cotrol systems_final draft_zuni.docx

8/14/2019 Cotrol systems_Final Draft_zuni.docx

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Control Systems

AST304EE

Lecturer: Qing Lu

AST304EE Assignment 1

Student No s.1089378

19-04-2013


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BSc (HONS) Electrical & Electronic Engineering AST304EE Control Systems

s1089378 2

Contents

Summary: ...................................................................................................................................................... 3

Introduction: ................................................................................................................................................ 4

Subject Matter: ............................................................................................................................................ 5

Task 1: RC First order system .................................................................................................................... 5

Determining First Order Differential Equation:............................................................................... 5

Using Differential equation to find system transfer function: ....................................................... 8

Output response at 10V step input using transfer function:......................................................... 9

Multisim simulation and comparison with theoretical prediction:.............................................. 10

Task 2: Solving Differential Equation ................................................................................................ 13

Transfer function of a thermocouple system:............................................................................... 13

Matlab simulation with respect to Unit Ramp Input and Discussion:....................................... 15Mathcad simulation and comparison of results and predictions:.............................................. 17

Task 3: Second Order System: .......................................................................................................... 19

Determining values of k, , n in standard second order equation for following circuit :

............................................................................................................................................................ 19

Multisim simulation and comparison with theoretical results:.................................................... 22

Modification for critically damped system, testing and discussion of results: ......................... 24

Predicting step input response for critically damped system and comparison with test

results: (MULTISIM) V/S (MATLAB): ............................................................................................ 27Task 4: Close Loop Feedback System: ............................................................................................ 30

Overall Transfer Function for closed loop feedback system:.................................................... 30

Overall Transfer Function for Second Order System: ................................................................ 31

Comparison with standard second order equation and determining key parameters when

unit input is applied: ......................................................................................................................... 32

Confirming predictions with simulation: ........................................................................................ 34

Results: ...................................................................................................................................................... 38

Conclusion: ................................................................................................................................................ 39

Bibliography: ............................................................................................................................................. 40

Appendices: .............................................................................................................................................. 41

Appendix A: Laplase table: ...................................................................................................................... 41

Appendix B: Block diagrams .................................................................................................................... 42


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Summary:

This assignment analysis the real world problems and allows the author the freedom to

play with them. Use of powerful software like Matlab, Multisim has been made available

to students to back their predictions with suitable calculations or simulations. The

following report emphasizes how circuits are designed and analyzed in different angles.

And gives an experience to solve industrial problems where as being self-critical to you.


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Introduction:

This assignment has been into four tasks. Task 1 deals with basic first order RC

systems and analysis the transient responses with respect to different inputs. Graphical

expressions are used to define time constants and simulations are performed in

multisim to back the results.

Task 2 deals with a general thermocouple, where differential equations are found for the

system followed by Matlab analysis of the system. Transfer functions are produced and

simulations are used to plot the results.

Task 3 with typical second order system where a given system is compared to standard

second order function. Simulations are done to confirm the prediction using multisim.

Furthermore circuits are tweaked to perform as critically damped and graphical

illustrations are shown to prove the predictions and discuss the results.

Last section of the report looks on the closed loop feedback systems where two

systems transfer functions are found manually. Their response to step input is recorded

and compared to normal working using suitable simulation such as Simulink modeling.


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Subject Matter:

1. Task 1: RC First order system

Figure 1: Typical First order RC circuit

1.1. Determining First Order Differential Equation:

Using KirchhoffsVoltage Law (KVL): RISE = DROP

So: VS = VR+ VC (equation 1)

Now using Ohms law we know: VR= iR

And as q = cVCso: VC =

So: VS= iR + (equation 2)

As i =so we replace it in (equation 2)

VS = R+ (equation 3)Now divide the whole (equation 3) by R so it becomes:

+ q = (equation 4)

V1

10 V C1

1F

R1

1.0k

1 2

0

VOUTVs


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Integration factor of (equation 4) is:

