel time halves! You range at this double speed is then (1 + 0,25) / 2 = 0,625. Less then two thirds of a range at a original speed!

This energy consumption DOES not stem from motor engineering and technical details but from basic laws of phisics. Every motor has its internal losses and with better engineering you can lower them. But you cannot cancel drag with different motor technologies. You cannot have something for nothing.

>> In a Tesla Roadster, or any other EV that uses a variable frequency drive, you are simply increasing the frequency of the current. You aren't using more power or draining the battery faster, so your range shouldn't change.

Wrong.

You are using more power and draing battery faster because at increased frequency the motor tries to turn faster. In doing so it has to move the car faster and overcome bigger drag forces. If at speed A these drag forces amount to B then at speed of 2*A those forces are greater then 2*B. Actualy closer to 4*B or even 8*B. It does not matter what exact number are behind A and B. General relation holds.

Every motor has its maximum power-out at any given RPM. Actual power could be lower but it cannot in any way be higher then its maximum capabilites at that speed. TM published what this max-power curve looks like for Tesla Roadster. We dont know aerodynamics and various other drag coeficients to calculate precise power required to drive at given constant speed. Be do have maximum available power. We know that required power to go 130mph does NOT exceed 105kW.

Again, in my calculations I presumed that at 130mph all available power (105kW) is consumed for overcoming drag forces. Actual power is lower, how much lower we do not know but we can estimate that not much lower, maybe 10kW, 20kW at most. Just look at other cas and their maximum speeds power.

This only means actual range numbers can even be a little higher than my numbers suggest.

 

[electrified]

I would guess that there would be significant losses in wheel resistance not just atmospheric drag, and those losses would be much more linear than the drag component.

 

[iisjsmith]

I was just reading the article about TM from WorldChanging.com and they say "It has a drag coefficient of around .3"

 

[Michael]

I'm surprised that Tesla has not posted the drag cooefficient of their Roadster. But, maybe if it's not so good, then maybe it is understandable.

Whatever it is, I would expect that their next version of the vehicle would have some focus on improving it since drag has such a negative impact on limiting range!

I hope their next vechicle aims for a drag in the mid .20's.

 

[iisjsmith]

>>>> In a Tesla Roadster, or any other EV that uses a variable frequency drive, you are simply increasing the frequency of the current. You aren't using more power or draining the battery faster, so your range shouldn't change.

>>Wrong.

Actually, not wrong. I've been researching this constantly for the past few days and my statement is correct. The variable frequency drive accepts a constant power level input and then varies the output so that the motor can perform differently.

When you slow down the VFD simply synthesizes a lower frequency and voltage to the motor...the line frequency and voltage does not change. So unlike a potentiometer or IEC system, you do NOT save power when you go slower.

However, the voltage also remains constant as you go above the rated frequency. There is a loss of torque as the motor goes into what is called "field weakening" but the horsepower remains constant. So going faster results in a loss of torque, but NOT an increase in battery drain.

Thus, range will stay the same regardless of your speed. The only factors that should decrease range are temperature and increased use of accessories that drain battery power.

I could do some complete calculations but we would need to know the rated voltage and frequency of Tesla's custom motor. Then we would know at what point the volts-per-hertz ratio starts to decrease...where we would see a decrease in torque. There's lots of info I'd like to have about their engine...I don't know why they are so tight-lipped about it. It's not like knowing the service life rating or calculated slip at max RPM is going to give a competitor enough info to manufacture their own!

 

[WarpedOne]

>> The variable frequency drive accepts a constant power level input and then varies the output so that the motor can perform differently.

Let's ASSUME you are correct. Tesla states that maximum power is 185kW. You claim that power draw is constantly at least 185kW. Telsa also states batteries store around 50kWh of energy. So, there is enough energy for 50kWh / 185kW = 16 minutes of driving. At maximum speed of 130mph this translates into 35 miles maximum range. At any lower speed, according to you, the range is even shorter.

But then again, Tesla claims the range is over 250miles according to EPA driving cycle.

You sir, are clearly mistaken.

 

[iisjsmith]

Okay, I have made more progress on understanding VFD technology. My problem is that variable speed drive systems work on converting AC line power to variable frequency and voltage outputs...they convert AC to DC back to AC. The Roadster doesn't need that, since the supply is already DC. So we only need to focus on the last stage of the system...converting input DC at a fixed voltage and current to a variable frequency/voltage 3-phase AC power using a pulse width modulation inverter.

Again, this is not a potentiometer throttle system, where you increase resistance to decrease the voltage that reaches the inverter. In that system you have tons of losses due to heat as you increase resistence. Also, this is not like an ICE where you increase or decrease the flow of fuel to increase or decrease engine power.

With the PWM inverter you are rapidly turning the circuit on and off to modulate the voltage. So when you are at full throttle you are supplying all of the input voltage to the motor. As you decrease the throttle the inverter starts to "stutter" the power. When the circuit is closed there is power flowing to the motor...when the circuit is open there is no power flowing. So at 50% throttle you are using 50% the electrical power you would use at full speed...without all of the losses due to resistance you would see when using potentiometers.

I guess a pretty close analogy would be an ICE vehicle that controls engine power not by adjusting the fuel flow rate, but turning the fuel flow on and off rapidly. When the fuel is flowing it is flowing at 100% rate. So to get 100% power you let the fuel flow completely. If you want to get 50% power you turn the flow on and off rapidly so the flow is on for 50% of the time instead of 100%.

So I was incorrect. Going slower will decrease the battery power used, giving you longer range. How much longer? Well you need to calculate the range based on full power. Note that this is NOT based on speed, but on power supplied to the battery...which is at maximum at 8000rpm. To get the speed ratings we need to know how fast the Roadster will go at 8000rpm when the motor is at maximum power. Then we can simply compare this to the speed we want to go.

Also note that this is assuming we are keeping a constant volts/hertz ratio with the inverter. If we want to take a hit on available torque we can increase the motor speed WITHOUT increasing the voltage supplied to the motor. Thus we would be able to go any speed without any effect on range. Available torque takes a big hit when we do this, though. Horsepower stays constant, however.

Interesting stuff. I'd like to know what speed the Roadster can get to when running at 8000rpm...then we can do all these calculations.

 

[WarpedOne]

>> Interesting stuff. I'd like to know what speed the Roadster can get to when running at 8000rpm...then we can do all these calculations.

In my second post, I've calculated just that. 80mph in second gear.

 

[iisjsmith]

So we can assume that as the Roadster accelerates from 0 to 80mph (0 to 8000rpm) the inverter is slowly increasing the frequency and voltage supplied to the motor. It increases the frequency to increase the motor speed, and increases the voltage to keep the volts/hertz ratio constant.

Now once we hit 80mph the motor is spinning at 8000rpm and we are at maximum power. As we accelerate past 80mph the inverter continues to increase the frequency supplied to the motor...but the voltage no longer increases. The volts/hertz ratio starts to change from the baseline and our available torque starts to suffer. But since the voltage is not increasing we are not draining the battery pack any faster.

So the range for the Roadster should be no different if you are going 80mph or 130mph. Driving slower than 80mph will increase your range.