1
INSTITUTE OF AERONAUTICAL ENGINEERING
(Autonomous)
Dundigal, Hyderabad - 500 043
Lecture Notes:
NETWORK ANALYSIS(AEEC05)
Drafted by :
Dr. D Shobharani (IARE 10507)
Professor
Department Of Electrical and Electronics Engineering
Institute of Aeronautical Engineering
October 22, 2021
Contents
Contents
1
List of Figures
3
Abbreviations
5
Symbols
6
1
ANALYSIS OF AC CIRCUITS
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
1.2 Analysis of AC circuits: . . . . . . . . . . . . . . .
1.2.1 AC Capacitance and Capacitive Reactance
1.3 AC Capacitance with a Sinusoidal Supply . . . . .
1.3.1 Sinusoidal Waveforms for AC Capacitance .
1.3.2 Phasor Diagram for AC Capacitance . . . .
2
SOLUTION OF FIRST AND SECOND ORDER NETWORKS
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Resistor . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Inductor . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 capacitor . . . . . . . . . . . . . . . . . . . . . . .
2.1.4 Initial Condition for (DC steady state solution) .
2.1.5 Response of Series RL Circuit (DC Excitation): .
2.2 Response of Series RL Circuit (AC Excitation): . . . . . .
2.3 Differential Equation and Laplace Transform Approach: .
3
LOCUS DIAGRAMS AND NETWORKS FUNCTIONS
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
3.2 RL Series Circuit . . . . . . . . . . . . . . . . . .
3.3 RC Series Circuit . . . . . . . . . . . . . . . . . .
3.4 RLC Series Circuit . . . . . . . . . . . . . . . . .
3.5 Network Functions . . . . . . . . . . . . . . . . . .
3.6 Properties of all Network Functions . . . . . . . .
3.7 The Pole Zero Plot . . . . . . . . . . . . . . . . .
4
THREE PHASE AC CIRCUITS
39
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
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1
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2
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38
38
Contents
4.2
4.3
4.4
4.5
4.6
5
Advantages of Three Phase is preferred over Single Phase .
Star Connection . . . . . . . . . . . . . . . . . . . . . . . .
Delta Connection . . . . . . . . . . . . . . . . . . . . . . . .
Phase Sequence . . . . . . . . . . . . . . . . . . . . . . . . .
Analysis of Balanced and Unbalanced Three Phase Circuits
2
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40
41
41
43
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FILTERS
45
5.1 Classification of filters, filter networks . . . . . . . . . . . . . . . . . . . . . 45
Bibliography
53
List of Figures
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1.12
1.13
1.14
1.15
1.16
1.17
1.18
1.19
1.20
1.21
1.22
1.23
DC Circuit and Waveform . . . . . . . . . . .
Phasor Diagram of a Sinusoidal Waveform .
Phase Difference of a Sinusoidal Waveform .
Vector Diagram of a Sinusoidal Waveform . .
Vector Diagram of a Sinusoidal Waveform . .
Phasor Addition of two Phasors . . . . . . . .
Phasor Subtraction . . . . . . . . . . . . . . .
Vector Subtraction of two Phasors . . . . . .
Vector Subtraction of two Phasors . . . . . .
Phase Difference of a Sinusoidal Waveform . .
Phasor Diagram of a Sinusoidal Waveform . .
Phasor Diagram of a Sinusoidal Waveform . .
Phase Relationship of a Sinusoidal Waveform
Phase Relationship of a Sinusoidal Waveform
AC Inductance with a Sinusoidal Supply . . .
Sinusoidal Waveforms for AC Inductance . .
Phasor Diagram for AC Inductance . . . . . .
Inductive Reactance against Frequency . . . .
Inductive Reactance against Frequency . . . .
AC Capacitance with a Sinusoidal Supply . .
Sinusoidal Waveforms for AC Capacitance . .
Phasor Diagram for AC Capacitance . . . . .
Capacitive Reactance against Frequency . . .
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3
6
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26
2.1
2.2
2.3
2.4
2.5
Resistor . .
Inductor . .
capacitor .
RL Circuit .
RL Circuit .
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28
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31
3.1
3.2
3.3
3.4
3.5
3.6
When
When
When
When
When
When
R is varied and XL fixed circuit . . . . . . . . . .
R is varied and XL fixed locus diagram . . . . .
XL is varied and R fixed circuit . . . . . . . . . .
XL is varied and R fixed locus diagram . . . . .
R is varied and XC fixed circuit . . . . . . . . .
R is varied and XC fixed locus diagram . . . . .
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34
34
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35
35
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List of Figures
4
3.7
3.8
3.9
3.10
3.11
When XC is varied and R fixed circuit . . .
When XC is varied and R fixed locus diagram
RLC series circuit . . . . . . . . . . . . . . .
RLC series circuit locus diagram . . . . . . .
Network . . . . . . . . . . . . . . . . . . . .
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35
35
36
36
37
4.1
4.2
4.3
4.4
4.5
4.6
Three Phase Waveform . . . . . .
Star connection . . . . . . . . . .
Star connection . . . . . . . . . .
Phase Sequence . . . . . . . . . .
Star connection vector diagram .
Delta connection vector diagram
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40
42
42
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44
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
Low Pass Filter . . . . . . . . . . . . . . .
High Pass Filter . . . . . . . . . . . . . .
Band Pass Filter . . . . . . . . . . . . . .
Band Stop Filter . . . . . . . . . . . . . .
Constant k T section filter . . . . . . . . .
m Derived Filter T section . . . . . . . . .
Constant k pi section filter . . . . . . . . .
m Derived Filter pi section . . . . . . . .
Eleiments m Derived Filter pi section . . .
m Derived Filter T section low pass filter
m Derived Filter pi section low pass filter
m Derived Filter T section high pass filter
m Derived Filter pi section high pass filter
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46
46
47
47
48
48
49
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50
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51
51
52
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Abbreviations
TVC
Thrust Vector Control
LOX
Liquid OXygen
LVDT
Liquid Propellant Rocket Engine
RC
Reinforced Concrete
5
Symbols
Del
Elasticity tensor
σ
Stress tensor
ε
Strain tensor
Veq
Equivalent velocity
ṁ
Mass flow rate
Isp
Specific Impulse
c
Effective exhaust velocity
It
Total impulse
υ
Exhaust velocity
mp
Propellant mass
me
Empty mass
Aex
Exit Area
pex
Exhaust pressure
PSL−a
Ambient pressure at sea level
FSL−a
Sea level thrust of the rocket
W˙sp
Specific propellant consumption rate
Cw
Weight flow coefficient
Cf
Thrust coefficient
6
Chapter 1
ANALYSIS OF AC CIRCUITS
Course Outcomes
After successful completion of this module, students should be able to:
CO 1
Identify various combinations for RL, RC and RLC Circuits with
Apply
sinusoidal excitation
CO 2
Explain Series and parallel resonance using resonant frequency
Understand
1.1 Introduction
The three basic passive components of: Resistance, Inductance, and Capacitance have
very different phase relationships to each other when connected to a sinusoidal alternating
supply. In a pure ohmic resistor the voltage waveforms are “in-phase” with the current. In
a pure inductance the voltage waveform “leads” the current by 90, giving us the expression
of: ELI. In a pure capacitance the voltage waveform “lags” the current by 90, giving us
the expression of: ICE. This Phase Difference, depends upon the reactive value of the
components being used and hopefully by now we know that reactance, ( X ) is zero if
the circuit element is resistive, positive if the circuit element is inductive and negative if
it is capacitive thus giving their resulting impedances. Instead of analysing each passive
element separately, we can combine all three together into a series RLC circuit. The
1
Chapter 1. ANALYSIS OF AC CIRCUITS
2
analysis of a series RLC circuit is the same as that for the dual series RL and RC circuits
we looked at previously, except this time we need to take into account the magnitudes of
both XL and XC to find the overall circuit reactance. Series RLC circuits are classed as
second-order circuits because they contain two energy storage elements, an inductance L
and a capacitance C. Consider the RLC circuit . The series RLC circuit above has a single
loop with the instantaneous current flowing through the loop being the same for each
circuit element. Since the inductive and capacitive reactance’s XL and XC are a function
of the supply frequency, the sinusoidal response of a series RLC circuit will therefore vary
with frequency, ƒ. Then the individual voltage drops across each circuit element of R, L
and C element will be “out-of-phase” with each other
1.2 Analysis of AC circuits:
Direct Current or D.C. as it is more commonly called, is a form of electrical current or
voltage that flows around an electrical circuit in one direction only, making it a “Unidirectional” supply.
