2.11: Conservation of Magnetoquasistatic Energy

Fundamentally, the energy stored in a magnetic field involves the same work done by moving a test charge from a reference position to the position of interest as was the starting point in Sec. 2.13. But, the same starting point leads to an entirely different form of energy storage. In a magnetoquasistatic system, the net free charge is a quantity evaluated after the fact. A self-consistent representation of the fields is built upon a statement of current continuity, Equation 2.3.26b, in which the free charge density is ignored altogether. Yet, the energy stored in a magnetic field is energy stored in charges transported against an electric field intensity. The apparent discrepancy in these statements is resolved by recognizing that the charges of interest in a magnetoquasistatic system are at least of two species, with the charge density of one species alone far outweighing the net charge density.

Thermodynamics: Because the free current density is solenoidal, a current "tube" can be define shown in Figure 2.14.1. This tube is defined as with a cross section having a normal \(\overrightarrow{i_n}\) in the direction of the local current density, and a surrounding surface having a normal perpendicular to the local current density. An example of a current tube is a wire surrounded by insulation and hence carrying a total current \(i\) which is the same at one cross section as at another.

Figure 2.14.1 Current tube defined as having cross-sectional area ds perpendicular to the local current density, and an outside surface with a normal vector perpendicular to the current density.

For bipolar conduction, and a stationary medium, the current density within the tube is related to the charge density by the expression

\[ \overrightarrow{J_f} = \rho_+ \overrightarrow{v_+} - \rho_{-}\overrightarrow{v_-} \label{1} \]

Here the conduction process is visualized as involving two types of carriers, one positive, with a charge density \(\rho_+\), and the other negative, with a magnitude \(\rho_-\). The carriers then have velocities which are respectively, \(\overrightarrow{v_+}\) and \(\overrightarrow{v_-}\). Even though there is a current density, in the magnetoquasistatic system there is essentially no net charge: \(\rho_f = \rho_+ - \rho_- \simeq 0\). In an increment of time \(\delta t\), the product of the respective charge densities and net displacements is \(\rho_+v_+\delta t\) and - \(\rho_-\overrightarrow{v_-}\delta t\). The work done on the charges as they undergo these displacements is the energy stored in magnetic form. This work is computed by recognizing that the force on each of the charged species is the product of the charge density and the electric field intensity. Hence, the energy stored in the field by a length of the current tube \(dl\) is to first order in differentials \(dl\) and \(ds\),

\[ -(\rho_+ \overrightarrow{v_+} - \rho_-\overrightarrow{v_-}) \cdot \overrightarrow{E} \delta t ds dl = -\overrightarrow{J_f} \cdot \overrightarrow{E} \delta t ds dl \label{2} \]

The expression for the free current density, Equation \ref{1}, is used on the right to restate the energy stored in the increment of time \(\delta t\). The unit vector \(\overrightarrow{i_n}\) is defined to be the direction of \(\overrightarrow{J_f}\). Thus, \(\overrightarrow{J_f} = (\overrightarrow{J_f} \cdot \overrightarrow{i_n}) \overrightarrow{i_n}\). Because the current density is solenoidal, it follows closed paths. The product \((\overrightarrow{J_f} \cdot \overrightarrow{i_n}) ds\) is, by definition, constant along one of these paths, and if \(\overrightarrow{i_n} dl\) is defined as an increment of the line integral, it then follows from Equation \ref{2} that the energy stored in a single current tube is

\[ -\overrightarrow{J_f} \cdot \overrightarrow{i_n} ds ( \oint_C \overrightarrow{E} \cdot \overrightarrow{i_n} dl) \delta t \label{3} \]

Figure 2.14.2 Schematic representation of a magnetoquasistatic energy storage system. Currents are either distributed in current loops throughout the volume of interest, or confined to one of \(n\) possible contours connected to the discrete terminal pairs.

By contrast with the electroquasistatic system, in which the electric field intensity is induced by the charge density (Gauss' law), the electric field intensity in Equation \ref{3} is clearly rotational. This emphasizes the essential role played by Faraday's law of magnetic induction.

It is helpful to have in mind at least the abstraction of a physical system. Figure 2.14.2 shows a volume of interest in which the currents are either distributed throughout the volume or confined to particular contours (coils), the latter case having been discussed in Sec. 2.12.

