Unit 1-Introduction of Control system.pptx

SVERI’s College of Engineering, Pandharpur
Department of Electrical Engineering
Linear Control Systems
Dr. Dipti A. Tamboli
HoD, Electrical Engg. Department

2
Unit 1- Introduction to Control systems
Control System – Definition and Practical
Examples
Definition, basic components & classification of
general control system,
Open loop & Close Loop control systems,
Advantages & disadvantages, examples,
Positive & negative feedback,
Transfer Function of open loop and closed loop
control system.
Content

“SYSTEM”
 A system is an arrangement of or a combination
of different physical components connected or
related in such a manner so as to form an entire
unit to attain a certain objective.
SYSTEM
Input Output
5

CONTROL
 It means to regulate, direct or command a system so
that the desired objective is attained.
Combining above definitions
System + Control = Control System
6

CONTROL SYSTEM
7
 It is an arrangement of different physical elements
connected in such a manner so as to regulate, direct or
command itself to achieve a certain objective. OR
 The control system is that by which any quantity of interest
in a machine, mechanism, or other equipment is maintained
or changed in accordance with a desired manner.
 Eg. Classroom with teacher & students, lamp with switch,
automobile system
CONTR
OL
SYSTE
M
Input Output
7

INPUT
8
 The stimulus, applied signal or excitation applied
to a control system from an external source in
order to produce the output is called input
Input
8

OUTPUT
9
 The actual response obtained from a system
is called output.
Output
Input
9

DIFFERENCE BETWEEN SYSTEM AND
CONTROL SYSTEM
10
System
Input Contro
l
Syste
m
Input Desired
Output
Proper
Output
(May or may not
be desired)
10

DIFFERENCE BETWEEN SYSTEM AND
CONTROL SYSTEM
11
An example :
Fan
Fan
(Syste
m)
230V/50Hz
AC Supply
Air Flow
Input Output
11

A FAN: CAN'T SAY SYSTEM
12
230V/50Hz
AC Supply
No Airflow
(No Proper/ Desired
Output)
 A Fan without blades cannot be a “SYSTEM”
Because it cannot provide a desired/proper output
i.e. airflow
Input Output
12

A FAN: CAN BE A
SYSTEM
13
 A Fan with blades but without regulator can be a “SYSTEM”
Because it can provide a proper output i.e. airflow
 But it cannot be a “Control System” Because it cannot
provide desired output i.e. controlled airflow
Input Output
230V/50Hz
AC Supply
Airflow
(Proper Output)
13

A FAN: CAN BE A CONTROL SYSTEM
14
 A Fan with blades and with regulator can be a “CONTROL
SYSTEM” Because it can provide a Desired output.
i.e. Controlled airflow
230V/50Hz
AC Supply
Controlled Airflow
(Desired Output)
Input Output
Control
Element
14

BLOCK DIAGRAM OF CLCS
15
Referenc
e
Transduc
er
Controller Plant
Feedback
Transducer
Command
I/p
Reference
I/p
Feedback
Signal
Manipulated
Signal
Error
Signal
Controlled
O/p
r(t)
e(t)
b(t) c(t)
c(t)
m(t)
Forward Path
Feedback Path
15

16
Plant:
The portion of the system to be controlled or regulated is
called the plant or process. OR
A set of machines parts functioning together to
perform a particular operation is known as a plant.
Process:
A process is to be natural, an artificial or voluntary,
progressively continuing operations that consist of
series of controlled actions or movement
systematically directed towards a particular result.
eg. Chemical, biological, economical process
16

17
Controller:
The element of the system itself or external to the system
which control the plant or process is called controller.
Disturbance:
Disturbance is a signal which tends to adversely
affect the value of the output of a syste. If such
disturbance is generated within the system itself, it is
called internal disturbance. If disturbance generated
outside the system acting as an extra input to the
system in addition to its normal input, then it is called
as external disturbance.
17

