For stable operation the loop should contain an odd nwnher of phase inversions, corresponding to negative feedback, Graphically, if the input signd vector is drawn as shown in Figure 3, the output vector should fall in the fint or fourth quadrant.
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Figure 3. Graphic representotion of the gain vecbr of 0 control loop in polar coordinates. If the sire of the input vector is defined as unity, the output vector represents the gain.
For signals of very low frequency, the transit time through the loop becomes insignificant, so that the ouput vector will be parallel to the input vector, and of oppposite direction. The presence of low-pass filters of the type shown in Figure 2 will give rise to a decrease in loop gain with increasing frequency, and to a concomitant phase lag. I t can be shown that 8. correlation exists between the gain-frequency curve and the phase lag, such that when dlnil/dlnw = n, the phase shift + = nr/2. The position of the break in the gain-frequency curve can be defined as the point at whioh the phase angle is r / 4 radians; a t that point w = l / in ~which w is the frequency in radians/second, and r = RC is the time constant of the filter. The gain-frequency curve far a single RC filter is shown in Figure 4.
Figure 5. Gain-frequency curve of h e entire control loop. The curves lobeled r r the response of the three loop elements with the longest time constants.
In a thermostat we can generally recognize at least 3 low-pass filters: one each due to the heat capacity of the heater, the hath, and the sensor. In addition, the control amphfier may have one or more low-pass filters, normally with much shorter time constants. At frequencies above the cut-off points of the three thermal low-pass filters we should expect 8. phase lag of 270'. This means that there will be some frequency at which the
A516
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Journal of Chemical Education
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phase shift due the low-pass filters equals thermostat cannot increase iithout limit 180°, which would produce a positive as the temperature error increases; neither loop gain. If at that frequency the loop can the heat output decrease below zero. gain has a m a g ~ t u d eof unity or larger, The presence of cooling devices changes the system will oscillate. this argument only slightly, to allow a We are thus faced with a set of conflicb finite, rather then zero cooling output. ing requirements: high loop gain is The proportional band is now defined as desirable to minimize the effect of external the temperrttnre interval within which factors such as changes in heat loss from the heat output varies linearly or nearly the heth to its surroundings; yet the gain linearly as a function of the deviation from must fall below unity at the frequency far the setpoint. Since an estimate of the which the phase shift reaches 180". To required range of heat output can usually make matters even more complicated, be made, knowledge of the proportional reasonable gain should be available at hand and the maximum available heat finite frequencies to insure adequate output allows one to predict the longresponse to transient disturbances. term stability of the system. This is The gain-frequency characteristic of the illostrated graphically in Figure 6. loop can he broken into three major regions When it is impossible to reduce offset (Fig. 5): at frequencies above w the errors of a proportional system to aceeptr gain should he less than unity to prevent able Levels without instability or oscillaoscillation; over a. range of frequencies up tion, an integral term may be introdneed to o, the gain should be essentially eoninto the loop gain, so that the output b e stant and adequate to handle transient comes a function of both the error signal disturbances, and toward zero frequency and its time integral. Mathematically the gain sho~ddbecome as high as possible it is easy to see that the integral term to minimize offset errors, i.e., constant corresponds to a. slope of -1 in the gaindeviations from the chosen setpoint. frequency plot. The resulting control In the middle region, the response of the system is referred to ss s. two-term, system is proportional to the instantaneous reset, or P-I system. The proportional error signal; we speak of proportional first two nemes have seniority, hut the control. The gain in this region is genlast one is the most concise and deserves to erallv made as laree as is consistent with become the preferred name. -rnhlr iqwrntianl. Pnwtirwl y~crifiwriotti Laboratory thermostats generally nrr rarely giwn i l l em. of pain, IIOWQVC~; operate in sheltered surroundings, so that ~ w r w d the , propmima1 I m d i. ntmnally heat losses vary only slightly with time. stated. Offset errors would then also remain I t is obvious that the heater output of a essentially constant, and they can be compensated by manual adjustment of the setpoint. (One might say that in this case the operator provides the integral response.) For this reason, integral response is rarely required for laboratory equipment. An exception might be the control of devices through a wide temperature range, e.g., the heating mantle around rs distillation column. On the other hand, fast response to transient disturbances is a desirable and often necessary property of laboratory equipment. This may require the use of an anticipating or derivative term in the control loop. If the gain of the two elements with the longest timeconstantsis Figure 6. Relotian$hip between pmportionol plotted as s function of frequency, the band, expected variation in heat load, and operating temperature range. (Conlinued n page A618)
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Figure 4. Gain-frequency curve for a single low-pox filter. Note that the goin- ond f r e quency axes hove logarithmic rcoler.
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