Physical model
A physical model is a collection of physical objects connected together to serve an
objective e.g. gear system. No physical model can be represented in its full
intricacies and idealized assumptions are always made for the purpose of analysis
and synthesis of the system.
An idealized physical system is called a model. A physical system can be modelled
in a number of ways depending upon the specific problem to be dealt with and the
desired accuracy.
Electronic equipment may be modelled as interconnections of linear lumped
elements e.g. inductors, capacitors and resistor. In analysis of the mechanical
systems it is convenient to use three idealized elements. These are mass, damper
(or dash pot) and the spring. The mass, damper and spring represents the three
essential phenomena which occurs in various ways in mechanical systems.
Ideal mass element is used to represent a particle of mass which is a lumped
approximation of the mass of the body, concentrated at the center of the mass.
Ideal spring is used to represent the concept of elastic deformation of a body. The
ideal damper is used to represent the concept of viscous friction.
Mathematical model
Mathematical model is a mathematical representation of the physical model and is
obtained through use of appropriate physical laws. Common physical laws used are
Kirchhoff’s and Newton laws. Normally differential equations are obtained.
Laplace transform
It is an operational method used advantageously for solving linear differential
equations. Laplace transform can be used to convert many common functions for
instance sinusoidal into algebraic functions of a complex variable 𝑠. operations
such as differentiation and integration can be replaced by algebraic equations in
the complex 𝑠 plane. In other words, linear differential equation can be
transformed into algebraic equation in complex variable 𝑠.
Let x be a variable. Then we have the following Laplace transforms assuming initial
conditions to be zero.
, 𝑑𝑥
ℒ( ) = 𝑠𝑋(𝑠)
𝑑𝑡
𝑑2𝑥
ℒ ( 2 ) = 𝑠 2 𝑋(𝑠)
𝑑𝑡
𝑋(𝑠)
ℒ (∫ 𝑥(𝑡) 𝑑𝑡) =
𝑠
ℒ(𝑥(𝑡) ) = 𝑋(𝑠)
Illustrations of mathematical modelling for some basic elements
Resistor
𝑣(𝑡)
𝑖(𝑡)
R
𝑣(𝑡) = 𝑖(𝑡) 𝑅
𝑉(𝑠) = 𝐼(𝑠) 𝑅
Capacitor
𝑣(𝑡)
𝑖(𝑡)
C
1
𝑣(𝑡) = ∫ 𝑖(𝑡) 𝑑𝑡
𝐶
𝐼(𝑠)
𝑉(𝑠) =
𝑠𝐶
Inductor
𝑣(𝑡)
𝑖(𝑡)
L
𝑑𝑖(𝑡)
𝑣(𝑡) = 𝐿
𝑑𝑡
A physical model is a collection of physical objects connected together to serve an
objective e.g. gear system. No physical model can be represented in its full
intricacies and idealized assumptions are always made for the purpose of analysis
and synthesis of the system.
An idealized physical system is called a model. A physical system can be modelled
in a number of ways depending upon the specific problem to be dealt with and the
desired accuracy.
Electronic equipment may be modelled as interconnections of linear lumped
elements e.g. inductors, capacitors and resistor. In analysis of the mechanical
systems it is convenient to use three idealized elements. These are mass, damper
(or dash pot) and the spring. The mass, damper and spring represents the three
essential phenomena which occurs in various ways in mechanical systems.
Ideal mass element is used to represent a particle of mass which is a lumped
approximation of the mass of the body, concentrated at the center of the mass.
Ideal spring is used to represent the concept of elastic deformation of a body. The
ideal damper is used to represent the concept of viscous friction.
Mathematical model
Mathematical model is a mathematical representation of the physical model and is
obtained through use of appropriate physical laws. Common physical laws used are
Kirchhoff’s and Newton laws. Normally differential equations are obtained.
Laplace transform
It is an operational method used advantageously for solving linear differential
equations. Laplace transform can be used to convert many common functions for
instance sinusoidal into algebraic functions of a complex variable 𝑠. operations
such as differentiation and integration can be replaced by algebraic equations in
the complex 𝑠 plane. In other words, linear differential equation can be
transformed into algebraic equation in complex variable 𝑠.
Let x be a variable. Then we have the following Laplace transforms assuming initial
conditions to be zero.
, 𝑑𝑥
ℒ( ) = 𝑠𝑋(𝑠)
𝑑𝑡
𝑑2𝑥
ℒ ( 2 ) = 𝑠 2 𝑋(𝑠)
𝑑𝑡
𝑋(𝑠)
ℒ (∫ 𝑥(𝑡) 𝑑𝑡) =
𝑠
ℒ(𝑥(𝑡) ) = 𝑋(𝑠)
Illustrations of mathematical modelling for some basic elements
Resistor
𝑣(𝑡)
𝑖(𝑡)
R
𝑣(𝑡) = 𝑖(𝑡) 𝑅
𝑉(𝑠) = 𝐼(𝑠) 𝑅
Capacitor
𝑣(𝑡)
𝑖(𝑡)
C
1
𝑣(𝑡) = ∫ 𝑖(𝑡) 𝑑𝑡
𝐶
𝐼(𝑠)
𝑉(𝑠) =
𝑠𝐶
Inductor
𝑣(𝑡)
𝑖(𝑡)
L
𝑑𝑖(𝑡)
𝑣(𝑡) = 𝐿
𝑑𝑡