18
Feedback control:
Feedback control is an operation which, in the
presence of disturbances, tend to reduce the
difference between output of the system and the
reference input. Only unpredictable disturbances are
to be designed, since for predictable/known
disturbances, it is always possible to include
compensation within the system, so that measurement
is not necessary. Thus,
A feedback control system is one which tend to
maintain a prescribed relationship between the output
and the reference input by comparing these and using
18

19
Referenc
e
Transduc
er
Controller Plant
Feedback
Transducer
Command
I/p
Reference
I/p
Feedback
Signal
Manipulated
Signal
Error
Signal
Controlled
O/p
r(t)
e(t)
b(t) c(t)
c(t)
m(t)
Forward Path
Feedback Path
19

Classification of Control
System
CLASSIFICATION OF CONTROL SYSTEM
20
20
Depending on
control action
• Open Loop Control System
• Closed Loop Control System
According to time
variation
• Time varying
• Time invariant
Mode of analysis
and design
• Linear
• Nonlinear
Mode of signal
• Continuous data (analog) control system
• Discrete (digital)data control system

System
(Depending on control action)
Open Loop Control
System
Closed Loop Control
System
21
21

OPEN LOOP CONTROL SYSTEM
22
Definition:
“A system in which output is dependent on input but
the control action is totally independent of the output
of the system is called as open loop system”. An
physical system which doesn’t automatically correct
for variation in its output is called open loop system
Fig. Block Diagram of Open loop Control System
Controller Process
Reference I/p
r(t) u(t) c(t)
Controlled o/p
22

OLCS EXAMPLES
23
 Electric hand drier –
Hot air (output) comes
out as long as you
keep your hand under
the machine,
irrespective of how
much your hand is
dried.
23

OLCS EXAMPLES
24
 Automatic washing
machine
– This machine runs
according to the pre-set
time irrespective of
washing is completed or
not.
24

 Bread toaster - This
machine runs as per
adjusted time
irrespective of toasting
is completed or not.
OLCS EXAMPLES
25
25

 Automatic tea/coffee
Vending Machine –
These machines also
function for pre
adjusted time only.
OLCS EXAMPLES
26
26

 Light switch – lamps glow whenever light switch
is on irrespective of light is required or not.
 Volume on stereo system – Volume
is manually irrespective of
output volume level.
adjuste
d
OLCS EXAMPLES
27
27

ADVANTAGES OF OLCS
28
 Simple in construction and design.
 Simple in design so economical.
 Easy to maintain.
 Generally stable.
 Very much convenient when output is difficult
to measure.
28

DISADVANTAGES OF OLCS
29
 They are inaccurate & unreliable because accuracy
of such system depends upon accurate
precalibration of the controller.
 Inaccurate for environmental variations.
 They can’t sense internal disturbances after
controller stage.
 Any change in output cannot be corrected
automatically.
 To maintain the quality and accuracy, recalibration of
the controller is regularly required. 29

CLOSED LOOP SYSTEM
30
Definition:
“A system in which the control action is
somehow dependent on the output or changes
in output is called as closed loop system”.
The information about the instantaneous state of the
output is feedback to the input and is used to modify
it in such a manner as to achieve the desired output.
30

31
Referenc
e
Transduc
er
Controller Plant
Feedback
Transducer
Command
I/p
Reference
I/p
Feedback
Signal
Manipulated
Signal
Error
Signal
Controlled
O/p
r(t)
e(t)
b(t) c(t)
c(t)
m(t)
Forward Path
Feedback Path
31
±

CLCS EXAMPLES
32
 Automatic Electric Iron- Heating elements
are controlled by output temperature
of the iron.
32

Servo voltage stabilizer – Voltage
controller operates depending upon output
voltage of the system.
CLCS EXAMPLES
33
33