Generally, both DC currents and voltages are produced by power supplies, batteries, dynamos and solar cells to name a few. A DC voltage or current has a fixed magnitude
(amplitude) and a definite direction associated with it. For example, +12V represents 12
volts in the positive direction, or -5V represents 5 volts in the negative direction.
We also know that DC power supplies do not change their value with regards to time,
they are a constant value flowing in a continuous steady state direction. In other words,
DC maintains the same value for all times and a constant uni-directional DC supply never
changes or becomes negative unless its connections are physically reversed. An example
of a simple DC or direct current circuit is shown below.
DC Circuit and Waveform
An alternating function or AC Waveform on the other hand is defined as one that varies in
both magnitude and direction in more or less an even manner with respect to time making
it a “Bi-directional” waveform. An AC function can represent either a power source or a
Chapter 1. ANALYSIS OF AC CIRCUITS
3
Figure 1.1: DC Circuit and Waveform
signal source with the shape of an AC waveform generally following that of a mathematical
sinusoid being defined as: A(t) = Amax sin(2πft).
The term AC or to give it its full description of Alternating Current, generally refers to a
time-varying waveform with the most common of all being called a Sinusoid better known
as a Sinusoidal Waveform. Sinusoidal waveforms are more generally called by their short
description as Sine Waves. Sine waves are by far one of the most important types of AC
waveform used in electrical engineering.
The shape obtained by plotting the instantaneous ordinate values of either voltage or
current against time is called an AC Waveform. An AC waveform is constantly changing
its polarity every half cycle alternating between a positive maximum value and a negative
maximum value respectively with regards to time with a common example of this being
the domestic mains voltage supply we use in our homes.
This means then that the AC Waveform is a “time-dependent signal” with the most common type of time-dependant signal being that of the Periodic Waveform. The periodic
or AC waveform is the resulting product of a rotating electrical generator. Generally,
the shape of any periodic waveform can be generated using a fundamental frequency and
superimposing it with harmonic signals of varying frequencies and amplitudes but that’s
for another tutorial. Alternating voltages and currents cannot be stored in batteries or
cells like direct current (DC) can, it is much easier and cheaper to generate these quantities using alternators or waveform generators when they are needed. The type and shape
of an AC waveform depends upon the generator or device producing them, but all AC
waveforms consist of a zero voltage line that divides the waveform into two symmetrical
halves. The main characteristics of an AC Waveform are defined as:
Chapter 1. ANALYSIS OF AC CIRCUITS
4
AC Waveform Characteristics
• The Period, (T) is the length of time in seconds that the waveform takes to repeat itself
from start to finish. This can also be called the Periodic Time of the waveform for sine
waves, or the Pulse Width for square waves.
• The Frequency, (ƒ) is the number of times the waveform repeats itself within a one
second time period. Frequency is the reciprocal of the time period, ( ƒ = 1/T ) with the
unit of frequency being the Hertz, (Hz).
• The Amplitude (A) is the magnitude or intensity of the signal waveform measured in
volts or amps.
In our tutorial about waveforms,we looked at different types of waveforms and said that
“Waveforms are basically a visual representation of the variation of a voltage or current
plotted to a base of time”. Generally, for AC waveforms this horizontal base line represents
a zero condition of either voltage or current. Any part of an AC type waveform which lies
above the horizontal zero axis represents a voltage or current flowing in one direction.
Likewise, any part of the waveform which lies below the horizontal zero axis represents a
voltage or current flowing in the opposite direction to the first. Generally for sinusoidal
AC waveforms the shape of the waveform above the zero axis is the same as the shape
below it. However, for most non-power AC signals including audio waveforms this is not
always the case.
The most common periodic signal waveforms that are used in Electrical and Electronic
Engineering are the Sinusoidal Waveforms. However, an alternating AC waveform may
not always take the shape of a smooth shape based around the trigonometric sine or cosine
function. AC waveforms can also take the shape of either Complex Waves, Square Waves
or Triangular Waves and these are shown below.
Phasor Diagrams and Phasor Algebra
Phasor Diagrams are a graphical way of representing the magnitude and directional relationship between two or more alternating quantities Sinusoidal waveforms of the same
frequency can have a Phase Difference between themselves which represents the angular
Chapter 1. ANALYSIS OF AC CIRCUITS
5
difference of the two sinusoidal waveforms. Also the terms “lead” and “lag” as well as “inphase” and “out-of-phase” are commonly used to indicate the relationship of one waveform
to the other with the generalized sinusoidal expression given as: A(t) = Am sin(�t ± Φ)
representing the sinusoid in the time-domain form.
But when presented mathematically in this way it is sometimes difficult to visualise this
angular or phasor difference between two or more sinusoidal waveforms. One way to
overcome this problem is to represent the sinusoids graphically within the spacial or phasordomain form by using Phasor Diagrams, and this is achieved by the rotating vector method.
Basically a rotating vector, simply called a “Phasor” is a scaled line whose length represents
an AC quantity that has both magnitude (“peak amplitude”) and direction (“phase”)
which is “frozen” at some point in time.
A phasor is a vector that has an arrow head at one end which signifies partly the maximum
value of the vector quantity ( V or I ) and partly the end of the vector that rotates.
Generally, vectors are assumed to pivot at one end around a fixed zero point known as
the “point of origin” while the arrowed end representing the quantity, freely rotates in an
anti-clockwise direction at an angular velocity, ( ω) of one full revolution for every cycle.
This anti-clockwise rotation of the vector is considered to be a positive rotation. Likewise,
a clockwise rotation is considered to be a negative rotation.
Although the both the terms vectors and phasors are used to describe a rotating line that
itself has both magnitude and direction, the main difference between the two is that a
vectors magnitude is the “peak value” of the sinusoid while a phasors magnitude is the
“rms value” of the sinusoid. In both cases the phase angle and direction remains the same.
The phase of an alternating quantity at any instant in time can be represented by a phasor
diagram, so phasor diagrams can be thought of as “functions of time”. A complete sine
wave can be constructed by a single vector rotating at an angular velocity of ω= 2πf,
where f is the frequency of the waveform. Then a Phasor is a quantity that has both
“Magnitude” and “Direction”.
Generally, when constructing a phasor diagram, angular velocity of a sine wave is always
assumed to be: ωin rad/sec. Consider the phasor diagram below.
Chapter 1. ANALYSIS OF AC CIRCUITS
6
Phasor Diagram of a Sinusoidal Waveform
Figure 1.2: Phasor Diagram of a Sinusoidal Waveform
As the single vector rotates in an anti-clockwise direction, its tip at point A will rotate
one complete revolution of 360o or 2π representing one complete cycle. If the length of its
moving tip is transferred at different angular intervals in time to a graph as shown above,
a sinusoidal waveform would be drawn starting at the left with zero time. Each position
along the horizontal axis indicates the time that has elapsed since zero time, t = 0. When
the vector is horizontal the tip of the vector represents the angles at 0o , 180o and at 360o .
Likewise, when the tip of the vector is vertical it represents the positive peak value, ( +Am
) at 90o or π/2 and the negative peak value, ( -Am ) at 270o or 3π/2. Then the time axis
of the waveform represents the angle either in degrees or radians through which the phasor
has moved. So we can say that a phasor represent a scaled voltage or current value of a
rotating vector which is “frozen” at some point in time, ( t ) and in our example above,
this is at an angle of 30o .
Sometimes when we are analysing alternating waveforms we may need to know the position of the phasor, representing the Alternating Quantity at some particular instant in
time especially when we want to compare two different waveforms on the same axis. For
example, voltage and current. We have assumed in the waveform above that the waveform
starts at time t = 0 with a corresponding phase angle in either degrees or radians.
But if a second waveform starts to the left or to the right of this zero point or we want
to represent in phasor notation the relationship between the two waveforms then we will
need to take into account this phase difference, φof the waveform.
Phase Difference of a Sinusoidal Waveform
Chapter 1. ANALYSIS OF AC CIRCUITS
7
Figure 1.3: Phase Difference of a Sinusoidal Waveform
The generalised mathematical expression to define these two sinusoidal quantities
The current, i is lagging the voltage, v by angle φand in our example above this is 30o . So
the difference between the two phasors representing the two sinusoidal quantities is angle
φand the resulting phasor diagram will be.
Phasor Diagram of a Sinusoidal Waveform
Figure 1.4: Vector Diagram of a Sinusoidal Waveform
The phasor diagram is drawn corresponding to time zero ( t = 0 ) on the horizontal
axis. The lengths of the phasors are proportional to the values of the voltage, ( V )
and the current, ( I ) at the instant in time that the phasor diagram is drawn. The
current phasor lags the voltage phasor by the angle, φ, as the two phasors rotate in an
anticlockwisedirection as stated earlier, therefore the angle, φis also measured in the same
anticlockwise direction.