Integration factor: Multiply the integrating factor to (equation 4)

+ = (equation 5)Reducing (equation 5):

(Equation 6)Now integrate both sides with respect to (t);

= where is integration constant (equation 7)Divide (equation 7) by so:

[

] Now: q = VsC +

At initial condition when (t) =0

q (0) =0= VsC + where so:

= - VsC hence equation becomes:

q= VsC (1- )This equation can be further solved to get current:


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As i =

The expression would be: i (t) = NOTE: As time constant ( ) =RC so after 5( ) the current will be =0 in the above

circuit which can be proved by solving the above equation hence the equation derived is

correct.


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1.2. Using Differential equation to find system transfer function:

Using KirchhoffsVoltage Law (KVL): RISE = DROP

So: Vin (t) = VR(t)+ Vo(t) (equation 1)

Now using Ohms law we know: VR(t)= i(t)R

So: Vin(t) = i(t)R + Vo(t) (equation 2)

As i = and q = CVo,then i(t) = C so we replace it in (equation 2)

Vin(t) = + Vo(t) (equation 3)Hence transfer function will be:

(Using Laplace transform for step function)

Vin(s)= CR(s)Vo(s) + Vo(s)

Vin(s)= Vo(s)[CR(s)+1]

= (Transfer funct ion of the system)As time constant ( ) =RC hence the equation will become:

= (Final transfer Func tion )


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1.3. Output response at 10V step input using transfer function:

Step Input: 10V

So Vin(s)=

Using the expression defined i.e. = so:

Matlab and multisim simulation are shown below to confirm the results.


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1.4. Multisim simulation and comparison with theoretical prediction:

Figure 1 as shown above was used to analyses the first order RC circuit and itsproperties.

Figure 2: transient analyses of RC circuit with 10V step input

Figure 2 above shows the transient response of a typical RC circuit to a step input

where the V(1) shows the step input and V(2) describes the transient response due to

capacitive circuit characteristics.

V1

10 V C1

1F

R1

1.0k

1 2

0


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As defined that ( ) =time constant and after 5 the circuit value = 0 or constant to the

step voltage. Figure below shows the simulation of predicted results.

Figure 3: transient nalysis at 63% of V(1)

As we know that ( ) =time constant =63% of input value hence it can be seen that

=1ms in this case and figure below shows that after five consecutive time constants i.e.

= 5ms the value becomes 10V hence the theoretical prediction is proved through

simulation.


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Figure 4: V(2) after five time constants


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2. Task 2: Solving Differential Equation

2.1. Transfer function of a thermocouple system:

Differential equation of the system: = k ( TmT1)Where k = constant, k =

=

, Time constantUsing Laplace transforms table the transfer function of the above defined differential

equation would be:

= k (equation 1) = =

Taking as common so: (equation 2)Dividing both numerator and denominator by k so (equation 2) becomes:

(equation 3)


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2.2. Matlab simulation with respect to Unit Ramp Input and Discussion:

Simulink model drawn in Matlab is shown below followed by graphical illustration

showing the results:

Figure 5: Mat lab Simulink model for the transfer function with unit ramp input

This Simulink model was designed in Matlab where the input was a unit ramp and the

output result was simulated through scope shown in the graph below.

The green line in the graph below shows the normal response of the circuit with a unit

ramp input excluding any transfer function. When a transfer function is added to the

existing system the graph shape changes due to the transient nature of the circuit and it

follows the pink curved line which is typical for second order systems. Hence the

simulation is similar to the prediction thus justified.


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Figure 6: graph showing variation of in unit ramp manner


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2.3. Mathcad simulation and comparison of results and predictions:

(Final Transfer Function)

For unit ramp Laplace transform is: (using laplase transformation table)So: Now: putting this equation into Mathcad the inverse Laplace can be found which comes

out to be:

(T1 normal function in time domain)

Figure below shows the graph drawn from

which is represented by y(t) against time

(t).