Perspirati
on
CLCS EXAMPLES
34
34

ADVANTAGES OF CLCS
35
 Closed loop control systems are more accurate even in
the presence of non-linearity.
 Highly accurateas any error arising is corrected due to
presence of feedback signal.
 Such system senses environmental changes as well as
internal disturbances and accordingly modifies the error.
 Bandwidth range is large.
 Facilitates automation.
 The sensitivity of system may be made small to make
system more stable.
 This system is less affected by noise. 35

DISADVANTAGES OF CLCS
36
 They are costlier.
 They are complicated to design.
 Required more maintenance.
 Feedback leads to oscillatory response.
 Overall gain is reduced due to presence of
feedback.
 Stability is the major problem and more care is
needed to design a stable closed loop system.
36

DIFFERENCE BETWEEN OLCS & CLCS
37
37
Sr
No.
Open Loop Control System Closed Loop Control System
1 Asystem in which output is dependent
on input but the control action is
totally independent of the output of
the system is called as open loop
system.
A system in which the control
action is somehow dependent on
the output or changes in output is
called as closed loop system.
2
3 The open loop systems are simple &
economical.
closed loopsystems are complex and
costlier.
4 They consume less power. They consume more power.

38
38
Sr
No.
11 The changes in the output due to
external disturbances are not
corrected automatically. So they
are more sensitive to noise and
other disturbances.
The changes in the output due to
external disturbances are
corrected automatically. So they
are less sensitive to noise and
other disturbances.
12 Examples: Coffee Maker,
Automatic Toaster, Hand Drier.
Examples: Guided Missile, Temp
control of oven, Servo voltage
stabilizer, Automatic Electric
Iron, Water Level Controller,
Missile Launched and Auto
Tracked by Radar, An Air
Cooling System in Car.

39
39
Sr
No.
5 The OL systems are easier to
construct because of less number
components required.
The CL systems are not easy
to construct because of more
number components required.
6 The open loop systems are
inaccurate & unreliable
The closed loop systems are
accurate & more reliable.
7 Stability is not a major problem
in OL control systems.
Stability is a major problem in
closed loop systems & more care
is needed to design a stable
closed loop system.
8 Small bandwidth. Large bandwidth.
9 Feedback element is absent. Feedback element is present.
10 Output measurement is not
necessary.
Output measurement is
necessary.

System
Linear Control
System
Non-linear Control
System
40
40

Linear Control System
 The systems which follow the principle of
homogeneity and additivity are called as linear
control systems.
 In order to understand the linear control system, we
should first understand the principle of
superposition. The principle of superposition
theorem includes two the important properties as
below:
 Homogeneity: A system is said to be
homogeneous, if we multiply input with some
41
41

Linear Control System
 Additivity: Suppose we have a system S and we are giving
the input to this system as a1 for the first time and we are
getting the output as b1 corresponding to input a1. On the
second time we are giving input a2 and correspond to this
we are getting the output as b2.
 Now suppose this time we are giving input as a summation
of the previous inputs (i.e. a1 + a2) and corresponding to this
input suppose we are getting the output as (b1 + b2) then we
can say that system S is following the property of additivity.
 Eg. a purely resistive network with a constant
DC source
42
42

 Non-linear systems do not obey law of
superposition.
 The stability of non-linear systemsdependson root
location as well as initial conditions & type of
input.
 Non-linearsystemsexhibit self sustained
oscillations of fixed frequency. Eg.
NON-LINEAR CONTROL SYSTEM
43
43

NON-LINEAR CONTROL SYSTEM
44
44
Magnetization curve or no load curve of a DC machine, hysteresis loop

DIFFERENCE BETWEEN LINEAR & NON-LINEAR
SYSTEM
45
Linear System Non-linear System
1. Obey superposition.
2. Can be analyzed by
standard test signals
3. Stability depends only
on
root location
4. Do not exhibit limit cycles
5. Do not exhibit
hysteresis/
jump resonance
6. Can be analyzed by
Laplace transform, z-
transform
1. Do not obey superposition
2. Cannot be analyzed by
standard test signals
3. Stability depends on root
locations, initial conditions &
type of input
4. Exhibits limit cycles
5. Exhibits hysteresis/
jump resonance
6. Cannot be analyzed by
Laplace transform, z-
transform
45

