Laplace operator ‘s’

Reminder from ENGR201

Know how components behave in an instant of time

Consider differential response

Combine and simplify into standard forms

Integration by lookup table determines long-term response

t domain s domain

Signals

Components are connected together by signals.

Signals have many different forms,

Must also have direction & name

Signals continue until interrupted!

Signals and components are considered ideal.

We add other signals and components to alter the properties

Voltage VElectrical

Force NMechanical

Flow FFluid

Temperature KThermal

We wish to know how the output signal varies with an input signal for a fixed (invariant) system

Consider a spring (constant k), with force F applied to stretch it a distance x

We may plot two signals against each other invariant of time (system relationship)

Signals

Distance xForce F

Distance x

Force F

System constants are time invariant for the given system

We now consider a different system as the spring has been changed.

However, the analysis stays the same

Constants

Distance x

Force F

Spring constant k = 1

Distance x

Force F

Now k = 2

Systems

In order to analyse a system:We identify an input signal

[a variable]Using block diagram components

[Basic block Summing junction Take-off point]

We combine internal signals[modified variables]

To produce the output signal[another variable].

The Input-Output relationship may then be determined

Flow F1 F1 + F2

Fluid F2

Components

Summing junction

+ and/or – the system signals

Voltage V1 V1 – V2

Electrical V2

+ _

++

Up to three inputs and only one output

Components

Block•or / or function of the system signal

Variables Constants

Input Direction Output

Only one input and only one output

Force F Distance x

Spring

Current I VR

1 / k

Components

Take-off point

The system signal can be used elsewhere,

but is not affected by the split

Voltage V1 V1

Electrical

V1

Only one input and many outputs

Mechanical Components

We know that distance (x) is related to velocity (v) is related to acceleration (a) through differentiation.

It gets messy writing all the time!Therefore, we will write ‘s’ instead.

x = xv = sxa = sv = s2x

NB, Both with respect to the variable

txx

dt

tdxv

2

2

dt

txd

dtdt

tdxd

dt

dva

dt

d

Mechanical Components

Spring

Damper

Mass

Source Nise 2004

Force - ImpedanceDistance

tKxtf

dt

tdxDtf

2

2

dt

txdMtf

tf

tx

tf

tx

tf

tx

sXsF

K

Ds

2Ms

Capacitor

Resistor

Inductor

Source Nise 2004

Electrical Components

Voltage - ImpedanceCurrent

tRitv

dt

tdiLtv

tv

ti

sIsV

R

Ls

tv

ti

tv

ti

tiC

tv1

sC

11

Assume armature inductance is negligible.

Armature resistor:

Back emf of motor:

Torque proportional to armature current:

Torque is opposed by the inertia torque:

Hint: apply Kirchhoff’s voltage law to the armature circuit

We need to form a relationship between input voltage and output velocity:

R2-D2 Motor System

Rba vvv

sVa

sIa

sVb

T

aTiKT eb Kv aaR Riv

dt

dJT

Combine components

Components of Motor System

sIRsV aR

sKsV eb

sJssT

bV

RV

eK

RI

Js

T

+_

bV

aV RVRba vvv

aTiKT

eb Kv

aaR Riv

dt

dJT

sIKsT TTI TK

Reduce block diagram:

Block Diagram of Motor System

bV

aV+_

1

1

sKKJR

K

te

a

e

1.

2.

3.

eKbV

1/R TK Js1

bV

aV+_

eKbV

sJR

K

a

T

aV

Output linked to input:

Can be expressed much more simply!:

Where:

Time Constant:

Gain:

Transfer Function of System

1

1

sKKJR

K

V

te

a

e

a

11

sT

K

Va

te

a

KK

JRT 1

eKK

1

Past Assignment question

Determine the block diagram for each system?

Source Nise 2004