Figure 1.5: Vector Diagram of a Sinusoidal Waveform
Chapter 1. ANALYSIS OF AC CIRCUITS
8
If however, the waveforms are frozen at time, t = 30o , the corresponding phasor diagram
would look like the one shown on the right. Once again the current phasor lags behind
the voltage phasor as the two waveforms are of the same frequency.
However, as the current waveform is now crossing the horizontal zero axis line at this
instant in time we can use the current phasor as our new reference and correctly say that
the voltage phasor is “leading” the current phasor by angle, φ. Either way, one phasor
is designated as the reference phasor and all the other phasors will be either leading or
lagging with respect to this reference.
Phasor Addition
Sometimes it is necessary when studying sinusoids to add together two alternating waveforms, for example in an AC series circuit, that are not in-phase with each other. If they
are in-phase that is, there is no phase shift then they can be added together in the same
way as DC values to find the algebraic sum of the two vectors. For example, if two voltages of say 50 volts and 25 volts respectively are together “in-phase”, they will add or sum
together to form one voltage of 75 volts (50 + 25).
If however, they are not in-phase that is, they do not have identical directions or starting
point then the phase angle between them needs to be taken into account so they are
added together using phasor diagrams to determine their Resultant Phasor or Vector Sum
by using the parallelogram law.
Consider two AC voltages, V1 having a peak voltage of 20 volts, and V2 having a peak
voltage of 30 volts where V1 leads V2 by 60o . The total voltage, VT of the two voltages
can be found by firstly drawing a phasor diagram representing the two vectors and then
constructing a parallelogram in which two of the sides are the voltages, V1 and V2 as
shown below.
Phasor Addition of two Phasors
By drawing out the two phasors to scale onto graph paper, their phasor sum V1 + V2 can
be easily found by measuring the length of the diagonal line, known as the “resultant rvector”, from the zero point to the intersection of the construction lines 0-A. The downside
of this graphical method is that it is time consuming when drawing the phasors to scale.
Chapter 1. ANALYSIS OF AC CIRCUITS
9
Figure 1.6: Phasor Addition of two Phasors
Also, while this graphical method gives an answer which is accurate enough for most
purposes, it may produce an error if not drawn accurately or correctly to scale. Then one
way to ensure that the correct answer is always obtained is by an analytical method.
Mathematically we can add the two voltages together by firstly finding their “vertical”
and “horizontal” directions, and from this we can then calculate both the “vertical” and
“horizontal” components for the resultant “r vector”, VT. This analytical method which
uses the cosine and sine rule to find this resultant value is commonly called the Rectangular
Form.
In the rectangular form, the phasor is divided up into a real part, x and an imaginary
part, yforming the generalised expression Z = x ± jy. ( we will discuss this in more detail
in the next tutorial ). This then gives us a mathematical expression that represents both
the magnitude and the phase of the sinusoidal voltage.
Definition of a Complex Sinusoid
Phasor Addition using Rectangular Form
Voltage, V2 of 30 volts points in the reference direction along the horizontal zero axis,
then it has a horizontal component but no vertical component as follows.
Chapter 1. ANALYSIS OF AC CIRCUITS
10
Horizontal Component = 30 cos 0o = 30 volts
Vertical Component = 30 sin 0o = 0 volts
This then gives us the rectangular expression for voltage V2 of: 30 + j0
Voltage, V1 of 20 volts leads voltage, V2 by 60o, then it has both horizontal and vertical
components as follows.
Horizontal Component = 20 cos 60o = 20 x 0.5 = 10 volts
Vertical Component = 20 sin 60o = 20 x 0.866 = 17.32 volts
This then gives us the rectangular expression for voltage V1 of: 10 + j17.32
The resultant voltage, VT is found by adding together the horizontal and vertical components as follows.
VHorizontal = sum of real parts of V1 and V2 = 30 + 10 = 40 volts
VVertical = sum of imaginary parts of V1 and V2 = 0 + 17.32 = 17.32 volts
Now that both the real and imaginary values have been found the magnitude of voltage,
VT is determined by simply using Pythagoras’s Theorem for a 90o triangle
Then the resulting phasor diagram will be:
Resultant Value of VT
Phasor subtraction is very similar to the above rectangular method of addition, except
this time the vector difference is the other diagonal of the parallelogram between the two
voltages of V1 and V2 as shown.
Vector Subtraction of two Phasors
Chapter 1. ANALYSIS OF AC CIRCUITS
11
Figure 1.7: Phasor Subtraction
Figure 1.8: Vector Subtraction of two Phasors
This time instead of “adding” together both the horizontal and vertical components we
take them away, subtraction.
Phasor Diagram of a Sinusoidal Waveform
Figure 1.9: Vector Subtraction of two Phasors
As the single vector rotates in an anti-clockwise direction, its tip at point A will rotate
one complete revolution of 360o or 2� representing one complete cycle. If the length of its
moving tip is transferred at different angular intervals in time to a graph as shown above,
a sinusoidal waveform would be drawn starting at the left with zero time. Each position
along the horizontal axis indicates the time that has elapsed since zero time, t = 0. When
the vector is horizontal the tip of the vector represents the angles at 0o, 180o and at 360o.
Chapter 1. ANALYSIS OF AC CIRCUITS
12
Likewise, when the tip of the vector is vertical it represents the positive peak value, (
+Am ) at 90o or π/2 and the negative peak value, ( -Am ) at 270o or 3π/2. Then the
time axis of the waveform represents the angle either in degrees or radians through which
the phasor has moved. So we can say that a phasor represent a scaled voltage or current
value of a rotating vector which is “frozen” at some point in time, ( t ) and in our example
above, this is at an angle of 30o.
Sometimes when we are analysing alternating waveforms we may need to know the position of the phasor, representing the Alternating Quantity at some particular instant in
time especially when we want to compare two different waveforms on the same axis. For
example, voltage and current. We have assumed in the waveform above that the waveform
starts at time t = 0 with a corresponding phase angle in either degrees or radians.
But if a second waveform starts to the left or to the right of this zero point or we want
to represent in phasor notation the relationship between the two waveforms then we will
need to take into account this phase difference, φof the waveform. Consider the diagram
below from the previous Phase Difference tutorial.
Phase Difference of a Sinusoidal Waveform
Figure 1.10: Phase Difference of a Sinusoidal Waveform
The generalised mathematical expression to define these two sinusoidal quantities will be
written as:
The current, i is lagging the voltage, v by angle φand in our example above this is 30o. So
the difference between the two phasors representing the two sinusoidal quantities is angle
φand the resulting phasor diagram will be.
Phasor Diagram of a Sinusoidal Waveform
The phasor diagram is drawn corresponding to time zero ( t = 0 ) on the horizontal
axis. The lengths of the phasors are proportional to the values of the voltage, ( V )
Chapter 1. ANALYSIS OF AC CIRCUITS
13
Figure 1.11: Phasor Diagram of a Sinusoidal Waveform
and the current, ( I ) at the instant in time that the phasor diagram is drawn. The
current phasor lags the voltage phasor by the angle, φ, as the two phasors rotate in an
anticlockwisedirection as stated earlier, therefore the angle, φis also measured in the same
anticlockwise direction.
Figure 1.12: Phasor Diagram of a Sinusoidal Waveform
If however, the waveforms are frozen at time, t = 30o, the corresponding phasor diagram
would look like the one shown on the right. Once again the current phasor lags behind
the voltage phasor as the two waveforms are of the same frequency.
However, as the current waveform is now crossing the horizontal zero axis line at this
instant in time we can use the current phasor as our new reference and correctly say that
the voltage phasor is “leading” the current phasor by angle, Φ. Either way, one phasor
is designated as the reference phasor and all the other phasors will be either leading or
lagging with respect to this reference.
Phase Difference and Phase Shift
Phase Difference is used to describe the difference in degrees or radians when two or
more alternating quantities reach their maximum or zero values Previously we saw that a
Sinusoidal Waveform is an alternating quantity that can be presented graphically in the
time domain along an horizontal zero axis. We also saw that as an alternating quantity,
Chapter 1. ANALYSIS OF AC CIRCUITS
14
sine waves have a positive maximum value at time �/2, a negative maximum value at time
3�/2, with zero values occurring along the baseline at 0, � and 2�.
However, not all sinusoidal waveforms will pass exactly through the zero axis point at the
same time, but may be “shifted” to the right or to the left of 0o by some value when
compared to another sine wave.
