This paper provides contributions to the study of weakly nonlinear electrical circuit oscillators. A method is introduced for the determination of the stable limit cycle and the dynamics of a family of oscillators, employed in the analysis of a nonlinear oscillator similar to the van der Pol circuit. The circuit studied includes the power dissipated in the inductor resistance and in a load resistor connected in parallel with the capacitor, along with a more generic nonlinear damping system.  It is demonstrated that the system has a stable limit cycle which is dependent on the internal power losses. The results of the theoretical analysis are verified with the use of computer simulation.