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

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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 1 (46.4 KiB) Viewed 42 times

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 come that is moved by an electro-magnet. The electrical and mechanical components of the speaker are modeled below. m.r(1) + kx() = () Lil) + Rg() = Cin(t) – 23(1) + ilt) (t) O 000 rell m sty, it) (a) (2 points) Find the transfer functions (1) from voltage V(s) = Vin(s) - V.(8) to current (6) and (2) from magnet force F(s) to speaker cone position X(s) where i(t) = (0). (b) (10 points) The back-voltage wy(t) = Ki(1) is proportional to the cone speed i(t) and the force f(t) = K-() is proportional to the current i(t) = 4 where K, and Ki 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 1, l, r, Din, 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 V..(s) to the cone position X(X). 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 radl/s to Hertz. Include your plot in your solution

HINT: One way to change the frequency is using the MATLAB commands options - bodeoptions, options. FreqUnits - 'Hz', and Table 2: Audio Frequency Ranges Table 1: Speaker Model Parameters Audio Subset Frequency Range Parameter Value Sub-Base 16-60 Hz m 4 x 10 Bass 25 x 104 60-250 Hz R Lower Midrange 250-500 Hz 10 Midrange 500 Hz - 2 kHz I 1 x 10-2 High Midrange 10 2-4 kHz Presence 4-6 kHz К. 10 Brillance 6-20 kHz If you were unable to find the transfer function P(8) in the previous part then use the transfer function below. 2.4325 R(S) = ** +990x2 +6.404e7s +6.0810 (e) (2 points) Using your Bode magnitude plot and Table 2. what andio subset is this speaker best for? Kj (I) (8 points) We are now going to design an open-koop controller to boost the less of the speaker. The two options for controllers are below. Draw the straight-line approximation of the Bode magnitude plots for both controllers CO)= (8 + 2 12.5)(8 + 2x x 1250) C®) (+27 x 1100)(8 +2x 110000) (8 + 2x x 125) (8+2x x 11000) (8) (2 points) Which of the controller C(s) or C(s) would you choose to boost the base? Justify your answer with a Bode magnitude plot of the resulting transfer function bodemag (P+Ci,options).