Dynamic Characteristics

Learning OutcomesAfter this lecture you should be able to ….

Explain what we mean by the word ‘dynamic’List the three different types of input to a systemCompare Zero, First and Second Order systemsRecognise the equation for a first order systemGive an example of a first order systemWrite the equation for a first order systemSketch the step response of a first order systemExplain what is meant by a time constant

Can you cook a Pizza?Don’t forget the basics -remove the plasticStep 2 - Preheat oven to 200 °C n Why?n How long does this

take?n Sketch how oven

temp changes in this time

DynamicDynamic means that something is changing (with time)Dynamic only relates to the period of time when the oven was heating upWe use differential equations to explain the behaviour of something like oven temperature that is changing with timeSteady state = no changeSteady state = oven temperature is constantAt steady state, differential equations are not needed

The First Order Type SystemDuring the dynamic period a first order differential equation issufficient to explain behaviourHere’s an example

TempKHeightdt

dHeight×=+τ

Solution to diff eqnWhere a step input is applied the solution to the above equation is:

Does this make sense? Enter some values:n For a step input of ice to boiling water, we have:

n Initial height = 0n Final height = 100

After 1 time constantHeight = 63.2After 5 time constantsHeight = 100

−∆+=

−τ

teHeightghtInitialHeiHeight 1

63.2

100

τ 5τTime

The differential equation that describes the first order system is:

This equation is easily solved using Laplace transforms. The solution looks like this:

ioo K

dtd

θθθ

τ =+

Don’t forget that τ is the time constant and has units of seconds.

)1( τθθt

io eK−

−=

Expand this equation and then think about it for a minute:

τθθθt

iio eKK−

−=This term varies with time, large initially, almost zero at t=5t

This term eventually dominates when the other dies away

In General

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 – Calculate exponential term

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

2 624 806 9010 9714 9918 100

Time

Activity - Thermometer

The typical response of the first order system to a step input is shown below:

)1)(( ,,,τθθθθ

t

initialofinaloinitialoo e−

−−+=

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.

Time

Output

?o,initial

?o,final

Step Response

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.

ωnt

Underdamped

Output θo %

Critically damped

Overdamped

The differential equation that describes the system is second order and has more variables than the first order one. Hence, the variations in damping.

Second Order System

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

Different Types of 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?

InputTemp

Time

Step Input

Step 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.

Time

Ramp Input

Input

Ramp 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.

Time

Sine Wave Input

Input

Sine Wave Input

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.

Classification of Systems

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 θiθo = kθi

where θo is the outputθi is the inputk 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.

How would a zero order thermometer behave? Does one exist?Rheostat/Potentiometer = zero order (input = movement, output = resistance)

Time

Input

Outputθi θok

Zero 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:

ioo cb

dtd

a θθθ

=+

where a, b and c are constants.

This equation is normally written as follows:

ioo K

dtd

θθθ

τ =+

where τ=a/b and is the time constant in secondsK=c/b and is the static sensitivity (units depending on application)

We will solve this equation later in the course.

First Order

The response of a first order system to a step input is as follows:

Time

Input

Output

63.2%

100%

τ 5τTime

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?

First Order Step Input