Most energy-efficient input voltage function for RC delay line

This paper presents the most energy-efficient input voltage function for RC delay line. The Euler-Lagrange's differential equation for the circuit is derived thoroughly. General solution of the differential equation is presented. However, determining the unique solution leads to an ill-conditioned problem easily. Here, a solution of the ill-conditioned problem is proposed. The general solution is manipulated in a form in terms of hyperbolic sine functions, a linear function and a constant. Using this form, the ill-conditioned problem can be alleviated with the condition numbers improved. Then, the unique solution is determined. The lowest energy dissipation and the lowest input energy are derived analytically. Finally, the energy dissipation of the most energy-efficient input voltage function is compared with ramp function for various final time.