65 Vehicle Architectures and Design dv a = = F F TR − RL dt m The motor power rating can be obtained by solving the above differential equation for a given force– velocity profile, such as one of the two shown in Figure 3.13, and the following boundary conditions: At t = 0, vehicle velocity v = 0. At t = t f , vehicle velocity v = v f . Integrating the differential equation within the interval t = 0 to t = t f for velocities 0 to v f f dv m = dt F F TR − RL ( ) v 0 0 v ∫ f t ∫ The vehicle rated velocity is higher than the motor rated velocity and lies in the constant power region of motor torque–speed characteristics.
Splitting the integral on the left side into two velocity regions of 0 –v rm for the constant torque mode and of v rm –v f for the constant power mode, one can write v f dv m ∫ ∫ v rm dv + m = t P f (3.1) m P m 0 − F v RL ( ) v v rm − F v RL ( ) rm v The road-load resistance force F RL can be expressed as a function of velocity as shown in Chapter 2 for given values of rolling resistance, aerodynamic drag force and roadway slope. Equation 3.1 can then be solved for motor power rating P m for specified vehicle rated velocity v f and rated motor speed.
Note that Equation 3.1 is a transcendental equation with F RL being a function of velocity and can be solved numerically to find the motor power rating P m . In fact, extensive computer computa- tion and simulation aids a practical design to derive the required motor power rating and gear ratio of the powertrain. An interesting analysis has been presented in [6] to stress the importance of extended constant power region of motor torque–speed characteristics.
It has been shown that for F RL = 0, the motor power rating is m P m = + ( ) v v 2 2 2 rm f (3.2) t f Equation 3.2 shows that the motor power rating will be minimum when v rm = 0, which means that the electric motor that operates entirely in the constant power mode is the smallest motor to satisfy the requirements. In the other extreme case, the motor power will be the double that of the smallest case if the motor operates entirely in the constant torque mode with v rm = v f . Of course, eliminating the constant torque region and operating entirely in the constant power region is not practically real- izable.
In a practical setting, the electric motor should be designed with a low base speed or rated speed and a wide constant power region. 3.5.1.2 Rated Vehicle Velocity The drivetrain designed to accelerate the vehicle from 0 to rated velocity will always have the suf- ficient power to cruise the vehicle at rated speed, provided the roadway slope specified for initial acceleration has not been raised for rated velocity cruising condition. 3.5.1.3 Maximum Velocity The traction power required to cruise the vehicle at maximum vehicle velocity v max is