# How can the Tesla car go up a steep hill without a gearbox?

• Jonathan212
In summary, the design engineer made a mistake by not including a gear lever on the Tesla Model 3. Without a gear lever, the car cannot reach its maximum speed.
Jonathan212
I thought electric cars had normal gearboxes like any other vehicle but the following does not. No gear lever.

It says at 6:00 that this is because the electric motor's torque is constant no matter the rpm. Presumably it means the maximum torque available is constant for all rpm. What if this maximum torque is not enough to drive a fully loaded car up a sufficiently steep slope?

If this torque is huge and can do a 45 degree slope or something, then wouldn't the fixed transmission ratio make it impossible to drive at 100 mph?

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The video shows an efficiency versus rpm curve, but not a torque versus rpm curve. I don't know what type of induction motor or the resistance of the rotor used in the video. Low rpm torque can be increased by using a higher resistance rotor. Link to article:

https://people.ucalgary.ca/~aknigh/electrical_machines/induction/im_trq_speed.html

The Tesla model 3 uses a permanent magnet (reluctance) motor instead of an induction motor. These type of motors date back to the 1800's but are more efficient now. Some variations of reluctance motors produce peak torque at 0 rpm and the torque versus rpm remains flat and at peak until rpm reaches the point where the motor transitions into constant power mode (torque decreases proportional to rpm) until near peak rpm.

For the Tesla Model S P100D, the battery and motor combine for an output of 588 hp, and 920 lb ft of torque, and the car weighs 4941 lbs. This translates into 1 g or more of acceleration, essentially traction limited, until about 45 mph, at which point it acceleration becomes power limited. Still it's 0 to 45 mph time is so quick that it carries this advantage on a 1/4 mile, about 10.75 seconds under ideal circumstances and waiting for the 10 minutes it takes to setup "ludicrous" mode (heating battery, cooling motors). The aerodynamic drag is low enough that without the speed limiter, the car could probably reach close to 200 mph, given sufficient distance (2 or 3 miles).

A comment about the video, which notes that the power to weight ratio of the Tesla motor is much greater than an internal combustion engine. It doesn't point out the fact that the specific energy (energy versus weight) of the battery is much less than gasoline. For the Model 3 battery, Tesla claims an energy density about 200 wh / kg, while gasoline has a specific energy of 12888 wh / kg, but this is offset by the 30% efficiency of a gasoline engine, versus 83% or so for the electric motor, after motor efficiencies, you have Tesla 3 battery 166 wh / kg, gasoline 3865 wh / kg. This why the electric cars with decent range are so heavy.

I started watching the video. I've only watched the first minute. I'll go back and watch the rest of it. They say it uses an induction motor. That is an AC motor. That's why it has an inverter. It operates with "slip" between the stator field and rotor field. If the slip increases, because of increased load, the force of the field increases.

Jonathan212 said:
I thought electric cars had normal gearboxes like any other vehicle but the following does not. No gear lever.

It says at 6:00 that this is because the electric motor's torque is constant no matter the rpm. Presumably it means the maximum torque available is constant for all rpm. What if this maximum torque is not enough to drive a fully loaded car up a sufficiently steep slope?
Then the design engineer gets fired for making a pretty basic mistake!
If this torque is huge and can do a 45 degree slope or something, then wouldn't the fixed transmission ratio make it impossible to drive at 100 mph?
Hopefully you recognize that photo is just rotated 45 degrees...

But anyway, it's all about rpm range. It's common for gas engine cars to idle at around 600 rpm and redline at 6,000, a 10:1 ratio, and difference in gear ratios of around 5x. Let's assume (conservative) that full torque is available throughout that range. So all that is needed from an electric motor is for full torque to be available over a 50:1 rpm range or more in order for the full speed range to be available to it. And they are, at essentially infinite.

In reality:
1. The RPM range is much broader on an electric car.
2. The horsepower available and therefore maximum speed is often lower than for gas cars because of the acceleration advantage of the constant torque and wider RPM range.

There's a caveat to the second, though: electric motors will output whatever power you want from them or fail trying. Typically you can output far more than a motor's nameplate rating for a short time, with little impact but a slightly reduced lifespan. My suspicion is that on a car like the Tesla Model S, the horsepower rating is based on rarely ever applying that horsepower, allowing them to claim a higher than really functional power rating for the motors. This would apply to a lesser extent for a gas engine, but a diesel truck needs every bit of its power, every time it accelerates.

jrmichler and scottdave
Here is a typical speed vs torque chart for a permanent magnet electric vehicle motor.

The motor designer specifies the torque for the constant torque portion, and the motor RPM at which the torque starts to fall off. The portion of the curve above the constant torque portion is called the constant horsepower portion, even though it is not exactly constant horsepower. If the motor designer does their job correctly, the finished motor will have, within a few perent, the exact performance that it was designed to have.

