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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 $L.$

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 $u \triangleq v_a.$