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A process responds via the process variable (PV) to a change in input (CV) in

a dynamic manner according to the characteristics of the process. These process characteristics, which include factors such as delay time and inherent physical responses of the process, are defined by a transfer function, represented by the term H T (see Figure 14-19). A transfer function is an equation that describes a process in terms of response over time, as well as calculates the outcome of the process variable. Therefore, the value of the term H T equals the value of the process variable at a particular control variable value and time, given the characteristics of the process. Every process has its own unique transfer function based on its particular characteristics, and for most processes, the transfer function equation is not known. Thus, certain

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assumptions must be made about the process to estimate H T . Experimentation can also be used to approximate the outcome of H T (i.e., the process variable response) to a forced change in the process input. This experimental change in process input is called a step test and the response is called a step response.

Process CV H PV

CV PV

Change in CV…through H T …affects change in PV

H T = PV CV

Figure 14-19. Transfer function.

The most important aspect of a transfer function is not so much its composi- tion or form, but its response to sudden process input changes created by disturbances. This behavioral response of a process is called a transient response , and it includes the time required for the output to reach a steady- state final value given a sudden change in input. Transient responses provide much information about the dynamics of a process and, therefore, about the transfer function.

As shown in Figure 14-20, a closed-loop control system includes two transfer functions—one that defines the controller (Hc) and another that defines the process (Hp). The input to the controller’s transfer function is the error signal (E), and its output is the control variable (CV). This control variable becomes the input to the process’s transfer function, whose output is the process variable (PV). In this chapter, we will discuss the process’s transfer function and its behavior. In the next chapter, we will discuss the controller’s transfer function and the different forms that it can take.

Figure 14-20.

A closed-loop control system with two transfer functions.

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To better understand transient responses and the information we can obtain from them, let’s explore a basic example of an open-loop hot-water heater system. Figure 14-21 shows this process, while Figure 14-22 shows the corresponding process block diagram. The transfer function of the process depends on many factors, such as the rate of flow of the steam, the temperature of the steam, the temperature of the incoming water at the inlet, the ambient temperature, and the inflow and outflow rates of the water. Regardless of all these process factors, the controller must maintain the temperature in the tank (the process variable) as close as possible to the user-defined set point by manipulating the control variable. For this example, let’s assume that the temperature in the tank (at steady state) is at a set point of 65 °

C, the

temperature range spans from 15 ° C to 93 °

C, and the steam control valve is at

55% of its open position.

Cold Water In CV

Temperature Transmitter

Hot Water Out

Figure 14-21. Hot-water heater system.

Hot-Water Heater

Hot Water

Temperature Hp is a function of steam flow, steam temperature, etc.

Flow

Figure 14-22. Water heater process diagram.

With a step change in valve position from 55% to 75% open, the temperature (PV) in the tank will begin to heat up (see Figure 14-23). After 15 minutes, the process variable will increase to 81 °

C. The process variable’s behavior during Industrial Text & Video Company 1-800-752-8398

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this 15-minute period is the transient response of the transfer function Hp. Note that the transient response is very smooth, heating the water slowly over the 15 minutes until steady state is once again achieved.

CV 100%

20% input change

C input change

Figure 14-23. (a) Control variable step change and (b) its corresponding process

variable change.

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