2.14: Energy Conservation and Quasistatic Transfer Relations

Applied to one of the three regions considered in Sec. 2.16, the incremental total electric energy given by Equation 2.13.6, can be written as

\[ \delta w = - \int_V \Delta \phi \cdot \delta \overrightarrow{D} dV = - \int_V \Delta \cdot (\phi \delta \overrightarrow{D} dV + \int_V \phi \Delta \cdot \delta \overrightarrow{D} dV \label{1} \]

Because \(\rho_f = 0\), the last integral is zero. The remaining integral is converted to a surface integral by Gauss' theorem, and the equation reduces to

\[ \delta w = - \oint_S \phi \delta \overrightarrow{D} \cdot \overrightarrow{n} da \label{2} \]

Similar arguments apply in the magnetic cases. Because there is no volume free current density, \(\overrightarrow{H} = - \Delta \psi\) and Equation 2.14.9 becomes

\[ \delta w = - \oint_S \psi \delta \overrightarrow{B} \cdot \overrightarrow{n} da \label{3} \]

Consider now the implications of these last two expressions for the transfer relations derived in Sec. 2.16. Discussion is in terms of the electrical relations, but the analogy made in Sec. 2.16 clearly pertains as well to Eqs. \ref{2} and \ref{3}, so that the arguments also apply to the magnetic transfer relations.

Suppose that the increment of energy \(\delta w\) is introduced through \(S\) to a volume bounded by sections o the \(\alpha\) and \(\beta\) surfaces extending one "wavelength" in the surface dimensions. In Cartesian coordinates, this volume is bounded by \((y,z)\) surfaces extending one wavelength in the \(y\) and \(z\) directions. In cylindrical coordinates, the volume is a pie-shaped cylinder subtended by outside and inside surfaces having length \(2 \pi /k\) in the \(z\) direction and \(2 \pi \alpha/m\) and \(2 \pi \beta/m\) respectively in the azimuthal direction. In spherical coordinates, the volume is a sector from a sphere with \(\theta = 2 \pi /m\) radians along the equator, \(\theta\) extending from \(0 \rightarrow \pi \) and the surfaces at \(r = \alpha\) and \(r = \beta\). In any of these cases, conservation of energy, as expressed by Equation \ref{2}, requires that

\[ \delta w = - a^{\alpha} \Bigg \langle \Bigg\langle \phi^{\alpha} \delta D_n^{\alpha} \Bigg\rangle \Bigg\rangle + a^{\beta} \Bigg \langle \Bigg \langle \phi^{\beta} \delta D_n^{\beta} \Bigg \rangle \Bigg \rangle \label{4} \]

The \(\langle\langle \, \rangle\rangle\) indicate averages over the respective surfaces of excitation. The areas \((a^{\alpha}, a^{\beta})\) are in particular

\[ a^{\alpha \atop \beta} = \begin{cases} (2 \pi)^2/ k_y k_z & \text{Cartesian} \\\\ [(2 \pi)^2/ mk] ({\alpha \atop \beta}) & \text{Cylindrical} \\\\ (4 \pi /m) ({\alpha^2 \atop \beta^2}) & \text{Spherical} \label{5} \end{cases} \]

In writing Equation \ref{2} as Equation \ref{4}, contributions of surfaces other than the \(\alpha\) and \(\beta\) surfaces cancel because of the spatial periodicity. It is assumed that \((k_y,k_z)\), \((m,k)\) and \(m\) are real numbers.

The transfer relations developed in Sec. 2.16 take the general form

\[\begin{bmatrix} \tilde{\phi}^{\alpha}\\ \tilde{\phi}^{\beta} \end{bmatrix} = \begin{bmatrix} -A_{11} & -A_{12} \\ -A_{21} & -A_{22} \end{bmatrix} \begin{bmatrix} \tilde{D}_n^{\alpha}\\ \tilde{D}_n^{\beta} \end{bmatrix} \label{6} \]

The coefficients \(A_{ij}\) are real. Hence, for the purpose of deducing properties of \(A_{ij}\), there is no loss in generality in taking \((\tilde{D}_n^{\alpha},D_n^{\beta})\) and hence \(\tilde{\phi}^{\alpha},\tilde{\phi}^{\beta}\) as being real. Then, Equation \ref{4} takes the form

\[ \delta w = C [ -a^{\alpha} \tilde{\phi}^{\alpha} \delta \tilde{D}_n^{\alpha} + a^{\beta} \tilde{\phi}^{\beta} \delta \tilde{D}_n^{\beta} ] \label{7} \]

where \(C\) is \(1/2\) in the Cartesian and cylindrical cases and is a positive constant in the spherical case.

With the assumption that \(w = w(\tilde{D}^{\alpha},\tilde{D}^{\beta} ), the incremental energy can also be written as

\[ \delta w = \frac{\partial{w}}{\partial{\tilde{D}_n^{\alpha}}} \delta \tilde{D}_n^{\alpha} + \frac{\partial{w}}{\partial{\tilde{D}_n^{\beta}}} \delta \tilde{D}_n^{\beta} \label{8} \]

where \((\tilde{D}^{\alpha}_n,\tilde{D}^{\beta}_n)\) constitute independent electrical "terminal" variables. Thus, from Eqs. \ref{7} and \ref{8},

\[ -a^{\alpha} \tilde{\phi}^{\alpha} = \frac{\partial{w}}{\partial{\tilde{D}_n^{\alpha}}}; \quad a^{\beta} \tilde{\beta}^{\alpha} = \frac{\partial{w}}{\partial{\tilde{D}_n^{\beta}}} \label{9} \]

A reciprocity condition is obtained by taking derivatives of these expressions with respect to \(\tilde{D}^{\beta}_n\) and \(\tilde{D}^{\alpha}_n\), respectively, and eliminating the energy function. In view of the transfer relations, Equation \ref{6},

\[ a^{\alpha} A_{12} = a^{\beta} A_{21} \label{10} \]

Thus, in the planar layer where the areas \(a^{\alpha}\) and \(a^{\beta}\) are equal, the mutual coupling terms \(A_{12} = A_{21}\). That the relations are related by Equation \ref{10} in the spherical case is easily checked, but the complicated expressions for the cylindrical case simplify the mutual terms (footnote to Table 2.16.2).

The energy can be evaluated by integrating Equation \ref{7} using the "constitutive" laws of Equation 6. The integration is first carried out with \(\tilde{D}^{\beta} = 0\), raising \(\tilde{D}^{\alpha}\) to its final value. Then, with \(\tilde{D}^{\alpha} = \tilde{D}^{\alpha}\), \(\tilde{D}^{\beta}\) is raised to its final value

\[ w = C \Bigg [ \frac{1}{2} a^{\alpha} A_{11} (\tilde{D}^{\alpha}_n)^2 - a^{\beta} A_{21} \tilde{D}^{\alpha}_n \tilde{D}^{\beta}_n + \frac{1}{2} a^{\beta} A_{22} (\tilde{D}^{\beta}_n)^2 \Bigg ] \label{11} \]

With either excitation alone, \(w\) must be positive and so from this relation it follows that

\[ A_{11} > 0 , \quad A_{22} > 0 \label{12} \]

These conditions are also met by the relations found in Sec. 2.16.