(Solved): 4 D-D-D-Drop the Base (30 Points) In this problem you will use Bode plots to design an open-loop c ...

4 D-D-D-Drop the Base (30 Points) In this problem you will use Bode plots to design an open-loop controller to boost the bass on a speaker. This speaker consists of a light-weight cone that is moved by an electro-magnet. The electrical and mechanical components of the speaker are modeled below. m(t) + kr(t) = f(t) L?(t)+ R?(t) = in(t) - () D(0) 000 L 000 fr m f(t), i?t) = (a) (2 points) Find the transfer functions (1) from voltage V(s) = Vin(8) - Vm(s) to current (3) and (2) from magnet force F(3) to speaker cone position X(s) where i(t) = (t). (b) (10 points) The back-voltage vy(t) = Ki(t) is proportional to the cone speed i(t) and the force f(t) = Kri(t) is proportional to the current i(t) = 9 where Ky and Ky are constants. Use these relationships to connect the transfer functions of the mechanical and electric parts of the motor. The block diagram should have the form below. In addition to adding the transfer functions, label the signals i, f. 1, Vin, vb, v in the block-diagram. (c) (4 points) Reduce the block-diagram above to obtain the transfer function P(s) for the speaker from the input voltage Vin (s) to the cone position X(s). HINT: This problem is similar to the problem from the previous exam, but not exactly the same. (d) (2 points) Using MATLAB plot the Bode magnitude plot for the speaker transfer function P(s). The parameters are given in Table 1. Change the frequency units in your Bode magnitude plot from rad/s to Hertz. Include your plot in your solution.