4.8: Constrained-Current Magnetoquasistatic Transfer Relations

By way of exemplifying how transfer relations can be used to represent fields in bulk regions,including volume distributions of known current density, these relations are derived in this section for one important class of physical situations. The current density (which is typically the result of exciting distributions of wire) is z-directed, while the magnetic field is in the \((r, \theta)\) plane.Thus, the relations are directly applicable to rotating machines with negligible end effects. Such an application is taken up in the next section.

In a broad sense, the objective in this section is to magnetic field systems what the objective in Sec. 4.5 was to electric field systems. But, the solution of the vector Poisson's equation,Eq. 2.19.2, is more demanding than the scalar Poisson's equation, Eq. 4.5.1, and hence the technique now illustrated is limited to certain configurations in which only one component of the vector potential describes the fields. Such cases are discussed in Sec. 2.18 and the associated transfer relations for a region of free space are derived in Sec. 2.19. The following discussion relates to the polar-coordinate situations of Tables 2.18.1 and 2.19.1.

In the two-dimensional cylindrical coordinates, the vector Poisson's equation (Eq. 2.19.2) has only a z component and the Laplacian is the same as the scalar Laplacian:

\[ \nabla^2 A = - \mu J_z \label{1} \]

Following the line of attack used in Sec. 4.5, the solution is divided into homogeneous and particular parts,

\[ A = A_H + A_P \label{2} \]

defined such that

\[ \nabla^2 A_P = - \mu J_z; \quad \nabla^2 A_H = 0 \label{3} \]

The imposed current is now represented in the complex amplitude form

\[ J_z = Re \tilde{J} (r,t) e^{-jm \theta} \label{4} \]

Of course, by superposition, such solutions could be the basis for a Fourier representation of an arbitrary current distribution. Substitution of Equation \ref{4} into Equation \ref{3} shows that \(\tilde{A}_P\) must satisfy the equation

\[ \frac{d^2 \tilde{A}_P}{dr^2} + \frac{1}{r} \frac{d \tilde{A}_P}{dr} - \frac{m^2}{r^2} \tilde{A}_P = - \mu \tilde{J}(r) \label{5} \]

The particular solution can be any solution to Equation \ref{5}. The magnetic field associated with this particular solution is, by the definition of the vector potential (Eq. 2.18.1),

\[ H_{\theta P} = -\frac{1}{\mu} \frac{d \tilde{A}_P}{dr}; \quad \tilde{B}_{rP} = - \frac{jm}{r} \tilde{A}_P \label{6} \]

From Equation \ref{2} it follows that the homogeneous solution is the total solution with the particular solution subtracted off. That is,

\[ \tilde{A}_H = \tilde{A} - \tilde{A}_P; \quad \tilde{H}_{\theta H} = \tilde{H}_{\theta} - \tilde{H}_{\theta P} \label{7} \]

The homogeneous parts are related by the transfer relations, Eqs. (d) of Table 2.19.1, so that substitution from Equation \ref{7} shows that

\[\begin{bmatrix} \tilde{A}^{\alpha} - \tilde{A}^{\alpha} _P \\ \tilde{A}^{\beta} - \tilde{A}^{\beta} _P \end{bmatrix} = \mu \begin{bmatrix} F_m(\beta, \alpha) & G_m(\alpha, \beta) \\ G_m(\beta, \alpha) & F_m(\alpha, \beta) \end{bmatrix} \begin{bmatrix} \tilde{H}^{\alpha}_{\theta} - \tilde{H}^{\alpha} _{\theta P} \\ \tilde{H}^{\beta}_{\theta} - \tilde{H}^{\beta} _{\theta P} \end{bmatrix} \label{8} \]

