ABSTRACT
In these circuits I;= 0 is achieved by severing a feedback path completely, rather than setting a component to a particular numerical value. Thinking of the circuit as an analog computer, it is set up to solve the second-order differential equation with zero damping. This approach is quite practical and is used to build sinusoidal oscillators of very pure waveform. In Fig. 6-9, for example, we would disconnect the feedback path through the coefficient B, making B exactly zero. The signal generator would also be disconnected. To generate an undamped sine or cosine wave we would just apply a nonzero initial condition to either the first integrator or the second integrator. These initial conditions would be just like giving an undamped spring-mass system some initial energy and then "letting it go." It oscillates sinusoidally "forever." (Practical oscillators that use this principle and must operate for long periods of time add a nonlinear feature which holds the amplitude constant indefinitely.)
Figure 8-48 displays some examples of fluid systems whose behavior, with respect to the labeled input and output quantities, will be found to follow the basic secondorder equation (8-2). In Fig. 8-48a we see an example of cascaded first-order systems in which no loading effect whatsoever occurs. That is, the addition of the second tank has no influence at all on the response of the first. When connected as in 8-48b, however, the usual loading effect does occur. Figure 8-48c and d again might represent pressure-measuring systems as did their first-order counterparts in Fig. 7-39; however, we now no longer neglect fluid inertance in the tube. The dynamic response of pressure sensing systems is of great practical interest and is covered in depth in specialist texts. II
Let us first analyze the two-tank system of Fig. 8-48b. In most such tank systems the inertance of the liquid can be neglected since flow accelerations are quite small. Since the pressure at the bottom of the tanks is determined by gravity effects, it is usually small enough to neglect liquid compressibility. Using these assumptions, the only significant fluid elements are the compliances of the tanks and the resistance in the "pipes." The "pipe" resistance can also include resistance effects of constrictions or valves, which commonly appear in tank systems. Interconnected tanks appear regularly in power plants, refineries, chemical processes, food processing, etc. and often are part of automatic control systems. The design of the overall control system requires knowledge of tank system dynamics.
