<P> The proportional band is the band of controller output over which the final control element (a control valve, for instance) will move from one extreme to another . Mathematically, it can be expressed as: </P> <P> P B = 100 K p (\ displaystyle PB = (\ frac (100) (K_ (p))) \) </P> <P> So if K p (\ displaystyle K_ (p)), the proportional gain, is very high, the proportional band is very small, which means that the band of controller output over which the final control element will go from minimum to maximum (or vice versa) is very small . This is the case with on--off controllers, where K p (\ displaystyle K_ (p)) is very high and hence, for even a small error, the controller output is driven from one extreme to another . </P> <P> The clear advantage of proportional over on--off control can be demonstrated by car speed control . An analogy to on--off control is driving a car by applying either full power or no power and varying the duty cycle, to control speed . The power would be on until the target speed is reached, and then the power would be removed, so the car reduces speed . When the speed falls below the target, with a certain hysteresis, full power would again be applied . It can be seen that this would obviously result in poor control and large variations in speed . The more powerful the engine; the greater the instability, the heavier the car; the greater the stability . Stability may be expressed as correlating to the power - to - weight ratio of the vehicle . </P>

In an automatic control system the controller output changes the position of the