(a) Relationship between Proportional Band and Gain:
Proportional band (PB) and gain are inversely proportional to each other. This means that as the PB increases, the gain decreases, and vice versa. Mathematically, it can be expressed as:
Gain = 1 / PB (expressed as a percentage)
For example, if the PB is 20%, the gain is 5 (100 / 20 = 5). Conversely, if the gain is 3, the PB is approximately 33% (100 / 3 = 33.33%).
(b) Level Control in a Water Tank:
(i) Increasing and Decreasing Gain:
In the given lever system, the gain can be adjusted by modifying the length of the lever arm. A shorter lever arm increases the gain because a smaller movement of the float translates to a larger movement of the control valve, resulting in a more significant response to level changes. Conversely, a longer lever arm decreases the gain, as a larger movement of the float produces a smaller movement of the valve, leading to a less pronounced response.
(ii) Increased Flow Out:
When the flow out of the tank increases, the water level will start to drop. The float will move down, and the lever will pivot, opening the control valve further to allow more water inflow. However, with a proportional-only system, the valve opening will only reach a point where the inflow matches the increased outflow, and the water level will stabilize at a new, lower level.
(iii) Increasing Gain with Increased Outflow:
Increasing the gain (shortening the lever arm) in this scenario will make the system more responsive to level changes. The valve will open wider for smaller float movements, attempting to compensate for the increased outflow more aggressively. However, this can lead to hunting, where the water level oscillates around the desired level due to the system’s overreaction.
(iv) Introducing Integral Action:
Adding integral action to the system can help address the limitations of proportional control. The integral term essentially remembers the past level deviations and continuously adjusts the valve position to eliminate them. This helps the system reach and maintain the desired level even with changes in outflow, reducing hunting and improving steady-state accuracy.
Remember that the specific behavior of the system will depend on various factors like the tank size, flow rates, and valve characteristics. This is a simplified explanation to illustrate the concepts.