BTEC HNC - Control Systems and Automation - Equations to Determine System Parameters

8/2/2019 BTEC HNC - Control Systems and Automation - Equations to Determine System Parameters

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Use Bode Standard OrderEquations to Determine

System ParametersControl Systems and AutomationBy Brendan Burr



Brendan Burr BTEC Higher National Certificate in ElectronicsControl Systems and Automation

Table of Contents

TABLE OF CONTENTS ........................................................... 2

TASK 1 ................................................................................ 6

1.1 Describe the following parameters that can be obtainedfrom Bode standard second order equations in the s domain.Identify the appropriate symbol in each case. ....................... 6

Damping ratio. .................................................................... 6Solution:- .........................................................................................6

Undamped natural frequency. .............................................. 7

Solution:- .........................................................................................7

Damped natural frequency. ................................................. 7Solution:- .........................................................................................7

The response time in terms of the following:- ....................... 8

(i) Rise time ........................................................................ 8Solution:- .........................................................................................8

(ii) Peak time ...................................................................... 8

Solution:- .........................................................................................8

(iii) Overshoot and percentage overshoot. ............................ 9Solution:- .........................................................................................9

(iv) Settling time ................................................................. 9Solution:- .........................................................................................9

1.2 A second order system has the following transfer functionin the s domain: ................................................................ 11

Determine the following parameters for this system:- ........ 11

(a) The undamped natural frequency in rads /sec. .............. 11Solution:- .......................................................................................11

....................................................................................... 11

(b) The damping ratio. ....................................................... 12Solution:- .......................................................................................12

(c) The damped natural frequency in rads/sec. .................... 13

Solution:- .......................................................................................13

TASK 2 .............................................................................. 14

2




2.1 A second order system has an undamped naturalfrequency of 10 rads/sec and a damped natural frequency of 9 rads/sec. .......................................................................... 14

Determine the following parameters for this system: .......... 14

(a) The damping ratio. ....................................................... 14Solution:- .......................................................................................14

(b) The equation for the transfer function in terms of the sdomain. ............................................................................ 15

Solution:- .......................................................................................15

(c) The 100% rise time. ...................................................... 15Solution:- .......................................................................................15

(d) The percentage maximum overshoot. ............................ 16Solution:- .......................................................................................16

(e) The 2 % settling time .................................................... 16Solution:- .......................................................................................16

(f) The peak time. .............................................................. 16Solution:- .......................................................................................16

2.2 A second order system has an overshoot of 20% and a risetime of 0.5 sec when subject to a step input. Determine the

following parameters for this system:- ............................... 17

(a) The damping ratio. ....................................................... 17Solution:- .......................................................................................17

(b) The damped natural angular frequency. ........................ 18Solution:- .......................................................................................18

(c) The undamped natural angular frequency. ..................... 18Solution:- .......................................................................................18

(d) The number of oscillations for a 2% settling time. .......... 18Solution:- .......................................................................................18

TASK 3 .............................................................................. 19

3.1 Construct the following series R-L-C circuit using the S/Wpackage Croc Clips or Electronic Workbench ....................... 19

3.2 Apply a step input voltage of 1 V to Vi by connecting aswitch in series with a 1V DC supply. .................................. 19

Solution:- .......................................................................................19

3.3 Connect an Oscilloscope probe to monitor the outputvoltage Vo. [You will also need to connect a switch across the

3




capacitor in order to discharge the Capacitor betweenreadings.] ......................................................................... 19

Solution:- .......................................................................................19

3.4 Connect a switch in parallel across R. Close the switch to

set R = 0 Ω........................................................................ 19Solution:- .......................................................................................19

3.5 Using the component values and with R = 0 Ω determineand tabulate a set of analytical results for the followingparameters:- ..................................................................... 20

(a) The Undamped natural frequency in Rads/sec. ............... 20Solution:- .......................................................................................20

