Page 811 - Basic Electrical Engineering

11.5 LINEAR AND NON-LINEAR SYSTEMS

A measurement system comprises a number of components or subsystems
connected together for the purpose of measurement and control of certain

variable quantity.
In studying the characteristics of such a system, we generally write

mathematical equations representing the system. Such representation in terms
of equations is called mathematical modelling. The mathematical model of

describing a system is generally expressed with the help of differential
equations. The coefficients of the differential equations can either be

constants or time variant. Mathematical equations describing a system can be
called linear if the laws of superposition and homogeneity are applicable to
the system. Suppose, x (t) and x (t) are the two inputs to the system and the
2
1
corresponding outputs are y (t) and y (t), respectively, then by applying the
1
2
principle of superposition we will be able to write

a x (t) + a x (t) = a y (t) + a y (t)
1
2
1
1
2
2
2
1
where a , a are constants.
2
1
Such a system is linear and time invariant. If the coefficients of the
differential equations describing a linear system are functions of time, then
the system is called linear time-variant system.
Linear systems can be analysed using Laplace transform and Fouriers
series. Since almost all measurement systems are nonlinear in nature, they are
first linearlized through approximations and then their analysis is done to
study the dynamic performance of systems.

11.6 DYNAMIC CHARACTERISTICS OF INSTRUMENTS
Systems are often subjected to inputs which are varying with time. The

output of such systems are to be measured. The behaviour of the system
under such varying input conditions are called dynamic response of the

system. Dynamic inputs can be of two types, viz transient type and steady-
state periodic type. Transient inputs die out with time where as periodic type