Five-port equivalent electric circuit of piezoelectric bimorph beam

Abstract

Electromechanical behavior of a three-layered piezoelectric bimorph beam is represented by an electrically equivalent circuit with impedance elements. Each impedance is determined from the components of the electrically equivalent 5×5 impedance matrix. The derivation of the impedance matrix is based on a bimorph beam theory where vertical and angular displacements at each end of the beam are considered. The impedance matrix thus consist of one electrical, four mechanical and one electromechanical coupling loops. Entire electromechanical flows and efforts accordingly have one electrical and four mechanical ports. For an electromechanical system where other mechanical or electrical components are attached to the bimorph, an equivalent electric circuit of the system can be thus generated by connecting the electrically equivalent impedances to the circuit according to the mechanical boundary conditions or external mechanical components. The characteristics of the electromechanical system, e.g., an electrical admittance, can be obtained from the equivalent circuit. As examples, three different types of mechanical boundary conditions — free, simply supported and cantilevered — are considered, and the equivalent circuits are drawn with each port opened or closed according to the boundary conditions. Two examples of external mechanical systems attached to the bimorph are also shown, i.e., a cantilevered bimorph beam with a mass–spring–damper system attached to its free end, and a segmented piezoelectric bimorph beam with an extended substrate arm. In each case, an equivalent circuit is also drawn and the electrical admittance is directly derived from the circuit. The resonance and antiresonance frequencies are accordingly calculated, and it is found that the five-port equivalent electric circuit of the piezoelectric bimorph beam presented in this paper yields an exact expression for the vibration of the piezoelectric bimorph.