2.16: Vector Potential Transfer Relations for Certain Laplacian Fields

Even in dealing with magnetic fields in regions where \(\overrightarrow{J_f} = 0\), if the flux linkages are of interest, it is often more convenient to develop a model in terms of transfer relations specified in terms of a vector rather than scalar potential. The objective in this section is to summarize these relations for the first three configurations identified in Table 2.18.1.

Table 2.18.1 Important configurations having solenoidal field \(\overrightarrow{B}\) represented by single components of vector potential \(\overrightarrow{A}\)

Two - dimensional Cartesian	Polar	Axisymmetric Cylindrical	Axisymmetric Spherical
\[ \overrightarrow{A} = A(x,y) \overrightarrow{i}_z \tag{a} \] \[ \overrightarrow{B} = \frac{\partial{A}}{\partial{y}} \overrightarrow{i}_x - \frac{\partial{A}}{\partial{x}} \overrightarrow{i}_y \tag{b} \] \[ \phi_{\lambda} = l [ A(a) - A(b)] \tag{c} \]	\[ \overrightarrow{A} = A(r,\theta) \overrightarrow{i}_z \tag{d} \] \[ \overrightarrow{B} = \frac{1}{r} \frac{\partial{A}}{\partial{\theta}} \overrightarrow{i}_r - \frac{\partial{A}}{\partial{r}} \overrightarrow{i}_{\theta} \tag{e} \] \[ \phi_{\lambda} = l [ A(a) - A(b)] \tag{f} \]	\[ \overrightarrow{A} = \frac{ \Lambda (r,z)}{r} \overrightarrow{i}_{\theta} \tag{g} \] \[ \overrightarrow{B} = \frac{1}{r} \frac{\partial{\Lambda}}{\partial{z}} \overrightarrow{i}_r - \frac{1}{r} \frac{\partial{\Lambda}}{\partial{r}} \overrightarrow{i}_{z} \tag{h} \] \[ \phi_{\lambda} = 2 \pi [ \Lambda(a) - \Lambda(b)] \tag{i} \]	\[ \overrightarrow{A} = \frac{ \Lambda (r,\theta)}{r\, sin \theta} \overrightarrow{i}_{\phi} \tag{j} \] \[ \overrightarrow{B} = \frac{1}{r\, sin \theta} \big [\frac{1}{r} \frac{\partial{\Lambda}}{\partial{\theta}} \overrightarrow{i}_r - \frac{\partial{\Lambda}}{\partial{r}} \overrightarrow{i}_{\theta} \big ] \tag{k} \] \[ \phi_{\lambda} = 2 \pi [ \Lambda(a) - \Lambda(b)] \tag{l} \]

Table 2.19.1 Vector potential transfer relations for two-dimensional or symmetric Laplacian fields

