The symmetry of capacitive radio frequency discharges can be controlled via the electrical asymmetry effect by driving one electrode with a fundamental frequency and its second harmonic. In such electrically asymmetric discharges, the mean ion energies at both electrodes are controlled separately from the ion flux by tuning the phase angle between the harmonics at fixed voltage amplitudes. Here, the question why the ion flux is nearly independent of is answered by investigating the power absorbed by the electrons P e as a function of and time experimentally, by a particle in cell simulation, and an analytical model. The dynamics of P e is understood by the model and is found to be strongly affected by the choice of. However, on time average, P e is nearly constant, independently of. Thus, the ion flux remains approximately constant. In addition, it is shown that the absolute value of the individual voltages across the powered and grounded electrode sheath vary linearly with the dc self-bias. However, their sum remains constant. This yields, in combination with the constancy of the ion flux, a constant power absorbed by the ions and, in conclusion, a total power absorption that is independent of .