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 Power System Transients

Department : EEE

Batch/Year : 2020-24/IV Year

Created by : Dr.J.Jayaudhaya
Associate Professor
RMDEC

Date : 7.8.2023
 Table of Contents
Page
S. No. TOPIC
No.
1 Course Objectives 7

2 Pre Requisites 8

3 Syllabus 9

4 Course Outcomes 10

5 Program Outcomes 11

6 Program Specific Outcomes 13

7 CO/PO Mapping 14

8 CO/PSO Mapping 15
UNIT – IV -TRAVELING WAVES ON TRANSMISSION LINE COMPUTATION OF
TRANSIENTS
9 Lecture Plan 16

10 Activity Based Learning 17

11 EBook 20

12 Video Links 21

13 Introduction - Travelling Wave Concept 22

14 Computation of transients 22
Transient response of systems with series and shunt lumped
15 27
parameters and distributed lines
16 Travelling Wave – Step Response 30

17 Transient response of systems with distributed parameters 35

18 Bewely’s lattice diagram 40

19 Standing waves and natural frequencies 45
 Table of Contents

Page
S. No. TOPIC
No.
20 Reflection and refraction of travelling waves 51

21 Assignment 57

22 Part A Question & Answers 58

23 Part B Questions 62

24 Part C Questions 63

25 Supportive Online Certification Courses 64

26 Real Time Applications in Day to Day Life and to Industry 65

27 Contents beyond the Syllabus 66

28 Assessment Schedule 67

29 Books 68

30 Mini Projects 69
 Course Objectives

Generation of switching transients and their control using circuit - theoretical

concept.
Mechanism of lighting strokes and the production of lighting surges.
Propagation, reflection and refraction of travelling waves.
Voltage transients caused by faults, circuit breaker action, load rejection on
integrated power system.
 Pre Requisites

EE8301- Electrical machines-I

EE8401 -Electrical machines-II
EE8402-Transmission & Distribution
EE8602-Protection & Switchgear
 Syllabus

EE8010 POWER SYSTEMS TRANSIENTS LTPC3003

UNIT I INTRODUCTION AND SURVEY 9
Review and importance of the study of transients - causes for transients. RL circuit
transient with sine wave excitation - double frequency transients - basic transforms
of the RLC circuit transients. Different types of power system transients - effect of
transients on power systems – role of the study of transients in system planning.
UNIT II SWITCHING TRANSIENTS 9
Over voltages due to switching transients - resistance switching and the equivalent
circuit for interrupting the resistor current - load switching and equivalent circuit -
waveforms for transient voltage across the load and the switch - normal and
abnormal switching transients. Current suppression - current chopping - effective
equivalent circuit. Capacitance switching - effect of source regulation - capacitance
switching with a restrike, with multiple restrikes. Illustration for multiple restriking
transients - ferro resonance.
UNIT III LIGHTNING TRANSIENTS 9
Review of the theories in the formation of clouds and charge formation - rate of
charging of thunder clouds – mechanism of lightning discharges and characteristics
of lightning strokes – model for lightning stroke - factors contributing to good line
design - protection using ground wires - tower footing resistance - Interaction
between lightning and power system.
UNIT IV TRAVELING WAVES ON TRANSMISSION LINE COMPUTATION
OF TRANSIENTS 9
Computation of transients - transient response of systems with series and shunt
lumped parameters and distributed lines. Traveling wave concept - step response -
Bewely’s lattice diagram - standing waves and natural frequencies - reflection and
refraction of travelling waves.
UNIT V TRANSIENTS IN INTEGRATED POWER SYSTEM 9
The short line and kilometric fault - distribution of voltages in a power system - Line
dropping and load rejection - voltage transients on closing and reclosing lines - over
voltage induced by faults -switching surges on integrated system Qualitative
application of EMTP for transient computation.

TOTAL : 45 PERIODS
 Course outcomes

Ability to understand and analyze switching and lightning transients.

