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Chapter 7

Q. 7.EX.4

Loudspeaker with Circuit in State-Variable Form
For the loudspeaker in Fig. 2.29 and the circuit driving it in Fig. 2.30 find the state-space equations relating the input voltage v_a to the output cone displacement x. Assume that the effective circuit resistance is R and the inductance is L.

Step-by-Step

Verified Solution

Recall the two coupled equations, (2.44) and (2.48), that constitute the dynamic model for the loudspeaker:

M \ddot{x} + b \dot{x} = 0.63i, \\ L \frac{di}{dt} + R_i = v_a − 0.63 \dot{x}.

M \ddot{x} + b \dot{x} = 0.63 i .             (2.44)

L \frac{di}{dt} + Ri = v_a – 0.63 \dot{x}.        (2.48)

A logical state vector for this third-order system would be x \triangleq [x  \dot{x}  i ]^T , which leads to the standard matrices

\pmb F = \left[\begin{matrix}0 & 1 & 0 \\ 0 & -b/M & 0.63/M \\ 0 & -0.63/L & -R/L \end{matrix} \right] , \ \ \ \pmb G = \left[\begin{matrix} 0 \\ 0 \\ 1/L \end{matrix} \right] , \ \ \ \pmb H = \left[\begin{matrix} 1 & 0 & 0 \end{matrix} \right] , \ \ \ J =0 .

where now the input u \triangleq v_a.