First, consider the energy stored in the current paths defined by coils having cross-sectional area ds. From Equation \ref{3}, this contribution to the total energy is conveniently written as

\[ -\overrightarrow{J_f} \cdot \overrightarrow{i_n} ds ( \oint_{C_i} \overrightarrow{E} \cdot \overrightarrow{i_n} dl) \delta t = i_i \delta \lambda_i \label{4} \]

Faraday's law and the definition of flux linkage, Eqs. 2.12.1 and 2.12.6, are the basis for representing the line integral as a change in the flux linkage.

Because the free current density is solenoidal, the distribution of free currents within the volume V excluded by the discrete coils can be represented as the superposition of current tubes. From Equation \ref{4} and the integral form of Faraday's law, Equation 2.7.3b with \(\overrightarrow{v_s} = v = 0\) (the medium is fixed), it follows that the energy stored in a current tube is

\[ \delta w_{current \quad tube} = \overrightarrow{J_f} \cdot \overrightarrow{i_n} ds (\int_{S_{tube}} \delta \overrightarrow{B} \cdot \overrightarrow{n} da) \label{5} \]

The magnetic flux density is also solenoidal, and for this reason it is convenient to introduce the magnetic vector potential \(\overrightarrow{A}\), defined such that \(\overrightarrow{B} = \nabla \times \overrightarrow{A}\), so that the magnetic flux density is automatically solenoidal. With this representation of the flux density in terms of the vector potential, Stokes's theorem, Equation 2.6.3, converts Equation \ref{5} to

\[ \overrightarrow{J_f} \cdot \overrightarrow{i_n} ds \oint_{C_{tube}} \delta \overrightarrow{A} \cdot \overrightarrow{i_n} dl = \oint_{C_{tube}} (\overrightarrow{J_f} \cdot \delta \overrightarrow{A}) ds dl = \int_{V_{tube}} \overrightarrow{J_f} \cdot \delta \overrightarrow{A} dV \label{6} \]

Here, \(\overrightarrow{J_f}\) is by definition in the direction of \(\overrightarrow{i_n}\), so that \(\overrightarrow{J_f} \cdot \delta \overrightarrow{A}\) takes the component of \(\delta \overrightarrow{A}\) in the \(\overrightarrow{i_n}\) direction. The second equality is based upon recognition that the product \(\overrightarrow{d_s} \cdot \overrightarrow{d_l}\) is a volume element of the current tube, and the line integration constitutes an integration over the volume, \(V_{tube}\), of the tube.

To include all of the energy stored in the distributed current loops, it is necessary only that the integral on the right in Equation \ref{6} be extended over all of the volume occupied by the tubes. The combination of the incremental energy stored in the discrete loops, Equation \ref{4}, and that from the distributed current loops, Equation \ref{6}, is the incremental total energy of the system

\[ \delta w = \sum_{i=1}^{n} i_i \delta \lambda_i + \int_V \overrightarrow{J_f} \cdot \delta \overrightarrow{A} dV \label{7} \]

In this expression, \(V\) is the volume excluded by the discrete current paths. This incremental magnetic energy storage is analogous to that for the electric field storage represented by Eq, 2.13.3.

In retrospect, it is apparent from the derivation that the division into discrete and distributed current paths, represented by the two terms in Equation \ref{7}, is a matter of convenience. In representing the incremental energy in terms of the magnetic fields alone, it is handy to extend the volume V over all of the currents within the volume of interest, including those that might be represented by discrete terminal pairs. With this understanding, the incremental change in energy, Equation \ref{7}, is the last term only, with \(V\) extended over the total volume. Moreover, Ampere's law represents the current density in terms of the magnetic field intensity, and, in turn, the integrand can be rewritten by use of a vector identity (Equation 8, Appendix B):

\[ \delta w = \int_{V} \nabla \times \overrightarrow{H} \cdot \delta \overrightarrow{A} dV = \oint_{V} \Big[ \overrightarrow{H} \cdot \nabla \times \delta \overrightarrow{A} + \nabla \cdot (\overrightarrow{H} \times \delta \overrightarrow{A}) \Big] dV \label{8} \]

The last term in Equation \ref{8} can be converted to a surface integral by using Gauss' theorem. With the understanding that the system is closed in the sense that the fields fall off rapidly enough at infinity so that the surface integration can be ignored, the remaining volume integration on the right in Equation \ref{8} can be used to obtain a field representation of the incremental energy change. With the curl of the vector potential converted back to a flux density, Equation \ref{8} becomes

\[ \delta w = \int_V \overrightarrow{H} \cdot \delta \overrightarrow{B} dV \label{9} \]