System
Time Varying
Control System
Time Invarying Control
System
46
46

Time varying/In-varying Control System
 Systems whose parameters vary
with time are called time varying
control systems. It is not depend on whether input
or output are function of time or not.
 Eg. Driving a vehicle, mass is a parameter of space
vehicle system. The mass of missile/rocket
reduces as fuel is burnt and hence the parameter
mass is time varying and the control system is
time varying type.
 If even though input or output are function of time
but the parameters of system are independent of
47

Time varying/In-varying Control System
 Eg. Control system made up of inductor, capacitor,
resistor only
 Let x(t) and y(t) be the input and output signals,
respectively, of a system shown in Fig. Then the
transformation of into is represented by the
mathematical notation y(t) = Tx(t)
 where T is the operator which defined rule by which
x(t) is transformed into y(t). A system is called time-
invariant if a time shift in the input signal x(t-t0) causes
the same time shift in the output signal y(t-t0).
48

System
Continuous data
Control System
Discrete data Control
System
49
49

Continuous/ Discrete Time/data Control System
 If all the system variables are function of a
continuous time t, then it is called as continuous data
control system.
 Eg. The speed control of a dc motor using
tachogenerator feedback
 A discrete time control system involves one or more
variables that are known only at discrete time
intervals.
 Eg. Microprocessor or computer based systems use
discrete time signals.
50

Continuous/ Discrete Time/data Control System
 Digital systems can handle nonlinear control systems
more effectively than the analog type of systems.
 Power requirement in case of a discrete or digital
system is less as compared to analog systems.
 Digital system has a higher rate of accuracy and can
perform various complex computations easily as
compared to analog systems.
 Reliability of the digital system is more as compared
to an analog system. They also have a small and
compact size.
 Digital system works on the logical operations which
increases their accuracy many times.
51

System
Lumped Parameter
Control System
Distributed Parameter
Control System
52
52

Lumped / Distributed Parameter Control System
 In these types of control systems, the various active and
passive components are assumed to be concentrated at a
point and that’s why these are called lumped parameter
type of system.
 Eg. R, L C parameters in electrical networks. Analysis of
such type of system is very easy which includes
differential equations.
 In these types of control systems, the various active (like
inductors and capacitors) and passive parameters
(resistor) are assumed to be distributed uniformly along
the length and that’s why these are called distributed
parameter type of system.
 Transmission line parameters R-L. Analysis of such type
of system is slightly difficult which includes partial
53

Deterministics/ Stochastic Control System
 A control system is said to be deterministic when its
response to input as well as behaviour to external
disturbances is predictable and repeatable.
 If the response to input is unpredictable and non-
repeatable, then it is called stochastic in nature.
SISO/ MIMO Control System
 A system having only one input and one output is
called as SISO systems. Eg. Position control system
 A system having multiple input and multiple output is
called as MIMO systems. Eg. Process control systems,
all complex systems
54

60
To achieve the required objective, a good control system must
satisfy the following requirements.
1. Accuracy:
 A good control system must be highly accurate.
 The open loop systems are generally less accurate and
hence feedback is introduced to reduce the error in the
system.
2. Sensitivity:
 A good control system should be very insensitive to
environmental changes, age etc. But, must be sensitive to
the input commands.
 The performance should not be affected by small changes in
the certain parameters of the system.
Requirement of good control system

61
3. External disturbance or noise:
 All the physical systems are subjected to external disturbances
and noise signals during operation.
 A requirement of a good control system is that system is
insensitive to noise and external disturbances but sensitive to
the input commands.
 It should be able to reduce the effects of undesirable
disturbances.
4. Stability:
 A concept of stability means output of system must follow
reference input and must produced bounded output for
bounded input.
 A good control system is one which is stable in nature.
5. Bandwidth:
 This requirement is related to the frequency response of the
system.
 For the input frequency range, it should give satisfactory output.