For example, comparing a voltage waveform to that of a current waveform. This then
produces an angular shift or Phase Difference between the two sinusoidal waveforms. Any
sine wave that does not pass through zero at t = 0 has a phase shift.
The phase difference or phase shift as it is also called of a Sinusoidal Waveform is the angle
φ(Greek letter Phi), in degrees or radians that the waveform has shifted from a certain
reference point along the horizontal zero axis. In other words phase shift is the lateral
difference between two or more waveforms along a common axis and sinusoidal waveforms
of the same frequency can have a phase difference.
The phase difference, φof an alternating waveform can vary from between 0 to its maximum
time period, T of the waveform during one complete cycle and this can be anywhere along
the horizontal axis between, φ= 0 to 2π(radians) or π= 0 to 360o depending upon the
angular units used. Phase difference can also be expressed as a time shift of T in seconds
representing a fraction of the time period, T for example, +10mS or – 50µS but generally
it is more common to express phase difference as an angular measurement.
Then the equation for the instantaneous value of a sinusoidal voltage or current waveform
we developed in the previous Sinusoidal Waveform will need to be modified to take account
of the phase angle of the waveform and this new general expression becomes.
Phase Difference Equation
Am - is the amplitude of the waveform.
ωt - is the angular frequency of the waveform in radian/sec.
φ- is the phase angle in degrees or radians that the waveform has shifted either left or right
from the reference point.
Chapter 1. ANALYSIS OF AC CIRCUITS
15
If the positive slope of the sinusoidal waveform passes through the horizontal axis “before”
t = 0 then the waveform has shifted to the left so φ>0, and the phase angle will be positive
in nature, +φgiving a leading phase angle. In other words it appears earlier in time than
0o producing an anticlockwise rotation of the vector.
Likewise, if the positive slope of the sinusoidal waveform passes through the horizontal
x-axis some time “after” t = 0 then the waveform has shifted to the right so φ<0, and
the phase angle will be negative in nature -φproducing a lagging phase angle as it appears
later in time than 0o producing a clockwise rotation of the vector. Both cases are shown
below.
Phase Relationship of a Sinusoidal Waveform
Figure 1.13: Phase Relationship of a Sinusoidal Waveform
Two Sinusoidal Waveforms – “in-phase”
Figure 1.14: Phase Relationship of a Sinusoidal Waveform
Now lets consider that the voltage, v and the current, i have a phase difference between
themselves of 30o, so (φ= 30o or �/6 radians). As both alternating quantities rotate at the
same speed, i.e. they have the same frequency, this phase difference will remain constant
for all instants in time, then the phase difference of 30o between the two quantities is
represented by phi, φas shown below.
Chapter 1. ANALYSIS OF AC CIRCUITS
16
Phase Difference of a Sinusoidal Waveform
The voltage waveform above starts at zero along the horizontal reference axis, but at that
same instant of time the current waveform is still negative in value and does not cross
this reference axis until 30o later. Then there exists a Phase difference between the two
waveforms as the current cross the horizontal reference axis reaching its maximum peak
and zero values after the voltage waveform.
As the two waveforms are no longer “in-phase”, they must therefore be “out-of-phase” by
an amount determined by phi, φand in our example this is 30o. So we can say that the
two waveforms are now 30o out-of phase. The current waveform can also be said to be
“lagging” behind the voltage waveform by the phase angle, φ. Then in our example above
the two waveforms have a Lagging Phase Difference so the expression for both the voltage
and current.
AC Inductance and Inductive Reactance:
The opposition to current flow through an AC Inductor is called Inductive Reactance and
which depends lineally on the supply frequency Inductors and chokes are basically coils
or loops of wire that are either wound around a hollow tube former (air cored) or wound
around some ferromagnetic material (iron cored) to increase their inductive value called
inductance.
Inductors store their energy in the form of a magnetic field that is created when a voltage
is applied across the terminals of an inductor. The growth of the current flowing through
the inductor is not instant but is determined by the inductors own self-induced or back
emf value. Then for an inductor coil, this back emf voltage VL is proportional to the rate
of change of the current flowing through it.
Chapter 1. ANALYSIS OF AC CIRCUITS
17
This current will continue to rise until it reaches its maximum steady state condition which
is around five time constants when this self-induced back emf has decayed to zero. At this
point a steady state current is flowing through the coil, no more back emf is induced
to oppose the current flow and therefore, the coil acts more like a short circuit allowing
maximum current to flow through it.
However, in an alternating current circuit which contains an AC Inductance, the flow of
current through an inductor behaves very differently to that of a steady state DC voltage.
Now in an AC circuit, the opposition to the current flowing through the coils windings not
only depends upon the inductance of the coil but also the frequency of the applied voltage
waveform as it varies from its positive to negative values.
The actual opposition to the current flowing through a coil in an AC circuit is determined
by the AC Resistance of the coil with this AC resistance being represented by a complex
number. But to distinguish a DC resistance value from an AC resistance value, which is
also known as Impedance, the term Reactance is used.
Like resistance, reactance is measured in Ohm’s but is given the symbol “X” to distinguish
it from a purely resistive “R” value and as the component in question is an inductor, the
reactance of an inductor is called Inductive Reactance, ( XL ) and is measured in Ohms.
Its value can be found from the formula.
Inductive Reactance
So whenever a sinusoidal voltage is applied to an inductive coil, the back emf opposes the
rise and fall of the current flowing through the coil and in a purely inductive coil which has
zero resistance or losses, this impedance (which can be a complex number) is equal to its
inductive reactance. Also reactance is represented by a vector as it has both a magnitude
and a direction (angle). Consider the circuit below.
AC Inductance with a Sinusoidal Supply
This simple circuit above consists of a pure inductance of L Henries ( H ), connected across
a sinusoidal voltage given by the expression: V(t) = Vmax sin ωt. When the switch is
closed this sinusoidal voltage will cause a current to flow and rise from zero to its maximum
Chapter 1. ANALYSIS OF AC CIRCUITS
18
Figure 1.15: AC Inductance with a Sinusoidal Supply
value. This rise or change in the current will induce a magnetic field within the coil which
in turn will oppose or restrict this change in the current.
But before the current has had time to reach its maximum value as it would in a DC
circuit, the voltage changes polarity causing the current to change direction. This change
in the other direction once again being delayed by the self-induced back emf in the coil,
and in a circuit containing a pure inductance only, the current is delayed by 90o.
The applied voltage reaches its maximum positive value a quarter ( 1/4f ) of a cycle earlier
than the current reaches its maximum positive value, in other words, a voltage applied to
a purely inductive circuit “LEADS” the current by a quarter of a cycle or 90o as shown
below.
Sinusoidal Waveforms for AC Inductance
Figure 1.16: Sinusoidal Waveforms for AC Inductance
This effect can also be represented by a phasor diagram were in a purely inductive circuit
the voltage “LEADS” the current by 90o. But by using the voltage as our reference, we
can also say that the current “LAGS” the voltage by one quarter of a cycle or 90o as shown
in the vector diagram below.
Chapter 1. ANALYSIS OF AC CIRCUITS
19
Phasor Diagram for AC Inductance
Figure 1.17: Phasor Diagram for AC Inductance
So for a pure loss less inductor, VL “leads” IL by 90o, or we can say that IL “lags” VL by
90o.
There are many different ways to remember the phase relationship between the voltage
and current flowing through a pure inductor circuit, but one very simple and easy to
remember way is to use the mnemonic expression “ELI” (pronounced Ellie as in the girls
name). ELI stands for Electromotive force first in an AC inductance, L before the current
I. In other words, voltage before the current in an inductor, E, L, I equals “ELI”, and
whichever phase angle the voltage starts at, this expression always holds true for a pure
inductor circuit.
The Effect of Frequency on Inductive Reactance
When a 50Hz supply is connected across a suitable AC Inductance, the current will be
delayed by 90o as described previously and will obtain a peak value of I amps before the
voltage reverses polarity at the end of each half cycle, i.e. the current rises up to its
maximum value in “T secs“.
If we now apply a 100Hz supply of the same peak voltage to the coil, the current will
still be delayed by 90o but its maximum value will be lower than the 50Hz value because
the time it requires to reach its maximum value has been reduced due to the increase in
frequency because now it only has “1/2 T secs” to reach its peak value. Also, the rate of
change of the flux within the coil has also increased due to the increase in frequency.
Then from the above equation for inductive reactance, it can be seen that if either the
Frequency OR the Inductance is increased the overall inductive reactance value of the
coil would also increase. As the frequency increases and approaches infinity, the inductors
Chapter 1. ANALYSIS OF AC CIRCUITS
20
reactance and therefore its impedance would also increase towards infinity acting like an
open circuit.
