Therefore, the room air capacity transferfunctionis: 1sKsqsrrtotalr(4.12) and 360036000paarvipaarpaarvfistpaardcVnAUscVscVnsAUcVs(4.13) can be considered as the disturbance.
117Equation (4.9) can be solved by using inverseLaplace transformation if the suppledtotalqis known. This is pretty much all there is to deriving a linear transfer function. Naturally, some are more complex than this case, but the procedural principles do not differ: write down the governing differential equation (s) in a convenient linear form; express the variables as deviations such that all other non-deviant terms become constants and disappear; finally, express in 'time constant' form. Of course few problems present us with a convenient set of linear differential equations and we shall see how to deal with these cases later. Meanwhile, we will pay further consideration to the linear case. 4.9.Analysis of P, PI, and PID Control Systems 4.9.1.PID Control Systems for HVAC The simple control system shown includes an air temperature sensor, a controller that compares the sensed temperature to the set point, a steam valve controlled by the controller, and the coil itself. (Two-position control applies to an actuator that is either fully open or fully closed.) Figure 4-4-16 Block Diagram of a Control System Equivalent control diagram for heating coil: In Figure 4-4-16 the G(s) represent functions relating the input to the output of each module (the expressions in parentheses are examples of the G(s) discussed in the last section of this chapter on Laplace transforms). Voltages V represent both temperatures (set point and coil outlet) and the controller output to the valve in electronic control systems. Proportional controlcorrects the controlled variable in proportion to the difference between the controlled variable and the set point. For example, a proportional controller would make a 10% increase in the coil heat output rate if a 10% decrease in the coil outlet air temperature were sensed. The proportionality constant between the error and the controller output is called the gainKp. Following equation shows the characteristic of a proportional controller:
118V = V0+ Kpe where V= controller output; the symbol Vis used since in electronic controls, the controller output is often a voltage V0= constant value of controller output when no error exists at the control range midpoint e= error In the case of the steam coil, the error eis the difference between the sensed air temperature Tsensedand the needed air temperature, the set point Tset: e= Tset- TsensedProportional controlis used with stable, slow systems that permit the use of a narrow throttling range and resulting small offset. Fast-acting systems need wide, Throttling rangesto avoid instability and large offsets results. In a later section, we will examine the quantitative criteria for determining the stability of proportional control systems.