## 2 Lossless Transmission Circuit Equations with Terminal Conditions
### Transmission Line Circuit Equation
\tag{16}\]  
\[0<D\chi_{k1}<1,D\chi_{k2}<-1,0<D\chi_{k3}<1,\]  
_which also holds when the following is the case:_  
\[\chi_{N1}(\psi(\tau-1)) = \frac{(1-\frac{\tau}{Z})\psi(\tau-1)}{\frac{\tau}{Z}+1}+\frac{((1 -\xi)+\xi)b_{p}}{\frac{\tau}{Z}+1},\] \[\chi_{N2}(\psi(\tau-1)) = \frac{(\frac{\zeta}{Z}+\frac{\tau}{Z})\psi(\tau-1)}{(\frac{\zeta }{Z}-\frac{\tau}{Z})}-\frac{\zeta b_{p}}{(\xi-\frac{\tau}{Z})},\] \[\chi_{N3}(\psi(\tau-1)) = \frac{(1-\frac{\tau}{Z})\psi(\tau-1)}{(\frac{\tau}{Z}+1)}-\frac{ (\psi(\zeta+1)+\xi)b_{p}}{(\frac{\tau}{Z}+1)},\] \[\psi_{N1} = \frac{1}{2}\Big{(}-(1-\xi)+\frac{Z\zeta}{r}\Big{)}b_{p},\] \[\psi_{N2} = \frac{1}{2}\Big{(}\eta+\xi-\frac{Z\zeta}{r}\Big{)}b_{p}(>\psi_{ N1}),\]_or we have the following:_  
\[\chi_{T1}(\psi(\tau-1)) = \frac{(1-\frac{r}{Z})\psi(\tau-1)}{(\frac{r}{Z}+1)}-\frac{\xi b_{p} }{(\frac{r}{Z}+1)},\] \[\chi_{T2}(\psi(\tau-1)) = \frac{(\zeta+\frac{r}{Z})\psi(\tau-1)}{(\zeta-\frac{r}{Z})}-\frac {(\zeta(1+\zeta)+1)b_{p}}{(\zeta-\frac{r}{Z})},\] \[\chi_{T3}(\psi(\tau-1)) = \frac{(2-\frac{r}{Z})\psi(\tau-1)}{(\frac{r}{Z}+2)}\] \[- \frac{((2+\zeta)(\eta-1)+1+2\zeta)b_{p}}{(\frac{r}{Z}+2)},\] \[\chi_{N1}(\psi(\tau-1)) = \frac{(1-\frac{\tau}{Z})\psi(\tau-1)}{\frac{\tau}{Z}+1}+\frac{((1 -\xi)+\xi)b_{p}}{\frac{\tau}{Z}+1},\]0 \[\chi_{N1}(\psi(\tau-1)) = \frac{(1-\frac{\tau}{Z})\psi(\tau-1)}{\frac{\tau}{Z}+1}+\frac{((1 -\xi)+\xi)b_{p}}{\frac{\tau}{Z}+1},\]1  
_where \(D_{Xk}\) is the slope of \(\chi_{ki}(\psi(\tau-1))\), \(1\leq i\leq 3\), and \(\mathcal{R}\) is the set of real numbers._  
Proof in Appendix A. **Corollary 1**. : _If \(1>\frac{r}{Z}>\zeta>0\) and Nagumo's 3 SPWL Resistor \((k=N)\) is \(\eta>-1\) or TD 3 SPWL Resistor \((k=T)\) is \(\eta>1\), then \(h_{k}\), \((k=N,T)\): \(\mathcal{R}\rightarrow\mathcal{R}\) is defined by the following:_  
\[\chi_{N1}(\psi(\tau-1)) = \frac{(1-\frac{\tau}{Z})\psi(\tau-1)}{\frac{\tau}{Z}+1}+\frac{((1 -\xi)+\xi)b_{p}}{\frac{\tau}{Z}+1},\]2 \[\chi_{N1}(\psi(\tau-1)) = \frac{(1-\frac{\tau}{Z})\psi(\tau-1)}{\frac{\tau}{Z}+1}+\frac{((1 -\xi)+\xi)b_{p}}{\frac{\tau}{Z}+1},\]3 \[\chi_{N1}(\psi(\tau-1)) = \frac{(1-\frac{\tau}{Z})\psi(\tau-1)}{\frac{\tau}{Z}+1}+\frac{((1 -\xi)+\xi)b_{p}}{\frac{\tau}{Z}+1},\]4 \[\chi_{N1}(\psi(\tau-1)) = \frac{(1-\frac{\tau}{Z})\psi(\tau-1)}{\frac{\tau}{Z}+1}+\frac{((1 -\xi)+\xi)b_{p}}{\frac{\tau}{Z}+1},\]5 \[= \left\{ \begin{array}{ll}h_{k1}(\psi(\tau-1))&,\psi(\tau-1)< \psi_{k1},\\ h_{k2}(\psi(\tau-1))&,\psi_{k1}\leq\psi(\tau-1)\leq\psi_{k2},\\ h_{k3}(\psi(\tau-1))&,\psi_{k2}<\psi(\tau-1).\end{array} \right. \tag{17}\]  
\[h_{ki} = \frac{(1-\frac{R}{Z})\chi_{ki}}{1+\frac{R}{Z}}+\frac{E}{1+\frac{ R}{Z}},\] \[\text{(in the case of $1>\frac{R}{Z}\geq 0$), or}\] \[= -\chi_{ki},\text{(in the case of $R\rightarrow\infty$).}\]  
_in the case where \(1>R\geq 0\): \(0<Dh_{k1}<1\), \(Dh_{k2}<-1\), \(0<Dh_{k3}<1\) holds, where \(Dh_{ki}=\frac{(1-\frac{R}{Z})\mathcal{D}h_{ki}}{1+\frac{R}{Z}}\): slope of \(h_{ki}(\psi(\tau-1))\), \(1\leq i\leq 3\)._  
_The other case is where \(R\rightarrow\infty\): \(-1<Dh_{k1}<0\), \(Dh_{k2}>1\), \(-1<Dh_{k3}<0\) holds, where \(Dh_{ki}=-D\chi_{ki}\) is the \(-\) slope of \(\chi_{ki}(\psi(\tau-1))\), \(1\leq i\leq 3\), and \(\mathcal{R}\) is the set of real numbers._  
Proof in Appendix A. **Corollary 2**.