# A New Modified Non-Approximative Method for Dynamic Systems Direct Calculation
## 4 Example from the Field of Electrical Engineering--Simulation and Validation
### Modelling given electric circuit  
Consider a system by the circuit diagram using the Steinmetz \(\pi\)-term [14; 15], Figure 5.  
A continuous dynamic model can be derived from the circuitry equations for \(u_{0}=U=\text{const. }\), and \(g_{1},~{}g_{2}\) is negligible:  
\[ \begin{gathered} U=r_{0}i_{0}(t)+L_{0}\frac{\text{d}i_{0}(t)}{ \text{d}t}+u_{1}(t)\\ u_{1}(t)=r_{1}i_{1}(t)+L_{1}\frac{\text{d}i_{1}(t)}{\text{d}t}+u_{2}(t) \\ i_{0}(t)=C_{1}\frac{\text{d}u_{1}(t)}{\text{d}t}+i_{1}(t)\\ i_{1}(t)=C_{2}\frac{\text{d}u_{2}(t)}{\text{d}t}+\frac{1}{\text{d}t} u_{2}(t)\end{gathered} , \tag{42a}\]  
Figure 5: Circuit diagram of the one Steinmetz \(\pi\)-term with input \(r_{0},~{}L_{0}\) and output R. \begin{table}
\begin{tabular}{c c c c c c c c} \hline Load & \(r_{0}\) [m\(\Omega\)] & \(r_{1}\) (\(\Omega\)) & \(L_{0}\) (mH) & \(L_{1}\) (mH) & \(C_{1}\) (μF) & \(C_{2}\) (μF) & \(R\) (\(\Omega\)) \\ \hline
1.00 & 1.035 & 1.00 & 2.62 & 70 & 70 & 1.00 \\ \hline \end{tabular}
\end{table} 
Table 9: Load parameters of the one Steinmetz \(\pi\)-term. Thus, the state-space form is:  
\[ \begin{array}{c}\frac{\mathrm{d}i_{0}(t)}{\mathrm{d}t}=-\frac{r_{0}}{L_{1}}i_{0 }(t)-\frac{1}{L_{0}}u_{1}(t)+\frac{1}{L_{0}}U\\ \frac{\mathrm{d}i_{1}(t)}{\mathrm{d}t}=-\frac{r_{1}}{L_{1}}i_{1}(t)+\frac{1}{L _{1}}u_{1}(t)-\frac{1}{L_{1}}u_{2}(t)\\ \frac{\mathrm{d}u_{1}(t)}{\mathrm{d}t}=\frac{1}{C_{1}}i_{0}(t)-\frac{1}{C_{1}} i_{1}(t)\\ \frac{\mathrm{d}u_{2}(t)}{\mathrm{d}t}=\frac{1}{C_{2}}i_{1}(t)-\frac{1}{RC_{2}} u_{2}(t)\end{array}  \tag{42b}\]  
and the shorter matrix form is:  
\[x_{k+1}=F_{\Delta}x_{n}+G_{\Delta}U_{k}. \tag{43}\]  
After time discretization with the \(h\)-step (see Appendix B), we obtain fundamental and transition matrices \(F_{\Delta},G_{\Delta}\). Then, we can calculate matrices \(F_{T/2},\ G_{T/2}\) (or \(F_{T/6},\ G_{T/6}\)), and we obtain:  
\[x_{n+1}=F_{\frac{T}{2}}x_{n}+G_{\frac{T}{2}}U_{n}\] (44a) or \[x_{n+1}=F_{\frac{T}{6}}x_{n}+G_{\frac{T}{6}}U_{n} \tag{44b}\]  
and, consequently, by the DNM method, we obtain the state variables vector \(x_{n}\) using Equations (38) or (A1), respectively. In the case of harmonic supply voltage or rectified harmonic voltage, instead of \(u_{0}=U\), it is in its discrete format:  
\[u_{0}=U\sin\biggl{(}\frac{2\pi}{T}h.k\biggr{)}\] (45a) or \[u_{0}=U\mathrm{abs}\biggl{[}\sin\biggl{(}\frac{2\pi}{T}h.k\biggr{)}\biggr{]}. \tag{45b}\]  
A similar method can be used for a nonharmonic impulse supply exciting voltage [12]:  
\[u_{0}=U\sin\biggl{(}\frac{\pi}{3}n+\frac{\pi}{6}\biggr{)}. \tag{46}\]  
where \(n\) is an order of the period of the impulse switching function (\(\frac{\pi}{3}\)); replacing \(n\) with a new variable \(n=\mathrm{integer}\bigl{(}\frac{6}{T}h.k\bigr{)}\), we obtain the following for the input exciting voltage:  
\[u_{0}=U\sin\biggl{[}\mathrm{integer}\biggl{(}\frac{6}{T}h.k\biggr{)}\frac{ \pi}{3}+\frac{\pi}{6}\biggr{]}. \tag{47}\]  
where \(h,k\) are the same as in previous cases. Simulations have been worked out for a given electric circuit model (Figure 4) with stationary matrices \(F_{\Delta}\), \(G_{\Delta}\) and \(F_{\tau}\), \(G_{\tau}\) or \(F_{T/2}\), \(G_{T/2}\), under  
\(u_{0}=U=\mathrm{const}\), Figure 6,  
\(u_{0}=U\mathrm{abs}\bigl{[}\sin\bigl{(}\frac{2\pi}{T}h.k\bigr{)}\bigr{]}\), Figure 7,  
\(u_{0}=U\sin\bigl{(}\frac{2\pi}{T}h.k\bigr{)}\), Figure 8, and  
\(u_{0}=U\sin\bigl{(}\mathrm{integer}\bigl{(}\frac{6}{T}h.k\bigr{)}\frac{\pi}{3 }+\frac{\pi}{6}\bigr{]}\), Figure 9.  
Figure 6: Time waveforms during the first three multiple \(t\) for \(u_{0}(t)=U=\text{const}\). Figure 7: Time waveforms during the first three half-periods for \(u_{0}(t)=U\text{abs}\left[\sin\left(\frac{2\pi}{T}h.k\right)\right]\). The results of each case using the direct numeric method are shown in Table 8, Table 10, Tables 12 and 14. The time waveforms during the first three multiple \(t\) under \(u_{0}(t)\)= \(U=\text{const.