Ability to acquire knowledge on generation of switching transients and their
control.
Ability to analyze the mechanism of lighting strokes.
Ability to understand the importance of propagation, reflection and refraction of
travelling waves.
Ability to find the voltage transients caused by faults.
Ability to understand the concept of circuit breaker action, load rejection on
integrated power system.
 PROGRAM OUTCOMES

The graduates will have the ability to

a. Apply the Mathematical knowledge and the basics of Science and Engineering to
solve the problems pertaining to Electrical and Electronics Engineering.

b. Identify and formulate Electrical and Electronics Engineering problems from

research literature and be ability to analyze the problem using first principles of
Mathematics and Engineering Sciences.

c. Come out with solutions for the complex problems and to design system
components or process that fulfill the particular needs taking into account public
health and safety and the social, cultural and environmental issues.

d. Draw well-founded conclusions applying the knowledge acquired from research

and research methods including design of experiments, analysis and
interpretation of data and synthesis of information and to arrive at significant
conclusion.

e. Form, select and apply relevant techniques, resources and Engineering and IT
tools for Engineering activities like electronic prototyping, modeling and control
of systems and also being conscious of the limitations.

f. Understand the role and responsibility of the Professional Electrical and

Electronics Engineer and to assess societal, health, safety issues based on the
reasoning received from the contextual knowledge.

g. Be aware of the impact of professional Engineering solutions in societal and

environmental contexts and exhibit the knowledge and the need for Sustainable
Development.

h. Apply the principles of Professional Ethics to adhere to the norms of the

engineering practice and to discharge ethical responsibilities.

i. Function actively and efficiently as an individual or a member/leader of different

teams and multidisciplinary projects.

11
 PROGRAM OUTCOMES

j.Analyze, design and implement control, instrumentation and power systems for
satisfying industry needs.
k.Use modern tools and appropriate solutions for the real time problems for
promoting energy conservation and sustainability.
l.Possess the capacity to embrace new opportunities of emerging technologies,
leadership and teamwork opportunities, all affording sustainable engineering career
in Electrical and Electronics related fields.

12
 PROGRAM SPECIFIC OUTCOMES

PSO 1
Analyze, design and implement control, instrumentation and power systems for
satisfying industry needs.

PSO 2
Use modern tools and appropriate solutions for the real time problems for promoting
energy conservation and sustainability.

PSO 3
Possess the capacity to embrace new opportunities of emerging technologies,
leadership and teamwork opportunities, all affording sustainable engineering career
in Electrical and Electronics related fields.
 CO/PO MAPPING

CO/PO a b c d e f g h i j k l

CO1 - 3 - 3 2 - - - - - - -

CO2 - 3 - 3 2 - - - - - - -

CO3 - 2 - - 2 - - - - - - -

CO4 - 3 - 3 2 - - - - - - -

CO5 - 3 - 3 2 - - - - - - -

CO6 - 3 - 3 - - - - - - - -
 CO/PSO Mapping

CO/PSO PSO1 PSO2 PSO3

CO1 3 - 2

CO2 3 - 2

CO3 2 - 2

CO4 3 - 3

CO5 3 - 3

CO6 3 - 2
 UNIT-IV
LECTURE PLAN

S. TOPIC No. Proposed Actual Per Taxo Mode

NO of date Lecture tai nomy of
Per Date nin level Deliver
i g y
ods CO
Introduction 1 4.10.2023 4.10.2023
1 - Travelling CO4 K2 BB &PPT
Wave
Concept
1 5.10.2023 5.10.2023
Computation
2 CO4 K3 BB &PPT
of transients
Transient 1 7.10.2023 7.10.2023
response
of systems with
3 series and shunt CO4 K3 BB &PPT
lumped
parameters and
distributed lines
1 9.10.2023 9.10.2023
Travelling
4 CO4 K2 BB &PPT
Wave
concept
1 10.10.202 10.10.202
Travelling Wave – 3 3
5 CO4 K2 BB &PPT
Step Response