Likewise, as the frequency approaches zero or DC, the inductors reactance would also
decrease to zero, acting like a short circuit. This means then that inductive reactance is
“directly proportional to frequency” and has a small value at low frequencies and a high
value at higher frequencies as shown.
Inductive Reactance against Frequency
Figure 1.18: Inductive Reactance against Frequency
The inductive reactance of an inductor increases as the frequency across it increases therefore inductive reactance is proportional to frequency ( XL αf) as the back emf generated
in the inductor is equal to its inductance multiplied by the rate of change of current in the
inductor.
Also as the frequency increases the current flowing through the inductor also reduces in
value.
We can present the effect of very low and very high frequencies on a the reactance of a
pure AC Inductance as follows:
Figure 1.19: Inductive Reactance against Frequency
Chapter 1. ANALYSIS OF AC CIRCUITS
21
So how did we arrive at this equation. Well the self induced emf in the inductor is
determined by Faraday’s Law that produces the effect of self-induction in the inductor
due to the rate of change of the current and the maximum value of the induced emf will
correspond to the maximum rate of change. Then the voltage in the inductor coil is given
as:
Where: VL = IωL which is the voltage amplitude and θ= + 90o which is the phase
difference or phase angle between the voltage and current.
In the Phasor Domain
In the phasor domain the voltage across the coil is given as:
1.2.1 AC Capacitance and Capacitive Reactance
The opposition to current flow through an AC Capacitor is called Capacitive Reactance and
which itself is inversely proportional to the supply frequency Capacitors store energy on
Chapter 1. ANALYSIS OF AC CIRCUITS
22
their conductive plates in the form of an electrical charge. When a capacitor is connected
across a DC supply voltage it charges up to the value of the applied voltage at a rate
determined by its time constant.
A capacitor will maintain or hold this charge indefinitely as long as the supply voltage
is present. During this charging process, a charging current, i flows into the capacitor
opposed by any changes to the voltage at a rate which is equal to the rate of change of the
electrical charge on the plates. A capacitor therefore has an opposition to current flowing
onto its plates.
The relationship between this charging current and the rate at which the capacitors supply
voltage changes can be defined mathematically as: i = C(dv/dt), where C is the capacitance value of the capacitor in farads and dv/dt is the rate of change of the supply voltage
with respect to time. Once it is “fully-charged” the capacitor blocks the flow of any more
electrons onto its plates as they have become saturated and the capacitor now acts like a
temporary storage device.
A pure capacitor will maintain this charge indefinitely on its plates even if the DC supply
voltage is removed. However, in a sinusoidal voltage circuit which contains “AC Capacitance”, the capacitor will alternately charge and discharge at a rate determined by the
frequency of the supply. Then capacitors in AC circuits are constantly charging and discharging respectively.
When an alternating sinusoidal voltage is applied to the plates of an AC capacitor, the
capacitor is charged firstly in one direction and then in the opposite direction changing
polarity at the same rate as the AC supply voltage. This instantaneous change in voltage
across the capacitor is opposed by the fact that it takes a certain amount of time to
deposit (or release) this charge onto the plates and is given by V = Q/C. Consider the
circuit below.
1.3 AC Capacitance with a Sinusoidal Supply
When the switch is closed in the circuit above, a high current will start to flow into the
capacitor as there is no charge on the plates at t = 0. The sinusoidal supply voltage, V
Chapter 1. ANALYSIS OF AC CIRCUITS
23
Figure 1.20: AC Capacitance with a Sinusoidal Supply
is increasing in a positive direction at its maximum rate as it crosses the zero reference
axis at an instant in time given as 0o. Since the rate of change of the potential difference
across the plates is now at its maximum value, the flow of current into the capacitor will
also be at its maximum rate as the maximum amount of electrons are moving from one
plate to the other.
As the sinusoidal supply voltage reaches its 90o point on the waveform it begins to slow
down and for a very brief instant in time the potential difference across the plates is
neither increasing nor decreasing therefore the current decreases to zero as there is no rate
of voltage change. At this 90o point the potential difference across the capacitor is at
its maximum ( Vmax ), no current flows into the capacitor as the capacitor is now fully
charged and its plates saturated with electrons.
At the end of this instant in time the supply voltage begins to decrease in a negative
direction down towards the zero reference line at 180o. Although the supply voltage is
still positive in nature the capacitor starts to discharge some of its excess electrons on its
plates in an effort to maintain a constant voltage. This results in the capacitor current
flowing in the opposite or negative direction.
When the supply voltage waveform crosses the zero reference axis point at instant 180othe
rate of change or slope of the sinusoidal supply voltage is at its maximum but in a negative
direction, consequently the current flowing into the capacitor is also at its maximum rate
at that instant. Also at this 180o point the potential difference across the plates is zero as
the amount of charge is equally distributed between the two plates.
Then during this first half cycle 0o to 180o the applied voltage reaches its maximum
positive value a quarter (1/4f) of a cycle after the current reaches its maximum positive
Chapter 1. ANALYSIS OF AC CIRCUITS
24
value, in other words, a voltage applied to a purely capacitive circuit “LAGS” the current
by a quarter of a cycle or 90o as shown below.
1.3.1 Sinusoidal Waveforms for AC Capacitance
Figure 1.21: Sinusoidal Waveforms for AC Capacitance
During the second half cycle 180o to 360o, the supply voltage reverses direction and heads
towards its negative peak value at 270o. At this point the potential difference across the
plates is neither decreasing nor increasing and the current decreases to zero. The potential
difference across the capacitor is at its maximum negative value, no current flows into the
capacitor and it becomes fully charged the same as at its 90o point but in the opposite
direction.
As the negative supply voltage begins to increase in a positive direction towards the
360opoint on the zero reference line, the fully charged capacitor must now loose some
of its excess electrons to maintain a constant voltage as before and starts to discharge
itself until the supply voltage reaches zero at 360o at which the process of charging and
discharging starts over again.
From the voltage and current waveforms and description above, we can see that the current
is always leading the voltage by 1/4 of a cycle or �/2 = 90o “out-of-phase” with the potential
difference across the capacitor because of this charging and discharging process. Then the
phase relationship between the voltage and current in an AC capacitance circuit is the
exact opposite to that of an AC Inductance we saw in the previous tutorial.
Chapter 1. ANALYSIS OF AC CIRCUITS
25
This effect can also be represented by a phasor diagram where in a purely capacitive circuit
the voltage “LAGS” the current by 90o. But by using the voltage as our reference, we can
also say that the current “LEADS” the voltage by one quarter of a cycle or 90o as shown
in the vector diagram below.
1.3.2 Phasor Diagram for AC Capacitance
Figure 1.22: Phasor Diagram for AC Capacitance
So for a pure capacitor, VC “lags” IC by 90o, or we can say that IC “leads” VC by 90o.
There are many different ways to remember the phase relationship between the voltage
and current flowing in a pure AC capacitance circuit, but one very simple and easy to
remember way is to use the mnemonic expression called “ICE”. ICE stands for current
Ifirst in an AC capacitance, C before Electromotive force. In other words, current before
the voltage in a capacitor, I, C, E equals “ICE”, and whichever phase angle the voltage
starts at, this expression always holds true for a pure AC capacitance circuit.
Capacitive Reactance
So we now know that capacitors oppose changes in voltage with the flow of electrons onto
the plates of the capacitor being directly proportional to the rate of voltage change across
its plates as the capacitor charges and discharges. Unlike a resistor where the opposition
to current flow is its actual resistance, the opposition to current flow in a capacitor is
called Reactance.
Like resistance, reactance is measured in Ohm’s but is given the symbol X to distinguish
it from a purely resistive R value and as the component in question is a capacitor, the
reactance of a capacitor is called Capacitive Reactance, ( XC ) which is measured in Ohms.
Chapter 1. ANALYSIS OF AC CIRCUITS
26
Since capacitors charge and discharge in proportion to the rate of voltage change across
them, the faster the voltage changes the more current will flow. Likewise, the slower the
voltage changes the less current will flow. This means then that the reactance of an AC
capacitor is “inversely proportional” to the frequency of the supply as shown.
Capacitive Reactance against Frequency
Figure 1.23: Capacitive Reactance against Frequency
Capacitive reactance of a capacitor decreases as the frequency across its plates increases.
Therefore, capacitive reactance is inversely proportional to frequency. Capacitive reactance
opposes current flow but the electrostatic charge on the plates (its AC capacitance value)
remains constant.
This means it becomes easier for the capacitor to fully absorb the change in charge on its
plates during each half cycle. Also as the frequency increases the current flowing into the
capacitor increases in value because the rate of voltage change across its plates increases.
