The proportional controller gives control signal in one to one correspondence with an error over some range. Each value of error about desired value will have corresponding controller output like controller output values are interpolated with respect to error values.
Mathematically this can be represented as
- P = Controller’s output
- ep (t)= Desired Value of controlled variable – Measured Value
- Kp = Proportional gain
- P0 = Controller output with no error
It indicates that, if the error is zero, the output is constant and equal to P0 and if there is an error, for every 1 % error, a correction of Kp % is added to or subtracted from P0, depending on the sign of error.
Thermostat used in room temperature control expands or compress it’s bimetallic spring as per temperature variation in the room and gives corresponding control signal to heater coil.
Another example is heat exchanger system,
- If there is sudden large flow of cold fluid enters into heat exchanger, since TT (temperature transmitter) detects this sudden change then,
- TC (Temperature controller) compares measured value of temperature with desired temperature value,
- a corresponding error is also large and if we multiply it with proper gain (proportional gain) and subtract from P0,
- Then control signal gradually opens steam flow through the control valve and maintains the temperature inside the heat exchanger.
- Processes with slower dynamics are controlled effectively.
- Helps to stabilize higher order processes which has transient instability.
- The residual error is always required to maintain desired value of the controlled variable.
E.g. If the controller’s output will increase with falling outlet temperature of the heat exchanger until there is enough steam flow admitted to the heat exchanger to prevent the temperature from falling any further. But in order to maintain this greater flow rate of steam (for greater heating effect), an error must develop between the measured temperature and desired temperature. In other words, the process variable (temperature) must deviate from desired value in order for the controller to call for more steam, in order that the process variable does not fall any further than this. This necessary error between the measured value and desired value is called ‘offset’ or residual error.