Transient 1 11.10.202 11.10.202

response 3 3
6 of systems CO4 K2 BB &PPT
with
distributed
parameters
1 12.10.202 12.10.202
Bewely’s 3 3
7. CO4 K2 BB &PPT
lattice
diagram
Standing 1 14.10.202 14.10.202
waves and 3 3
8. CO4 K2 BB &PPT
natural
frequencies
Reflection and 1 17.10.202 17.10.202
refraction of 3 3
9. CO4 K2 BB &PPT
travelling
waves
 Activity based learning

Activity: Solving MCQs in Google classroom

Travelling Wave Concepts
https://forms.gle/7fxo6rcmf2mN6eD18

1. The transmitted power in a transmission line, when the reflection coefficient and
the incident power are 0.6 and 24V respectively, is
a) 15.36
b) 51.63
c) 15.63
d) 51.36

2. The reflection coefficient of a short circuit transmission line is -1.

a) True
b) False

3. Find the power reflected in a transmission line, when the reflection coefficient and
input power are 0.45 and 18V respectively.
a) 3.645
b) 6.453
c) 4.563
d) 5.463

4. The transmission coefficient in a wave travelling through two media having

permittivities 4 and 1 is
a) 1/4
b) 3/2
c) 3/4
d) 2/3

5. What is the resistive load if SWR= 3.05 and Zo =75Ω?

a) 1.23Ω
b) 51.23Ω
c) 254.2Ω
d) 24.59Ω
 Activity based learning

6. What is the Standing wave ratio if a 75Ω antenna load is connected to a 50Ω
transmission line?
a) 1
b) 2
c) 1.5
d) 1.43

7. Which of the following is not true regarding standing wave?

a) In a standing wave the energy moves towards the power source
b) In a standing wave power loss occurs
c) Standing waves do not affect signal strength
d) Standing waves are not desirable Other:

8. Two waves are propagating with the same amplitude and nearly same frequency
in opposite direction, they result in
a) Beats
b) Stationary wave
c) Resonance
d) Wave packet

9. Reflection coefficient at the load end of short circuited line is

(A) Zero
(B) 10°
(C) 190°
(D) 180°

10. A surge of 260 kV traveling in a line of natural impedance of 500 Ω arrives at the
junction with two lines of natural impedance of 250 Ω and 15 Ω respectively. The
voltage transmitted in the branch line is
(A) 400 kV
(B) 260 kV
(C) 80 kV
(D) 40 kV
 Activity based learning

11. Steepness of the traveling wave is attenuated by

(A) Line resistance
(B) Line inductance
(C) Line capacitance
(D) Both (b) and (c)

12. The relation between traveling voltage wave and current wave is given as
(A) ei=√(L/C)
(B) e/i=√(L/C)
(C) ei=√(LC)
(D) e/i=√(LC)

13. For a transmission line the standing wave ratio is the ratio of
(A) Peak voltage to RMS voltage
(B) Maximum current to minimum current
(C) Maximum voltage to minimum voltage
(D) Maximum impedance to minimum impedance

14.The propagation constant of a transmission line is 0.15×10–3 + j1.5×10– 3. The

wavelength of the traveling wave is
A) 15×10–3/2π
B) 2π/15×10–3
C) 15×10–3/π
D) π/15×10–3

15.Reflection coefficient of the wave of load connected to a transmission line of

surge impedance equal to that of transmission line is
(A) 1
(B) -1
(C) Zero
(D) Infinity
 E-Book

1) Electrical Transients in Power Systems-Allan Greenwood

2) Transients in Power Systems-Lou van der Sluis, Wiley
3)Transient Analysis of Power Systems: Solution Techniques, Tools and Applications-
Ametani, Akihiro, Baba, Yoshihiro, Nagaoka, Naoto, Ohno, Teruo CRC Press
 VIDEO LINKS

TITLE LINK
Reflection and refraction of https://www.youtube.com/watch?v=BdR3mvvqFuQ
travelling waves
Standing waves and natural https://www.youtube.com/watch?v=Nv7gvrYPbik
frequency
Bewely’s lattice diagram https://www.youtube.com/watch?v=BXfPchdJIQs

21
 Travelling Waves on Transmission Line

The transmission line consists of distributed line parameters R,L and C.

