As an introduction, consider dynamic systems:

1
Dynamic Systems

• As an introduction, consider dynamic systems
• these change with time
• As an example consider water system with two
  tanks
• Water will flow from first tank to second
• When will it stop following?

Stops when levels equal.
Why stop then? Why flow at all? Plot time
response of system.?
2
Dynamic Systems
Pressure
F
Pressure

• Water flows because of pressure difference
• (Ignore atmospheric pressure - approx equal at
  both ends of pipe)
• If have water at one end - what is its pressure?
• (Tanks with constant cross sectional area A)
• Pressure is force per unit area, Force / A
• Force is mass of water g
• Mass of water is volume of water density
• M V ?
• Volume is height of water, h, times its area A
• V h A
• Combining
• pressure is

3
Dynamic Systems

• For first tank, pressure is I ? g
• For second tank, pressure is L ? g
• Thus flow depends on (I-L) ? g as well as on
  the pipe (its restrictance, R)
• Flow changes volume of tanks
• Volume change A change in height (L) Flow
• Thus change in L
• A tank has a capacitance, C
• Thus change in height L is
• Flow stops, and there is no change in height,
• when I L

4
Dynamic Systems

• For first tank, pressure is I ? g
• For second tank, pressure is L ? g
• Thus flow depends on (I-L) ? g as well as on
  the pipe (its restrictance, R)
• Flow changes volume of tanks
• Volume change A change in height (L) Flow
• Thus change in L
• A tank has a capacitance, C
• Thus change in height L is
• Flow stops, and there is no change in height,
• when I L

5
Dynamic Flow

• Level change not instantaneous
• Initially
• Large height difference ? Large flow ? L up a lot
• Then
• Height difference less ? Less flow ? L increases,
  but by less
• Later
• Height difference lesser ? Less flow ? L up,
  but by less, etc
• Graphically we can thus argue the variation of
  level L and flow F is

L
I
t
T
6
Time Response

• We have dynamic equation
• Put into s domain and rearrange
• Integrate a step input with respect to time
• Variation of L is
• As t gets larger, exponential term disappears,
• L tends to I.

7
Time Response

• The variation of F is inverse exponential,
• T is the time constant of the system C R.
• At t T, F down to 37 of initial value, I/R
• At t 5T, F is less than 1 of initial value.
• As t gets larger, so F tends to 0.
• Variation of L is
• As t gets larger, exponential term disappears,
  and L thus tends to I.
• At t T, L is 63 of final value, I
• At t 5T, L is within 1 of final value.

8
Differentiation

• Alternative method
• Level, L, changes because of the flow of liquid
• Mathematically, change of L with time is
• In fact, the change in L is proportional to flow,
  F
• Flow related to difference in levels
• Thus
• CR is the time constant, T.
• Note, the above is a differential equation

9
Differentiation Slopes

• Consider the graph of L
• At any instant of time we can see value of L
• The change in L is the slope of the graph, which
  varies with time.
• Initially steep (high value), then less, then
  less
• But the flow is initially high, then less, then
  less
• Thus slope of L is like F, but slope is change of
  L.
• In fact F is proportional to

10
Integration Area

• The reverse process is integration
• Graphical interpretation area under a graph.
• Consider the flow graph
• the area at different times is shown.
• After a short time, area is as shown. Later, area
  has grown, but by less, etc.
• Consider the height of water in the tank
• Thus L like Area under F
• L is prop to integral of F with time.

11
Integration Area

• In fact, for this system we have
• and
• F is differential of L and L is integral of F.
• Differentiation and integration are opposites.
• Note, here they are used to model a water
  systems.
• Can also model electronic circuits, mechanical
  systems, motors, etc.
• In fact, the differential equation has the same
  form and hence the same exponential response as
  that for many systems.
• Note also there are analogies between water
  systems and electronics
• pipe like a resistor, tank like a capacitor
• Also, for thermals, walls have thermal
  resistance, rooms have capacity.

12
R2-D2 Motor System

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

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 Kirchhoffs voltage
law to the armature circuit
13
Components of Motor System

• Combine components

R
_

14
Block Diagram of Motor System

• Reduce block diagram

1. 2. 3.
_

1/R
_

15
Transfer Function of System

• Output linked to input
• Can be expressed much more simply!
• Where
• Time Constant
• Gain

16
Time Response of System

• Any system of the form
• Has a time response (depending on input)

Output
Exponential
Time
17
Unit Step Response of System

• Has a time response to a unit step input

Output
Input
Output when K 1
Time
Output
Input
Output when K 1.6
18
Unit Step Response of System

• Has a time response to a unit step input

Output
Input
Output when K 1, T 0.01
Time
Output
Input
Output when K 1, T 0.02
19
Exercises

• Exercises
• Two tanks are connected by pipe.
• Heights of liquid are 0.1m and 0.02m
• Liquid density, ? 2 kg/m3, g 9.8 ms-2
• What is the pressure difference?
• If restrictance is 0.1 N s / m5, what is flow?
• Write down differential equation for system if
  the area of the second tank is 0.5 m2
• Write down equations showing variation of F and
  L.
• (In each case, use equations given in notes, and
  put in appropriate component values)

20
Antenna example
Source Nise 2004
21
Block diagram
Source Nise 2004