## 2 Theory and Models  
### Voltage, Current and Instantentious Power Definitions  
The instantaneous power delivered to any element in a circuit is considered as a time-varying signal given by the product of the instantaneous voltage \(v(t)\) across the element and the instantaneous current \(i(t)\) through it:  
\[p(t)=v(t)i(t). \tag{1}\]  
In this research, we focus on strictly periodic waveforms representing the voltage and currents  
\[v(t)=v(t+T),\;i(t)=i(t+T), \tag{2}\]  
which can be expressed as their Fourier series, written both in real or complex forms  
\[v(t)=V_{0}+\sum_{k=1}^{+\infty}V_{k}\cos\left(2\pi\frac{k}{T}t+\varphi_{Vk} \right)=\sum_{k=-\infty}^{+\infty}M_{v}^{(RF)}\exp(j2\pi kFt) \tag{3}\]  
and  
\[i(t)=I_{0}+\sum_{k=1}^{+\infty}I_{k}\cos\left(2\pi\frac{k}{T}t+\varphi_{lk} \right)=\sum_{k=-\infty}^{+\infty}M_{i}^{(RF)}\exp(j2\pi kFt) \tag{4}\]  
where \(T\) is the period, which is the same for the voltage and current, \(F=1/T\) is the frequency, \(V_{k}\) and \(\varphi_{Vk}\) are the voltage amplitude and phase, respectively, \(I_{k}\) and \(\varphi_{lk}\) are the current amplitude and phase, respectively. \(M_{v}^{(RF)}\) and \(M_{l}^{(RF)}\) denote the complex valued Fourier coefficients for the complex exponential of frequency, \(kF\), which is written in parentheses in the superscript, in the series of the voltage and current, respectively. These coefficients can be evaluated directly as soon as the real form parameters are known  
\[M_{x}^{(RF)}=\left\{\begin{array}{c}\frac{X_{k}}{2}\exp(j\varphi_{Xk}),\;k >0;\\ X_{k},\;k=0;\\ \frac{X_{-k}}{2}\exp(-j\varphi_{X(-k)}),\;k<0,\end{array}\right. \tag{5}\]  
where the term \(x\) takes one of the symbolic values, \(v\) or \(i\), denoting either the voltage or the current.  
It is important to highlight that the right-hand sides of Equations (3) and (4) are written as sums whose summands are related to frequencies; \(f=kF\). These frequencies can take both positive and negative values, since they are related to the integral Fourier transform:  
\[X(f)=\int\limits_{-\infty}^{+\infty}x(t)\exp(-j2\pi ft)dt, \tag{6}\]  
where the complex exponentials \(\exp(j2\pi f)\) are involved.  
In other words, each harmonic of nonzero physical frequency \(kF\) is represented by two complex exponentials, \(\exp(j2\pi f)\), of frequencies \(\pm kF\), whose complex valued amplitude are defined using (5) as soon as physical amplitude and phase are defined. These complex exponentials provide a reasonable mathematical model, which is often more convenient than an amplitude-phase form for performing the vast majority of formal manipulation.  
Strictly speaking, the Fourier transform (6) does not exist as an ordinary function of frequency \(f\) if the signal \(x(t)\) is periodic. However, the complex form of the Fourier series, written for the voltage (3) and current (4), can be Fourier transformed using (6), if one treats the formal expression of their spectra as generalized functions, or distributions [44] in frequency domain, assembled of Dirac delta functions only:  
\[V(f)=\sum\limits_{k=-\infty}^{+\infty}M_{v}^{(kF)}\delta(f-kF), \tag{7}\]  
\[I(f)=\sum\limits_{k=-\infty}^{+\infty}M_{i}^{(kF)}\delta(f-kF). \tag{8}\]