Coupled \(A-V-J-E-T\) model for resistive magnets

This page summarizes the coupled model used for high-field resistive magnets where electrical and magnetic quantities are solved together with temperature.

1. Scope and assumptions

We use a Maxwell quasi-static (MQS) setting:

  • displacement currents are neglected,

  • magnetic flux is represented by the vector potential \(\mathbf{A}\),

  • electric field is represented by \(\mathbf{A}\) and scalar potential \(V\),

  • material properties depend on temperature \(T\) (and optionally on field magnitude).

We denote by \(\Omega_c\) the conducting regions (coils, inserts) and \(\Omega_n\) the non-conducting regions (air, insulation), with \(\Omega=\Omega_c\cup\Omega_n\).

For resistive high-field magnets considered here, surrounding materials are a-magnetic, hence in void/non-magnetic regions: \(\mu=\mu_0\) (equivalently \(\nu=\nu_0=1/\mu_0\)).

2. Kinematics and constitutive relations

\[\mathbf{B} = \nabla \times \mathbf{A}, \qquad \mathbf{E} = -\frac{\partial \mathbf{A}}{\partial t} - \nabla V\]
\[\mathbf{H} = \nu(\mathbf{x},T,\mathbf{B})\,\mathbf{B}, \qquad \mathbf{J} = \chi_c \,\sigma(T,\mathbf{B})\,\mathbf{E}\]

with

\[\nu(\mathbf{x},T,\mathbf{B})= \begin{cases} \nu_0 & \text{in void / non-magnetic regions},\\ \nu_c(T) & \text{in a-magnetic conductors (coils, inserts)},\\ \nu(\mathbf{B},T) & \text{in magnetic materials (if present)}. \end{cases}\]

where \(\chi_c\) is the indicator of conducting regions (1 in \(\Omega_c\), 0 otherwise).

Typical temperature-dependent laws for resistive magnets are:

  • \(\sigma=\sigma(T)\) in copper alloys,

  • \(\mu=\mu_0\) (or \(\nu=\nu_0\)) in void/a-magnetic regions,

  • \(k=k(T)\) and \(C_p=C_p(T)\) in the thermal model.

3. Coupled electromagnetic model (MQS)

Ampere’s law with impressed/source current \(\mathbf{J}_{src}\) gives

\[\nabla\times\left(\nu(\mathbf{x},T,\mathbf{B})\nabla\times\mathbf{A}\right) + \chi_c\,\sigma(T,\mathbf{B})\left(\frac{\partial \mathbf{A}}{\partial t}+\nabla V\right) = \mathbf{J}_{src} \qquad \text{in } \Omega\]

Charge conservation in conducting regions gives

\[-\nabla\cdot\left( \chi_c\,\sigma(T,\mathbf{B})\left(\frac{\partial \mathbf{A}}{\partial t}+\nabla V\right) \right)=0 \qquad \text{in } \Omega\]

Useful derived quantities:

\[\mathbf{J} = -\chi_c\,\sigma(T,\mathbf{B})\left(\frac{\partial \mathbf{A}}{\partial t}+\nabla V\right), \qquad \mathbf{B}=\nabla\times\mathbf{A}, \qquad \mathbf{E}=-\frac{\partial \mathbf{A}}{\partial t}-\nabla V\]

4. Thermal model and Joule coupling

The temperature is driven by Joule heating and cooling conditions:

\[\rho(T)\,C_p(T)\,\frac{\partial T}{\partial t} - \nabla\cdot\left(k(T)\nabla T\right) = Q_J + Q_{ext} \qquad \text{in } \Omega_T\]

with

\[Q_J = \mathbf{J}\cdot\mathbf{E} = \chi_c\,\sigma(T,\mathbf{B}) \left\lVert \frac{\partial \mathbf{A}}{\partial t}+\nabla V \right\rVert^2\]

and typical cooling boundary condition on water-cooled boundaries \(\Gamma_{cool}\):

\[-k(T)\nabla T\cdot\mathbf{n}=h\,(T-T_{cool}) \qquad \text{on } \Gamma_{cool}\]

5. DC/slow-ramp simplification

For slowly varying or quasi-stationary operation, \(\partial\mathbf{A}/\partial t \approx 0\):

\[\nabla\times\left(\nu(\mathbf{x},T,\mathbf{B})\nabla\times\mathbf{A}\right) + \chi_c\,\sigma(T,\mathbf{B})\,\nabla V = \mathbf{J}_{src}\]
\[-\nabla\cdot\left(\chi_c\,\sigma(T,\mathbf{B})\,\nabla V\right)=0\]
\[- \nabla\cdot\left(k(T)\nabla T\right)=\chi_c\,\sigma(T,\mathbf{B})\left\lVert\nabla V\right\rVert^2 + Q_{ext}\]

6. Boundary and excitation examples

Electromagnetic side:

  • magnetic insulation \(\mathbf{n}\times\mathbf{A}=0\) on outer truncation boundaries,

  • electric insulation \(\mathbf{J}\cdot\mathbf{n}=0\) on insulated conductor boundaries,

  • terminal excitation by prescribed voltage/current on electrode boundaries for coil powering.

Thermal side:

  • convection on cooling channels or cooled surfaces,

  • prescribed temperature on selected supports/interfaces,

  • adiabatic boundaries when heat exchange is negligible.

7. Far-field boundary condition using Biot-Savart

For resistive magnets, a better truncation condition can be obtained by prescribing the far magnetic field from source currents using a Biot-Savart background field.

Given the source current density \(\mathbf{J}_{src}\) in the coil region \(\Omega_{src}\), the far magnetic field is

\[\mathbf{B}_{BS}(\mathbf{x}) = \frac{\mu_0}{4\pi} \int_{\Omega_{src}} \mathbf{J}_{src}(\mathbf{x}') \times \frac{\mathbf{x}-\mathbf{x}'}{\lVert \mathbf{x}-\mathbf{x}' \rVert^3} \,d\Omega'\]

Introduce a background potential \(\mathbf{A}_{BS}\) such that \(\nabla\times\mathbf{A}_{BS}=\mathbf{B}_{BS}\), and decompose

\[\mathbf{A} = \mathbf{A}_{BS} + \mathbf{A}_{c}\]

Then on the truncation boundary \(\Gamma_{\infty}\), apply homogeneous boundary conditions to the correction field:

\[\mathbf{n}\times\mathbf{A}_{c}=0 \qquad \text{on } \Gamma_{\infty}\]

This is equivalent to enforcing the Biot-Savart far field on \(\Gamma_{\infty}\) through the lifted/background part, and usually reduces boundary-condition error compared to a pure \(\mathbf{n}\times\mathbf{A}=0\) condition.

8. Practical coupling strategy

A robust workflow for resistive magnets uses staggered coupling at each time step:

  1. solve \((\mathbf{A},V)\) with current material fields \(\sigma(T),\nu(\mathbf{x},T,\mathbf{B})\);

  2. compute \(\mathbf{E}\), \(\mathbf{J}\), and \(Q_J=\mathbf{J}\cdot\mathbf{E}\);

  3. solve temperature \(T\) using \(Q_J\) and cooling boundary conditions;

  4. update \(\sigma,k,C_p,\nu\) from the new \(T\) (and \(\mathbf{B}\) if needed);

  5. iterate until coupled convergence.

For complete MQS details, see \(A-V\) formulation. For nonlinear solver and variational strategies, see Variational formulations and resolution strategy.