The use of advanced dielectric liquids and manufacturing techniques for enhanced surface area per unit volume, along with other multiplicative gains, is enabling macro-scale electrostatic machines with competitive torque densities. This emergence prompts the need for circuit modeling-that guides machine design and drive controls-rooted in the canon of well-established electromagnetic machinery practices. In this study, prior dielectric and manufacturing innovations are combined with a newly developed unified electrostatic machine d−q-axis framework to form a machine that demonstrates industrial utility. Design, equivalent circuits, and performance are validated by an experimental prototype intended for low-speed direct-drive servo actuators. A prototype electrostatic machine was constructed entirely of aluminum and printed circuit boards, and possesses active material torque densities ࣙ1.4 Nm/kg and ࣙ2.65 Nm/L without the need for forced air or liquid cooling. Additional features include near zero loss at stall and low torque ripple. Index Terms-Capacitance, d-q equivalent circuits, electric machines, electrostatic machines, medium voltage. NOMENCLATURE Symbols Meaning j, a Ý(−1), constant complex vector e j 2π /3 . v, i Transient voltage (V) and current (A). V, I Steady state voltage (V) and current (A). d s , d r Duty ratio of stator and rotor traces. N Number of rotor plates. C, G Capacitance (F), conductance (S). r Resistance (Ω) or radius (m).
P, λPole number, angular period (rad). r i , r o , gActive inner radius, outer radius, gap (m). ε 0 , ε x Vacuum and material permittivities (F/m). γ Torque angle (rad).
Advances in electrostatic machine design have enhanced the torque density of macroscale electrostatic machines toward practical use. A recently developed fractional horsepower three-phase separately excited synchronous electrostatic machine (SEM) demonstrates torque densities comparable to those of air-cooled permanent-magnet-based electromagnetic machines (1.5 Nm/kg) when excited with a medium voltage (5 kV). SEMs develop torque from voltage, not from current, and therefore incur nearly zero losses at low speeds or stall. However, there is no off-the-shelf medium-voltage drive at this power level, and the appropriate control framework for these machines has yet to be established. This article presents a complex vector voltage regulator control approach as a means for modulating torque in an SEM. Ampere-second (charge) is sourced from a current source inverter (CSI) serving as the drive electronics for voltage regulation. Together, the control approach and the CSI hardware form the first high-performance electrostatic drive. Key research outcomes include the theoretical development and experimental verification of charge-oriented control via voltage regulation. Experimental results are presented for rotational and stall conditions, which are reflective of the "position and hold" applications suited to electrostatic machines. The dynamic performance of the voltage regulator is verified by measuring the controller frequency response function, dynamic stiffness, and command tracking on a separately excited SEM.
Index Terms-Complex vector control, current source inverter (CSI), electrostatic machines, torque control. NOMENCLATURE Symbols Meaning C s Stator capacitance [F]. R s Stator resistance []. C md Mutual capacitance [F]. V fd Field excitation [V]. M Back MMF or current [A]. P Pole number.
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