These relations, multiplied out, are the transfer relations for the cylindrical annulus supporting a given distribution of z-directed current density:

\[\begin{bmatrix} \tilde{A}^{\alpha} \\ \tilde{A}^{\beta} \end{bmatrix} = \mu \begin{bmatrix} F_m(\beta, \alpha) & G_m(\alpha, \beta) \\ G_m(\beta, \alpha) & F_m(\alpha, \beta) \end{bmatrix} \begin{bmatrix} \tilde{H}^{\alpha}_{\theta} \\ \tilde{H}^{\beta}_{\theta} \end{bmatrix} + \begin{bmatrix} \tilde{A}^{\alpha}_P \\ \tilde{A}^{\beta}_P \end{bmatrix} - \mu \begin{bmatrix} F_m(\beta, \alpha) & G_m(\alpha, \beta) \\ G_m(\beta, \alpha) & F_m(\alpha, \beta) \end{bmatrix} \begin{bmatrix} \tilde{H}^{\alpha}_{\theta P} \\ \tilde{H}^{\beta}_{\theta P} \end{bmatrix} \label{9} \]

Following the format used in Sec. 4.5, it would be natural to now proceed to generate particular solutions that form a complete set of orthogonal functions which are solutions to the Helmholtz equation. Such an approach to evaluating the particular solutions in Equation \ref{9} is required if an arbitrary radial distribution of current density is to be represented. The approach parallels that presented in Sec. 4.5.

In important physical configurations, to which the remainder of this section is confined, the radial distribution is uniform:

\[ \tilde{J}(r) = \tilde{J} \label{10} \]

Fortunately, inspection of Equation \ref{5} in this case yields simple particular solutions:

\[ \tilde{A}_P = \mu \tilde{J} \begin{cases} \frac{r^2}{m^2 – 4}; & \text{m \(\neq\) 2} \\\\ - \frac{1}{4} r^2 \, ln \, r;& \text{m = \(\pm\) 2} \end{cases} \label{11} \]

Thus, for the case of a radially uniform current density distribution, substitution of Equation \ref{11} into Equation \ref{9} yields the transfer relations

\[\begin{bmatrix} \tilde{A}^{\alpha} \\ \tilde{A}^{\beta} \end{bmatrix} = \mu_o \begin{bmatrix} F_m(\beta, \alpha) & G_m(\alpha, \beta) \\ G_m(\beta, \alpha) & F_m(\alpha, \beta) \end{bmatrix} \begin{bmatrix} \tilde{H}^{\alpha}_{\theta} \\ \tilde{H}^{\beta}_{\theta} \end{bmatrix} + \mu_o \tilde{J} \begin{bmatrix} h_m(\alpha, \beta) \\ h_m(\beta, \alpha) \end{bmatrix} \label{12} \]

Where

\[ h_m(x,y) = \begin{cases} \frac{1}{m^2 – 4} [ x^2 + 2xF_m (y,x) + 2y G_m(x,y); & \text{m \(\neq\) 2} \\\\ \frac{x}{8} [ x + g_m (x,y) y^2 ln \, (\frac{x}{y})]; & \text{m = \(\pm\) 2} \end{cases} \nonumber \]

And the functions \(F_m, G_m\), and \(g_m\) are defined in Table 2.16.2 with \(k = 0\).

The radial distribution of \(A\) within the volume of the annular region described by Equation \ref{12} is obtained by adding to the homogeneous solution, which is Eq. 2.19.5 with \(\tilde{A}^{\alpha} \rightarrow \tilde{A}^{\alpha}\), and \(\tilde{A}^{\beta} \rightarrow \tilde{A}^{\beta} - \tilde{A}^{\beta}_P\),the particular solution \(\tilde{A}_p\):

\[ \tilde{A} = ( \tilde{A}^{\alpha} - \tilde{A}^{\alpha}_P) \frac{\Big [ (\frac{\beta}{r})^m -(\frac{r}{\beta})^m \Big] }{ \Big [ (\frac{\beta}{\alpha})^m - (\frac{\alpha}{\beta})^m \Big ] } + ( \tilde{A}^{\beta} - \tilde{A}^{\beta}_P) \frac{\Big [ (\frac{r}{\alpha})^m -(\frac{\alpha}{r})^m \Big] }{ \Big [ (\frac{\beta}{\alpha})^m - (\frac{\alpha}{\beta})^m \Big ] } + \tilde{A}_P \label{13} \]

For Equation \ref{12}, the particular solution is given by Equation \ref{11}, so the associated volume distribution is evaluated using Equation \ref{11}.

The constrained-current transfer relations are applied to a specific problem in the next section.