0195136047.pdf

166 TIME-DEPENDENT CIRCUIT ANALYSIS

Block Diagrams

The mathematical relationships of control systems are usually represented byblock diagrams,

which show the role of various components of the system and the interaction of variables in it.

It is common to use a block diagram in which each component in the system (or sometimes a

group of components) is represented by a block. An entire system may, then, be represented by the

interconnection of the blocks of the individual elements, so that their contributions to the overall

performance of the system may be evaluated. The simple configuration shown in Figure 3.4.11

is actually the basic building block of a complex block diagram. In the case of linear systems,

the input–output relationship is expressed as a transfer function, which is the ratio of the Laplace

transform of the output to the Laplace transform of the input with initial conditions of the system

set to zero. The arrows on the diagram imply that the block diagram has a unilateral property. In

other words, signal can only pass in the direction of the arrows.

A box is the symbol for multiplication; the input quantity is multiplied by the function in the

box to obtain the output. With circles indicating summing points (in an algebraic sense) and with

boxes or blocks denoting multiplication, any linear mathematical expression may be represented

by block-diagram notation, as in Figure 3.4.12 for the case of an elementary feedback control

system.

The block diagrams of complex feedback control systems usually contain several feedback

loops, and they may have to be simplified in order to evaluate an overall transfer function for the

system. A few of the block diagram reduction manipulations are given in Table 3.4.1; no attempt

is made here to cover all the possibilities.

R(s)

Input

C(s)

G(s) Output

Figure 3.4.11Basic building block of a block diagram.

R(s) E(s)

r(t) e(t)

B(s)

Feedforward

elements

Controlled

variable

(output signal)

Actuating

signal

Feedback

signal

Feedback elements

R(s) Reference input

C(s) Output signal (controlled variable)

B(s) Feedback signal = H(s)C(s)

E(s) Actuating signal (error) = [R(s) − B(s)]

H(s) Feedback path transfer function

G(s)H(s) Loop gain

R(s) Error-response transfer function

E(s)

Forward path transfer function or

open-loop transfer function =

G(s)

M(s) C(s)/R(s) = G(s)/[1 + G(s)H(s)]

Reference

input

C(s)

c(t)

G(s)

H(s)

1

= 1 + G(s)H(s)

Closed-loop transfer function =

C(s)/E(s)

+

−

b(t)

Figure 3.4.12Block diagram of an el-

ementary feedback control system.