5.1.3 Quasi-Magnetostatics

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5.1.3 Quasi-Magnetostatics

If the characteristic lengths of the analyzed structures are much smaller than the considered wave lengths and for conducting areas $\epsilon / \gamma \ll T$ , the displacement current can be neglected [40,69]. is the characteristic period of the time change rate. In this case the Maxwell equations (4.1) to (4.4) can be simplified and the so called dominant magnetic field model is achieved:

	$\displaystyle \vec{\nabla}\times\vec{E} = -\mu\partial_t\vec{H}$	(5.11)
	$\displaystyle \vec{\nabla}\cdot\vec{B} = 0$	(5.12)
	$\displaystyle \vec{\nabla}\times\vec{H} = \vec{J}.$	(5.13)

After transforming (5.13)

$\displaystyle \vec{\nabla}\times\vec{H} = \vec{J} = \gamma\vec{E} \Rightarrow \frac{1}{\gamma}\vec{\nabla}\times\vec{H} = \vec{E}$

(5.14)

the rotor operator is applied and the right hand side is substituted by (5.11)

$\displaystyle \vec{\nabla}\times\left(\frac{1}{\gamma}\vec{\nabla}\times\vec{H}\right) = \vec{\nabla}\times\vec{E} = -\partial_t\vec{B} = -\mu\partial_t\vec{H}.$

(5.15)

Thus, in the frequency domain the quasi-magnetostatic case is described by the following partial differential equation for the magnetic field $\vec{H}$

$\displaystyle \vec{\nabla}\times\left(\frac{1}{\gamma}\vec{\nabla}\times\vec{H}\right) + \jmath\omega\mu\vec{H} = \vec{0}.$

(5.16)

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A. Nentchev: Numerical Analysis and Simulation in Microelectronics by Vector Finite Elements