Nowadays capacity electromechanical systems are widely used in such applications as microactuators for high-accuracy drives, accelerometers for automotive control and safety, etc. It is due to the development of thin-film technologies. The theory of capacity devices as a whole has also received a development. A new branch of science - film electromechanics - has appeared [1]. The systems under consideration have small sizes - at least some microns in one direction. It is caused by their rather high value of power capacity at given sizes. This fact determines rather perspective branch of systems´ application - transducers of microtransferences.

This paper deals with a mathematical model of the step-type capacity motor with a rolling rotor. The system is represented in figure 1.

**Figure 1.** The model of the capacity electromechanical system

*R* - the radius of the rotor, *d _{i}* - the thickness of the thin dielectric film on the stator surface, ε

_{i}-permittivity of the dielectric,

*p*- electrostatic pressure applied to an element of the rotor surface

_{es}The stator surface contains electrodes and is covered by dielectric thin film. The rotor represents metal hollow thin-walled cylinder.

Electrostatic pressure is defined as:

. (1)

The determinative factor of capacity devices operation is mutual capacitance, i.e. the capacitance between a rotor and a stator. With the purpose of its increase the choice of a design of the motor with a rolling rotor [2] was made. Between the rotor and the stator electrodes the potential differences are applied by turns, and the rotor rolls along the stator´s surface under action of the force created by the interaction of electric charges.

The following assumptions were made: 1) the stator and rotor surfaces have no any defects, i.e. the stator surface is a plane and the rotor surface is a cylindrical surface; 2) electric losses (losses in the dielectric) and mechanical losses (losses on friction) are not taken into account; 3) the phenomenon of charge accumulation on the border gas - solid dielectric is not taken into account; 4) the external factors change influence on the system´s operation is negligible, i.e. temperature, pressure and humidity practically do not vary.

Thus, it is supposed, that any energy change of the system under consideration is converted into mechanical work.

Let´s acquaint the reference frame *xoy*. Assume, that abscissa *x* is the generalized coordinate. Hence, the force exerted to the rotor in a direction of *x* will be directly proportional to the generalized coordinate partial derivative of the system´s energy [3], in compliance with the following expression:

, (2)

where *W* - energy of an electric field, *C* - mutual capacitance.

Thus, the more mutual capacitance change, the more force will be exerted to the rotor at .

In its turn, mutual capacitance is defined by the following expression:

(3)

and is also the function of *x*. Here *y(x)* is the function describing distance between the rotor surface and the stator one.

,

and the expression in the denominator is the function describing the change of a work gap.

The developed mathematical model allows to estimate one of the basic geometrical parameters of the capacitor electromechanical system of the offered design (the width of the stator electrode) and it will be possible to apply this model to estimate parameters of any other design of the systems under consideration.

**References:**

- Dyatlov V.L., Konyashkin V.V., Potapov B.S., Fadeev S.I. Film electromechanics. - Novosibirsk: Science, 1991. - 247 p.
- Bekishev R.F., Semenova L.N., Lyapunov D.Y. Capacitance wobble micromotors // The News of Tomsk polytechnic university, №1, 2004.
- Kaplyancky A.Y. Introduction to the general theory of electric machines. - Moscow: the State power publishing house, 1941. - 96 p.

*The work was submitted to III international scientific conference «Basic Research», Dominican Republic, April, 10-20, 2008, came to the editorial office 14.02.2008.*