Two-dimensional Cartesian	Polar	Axisymmetric Cylindrical
\[ \overrightarrow{A} = \overrightarrow{i}_z\, Re\, \tilde{A}(x) \, exp(-jky) \nonumber \] \[ \begin{bmatrix} \tilde{H}^{\alpha}_y \\ \tilde{H}^{\beta}_y \end{bmatrix} = \frac{k}{\mu} \begin{bmatrix} -coth \, (k \Delta) & \frac{1}{sinh \, (k \Delta)} \\ \frac{-1}{sinh \, (k \Delta)} & coth \, (k \Delta) \end{bmatrix} \begin{bmatrix} \tilde{A}^{\alpha}\\ \tilde{A}^{\beta} \end{bmatrix} \tag{a} \] \[\begin{bmatrix} \tilde{A}^{\alpha}\\ \tilde{A}^{\beta} \end{bmatrix} = \frac{k}{\mu} \begin{bmatrix} -coth \, (k \Delta) & \frac{1}{sinh \, (k \Delta)} \\ \frac{-1}{sinh \, (k \Delta)} & coth \, (k \Delta) \end{bmatrix} \begin{bmatrix} \tilde{H}^{\alpha}_y\\ \tilde{H}^{\beta}_y \end{bmatrix} \tag{b} \]	\[ \overrightarrow{A} = \overrightarrow{i}_z\, Re\, \tilde{A}(r) \, exp(-jm \theta) \nonumber \] \[ \begin{bmatrix} \tilde{H}^{\alpha}_{\theta} \\ \tilde{H}^{\beta}_{\theta} \end{bmatrix} = \frac{1}{\mu} \begin{bmatrix} f_m (\beta, \alpha) & g_m( \alpha, \beta) \\ g_m(\beta, \alpha) & f_m (\alpha, \beta) \end{bmatrix} \begin{bmatrix} \tilde{A}^{\alpha}\\ \tilde{A}^{\beta} \end{bmatrix} \tag{c} \] \[\begin{bmatrix} \tilde{A}^{\alpha}\\ \tilde{A}^{\beta} \end{bmatrix} = \mu \begin{bmatrix} F_m (\beta, \alpha) & G_m( \alpha, \beta) \\ G_m(\beta, \alpha) & F_m (\alpha, \beta) \end{bmatrix} \begin{bmatrix} \tilde{H}^{\alpha}_{\theta} \\ \tilde{H}^{\beta}_{\theta} \end{bmatrix} \tag{d} \] For \(f_m,\, g_m,\, F_m,\, G_m,\, \) see Table 2.16.2, \(k = 0, m \neq 0 \)	\[ \overrightarrow{A} = \overrightarrow{i}_{\theta}\, Re\, \tilde{A}(r) \, exp(-jkz); \quad \tilde{\Lambda} = \tilde{A} r \nonumber \] \[ \begin{bmatrix} \tilde{H}^{\alpha}_{z} \\ \tilde{H}^{\beta}_{z} \end{bmatrix} = \frac{-k^2}{\mu} \begin{bmatrix} F_o (\beta, \alpha) & G_o( \alpha, \beta) \\ G_o (\beta, \alpha) & F_o (\alpha, \beta) \end{bmatrix} \begin{bmatrix} \frac{\tilde{\Lambda}^{\alpha}}{\alpha} \\ \frac{\tilde{\Lambda}^{\beta}}{\beta} \end{bmatrix} \tag{e} \] \[\begin{bmatrix} \frac{\tilde{\Lambda}^{\alpha}}{\alpha} \\ \frac{\tilde{\Lambda}^{\beta}}{\beta} \end{bmatrix} = -(\frac{\mu}{k^2}) \begin{bmatrix} f_o (\beta, \alpha) & g_o( \alpha, \beta) \\ g_o (\beta, \alpha) & f_o (\alpha, \beta) \end{bmatrix} \begin{bmatrix} \tilde{H}^{\alpha}_{z} \\ \tilde{H}^{\beta}_{z} \end{bmatrix} \tag{f} \] For \(F_o,\, G_o,\, f_o,\, g_o,\, \) see Table 2.16.2, \(m = 0, k \neq 0 \)
\[\begin{bmatrix} \hat{A}^{\alpha}\\ \hat{A}^{\beta} \end{bmatrix} = \frac{j}{k} \begin{bmatrix} \hat{B}^{\alpha}_x\\ \hat{H}^{\beta}_x \end{bmatrix} \nonumber \]	\[\begin{bmatrix} \hat{A}^{\alpha}\\ \hat{A}^{\beta} \end{bmatrix} = \frac{j}{m} \begin{bmatrix} \alpha \hat{B}^{\alpha}_r\\ \beta \hat{H}^{\beta}_r \end{bmatrix} \nonumber \]	\[\begin{bmatrix} \frac{\hat{\Lambda}^{\alpha}}{\alpha}\\ \frac{\hat{\Lambda}^{\beta}}{\beta} \end{bmatrix} = \frac{-j}{k} \begin{bmatrix} \hat{B}^{\alpha}_r\\ \hat{H}^{\beta}_r \end{bmatrix} \nonumber \]

With \(\overrightarrow{B}\) represented in terms of \(\overrightarrow{A}\) by Equation 2.18.1, Ampere's law (Equation 2.3.23) requires that in a region of uniform permeability \(\mu\),

\[ \nabla \times \nabla \times \overrightarrow{A} = \mu \overrightarrow{J}_f \label{1} \]

For a given magnetic flux density \overrightarrow{B}, curl \overrightarrow{A} is specified. But to make \overrightarrow{A} unique, its divergence must also be specified. Here, the divergence of \overrightarrow{A} is defined as zero. Thus, the vector identity \(\nabla \times \nabla \times \overrightarrow{A} = \nabla (\nabla \cdot \overrightarrow{A}) - \nabla^2 \overrightarrow{A}\) reduces Equation \ref{1} to the vector Poisson's equation:

\[ \nabla^2 \overrightarrow{A} = -\mu \overrightarrow{J}_f; \quad \nabla \cdot \overrightarrow{A} = 0 \label{2} \]

The vector Laplacian is summarized in Appendix A for the three coordinate system of Table 2.18.1. Even though the region described in the following developments is one where \(\overrightarrow{J}_f = 0\), the source term on the right has been carried along for later reference.

Cartesian Coordinates: In the Cartesian coordinate system of Table 2.18.1 it is the z component of Equation \ref{2} that is of interest. The \(z\) component of the vector Laplacian is the same operator as for the scalar Laplacian. Thus, the situation is analogous to that outlined by Eqs. 2.16.11 to 2.16.16 with \(\phi \rightarrow A \). With solutions of the form \(A = Re\, A(x,t)\, exp(-jky)\) so that \(\gamma \rightarrow k_y \equiv k\), the appropriate linear combination of solutions is

\[ \tilde{A} = \tilde{A}^{\alpha} \frac{sinh\, kx}{sinh\, k\Delta} - \tilde{A}^{\beta} \frac{sinh\, k(x-\Delta)}{sinh\, k\Delta} \label{3} \]

Because \(\overrightarrow{H} = \overrightarrow{B}/\mu\), the associated tangential field intensity is given by Equation (b), Table 2.18.1,

\[ H_y = - \frac{1}{\mu} \frac{\partial{A}}{\partial{x}} \label{4} \]

Expressed in terms of Equation \ref{3} and evaluated at the surfaces \(x = \alpha\) and \(x = \beta\) respectively, Equation \ref{4} gives the first transfer relations, Equation (a, of Table 2.19.1. Inversion of these relations gives Eqs. (b).

