Library / Automotive and vehicle dynamics / VEH-11
Automotive and vehicle dynamics
Electric motor torque envelope
Idealised torque an electric motor can deliver at a given speed: constant at its maximum below the base speed, then falling as 1/speed in the constant-power region.
- Ref
- VEH-11
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- Tm [N·m]
Tmax is the maximum torque the motor can deliver at low speed, Pmax the maximum power it can deliver, and Nm the motor speed in rpm (converted to ωm = 2πNm/60). Below the base speed, Pmax/Tmax, the torque is limited to Tmax. Above it the power is the limit, so the torque falls as 1/ωm. It is an idealised envelope: real limits also come from inverter current, battery voltage, temperature and the operating map.
Idealised torque an electric motor can deliver at a given speed: constant at its maximum below the base speed, then falling as 1/speed in the constant-power region. Real limits also come from current, voltage, temperature and the battery.
Calculator
- Base speed, Pmax / Tmax3,820 rpm
- Motor power at this speed100 kW
- Torque if the power limit alone applied, Pmax / ωm159.2 N·m
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The base speed is 3,820 rpm. Below it the motor gives its full 250 N·m; at the default 6,000 rpm the torque is down to 159.2 N·m, and at 12,000 rpm to 79.58 N·m.
- The envelope is why electric vehicles feel strong from a standstill: the full torque is available from zero speed, and why a single gear is usually enough, because the constant-power region covers the upper speed range.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N·m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Pmax Maximum power | 10 to 500 kW | 15.92 | 250 | 234.1 | +10% | +10 % |
| Nm Motor speed | 100 to 15,000 rpm | 63.66 | 250 | 186.3 | +10% | −9.09 % |
| Tmax Maximum (low-speed) torque | 50 to 1,000 N·m | 50 | 159.2 | 109.2 | +10% | 0 % |
Sweep of Motor speed
| Motor speed (rpm) | Motor torque (N·m) |
|---|---|
| 100 | 250 |
| 1,963 | 250 |
| 3,825 | 249.7 |
| 5,688 | 167.9 |
| 7,550 | 126.5 |
| 9,413 | 101.5 |
| 11,280 | 84.69 |
| 13,140 | 72.69 |
| 15,000 | 63.66 |
Sweep one variable
Variables and how each one contributes
Motor torque
What the equation gives you. Everything else feeds it.
Maximum (low-speed) torque
The maximum torque the motor can produce at low speed, in N·m. Raises the torque only below the base speed. At the default 6000 rpm the motor is in the constant-power region, so it has no effect on the result there; it would matter below the base speed. Raising it by 10 % (to 275) moves the result by +0 %, from 159.2 to 159.2 N·m.
Maximum power
The maximum power of the motor, in kW. At the default speed the torque is set by the power limit, so the result rises in direct proportion. Raising it by 10 % (to 110) moves the result by +10 %, from 159.2 to 175.1 N·m.
Motor speed
The motor speed, in rpm. Above the base speed the torque is inversely proportional to the speed, so spinning the motor faster reduces the torque available. Raising it by 10 % (to 6,600) moves the result by −9.09 %, from 159.2 to 144.7 N·m.
Worked example with the values above
- Motor speed: ωm = 2π × 6,000 / 60 = 628.3 rad/s.
- Power-limited torque: Pmax / ωm = 100,000 / 628.3 = 159.2 N·m, against the low-speed limit Tmax = 250 N·m.
- Tm = the smaller = 159.2 N·m (constant-power region).
Assumptions and limits
- An ideal two-region envelope with a sharp corner at the base speed. Real motors round off the corner and lose torque from field weakening and voltage limits.
- Peak ratings available at all speeds, ignoring the time limit and the battery state of charge.
Common mistakes
- Entering the speed in rad/s instead of rpm.
- Treating the peak torque as continuous. A motor can usually give it for seconds only.
Statistics and uncertainty
The torque and power ratings are manufacturer figures with tolerances of a few per cent, but they are peak values that apply for short periods. The continuous rating is much lower, since it is limited by heating, so the useful envelope depends on how long the load lasts. The base speed is the ratio of two ratings, so its error is about the sum of the two relative errors.
Where it is used
- Estimating the acceleration of an electric vehicle across its speed range.
- Choosing the gear ratio to match the base speed to the vehicle’s needs.
Maths you need
- Power, torque and speed: P = T ω
- Units: rpm to rad/s