Dynamic Characteristics
Learning Outcomes
After this lecture you should be able to ….
Explain what we mean by the word ‘dynamic’
List the three different types of input to a system
Compare Zero, First and Second Order systems
Recognise the equation for a first order system
Give an example of a first order system
Write the equation for a first order system
Sketch the step response of a first order system
Explain what is meant by a time constant
Can you cook a Pizza?
Don’t forget the basics remove the plastic
Step 2 - Preheat oven to
200 °C
n Why?
n How long does this
take?
n Sketch how oven
temp changes in this
time
Dynamic
Dynamic means that something is changing (with time)
Dynamic only relates to the period of time when the oven was
heating up
We use differential equations to explain the behaviour of
something like oven temperature that is changing with time
Steady state = no change
Steady state = oven temperature is constant
At steady state, differential equations are not needed
The First Order Type System
During the dynamic period a first order differential equation is
sufficient to explain behaviour
Here’s an example
dHeight
τ
+ Height = K × Temp
dt
Solution to diff eqn
Where a step input is applied the solution to the above
equation is:
−t

Height = InitialHeight + ∆Height 1 − e τ 


Does this make sense? Enter some values:
n For a step input of ice to boiling water, we have:
n Initial height = 0
n Final height = 100
100
After 1 time constant
63.2
Height = 63.2
After 5 time constants
Height = 100
τ
Time
5τ
In General
The differential equation that describes the first order system is:
dθ o
τ
+ θ o = Kθ i
dt
This equation is easily solved using Laplace transforms. The solution looks like this:
θ o = Kθi (1 − e
−t
τ
)
Expand this equation and then think about it for a minute:
θ o = Kθi − Kθ i e
−t
τ
This term varies with time, large
initially, almost zero at t=5t
This term eventually dominates
when the other dies away
Don’t forget that τ is the time constant and has units of seconds.
Activity – Calculate exponential term
Think of the exponential term and, using a calculator, work out its magnitude at
different time values. Use the table and then plot the data below.
Time t =
exp(-t/t )
1 – exp(-t/t )
t
2t
3t
4t
5t
Time
Activity - Thermometer
Determine the time constant of a system
A thermometer was placed in beaker of water and left for a while to reach steady state.
A reading of 25ºC was obtained. The thermometer was then quickly placed in a beaker
of boiling water and the changing reading was recorded in the table below.
Determine the time constant
Time t =
Temp ºC
2
4
6
10
14
18
62
80
90
97
99
100
Time
Step Response
The typical response of the first order system to a step input is shown below:
? o,final
Output
? o,initial
Time
An alternative equation that relates the output to time is shown below (slightly different
to the last equation which had both the output and the input). The initial and final
values of the output must be known to use this equation but knowledge of the input is
not necessary.
θ o = θ o,initial + (θ o, final − θ o ,initial )(1 − e
−t
τ
)
Second Order System
A second order system is similar to the first order one in that it takes time for the output
to settle down to a new steady state value after a change is made to the input. The
difference is that with the second order system overshoot and undershoot are often
observed.
Underdamped
Output θo %
Critically damped
Overdamped
ω nt
The differential equation that describes the system is second order and has more
variables than the first order one. Hence, the variations in damping.
Different Types of Input
The static characteristics refer to the results when a constant input is applied. What
happens if the input is not constant but is changing? How does the instrument
respond? That depends on the dynamic characteristics.
For example, a standard thermometer is suitable for measuring the temperature in this
room. This changes slowly during the day and night, without sudden changes.
Compare this to the measurement of cylinder temperature in a combustion engine.
This change extremely suddenly and by a large amount. The output from the standard
thermometer to this type of input would be useless.
Different instruments handle changing inputs in different ways.
To compare instruments fairly, we should apply the same type of input to each and
measure the result. An easily repeatable changing input is required.
There are three standard types of changing inputs
Step Input
Ramp Input
Sinusoidal Input
Step Input
This is a an abrupt change from one steady input value to another. The response of the
system to it is called the transient response and is a measure of how well the system
can respond to sudden changes.
Think of the situation where a thermometer is suddenly moved from a beaker of ice and
water into a bath of boiling water.
Is this a step change in the input?
Describe what happens to the reading on the thermometer.
Does it take a long time to get to the new value?
Step Input
Input
Temp
Time
Ramp Input
The ramp input varies linearly with time and the ramp response of the system is
observed to give the steady state error between the output and the input.
For example a thermometer is placed in a bath of water and ice and a constant heat is
applied to the bath. The thermometer reading is recorded as the bath temperature is
ramped from 0 to 100degC.
Ramp Input
Input
Time
Sine Wave Input
The sine wave input is used to provide the frequency response of the system. It shows
how the system responds to inputs of cyclic nature at different frequencies.
How could this be implemented for the thermometer?
Is this easy or difficult?
What type of system is this most suited to?
The laboratory experiment with the LVDT uses a sine wave input.
Sine Wave Input
Input
Time
Classification of Systems
A measuring system can be characterised by examining its behaviour to each of the
three test inputs.
What has been found is that different systems can produce identical forms of response.
For example, the response of the thermometer to the step change in temperature might
have an identical pattern to a pressure sensor that is exposed to a step change in
pressure.
Measurement systems can be classified based on their response into one of three groups
•
Zero Order
•
First Order
•
Second Order
Each type of system has a different response to each of the three types of input we have
mentioned.
We will next look at these three groups.
Zero Order
The zero order system is one whose output is proportional to the input no matter how
the input varies. The equation that describes this behaviour is:
Output θo = k x Input θi
θo = kθi
θo is the output
θi is the input
k is a proportionality constant (= sensitivity)
Another way to describe a zero order system is that a new steady state output is
immediately provided when the input is changed.
Input
where
θi
k
Output
θo
Time
How would a zero order thermometer behave? Does one exist?
Rheostat/Potentiometer = zero order (input = movement, output = resistance)
First Order
Many systems take time to reach a new steady state value. The definition of Steady
State exists is that the output stops changing - Important definition!
If the behaviour when a step input is applied is such that the output responds quickly
and then slackens as it reaches the new steady state value the system is first order.
The term first order is used because the relationship between the output and the input for
these systems is described by first order differential equation:
dθ o
a
+ b θ o = cθ i
dt
where a, b and c are constants.
This equation is normally written as follows:
τ
dθ o
+ θ o = Kθ i
dt
τ=a/b and is the time constant in seconds
K=c/b and is the static sensitivity (units depending on application)
We will solve this equation later in the course.
where
First Order Step Input
The response of a first order system to a step input is as follows:
Input
100%
63.2%
Output
Time
τ
Time
5τ
A thermometer is an example of a first order system. When a step change is applied to
it, the output follows the curve above.
Test it. Time how long it takes the thermometer to reach 63degC and time how long it
takes to reach 100degC (should take five times longer).
Can you think of any other first order systems?