A general theory for compensating acoustic transducers

Electromechanical and piezoelectric acoustic transducers are shown to be mathematically model-able by a common state-variable formalism. In the three cases considered the matrix differential equations are completely controllable and completely observable, so state-variable compensation is possible. A simplified derivation of the Kalman Observer(1) is presented, illustrating how the internal transducer state-variables are obtained from the observer filter even though they are not directly available from the transducer. These variables which are signals within the Kalman Observer(l), are used in a state-variable feedback compensation scheme which has the capability of placing the closed-loop transducer-observer filter system poles arbitrarily in the complex plane. References for the Luenberger Observer(2) approach are given. Theoretically therefore, at least in the linear mode, transducers may be compensated to have arbitrarily sharp transient response without ringing, as well as a very wide-band frequency response. Similarly, the Q and resonant frequency are also arbitrarily compensatable. This compensation scheme may also be viewed as a generalization of the well-known negative damping-factor idea to obtain not only an effective negative source resistance, but also negative mass, viscous damping, spring stiffness, resistance, capacitance, and negative inductance in desired places in the transducer equivalent circuit. However, extreme response improvement also requires large input power, which leads to a violation of the linearity assumption; so the practical improvement possible depends on the linearity of the transducer until nonlinear transducer models and nonlinear observer theory is improved.