Second Order Elements

Second-Order Element Because of their linear nature, transfer functions can be combined in a straightforward manner. Consider the two tank system shown in Fig. 8-12. For tank 1, the transfer funcdion relating changes in/i to changes in can be obtained by combining two first order transfer functions to give ... 

General Second-Order Element Figure 8-3 illustrates the fact that closed loop systems often exhibit oscillatory behavior. A general... 

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 situation for the IP of Ne and the EA of F is somewhat different, however. It appears that the extensive EOM, though a definite improvement over the primitive repartitioned EOM, is not converging to the same limit as the Cl calculations. At the lowest selection tolerance of 0.00125 au, the extensive EOM results are 0.43 eV above our best Cl calculation for the ionization potential of Ne and 0.79 eV above the SDT relaxed Cl EA for F. These extensive EOM treatments produce a 0.33 eV lowering of the IP of Ne and a 0.91 eV lowering of the EA of F over the primitive repartitioned EOM values at the same tolerances. In both cases, this lowering is due almost entirely to the inclusion of the 5-block basis operators in the Q-space. The inclusion of the second-order elements in App results in only about a 0.01 eV shift in both cases. 

The second-order elements thus are related to a product of two 3C symbols. A special element arises when Q is totally symmetric. In this case, the coupling coefficients are given by ... 

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