Second-order wave maker theory has long been established; the most extensive and detailed approach given by Schäffer [1996. Second-order wave maker theory for irregular waves. Ocean Engineering 23, 47–88]. However, all existing theories assume the wave paddle is driven by a position-feedback motion controller. Early research in the wave power field led to the design of a force-controlled absorbing wave machine [Salter, S., 1982. Absorbing wave-makers and wide tanks. In: Directional Wave Spectra Applications, pp. 185–200]. In addition to operating as an excellent absorber, this machine seemed to introduce very little spurious harmonic content when driven with a first-order command signal. The present paper provides a mathematical model for the operation of wave makers using force-feedback control and seeks to explain this apparent advantage. The model is developed to second-order so that a command signal compensating for the remaining spurious wave is also provided. Due to the complexity of the problem, the model has been limited to flap-type wave machines and the generation of regular waves. A variety of numerical tests in force-control mode have been conducted, indicating that the spurious wave content is greatly reduced when compared to the position-control mode. A separate experimental study validating the theory is presented in a part II paper by the same authors.