Chapter 8, Problem 60P

Wind turbines, such as the one shown in Figure P8.17 (a), are becoming popular as a way of generating electricity. Feedback control loops are designed to control the output power of the turbine, given an input power demand. Blade-pitch control may be used as part of the control loop for a constant-speed, pitch-controlled wind turbine, as shown in figure P8.17 (b). The drivetrain, consisting of the windmill rotor, gearbox, and electric generator (see Figure P8.17(c)), is part of the control loop. The torque created by the wind drives the rotor. The windmill rotor is connected to the generator through a gearbox.
The transfer function of the drivetrain is

$\begin{array}{l}\frac{P_O\left(s\right)}{T_R\left(s\right)}=G_{dt}\left(s\right)\\ =\frac{3.92K_{LSS}{K}_{HSS}{K}_G{N}^{2}s}{\left\{N^2{K}_{Hss}\left(J_R{s}^2+K_{Lss}\right)\left(J_G{s}^2\left[{\tau }_{el}s+1\right]+K_{G}s\right)+J_R{s}^2{K}_{LSS}\left[\left(J_G{s}^2+K_{HSS}\right)+\left({\tau }_{el}s+1\right)+K_{G}s\right]\right\}}\end{array}$

where $P_o\left(s\right)$ is the Laplace transform of the output power from the generator and $T_R\left(s\right)$ is the Laplace transform of the input torque on the rotor. Substituting typical numerical values into the transfer function yields
$\begin{array}{l}\frac{P_o\left(s\right)}{T_R\left(s\right)}=G_{dt}\left(s\right)\\ =\frac{\left(3.92\right)\left(12.6\times {10}^6\right)\left(301\times {10}^3\right)\left(688\right)N^{2}s}{\left\{N^2\left(301\times {10}^3\right)\left(190,120s^2+12.6\times {10}^6\right)\times \left(3.8s^2\left[20\times {10}^{-3}s+1\right]+668s\right)+190,120s^2\left(12.6\times {10}^6\right)\times \left[\left(3.8s^2+301\times {10}^3\right)\times \left(20\times {10}^{-3}s+1\right)+668s\right]\right\}}\end{array}$

(Anderson, 1998). Do the following for the drivetrain dynamics, making use of any computational aids at your disposal:

a. Sketch a root locus that shows the pole locations of $G_{dt}\left(s\right)$ for different values of gear ratio, N.

b. Find the value of N that yields a pair of complex poles of $G_{dt}\left(s\right)$ with a damping ratio of 0.5.