Polar Coordinates: In cylindrical coordinates with no z dependence, it is again the z component of Equation \ref{2} that is pertinent. The configuration is summarized in Table 2.18.1. Solutions take the form \(A = Re\, A(r,t)\, exp(-jm \theta)\) and are analogous to Equation 2.16.21 with \(\phi\) replaced by \(A\):

\[ \tilde{A} = \tilde{A}^{\alpha} \frac{[ (\frac{\beta}{r})^m - (\frac{r}{\beta})^m ]}{[(\frac{\beta}{\alpha})^m - (\frac{\alpha}{\beta})^m ]} + \tilde{A}^{\beta} \frac{[ (\frac{r}{\alpha})^m - (\frac{\alpha}{r})^m ]}{[(\frac{\beta}{\alpha})^m - (\frac{\alpha}{\beta})^m ]} \label{5} \]

The tangential field is then evaluated from Equation (e), Table 2.18.1:

\[ H_{\theta} = - \frac{1}{\mu} \frac{\partial{A}}{\partial{r}} \label{6} \]

Evaluation at the respective surfaces \(r = \alpha\) and \( r = \beta\) gives the transfer relations, Eqs. (c) of Table 2.19.1. Inversion of these relations gives Eqs. (d).

Axisymmetric Cylindrical Coordinates: By contrast with the two-dimensional configurations so far considered, where the vector Laplacian of \(A_z\) is the same as the scalar Laplacian, the vector nature of Equation \ref{2} becomes apparent in the axisymmetric cylindrical configuration. The \(\theta\) component of Equation \ref{2} is the scalar Laplacian of \(A_{\theta}\) plus \((-A_{\theta} /r^2 )\) (see Appendix A). With \(A_{\theta} \equiv A\),

\[ \frac{\partial{^2 A}}{\partial{r^2}} + \frac{1}{r} \frac{\partial{A}}{\partial{r}} - \frac{A}{r^2} + \frac{\partial{^2 A}}{\partial{z^2}} = -\mu J_{\theta} \label{7} \]

Even though solutions do not have a \(\theta\) dependence, so that

\[ A = Re\, \tilde{A}(r,t)\, e^{-jkz} \label{8} \]

Equation \ref{7} reduces to a form of Bessel's equation to which solutions are Bessel's and Hankel's functions of order unity:

\[ \frac{\partial{^2 \tilde{A}}}{\partial{r^2}} + \frac{1}{r} \frac{\partial{\tilde{A}}}{\partial{r}} - (k^2 + \frac{1}{r^2}) \tilde{A} = -\mu \tilde{J} \label{9} \]

(Compare Equation \ref{9} to Equation 2.16.19.) It follows that solutions are of the form of Equation 2.16.25 with \(\tilde{\phi} \rightarrow \tilde{A} \) and \(m = 1\):

\[ \begin{align} \tilde{\Lambda} \equiv r \tilde{A} &= \tilde{A}^{\alpha} \frac{H_1 (jk \beta) [ rJ_1 (jkr)] - J_1 (jk \beta) [ rH_1 (jkr)]}{ H_1 (jk \beta) J_1(jk \alpha) - J_1 (jk \beta) H_1 (jk \alpha) } \nonumber \\ \nonumber \\ &+ \tilde{A}^{\beta} \frac{J_1 (jk \alpha) [ rH_1 (jkr)] - H_1 (jk \alpha) [ rJ_1 (jkr)]}{ J_1 (jk \alpha) H_1(jk \beta) - H_1 (jk \alpha) J_1 (jk \beta) } \nonumber \end{align} \label{10} \]

The tangential field intensity follows from Equation \ref{10} and Equation (h) of Table 2.18.1:

\[ H_z = \frac{1}{\mu r} \frac{\partial{\Lambda}}{\partial{r}} \label{11} \]

In performing the differentiation, observe from Equation 2.16.26d that whether \(R_m\) is \(J_m\) or \(H_m\)

\[ \frac{d}{dr} [r R_1 (jkr)] = jkrR_o (jkr) \label{12} \]

Evaluation of \(H_z\) at the respective surfaces \(r = \alpha\) and \(r = \beta\) gives the transfer relations, Eqs. (e) of Table 2.19.1. Inversion of these relations gives Eqs. (f).