(b) The Undamped natural frequency in Hz. ........................ 20Solution:- .......................................................................................20

(c) The Undamped periodic time T in sec. ............................ 20Solution:- .......................................................................................20

3.6 Run the simulation. ...................................................... 21Solution:- .......................................................................................21

3.7 From the simulation determine and tabulate a set of measured results for the following parameters:- ................. 21

(a) The Undamped natural frequency in Rads/sec. ............... 21Solution:- .......................................................................................21

(b) The Undamped natural frequency in Hz. ........................ 21Solution:- .......................................................................................21

(c) The Undamped periodic time T in sec. ............................ 21Solution:- .......................................................................................21

3.8 Compare the analytical results obtained in 3.5 with themeasured results from the practical simulation in 3.7 ......... 23

Solution:- .......................................................................................23

3.9 Using relevant theory and techniques produce a table of analytical results for the following parameters. Use thevalues of the damping ratio given and suggest you use anExcel spreadsheet to enter the appropriate formulae. ......... 24

Solution:- .......................................................................................24

4.3 For each simulation produce a printout of the waveformresponse for Vo. ................................................................ 26

(0.1) ...............................................................................................26

(0.2) ...............................................................................................27(0.3) ...............................................................................................28(0.4) ...............................................................................................29

4




(0.5) ...............................................................................................30(0.6) ...............................................................................................31(0.8) ...............................................................................................32(1.0) ...............................................................................................33(1.6) ...............................................................................................34

4.5 Produce a printout of the complete simulation circuitused. ................................................................................ 35

CONCLUSION ..................................................................... 36

Books ............................................................................... 37

Catalogues ........................................................................ 37

Websites ........................................................................... 37

5




Task 1

1.1 Describe the following parameters that can be obtained from Bodestandard second order equations in the s domain. Identify theappropriate symbol in each case.

Damping ratio.

Solution:-

As you can see from the graph above, the greater the damping ratio,the less steep the increase of y(t) is. Any damping ratio greater than 1results in an over damped waveform, and will never reach the value of 1 (on the y axis). This is exampled by zeta = 2 above.If the damping ratio is equal to 1 then this is known as criticallydamped, i.e the waveform is non oscillatory and actually exponential.This is exampled by zeta = 1 above.

A damping ratio less than 1 makes the waveform oscillatory and itovershoots 1 (on the y axis), this is known as under damped. You cansee this on the above graph where zeta = 0.7, 0.4, 0.2 and 0.1. Theoutput is clearly decaying with its peak losing amplitude with eachoscillation.When the waveform is oscillatory without any form of decay, thewaveform is called undamped. This is shown above where zeta = 0,and the waveform does not decay over multiple cycles.The formula to work out the damping ratio is as follows:

L

C R

2=

ζ

6




Undamped natural frequency.

Solution:-

The undamped natural frequency of a waveform is where there is nodecay in the amplitude of the waveform over time. For this to happenin an R-L-C circuit, the value of the resistor must equal 0 Ohms.This effectively makes the circuit simply an LC circuit with just theInductor and Capacitor, and creates a resonance effect which in theorydoes not decay.

The formula to work out the undamped natural frequency is as follows:

LC

1=

The symbol used for undamped natural frequency is nω and is

measured in Radians/Sec.

Damped natural frequency.

Solution:-

The damped natural frequency occurs when the resistor in an R-L-Ccircuit has a value greater than 0. As I have explained above, if the

damping ratio is greater than 1 then the waveform will be over damped,if under 1 then it will be under damped, if the damping ratio calculatesto be equal to 1 then the waveform will be critically damped.You should expect to see a waveform with an overshoot, if thedamping ratio is less than 1, likewise you would expect to see noovershoot if the damping ratio is greater than or equal to 1.The formula required to work out the damped natural frequency is asfollows:

21 ζ ω −= n

The symbol used for damped natural frequency is d ω and is measured

in Radians/Sec.

