9. Electromagnetic Circuits#

Under appropriate assumptions, it is possible to use a circuit model for electromagnetic systems, such as transformers or electric motors.

  • Gauss’s Law for Magnetic Fields:

    \[\nabla \cdot \vec{b} = 0\]

  • Ampère-Maxwell’s Law:

    \[\nabla \times \vec{h} - \partial_t \vec{d} = \vec{j}\]

Additional assumptions include:

  • Linear, non-dissipative, and non-dispersive materials: \(\vec{b} = \mu \vec{h}\) todo discuss this assumption, along with material hysteresis, magnetization cycles, etc..

  • Negligible time variations of the field \(\vec{d}\), i.e., \(\partial_t \vec{d} = \vec{0}\).

The integral form of Gauss’s law for the magnetic field allows writing the node law for the magnetic field flux in magnetic circuits:

\[0 = \oint_{\partial V} \vec{b} \cdot \hat{\vec{n}} = \sum_k \phi_k \ .\]

The integral form of Ampère-Maxwell’s law, considering:

  • A path linked only with the inductor:

    \[\int_{\ell_{ind}} \vec{h} \cdot \hat{\vec{t}} + \int_{\ell_{12}} \vec{h} \cdot \hat{\vec{t}} = \oint_{\ell_{1}} \vec{h} \cdot \hat{\vec{t}} = \int_{S^{ind}} \vec{j} \cdot \hat{\vec{n}} = N i =: m\]

  • A path linked with the air gap, bypassing the inductor:

    \[0 = \int_{\ell_{traf}} \vec{h} \cdot \hat{\vec{t}} + \int_{\ell_{21}} \hat{h} \cdot \hat{\vec{t}} = \sum_{k} h_k \ell_k + \int_{\ell_{21}} \hat{h} \cdot \hat{\vec{t}}\]

By summing these two equations and recognizing that the two line integrals over the same path in opposite directions cancel each other out, we obtain the loop law for magnetic circuits:

\[\begin{split}\begin{aligned} m & = \int_{\ell_{ind}} \vec{h} \cdot \hat{\vec{t}} + \int_{\ell_{traf}} \vec{h} \cdot \hat{\vec{t}} \\ & \approx \sum_{k \in \ell} h_k \, \ell_k \\ & = \sum_{k \in \ell} \frac{b_k}{\mu_k} \, \ell_k \\ & = \sum_{k \in \ell} \frac{\ell_k}{\mu_k \, A_k} \, \phi_k \ . \end{aligned}\end{split}\]

Kirchhoff’s laws for magnetic circuits are therefore:

\[\begin{split}\begin{cases} \sum_{k \in N_j} \phi_k = 0 \\ m_{\ell_i} = \sum_{k \in \ell_i} \theta_k \phi_k \ , \end{cases}\end{split}\]

where \(\theta_k = \frac{\ell_k}{\mu_k \, A_k}\) is the reluctance, the inverse of the permeance \(\Lambda_k = \theta_k^{-1}\).