The mechanical engineer works with the motor engineer and drive engineer and the battery engineer and the chassis engineer and the marketing people and top management to determine the drive gear ratio, peak motor torque, RMS (average) motor torque, maximum RPM of the constant torque portion, and maximum RPM. If the boss wants ludicrous acceleration, the boss gets ludicrous acceleration.

russ_watters said:
electric motors will output whatever power you want from them

I once worked for a company that designed and built their own motors. One of their motors fit in the palm of ones hand, and delivered almost 10 horsepower. But it only had to do that for 300 milliseconds at a time. The worst case duty cycle was about 1%.

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russ_watters
For the Tesla Model S P100D, the battery and motor combine for an output of 588 hp, and 920 lb ft of torque, and the car weighs 4941 lbs. This translates into 1 g or more of acceleration, essentially traction limited, until about 45 mph, at which point it acceleration becomes power limited.

Where did you get that 1 g number? Don't the specs give the transmission ratio from motor to wheels and the radius of the wheels so one can convert motor torque to force in the direction of travel?

opposing force = WEIGHT * 9.81 * sin(45)

POWER = opposing force * speed
=> speed = POWER / opposing force

speed = WHEEL RADIUS * wheel angular velocity
=> wheel angular velocity = speed / WHEEL RADIUS

rpm = 60/(2*pi) * wheel angular velocity * TRANSMISSION RATIO

motor torque = opposing force * WHEEL RADIUS / TRANSMISSION RATIO

Then you calculate the ones in bold and look them up in the torque diagram. If torque is not enough, forget it, can't do 45 degree slopes.

If torque is enough by a margin, that may be because of the high transmission ratio, in which case I bet the motor would have to spin insanely fast for the car to do 100 mph.

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Back of the envelope calculation...

Tesla claim a 0-60 time of under 2.8 seconds for some models. 60mph is about 27m/s. A=V/t =27/2.8=10m/s/s or about 1g.

scottdave
I get speed = 0.028 m/s, rpm = 89 and torque = 46 Nm. Seems ok.

But then at 200 km/h the motor should be doing 177,000 rpm. Doesn't that sound too much?

Jonathan212 said:
Can someone interpret these specs? What's the transmission ratio from motor to wheel? 8.28 * 9.73?

https://www.autoguide.com/new-cars/2017/tesla/model-s/p100d/awd/specs.html
Either 8.28 or 9.73 would be a reasonable gear reduction from motor to axle speed, but as you suspect, multiplying both is not. The top speed for electric motors is in the 12,000 rpm range, more or less, not 177,000! I suspect that those ratios might be overall ratios for different versions of the vehicle (just guessing).

The specifications in that link are incorrect - probably copied down wrong from somewhere. In a conventional ICE powertrain there is a distinct set of transmission reduction ratios (like a 2.73 first gear) to select from, each of which would be combined with a final drive reduction (like 4.11). With an electric motor there is no reason to split the reduction into two parts (although it is multi-stage), since the overall reduction is constant. Sometimes websites and the people filling them out don't keep up with changes in technology. ;-)

The Tesla Model S P100D tires are spec'ed at 727 revolutions per mile. At the speed limited 155 mph, that's 727*155 / 60 ~= 1878 revolutions per minute. There is a twin gear reduction, but the overall reduction at the rear end is 9.73, so 155 mph translates into 1878 * 9.73 = 18273 rpm. The 8.28 reduction ratio is for the front end. Based on this, although there may be enough power to reach 200 mph as claimed as a possibility in some articles, the car would require a lower reduction factor to achieve this.

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Since 8.28 is not to be included in the math, then the necessary torque for a given slope gets 8.28 times higher. That's 381 Nm for the 45 degree slope. Above the available in the diagram.

Randy Beikmann said:
With an electric motor there is no reason to split the reduction into two parts.
With a reduction ratio of 9.73, the difference in diameter in a single stage reduction would put too much stress on the few engaged teeth of the smaller gear or require much larger gears. Using a two stage reduction spreads the stress on a higher percentage of teeth of the smaller gears, while allowing the gear set to be smaller.

rcgldr said:
With a reduction ratio of 9.73, the difference in diameter in a single stage reduction would put too much stress on the few engaged teeth of the smaller gear or require much larger gears. Using a two stage reduction spreads the stress on a higher percentage of teeth of the smaller gears, while allowing the gear set to be smaller.
That's why I said "...it is multi-stage." There is just no reason to list them separately in a chart. ;-)

Randy Beikmann said:
That's why I said "...it is multi-stage." There is just no reason to list them separately in a chart.
Tesla doesn't. For the Model S P100D, the rear end ratio is 9.73, while the front end ratio is 8.28, or you're looking at a chart I haven't seen yet.

By construction, the power output of an ideal internal combustion engine (ICE) is proportional to its rpm. The cylinders pulls in the same quantity of air-fuel mixture for every revolution. Therefore, the torque is constant throughout the rpm range and the power input variation (i.e. the mass flow rate of air-fuel mixture) depends only on the number of revolutions per minute.