7

http://upload.wikimedia.org/wikipedia/commons/7/70/Lc_circuit.svg

http://upload.wikimedia.org/wikipedia/commons/4/4e/RLC_series_circuit.png




The response time in terms of the following:-

(i) Rise time

Solution:-

Rise time is the amount time it takes for the response output to reachthe steady state line. This is the first point at which the responseoutput reaches the steady state line, so even if under damping occursthe rise time can still be calculated.The formula to calculate the rise time is as follows:

d ω

π

2=

It requires the damped natural frequency which can be determined asshown above.If the system is over damped then the response output will never reachthe steady state line and therefore the rise time will be immeasurable.

The symbol used for rise time is r t and is measured in Seconds.

(ii) Peak time

Solution:-

The peak time is the time it takes for the response output to rise from 0to the first peak point, which is also the time it takes the response

output to reach half a cycle. Therefore pd t ω π = so to work out the

peak time the equation would be:

d ω

π =

The symbol used for peak time is pt and is measured in Seconds.

8




(iii) Overshoot and percentage overshoot.

Solution:-

Overshoot is the maximum amount by which the output responseovershoots the steady state line. It is calculated at the first peak point,

i.e when pd t ω π = .

The overshoot can be calculated by using the formula:

21 ζ ω

π ζω

θ −

−

×=n

n

e ss

To calculate the percentage overshoot the formula is the same butsimply:

%10021××=

−

−

ζ

ζπ

θ e ss

Percentage overshoot does not have a symbol to express it, however itis measured as a percentage of the steady state value so should havea percentage sign at the end.

(iv) Settling time

Solution:-

The settling time is used to determine how long it takes for theoscillations to die away and for the output response to reach acontinuous steady state. The continuous steady state is accomplishedwhen the waveform falls within a percentage of the steady state line.To work out the settling time for 2% you would use the followingformula:

snt ζω −=02.0ln

912023005.302.0ln −=

n

st

ζω

912023005.3= However we use the formula

n

st ζω

4= for ease.

To work out the settling time for 5% you would use the followingformula:

snt ζω −=05.0ln

995732274.205.0ln −=n

st

ζω

995732274.2= However we use the formula

n

st ζω

3= for ease.

9




The symbol used for settling time is st and is measured in Seconds.

Using this information you could then go back and work out the number of overshoots there would be before the output response reaches thesettling time boundaries.

time Periodic

timeSettling nsoscillatioof Number

_

_ _ _ =

n

n

n

d

d

n

nsoscillatioof Number

nsoscillatioof Number

πζω

ζ ω

πζω

ω

ω π

ζω

212 _ _

2

2

4

_ _

−=

==

112

_ _ 2−=

ζ π nsoscillatioof Number

10




1.2 A second order system has the following transfer function in the sdomain:

Determine the following parameters for this system:-

(a) The undamped natural frequency in rads /sec.

Solution:-

22

2

2nn

n

i

o

s s ω ζω

ω

θ

θ

++

=

If:

165

162 ++

= s si

o

θ

θ

Then:

16

162

=

=

n

n

ω

ω

4=n

ω Rads/sec

11

165

162 +=θ

θ

s

o




(b) The damping ratio.

Solution:-

So:22

2nn

s s ω ζω ++

If:

1652 ++ s s

Then:

8

5

58

542

52

52

=

=

=

=

=

ζ

ζ

ζ

ζω

ζω

n

ns s

625.0=ζ

12




(c) The damped natural frequency in rads/sec.

Solution:-

So:4=

nω

Then:

122498999.3

625.014 2

=

−=

d

d

ω

ω

12.3=d ω Rads/Sec

13




Task 2

2.1 A second order system has an undamped natural frequency of 10 rads/sec and a damped natural frequency of 9 rads/sec.

Determine the following parameters for this system:

(a) The damping ratio.

