Single-Speed Gear Ratio Selection for Acceleration, Top Speed and Range

The problem

Most EVs use a single fixed reduction, so the ratio sets both launch torque at the wheels and the maximum speed the motor can reach. A short ratio gives strong acceleration but caps top speed and can push the motor into less efficient regions on the highway; a tall ratio does the opposite. A systematic study shows the trade-off and the feasible range.

Objectives

• Derive the bounds on the ratio from top speed and launch-torque requirements.
• Build a Simulink model with a motor efficiency map and a parametrised ratio.
• Sweep the ratio and record acceleration, top speed, gradeability and Wh/km on several cycles.
• Solve a constrained optimisation for minimum energy subject to performance limits.
• Present a Pareto view of performance against energy.

What you need

Key concepts

Ratio bounds

The upper bound comes from top speed: G ≤ ω_max·r / v_max. The lower bound comes from the wheel torque needed for launch and grade: G ≥ T_wheel,req / T_motor,peak.

Efficiency map

Motor plus inverter efficiency against speed and torque. Changing the ratio moves the cycle's operating points across this map.

Constrained optimisation

Minimise an objective (here Wh/km on a weighted cycle mix) while keeping acceleration time and top speed within limits.

Pareto front

The set of designs where improving one objective (acceleration) worsens another (energy). It shows the trade-off rather than one answer.

Step-by-step guide

1. 1

   Define targets and parameters

   Use an example car: mass 1600 kg, CdA 0.65 m², Crr 0.010, wheel radius 0.31 m, motor 250 N·m peak, 100 kW peak, maximum speed 12,000 rpm. Targets: top speed at least 150 km/h, 0–100 km/h in 9 s or less, a 25 per cent grade at low speed. Put these in a parameter script and label them as example values. Choose a weighted cycle mix, for example 50 per cent MIDC and 50 per cent WLTP.

   Checkpoint: A targets table and parameter script, and a written definition of the weighted energy objective.

2. 2

   Compute analytical bounds

   From top speed, G_max = ω_max·r / v_max, which is about 9.4 for the example. From the grade target, find the wheel torque on a 25 per cent slope including rolling resistance and get G_min = T_wheel / T_peak, allowing for driveline efficiency. Also check that at G_max the motor's power at top speed covers road load. Plot both bounds on a number line.

   Checkpoint: Numerical G_min and G_max with the feasible interval clearly identified.

3. 3

   Build the parametrised model

   Build a forward-facing Simulink model with a PI driver, a torque–speed envelope, a 2-D Lookup Table for motor and inverter efficiency (speed by torque, with regenerating quadrants), a gear ratio variable G and a battery energy integrator. Use a fixed-step ode4 solver at 0.01 s. Make G, the cycle and the test type model arguments so one model runs every test.

   Checkpoint: The model runs on WLTP with G = 9 and gives consumption in a sensible range for the example, around 130–180 Wh/km.

4. 4

   Build the performance tests

   Create three test cases inside the model or as separate harnesses: a full-throttle launch that records the 0–100 km/h time, a long full-throttle run that records maximum speed, and a slope test that records the maximum grade climbable at 10 km/h. Drive the test choice from a variable so a script can loop over them. Use the same solver settings for every test so the results are directly comparable across ratios.

   Checkpoint: For G = 9, each test returns a single number and the numbers agree with quick hand estimates.

5. 5

   Verify the efficiency map use

   Plot the cycle operating points over the efficiency map for two ratios, for example 7 and 11. Confirm that a taller ratio (smaller G) pushes urban operation to higher torque and lower speed, and that interpolation never leaves the table range. Clip inputs to the table limits and log how often clipping happens. Also check that regenerating operating points use the negative-torque half of the map, not a mirrored motoring value.

   Checkpoint: Operating-point plots for two ratios with no out-of-range lookups.

6. 6

   Sweep the ratio

   Create a Simulink.SimulationInput array for G from G_min to G_max in steps of 0.25 and every test and cycle, and run them with sim() (or parsim). Collect 0–100 time, top speed, gradeability, and Wh/km on each cycle into a table. Plot each metric against G with the target limits shown as horizontal lines. Save the results table to a MAT file so the optimisation and plots can reuse it without rerunning hundreds of simulations.

   Checkpoint: Four plots against G with feasible regions shaded where all targets are met.

7. 7

   Optimise

   Define the objective as the weighted Wh/km and the constraints as 0–100 time and top speed limits. Because one simulation takes seconds, wrap sim() in an objective function and call fminbnd or fmincon over the feasible interval, or use the sweep results with interpolation. Check that the optimum is not simply at a bound. Report the number of simulations the optimiser used and compare its answer with the best point from the sweep.

   Checkpoint: An optimum G with its metrics, confirmed by a separate simulation at that ratio.

8. 8

   Pareto and sensitivity

   Plot weighted Wh/km against 0–100 time for every G to show the trade-off front. Repeat the optimisation with a different cycle weighting (urban-heavy and highway-heavy) and with a ±10 per cent change in vehicle mass. Report how much the optimal ratio moves and how flat the energy curve is near the optimum. Highlight the designs that meet all targets on the Pareto plot, so readers see which part of the front is feasible.

   Checkpoint: A Pareto plot and a table of optimal G for each scenario.

9. 9

   Report

   Document the bounds, model, verification, sweep, optimisation and sensitivity. Explain the physical reasons for the shape of each curve. State the limits: one generic efficiency map, no gearbox loss variation with ratio, and no thermal limits on continuous operation. Recommend a ratio range, not just one number, and say why. Include the sweep script and saved results so a reader can repeat the study with their own motor map or vehicle parameters.

   Checkpoint: A report with a recommended ratio range backed by sweep and optimisation results.

Testing & validation

• Analytical bounds agree with the sweep (top speed falls below target above G_max).
• Efficiency lookups stay within table limits.
• Optimum is confirmed by an independent simulation.
• Results are repeatable with a different solver step size.

Expected results

A recommended single-speed ratio range for an example electric car with plots of acceleration, top speed, gradeability and energy against ratio, an optimisation result and a Pareto front.

Take it further

• Make motor size a second design variable.
• Add a gearbox loss model that depends on torque and speed.
• Compare against a two-speed gearbox.
• Include continuous-power thermal limits for long highway climbs.

Viva and review questions

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