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TRANSCRIPT
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Control Systems
AST304EE
Lecturer: Qing Lu
AST304EE Assignment 1
Student No s.1089378
19-04-2013
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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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