Control engineering approach to magnetohydrodynamic stability

The well established transfer-function method of determining the stability of lumped-parameter dynamic systems is applied in the paper to a study of the stability of an infinitely conducting fluid immersed in a magnetic field. Such a system is both distributed in space and multidimensional, and it is shown how, by the application of a modal energisation technique, the whole may be replaced by a number of simpler systems each subjected to a spatial energisation in the fundamental or a higher-order (harmonic) mode, and each amenable to a one-dimensional analytical treatment. Complete stability is obtained only if all the component systems are themselves individually stable. To accommodate the distributed nature of the system the concept of field impedance is introduced, and, to help the electrical engineer to understand and appreciate better the physical significance of the problem, the whole is converted to an equivalent circuit or analogue, involving electrical transmission lines. A further advantage of this approach is, of course, the wealth of analytical solutions and tabulated transforms which are applicable to the electrical network field. Two particular geometrical configurations are then discussed in detail. In the first, a cylindrical fluid is contained by an axially directed magnetic field, and, in the second, a cylindrical fluid is contained by a circumferentially directed magnetic field. These are two geometries which have been intensively studied in the controlled thermonuclear fusion field and are usually referred to as the ¿pinch and the Zpinch, respectively.