Analysis and Control of Electric Drives. Ned Mohan

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      1 2‐11 In wind turbines, the shaft power available is given as follows, where the pitch‐angle θ is nearly zero to “catch” all the wind energy available: where Cp is the wind‐turbine Coefficient of Performance (a unit‐less quantity), ρ is the air density, Ar is the area swept by the rotor‐blades, and VW is the wind speed, all in MKS units. The rotational speed of the wind turbine is controlled, such that it is operating near its optimum value of the coefficient of performance with Cp = 0.48. Assume the combined efficiency of the gear‐box, the generator, and the power electronic converter to be 90%, and the air density to be 1.2 kg/m3, Ar = 4000 m2. Calculate the electrical power output of such a wind turbine at its rated wind speed of 13 m/s.

      2 2‐12 A wind turbine is rotating at 22 rpm in steady state at a wind speed of 13 m/s and producing 1.5 MW of power. The inertia of the mechanism is 3.4 × 106kg ⋅ m2. Suddenly, there is a short‐circuit on the electric grid and the electrical output goes to zero for two seconds. Calculate the increase in speed in rpm during this interval. Assume that the shaft‐torque remains constant and all other efficiencies to be 100% for the purpose of this calculation.

      3 2‐13 In an electric vehicle, each wheel is powered by its own motor. The vehicle weight is 2000 kg. This vehicle increases in its speed linearly from 0 to 60 mph in 10 seconds. The tire diameter is 70 cm. Calculate the maximum power required from each motor in kW.

      4 2‐14 In an electric vehicle, each of the four wheels is supplied by its own motor. This EV weighs 1000 kg, and the tire diameter is 50 cm. Using regenerative braking, its speed is brought from 20 m/s (72 km/h) to zero in 10 seconds, linearly with time. Neglect all losses. Calculate and plot, as a function of time for each wheel, the following: (a) the electromagnetic deceleration torque Tem in Nm, (b) rotation speed ωm in rad/s, and (c) power Pm recovered in kW. Label the plots.

      1 2‐15 Making an electrical analogy, solve Problem 2-4 .

      2 2‐16 Making an electrical analogy, solve Problem 2-6 .

      1 (Adapted from chapter 1 of Electric Machines and Drives: A First Course ISBN: 978‐1‐118‐07481‐7 by Ned Mohan, January 2012)