Fractor

A fractor is an elementary electronic component, with characteristics of both capacitors and resistors. The fractor is most useful in the field of control, where the phase properties of the fractor help naturally stabilize systems. The fractor was invented during the exploration of materials that show power-law relaxation behavior, such as piezoelectric actuators. The most accurate model for the relaxation behavior turned out to be based in fractional calculus.
The continuous nature of the differintegral allows a control systems designer to select an arbitrary phase angle of the response of their systems. Integer order calculous only allows 90 degree phase shifts, in contrast. Before the invention of the fractor, control systems designers were required to scale a 90 degree phase lag with a 90 degree phase lead, in what is commonly known as a PID controller.

The fractor replaces the ID (integer/derivative) terms of the PID with a single F term of the desired phase.

Principle of Operation
A fractor device depends on the non exponential nature shared by most dielectric materials. The dielectric material used in capacitors are selected to behave in such a way that the exponential model used to predict the behaviour of the devices is accurate. A perfect capacitor is expected to demonstrate an exponential decay in potential when it is discharged through a resistor. Unfortunatly, a perfect capacitor does not exist.

A perfect capacitor would not retain charge after it had been discharged, as can be seen by letting a freshly discharged capacitor sit for some time with its leads insulated from each other. A memory effect within the capacitor can be observed by measuring the voltage on the leads. The precise measurement of the voltage after a varying amount of time is effected by the amount voltage and the duration of time it was applied. This memory effect can be seen much more dramatically in chemical batteries, which are very sensitive to the way in which they are charged.

The memory effect of a fractor is carefully controlled to make an industrially usefully device, i.e. one which behaves consistently. A dielectric material with a desired fractional impedance is applied to roughened conductive plates, which are pressed together to build a capacitor like device. The plates are then clamped together, compressing the dielectric. Subtle performance changes can be seen by varying the pressure applied to the plates.

State of Existence
Only a few fractors have actually ever been made, and they were made by hand. A sample control circuit was designed and built largely from NSF funds. The circuit was used to stabilize a remote spring arm which was operated from a PC joystick. This sample control system known as the "Fractolizer". Unfortunately, after the Fractolizer was constructed and demonstrated at Montana State University to work exactly as expected, all funding for further development was lost. Research continues at Utah State University, in Logan, Utah.

The remaining devices have been tested periodically since their first construction, and most have shown a consistent decline in performance characteristically. Only the few devices that were not sealed after being clamped have stayed somewhat consistent. Ironically, it appears the dielectric reacts more to the glue used to protect the material than the water vapor it meant to defend against.

Publications
AIP Conference Proceedings -- August 3, 2001 -- Volume 582, pp. 175-184
The 11th williamsburg workshop on fundamental physics of ferroelectrics

Fractional Calculus and Biomimetic Control - http://www.csois.usu.edu/people/yqchen/paper/04C16_robio04d.pdf

Links
YanQuan Chen's Group at USU - http://www.ece.usu.edu/csois/people/yqchen/

Fractional Order Control - http://mechatronics.ece.usu.edu/foc/

Digital Fractor in C++ - http://www.cs.montana.edu/~bohannan/Fractor/index.htm

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