A LEAD COMPENSATOR CHARACTERISTICS USING BODE DIAGRAM FOR MAXIMUM OF 50 DEG PHASE ANGLE
THIS PPT IS SO USEFUL FOR THE ENGINEERING STUDENTS FOR CONTROL SYSTEMS STUDENTS AND THIS PPT ALSO CONTAINS A MATLAB CODING FOR THE LEAD COMPENSATOR AND THE RESULTS ARE ALSO PLOTTED IN THAT PPT AND THE PROBLEM CAN ALSO BE SOLVED BY USING THE DATA IN PPT AND IT IS SO USEFUL PPT
Necessary of Compensation, Methods of Compensation, Phase Lead Compensation, Phase Lag Compensation, Phase Lag Lead Compensation, and Comparison between lead and lag compensators.
Poles and Zeros of a transfer function are the frequencies for which the value of the denominator and numerator of transfer function becomes zero respectively
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This Presentation explains about the introduction of Frequency Response Analysis. This video clearly shows advantages and disadvantages of Frequency Response Analysis and also explains frequency domain specifications and derivations of Resonant Peak, Resonant Frequency and Bandwidth.
Root locus is a graphical representation of the closed-loop poles as a system parameter is varied.
It can be used to describe qualitatively the performance of a system as various parameters are changed.
It gives graphic representation of a system’s transient response and also stability.
We can see the range of stability, instability, and the conditions that cause a system to break into oscillation.
It gives how states are representing in various canonical forms and how it it is different from transfer function approach. and finally test the system controllability and observability by kalman's test
state space modeling of electrical systemMirza Baig
Introduction
As systems become more complex, representing them with differential equations or transfer functions becomes cumbersome. This is even more true if the system has multiple inputs and outputs. This document introduces the state space method which largely alleviates this problem. The state space representation of a system replaces an nth order differential equation with a single first order matrix differential equation. The state space representation of a system is given by two equations :
The first equation is called the state equation, the second equation is called the output equation. For an nth order system (i.e., it can be represented by an nth order differential equation) with r inputs and m outputs the size of each of the matrices is as follows:
Several features:The state equation has a single first order derivative of the state vector on the left, and the state vector, q(t), and the input u(t) on the right. There are no derivatives on the right hand side.The output equation has the output on the left, and the state vector, q(t), and the input u(t) on the right. There are no derivatives on the right hand side.
q is nx1 (n rows by 1 column)q is called the state vector, it is a function of timeA is nxn; A is the state matrix, a constantB is nxr; B is the input matrix, a constant u is rx1; u is the input, a function of time C is mxn; C is the output matrix, a constant D is mxr; D is the direct transition matrix, a constant y is mx1; y is the output, a function of time
Derivation of of State Space Model (Electrical)
To develop a state space system for an electrical system, they choosing the voltage across capacitors, and current through inductors as state variables. Recall that
so if we can write equations for the voltage across an inductor, it becomes a state equation when we divide by the inductance (i.e., if we have an equation for einductor and divide by L, it becomes an equation for diinductor/dt which is one of our state variable). Likewise if we can write an equation for the current through the capacitor and divide by the capacitance it becomes a state equation for ecapacitor
There are three energy storage elements, so we expect three state equations. Try choosing i1, i2 and e1 as state variables. Now we want equations for their derivatives. The voltage across the inductor L2 is e1 (which is one of our state variables)so our first state variable equation is
This equation has our input (ia) and two state variable (iL2 and iL1) and the current through the capacitor. So from this we can get our second state equation
Our third, and final, state equation we get by writing an equation for the voltage across L1 (which is e2) in terms of our other state variables
references:
http://lpsa.swarthmore.edu/Representations/SysRepSS.html
https://en.wikipedia.org/wiki/State-space_representation
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Root locus is a graphical representation of the closed-loop poles as a system parameter is varied.
It can be used to describe qualitatively the performance of a system as various parameters are changed.
It gives graphic representation of a system’s transient response and also stability.
We can see the range of stability, instability, and the conditions that cause a system to break into oscillation.