We can present the effect of very low and very high frequencies on the reactance of a pure
AC Capacitance as follows:
Where: IC = V/(1/�C) (or IC = V/XC) is the current magnitude and � = + 90o which
is the phase difference or phase angle between the voltage and current. For a purely
capacitive circuit, Ic leads Vc by 90o, or Vc lags Ic by 90.
Chapter 2
SOLUTION OF FIRST AND SECOND
ORDER NETWORKS
Course Outcomes
After successful completion of this module, students should be able to:
CO 3
CO 4
Illustrate the initial and steady state conditions of R L and C
Under-
parameters
stand
Solve the transient response of first and second order electric
Apply
circuits using differential equation and Laplace transform techniques.
2.1 Introduction
If the output of an electric circuit for an input varies with respect to time, then it is called
as time response. The time response consists of following two parts.
Transient Response
Steady state Response
In this chapter, first let us discuss about these two responses and then observe these two
responses in a series RL circuit, when it is excited by a DC voltage source.
Transient Response
27
Chapter 2. SOLUTION OF FIRST AND SECOND ORDER NETWORKS
28
After applying an input to an electric circuit, the output takes certain time to reach steady
state. So, the output will be in transient state till it goes to a steady state. Therefore, the
response of the electric circuit during the transient state is known as transient response.
The transient response will be zero for large values of ‘t’. Ideally, this value of ‘t’ should
be infinity. But, practically five time constants are sufficient. Presence or Absence of
Transients
Transients occur in the response due to sudden change in the sources that are applied to
the electric circuit and / or due to switching action. There are two possible switching
actions. Those are opening switch and closing switch. The transient part will not present
in the response of an electrical circuit or network, if it contains only resistances. Because
resistor is having the ability to adjust any amount of voltage and current. The transient
part occurs in the response of an electrical circuit or network due to the presence of energy
storing elements such as inductor and capacitor. Because they can’t change the energy
stored in those elements instantly. For higher order differential equation, the number of
arbitrary constants equals the order of the equation. If these unknowns are to be evaluated
for particular solution, other conditions in network must be known. A set of simultaneous
equations must be formed containing general solution and some other equations to match
number of unknown with equations. We assume that at reference time t=0, network
condition is changed by switching action. Assume that switch operates in zero time. The
network conditions at this instant are called initial conditions in network
2.1.1 Resistor
Figure 2.1: Resistor
Equation 1 is linear and also time dependent. This indicates that current through resistor
changes if applied voltage changes instantaneously. Thus in resistor, change in current is
instantaneous as there is no storage of energy in it.
Chapter 2. SOLUTION OF FIRST AND SECOND ORDER NETWORKS
2.1.2
29
Inductor
If dc current flows through inductor, dil/dt becomes zero as dc current is constant with
respect to time. Hence voltage across inductor, VL becomes zero. Thus, as for as dc
quantities are considered, in steady stake, inductor acts as short circuit.
Figure 2.2: Inductor
2.1.3
capacitor
Figure 2.3: capacitor
If dc voltage is applied to capacitor, dVC / dt becomes zero as dc voltage is constant with
respect to time.
Hence the current through capacitor iC becomes zero, Thus as far as dc quantities are
considered capacitor acts as open circuit.
Chapter 2. SOLUTION OF FIRST AND SECOND ORDER NETWORKS
2.1.4
30
Initial Condition for (DC steady state solution)
• Initial condition: response of a circuit before a switch is first activated. – Since power
equals energy per unit time, finite power requires continuous change in energy.
• Primary variables: capacitor voltages and inductor currents-> energy storage elements
� Capacitor voltages and inductor currents cannot change instantaneously but should be
continuous. -> continuity of capacitor voltages and inductor currents
� The value of an inductor current or a capacitor voltage just prior to the closing (or
opening) of a switch is equal to the value just after the switch has been closed (or opened).
2.1.5
Response of Series RL Circuit (DC Excitation):
In the preceding lesson, our discussion focused extensively on dc circuits having resistances
with either inductor (L) or capacitor (C) (i.e., single storage element) but not both. Dynamic response of such first order system has been studied and discussed in detail. The
presence of resistance, inductance, and capacitance in the dc circuit introduces at least a
second order differential equation or by two simultaneous coupled linear first order differential equations. We shall see in next section that the complexity of analysis of second
order circuits increases significantly when compared with that encountered with first order
circuits. Initial conditions for the circuit variables and their derivatives play an important
role and this is very crucial to analyze a second order dynamic system.
Chapter 2. SOLUTION OF FIRST AND SECOND ORDER NETWORKS
31
Figure 2.4: RL Circuit
2.2 Response of Series RL Circuit (AC Excitation):
In the below circuit, the switch was kept open up to t = 0 and it was closed at t = 0.
So, the AC voltage source having a peak voltage of Vm volts is not connected to the
series RL circuit up to this instant. Therefore, there is no initial current flows through
the inductor.Now, the current i(t) flows in the entire circuit, since the AC voltage source
having a peak voltage of Vm volts is connected to the series RL circuit. We know that the
current i(t) flowing through the above circuit will have two terms, one that represents the
transient part and other term represents the steady state.
If a sinusoidal signal is applied as an input to a Linear electric circuit, then it produces
Figure 2.5: RL Circuit
a steady state output, which is also a sinusoidal signal. Both the input and output sinusoidal signals will be having the same frequency, but different amplitudes and phase
angles. We can calculate the steady state response of an electric circuit, when it is excited by a sinusoidal voltage source using Laplace Transform approach. The s-domain
circuit diagram, when the switch is in closed position, is shown in the following figure.In
the above circuit, all the quantities and parameters are represented in s-domain. These
are the Laplace transforms of time-domain quantities and parameters.the current flowing
through the series RL circuit, when it is excited by a sinusoidal voltage source. It is having
two terms. The first and second terms represent the transient response and steady state
Chapter 2. SOLUTION OF FIRST AND SECOND ORDER NETWORKS
32
response of the current respectively. We can neglect the first term of Equation 4 because
its value will be very much less than one.
2.3 Differential Equation and Laplace Transform Approach:
Using the Laplace transform as part of your circuit analysis provides you with a prediction
of circuit response. Analyze the poles of the Laplace transform to get a general idea of
output behavior. Real poles, for instance, indicate exponential output behavior. Follow
these basic steps to analyze a circuit using Laplace techniques: Develop the differential
equation in the time-domain using Kirchhoff’s laws and element equations. Apply the
Laplace transformation of the differential equation to put the equation in the s-domain.
Algebraically solve for the solution, or response transform. Apply the inverse Laplace
transformation to produce the solution to the original differential equation described in the
time-domain. To get comfortable with this process, you simply need to practice applying
it to different types of circuits such as an RC (resistor-capacitor) circuit, an RL (resistorinductor) circuit, and an RLC (resistor-inductor-capacitor) circuit.
Chapter 3
LOCUS DIAGRAMS AND NETWORKS
FUNCTIONS
Course Outcomes
After successful completion of this module, students should be able to:
CO 5
Recall the concept of locus diagram for series and parallel circuits
Remember
CO 6
Illustrate the concept of network functions for one port and two
Under-
port networks.
stand
3.1 Introduction
Locus diagrams are the graphical representations of the way in which the response of
electrical circuits vary, when one or more parameters are continuously changing. They
help us to study the way in which a. Current / power factor vary, when voltage is kept
constant, b. Voltage / power factor vary, when current is kept constant, when one of the
parameters of the circuit (whether series or parallel) is varied. The Locus diagrams yield
such important information as Imax, Imin , Vmax ,Vmin the power factor‘s at which they
occur. In some parallel circuits, they will also indicate whether or not, a condition for
response is possible.
