Define proportional band (pb) or gain or band width? Example problem for proportional band calculation?

Definition of ON-OFF controller:

The term proportional band is commonly used among process control engineers in place of the term gain. Proportional band (pb) is defined as the error (expressed as a percentage of the range of measured variable) required to move the valve from fully closed to fully open. A frequently used synonym is bandwidth.

Example 1.  A pneumatic proportional controller is used to control temperature within the range of 60 to 1OO”E The controller is adjusted so that the output pressure goes from 3 psi (valve fully open) to 15 psi (valve fully closed) as the measured temperature goes for 71 to 75’ F with the set point held constant. Find the gain and

the proportional band.

Answer: 

 

Now assume that the proportional band of the controller is changed to 75 percent. Find the gain and the temperature change necessary to cause a valve to go from fully open to fully closed.