## 3 Simulation Results
### Transmission Line Example  
The third example is a lossless transmission line (Figure 14a) with a load, which coincides with the load of the circuit in Figure 1a.  
The input impedance of transmission line can be evaluated as [48]:  
\[Z_{in}(f)=\frac{Z(f)\cos\left(\frac{2\pi f}{c}\right)+jZ_{0}\sin\left(\frac{2 \pi f}{c}\right)}{\cos\left(\frac{2\pi f}{c}\right)+j\frac{Z(f)}{Z_{0}}\sin \left(\frac{2\pi f}{c}\right)}, \tag{62}\]  
where \(c\) is the speed of wave propagation, \(l\) is the length of transmission line, \(Z\) is defined by the formula (1), and \(Z_{0}\) is characteristic impedance. For numerical simulation, the following parameters were chosen: \(c=3\times 10^{8}\) m/s, \(Z_{0}=50\)\(\Omega\), \(l=12\) m.  
Consider the current source, which is the same as in the second example, defined by formulas (60) and (61). The input impedance versus frequency is presented in Figure 14b, where its absolute value and phase are combined in the same double sided plot.  
The current, \(i(t)\), voltage, \(v(t)\), and power, \(p(t)\), signals are depicted in Figure 15. They are periodic functions with period \(T\).  
Figure 14: Transmission line loaded by the resonant circuit (**a**) and the frequency characteristic of its input impedance as impedance module and phase versus the excitation current frequency (**b**).  
Figure 13: The spectrum of instantaneous power for the case of pulse train current.  
The spectrum of current \(i(t)\) is the same as in Figure 9, the spectrum of voltage \(v(t)\) is presented in Figure 16. It can be seen that the maximum by absolute weight value delta function are at frequencies \(\pm\)20 MHz. This can be explained by the higher absolute value of the input impedance (62) at frequencies \(\pm\)20 MHz, compared to \(\pm\)10 MHz.  
Figure 17 shows the diagram of CSCD \(S_{\rm ori}(\alpha f)\) (21). It has even more components than diagram in Figure 11 because more spectral components of \(\mid V(f)\mid\) in Figure 16 have sufficient value, compared to \(\mid V(f)\mid\) in Figure 10.  
Figure 16: The voltage spectrum for the case of transmission line and pulse train current.  
Figure 15: The current, voltage and power for the case of transmission line and pulse train current.  
Figure 18 shows the distribution of the real and imaginary parts of average power. The weights of the delta function of the real part are maximum at frequencies \(\pm 10\) MHz, which correspond to fundamental frequency, \(F\), of the considered signals. The imaginary part of delta function weights is maximum by its absolute value at frequencies \(\pm 30\) MHz.  
Figure 19 shows the spectrum of \(p(t)\). The delta functions forming the spectrum of instantaneous power \(p(t)\) are at the same frequencies as the delta functions forming the spectrum of instantaneous current \(i(t)\) and voltage \(v(t)\). The first five spectral components are higher with respect to the dc component than took place in the case of the resonant circuit without transmission line (Figure 13).  
The following quantitative power characteristics were calculated: the average power (33) \(P_{av}\) = 297.1 uW, apparent power (39) \(P_{app}\) = 537.2 uVA, power factor (43) \(PF\) = 0.55.