## 5 Conclusions  
The approach proposed in this paper to power analysis in electric circuits under arbitrary periodic excitation suggests a systematic framework, which is mainly based on two ideas. The first is expressing the voltage across, and current through, an element by their complex valued Fourier series; this allows one to consider them as first order CS processes. The second idea consists in applying the set of conventional CS tools, such as CCF and SCF, to the voltage-current pair in the element under analysis. The main advantages of the proposed method are generally based on its methodology. The method reveals the elementary, so called \"atomic\", power components, which still obey the conservation law. Proper combinations of these components, including nonlinear, as for apparent power by (42), can be used for representing any quantity, characterizing power. As it is shown in this paper, the description based on CS properties perfectly matches the nature of instantaneous power as a product of the voltage and current related to the same element of electric circuit, or one port. On the one hand, such an approach may look overly complicated since there exist alternative approaches to the same problem based on real valued, or amplitude and phase representation. The introduction of complex numbers for describing power components may be looking excessive, since complex Fourier series are more common for researchers dealing with signal processing tasks than for researchers working in the fields of electric machines and power delivery. At the current development stage, the proposed method was implemented as a specialized software utility used for postprocessing the data obtained as the output of an applied software performing circuit analysis in steady state mode. One of the classes of the problems typically solved by means of this software is the numerical optimization of the power factor, which is to be achieved by the allowed variation of the load, e.g., by adding extra reactance or changing the length of the transmission line. However, the classical optimization approach is usually performed for a single harmonic, whereas the other harmonics are considered primarily as the distortion. In contrast, the proposed method utilizes the compact component wise representation of the full instantaneous power in the element of the circuit excited by an arbitrary periodic waveform. This paves the way to more accurate optimization procedures, taking into account higher harmonics of the signal. Although, from an applied mathematician's point of view, CCF can be treated as an extended model of the time varying instantaneous power, in which the auxiliary lag parameter has been introduced, the physical meaning of CCF may still require deeper clarification. Thus, although each of the components given by (22) obeys conservation law (70) separately, they will contribute to the Fourier series representing the instantaneous power (30) only in aggregated form (31). Other aggregations are carried out to form the average and reactive powers. In addition, the question of the realizability of the circuit, which can implement the product of the time-translated voltage and currents, is omitted in this paper, leaving this issue open. Future development of the CS framework proposed in this paper can be carried out in several directions. The first one consists in its application to the circuits containing nonlinear or time varying elements. As soon as the simplified model of a nonlinear element is involved, the relation between the voltage and current in such an element can be described by its volt-ampere characteristics. In contrast, the time varying behavior of an element can be described by linear periodically time-varying, which are perfectly covered by the CS theory. In both cases, instantaneous power still remains periodic and keeps its representative ability as the quantitative measure of the voltage-current interaction.