Nth-order system

Now let us prove that this simple substitution 5 = toi really works. Let be the transfer function of any arbitrary Nth-order system. The only restriction... 

Several important features should be noted. The first-order process considered in Example 19.1 gave a pulse transfer function that was also first-order, i.e., the denominator of the transfer function was first-order in z. The second-order process considered in this example gave a sampled-data pulse transfer function that had a second-order denominator polynomial. These results can be generalized to an Nth-order system. The order of s in the continuous transfer function is the same as the order of z in the corresponding sampled-data transfer function. 

In this case, for certain values of car with f < 1/VT, the amplitude ratio is greater than unity (equation 7.946 and Fig. 7.46). Also yr-t-90° as tor- 1 (equation 7.95). As tor increases above unity tan yr approaches zero but is always positive. Thus, as tor oo, ys- -180°. The maximum phase lag is therefore 180°, whereas it is 90° for a first-order system. For an nth-order system the maximum lag obtainable is 90n degrees (n/r/2 radians). 

It is possible in many cases to predict highly accurate phase equihbria in multi-component systems by extrapolation. Experience has shown extrapolation of assessed (n — 1) data into an nth order system works well for n < 4, at least with metallurgical systems. Thus, the assessment of unary and binary systems is especially critical in the CALPHAD method. A thermodynamic assessment involves the optimization of aU the parameters in the thermodynamic description of a system, so that it reproduces the most accurate experimental phase diagram available. Even with experimental determinations of phase diagrams, one has to sample compositions at sufficiently small intervals to ensure accurate reflection of the phase boundaries. 

Common approaches for the tailoring of nonmetallic (ceramic) materials properties involve topochemical methods (those where the crystal structure remains largely unaffected) and the preparation of phases in which one or more sublattices are alloyed. In principle, such materials are within the realm of CALPHAD. On the other hand, as has already been stated, extrapolation does not really aid the discovery of new or novel phases, with unique crystal structures. Furthermore, assessed thermochemical data for the vast majority of ceramic systems, particularly transition metal compounds, are presently not available in commercial databases for use with phase diagram software. This does not necessarily preclude the use of the CALPHAD method on these systems However, it does require the user to carry out their own thermodynamic assessments of the (n — 1 )th-order subsystems and to import that data into a database for extrapolation to nth-order systems, which is not a trivial task. 

Some terminology the phase space for the general system (2) is the space with coordinates x, x . Because this space is n-dimensional, we will refer to (2) as an n-dimensional system or an nth-order system. Thus n represents the dimension of the phase space. 

Now let us prove that this simple substitution s = io) really works. Let G(s) be the transfer function of any arbitrary Nth-order system. The only restriction we place on the system is that it is stable. If it were unstable and we forced it with a sine wave input, the output would go off to infinity. So we cannot experimentally get the frequency response of an unstable system. This does not mean that we cannot use frequency-domain methods for openloop-unstable systems. We return to this subject in Chapter 11. 

The remaining terms in Eq. (4-24) are the nth-order corrections to approximate the real system, in which the expectation value ( c is called cumulant, which can be written in terms of the standard expectation value ( by cumulant expansion in terms of Gaussian smearing convolution integrals ... 

The fractional life approach is most useful as a means of obtaining a preliminary estimate of the reaction order. It is not recommended for the accurate determination of rate constants. Moreover, it cannot be used for systems that do not obey nth order rate expressions. 

If one combines the definition of the reaction rate in variable volume systems with a general nth-order rate expression, he finds that the time necessary to achieve a specified fraction conversion is given by... 

Equation 3.1.32 applies to a constant volume system that follows nth-order kinetics. If we take vA = — 1 it can be rewritten as... 

TANK and TANKDIM - Single Tank with Nth-Order Reaction System... 

The characteristic equation of any system, closedloop or openloop, is the equation that you get when you take the denominator of the transfer function describing the system and set it equal to zero. The resulting Nth-order polynomial... 

For closedloop systems, the denominator of the transfer functions in the closedloop servo and load transfer function matrices gives the closed-loop characteristic equation. This denominator was shown in Chap. 15 to be [I + which is a scalar Nth-order polynomial in s. Therefore, the... 

An INA plot for an Nth-order multivariable system consists of N curves, one for each of the diagonal elements of the matrix that is the inverse of the... 

The energy of a quantum system is invariant to permutations of identical particles in the system. Thus, the Hamiltonian for a system with n identical particles can be said to commute with the elements of the nth-order symmetric group ... 

In plug flow, the concentration of reactant decreases progressively through the system in mixed flow, the concentration drops immediately to a low value. Because of this fact, a plug flow reactor is more efficient than a mixed flow reactor for reactions whose rates increase with reactant concentration, such as nth-order irreversible reactions, n > 0. 

The previous discussion shows that the relaxation processes emerge from the quantum dynamics under appropriate circumstances leading to the formation of time-dependent quasiclassical parts in the observable quantities. Let us add that quasiclassical and semiclassical methods have been recently applied to the optical response of quantum systems in several works [65, 66] where the relation to the Liouville formulation of quantum mechanics has been discussed, without however pointing out the existence of Liouvillian resonances as we discussed here above. The connection between the property of chaos and n-time correlation functions or the nth-order response of a system in multiple-pulse experiments has also been discussed [67, 68]. 

Suppose that a chemical reaction takes place in a dispersed system between the two reactants A and B, where A is dissolved in the dispersed phase and B in the continuous phase. Suppose further that at a certain place in the reactor the concentration of B is equal to b, while the dispersed drops at this place have different concentrations a of the reactant A caused by segregation. When the chemical conversion for each isolated drop can be described as being of the nth order in the reactant A, then the amount reacting per second in each drop of volume v equals2... 

A survey of the extensive literature of kinetic results reported for epoxy-anhydride-tertiary amine systems is surprising. Both nth order and autocatalytic expressions have been reported for the same system. As an example, we analyze the results reported for the copolymerization of a diepoxide based on diglycidyl ether of bisphenol A (DGEBA) with methyl-tetrahydrophthalic anhydride (MTHPA), initiated by benzyldimethylamine (BDMA). 

To add to the confusion noted for conventions of polarizabilities, both cgs and recommended SI units for linear and non-linear optical polarizabilities coexist in the literature. We strongly advocate the use of SI units. The SI unit of the electric dipole moment is Cm (Cohen and Giacomo, 1987). Thus, consistent SI units of an nth-order polarizability are defined as C m(mV )" = C m " V ", cf. (34)-(37). Conversions from the SI to the esu system for the dipole moment, the first-, second-, and third-order polarizability, are given in (38)-(41). 

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