Hysteresis operator

The next step to extend the validity of the system model is carried out by the modeling of complex hysteretic nonlinearities by the so called complex hysteresis operators. These complex hysteretic nonlinearities are present in varying degrees in virtually all sohd-state actuators provided that they are driven with sufEciently high amphtudes [349],... 

If the mappings T in the sensor equation (6.75) and the actuator equation (6.76) are purely hysteretic they can be modeled by a Prandtl-Ishhnskii operator H, a modified Prandtl-Ishlinskii operator M or a Preisach hysteresis operator R depending on the degree of symmetry of the branching behaviour. The calculation of these hysteresis operators and the corresponding compensators from the measured output-input characteristic requires special computer-aided synthesis procedures which is based on system identification methods. Due to a lack of space, this article cannot further comment on these synthesis methods. However, a detailed description of both the synthesis method and the mathematical basics can be found in the literature [332,341,350-352,356]. 

A scalar operator which considers simultaneously complex hysteresis effects, log(t)-type creep effects as well as saturation effects can be constructed by the parallel connection of a Prandtl-Ishlinskii hysteresis operator H and a Prandtl-Ishlinskii log(t)-type creep operator K followed by a concatenation with a memory-free scalar nonlinearity S. In this case the mapping T in (6.75) and (6.76) is given by a so-called modified Prandtl-Ishlinskii creep extension Mk. The corresponding reconstruction model is then given by (6.77) and (6.78) with the compensator Tg... 

The self-sensing actuator concept requires the powerful mathematical machinery of complex hysteresis operators - first for reconstructing the mechanical quantities by means of the measured values of electrical quantities and second for compensating the hysteretic nonlinearities and the load dependency. Whereas robust software tools exist for modeling, identifying and compensating scalar complex hysteretic nonlinearities in practical applications, a considerable amount of research activities is necessary in the field of vectorial hysteresis phenomena to obtain a similar status. 

Ferrites aHowing for operation at frequencies well above 1 MH2 have also become available, eg, 3F4 and 4F1 (Table 6). Other newer industrial power ferrites are the Siemens-Matsushita N-series (28,97) the TDK PC-series (28,100), and the Thomson B-series (28,103). While moving to higher frequencies, the ferrites have been optimized for different loss contributions, eg, hysteresis losses, eddy current losses, and resonance losses. Loss levels are specified at 100°C because ambient temperature in power appHcations is about 60°C plus an increase caused by internal heat dissipation of about 40°C. 

Static performance measurements related to positioner/ac tuator operation are conformity, measured accuracy, hysteresis, dead baud, repeatability, and locked stem-pressure gain. Definitions and standardized test procedures for determining these measurements can be found in ISA-S75.13-1989, Method of Evaluating the Performance of Positioners with Analog Input Signals and Pneumatic Output . 

There is one obvious drawback of high-hysteresis rubber. In normal rolling operation, considerable elastic deformations still take place in the tyre wall, and high-loss tyres will consume fuel and generate considerable heat. The way out is to use a low-loss tyre covered with a high-loss tread - another example of design using composite materials (Fig. 26.9). 

The eddy current loss is a much smaller loss than the hysteresis loss, but increases significantly with the operating frequency. It is shown in Equation 4.7. 

The major losses within any core material are the hysteresis loss and eddy current loss. These losses are typically lumped together by the core manufacturer and given in a graph of watts lost per unit volume V5. the peak operational flux density (5max) and frequency of operation. Hysteresis loss is given as... 

The problem of ignition and extinction of reactions is basic to that of controlling the process. It is interesting to consider this problem in terms of the variables used in the earlier discussion of stability. When multiple steady-state solutions exist, the transitions between the various stable operating points are essentially discontinuous, and hysteresis effects can be observed in these situations. 

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