7. Energy balance in circuit approximation#

Integral balance of electromagnetic energy (4.7) reads

\[\dfrac{d}{dt} \int_V u + \int_{V} \vec{e} \cdot \vec{j} = - \oint_{\partial V} \, \phi \vec{j} \cdot \hat{n} + \oint_{\partial V} \hat{n} \cdot \left[ \frac{1}{\mu_0} \partial_t \vec{a} \times \vec{b} + \varepsilon_0 \, \phi\, \partial_t \vec{e} \right] \ .\]

Volume terms represent

  • time derivative of the electromagnetic energy stored in the system, as an example in capacitors, inductors, air gaps in magnetic components,

    \[U = \sum_{k \in \text{Capacitors}} \dfrac{1}{2} C_k v_k^2 + \sum_{k \in \text{Inductors}} \dfrac{1}{2} L_k i_k^2 + \sum_{k \in \text{Gaps}} \dfrac{1}{2} \theta_k \phi_k^2 \ ,\]

    …

  • other contributions to electric power, like power dissipated in resistors

    \[\int_{V_k} \vec{e} \cdot \vec{j} = \int_{V_k} \rho_R \, |\vec{j}|^2 = \rho_{R_k} A_k \ell_k \dfrac{i_k^2}{A_k^2} = \dfrac{\rho_{R_k} \ell_k}{A_k} i_k^2 = R_k i_k^2 \ , \]

    with the constitutive law of Ohm resistors \(\vec{e} = \rho_R \vec{j}\), the definition of electric current \(i = \int_S \vec{j} \cdot \hat{n} \sim j A\) and resistance \(R = \frac{\rho_R \ell}{A}\)

Boundary terms represent:

  • the “vi” contribution, that can be re-written as the product of voltage and current intensity at wires of the electric ports, the only “active” interface in circuit approximation

    \[\oint_{\partial V} \phi \vec{j} \cdot \hat{n} = - \sum_{k \in \text{wires}} \phi_k \int_{S_k} \hat{j} \cdot \hat{n} = \sum_{k \in \text{wires}} v_k \, i_k \ ,\]

    having defined the current current entering the system through wire \(k\) (assuming equipotential section of the wire, constant \(\phi = v_k\) on section \(S_k\) of the \(k^{th}\) wire),

    \[i_k = - \int_{S_k} \vec{j} \cdot \hat{n} \ ,\]

    as the unit vector \(\hat{n}\) is pointing outwards the boundary of the system.

  • a radiation term, due to radiation of electromagnetic energy in free-space through the boundary of the domain; this contribution is the dominant contribution making an antenna work, and it’s usually negligible for slow regimes of systems of moderate dimensions, as discussed below comparing the order of magnitude of these contributions.

7.1. Boundary terms in circuit approximation#

In the limit of slow regime, \(\frac{f L}{c_0} \ll 1\), the comparison of the characteristic dimensions of the three boundary contributions gives (with the same abuse of notation used for non-dimensional equations in electromagnetism, indicating non-dimensional quantities with the same symbols as the dimensional quantities),

\[\begin{split}\begin{aligned} - \oint_{\partial V} \phi \, \vec{j} \cdot \hat{n} & = \sum_{k \in \text{wires}} v_k i_k = \Phi I \sum_{k \in \text{wires}} v_k i_k && (1) \\ \\ \oint_{\partial V} \hat{n} \cdot \frac{1}{\mu_0} \partial_t \vec{a} \times \vec{b} & = S \dfrac{ F A B }{\mu_0} \oint_{\partial \widetilde{V}} \hat{n} \cdot \partial_t \vec{a} \times \vec{b} = S \frac{\Phi B F^2 L}{\mu c^2} \oint_{\partial \widetilde{V}} \hat{n} \cdot \partial_t \vec{a} \times \vec{b} && (2) \\ \\ \oint_{\partial V} \hat{n} \cdot \varepsilon_0 \, \phi\, \partial_t \vec{e} & = S \, \varepsilon F E \Phi \oint_{\partial \widetilde{V}} \hat{n} \cdot \phi\, \partial_t \vec{e} = S \varepsilon \Phi B F^2 L \oint_{\partial \widetilde{V}} \hat{n} \cdot \phi\, \partial_t \vec{e} && (3) \\ \end{aligned}\end{split}\]

having used results from (5.1), namely in (2) \(A = \frac{\Phi F L}{c^2}\), and in (3) \( \Phi = E L\), \(E = B F L\). As \(\varepsilon \mu = \frac{1}{c^2}\), terms (2) and (3) have the same order of magnitude, \(\sim S \frac{\Phi B F^2 L}{\mu c^2}\).

Comparing the order of magnitude of terms (1) and (2) (equal to (3)), radiation contributions (2), (3) are negligible if

\[1 \gg \dfrac{S \frac{\Phi B F^2 L}{\mu c^2}}{\Phi I} = \left(\frac{FL}{c}\right)^2 \ ,\]

having used the definition of the characteristic intensity of the magnetic field \(B = \mu J L = \mu \frac{I}{S} L\), from (5.1). In the limit of slow operating regime for systems of modest dimensions, \(\frac{FL}{c} \ll 1\), radiation is negligible if compared with the “vi” power flux at the electric port. This conclusion is a strong argument in favor of circuit approximation of electromagnetic systems of modest dimensions with low characteristic frequencies.