## Electromagnetic Circuits

The electromagnetic performance of the displacer piston actuator and the power pis­ton generator is characterized by the finite-element method (FEM) and numerical analysis. COMSOL is the FEM engine which is used to determine magnetic flux density data. A two dimensional axial-symmetric model is considered for both pistons, which assumes ring mag­nets for the magnetic poles. However, as explained before, the magnetic poles are realized by block magnets in the prototype to eliminate corresponding eddy losses. Therefore, the magnet poles of the fabricated prototype will have lower surface area than the simulated model. Hence, to compensate for this effect, appropriate reduction factors are applied to the remanent flux density data in COMSOL. Since the motion of the displacer piston relative to stationary ferromagnetic components produces different spatial fields at different points along its trajectory, finite-element simulations are repeated for several different positions spanning the full excursion of the piston to obtain a complete picture. The magnetic flux density data is then interpolated over independent spatial dimensions to generate a finer data set appropriate for further calculations.

Numerical computation utilizes the flux density data from the FEM analysis to produce the electro-motive force (EMF) data. The EMF generated by each piston is calculated based on the change in flux linkage, A, i. e.,

DX dX dzP

EMF = — m = — azrp DT (41)

Where the flux linkage is a function of the piston position, zP. In order to produce a full waveform, the EMF is calculated by this method over one period of the nominal piston trajectory, which is considered to be a single-frequency sinusoid as below,

ZP = zm sin(ut) (4.2)

Where zm and u are, respectively, the amplitude and angular velocity of the piston oscilla­tion. The z-th wire loop of the winding is identified by its radial, r*, and axial, z*, distances
from the origin, designated as the neutral resting position of the piston. Therefore, the flux linkage of each wire loop is calculated by numerical integration over the relevant enclosure area, then summed over all wires in the winding configuration. For the concentric winding configuration in the prototype, only axially oriented flux density, Bz, is relevant. Therefore, assuming that each winding has N turns, the flux linkage, X(zP), is calculated as,

(4.3)

Where Bz (zP, Zi, r) is the axial flux density at position (r, Zi) while the piston is positioned

At zP.

Since EMF data obtained thus far is discrete, frequency domain analysis (the Fourier Series, in this case) is the most effective method to determine the output power of the generator or the required input power of the actuator. At each harmonic frequency, the corresponding equivalent electric circuit is solved based on the loading and EMF component at that particular frequency. For the displacer piston, the fundamental frequency compo­nent of the EMF gives an indication of the power developed into useful kinetic energy, while the power contained in all higher harmonics contributes to losses in the windings. For the power piston, the power in the fundamental frequency component gives an indication of the useful power developed at the Stirling engine operating frequency. Note that if the output is rectified, the total RMS power is a more appropriate metric.

The results produced by numerical analysis are verified by a ring-down test. In the ring-down test, the EMF sinusoidal signal has a decaying envelope (refer to appendix A). Therefore, only the measured EMF during the first half-period is expected to closely approx-

 Figure 4.9: Simulated and measured waveforms of the displacer actuator EMF with no separation between the two windings. Refer to Figure 4.4

Imate the numerically simulated waveform. The calculated and measured EMF waveforms for this displacer are depicted in Figure 4.9, for the case where there is no spacing between the two windings. For the fabricated prototype, the position of the windings is optimized to achieve the best electromagnetic performance for the displacer and power pistons magnetic circuits. The optimum spacing of the two windings is 2 inches and corresponding simulated and measured EMF waveforms are illustrated in Figure 4.10. Although the fundamental frequency component of EMF is smaller in the optimal case, its harmonic distortion is the smallest in this case which, in turn, generates the lowest copper loss and the highest actuator efficiency.

 Figure 4.10: Simulated and measured waveforms of the displacer actuator EMF with the optimal separation (2 in.) between the two windings.