## 4 Discussion on Conservation Law  
From its definition (18), CCF suggests performing the multiplication of time shifted versions of the voltage and current. We are going to show, in this section, that this can lead us to important properties linking the components of CSCD (22) over all the elements of which any circuit consists. Let us consider an electric circuit assembled of \(N\) elements, whose currents and voltages are denoted by \(i_{n}(t)\) and \(v_{n}(t)\), respectively. [PERSON]'s theorem [10] states that, whenever the currents obey the equation set constituting [PERSON]'s current law and the voltages obey the equation set constituting [PERSON]'s voltage law (KVL), the instantaneous powers \(p_{n}(t)=v_{n}(t)i_{n}(t)\) will submit to the balance equation:  
\[\sum_{n=1}^{N}p_{n}(t)=\sum_{n=1}^{N}v_{n}(t)i_{n}(t)=0. \tag{63}\]  
Figure 19: The spectrum of \(p(t)\) for the case of transmission line and pulse train current. Figure 18: Average power distribution for the case of transmission line and pulse train current. The equation expressing KVL in the \(l\)th loop looks like the linear combination of the voltages:  
\[\sum\limits_{n=1}^{N}\lambda_{ln}v_{n}(t)=0, \tag{64}\]  
where \(\lambda_{ln}\) is a coefficient taking one of three possible values: \(1\), \(-1\) or \(0\). If the element has not been included in the \(l\)th loop, \(\lambda_{ln}=0\). If the \(n\)th element is a part of the loop, the mutual directions of the element voltage and the loop trace determine \(\lambda_{ln}=1\) if they coincide, and \(\lambda_{ln}=-1\) otherwise. The structure of equation set describing KVL establishes the fact that they are invariant to time shift \(\theta\) applied to all voltages, \(v_{n}(t+\theta)\), simultaneously. Similarly, the equation for KCL in the \(p\)th node is the linear combination of the currents:  
\[\sum\limits_{n=1}^{N}\beta_{pn}i_{n}(t)=0, \tag{65}\]  
where \(\beta_{pn}\) is a coefficient taking values out of the set \(\{1,\,-1,\,0\}\). If the \(n\)th element is not adjacent to the \(p\)th loop, \(\beta_{pn}=0\). If the current associated with the \(n\)th element is leaving the node, \(\beta_{pn}=1\), and if the current is entering the node, \(\beta_{pn}=-1\). Provided the time shift \(\sigma\) applied to all currents, \(i_{n}(t+\sigma)\), simultaneously, the KCL equation set remains valid. More importantly, time shifts applied to voltages and currents do not need to be equal to each other. Returning to the genuine [PERSON]'s theorem, we may conclude that:  
\[\forall(t,\tau)\in\mathbb{R}^{2}:\sum\limits_{n=1}^{N}\mathcal{R}_{\tau i}\, (t,\tau)=\sum\limits_{n=1}^{N}v_{n}(t+\tau/2)i_{n}^{*}(t-\tau/2)=0, \tag{66}\]  
where the additional index, \(n\), written in the subscript of \(\mathcal{R}_{\tau i}\,n(t,\tau)\), highlights the possession of CCF by the \(n\)th element of the circuit. Equation (66) provides us with a generalized version of [PERSON]'s theorem in a time domain. However, the series made of Dirac delta functions in (21) forms the frequency domain counterpart of orthogonal basis for formal expansion of CCF into double Fourier series:  
\[\mathcal{R}_{\tau i}(t,\tau)=\sum\limits_{m=-\infty}^{+\infty}\sum\limits_{ k=-\infty}^{+\infty}\mathcal{S}_{\tau i}^{(m,k)}\exp\bigg{\{}j2\pi\bigg{[}mFt+ \bigg{(}k+\frac{((m))_{2}}{2}\bigg{)}F\tau\bigg{]}\bigg{\}}. \tag{67}\]  
In other words, provided the complex exponential term is rewritten into a concise expression:  
\[E_{m,k}(t,\tau)=\exp\bigg{\{}j2\pi\bigg{[}mFt+\bigg{(}k+\frac{((m))_{2}}{2} \bigg{)}F\tau\bigg{]}\bigg{\}}, \tag{68}\]  
the orthogonality condition will hold:  
\[\frac{1}{2T^{2}}\int\limits_{T}\int\limits_{2T}E_{m_{1},k_{1}}(t,\tau)E_{m_{ 2},k_{2}}^{*}(t,\tau)d\tau dt=\delta_{k_{1}k_{2}}\delta_{m_{1}m_{2}}, \tag{69}\]  
where \(\delta_{pq}\) denotes the Kronecker delta, which is equal to \(1\) if \(p=q\) and \(0\) otherwise.