, that is, the arithmetic sum of the phase apparent powers; and , that is, the magnitude of total three-phase complex power. The 1920 committee found no consensus and the topic continued to dominate discussions. In 1930, another committee formed and once again failed to resolve the question. The transcripts of their discussions are the lengthiest and most controversial ever published by the AIEE. Further resolution of this debate did not come until the late 1990s. A new definition based on symmetrical components theory was proposed in 1993 by Alexander Emanuel for unbalanced linear load supplied with asymmetrical sinusoidal voltages: , that is, the root of squared sums of line voltages multiplied by the root of squared sums of line currents. denotes the positive sequence power: denotes the positive sequence voltage phasor, and denotes the positive sequence current phasor. Real number formulas A perfect resistor stores no energy; so current and voltage are in phase. Therefore, there is no reactive power and (using the passive sign convention). Therefore, for a perfect resistor . For a perfect capacitor or inductor, there is no net power transfer; so all power is reactive. Therefore, for a perfect capacitor or inductor: . where is the reactance of the capacitor or inductor. If is defined as being positive for an inductor and negative for a capacitor, then the modulus signs can be removed from S and X and get . Instantaneous power is defined as: , where and are the time-varying voltage and current waveforms. This definition is useful because it applies to all waveforms, whether they are sinusoidal or not. This is particularly useful in power electronics, where non-sinusoidal waveforms are common. In general, engineers are interested in the active power averaged over a period of time, whether it is a low frequency line cycle or a high frequency power converter switching period. The simplest way to get that result is to take the integral of the instantaneous calculation over the desired period: . This method of calculating the average power gives the active power regardless of harmonic content of the waveform. In practical applications, this would be done in the digital domain, where the calculation becomes trivial when compared to the use of rms and phase to determine active power: . Multiple frequency systems Since an RMS value can be calculated for any waveform, apparent power can be calculated from this. For active