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5.3.1 Proportional band

Note that it is customary to express the output of a controller as a

percentage of the full range of output that it is capable of passing on to

the correction element. Thus, with a valve as a correction element, as in

the float operated control of level in Figure 5.9, we might require it to becompletely closed when the output firom the controller is 0% and fully

open when it is 100% (Figure 5.10). Because the controller output is

proportional to the error, these percentages correspond to a zero value for

the error and the maximum possible error value. When the error is 50%

of its maximum value then the controller output will be 50% of its full

range.

Some terminology that is used in describing controllers:

1 Range

The range is the two extreme values between which the system

operates. A common controller output range is 4 to 20 mA.

2 Span

The span is the difference between the two extreme values within

which the system operates, e.g. a temperature control system mightoperate between 0C and 30C and so have a span of 30C.

Absolute deviati on

The set-point is compared to the measured value to give the error

signal, this being generally termed the deviation. The term absolute

deviation is used when the deviation is just quoted as the difference

between the measured value and the set value, e.g. a temperature

control system might operate between (fC and 30C and have anabsolute deviation of 3T.

4 Fractional deviation

The deviation is often quoted as a fractional or percentage

deviation, this being the absolute deviation as a fraction or

percentage of the span. Thus, a temperature control system

operating between 0C and 30C with an error of 3C has a

percentage deviation of (3/30) x 100 = 10%. When there is no

deviation then the percentage deviation is 0% and when the

deviation is the maximum permitted by the span it is 100%.

Generally with process controllers, the proportional gain is described

in terms of its proportional band (PB). The proportional band is the

fractional or percentage deviation that will produce a 100% change in

controller output (Figure 5.11)


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The 100% controller output might be a signal that fully opens a valve,

the 0% being when it fully closes it. A 50% proportional band means

that a 50% error will produce a 100% change in controller output; 100%

proportional band means that a 100% error will produce a 100% change

in controller output.Since the percentage deviation is the error e as a percentage of the

span and the percentage change in the controller output is the controller

output >^c as a percentage of the output span of the controller:

5.3.2 Limitations of proportional control

Proportional controllers have limitations. Consider the above example inFigure 5.8 of the amplifier as the proportional controller. Initially, takethe temperature of the liquid in the bath to be at the set value. There isthen no error signal and consequently no current to the heating element.

Now suppose the temperature of tlie inflowing liquid changes to aconstant lower value (Figure 5.12). The temperature sensor will, after a


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time lag, indicate a temperature value which differs from the set value.

The greater the mass of the liquid in the tank, i.e. the capacitance, thelonger will be the time taken for the sensor to react to the change. This is

because it will take longer for the colder liquid to have mixed with theliquid in the tank and reached the sensor. The differential amplifier willthen give an error signal and the power amplifier a signal to the heater

which is proportional to the error. The current to the heater will beproportional to the error, the constant of proportionality being the gain of

the amplifier. The higher the gain the larger will be the current to theheater for a particular error and thus the faster the system will respond tothe temperature change. As indicated in Figure 5.12, the inflow isconstantly at this lower temperature. Thus, when steady state conditionsprevail, we always need current passing through the heater. Thus there

must be a continuing error signal and so the temperature can never quitebe the set value. This error signal which persists under steady state

conditions is termed tliesteady state error or theproportional offset.The higher the gain of the amplifier the lower will be the steady state

error because the system reacts more quickly.In the above example, we could have obtained the same type ofresponse if, instead of changing the temperature of the input liquid, wehad made a sudden change of the set value to a new constant value.There would need to be a steady state error or proportional offset fromthe original value. We can also obtain steady state errors in the case of acontrol system which has to, say, give an output of an output shaftrotating at a constant rate, the error results in a velocity-lag.All proportional control systems have a steady state error. Theproportional mode of control tends to be used in processeswhere the gain AT? can be made large enough to reduce thesteady state error to an acceptable level. However, the larger the

gain the greater the chance of the system oscillating. Theoscillations occur because of time lags in the system, the higherthe gain the bigger will be the controlling action for a particularerror and so the greater the chance that the system will

overshoot the set value and oscillations occur.

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