62
6. Speed:
 A system should have good speed. This means output
of the system should approach to its desired value as
quickly as possible.
 System should settled down to its final, value as
quickly as possible.
7. Oscillations:
 The system should exhibits suitable damping i.e. the
controlled output should follow the changes in the
reference input without unduly large oscillations or
overshoots.

 If either the output or some part of the output is returned to
the input side and utilized as part of the system input, then
it is known as feedback.
 Feedback plays an important role in order to improve the
performance of the control systems.
 Types of feedback:
1. Positive feedback
2. Negative feedback
POSITIVE AND NEGATIVE FEEDBACK
63
63

 Positive feedback
1. The positive feedback adds the reference input, R(s) and
feedback output. The following figure shows the block diagram
of positive feedback control system.
2. Consider the transfer function of positive feedback control
system is, T=
𝐺
1−𝐺 𝐻
(1)
Where,
T = the transfer function or overall gain of positive feedback
control system.
G is the open loop gain, which is function of frequency.
H is the gain of feedback path, which is function of frequency.
64

 Negative feedback
1. Negative feedback reduces the error between the reference
input, R(s) and system output. The following figure shows the
block diagram of the negative feedback control system.
2. Transfer function of negative feedback control system is,
T=
𝐺
1+ 𝐺 𝐻
(2)
Where,
T = the transfer function or overall gain of negative feedback
control system.
G is the open loop gain, which is function of frequency.
H is the gain of feedback path, which is function of frequency.
65

 Effect of Feedback on Overall Gain
 From Equation 2, we can say that the overall gain of
negative feedback closed loop control system is the ratio of
'G' and (1+GH). So, the overall gain may increase or
decrease depending on the value of (1+GH).
 If the value of (1+GH) is less than 1, then the overall gain
increases. In this case, 'GH' value is negative because the
gain of the feedback path is negative.
 If the value of (1+GH) is greater than 1, then the overall gain
decreases. In this case, 'GH' value is positive because the
gain of the feedback path is positive.
EFFECTS OF FEEDBACK
66

 In general, 'G' and 'H' are functions of frequency. So, the
feedback will increase the overall gain of the system in one
frequency range and decrease in the other frequency range.
 Effect of Feedback on Stability
 A system is said to be stable, if its output is under control.
Otherwise, it is said to be unstable.
 In Equation 2, if the denominator value is zero (i.e., GH = -
1), then the output of the control system will be infinite. So,
the control system becomes unstable.
 Therefore, we have to properly choose the feedback in order
to make the control system stable.
EFFECTS OF FEEDBACK
67

EFFECTS OF FEEDBACK
68
 Effect of Feedback on Noise
 To know the effect of feedback on noise, let us compare the
transfer function relations with and without feedback due to
noise signal alone.
 Consider an open loop control system with noise signal as
shown below.
 If input R(s) equal to zero.
 The open loop transfer
function due to noise
signal alone is

C(s)
N(s)
= 𝐺𝑏 (3)

69
 Consider a closed loop control system with noise signal as
shown below.
 The closed loop transfer function due to noise signal alone
is
C(s)
N(s)
=
𝐺𝑏
1+𝐺𝑎𝐺𝑏H
(4)
 It is obtained by making the other input R(s) equal to zero.
Compare Equation (3) and (4),
 In the closed loop control system, the gain due to noise
signal is decreased by a factor of (1+𝐺𝑎𝐺𝑏𝐻) provided
that the term (1+𝐺𝑎𝐺𝑏𝐻) is greater than one.

