30 Electric and Hybrid Vehicles F TR v xT β β (s) or (x (Roadway Input) s(t) or x F (t) FIGURE 2.9 Modeling of vehicle kinetics and roadway. If β is given in terms of x F , then the second state variable equation is dx v F x = T . (2.7) dt df 2 1 + dx F The input–output relational diagram for simulating the vehicle kinetics is shown in Figure 2.9. 2.5 PROPULSION POWER The desired power rating of the electric motor or the power required from the combination of elec- tric motor and internal combustion engine (i.e.
the propulsion unit) can be calculated from the above equations based on the system constraints of starting acceleration, vehicle rated and maximum velocity and vehicle gradability. The torque at the wheels of the vehicle can be obtained from the power relation Power = ⋅ T F TR ω wh = ⋅ TR v xT [ ] W (2.8) where T TR is the tractive torque in Nm and w wh is the angular velocity of the wheel in rads/s. F TR is in N and v xT is in m/s. The angular velocity and the vehicle speed is related by v r xT = ⋅ ω wh wh (2.9) where r wh is the radius of the wheel in meters.
The losses between the propulsion unit and wheels in the transmission and the differential have to be appropriately accounted for while specifying the power requirement of the propulsion unit. A big advantage of an electrically driven propulsion system is the elimination of multiple gears to match the vehicle speed and the engine speed. The wide speed range operation of electric motors enabled by power electronics control makes it possible to use a single gear-ratio transmission for instan- taneous matching of the available motor torque T motor with the desired tractive torque T TR .
The gear ratio and the size depend on the maximum motor speed, the maximum vehicle speed and the wheel radius. A higher motor speed relative to the vehicle speed means a higher gear ratio, larger size and higher cost. However, higher motor speed is also desired in order to increase the power density of the motor. Therefore, a compromise is necessary between the maximum motor speed and the gear ratio to optimize the cost. Planetary gears are typically used for EVs with the gear ratio rarely exceeding 10.
2.5.1 f orce –V elociTy c haracTerisTics Having identified the fundamental forces and the associated dynamics for EV and HEVs, let us now attempt to relate these equations to a vehicle design specification and requirements. For an efficient design of the propulsion unit, the designer must know the force required to accelerate the vehicle to a cruising speed within a certain time and then to propel the vehicle forward at the rated steady-state ulsion Unit) F) nput) Vehicle Kinetics Model Eqs. (2.5) and (2.6) or Eqs. (2.5) and (2.7)