1

s 1( ) s2

invlaplace t e t

1

y t( ) t e t

1


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0 2 4 6 8 100

2

4

6

8

10

y t( )

t

Figure 7: graph of y(t) vs. t in Mathcad for time domain function

When compared, the simulation results in the last section and the current graph is

exactly the same which justifies the results. The exponential rise of both the graphs is

the prove for the right solution to the problem.


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3. Task 3: Second Order System:

3.1. Determining values of k,,nin standard second order

equation for following circuit :

Figure 8: standard RLC circuit

Standard second order equation:

Where;

Kis system gain,

is system damping factor,

is the undamped natural frequency.AndYo, Xoare output and input signals.

Applying KirchhoffsVoltage Law (KVL) on the above circuit:


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(equation1)Now:

Note:

Put the values in (equation 1),it becomes:

(equation 2)Divide (equation 2) by LC:

(equation 3)Using Laplace Transform (equation 3) becomes: Taking as common equation becomes:

[ ]

Final transfer function comes out to be:

(equation 4)


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3.2. Multisim simulation and comparison with theoretical results:

Figure 9: Multisim assembly of RLC circuit with components values

Figure above shows the wiring diagram assembled in Multisim with values associated

with each and every component to analyses a suitable simulation.it is predicted thatwhen the switch (S1) is turned ON, the capacitor will start charging until it reaches the

max value of its charging capacity which here is 10V as given through (V1).as the

nature of the circuit is capacitive so it will observe a transient response until it reaches

the same value as step input which should be equal to five time constants 5( ).

Figure below shows the results of the simulation in Multisim Transient Analysis.it can be

observed that the channel B which the voltage through capacitor rises when switch is

turned and capacitor charges charging. Maintaining transient response it reaches thevalue of 10V and follows the same unit step voltage as it cannot be charged more.

Hence prediction is proved and simulation analysis proves the results expected.

V1

10 V

C1

1mF

L1

100mH

R1

1

1

XSC1

A B

Ext Trig+

+

_

_ + _

S1

Key = A

4

2

R2

1.0k

0

3


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Figure 10 : transient analysis of RLC circuit shown above.


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3.3. Modification for critically damped system, testing and discussion of

results:

The equation proved earlier is concluded below :

k = gainwhich is = 1 in this case as found by comparing the two equations.

Numerical value can be found using specific values for L,C as calculatedfurther in the report.

Now for critically damped system the value for which can be used to calculatethe resistance (R) value which can be used for simulation purposes to prove thetheoretical prediction of results.

So using:

(equation 1)

(for critical damping)

Replacing values of and in (equation 1) (equation 2)Put values of L,C in (equation 2) we get:

R = 60 would be used in the modified multisim circuit to analyse critical damping.thefigures below the modified assemble and the simulation results for critically damped

circuit.


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Figure11: Multisim circuit for critical damping using R = 60

Figure 12: Critically Damped Circuit Simulation

V1

10 V

C1

1mF

L1

100mH

R1

60

1

XSC1

A B

Ext Trig

+

+

_

_ + _

S1

Key = A

4

2

R2

1.0k

0

3


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As predicted through theoretical calculation that when we use R = 60 the simulationshows a critically damped behavior which means that the component which is capacitor

in this case takes minimum possible time to reach the maximum voltage value i.e. 10V.

The switch is turned ON at T(s) = 200 ms and it takes capacitor 200 ms to reach the

peak value and it continues with the step voltage as it becomes fully charged and

cannot be charged more.


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3.4. Predicting step input response for critically damped system and

comparison with test results: (MULTISIM) V/S (MATLAB):

Multisim Analysis:

Figure 13: RLC circuit wiring diagram using step input

The figure above was used to analyses circuit response to step input for a critically

damped circuit. For this the function of Device Parameter Sweep was used in multisim

simulation software to analyses the response of the circuit to a step input with respect to

different values of resistors. Results are illustrated in the figure below.