 A control system consists of an output as well as an input
signal. The output is related to the input through a function call
transfer function.
 Defination:
 A transfer function is expressed as the ratio of Laplace
transform of output to the Laplace transform of input
assuming all initial condition to be zero.
 Consider a system whose time domain block diagram is
 Now the transfer function G(s) is given by
 A transfer function (TF) relates one
input and one output.
TRANSFER FUNCTION
70

 Transfer function for any input is same for a given circuit and
it is convenient representation of a linear, dynamic model.
 While taking the Laplace transform for determining the
transfer function of a control system, it is assumed that all
initial condition concerning the differential equations are zero.
 The transfer function of a given system is a fixed quantity.
 The output of the system can be evaluated simply by taking
the inverse Laplace of the product of transfer function and the
given input. Therefore, output is given by
C(s) = G(s) x R(s)
 Then, output in the time domain can be evaluated by taking
inverse Laplace as
TRANSFER FUNCTION
71

 Why input, output and other signals are represented in Laplace
form in a control system:
 The input and output of a control system can be different types.
 For mathematical analysis of a system, all kinds of signal should
be represented in similar form.
 Properties of Transfer Function:
 The transfer function of a system is the Laplace transform of its
impulse response for zero initial conditions.
 The transfer function can be determined from system input-output
pair by taking ratio Laplace of output to Laplace of input.
 The transfer function is independent of the inputs to the system.
 The system poles/zeros can be found out from transfer function.
 The transfer function is defined only for linear time invariant
systems. It is not defined for non-linear systems. 72

73
 Advantages of Transfer function
1. If transfer function of a system is known, the response of the
system to any input can be determined very easily.
2. A transfer function is a mathematical model and it gives the
gain of the system.
3. The value of transfer function is dependent on the parameters of
the system and independent of the input applied.
4. Poles and zeroes of a system can be determined from the
knowledge of the transfer function of the system.
5. It helps in the stability analysis of the system.
6. Since it involves the Laplace transform, the terms are simple
algebraic expressions and no differential terms are present.

74
7. Integral and differential equations are converted to simple
algebraic equations.
8. System differential equation can be obtained by replacement
of variable ‘s’ by ‘d/dt’.
 Disadvantages of Transfer function
1. Transfer function does not take into account the initial
conditions, so effects arising due to initial conditions are
neglected. Hence initial conditions lose their importance.
2. The transfer function can be defined for linear systems only.
3. From transfer function, physical nature of the system
whether it is electrical, mechanical, thermal or hydraulic,
cannot be judged.

75
Procedure for determining the Transfer Function of a
control system:
 Write down the time domain equation for the system
introducing all the variables (using KCL &KVL).
 Taking the Laplace transform of the system equations,
assuming initial conditions as zero.
 Identify system input and output variables.
 Eliminating introduced variables get the resultant
equation in term of input and output variables.
 Lastly we take the ratio of the Laplace transform of the
output and the Laplace transform of the input which is the
required transfer function

TRANSFER FUNCTION OF CLOSED LOOP SYSTEM
R(s)
G(s) C(s)
Output
H(s)
B(s)
+-
Error
Signal
E(s)
Feedback
Signal
Input
Error signal is given by;
E(s)  R(s)  B(s)      (1)
 R(s)  E(s) B(s)
Gain of feedback network is given by;
H(s) 
B(s)
C(s)
B(s) =H(s).C(s) -- (2)
Gain for CL system is given by;
G(s) 
C(s)
E(s)
C(s)  G(s).E(s)      (3)
Substitute value of E(s) from eq. 1 to 3
C(s)G(s).(R(s) B(s))
C(s) G(s).R(s) G(s).B(s) (4)
Substitute value of B(s) from eq. 2 to 4
C(s)  G(s) R(s)  G(s).H(s).C(s)
G(s).R(s)  C(s)  G(s).H(s).C(s)
G(s).R(s)  C(s)(1  G(s).H(s))
Transfer function is given by;
C(s) G(s)

