## Proportional Controller

Proportional controller is called as gain controller, where the output position is related to the deviation of the set point and measured control process value. As the name specifies its output will be error deviation multiplied a factor (electronic circuit gain) Proportional controller can be used where the processing time of equipment or process is very large or where error magnitude is not needed to minimize to zero. The setting of proportional constant can be expressed as

1. Proportional Gain
2. Proportional Band

Proportional Gain: This concept is basically used in electronic controllers where the circuit gain acts as proportional constant. The proportional gain is the percentage change of controller output related to percentage change in controller input.

Proportional Gain =   Δoutput (%) / Δinput (%).

Figure shows the electronic controller with gain element is resistor. The resistor value can be changed to control the gain of the circuit and hence output of the system.

Proportional Band:  It is also known as throttling range or throttling band, is defined as percentage change in input controlled variable span will cause a 100 % change in controller output. It also defined as the change in control variable that will drive the controller output for full scale.

Proportional Band =  Δinput (% span) / Δoutput (%).

Output = 100/PB (error) + offset.

Normally this concept will be used in hydraulic controller and digital controller. For example a hydraulic controller, to control the level of the tank with down stream valve. A band of level points will be given to controller saying that the controller should give minimum valve position (0%) when level reaches minimum point in the band, and it should give the maximum value (100%) when level reaches Maximum point. This is ensure the level should not cross the band points in both sides.

Relation between PB and Proportional Gain:

PB = 100/Gain

Gain = 100 % /PB

Offset: In proportional controller, if the control variable reaches the set point then the deviation will become zero and hence the output valve which is multiplication of error will become zero. Consider the above example of level controller, If the tank level reaches the set-point then valve will be closed by the controller and hence level raises again. to avoid this situation a proportional offset is added to the controller. This proportional offset also helps in bringing the controller variable near to the set-point which helps in reducing the error in steady state.

This offset is changed manually by the operator for adjusting the controlled variable near to the set-point. This is called manual resetting the controller to reduce the steady state error. Further this automatic resetting is envisaged in controller concepts by adding integral controller to the proportional controller.

Practical example of proportional controller:

Consider a tank with level set point of 2 mm and controller with proportional band is 50 %. i.e lower is 1mm and higher is 3 mm. If the set-point suddenly raised to 2.2 mm then the output of the control will be as follows.

Change in Set point (%)= 2.2/2 * 100 = 10%

Proportional Gain = 100 % / PB= 100 % / 50% =2

Controller output = Change in input * Gain

= 10 % * 2= 20%

Total the output will be increased by 20 % to reach the set-point as near as possible,

Transfer function of Proportional controller

The Proportional is used improve the transient response of the system. The transfer function of the proportional controller is ‘Kp’.

The open loop transfer function is of above system is

Closed loop Transfer function of the system

By Comparing this with standard closed loop transfer function

In the above transfer function \$= 5 and Wn =1. by the basic control system concepts if \$>1 then the system is unstable. Now if u add Proportional controller to the system

Then transfer function of closed loop system with proportional controller is

here Wn= √Kp   and \$= 10/√Kp

So By changing the  Kp Valve we can change the nature of the system.

Tuning of Proportional Controller:

Application of proportional controller

A very important application of proportional controller with fixed bias or offset id the zero load process. It means the dynamic characteristics of process will not get any disturbance even if there is no flow through the controller for small duration.This introduces the proportional controller can be used for temperature control of any material or fluid.

Advantages

• Easy to implement
• low cost
• Easy to tune the proportional constant

Disadvantages / Limits

• Responds only change in error.
• Error can be reduced to zero (i.e controlled value cannot reach Set point)
• Fine controlling is not possible.