This quantity PB is defined as the error required to move the final control element over the whole of its range (e.g. from fully open to fully shut) and is expressed as a percentage of the total range of the measured variable (e.g. two extremes of temperature).

A fermenter with a heating jacket will be used as an example (Fig. 8.18). A thermocouple is connected to a temperature controller which has a span of 10° covering the range 25° to 35°, with a set point at 30°. The controller valve which is controlled by a pressure regulator, is fully open at 5 psig (46 kN m~2) and fully closed at 15 psig (138 kN m~2), while the set point of 30° corresponds to a control pressure on the valve of 10 psig (92 kN nr2). When the controller gain is 1, a change of 10° will cause a pressure change of 10 psig when the valve will be fully open at 25° and fully closed at 35°. Thus, the proportional band is 100%. If the controller gain is 2, a 5° change will cause the valve to go from fully open to fully closed, i.e. 27.5° to 32.5°. In this case the proportional band width will be:

Actual band width

Total band range

Either side of this band the pressure will be constant. In the case of a controller gain of 4, a 2.5° change will cause a pressure change of 10 psig (92 kN m~2), which will cause the control valve to go from fully open to fully closed (28.75° to 31.25°). In this case the proportional band will be

These results are summarized in Table 8.2 for controller gains of 1, 2 and 4.

When the proportional band is very small (the controller gain is high) the control mode can be likened to simple ON/OFF, with a high degree of oscillation but no offset. As the proportional band is increased (low controller gain) the oscillations are reduced but the offset is increased. Settings for proportional band width are normally a compromise between degree of oscillation and offset. If the offset is not desirable it can be eliminated by the use of proportional control in association with integral control (see Fig. 8.17 and later section).

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