R(s) 1  G(s).H(s)
T
.F
.=

 The Laplace transform can be used independently on different
circuit elements, and then the circuit can be solved entirely in the
S Domain (Which is much easier).
 Let's take a look at some of the circuit elements.
 Resistors are time and frequency invariant. Therefore, the
transform of a resistor is the same as the resistance of the resistor.
L{Resistor}=R(s)
 Let us look at the relationship between voltage, current, and
capacitance, in the time domain:
i (t)=C
𝑑𝑣(𝑡)
𝑑𝑡
 Solving for voltage, we get the following integral:
V 𝑡 =
1
𝐶 0
∞
𝑖 𝑡 𝑑𝑡
LAPLACE TRANSFORM OF PASSIVE ELEMENT (R,L & C)
77

 Then, transforming this equation into the Laplace domain, we get
the following:
V 𝑠 =
1
𝐶 𝑆
I(s)
L{capacitor} =
V 𝑠
I(s)
=
1
𝐶 𝑆
 Let us look at the relationship between voltage, current, and
inductance, in the time domain:
V (t)=L
𝑑𝑖(𝑡)
𝑑𝑡
 Putting this into the Laplace domain, we get the formula:
V (s)=s L I(s)
 And solving for our ratio
L{inductor} =
V 𝑠
I(s)
= s L
78

Transfer Function of RC and RLC electrical circuits
Example: Find the TF of given RC network
C
Vo(t)
Vi(t) i(t)
Apply KVL for input loop,
1 t
Vi(t)  Ri(t)  i(t)dt
C
1
0
Taking Laplace transform above equation
Vi(s)  RI(s) 
1
I(s)      (1)
sC
Apply KVL for output loop,
t
vo(t)  i(t)dt
C
Taking Laplace transform above equation
V
o(s)
1
I(s)(2)
sC
From equation 3 and 4,
sC
I(s)  sC.Vo(s)(3)
From equation 1,
Vi(s)  I (s)(R
1
)      (4)
1
)
Vi(s)  Vo(s).sC.(R
sC
79
0

Vo(s)

1
Vi(s)
sC
sC.(R
1
)
Vo(s)

1
Vi(s)
)
sC.(
sCR 1
sC
Vo(s)

1
Vi(s) sCR 1
Transfer Function= G(s)=
1
sCR 1
Vi(s) Vo(s)
80

TRANSFER FUNCTION OF RC AND RLC ELECTRICAL
CIRCUITS
Vo(t)
Vi(t) i(t)
Example: Find the TF of given RLC network
L
Apply KVL for input loop,
C
Vo(s)
Vi(s) I(s)
sL
1
sC
1
Vi(s)  RI(s)  sLI(s) I(s)
sC
sC
Taking Laplace transform above network 1
Vi(s) [RsL ]I(s)(1)
Apply KVL for output loop,
1
Vo(s) I(s)(2)
sC
81

From equation 1 and 2,
Transfer Function=
1 I ( s )
s C
s C
Vo ( s )

Vi(s)
[ R  s L 
1
] I(s)
1
]
1
 s C
[ R  s L 
s C
1
sC sCR  s2
LC  1
sC
sCR  s2
LC  1
1

s2
LC  sCR1
1

80

Higher-order Transfer Functions
Consider a general n-th order, linear ODE:
1
1 1 0
1
1
1 1 0
1
(4-39)
n n m
n n m
n n m
m
m m
d y dy dy d u
a a a a y b
dt
dt dt dt
d u du
b b b u
dt
dt

 

 
    
  
Take LT, assuming the initial conditions are all zero. Rearranging
gives the TF:
The order of the TF is defined to be the order of the denominator
polynomial.
The order of the TF is equal to the order of the ODE.
 
 
 
0
0
(4-40)
m
i
i
i
n
i
i
i
b s
Y s
G s
U s
a s


 



Unit 1-Introduction of Control system.pptx

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