1. Introduction
Figure 1. A generalized 2 × n resistor network with an arbitrary intermediate horizontal axis; there are three resistance elements {${r}_{0},r,{r}_{1}$} arranged in the network model. |
2. Main results
2.1. Necessary parameter definitions
2.2. Analytic expression of the potential function
a | (a)The potential function of the node ${A}_{x}$ on the axis ${A}_{0}{A}_{n}$ can be expressed as $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{U\left({A}_{x}\right)}{J}=\displaystyle \frac{{r}_{1}r}{r+2{r}_{1}}\left({x}_{1}-{x}_{s}\right)\\ \,+\,\displaystyle \frac{r}{2h}\left(\displaystyle \frac{{\beta }_{{x}_{1}\vee x}^{\left(1\right)}+{\beta }_{{x}_{2}\vee x}^{\left(1\right)}}{{F}_{n+1}^{\left(1\right)}}+{r}^{2}\displaystyle \frac{{\beta }_{{x}_{1}\vee x}^{\left(2\right)}-{\beta }_{{x}_{2}\vee x}^{\left(2\right)}}{{\left(r+2{r}_{1}\right)}^{2}{F}_{n+1}^{\left(2\right)}}\right),\end{array}\end{eqnarray}$ where $h=r/{r}_{0},$ ${h}_{1}={r}_{1}/{r}_{0},$ ${\beta }_{{x}_{s}\vee x}^{(i)}$ is defined in equation ( $\begin{eqnarray}\begin{array}{l}{x}_{{\rm{s}}}=\left\{{x}_{1}:\,0\leqslant x\leqslant {x}_{1}\right\}\cup \left\{x:\,{x}_{1}\leqslant x\leqslant {x}_{2}\right\}\cup \\ \,\,\left\{{x}_{2}:\,{x}_{2}\leqslant x\leqslant n\right\}.\end{array}\end{eqnarray}$ |
b | (b)The potential function of node ${B}_{x}$ on the axis ${B}_{0}{B}_{n}$ can be expressed as $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{U\left({B}_{x}\right)}{J}=\displaystyle \frac{{r}_{1}r}{r+2{r}_{1}}\left({x}_{1}-{x}_{s}\right)\\ \,+\,\displaystyle \frac{h{r}_{1}}{{\left(h+2{h}_{1}\right)}^{2}}\left(\displaystyle \frac{{\beta }_{{x}_{2}\vee x}^{\left(2\right)}-{\beta }_{{x}_{1}\vee x}^{\left(2\right)}}{{F}_{n+1}^{(2)}}\right).\end{array}\end{eqnarray}$ In particular, when $x=0,$ the potential of the reference point $U({B}_{0})={U}_{0}$ is obtained as $\begin{eqnarray}\displaystyle \frac{U\left({B}_{0}\right)}{J}=\displaystyle \frac{h{r}_{1}}{{\left(h+2{h}_{1}\right)}^{2}}\left(\displaystyle \frac{{\rm{\Delta }}{F}_{n-{x}_{2}}^{\left(2\right)}-{\rm{\Delta }}{F}_{n-{x}_{1}}^{\left(2\right)}}{{F}_{n+1}^{\left(2\right)}}\right).\end{eqnarray}$ |
c | (c)The potential function of node ${C}_{x}$ on the axis ${C}_{0}{C}_{n}$ can be expressed as |
2.3. Equivalent resistance formulae
3. Methods and calculations
3.1. Modeling general differential equations
Figure 2. Sub-network of a 2 × n resistor network with current parameters. |
3.2. Constraint equations on the left and right boundaries
3.3. Constraint equation of the current input and output
Figure 3. Current input image with the current parameters of the 2 × n resistor network. |