33
Chapter 3. LOCUS DIAGRAMS AND NETWORKS FUNCTIONS
3.2
34
RL Series Circuit
There are two categories, namely Booster propulsion and Auxiliary propulsion Booster
propulsion are used for boosting a Consider an R – XL series circuit as shown below,
across which a constant voltage is applied. By varying R or XL, a wide range of currents
and potential differences can be obtained. ‘R ‘ can be varied by the rheostatic adjustment
and XL can be varied by using a variable inductor or by applying a variable frequency
source. When the variations are uniform and lie between 0 and infinity, the resulting locus
diagrams are circles
Figure 3.1: When R is varied and XL fixed circuit
Figure 3.2: When R is varied and XL fixed locus diagram
Figure 3.3: When XL is varied and R fixed circuit
Figure 3.4: When XL is varied and R fixed locus diagram
Chapter 3. LOCUS DIAGRAMS AND NETWORKS FUNCTIONS
3.3
35
RC Series Circuit
There are two categories, namely Booster propulsion and Auxiliary propulsion Booster
propulsion are used for boosting a Consider an R – XC series circuit as shown below,
across which a constant voltage is applied. By varying R or XC, a wide range of currents
and potential differences can be obtained. ‘R ‘ can be varied by the rheostatic adjustment
and XC can be varied by using a variable inductor or by applying a variable frequency
source. When the variations are uniform and lie between infinity and 0, the resulting locus
diagrams are circles
Figure 3.5: When R is varied and XC fixed circuit
Figure 3.6: When R is varied and XC fixed locus diagram
Figure 3.7: When XC is varied and R fixed circuit
Figure 3.8: When XC is varied and R fixed locus diagram
Chapter 3. LOCUS DIAGRAMS AND NETWORKS FUNCTIONS
3.4
36
RLC Series Circuit
There are two categories, namely Booster propulsion and Auxiliary propulsion Booster
propulsion are used for boosting a Consider an R – XC-XL series circuit as shown below,
across which a constant voltage is applied. By varying R or XC or XL, a wide range of
currents and potential differences can be obtained. ‘R ‘ can be varied by the rheostatic
adjustment and XC or XL can be varied by using a variable inductor or by applying a
variable frequency source. When the variations are uniform and lie between 0 and 0, the
resulting locus diagrams are circles
Figure 3.9: RLC series circuit
Figure 3.10: RLC series circuit locus diagram
3.5 Network Functions
A network function is the Laplace transform of an impulse response. Its format is a ratio
of two polynomials of the complex frequencies. Consider the general two-port network
shown in Figure 2.2a. The terminal voltages and currents of the two-port can be related
by two classes of network functions, namely, the driving point functions and the transfer
functionsThe driving point functions relate the voltage at a port to the current at the
same port. Thus, these functions are a property of a single port. For the input port the
driving point impedance function ZIN(s) is defined as: This function can be measured by
Chapter 3. LOCUS DIAGRAMS AND NETWORKS FUNCTIONS
37
Figure 3.11: Network
observing the current IIN when the input port is driven by a voltage source VIN . The
driving point admittance function YIN(s) is the reciprocal of the impedance function, and
is given by: The output port driving point functions are denned in a similar way. The
transfer functions of the two-port relate the voltage (or current) at one port to the voltage
(or current) at the other port. The possible forms of transfer functions are:
1. The voltage transfer function, which is a ratio of one voltage to another voltage.
2. The current transfer function, which is a ratio of one current to another current.
3. The transfer impedance function, which is the ratio of a voltage to a current.
4. The transfer admittance function, which is the ratio of a current to a voltage.
The voltage transfer functions are defined with the output port an open circuit, as:
To evaluate the voltage gain, for example, the output voltage VO is measured with the
input port driven by a voltage source VIN . The other three types of transfer functions
can be defined in a similar manner. Of the four types of transfer functions, the voltage
transfer function is the one most often specified in the design of filters. The functions
defined above, when realized using resistors, inductors, capacitors, and active devices, can
be shown to be the ratios of polynomials in s with real coefficients. This is so because the
network functions are obtained by solving simple algebraic node equations, which involve
at most the terms R, sL, sC and their reciprocals. The active device, if one exists, the
solution still involves only the addition and multiplication of simple terms, which can only
lead to a ratio of polynomials in s. In addition, all the coefficients of the numerator and
denominator polynomials will be real. Thus, the general form of a network function
Chapter 3. LOCUS DIAGRAMS AND NETWORKS FUNCTIONS
3.6
38
Properties of all Network Functions
We have already seen that network functions are ratios of polynomials in s with real
coefficients. A consequence of this property is that complex poles (and zeros) must occur
in conjugate pairs. To demonstrate this fact consider a complex root at (s = -a – jb)
which leads to the factor (s + a + jb) in the network function. The jb term will make
some of the coefficients complex in the polynomial, unless the conjugate of the complex
root at (s = -a + jb) is also present in the polynomial. The product of a complex factor
and its conjugate is Further important properties of network functions are obtained by
restricting the networks to be stable, by which we mean that a bounded input excitation
to the network must yield a bounded response. Put differently, the output of a stable
network cannot be made to increase indefinitely by the application of a bounded input
excitation. Passive networks are stable by their very nature, since they do not contain
energy sources that might inject additional energy into the network. Active networks,
however, do contain energy sources that could join forces with the input excitation to
make the output increase indefinitely. Such unstable networks, however, have no use in
the world of practical filters and are therefore precluded from all our future discussions.
A convenient way of determining the stability of the general network function H(s)
is by considering its response to an impulse function, which is obtained by taking the
inverse Laplace transform of the partial fraction expansion of the function.
3.7 The Pole Zero Plot
A system is characterized by its poles and zeros in the sense that they allow reconstruction of the input/output differential equation. In general, the poles and zeros of a transfer
function may be complex, and the system dynamics may be represented graphically by
plotting their locations on the complex s-plane, whose axes represent the real and imaginary parts of the complex variable s. Such plots are known as pole-zero plots. It is usual
to mark a zero location by a circle and a pole location a cross . The location of the poles
and zeros provide qualitative insights into the response characteristics of a system.
Chapter 4
THREE PHASE AC CIRCUITS
Course Outcomes
After successful completion of this module, students should be able to:
CO 7
CO 8
Explain the importance of three phase circuits and analyze the
Under-
star delta connected balanced and unbalanced loads. .
stand
Demonstrate the three-phase active power and reactive power
Under-
using two wattmeter and one wattmeter methods, respectively.
stand
4.1 Introduction
Three-phase systems are commonly used in generation, transmission and distribution of
electric power. Power in a three-phase system is constant rather than pulsating and threephase motors start and run much better than single-phase motors. A three-phase system
is a generator-load pair in which the generator produces three sinusoidal voltages of equal
amplitude and frequency but differing in phase by 120 from each other.
There are two types of system available in electric circuit, single phase and three phase
system. In single phase circuit, there will be only one phase, i.e the current will flow
through only one wire and there will be one return path called neutral line to complete
the circuit. So in single phase minimum amount of power can be transported. Here the
generating station and load station will also be single phase. This is an old system using
39
Chapter 1. THREE PHASE AC CIRCUITS
40
from previous time. In polyphase system, that more than one phase can be used for
generating, transmitting and for load system. Three phase circuit is the polyphase system
where three phases are send together from the generator to the load. Each phase are
having a phase difference of 120, i.e 120 angle electrically. So from the total of 360, three
phases are equally divided into 120° each. The power in three phase system is continuous
as all the three phases are involved in generating the total power. The sinusoidal waves
for 3 phase system is shown below The three phases can be used as single phase each. So
if the load is single phase, then one phase can be taken from the three phase circuit and
the neutral can be used as ground to complete the circuit.
Figure 4.1: Three Phase Waveform
4.2 Advantages of Three Phase is preferred over Single Phase
The three phase system can be used as three single phase line so it can act as three single
phase system. The three phase generation and single phase generation is same in the
generator except the arrangement of coil in the generator to get 120° phase difference.
The conductor needed in three phase circuit is 75 that of conductor needed in single phase
circuit. And also the instantaneous power in single phase system falls down to zero as in
single phase we can see from the sinusoidal curve but in three phase system the net power
from all the phases gives a continuous power to the load. the will have better and higher
efficiency compared to the single phase system.
Chapter 1. THREE PHASE AC CIRCUITS
41
In three phase circuit, connections can be given in two types:
1. Star connection
2. Delta connection
4.3 Star Connection
In star connection, there is four wire, three wires are phase wire and fourth is neutral
which is taken from the star point.
Star connection is preferred for long distance power transmission because it is having the
neutral point. In this we need to come to the concept of balanced and unbalanced current
in power system.When equal current will flow through all the three phases, then it is called
as balanced current. And when the current will not be equal in any of the phase, then
it is unbalanced current. In this case, during balanced condition there will be no current
flowing through the neutral line and hence there is no use of the neutral terminal. But
when there will be unbalanced current flowing in the three phase circuit, neutral is having
a vital role. It will take the unbalanced current through to the ground and protect the
transformer. Unbalanced current affects transformer and it may also cause damage to the
transformer and for this star connection is preferred for long distance transmission. In star
connection, the line voltage is 1.7321times of phase voltage. Line voltage is the voltage
between two phases in three phase circuit and phase voltage is the voltage between one
phase to the neutral line. And the current is same for both line and phase.
the finishing ends or starting ends of the three phase windings are connected to a common
point as shown in. A’, B’, C’ are connected to a common point called neutral point. The
other ends A, B, C are called line terminals and the common terminal neutral are brought
outside. Then it is called a 3 phase 4 wire star connected systems. If neutral point is not
available, then it is called 3 phase, 3 wire star connection
4.4 Delta Connection
In delta connection, there are three wires alone and no neutral terminal is taken. Normally
delta connection is preferred for short distance due to the problem of unbalanced current
Chapter 1. THREE PHASE AC CIRCUITS
42
Figure 4.2: Star connection
in the circuit. The figure is shown below for delta connection. In the load station, ground
can be used as neutral path if required. In delta connection, the line voltage is same with
that of phase voltage. And the line current is 1.7321 times of phase current.
