General second-order transfer function

Thus, for a general second-order transfer function... 

General Second-Order Element Figure 8-3 illustrates the fact that closed-loop systems can exhibit oscillatory behavior. A general second-order transfer function that can exhibit oscillatory behavior is important for the study of automatic control systems. Such a transfer function is given in Fig. 8-15. For a unit step input, the transient responses shown in Fig. 8-16 result. As can be seen, when t, < 1, the response oscillates and when t, < 1, the response is S-shaped. Few open-loop chemical processes exhibit an oscillating response most exhibit an S-shaped step response. 

The general formula of a second-order transfer function in the frequency domain is... 

We start with the response properties of a first-order process when forced by a sinusoidal input and show how the output response characteristics depend on the frequency of the input signal. This is the origin of the term frequency response. The responses for first- and second-order processes forced by a sinusoidal input were presented in Chapter 5. Recall that these responses consisted of sine, cosine, and exponential terms. Specifically, for a first-order transfer function with gain K and time constant t, the response to a general sinusoidal input, x t) = A sin (o, is... 

Take note (again ) that the characteristic polynomials in the denominators of both transfer functions are identical. The roots of the characteristic polynomial (the poles) are independent of the inputs. It is obvious since they come from the same differential equation (same process or system). The poles tell us what the time-domain solution, y(t), generally would "look" like. A final reminder no matter how high the order of n may be in Eq. (3-4), we can always use partial fractions to break up the transfer functions into first and second order terms. 

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. 

A relationship is expected between Hr and Ha. However, if the packing density depends on the radial position, the bed tortuosity and eddy diffusion may be different in the axial and radial directions. Furthermore, the mass transfer resistances do not affect Hf. Although, in the general case. Da and Dr could both be functions of the coordinates z and r and of the concentration, we assumed in writing Eq. 2.18 that they are constant. This would constitute the second order approximation of a model of physical columns, the other models discussed here being first order approximations since they all assume a homogeneous column. 

This example demonstrates that the inverse response is the result of two opposing effects. Table 12.1 shows several such opposing effects between first- or second-order systems. In all cases we notice that when the system possesses an inverse response, its transfer function has a positive zero. In general, the transfer function of a system with inverse response... 

We will now show how polynomial analysis can be applied to the transfer function. A polynomial defined in terms of the complex variable z takes on just the same form as when defined in terms ofx. The z form is actually less misleading, because in general the roots will be complex (e.g. /(z) = z + z - 0.5 has roots 0.5 + 0.5J and 0.5 - 0.5J.). The transfer function is defined in terms of negative powers of z - we can convert a normal polynomial into one in negative powers by multiplying ly z. So a second-order polynomial is... 

The general equations of change given in the previous chapter show that the property flux vectors P, q, and s depend on the nonequi-lihrium behavior of the lower-order distribution functions g(r, R, t), f2(r, rf, p, p, t), and fi(r, P, t). These functions are, in turn, obtained from solutions to the reduced Liouville equation (RLE) given in Chap. 3. Unfortunately, this equation is difficult to solve without a significant number of approximations. On the other hand, these approximate solutions have led to the theoretical basis of the so-called phenomenological laws, such as Newton s law of viscosity, Fourier s law of heat conduction, and Boltzmann s entropy generation, and have consequently provided a firm molecular, theoretical basis for such well-known equations as the Navier-Stokes equation in fluid mechanics, Laplace s equation in heat transfer, and the second law of thermodynamics, respectively. Furthermore, theoretical expressions to quantitatively predict fluid transport properties, such as the coefficient of viscosity and thermal... 

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