Transfer Functions

Unit 1: Process Control LoopProcess control loopProcessSensorTransmitterControllerTransducerControl valve Abdul Aziz Ishak, Universiti Teknologi MARA Malaysia (2009)

Unit 1: Process Control LoopProcessSensorTransmitterControllerTransducerControl valvePressureFlowLevelTemperaturepHdP cellCapacitanceRadar, SonicMagneticResistanceIR/Laser4-20 mA1-5 VdcField/profibusPIDFuzzy logic4-20 mA3-15 psigLinearEqual percentageProcess control loop Abdul Aziz Ishak, Universiti Teknologi MARA Malaysia (2009)*SP = set point*PV = process value

Unit 1: Process Control LoopCONTROLLERCONTROLVALVEPROCESSPVSPSimulation modeProcess control loop: The Block Diagram Abdul Aziz Ishak, Universiti Teknologi MARA Malaysia (2009)TRANSMITTER

Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output:The following terminology is used:xinputforcing functioncauseyoutputresponseeffectChapter 4Transfer FunctionsThe TF model enables us to determine the output response to any change in an input.

Definition of the transfer function:Let G(s) denote the transfer function between an input, x, and an output, y. Then, by definitionwhere:Chapter 4

Figure 2.3 Stirred-tank heating process with constant holdup, V.Chapter 4Example: Stirred Tank Heating SystemTransfer Functions for a Process

Equation (1) is the energy balance of the stirred-tank heating system, assuming constant liquid holdup and flow rates:Suppose the process is at steady state:Subtract (2) from (1): Chapter 4(1)

But,where the deviation variables areTake L of (4):Chapter 4At the initial steady state, T(0) = 0.

Rearrange (5) to solve for Chapter 4where

K (gain) it describes how far the output will travel with the change of the input.

(time constant) describes how fast the output moves in response to a change in the input.*The time constant must be positive and it must have units of time

*If a process has a large K, then a small change in the input will cause the output to move a large amount. If a process has a small K, the same input change will move the output a small amount

Chapter 4Order of transfer function General first order transfer functionGeneral second order transfer function First-order-plus-dead-time (FOPDT)

Response with time delay

to=Time delay/dead time

All first order systems forced by a step function will have a response of this same shape. Step response for first order system

To calculate the gain and time constant from the graph

Gain, Time constant, value of t which the response is 63.2% complete

Transfer Functions for a TransmitterKT = transmitter gain

= transmitter time constant

For proportional control, the controller output is proportional to the error signal,where:Proportional Control

is controller output when the error is zero

-The proportionality is given by the controller gain, Kc-The controller gain determines how much the output from the controller changes for a given change in error

Transfer Functions for a ControllerTransfer function

*Integral ControlFor integral control action, the controller output depends on the integral of the error signal over time,where , an adjustable parameter referred to as the integral time or reset time, has units of time.Integral control action is normally used in conjunction with proportional control as the proportional-integral (PI) controller :

The corresponding transfer function for the PI controller in is given byThe PI controller has two parameters, Kc andTransfer function

Derivative ControlThe function of derivative control gives the controller the capability to anticipate where the process is heading by calculating the derivative errorThus, for ideal derivative action,where , the derivative time, has units of time.

Proportional-Integral-Derivative (PID) ControlNow we consider the combination of the proportional, integral, and derivative control modes as a PID controller.Form of PID ControlThe form of the PID control algorithm is given byThe corresponding transfer function is:Transfer function

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