Mathematic backgrounds

QUADCOIL uses the Simsopt angle convention, where the toroidal angle is \(\phi\) and the poloidal angle is \(\theta\). The angles go from 0 to 1 over all field perods. (0 to \(1/n_{FP}\) for one field period.)

QUADCOIL represents a sheet current approximating a coil set, \(\mathbf{K}\), with a “current potential” \(\Phi\):

\[\mathbf{K} = \hat{\mathbf{n}} \cdot \nabla \Phi.\]

\(\Phi\) can be split into 3 parts:

\[\Phi = \Phi_{sv} + \frac{G\phi'}{2\pi} + \frac{I\theta'}{2\pi}\]

The first term represents contributions from a “single-valued” component, \(\Phi_{sv}\). \(\Phi_{sv}\) is the degree of freedom QUADCOIL solves for. It can also be thought of as the density of a dipole array pointing perpendicular to the winding surface.

The second and third terms represent contributions from the net poloidal current \(G\) and the net toroidal current \(I\). \(G\) is determined by the equilibrium, and \(I\) is a free variable.

QUADCOIL solves the constrained optimization problem:

\[\begin{split}\Phi^*_{sv} = &\text{argmin}_{\Phi_{sv}} \Sigma_i\lambda_i f_i(\Phi_{sv}), &\text{subject to } g_j(\Phi_{sv}) \leq \text{ or } \geq\text{ or } = p_j \\ ...\end{split}\]

Here, \(f_i\) can be any supported scalar quantity(ies), and \(g_j\) can be any supported scalars or fields. See Available Quantities for the list of supported quantities.