# Proportional Control

(Controller output can go directly to a valve or to the setpoint of another controller. In the following discussions, it is assumed that controllers send their output directly to a valve.)

Figure 300-7 shows the relationship between valve position and error that is characteristic of proportional control: The valve position changes in exact proportion to the amount of error, not to its rate or duration. The response is almost instantaneous, and the valve returns to its initial value when the error returns to zero. Control Algorithm
The linear relationship between the setpoint deviation (error) and the valve position (controller output) for proportional action can be expressed as follows:

O = KcE
(Eq. 300-1)

where:
O = Controller output
Kc = Controller Gain = DOutput / DError
E = Error = (Setpoint – Measurement)

This equation is called the control algorithm. The gain, Kc, is also called the controller sensitivity. It represents the proportionality constant between the control valve position and controller error.

Proportional Band
Another way of characterizing a proportional controller is to describe its proportional band. The proportional band is the percent change in value of the controlled variable necessary to cause full travel of the final control element. The proportional band, PB, is related to its gain as follows:

Kc= 100/PB
(Eq. 300-2)

Both proportional band and gain are expressions of proportionality. Manufacturers may call their adjustments gain, sensitivity, or proportional band. Figure 300-8 shows the relationship between valve opening and proportional bands of different percentages. High percentage proportional bands (wide bands) have a less sensitive response than low percentage proportional bands (narrow bands). Bias
Bias is the amount of output from a proportional controller when the error is zero. Equation 300-1 implies that when the error is zero, controller output is zero. The valve is either fully open or fully closed and provides no throttling action. Adding a bias provides this throttling action. Equation 300-1 then becomes:

O = KcE + B
(Eq. 300-3)

where:
B = Bias (percent of full output)

Typically, manufacturers set the bias at 50%. To prevent a process bump, the operator is sometimes allowed to adjust the bias before putting the controller in automatic. Figure 300-9 shows controller output versus error at different proportional bands with a 50% bias. At zero error, the controller output is 50% of full range for any proportional band.

Offset
A controller’s error is the difference between its setpoint and measurement. In a proportional-only controller, a change in setpoint or load introduces a permanent error called offset (see Figure 300-10). It is impossible for a proportional-only controller to return the measurement exactly to its setpoint, because proportional output only changes in response to a change in the error, not to the error’s duration. Assume that a proportional-only controller controls the outlet temperature of a furnace and that the temperature is at the setpoint. If the feed rate to the furnace increases, more fuel will be needed. This disturbance represents a load change to the furnace. To get more fuel, the fuel valve must be opened more. As is suggested by Equation 300-3, the only way that the valve can be at some value other than its starting point is for an error to exist. Thus, the proportional controller alone cannot return the outlet temperature to its setpoint. As mentioned, some controllers allow the operator to adjust the bias until the value of E (the error, or offset) is zero.

Offset is determined by the proportional band value for the controller and the change in valve position that occurs when a disturbance takes place: The proportional-only controller is the easiest continuous controller to tune. It provides rapid response and is relatively stable. If offset can be tolerated (loose control), proportional-only control can be used.