TWELVE in Software

Generating PDF-417 2d barcode in Software TWELVE

equation are usually independent of time Therefore, for process control the basic process differential equation is usually of the form mn(t) dny(t) dtn mn

m0y(t)

x(t)

2 Transfer Functions* To illustrate the convenience of the Laplace transfor-

mation, consider a differential equation given by T dc(t) dt c(t) RMu(t) c(0) 0 (121)

where T represents the time constant of the system, and Mu(t) describes a step input to the system of magnitude M We wish to nd the response of the output c(t) The Laplace transforms needed are taken from tables We have

sC(s)

c(o )

c(0 )

L [c(t)] L [Mu(t)]

The Laplace transform of Eq (121) is then TsC(s) C(s) C(s) R M s (122)

RM s(Ts 1)

To transform this equation back into the time domain, we look up inverse transform L 1 and nd that c(t)

[C(s)]

RM(1

In a system of many elements the transformed equation corresponding to Eq (122) may be quite complicated However, it can usually be manipulated into a form for which the inverse transform can be found in a table In process-control work Laplace transforms are used to determine responses to disturbances Steady-state or constant terms will usually drop out of the solutions of the differential equations because initial conditions will usually be assumed to be zero

* This section taken in part from Perry s Chemical Engineers Handbook, 6th ed, by R H Perry D W 1984 Used by permission of McGraw-Hill, Inc All rights Green, and J O Maloney (eds) Copyright reserved

The transfer function is de ned as the ratio of the Laplace transform of the responding variable (output) to the Laplace transform of the disturbing variable (input) From Eq (122), we get for the transfer function KG(s) KG(s) output input C(s) M/ s R Ts 1 (123)

The convention for designating transfer functions in a control diagram is the expression KG(s) Capital letters are used when the functions are in the s domain, and small letters are used in the time domain G(s) represents the dynamic portion of transfer function, and K is related to the steady-state gain through an element In the transfer function, Eq (123), K R and G(s) 1 / (Ts 1) It is common practice in drawing block diagrams in the s domain to omit the (s) from the F(s) s Instead only the capital-letter designation is used to represent the s-domain transforms Combining Transfer Functions Fluid- and thermal-process systems exhibit many different dynamic characteristics, but many systems may be described by combinations of ve transfer functions: Proportional element: Capacitance element: First-order element: Second-order element: Dead-time element: K 1 Ts 1 Ts Ts

Transfer functions are important tools in the analysis of control systems Each block or element of the control system has its own characteristic transfer function If the s-domain transfer function notation KG(s) is used for each block, the system elements can be combined by algebraic procedures into an overall expression for the entire control system

de ning the ow of information and energy through the control system is by means of a block diagram In the diagram the components of the control system are considered as functional blocks in series and parallel arrangements according to their position in the actual control system Each component is represented by its transfer function, the ratio of the Laplace transform of the output variable to the input variable with all initial conditions taken as zero The block diagram of a single-loop feedback-control system subjected to a command input R(s) and a disturbance U(s) is shown in Fig 125

* This Section taken in part from Marks Standard Handbook for Mechanical Engineers 9th ed, by E A Avallone and T Baumeister III (eds) Copyright 1987 Used by permission of McGraw-Hill, Inc All rights reserved