diff --git a/docs/formulation.tex b/docs/formulation.tex index 84cb4ac06177891c1f68d3009c1dc48a6ac8dedd..1ecc9843e32627e7a05cb77986b85e12f41bce08 100644 --- a/docs/formulation.tex +++ b/docs/formulation.tex @@ -235,7 +235,7 @@ The equations can be rearranged to have all unknown voltage values on the lhs, a \end{align} where the value \begin{equation} - \alpha_{ij} = \alpha_{ji} = \frac{\Delta t \sigma_{ij}}{ c_m \Delta x_{ij}} + \alpha_{ij} = \alpha_{ji} = \frac{\Delta t \sigma_{ij}}{ c_m r_L \Delta x_{ij}} \label{eq:alpha_linsys} \end{equation} is a constant that can be computed for each interface between adjacent compartments during set up. @@ -248,9 +248,9 @@ For an unrbanched uniform cable of constant radius $a$, with length $L$ and $n$ \Delta x_{ij} &= \Delta x = \frac{L}{n-1}, \nonumber \\ \sigma_{ij} &= \pi a^2, \nonumber \\ \sigma_{i} &= 2 \pi a \Delta x, \nonumber \\ - \alpha_{ij} &= \frac{\pi a^2\Delta t}{c_m\Delta x}, \nonumber \\ + \alpha_{ij} &= \frac{\pi a^2\Delta t}{c_m r_L\Delta x}, \nonumber \\ \frac{\alpha_{ij}}{\sigma_i} - &= \frac{a\Delta t}{2c_m\Delta x^2}. \nonumber + &= \frac{a\Delta t}{2c_m r_L\Delta x^2}. \nonumber \end{align} With these simplifications, the lhs of the linear system is \begin{align} @@ -258,7 +258,7 @@ With these simplifications, the lhs of the linear system is \nonumber \\ = & (1+2\beta)V_i^{k+1} - \beta V_{i+1}^{k+1} - \beta V_{i-1}^{k+1}. \end{align} -where $\beta=\frac{a\Delta t}{2c_m\Delta x^2}$. +where $\beta=\frac{a\Delta t}{2c_m r_L\Delta x^2}$. The end points of the cable, i.e. the compartments for $x_1$ and $x_n$, have to be handled differently. If we assume that a no-flux boundary condition, i.e. $\vv{J}\cdot\vv{n}=0$, is imposed at the end of the cable, the lhs of the linear system are diff --git a/docs/symbols.tex b/docs/symbols.tex index b97daf6986320fda5b911fe5f906b67575b362b8..90ac0ac5a823652d0e42d97b2167ac9f2bd84ce0 100644 --- a/docs/symbols.tex +++ b/docs/symbols.tex @@ -11,6 +11,7 @@ voltage & $V$ & volt $V$ & potential work per unit charge \\ resistance & $R$ & ohm $\Omega$ & recall Ohm's law $V=IR$ \\ capacitance& $C$ & farad $F$ & $C=\frac{q}{V}$, $[J\cdot C^{2}]$\\ + conductance& $g$ & siemens $S$ & \\ \hline \end{tabular} @@ -35,6 +36,8 @@ $C$ & $F$ & $C\cdot V^{-1}$ & $kg^{-1}\cdot m^{-2}\cdot s^{4}\cdot A^{2}$ \\ + $g$ & $S$ & $A\cdot V^{-1}$ & + $kg^{-1}\cdot m^{-2}\cdot s^3\cdot A^2$ \\ \hline \end{tabular}