Second-order transfer functions

Equations (8-23) and (8-24) can be multiphed together to give the final transfer function relating changes in ho to changes in as shown in Fig. 8-13. This is an example of a second-order transfer function. This transfer function has a gain R Ro and two time constants, R A and RoAo. For two equal tanks, a step change in fi produces the S-shaped response in level in the second tank shown in Fig. 8-14. 

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

Determine the values of Wn and ( and also expressions for the unit step response for the systems represented by the following second-order transfer functions... 

We do not need to carry the algebra further. The points that we want to make are clear. First, even the first vessel has a second order transfer function it arises from the interaction with the second tank. Second, if we expand Eq. (3-46), we should see that the interaction introduces an extra term in the characteristic polynomial, but the poles should remain real and negative.1 That is, the tank responses remain overdamped. Finally, we may be afraid( ) that the algebra might become hopelessly tangled with more complex models. Indeed, we d prefer to use state space representation based on Eqs. (3-41) and (3-42). After Chapters 4 and 9, you can try this problem in Homework Problem 11.39. 

Example 8.3. What are the Bode and Nyquist plots of a second order transfer function ... 

Equation 7.52 is the standard form of a second-order transfer function arising from the second-order differential equation representing the model of the process. Note that two parameters are now necessary to define the system, viz. r (the time constant) and (the damping coefficient). The steady-state gain KMT represents the steady-state relationship between the input to the system AP and the output of the system z (cf. equation 7.50). 

There are distinct similarities between second order systems and two first-order systems in series. However, in the latter case, it is possible physically to separate the two lags involved. This is not so with a true second order system and the mathematical representation of the latter always contains an acceleration term (i.e. a second-order differential of displacement with respect to time). A second-order transfer function can be separated theoretically into two first-order lags having the same time constant by factorising the denominator of the transfer function e.g. from equation 7.52, for a system with unit steady-state gain ... 

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 two poles of the second-order transfer function are given by the roots of the characteristic polynomial,... 

The cascade design procedure begins with a transfer function in factored form such as is shown in Eq. (7.156) and realizes the transfer function as a product of second-order transfer functions and, if the order is odd, a first-order transfer function. These constituent transfer functions are termed sections. For the transfer function in Eq. (7.156), we have three sections so that T = T TiT where Ti = l/(s -F 1), and Ti and T3 have the form l/(s + s (undamped natural frequency and is the radius of the circle centered at the origin of the s-plane on which the pair of complex poles lie. To have poles that are complex and in the open left-halfs -plane, 1 /2 <... 

The Second-Order Transfer Function for a System at Resonance... 

Figure 5. A double RC filter circuit, whose second-order transfer function representation models the response of a spin system in a pulsed saturation experiment.

Parameters of the Second-Order Transfer Function for Magnetic Resonance Calculations... 

It can be seen that control of the residence time Tr is important. When the residence time changes, the concentration of component B changes. Control can be achieved by measuring the outlet concentration of component B and manipulating the reactor throughput F. As can be seen from Fig. 12.7, the relationship between the two variables will be a combination of a first- and second-order transfer function. The overall transfer function can easily be derived from Eqn. (12.27). Equation (12.27a) is the same as Eqn. (12.1), therefore linearization of Eqn. (12.27a) results in Eqn. (12.4). Equation (12.27b) is the same as Eqn. (12.12), therefore linearization of Eqn. (12.27b) results in Eqn. (12.15). Combination of Eqns. (12.4) and (12.15) results in ... 

The major difference between the second term in Eqn (14.37) and the second term in Eqn. (14.28) is that the term between brackets is multiplied by a first-order transfer function in Eqn. (14.28) and a second order transfer function in Eqn (14.37). 

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