After cogitating on the original answer, I believe that answer is incorrect. On page 112 of the book "Field Theory Handbook" by Moon and Spencer (second edition) is the toroidal coordinate system which has the equation relating the toroidal variable $\eta$ to the cartesian coordinates $(x,y,z)$ as follows:$$x^2+y^2+z^2+a^2= 2a\sqrt{x^2+y^2}\coth\eta$$ The curves $\eta= constant$ in the $\phi=constant$ plane are nested circles with $0\leqslant \eta \lt \infty$. $\quad$We have $$x^2+y^2=\rho^2$$Choosing $a=1$ for use in Mathematica is equivalent to replacing $\rho$ and $z$ with $\frac{\rho}{a}$ and $\frac za$ , so equivalently we have$$ \coth\eta=\frac{\rho^2+z^2+1}{2\rho}$$where $(\rho,\phi,z)$ are the cylindrical coordinates with $\rho\geqslant0$ and $-\infty\lt z\lt+\infty$.$\quad$ In spherical coordinates this is $$\coth\eta=\frac{r^2+1}{2r\sin\theta}$$where $\rho = r\sin\theta$ and $\rho^2+z^2=r^2$.$\quad$We want to study the curves $\eta = constant$, so any function of $\eta$ will give the same curve shape, i.e. the nested circles of toroidal coordinates. If we choose the function $V$ such that $$V=\frac{2r\sin\theta}{r^2+1}$$ then $0\leqslant V\leqslant 1$, or in cylindrical coordinates$$V=\frac{2\rho}{\rho^2+z^2+1}$$This is the formula that is needed for plotting the originally stated problem, but the domain size of $\rho$ must be equal to the domain size of $z$ (like 8 for the ContourPlot below), otherwise the plotted curves will appear as nested ellipses instead of nested circles. I believe that ContourPlot can only be used for 2D cartesian coordinates or cylindrical coordinates $(\rho,z)$ in the $\phi = constant$ plane because angles on one axis like $\theta$ and distances like $r$ on the other makes no sense. 
ContourPlotof that, in the $xz$ coordinate system? $\endgroup$