It gives how states are representing in various canonical forms and how it it is different from transfer function approach. and finally test the system controllability and observability by kalman's test
state space modeling of electrical systemMirza Baig
Introduction
As systems become more complex, representing them with differential equations or transfer functions becomes cumbersome. This is even more true if the system has multiple inputs and outputs. This document introduces the state space method which largely alleviates this problem. The state space representation of a system replaces an nth order differential equation with a single first order matrix differential equation. The state space representation of a system is given by two equations :
The first equation is called the state equation, the second equation is called the output equation. For an nth order system (i.e., it can be represented by an nth order differential equation) with r inputs and m outputs the size of each of the matrices is as follows:
Several features:The state equation has a single first order derivative of the state vector on the left, and the state vector, q(t), and the input u(t) on the right. There are no derivatives on the right hand side.The output equation has the output on the left, and the state vector, q(t), and the input u(t) on the right. There are no derivatives on the right hand side.
q is nx1 (n rows by 1 column)q is called the state vector, it is a function of timeA is nxn; A is the state matrix, a constantB is nxr; B is the input matrix, a constant u is rx1; u is the input, a function of time C is mxn; C is the output matrix, a constant D is mxr; D is the direct transition matrix, a constant y is mx1; y is the output, a function of time
Derivation of of State Space Model (Electrical)
To develop a state space system for an electrical system, they choosing the voltage across capacitors, and current through inductors as state variables. Recall that
so if we can write equations for the voltage across an inductor, it becomes a state equation when we divide by the inductance (i.e., if we have an equation for einductor and divide by L, it becomes an equation for diinductor/dt which is one of our state variable). Likewise if we can write an equation for the current through the capacitor and divide by the capacitance it becomes a state equation for ecapacitor
There are three energy storage elements, so we expect three state equations. Try choosing i1, i2 and e1 as state variables. Now we want equations for their derivatives. The voltage across the inductor L2 is e1 (which is one of our state variables)so our first state variable equation is
This equation has our input (ia) and two state variable (iL2 and iL1) and the current through the capacitor. So from this we can get our second state equation
Our third, and final, state equation we get by writing an equation for the voltage across L1 (which is e2) in terms of our other state variables
references:
http://lpsa.swarthmore.edu/Representations/SysRepSS.html
https://en.wikipedia.org/wiki/State-space_representation
This presentation explains about the introduction of Polar Plot, advantages and disadvantages of polar plot and also steps to draw polar plot. and also explains about how to draw polar plot with an examples. It also explains how to draw polar plot with numerous examples and stability analysis by using polar plot.
State space analysis, eign values and eign vectorsShilpa Shukla
State space analysis concept, state space model to transfer function model in first and second companion forms jordan canonical forms, Concept of eign values eign vector and its physical meaning,characteristic equation derivation is presented from the control system subject area.
Two port network parameters, Z, Y, ABCD, h and g parameters, Characteristic impedance,
Image transfer constant, image and iterative impedance, network function, driving point and
transfer functions – using transformed (S) variables, Poles and Zeros.
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CONTROL SYSTEMS PPT ON A LEAD COMPENSATOR CHARACTERISTICS USING BODE DIAGRAM FOR MAXIMUM OF 50 DEG PHASE ANGLE
1. Raghu Engineering College
Department of Electrical & Electronics Engineering(Autonomous)
Accredited by NBA & NAAC with ‘A Grade, Permanently Affiliated JNTU Kakinada
Dakamarri (v), Bheemunipatnam Mandal, Visakhapatnam, Andhra Pradesh 531162
A CASE STUDY ON
A LEAD COMPENSATOR CHARACTERISTICS USING BODE DIAGRAM
FOR MAXIMUM OF 50 DEG PHASE ANGLE
UNDER THE GUIDANCE
Dr.B.SIVA RAMA RAO
ASSOCIATE PROFESSOR
BY
P. ESWAR SAI 18981A0237
P.MOHAN 18981A0238
P.SUPRIYA 18981A0239
P.NITHESH KUMAR 18981A0240
P.SANJAY KUMAR 18981A0241
BACHELOR OF TECHNOLOGY
IN
ELECTRICAL AND ELECTRONICS ENGINEERING
3. INTRODUCTION
• A feedback control system that provides an optimum Performance
without any necessary adjustment is rare
• In building a control system , we know that proper Modification of
the plant dynamics may be simple way to meet the performance
specifications.
• This however, may not be possible in many practical situations
because the plant may be fixed and not modifiable.
• Then we must adjust parameters other than those in the fixed plant.
5. • The choice between series compensation and parallel
compensation
depends on
The nature of the signals
The power levels at various points
Available components
The designer’s experience
Economic considerations and so on.
6. PHASE LEAD COMPENSATOR :-
The magnitude of the compensator continuously grows with
increasing frequency.
• The feature is undesirable because it amplifies high frequency
noise that is typically present in an real system.
• A typical lead compensator has the following transfer function.