Delta connection: in this form of interconnection the dissimilar ends of the three coils i.e
A and B’, B and C’, and C and A’ are connected to form a closed Δ circuit (starting
end of one phase is connected to finishing end of the next phase). The three junction are
brought outside as line terminal A, B, C. the three phase windings are connected in series
and form a closed path. The sum of the voltages in the closed path for balanced system of
voltages at any instant will be zero fig. The main advantage of star connection is that we
can have two different 3phase voltages. The voltage that was the line terminals between A
B, BC, and C A are called line voltages and form a balanced three phase voltage. Another
voltage is between the terminals A N, B N, and C N are called phase voltage and form
another balanced three phase voltage (line to neutral voltage or wye voltage).
Figure 4.3: Star connection
In three phase circuit, star and delta connection can be arranged in four different ways
1. Star-Star connection
2. Star-Delta connection
3. Delta-Star connection
4. Delta-Delta connection
Chapter 1. THREE PHASE AC CIRCUITS
43
4.5 Phase Sequence
But the power is independent of the circuit arrangement of the three phase system. The
net power in the circuit will be same in both star and delta connection. The power in
three phase circuit can be calculated from the equation Since there is three phases, so the
multiple of 3 is made in the normal power equation and the PF is power factor. Power
factor is a very important factor in three phase system and sometimes due to certain error,
it is corrected by using capacitors.
It is the order in which the phase voltages will attain their maximum values. From the
fig it is seen that the voltage in A phase will attain maximum value first and followed by
B and C phases. Hence three phase sequence is ABC. This is also evident from phasor
diagram in which the phasors with its +ve direction of anti-clockwise rotation passes a
fixed point is the order ABC, ABC and so on. The phase sequence depends on the direction
of rotation of the coils in the magnetic field. If the coils rotate in the opposite direction
then the phase voltages attains maximum value in the order ACB. The phase sequence
gets reversed with direction of rotation.
Figure 4.4: Phase Sequence
4.6 Analysis of Balanced and Unbalanced Three Phase Circuits
In a balanced system, each of the three instantaneous voltages has equal amplitudes, but is
separated from the other voltages by a phase angle of 120. The three voltages (or phases)
are typically labeled a, b and c. The common reference point for the three phase voltages
is designated as the neutral connection and is labeled as n. Three-phase systems deliver
power in enormous amounts to single-phase loads such as lamps, heaters, air-conditioners,
Chapter 1. THREE PHASE AC CIRCUITS
44
and small motors. It is the responsibility of the power systems engineer to distribute these
loads equally among the three-phases to maintain the demand for power fairly balanced
at all times. While good balance can be achieved on large power systems, individual loads
on smaller systems are generally unbalanced and must be analyzed as unbalanced three
phase systems. When the three phases of the load are not identical, an unbalanced system
is produced. An unbalanced Y-connected system is shown in Fig.1. The system of Fig.1
contains perfectly conducting wires connecting the source to the load. Now we consider
a more realistic case where the wires are represented by impedances Zp and the neutral
wire connecting n and n’ is represented by impedance Zn.
Figure 4.5: Star connection vector diagram
The currents flowing through the phase windings IAA’, IBB’, and ICC’ or IAB, IBC, and
ICA are called phase currents and are balanced as shown in phase diagram.The line current
IA, IB, IC and also equal and differ in phase by 1200. They form a balanced system of
currents. The line and phase currents differ in phase by 300
Figure 4.6: Delta connection vector diagram
Chapter 5
FILTERS
An electrical filter is a circuit which can be designed to modify, reshape or reject all the
undesired frequencies of an electrical signal and pass only the desired signals.
In other words we can say that an electrical filter is usually a frequency selective network
that passes a specified band of frequencies and blocks signals of frequencies outside this
band.
5.1 Classification of filters, filter networks
Depending on the type of element used in their construction, filters are classified into two
types, such as:
Passive Filters : A passive filter is built with passive components such as resistors, capacitors and inductors.
Active Filters : An active filter makes use of active elements such as transistors, op-amps
in addition to resistor and capacitors.
According to the operating frequency range, the filters may be classified as audio frequency
(AF) or radio frequency (RF) filters. Filters may also be classified as :
Low Pass Filter : The low pass filter only allows low frequency signals from 0 Hz to its
cut-off frequency, ƒc point to pass while blocking any higher frequency signals.It is the
order in which the phase voltages will attain their maximum values. From the fig it is
seen that the voltage in A phase will attain maximum value first and followed by B and C
45
Chapter 5. FILTERS
46
phases. Hence three phase sequence is ABC. This is also evident from phasor diagram in
which the phasors with its +ve direction of anti-clockwise rotation passes a fixed point is
the order ABC, ABC and so on. The phase sequence depends on the direction of rotation
of the coils in the magnetic field. If the coils rotate in the opposite direction then the phase
voltages attains maximum value in the order ACB. The phase sequence gets reversed with
direction of rotation. High Pass Filter : The high pass filter only allows high frequency
Figure 5.1: Low Pass Filter
signals from its cut-off frequency, ƒc point and higher to infinity to pass through while
blocking those any lower.High pass filter as the name suggests, it allows (passes) only high
frequency components. That means, it rejects (blocks) all low frequency components The
s-domain circuit diagram (network) of High pass filter is shown in the following figure Band
Figure 5.2: High Pass Filter
Pass Filter : The band pass filter allows signals falling within a certain frequency band
set up between two points to pass through while blocking both the lower and higher frequencies either side of this frequency band.Band pass filter as the name suggests, it allows
(passes) only one band of frequencies. In general, this frequency band lies in between low
frequency range and high frequency range. That means, this filter rejects (blocks) both
Chapter 5. FILTERS
47
low and high frequency components The s-domain circuit diagram (network) of Band pass
filter is shown in the following It consists of three passive elements inductor, capacitor and
resistor, which are connected in series. Input voltage is applied across this entire combination and the output is considered as the voltage across resistor. Band Stop Filter : The
Figure 5.3: Band Pass Filter
band stop filter blocks signals falling within a certain frequency band set up between two
points while allowing both the lower and higher frequencies either side of this frequency
band.Band stop filter as the name suggests, it rejects (blocks) only one band of frequencies.
In general, this frequency band lies in between low frequency range and high frequency
range. That means, this filter allows (passes) both low and high frequency components.
In the attenuation band, the attenuation does not increase sharply with frequency beyond
Figure 5.4: Band Stop Filter
the cutoff frequency. In the pass band, Z0 does not remain constant but varies widely from
the nominal value R0. Hence when constant-k filter is terminated in a fixed resistive load,
mismatch and associated reflection occur at different frequencies. Regarding attenuation
characteristic, some improvement in attenuation band beyond cutoff may be achieved by
Chapter 5. FILTERS
48
connecting two or more similar constant-k sections in tandem. Use of m-derived filter
sections makes it possible to get vary rapid attenuation characteristic in the attenuation
band. However, even use of m-derived filter section fails to provide constant Z0 over the
entire pass band. Almost constant Z0 may be achieved only by use of composite filter
which consists of one or more constant-k filter section, one or more m-derived section and
a terminating m-derived half-section at each end.
a simple constant-k T-section filter while figure 2 gives the corresponding m-derived filter
having the same characteristic impedance. The series arm of the m-derived section has
been obtained by multiplying the series arm element of constant-k section by a constant
“m”. Simultaneously the shunt arm element Z2 is also modified in such a way that the
m-derived section has the same characteristic impedance as the constant-k section.
Figure 5.5: Constant k T section filter
Figure 5.6: m Derived Filter T section
Chapter 5. FILTERS
49
Figure 5.7: Constant k pi section filter
Figure 5.8: m Derived Filter pi section
Chapter 5. FILTERS
50
Figure 5.9: Eleiments m Derived Filter pi section
Figure 5.10: m Derived Filter T section low pass filter
Chapter 5. FILTERS
Figure 5.11: m Derived Filter pi section low pass filter
Figure 5.12: m Derived Filter T section high pass filter
51
Chapter 5. FILTERS
Figure 5.13: m Derived Filter pi section high pass filter
52
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