Solution:-

21 ζ ω ω −= nd

ζ

ζ

ζ

ζ

ζ

ζ

=

=

−=

−=

−=

−=

19.0

19.0

181.0

19.0

110

9

1109

2

2

22

2

2

10

19=ζ or 4358898944.0=ζ

14




(d) The percentage maximum overshoot.

Solution:-

%10021×=

−

−

ζ

ζπ

e

%83747257.21

100

100

19.01

19.0

19.01

19.0

2

=

×=

×=

−

−

−

−

π

π

e

e

%84.21=

(e) The 2 % settling time

Solution:-

n

st ζω

4=

9176629355.0

19

194

1019.0

4

=

=

×

=

s

s

s

t

t

t

92.0= st Seconds

(f) The peak time.

Solution:-

d

pt ω

π =

3490658504.0

9

=

=

p

p

t

t π

35.0= pt Seconds

16




2.2 A second order system has an overshoot of 20% and a rise time of 0.5 sec when subject to a step input . Determine the following parameters for this system:-

(a) The damping ratio.

Solution:-

%100%21×=

−

−

ζ

ζπ

eOvershoot

2

2

1

1

2.0

%100%20

ζ ζπ

ζ

ζπ

−−

−−

=

×=

e

e

Natural Log both sides.Giving:

( )

( )2.01

2.01

2

2

LN

LN

×−−=

−=−

ζ ζπ

ζ

ζπ

Square both sides:

( ) ( ) ( )( ) 2222.01 LN ×−= ζ ζπ

Multiply out the brackets:

[ ] ( )[ ] ( )[ ]

[ ] ( )[ ] ( )[ ]

( )[ ] ( )[ ][ ] ( )[ ][ ] ( )[ ]

( )[ ]

[ ] ( )[ ][ ]( )[ ]

[ ] ( )[ ][ ]4559498108.0

2.0

2.0

2.0

2.0

2.02.02.02.0

2.02.0

2.02.0

22

2

22

2

2

2222

2222

2222

2222

=

+=

+=

=+ =+×

=+

−=

ζ

π ζ

π ζ

π ζ ζ π ζ

ζ ζπ

ζ ζπ

LN

LN

LN

LN

LN LN LN LN

LN LN

LN LN

46.0=ζ

17




(b) The damped natural angular frequency .

Solution:-

d

r t ω

π

2=

d

d

d

ω π

ω π

ω

π

22

25.0

25.0

=

=

=

d ω π = Rads/sec

(c) The undamped natural angular frequency .

Solution:-

21 ζ ω ω −= nd

529857617.3

46.01

46.01

2

2

=

=−

−=

n

n

n

ω

ω π

ω π

53.3=nω Rads/sec

(d) The number of oscillations for a 2% settling time.

Solution:-

1122−=

ζ π

242669869.1

146.0

12

2

=

−=

π

24.1= Oscillations

18




Task 3

3.1 Construct the following series R-L-C circuit using the S/W package Croc Clips or Electronic Workbench

R = variable Ω, L = 1 mH, C = 1 µF.

3.2 Apply a step input voltage of 1 V to Vi by connecting a switch in serieswith a 1V DC supply .

Solution:-

Complete

3.3 Connect an Oscilloscope probe to monitor the output voltage Vo. [Youwill also need to connect a switch across the capacitor in order todischarge the Capacitor between readings.]

Solution:-

Complete

3.4 Connect a switch in parallel across R. Close the switch to set R = 0 Ω

Solution:-

Complete

19




3.5 Using the component values and with R = 0 Ω determine and tabulate aset of analytical results for the following parameters:-

(a) The Undamped natural frequency in Rads/sec .

Solution:-

LC n

1=ω

( ) ( )

9

63

63

10

1

1010

1

101101

1

−

−−

−−

=

×=

×××=

n

n

n

ω

ω

ω

7766.31622=nω Rads/sec

(b) The Undamped natural frequency in Hz .