𝐶 𝑠 = 𝐾
𝜏𝑠+1
𝛼𝜏𝑠+1
where , 𝛼<1
•
1
𝛼
is the ratio between the pole zero break point frequencies.
• Magnitude of the lead compensator is 𝐾
1+𝜔2 𝜏2
1+𝛼2 𝜔2 𝜏2
7. • The phase contributed by the lead compensator is given by
𝜑 = tan−1 𝜔𝜏 − tan−1 𝛼𝜔𝜏
• Thus a significant amount of phase is still provided with much less amplitude at
high frequencies.
• The frequencies response of a typical lead compensator magnitude varies from
20 log10 𝐾 to 20 log10
𝐾
𝛼
and maximum phase is always less than 900
.
• The frequency can be shown where the phase is maximum
𝜔 𝑚𝑎𝑥 =
1
𝜏 𝛼
• The maximum phase corresponds to sin∅ 𝑚𝑎𝑥 = (
1−sin (∅ 𝑚𝑎𝑥)
1+sin(∅ 𝑚𝑎𝑥)
)
• The magnitude of C(s) at 𝜔 𝑀𝐴𝑋 is
𝐾
𝛼
8. Analyze the stability of the System with transfer function using
Bode Diagram
Given transfer function:
𝟒
𝑺(𝑺+𝟏) 𝟐
Simplify the given transfer function
Transfer function =
4
𝒔 𝟑+𝟐𝒔 𝟐+𝒔
Use the MATLAB codding to find out the stability of a given transfer function
by obtaining the bode graph.
9. •Then run the codding, this is the bode graph of the given system.
10. •From the above bode graph of the given transfer function is a unstable system because of
the phase margin and gain margin are the negative values.
•Gain margin = -6.02 dB
•Phase margin = -18.1 degree
•Natural frequency(Wn) = 1.38 rad/sec
•The gain margin and phase margin are negative value, then the system becomes
unstable.
•So by adding the some phase to the system then the phase margin of the system becomes
positive values. Then system becomes stable.
•Use the lead compensator for adding phase margin to the system.
11. 2.Design of lead compensator :-
This is the lead compensator network. It consists of resistors and capacitor.
The capacitor(c) is connected parallel to the R1.
The output is taken across the R2.
Before going to design the lead compensator we have to know the ‘α’ value.
If we add the ‘500’ to the system then phase margin become positive.
If we add more phase angle to the system then the system performance will
decrease. Else we add less phase angle to the system then the system stability
will decrease
We already determine the formula for knowing the ‘α’ value.
12. α=
𝟏−𝒔𝒊𝒏∅
𝟏+𝒔𝒊𝒏∅
here ∅ is the how much have to added to the system.
Then ∅ = 500
α=
𝟏−𝒔𝒊𝒏(𝟓𝟎)
𝟏+𝒔𝒊𝒏(𝟓𝟎)
=
𝟏−𝟎.𝟕𝟔𝟔
𝟏+𝟎.𝟕𝟔𝟔
= 0.132
α=
R2
R1+
R2
-----------------(1)
Assuming the R2 value = 250ohm, substitute value in equation (1)
0.132 = 250
R1+25
0
0.132(R1+250 ) = 25
R1 =
250
−(
250
∗
0
.
132
)
0.132
R1 = 1643.93 ohm
13. R1 = 1643.93 ohm, R2 = 250 ohm
for determine the capacitor value
T = C * R1
C =
T
R1
----------(2)
We know the relation for determine the T value
Wn =
1
T√∝
----------(3)
Wn = 1.38 rad/sec
Substitute value in equation (3)
T=1Wn √∝
T=11.38 √0.198
T=1.99
T=1.99
14. Then substitute this value in equation (2)
C = T
R1
C =
1
.
99
1643.93
C = 1.2105 mF
After knowing the values of lead compensation then design the circuit. Use the
MATLAB codding for find out the bode graph of the lead compensator with the
above values.
15. Bode graph to the lead compensator.
Above graph is for compensator , here the phase margin is 50 degree angle. So,
add this compensator to our system. Then system becomes stable.
16. Compensated system:
connect the compensator to the system in cascade connection.
Phase margin of the system without compensator = -18.1 degree
Compensator maximum phase angle = 50.1 degree
Values of phase margin and gain margin when compensator added to the system by
using MATLAB. Shown in the below fig. is codding for compensated system in MATLAB.
17. Corresponding Bode Graph
By the bode graph
Gain margin = 0.23 dB
Phase margin = 0.711 degree
The gain margin and phase margin are positive values. Then the system become
stable.