Linear operating regime

Thus, while the contact resistance values summarized in Tables 2.4.1 and 2.4.2 are generally reported at large values of Vg in the linear operating regime, not all of the measurements have been made at the same induced charge density in the channel. Additionally, there has been some evidence that, even with good contacts, contact resistance is also inversely proportional to the semiconductor mobihty [9,12,17]. 

Johansen, T.A. and Foss, B.A. (1995). Semi-empirical modeling of non-linear dynamic systems through identification of operating regimes and locals models. In Neural Network Engineering in Control Systems, K Hunt, G Irwin and K Warwick, Eds., pp. 105-126, Springer-Verlag. 

We turn first to computation of thermal transport coefficients, which provides a description of heat flow in the linear response regime. We compute the coefficient of thermal conductivity, from which we obtain the thermal diffusivity that appears in Fourier s heat law. Starting with the kinetic theory of gases, the main focus of the computation of the thermal conductivity is the frequency-dependent energy diffusion coefficient, or mode diffusivity. In previous woik, we computed this quantity by propagating wave packets filtered to contain only vibrational modes around a particular mode frequency [26]. This approach has the advantage that one can place the wave packets in a particular region of interest, for instance the core of the protein to avoid surface effects. Another approach, which we apply in this chapter, is via the heat current operator [27], and this method is detailed in Section 11.2. 

Figure 16.1 Operating regimes of a unipolar FET schematic of a typical FET and graph representation of the linear regime (a), the pinch off point (b), and the saturation regime (c). (Reprinted with permission from Ref [18]).

Figure 25.7 Dlustrations of operation regimes of field-effect transistors (a) generation of the charges induced, (b) linear regime, (c) start of saturation regime at pinch-off, and (d) saturation regime.

Now we want to find out whether the process shows highly nonlinear behaviour or whether the process is only mildly nonlinear, indicating that no severe problems are expected to occur during process operation using linear control techniques. The value of the nonlinearity measure of the plant without controller even for a very small operating regime around the operating point OP is computed to be... 

For this situation, it can be shown that this is the exact value even for an arbitrarily small operating regime [20]. This highest possible value for the nonlinearity measure indicates that problems are likely to occur in the case that this process is operated by a linear controller. And... 

The fluorescence signal is linearly proportional to the fraction/of molecules excited. The absorption rate and the stimulated emission rate 1 2 are proportional to the laser power. In the limit of low laser power,/is proportional to the laser power, while this is no longer true at high powers 1 2 <42 j). Care must thus be taken in a laser fluorescence experiment to be sure that one is operating in the linear regime, or that proper account of saturation effects is taken, since transitions with different strengdis reach saturation at different laser powers. 

On the other hand, for aircraft and spacecraft structures, real laminate behavior is pretty typically linear. Laminate behavior is reasonably linear even with some 45° layers which you would expect to contribute their nonlinear shear deformation characteristic to the overall laminate and degrade its relative performance. If you go beyond the behavior of a laminate and look at a large structure, typically the load-response characteristics are linear. Even around a cutout, linear behavior exists. Beyond that apparent linear performance of many laminates, you might not like to operate in some kind of a nonlinear response regime. Certainly not when in a fatigue environment and probably not in a creep environment either would you like to operate in a nonlinear behavior range. 

Linear control theory will be of limited use for operational transitions from one batch regime to the next and for the control of batch plants. Too many of the processes are unstable and exhibit nonlinear behavior, such as multiple steady states or limit cycles. Such problems often arise in the batch production of polymers. The feasibility of precisely controlling many batch processes will depend on the development of an appropriate nonlinear control theory with a high level of robustness. 

A reaction at steady state is not in equilibrium. Nor is it a closed system, as it is continuously fed by fresh reactants, which keep the entropy lower than it would be at equilibrium. In this case the deviation from equilibrium is described by the rate of entropy increase, dS/dt, also referred to as entropy production. It can be shown that a reaction at steady state possesses a minimum rate of entropy production, and, when perturbed, it will return to this state, which is dictated by the rate at which reactants are fed to the system [R.A. van Santen and J.W. Niemantsverdriet, Chemical Kinetics and Catalysis (1995), Plenum, New York]. Hence, steady states settle for the smallest deviation from equilibrium possible under the given conditions. Steady state reactions in industry satisfy these conditions and are operated in a regime where linear non-equilibrium thermodynamics holds. Nonlinear non-equilibrium thermodynamics, however, represents a regime where explosions and uncontrolled oscillations may arise. Obviously, industry wants to avoid such situations ... 

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