Transfer functions, definitions

Modulation Transfer Function, Definition of MTFIO and MTF50... 

Functional Definitions Heat-transfer equipment can be designated by type (e.g., fixed tube sheet, outside packed head, etc.) or by... 

Hence, we can view the transfer function as how the Laplace transform of the state transition matrix O mediates the input B and the output C matrices. We may wonder how this output equation is tied to the matrix A. With linear algebra, we can rewrite the definition of O in Eq. (4-5) as... 

Consistent with these definitions, DNS data (Yeung 1994 Yeung 1996) show that Taa is always positive, while Taa is always negative. Analogous definitions and remarks hold for the scalar-covariance transfer function, i.e., for Tap and T p. 

Consider a process with an input m, and an output x, . By definition, the transfer function of the process is... 

In practice, however, this extension is not as straightforward as in DMC. In multivariable DMC, there is a definite design procedure to follow. In multi-variable IMC, there are steps in the design procedure that are not quantitative but involve some art. The problem is in the selection of the invertible part of the process transfer function matrix. Since there are many possible choices, the design procedure becomes cloudy. 

Since all tracer entered the system at the same time, t = 0, the response gives the distribution or range of residence times the tracer has spent in the system. Thus, by definition, eqn. (8) is the RTD of the tracer because the tracer behaves identically to the process fluid, it is also the system RTD. This was depicted previously in Fig. 3. Furthermore, eqn. (8) is general in that it shows that the inverse of a system transfer function is equal to the RTD of that system. To create a pulse of tracer which approximates to a dirac delta function may be difficult to achieve in practice, but the simplicity of the test and ease of interpreting results is a strong incentive for using impulse response testing methods. 

Although by definition the system is stable, the phase margin is so close to zero and the gain margin so close to unity that any slight variation in any of the control system parameters or, indeed, in the process conditions, could make the system unstable, i.e. could cause a pole or poles of the system closed-loop transfer function to move into the right half of the complex plane (Section 7.10.1). 

A more general definition of magnetization-transfer efficiency e, , that includes only coherent effects and that is independent of structural or motional properties of a specific sample can be defined based on the idealized transfer function Lfir) ... 

Alternatively, quality factors can also be defined based on the maximum value of damped magnetizatrion-transfer functions, in analogy to the definition of the transfer efficiencies e, or 17," [see Section VI, Eqs. (205) and (206) Glaser, 1993c). A quality factor that is based on the initial buildup of the transfer function was used by Briand and Ernst (1991) and by Ernst et al. (1991). 

Functional Definitions Heat-transfer equipment can be designated by type (e.g., fixed tube sheet, outside packed head, etc.) or by function (chiller, condenser, cooler, etc.). Almost any type of unit can be used to perform any or all of the Usted functions. Many of these terms have been defined by Donahue [Pet. Process., 103 (March 1956)]. 

Let us try to connect the dots now. Both the first- and second-order filters we have discussed gave us poles. That is because they both had s in the denominators of their transfer functions — if s takes on specific values, it can force the denominator to become zero, and the transfer function then becomes infinite, and we get a pole by definition. The values of 5 at which the denominator becomes zero are the resonant (or break) frequencies, that is, the locations of the poles. For example, a hypothetical transfer function 1/s will give us a pole at zero frequency (the pole-at-zero we talked about earlier). 

Basic functional controls will include those in Table 11.19. Most of the values can be derived from a straightforward transfer function but several, particularly those determining maximum charge current, discharge current, and state-of-charge, will require careful scrutiny of the definition, calculation, and error analysis. 

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