C1

1F

L1

10mH

R1

40

1

XSC1

A B

Ext Trig+

+

_

_ + _

S1

Key = A

4

V2

1ms

2

0

3


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Mathcad Analysis:

I s( ) 1

s

O s( ) H s( ) I s( ) simplify

1

s s 4( )2

O s( ) parfrac 1

16 s

1

16 s 4( )

1

4 s 4( )2

O s( ) invlaplace 1

16

t e 4 t

4

e 4 t

16

O t( ) 1

16

t e 4 t

4

e

O1 t( )1

16

O2 t( ) t

exp 4 t( )

4

O3 t( )1

16exp 4 t( )

0 0.375 0.75 1.125 1.5 1.875 2.25 2.625 3

0.013

0.025

0.038

0.05

0.063

0.075

0.088

0.1

O t( )

O1 t( )

O2 t( )

O3 t( )

t


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4. Task 4: Close Loop Feedback System:

4.1. Overall Transfer Function for closed loop feedback system:

Figure11: closed loop negative feedback system

Overall transfer function would be:


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4.2. Overall Transfer Function for Second Order System:

Figure 12: second order system

Transfer function for the system shown in figure above is calculated as follows:

Taking common from (equation 2) we get: Now substitute the values of in (equation 3): [

] [

]

Putting into main equation we get:


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of numerator deducts in the denominator so the overall function for thesystem would be:

4.3. Comparison with standard second order equation and determining key

parameters when unit input is applied:

Standard second order equation:

Overall Transfer function:

Comparing the two equations the key parameters found are as follows:

Now we get:

From (Equation 4) we get:


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Replacing value of in (Equation 2) we get:

So Replacing value of in (Equation 3) we get:


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4.4. Confirming predictions with simulation:

Figure13: Simulink model with unit step input

The figure above shows the simulink model designed for transfer function analysis.the

scope results are shown in the graph below which shows an initial rise in the voltage as

the circuit charghes and the value keeps on decreasing following a transient path

showing the damping in the circuit.


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Figure: scope results of simulink model

In the following set of illustrations a multiplex is used to compare the response of the

circuit with normal step input vs step input through the transfer function.figure below

shows the simulink model followed by its graphical analysis where the pinl line shows

the normal step input and the blue curve is the response of the circuit transfer function

to step input.

Figure14: Simulink model with unit step using multiplex


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Figure15: Simulation with unit step and transfer function using multiplex

In this last set of analysis the overall transfer function calculated manually in part 2 wasused to analyses the response of the circuit. The graph below confirms the predictions

as the graphical illustration is exactly the same as shown in the last graphical analysis.

Figure16: Overall transfer function


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Figure17: Simulation of overall transfer function


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5. Results:

In this assignment author looked over the first order systems followed by its response

which was achieved by simulations using Multisim software. Second order systems

were analyzed and compared with theoretical predictions with the help of powerful

mathematical sofwares such as Matlab and Mathcad. Thermocouple system and closed

loop feedback system were analyzed lastly, first with manual calculations and

predictions were counter checked building Simulink models and circuits were made to

perform different functions to support the findings in order to accomplish results basis

on predictions.


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6. Conclusion:

The assignment was a healthy exercise to get used to the electrical, mathematical and

simulation softwares and apply them to general applications. Building circuits in multisim

and using them to do different things while comparing them with theoretical predictions

was a tough task. Mathcad and Matlab were used to perform all the calculations and

graphical analysis were done to back every prediction which gave an idea to the author

regarding real world problems and countering strategies.

Real world problems were assigned as a task for this assignment which was interesting

to solve and self-analysis of predictions helped to justify the finding was quite an

amazing experience.


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8. Appendices:

8.1. Appendix A: Laplase table:


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cotrol systems_final draft_zuni.docx

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