Solution:-

π

ω

2

nn f =

π 2

7766.31622=n f

92121.5032=n f Hz

(c) The Undamped periodic time T in sec.

Solution:-

n

T f

P 1

=

92121.5032

1=T P

6106917653.198−

×=T P Sec

20




3.6 Run the simulation.

Solution:-

Complete

3.7 From the simulation determine and tabulate a set of measured resultsfor the following parameters:-

(a) The Undamped natural frequency in Rads/sec .

Solution:-

92654.31415=nω

ω 000,10=n Rads/sec

(b) The Undamped natural frequency in Hz .

Solution:-

5000=n f Hz

(c) The Undamped periodic time T in sec .

Solution:-

610200 −

×=T

P Sec

21




22




3.8 Compare the analytical results obtained in 3.5 with the measured results from the practical simulation in 3.7

Solution:-

For the analytical result for a) I got 7766.31622=nω rads/sec. This is

compared to the π ω 000,10=n rads/sec result which was measured.

The difference in this is actually quite small considering the possibleimperfections of manually measuring the graphs.

I also managed to get the undamped natural frequency measured andcalculated reasonably close to one another. The undamped natural

frequency which I measured was 92121.5032=n f Hz compared to

5000=n f Hz, which was calculated. The difference is down to anapproximation from the graph, a greater level of accuracy could havebeen obtained if I was able to zoom in, however crop clips doesn’tallow this.

The frequency was calculated using the Periodic Time, by taking thereciprocal value of it. This meant that any discrepancies in the value of the Periodic Time also ran through to undamped natural frequency.

The measured periodic time was6106917653.198 −×=T P Sec, compared

to the calculated periodic time of 610200 −×=T P Sec.

Overall the calculated and the measured simulation results are verysimilar, enforcing the chance of it being correct and accurate.

23



3.9 Using relevant theory and techniques produce a table of analytical results for the following parameters. Use the valuesof the damping ratio given and suggest you use an Excel spreadsheet to enter the appropriate formulae.

Solution:-

4.4 Measured Results

Damping Ratio (y) 0.10 0.20 0.30 0.40 0.50 0.60 0.80 1.00 1.60

Value of R 6.32 12.65 18.97 25.30 31.62 37.95 50.60 63.25 101.19

Damped naturalfrequency (Wd) 31415.93 31415.93 31415.93 27925.27 27925.27 27925.27 27925.27 0.00 No Cycle

Damped naturalfrequency (f) 5000.00 5000.00 5000.00 4444.44 4444.44 4444.44 4444.44 0.00 No Cycle

Peak time 1.00E-04 1.00E-04 1.00E-04 1.00E-04 1.25E-04 1.25E-04 1.50E-04 Exponential Exponential

% Overshoot 75.00% 50.00% 35.00% 25.00% 18.00% 10.00% 1.00% 0.00% 0.00%

3.9 Analytical Results

Damping Ratio (y) 0.10 0.20 0.30 0.40 0.50 0.60 0.80 1.00 1.60

Value of R 6.32 12.65 18.97 25.30 31.62 37.95 50.60 63.25 101.19

Damped naturalfrequency (Wd) 31464.27 30983.87 30166.21

28982.75

27386.13 25298.22 18973.67 0.00 #NUM!

Damped naturalfrequency (f) 5007.69 4931.24 4801.10 4612.75 4358.64 4026.34 3019.75 0.00 #NUM!

Peak time 9.98E-05 1.01E-04 1.04E-04 1.08E-04 1.15E-04 1.24E-04 1.66E-04 #DIV/0! #NUM!

% Overshoot 72.92% 52.66% 37.23% 25.38% 16.30% 9.48% 1.52% #DIV/0! #NUM!

2% Settling time 1.265E-03 6.325E-04 4.216E-04 3.162E-04 2.530E-04 2.108E-04 1.581E-04 1.265E-04 7.906E-05

Peak voltage of Vo 1.72925 1.52662 1.37233 1.25383 1.16303 1.09478 1.01516 #DIV/0! #NUM!

