We present a new technique for the design and test of coaxial switched oscillators (SWOs). In this paper (Part I of a two-part paper), we concentrate on the electrostatic design of coaxial SWOs. We introduce a new design for the electrodes, which have been conceived as conformal surfaces in a geometrically convenient curvilinear space where Laplace equation is solvable. Analytical solution of the electrostatic fields is presented, discussed, and validated using numerical simulation. A prototype of an SWO based on this design is built and validated using experimental data.