ASHRAE Journal - August 2009 - 60

ASHRAE 19 50 59–2009 YE A R S JOURNAL tances. The number of these products that exist determines the order of the system. Let us consider an example of a stem thermometer in thermal equilibrium suddenly being immersed in a bath at a higher temperature. The thermal resistance retards the heat flow from the bath into the thermometer and the capacitance (mass and specific heat) determines the rate of temperature rise. As the thermometer warms, the rate of heat flow diminishes, due to the reduced thermal potential, and finally reaches zero when the bath and thermometer are at the same temperature. At any instant the thermometer temperature is E = I (l – e –t/T) , where I is the initial temperature difference and t is the time after immersion. If T is known, the entire time response curve can be computed for this first-order system. A typical curve for a firstorder system is shown in Fig. 2. However, if the thermometer were placed in a well, then the resistance and capacitance of the well would enter into the response curve. If it is assumed that Tl represents the response time of the thermometer and T2 the well, then the above example would be E = I (I – Ae –t/T1 + Be –t/T2). (Note: A and B are constants determined by Tl and T2.) This is a second-order system, and no single value such as T suffices to completely explain the system response. Another complication arises if the thermometer signal is transmitted sufficiently far to cause a measurable delay. For example, assume that the thermometer is replaced by an instrument called a temperature transmitter, and that this transmitter is placed some 500 ft away from the receiver. The 500 ft will be connected in this example by a copper tube that will transmit a pneumatic signal. The signal (in this case a pressure) will now be restricted in the tube and the capacitance of the receiver will add a time constant term to the response equation (remember the transmitter will also have a response time,). In addition to these two response times, the signal will take a certain amount of time to travel the 500 ft. This lag is called transportation lag, or dead time. This delay is not serious in an indicating system, but can be significant and worth considering in a controlling system. Fig. 2 illustrates three typical response curves for the examples that have been discussed. It can be shown mathematically that a single-order system cannot overshoot; hence, it is always stable. A second-order system may oscillate but is always stable from a mathematical standpoint. A third or higher order system 60 ASHRAE Journal F I G . 2 Response of overdamped systems to a step change. may be truly unstable. Most systems encountered in the field are much higher than third order but, since many of their characteristics can be neglected, it is possible to approximate the response of the system by using an equivalent third or even second-order system for the purpose of analysis. SYSTEM ANALYSIS Armed with these concepts and tools, we are now ready to consider system analysis. The acceptability of a control system usually can be described by two interdependent criteria: (1) steady state and (2) dynamic performance. Generally, one must be compromised somewhat by the other. We will now try to see how these two criteria are related, and also to see their relative importance in describing the complete system. Steady-State Criteria As indicated previously, an actual system never truly reaches equilibrium, however, for all practical purposes systems do settle out so that the output falls within the offset band as shown in Fig. 3. The width of this band is specified and is a yardstick with which to measure control accuracy. One way to reduce offset is to increase controller gain, another is to use reset (more properly called integral action). An increase in gain increases the chances of instability and the introduction of reset action decreases the speed of response, in other words, each has its disadvantage and the improvement of the steady-state performance is made at the expense of dynamic performance. a s h r a e. o rg August 2009

ASHRAE Journal - August 2009

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