Derivation of the Transfer Function

To investigate how the bottom concentration xb changes as a result of a change in the feed flowF, the component balance, Eqn. (15.36), is combined with Eqn. (15.38) to give  

The response of xb to changes in the feed F can easily be obtained from Eqn. (15.41). To get more insight, it is rewritten to the following equation  

The overall response of xb to changes in F depends on the sign of the term o xpo-Xgp)-TT. Tgp-Tp)yD - If Tg, -r 0 then the term is positive and the response of Xb to changes in F will be psendo-first-order, since ris positive. If, however, o, the 

As can be seen, the response increasingly gets an inverse character when the time constant r becomes more negative. 

The response is shown in Fig. 15.9, for a value of Tc = 2, Tt = 2 and =1. It is obvious that for different values of K, responses with a different peak height will be obtained. 

---

Although a transfer function relation may not be always invertible analytically, it has value in that the moments of the RTD may be derived from it, and it is thus able to represent an RTD curve. For instance, if Gq and Gq are the limits of the first and second derivatives of the transfer function G(.s) as. s 0, the variance is... 

After error computation and transformation, a PE may pass information to PEs in a previous layer. This information is either the transformed error, the transformed error scaled by the derivative of the transfer function, or the desired output. Finally, the weights of a PE are modified according to a learning rule or learning function. (Recall that these weights are those associated with... 

Our question is to formulate this model under two circumstances (1) when we only vary the dilution rate, and (2) when we vary both the dilution rate and the amount of glucose input. Derive also the transfer function model in the second case. In both cases, C, and C2 are the two outputs. 

Example 8.10. Derive the magnitude and phase lag of the transfer function of a PI controller. 

X Example 8.13. Derive the magnitude and phase lag of the transfer functions of phase-lead and phase-lag compensators. In many electromechanical control systems, the controller Gc is built with relatively simple R-C circuits and takes the form of a lead-lag element ... 

The densities and volumetric heat capacities of the binary systems, which are required for the calculation of the transfer functions, were measured at the same time as those of the ternary systems. The derived apparent molar quantities of the binaries were In excellent agreement with those In the literature (11,16). 

Michelsen and Ostergaard68 subsequently proposed three additional methods for calculating mean residence time and Peclet number, based upon numerical evaluation of the transfer function and its derivative for a number of values of the Laplace transform parameter sfm. They defined... 

This is certainly a very desirable form of solution since the temperature at any point within the sphere can be determined simply by substituting the r-coordinate of the point into the analytical solution function above. The analytical solution of a problem is also referred to as the exact solution since it satisfies the differential equation and the boundary conditions. This can be verified by substituting the solution function into the differential equation and the boundary conditions. Further, the rate of heat transfer at any location within the sphere or its surface can be determined by taking the derivative of the solution function T r) and substituting it into Fourier s law as... 

Table 2-1 Derivation of dc transfer functions of the three topologies...

Note that this is a simplified transfer function where the high gain OP does not appear, because during the derivation of this transfer function the high-gain terms get eliminated. 

In Chapter 5 is was stated that a linear system based on deviation variables, is asymptotically stable if the roots of the characteristic eqnation of an inpnt-output relationship have negative real parts. The roots of the characteristic eqnation correspond to the poles of the transfer function or the eigen valnes of the homogenons part of the differential equation. Therefore, it is necessary to derive the characteristic linear input-output relationship for the process in an operating point. Because only the homogenous part of the differential equation is of interest, the inputs of the differential equations can be ignored. Next, fiom the roots of the characteristic equation, conditions for stability can be formulated. The following three steps will be performed ... 

The discussions of the equation of transfer and the solution of this equation in Chapter 2 rest entirely on concepts of classical physics. Such treatment was possible because we considered a large number of photons interacting with a volume element that, although it was assumed to be small, was still of sufficient size to contain a large number of individual molecules. But with the assumption of many photons acting on many molecules we have only postponed the need to introduce quantum theory. Single photons do interact with individual atoms and molecules. The optical depth, r (v), depends on the absorption coefficients of the matter present, which must fully reflect quantum mechanical concepts. The role of quantum physics in the derivation of the Planck function has already been discussed in Section 1.7. Both the optical depth and the Planck function appear in the radiative transfer equation (2.1.47). 

Various mathematical concepts and techniques have been used to derive the functions that describe the different types of dispersion and to simplify further development of the rate theory two of these procedures will be discussed in some detail. The two processes are, firstly, the Random Walk Concept [1] which was introduced to the rate theory by Giddings [2] and, secondly, the mathematics of diffusion which is both critical in the study of dispersion due to longitudinal diffusion and that due to solute mass transfer between the two phases. The random walk model allows the relatively simple derivation of the variance contributions from two of the dispersion processes that occur in the column and, so, this model will be the first to be discussed. 

A sigmoid (s-shaped) is a continuous function that has a derivative at all points and is a monotonically increasing function. Here 5,p is the transformed output asymptotic to 0 < 5/,p I and w,.p is the summed total of the inputs (- 00 < Ui p < -I- 00) for pattern p. Hence, when the neural network is presented with a set of input data, each neuron sums up all the inputs modified by the corresponding connection weights and applies the transfer function to the summed total. This process is repeated until the network outputs are obtained. 

---