Potential Formulation of the Governing Equations

To reduce the number of equations, it is convenient to reformulate the governing equations in terms of two auxiliary fields, the electric scalar potential $ϕ$ and the magnetic vector potential $A$ .

Eqn. (659) and Eqn. (662) allow you to express the electric field $E$ and the magnetic flux density $B$ in terms of a scalar field $ϕ$ and a vector field $A$ :

图 1. EQUATION_DISPLAY

E = - \nabla ϕ - \frac{\partial A}{\partial t}

(4232)

图 2. EQUATION_DISPLAY

B = \nabla \times A

(4233)

where $ϕ$ is known as the electric potential and $A$ is known as the magnetic vector potential.

Writing Maxwell's equations in terms of the potentials reduces the number of independent equations, as Eqn. (659) and Eqn. (662) become identities. However, the potentials are not unique. To uniquely define the potential $A$ , it is necessary to prescribe both its curl and its divergence. The divergence of $A$ is defined by an arbitrary gauge condition. For low-frequency electromagnetic fields, a common choice is the Coulomb gauge, $\nabla \cdot A = 0$ .

Ohm's law (Eqn. (4228)) can be written in terms of the potentials as:

图 3. EQUATION_DISPLAY

J = J_{ϕ} + J_{e d d y} + J_{e x} = - σ \nabla ϕ - σ \frac{\partial A}{\partial t} + J_{e x}

(4234)

where $J_{ϕ} = - σ \nabla ϕ$ is the contribution from the spatial variation of the electric potential, $J_{eddy} = - σ \partial A / \partial t$ is the eddy current density, which is related to the time variation of the magnetic vector potential, and $J_{e x}$ accounts for external sources of electric current density. External sources include user-defined sources and sources from additional physics—for example, excitation coils (see Eqn. (4332)) and conducting fluids in motion (see Eqn. (4369)).

In Eqn. (4234), the direction of the electric current density is from the areas of high values to the areas of low values of the electric potential, which is known as the technical direction of the electric current density.

The electric current $I$ flowing through a surface $A$ is defined as:

图 4. EQUATION_DISPLAY

I = \int_{A} J \cdot d a

(4235)

Statics

For fields that do not vary with time, the transient terms in Maxwell's equations can be neglected, giving decoupled equations for the magnetic vector potential and the electric potential.

Electrostatics: For static fields, the transient term in Eqn. (4232) can be neglected. Therefore, the electric field only depends on the electric potential:; 图 5. EQUATION_DISPLAY

$E = - \nabla ϕ$
(4236); The electrostatic force acting on an electric charge $q$ is $F_{e} = q E$ . The electric charge has a quantized nature and appears only in multiples of the elementary electric charge, $e = 1.602176487 \times 10^{- 19}$ C.; The electric potential can be calculated from Eqn. (661), which, together with Eqn. (4219), gives:; 图 6. EQUATION_DISPLAY

$- \nabla \cdot (ε \nabla ϕ) = ρ$
(4237); The Simcenter STAR-CCM+ implementation of Eqn. (4237) is discussed in the section, 静电势模型.
Magnetostatics: The magnetic vector potential can be calculated from Eqn. (660), which, when neglecting the transient term, reduces to Ampère’s law:
图 7. EQUATION_DISPLAY

$\begin{array}{l} \nabla\times \frac{1}{μ} B = J \end{array}$
(4238); In magnetostatic applications, the transient term and the electric potential contribution in Eqn. (4234) can be neglected. By using Eqn. (4220) and Eqn. (4233), Ampère’s law can then be written as:; 图 8. EQUATION_DISPLAY

$\nabla \times \frac{1}{μ} \nabla\times A = J_{e x}$
(4239); The Simcenter STAR-CCM+ implementation of Eqn. (4239) is discussed in the section, 磁矢势模型.

Low-Frequency Electromagnetics in Conducting Media

For fields that vary with time, magnetic and electric fields are mutually coupled. Specifically, a time-varying magnetic field induces eddy currents which contribute to the electric field (see Eqn. (659)). Similarly, the displacement and conduction currents associated with the electric field contribute to the magnetic field (see Eqn. (660)). This formulation considers low-frequency electromagnetic fields in conducting materials, where the conduction currents are several orders of magnitude greater than the displacement currents, $J ≫ \partial D/ \partial t$ . In many industrial applications, such as electric machines, transformers, and electric switches, the time scales of electromagnetic phenomena are relatively long, and a low-frequency assumption is appropriate.

Neglecting the displacement currents reduces Eqn. (660) to Ampère’s law (Eqn. (4238)). Also, for conducting materials, the electric current density is incompressible in nature and Eqn. (663) simplifies to:

图 9. EQUATION_DISPLAY

\nabla \cdot J = 0

(4240)

Inserting Eqn. (4220), Eqn. (4233), and Eqn. (4234) into Eqn. (4238) and Eqn. (4240), gives:

图 10. EQUATION_DISPLAY

\nabla \times \frac{1}{μ} \nabla\times A + σ \frac{\partial A}{\partial t} = - σ \nabla ϕ + J_{e x}

(4241)

图 11. EQUATION_DISPLAY

- \nabla \cdot (σ \nabla ϕ) = \nabla \cdot (σ \frac{\partial A}{\partial t}) - \nabla \cdot J_{e x}

(4242)

Eqn. (4241) and Eqn. (4242) describe the response of conducting media to low-frequency electromagnetic fields. These equations are a set of mutually coupled equations with two unknown fields, $A$ and $ϕ$ . In Simcenter STAR-CCM+, two separate solvers compute $A$ and $ϕ$ . Based on analysis requirements, you can solve the coupled system of equations, or solve either equation singularly to compute one of the potentials in a decoupled manner. The Simcenter STAR-CCM+ implementation of Eqn. (4241) and Eqn. (4242) is discussed in two separate sections, 电动势模型 and 磁矢势模型.

After solving for the potentials, the electromagnetic fields $E$ , $D$ , $H$ , and $B$ can be calculated from Eqn. (4232), Eqn. (4233), and the constitutive relations.

Transverse Magnetic Modes

Eqn. (4241) accounts for 3D modes and 2D Transverse Electric (TE) modes, where the magnetic vector potential lies on the plane defined by the 2D domain. For 2D Transverse Magnetic (TM) modes, where the magnetic vector potential is normal to the 2D domain, Simcenter STAR-CCM+ solves the decoupled equation:

图 12. EQUATION_DISPLAY

- \nabla\cdot \frac{1}{μ} \nabla A_{n} + σ \frac{\partial A_{n}}{\partial t} = J_{n, e x}

(4243)

$A_{n}$ and $J_{n, e x}$ are the transverse magnetic potential and electric current density. The magnetic flux density $B$ is calculated from Eqn. (4233), with the tangential components $A_{t_{1, 2}}$ always zero.

The implementation of Eqn. (4243) is discussed in the section, 横向磁模式.