3D mathematical model of contact-separation and single-electrode mode triboelectric nanogenerators

Highlights: • A time-dependent three dimensional spatial model is proposed to assert triboelectric nanogenerator operation. • The model describes 3D geometry effects for the evaluation of the displacement current and nanogenerator operation. • New and more accurate capacitance equations for contact-separation and single-electron mode nanogenerators. -- Abstract: Based on a set of finite-sized charged planes (FSCP), a simple time-dependent three-dimensional spatial model for the electric potential and electric field in an inhomogeneous medium composed of dielectric materials and metal contacts is proposed and used to assert triboelectric nanogenerator operation. Solving the problem of FSCP makes the three-dimensional spatial model relevant for practical TENG applications and allow for accurate and reliable results. Connecting the metal contacts to an electric resistance, Kirchhoff's law is used to derive a first-order time-dependent differential equation for the mobile charges on the metal contacts and the displacement current. Specially, the displacement current (Maxwell's displacement current) in a TENG equals to the conduction current in the external circuit is obtained. We then consider two important types of triboelectric nanogenerators: the contact-separation (CS) mode and the single-electrode (SEC) mode. A forced movement of the dielectric materials and/or the metal contacts leads to currents flowing in the system and a time-varying electrical potential, and therefore the generation of electrical power. Then, new and more accurate capacitance equations for CS and SEC modes of TENGs are extracted. Several examples of energy harvesting scenarios are finally analyzed.