The integrand of Equation \ref{9} is defined as an incremental magnetic energy density

\[ \delta W = \overrightarrow{H} \cdot \delta \overrightarrow{B} \label{10} \]

It is helpful to note the clear analogy between this energy density and the incremental total energy represented by lumped parameters. In the absence of volume free current densities that cannot be represented by discrete terminal pairs, Equation \ref{7} reduces to the lumped parameter form

\[ \delta w = \sum_{i=1}^{n} i_i \delta \lambda_i \label{11} \]

The magnetic field intensity plays the continuum role of the discrete terminal currents, and the magnetic flux density is the continuum analog of the lumped parameter flux linkages. The situation in this magnetic case is, of course, analogous to the electrical incremental energy storages in continuum and in lumped parameter cases, as discussed with Eqs. \ref{7} and \ref{8} of Sec. 2.13.

Just as it is often convenient in dealing with electrical lumped parameters to use the voltage as an independent variable, so also in magnetic field systems it is helpful to use the terminal currents as independent variables. In that case, the coenergy function \(w^{'}\) is conveniently introduced as an energy function

\[ \delta w^{'} = \sum_{i=1}^{n} \lambda_i \delta i_i \label{12} \]

In an analogous way, the co-energy density, \(w^{'}\), is defined such that

\[ \delta W^{'} = \overrightarrow{B} \cdot \overrightarrow{H}; \quad W^{'} = \overrightarrow{H} \cdot \overrightarrow{B} - W \label{13} \]

Power Flow: Thus far, the storage of energy in magnetic form has been examined. The postulate has been that all work done in moving the charges against an electric field is stored. In any system as a whole this is not likely to be the case. The general magnetoquasistatic laws enable a deduction of an equation representing the flow of power, and the rate of change of the stored energy. This places the energy storage in the context of a more general system.

A clue as to how an energy conservation statement might be constructed from the differential magnetoquasistatic laws is obtained from Equation \ref{2}, which makes it clear that the product of the free current density and the electric field intensity are closely connected with the statement of conservation of energy. The dot product of the electric field and Ampere's law, Equation 2.3.23b, is

\[ \overrightarrow{E} \cdot \Big[ \nabla \times \overrightarrow{H} - \overrightarrow{J_f} \Big] = 0 \label{14} \]

Use of a vector identity (Equation 8, Appendix B) makes it possible to rewrite this expression as

\[ \overrightarrow{H} \cdot \nabla \times \overrightarrow{E} - \nabla \cdot (\overrightarrow{E} \times \overrightarrow{H}) = \overrightarrow{E} \cdot \overrightarrow{J_f} \label{15} \]

With the additional use of Faraday's law to represent \( \nabla \times \overrightarrow{E} \), Equation \ref{15} takes the form of Equation 2.13.16, with

\[ \begin{align} S_e &\equiv \overrightarrow{E} \times \overrightarrow{H} \nonumber \\ W_e &\equiv \frac{1}{2} \mu_o \overrightarrow{H} \cdot \overrightarrow{H} \nonumber \\ \phi_e &\equiv -\overrightarrow{E} \cdot \overrightarrow{J_f} - \overrightarrow{H} \cdot \frac{\partial{\mu_o \overrightarrow{M}}}{\partial{t}} - \overrightarrow{H} \cdot \nabla \times (\mu_o \overrightarrow{M} \times \overrightarrow{v}) \nonumber \end{align} \label{16} \]

These quantities have much the same physical significances discussed in connection with Equation 2.13.16.

To place the magnetic energy storage identified with the thermodynamic arguments in the context of an actual system, consider a material which is ohmic and fixed so that \(\overrightarrow{v} = 0 \) and \(\overrightarrow{J_f} = \sigma \overrightarrow{E} \). Then the second term on the right in Equation \ref{16}c is in the form of a time rate of change of magnetization energy density. Hence, the power flow equation assumes the form of Equation 2.13.17, with

\[ \begin{align} \overrightarrow{S_E} &= \overrightarrow{E} \times \overrightarrow{H} \nonumber \\ \overrightarrow{W_E} &=\int_{o}^{\overrightarrow{B}} \overrightarrow{H} \cdot \delta \overrightarrow{B} \nonumber \\ \phi_E &= -\sigma \overrightarrow{E} \cdot \overrightarrow{E} \nonumber \end{align} \label{17} \]

Implicit is the assumption that H is a single-valued function of the instantaneous \(\overrightarrow{B}\). The resulting energy density includes magnetization energy and is consistent with Equation 2.14.10.