The Onde Martenot is a classic electronic musical instrument. This paper focuses on the power-balanced simulation of its ribbon-controlled oscillator, composed of linear and non-linear as well as time-varying components. To this end, the proposed approach consists of formulating the circuit as a Port-Hamiltonian System for which power-balanced numerical methods are available. A specificity of the Martenot oscillator is to involve parallel capacitors, one of them having a capacitance that non-linearly depends on the time-varying ribbon position state. In the case of linear time-invariant (LTI) capacitors in parallel, an equivalent component can be deduced using the classic impedance approach. Such a reformulation into a single equivalent component is required to derive a state-space Port-Hamiltonian representation of a circuit. One technical result of this paper is to propose a method to determine such an equivalent component in the non-LTI case. This method is applied to the present Martenot oscillator. Then, power-balanced numerical experiments are presented for several configurations: fixed ribbon position, realistic, and over-speed movements. These results are examined and interpreted from both the electronic and mechanical points of view.