What is proportional band and how does it relate to controller gain?

Proportional band (PB) is the percentage of input range that causes 100% change in output. Controller gain (Kp) is inversely related: Kp = 100/PB. A 50% PB means Kp = 2 (50% input change causes 100% output change). Narrow PB (high gain) gives more aggressive response but may cause instability; wide PB (low gain) gives sluggish but stable response. Modern controllers typically use gain directly; traditional pneumatic controllers used proportional band.