2.9: Lumped Parameter Magnetoquasistatic Elements

An extremely practical idealization of lumped parameter magnetoquasistatic systems is sketched schematically in Figure 2.12.1. Perfectly conducting coils are excited at their terminals by currents \(i_i\) and, in general, coupled together by the induced magnetic flux. The surrounding medium is magnetizable but free of electrical losses. The total flux Ai linked by the ith coil is a terminal variable, defined such that

\[ \lambda_i = \int_{S_i} \overrightarrow{B} \cdot \overrightarrow{n} da \label{1} \]

Figure 2.12.1 Schematic representation of a system of perfectly conducting coils. The ith coil is shown with the wire assuming the contour \(C_i\) enclosing a surface \(S_i\). There is a total of \(n\) coils in the system.

A positive \(\lambda\) is determined by first assigning the direction of a positive current \(i_i\). Then, the direction of the normal vector (and hence the positive flux) to the surface \(S_i\) enclosed by the contour \(C_i\) followed by the current \(i_i\),has a direction consistent with the right-hand rule, as Figure 2.12.1 illustrates.

Because the MQS current density is solenoidal, the same current flows through the cross section of the wire at any point. Thus, the terminal current is defined by

\[ i_i = \int_{s_i} \overrightarrow{J_f} \cdot \overrightarrow{i_n} da \label{2} \]

where the surface \(s_i\) intersects all of the cross section of the wire at any point, as illustrated in the figure.

The first two MQS equations are sufficient to determine the flux linkages as a function of the current excitations and the geometry of the coil. Thus, Ampere's law and the condition that the magnetic flux density be solenoidal are solved to obtain relations having the form

\[ \lambda_i = \lambda_i (i_1...i_n, \xi_1...\xi_m) \label{3} \]

If the materials involved are magnetically linear, so that \(\overrightarrow{B} = \mu \overrightarrow{H}\), where \(\mu\) is a function of position but not of time or the fields, then it is convenient to define inductance parameters which depend only on the geometry:

\[ L_{ij} = \frac{\lambda_i}{i_j} \Big|_{i_{i \neq j} = 0} = \frac{\int_{S_i} \mu \overrightarrow{H} \cdot \overrightarrow{n} da}{\int_{s_j} \overrightarrow{J_f} \cdot \overrightarrow{i_n} da} \label{4} \]

The inductance \(L_{ij}\) is the flux linked by the ith coil per unit current in the jth coil, with all other currents zero. For the particular cases in which an inductance can be defined, Equation \ref{3} becomes

\[ \lambda_i = \sum_{j=1}^n L_{ij} i_j, \quad L_{ij} = L_{ij}(\xi_1...\xi_m) \label{5} \]

The dynamics of a lumped parameter system arise through Faraday's integral Law of induction, Equation 2.7.3b, which can be written for the ith coil as

\[ \oint_{C_1} \overrightarrow{E^{'}} \cdot \overrightarrow{d}l = - \frac{d}{dt} \int_{S_1} \overrightarrow{B} \cdot \overrightarrow{n} da \label{6} \]

Here the contour is one attached to the wire and so \(\overrightarrow{v_s} = \overrightarrow{v}\) in Equation 2.7.3b. The line integration can be broken into two parts, one of which follows the wire from the positive terminal at (a) to (b), while the other follows a path from (b) to (a) in the insulating region outside the wire

\[ \oint_{C_1} \overrightarrow{E^{'}} \cdot \overrightarrow{d}l = \curvearrowleft \int_a^b \overrightarrow{E{'}} \cdot \overrightarrow{d}l + \uparrow \int_b^a \overrightarrow{E^{'}} \cdot \overrightarrow{d}l \label{7} \]

Even though the wire is in general deforming and moving, because it is perfectly conducting, the electric field intensity \(\overrightarrow{E^{'}}\) must vanish in the conductor, and so the first integral called for on the right in Equation \ref{7} must vanish. By contrast with the EQS fields, the electric field here is not irrotational. This means that the remaining integration of the electric field intensity between the terminals must be carefully defined. Usually, the terminals are located in a region in which the magnetic field is sufficiently small to take the electric field intensity as being irrotational, and therefore definable in terms of the gradient of the potential. With the assumption that such is the case, the remaining integral of Equation \ref{7} is written as

\[ \uparrow \int_b^a \overrightarrow{E^{'}} \cdot \overrightarrow{d}l =- \int_b^a \nabla \phi \cdot \overrightarrow{d}l = -(\phi_a - \phi_b) \equiv - v_i \label{8} \]

Thus it follows from Equation \ref{6}, combined with Eqs. \ref{1} and \ref{8}, that the voltage at the coil terminals is the time rate of change of the associated flux linked:

\[ v_i = \frac{d\lambda_i}{dt} \label{9} \]

With \(\lambda_i\) given by Equation \ref{3} or Equation \ref{5}, the terminal voltage follows from Equation \ref{9}.