Circuits without wires

Kevin C Craig, PhD - March 15, 2012

According to novelist and physicist Charles P Snow in his influential 1959 lecture entitled “The two cultures,” not knowing the second law of thermodynamics is equivalent to having never read a work by Shakespeare. Two cultures also appear to exist in the engineering community, and the situation is worsening. At one extreme, engineers use the trial-and-error, plug-and-chug approach of just getting an answer. The other approach embraces a world of understanding through modeling and the application of the laws of nature through their language, mathematics.

The mathematical statements of Maxwell’s equations specify the divergence and curl of E (electric)- and B (magnetic)- vector fields. They include the laws of Gauss, Ampère, and Faraday. Maxwell’s four equations simply state that E diverges outward from positive charges and inward to negative charges; E curls around changing B fields; B never diverges, always looping around; and B curls around currents and changing E fields. Magnetic-circuit analysis represents algebraic approximations to exact field-theory solutions. Mechanical motion must occur in all electromechanical devices; this motion changes flux linkages. In a linear electromagnetic system, inductances are functions of mechanical motion.

The figure shows in cross-section the dynamics of motion for electromechanical systems. In a cylindrical solenoid magnet, the cylindrical plunger of mass, M, moves vertically in brass guide rings of thickness, g, and mean diameter, d. The permeability of brass is the same as that of free space. A spring with a constant of K supports the plunger. Its unstretched length is ℓ₀. The mechanical system that connects to the plunger applies a mechanical load force, f_t, to the plunger.

Assume that the frictional force is linearly proportional to the velocity and that the damping coefficient is B. The coil has N turns and exhibits resistance. Its terminal voltage is e_t, and its current is i. The effects of magnetic leakage and reluctance of the steel are negligible. The reluctance of the magnetic circuit is that of the two guide rings in series, with the flux directed radially through them. Assume constant flux density in the guide rings with respect to the radial distance because the length of the flux path in the direction of the field is much less than the diameter. The upper and lower areas of the flux path are perpendicular to the field.

For the upper gap’s reluctance expression, assume that the field is concentrated in the area between the upper end of the plunger and the lower end of the upper guide ring. As the electrical resistance of a wire, ℓ/σA, the reluctances of the upper and lower gaps are g/μ₀πxd and g/μ₀πad, respectively, which add together to give the total reluctance. The inductance, L(x), is equal to N² divided by the total reluctance, and the magnetic force acting upward on the plunger is given by ½i²(dL/dx). The induced voltage in the coil is given by d(Li)/dt. Application of Newton’s second law and Kirchhoff’s voltage law results in the following two dynamic equations of motion for the system:

and

where

These equations provide a better understanding of the concept of circuits without wires.

Kevin C Craig, PhD, is the Robert C Greenheck chairman in engineering design and a professor of mechanical engineering at the College of Engineering at Marquette University. For more mechatronic news, visit mechatronics zone.com.