On the other hand, the power input of an ideal electric motor is proportional to its voltage and current input. It is independent of the motor rpm. And since torque is power divided by rpm, you can ideally get any torque value at any rpm. The only limit is how much power the motor can support before breaking up.

To get that high power in an ICE, you need to keep the engine rpm where it produces its most power by constantly adjusting (with a gearbox) the wheel rpm with respect to the engine rpm.

You could theoretically make an ideal ICE with rpm-independent power characteristics by deliberately cramming more air-fuel mixture in the cylinder at low rpm (i.e. keeping the same mass flow rate as when in high rpm, therefore creating a 'constant power' engine). Technically, this means using a very large engine that would be underused at high rpm.

I found this image online that might help visualized the concept:

The downward part of the yellow curve represents more or less a 'constant power' curve, i.e. the maximum power the Tesla electrical system can support. The flat portion is limited by design to match the traction limit of the tires.

With the white curve, the first gear is designed such that the peak torque matches the traction limit of the tires, and the last gear is designed to match the desired maximum speed. Then other gears are inserted in between to maintain the engine as close as possible to its maximum power, therefore simulating a 'constant power' curve (i.e. torque decreasing as speed increases).

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scottdave and jrmichler
Shouldn't the torque go to zero during gear changes in that diagram?

Well, technically, you are not accelerating when shifting so your speed stays constant. You should then see the value at that speed to be the maximum possible value. Although not ideal, it may also be different from one gear compared to the other.

jack action said:
By construction, the power output of an ideal internal combustion engine (ICE) is proportional to its rpm. The cylinders pulls in the same quantity of air-fuel mixture for every revolution.
Do you really mean this as written? It sounds like you are ignoring the throttle or specifying a wide open throttle condition. Obviously the fuel/air quantity must be varied otherwise the engine would just uncontrollably accelerate as soon as it started.

In this way, it's not really any different from "throttling" an electric by adjusting the voltage.

I specified the word ideal to emphasize the more theoretical approach to illustrate the point. The quantity of energy going into an ICE - given the same conditions - is proportional to its rpm. Whatever air-fuel mixture going in at rpm X, twice as much goes in at rpm 2X. Of course, some other limits will apply at some point that will restrict the flow at higher rpm. For a theoretical electric motor, there are not really such limitations. You can increase the voltage or current as you wish, at any rpm. The ability of real electric to fully convert that power to mechanical power or to support it is another story.

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jack action said:
I specified the word ideal to emphasize the more theoretical approach to illustrate the point. The quantity of energy going into an ICE - given the same conditions - is proportional to its rpm. Whatever air-fuel mixture going in at rpm X, twice as much goes in at rpm 2X. Of course, some other limits will apply at some point that will restrict the flow at higher rpm. For a theoretical electric motor, there are not really such limitations. You can increase the voltage or current as you wish, at any rpm. The ability of real electric to fully convert that power to mechanical power or to support it is another story.
For a naturally aspirated ICE, there will be a peak torque at some rpm, due to momentum and resistance to the flow going into and out of the engine. A supercharger where the pressure is modulated could be used to produce a flat or nearly flat torque versus rpm curve.

Not all electrical motors have the same type of torque curves. A typical DC motor has peak torque at 0 rpm, and the torque decreases linearly to zero as rpm increases to the max no load rpm. Peak power occurs at 1/2 the max rpm. Reluctance AC type motors require some resistance in the rotor to produce a nearly flat torque curve over part of the rpm range.

## 1. How does the Tesla car go up a steep hill without a gearbox?

The Tesla car is equipped with an electric motor that produces instant torque, allowing it to climb steep hills without the need for a traditional gearbox. The electric motor delivers power directly to the wheels, eliminating the need for multiple gears to transfer power.

## 2. What makes the Tesla car's electric motor different from a traditional gas engine?

The Tesla car's electric motor is powered by a battery pack, rather than burning fuel like a traditional gas engine. This battery-powered motor provides a smoother and more efficient power delivery, making it ideal for climbing steep hills without a gearbox.

## 3. Can the Tesla car go up any steep hill without a gearbox?

The Tesla car's ability to climb steep hills without a gearbox depends on several factors, such as the grade of the hill, the weight of the vehicle, and the condition of the battery. In most cases, the Tesla car can easily handle steep hills without the need for a gearbox.

## 4. Are there any downsides to not having a gearbox in the Tesla car?

One potential downside of not having a gearbox in the Tesla car is that it may not have as much top speed compared to vehicles with a traditional gearbox. Additionally, some drivers may prefer the feeling of shifting gears while driving.

## 5. How does the Tesla car's lack of a gearbox impact its maintenance and repair?

Since the Tesla car does not have a traditional gearbox, it requires less maintenance and has fewer parts that can break down. This can lead to lower repair costs and a longer lifespan for the vehicle.

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