Number of overshoots 6.33 3.12 2.02 1.46 1.10 0.85 0.48 0.00 #NUM!



ndan Burr BTEC Higher National Certificate in ElectronicsControl Systems and Automation

2% Settling time 1.100E-03 7.000E-04 5.500E-04 3.500E-04 2.500E-04 2.250E-04 1.750E-04 1.750E-04 3.500E-04

Peak voltage of Vo 1.75 1.50 1.35 1.25 1.18 1.10 1.10 1.00 1.00

Number of overshoots 6.00 3.00 2.00 1.00 1.00 1.00 0.00 0.00 0.00

25



4.3 For each simulation produce a printout of the waveform response for Vo.

(0.1)



(0.2)



(0.3)




(0.4)

29




(0.5)

30




(0.6)

31




(0.8)

32




(1.0)

33




(1.6)

34




4.5 Produce a printout of the complete simulation circuit used.

35




Evaluation

I began this assignment by using my notes from class to help me understandthe basic principles of Bode standard second order equations. Once I haddone this I was able to use the equations given to me in the assignment tocalculate the various parameters of the systems in the questions.I gained my underlying knowledge by answering Task 1 using notes and doinga bit of research in the Control Engineering Book by W. Bolton. I was able tosee the mathematical routes taken to derive the formula and thereforeunderstand its relationship to the waveform plotted on the graph. For example the rise time uses a direct relationship between the x and y axis andthe explanations I had previously received allowed me to visualise this for myself.For Task 1.2, I performed some basic equations where I simply had to extractinformation from the question and then put that information into a predefined

formula which then gave me the answer I required.Task 2.1 and 2.2 were similar in principle as they required me to extractinformation from the question and use the correct formula to calculate theanswer.Task 3 I managed to complete in class when we originally went through it. Imanaged to create an excel spreadsheet with all of the necessary formula,which meant all I had to do was determine the measured values. This didn’ttake too long, however it was very repetitive, as there were numerous resistor values to compare.Overall I managed to complete the assignment without finding it too taxing.

Conclusion

I had a problem with my assignment as the word file became corrupt after Ihad put the excel spreadsheet copy in for Tasks 3.8 and 4.4. This hashappened to me before in a previous year and it involves redoing the entireassignment as there is no solution to it. I tried to work from a back-up copy Imade last month, but it would seem that I hadn’t completed much of Tasks 1and 2, so a lot of work had to be retyped. In future I think I will make a backupcopy every time I successfully finish a task, and then delete the copies once Iknow the final copy is safe. It was especially annoying because of the timeconsuming process of typing up the formula using Microsoft Equation Editor 3.0.

36




Bibliography

Through guidance from my lecturer, the following text books, catalogues andwebsites I was able to complete this assignment:

Books

Control Engineering (W.Bolton)ISBN: 0-582-32773-3

Catalogues

N/A

Websites

1. http://wapedia.mobi/thumb/3ad1500/en/fixed/470/300/2nd_Order_Damping_Ratios.svg?format=jpg

2. http://en.wikipedia.org/wiki/File:RLC_series_circuit.png3. http://en.wikipedia.org/wiki/File:Lc_circuit.svg

http://wapedia.mobi/thumb/3ad1500/en/fixed/470/300/2nd_Order_Damping_Ratios.svg?format=jpg


http://en.wikipedia.org/wiki/File:RLC_series_circuit.png

http://en.wikipedia.org/wiki/File:Lc_circuit.svg



http://en.wikipedia.org/wiki/File:RLC_series_circuit.png

http://en.wikipedia.org/wiki/File:Lc_circuit.svg

BTEC HNC - Control Systems and Automation - Equations to Determine System Parameters

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Transcript of BTEC HNC - Control Systems and Automation - Equations to Determine System Parameters