ELEC2400 Lecture 3 System Modelling, Differentiation Equations and System Properties

ELEC2400-2008-T3-PSB ACADEMY-Chapter 3 1

ELEC2400 Signals and Systems

Chapter 3: Modelling, Differential Equation

and System Properties

• Electrical Circuit and Mechanical System Modelling

• 1st Order Differential Equations

• Impulse Response

• Convolution and Properties of Convolution

• System Properties


Introduction


System

• A system is an interconnected set of components

(devices, processes, subsystems, …) with terminals or

ports through which matter, energy, information or signal

can be applied, transferred and extracted

• The block diagram representation of a system:

• Examples of a system: electrical circuit, spring, …


Electrical Circuit Modelling


Mechanical System Modelling



Translational Mass

Fm

x

• Applying Newton’s 2nd Law gives

2

2

dv d xF ma m mdt dt

where F = applied force [N]

m = mass (of object) [g]

a = acceleration [m/s2]

v= velocity [m/s]

x= displacement [m]



Translational Spring

• From Hooke’s Law,

F

x

k

s s s

dF dxF k x k k v

dt dt

where ks = spring coefficient [N/m]

Translational Damper

• From Newton’s Law,

where kd = fiction constant or stiffness [N-s/m]

F

b

xd d

dxF k v k

dt


First Order Differential Equations


Solutions of Differential Equations


Impulse Response

Impulse Response: Example 1


Impulse Response: Example 1

Impulse Response


Convolution


Convolution

Properties of Convolution



Convolution


Convolution Example: RC Circuit

Convolution



Convolution


Convolution Example: Rectangular Signals

Convolution



Convolution


Convolution Example: Other Signals

Convolution


System Properties


System Properties

• An example: data smoothing

• Given a batch of data: …, x[0], x[1], x[2], … which can be

smoothened using the following expression:

• Is this process causal or non-causal?

x i 1 x i x i 1

y i3


System Properties

ELEC2400 Lecture 3 System Modelling, Differentiation Equations and System Properties

Documents

Transcript of ELEC2400 Lecture 3 System Modelling, Differentiation Equations and System Properties