These parameters are distributed throughout the length of transmission line.
Travelling wave on transmission line is the voltage / current waves from the
source end to the load end during the transient condition propagating as
electromagnetic waves with a finite velocity. Hence it takes a short time for load
to receive the Power.
This gives rise to concept of travelling waves on transmission lines.
The current flow is governed by the load impedance, line charging current at
power frequency and voltage.
If the load impedance is not matched with the line impedance, some of the
energy transmitted by the source is not absorbed by the load and is reflected
back to the source. Since load can vary from no-load to rated value the load
impedance is not equal to the line impedance always. Therefore there always
exist transmitted wave from the source and reflected waves from the load end.
Due to the presence of transmitted and reflected waves the resulting voltage and
current is equal to the sum of transmitted and reflected quantities. The polarity of
the voltage is same for both transmitted and reflected waves. But the direction of
current is opposite, so that the ratio of voltage to current will be positive for the
transmitted wave and negative for the reflected waves.

These waves travel along the line with the velocity equal to velocity of light if line
losses are neglected. But practically there always exists some line loss and hence
these waves propagate along the line with velocity somewhat lower than the
velocity of light.
Computation of Transients in Transmission Line
Let us consider a two wire lossless transmission line.
Let L and C be the inductance and capacitance per unit length of the line.

Two wire Transmission Line

 Computation of Transients in Transmission Line

The above transmission line can be represented by its equivalent circuit having L and C
distributed over the whole line as shown in fig.

Two wire Transmission Line with lumped parameter

When the switch is closed at the transmission line's starting end, voltage will not appear
instantaneously at the other end. This is caused by the transient behavior of inductor and
capacitors that are present in the transmission line. The transmission lines may not have
physical inductor and capacitor elements but the effects of inductance and capacitance
exists in a line. Therefore, when the switch is closed the voltage will build up gradually over
the line conductors. This phenomenon is usually called as the voltage wave is travelling from
transmission line's sending end to the other end. And similarly the gradual charging of the
capacitances happens due to the associated current wave.

In the circuit by closing the switch S the transmission line is connected to a source of
voltage V, which is assumed for the time being to have zero impedance. In the above fig.
the line is divided into large number of lumped sections. Each section contains certain
inductane(L) and certain capacitance(C).

When switch S1 is closed, current flows through the first inductance to charge the first
capacitance C1.After charging of C1 there is a voltage across the second section of the line
and current commences to flow through the second L2 to charge the second capacitor
C2.This process is followed throughout the length of the line.
Thus with a lumped circuit even the very small disturbance is felt in the nth section of the
line.
 Relationship between Voltage and Current Wave

Let it be assumed that after a time Δt a length Δx of line has been so

charged. If the capacitance of the line is C farads per meter, a charge Q will have
been imparted to the line.
Hence the stored charge in shunt capacitance Q = CVΔx
The consequences are
(i)An electric field is created between the conductors of the first Δx meters of the
line.
(ii) A magnetic field is created around the conductors by virtue of the currents
flowing in them.
The current is determined from the rate at which charge flows into and out of the
line.
But I = dQ/dt
= CV Δx/Δt
=CV dx/dt
But dx/dt = velocity of travelling wave = ν (say)
Therefore, I = CVν (1)
The Fluxlinkages can be defined interms of the line inductance.
ɸ= LΔxI=LΔx CVν (2)
The induced emf is the rate of change of flux linkage is being induced in the loop
formed by conductors and the wavefront.
The induced emf can be expressed as
V = dɸ /dt
=LCVv Δx/Δt
= LCVv dx/dt
= LCVv2
Therefore V= LCv2V
=√(1/LC (3)
The velocity of propagation of these waves of current and voltage along the line
appears to depend upon the line geometry and electromagnetic properties of the
surrounding medium.
 Relationship between Voltage and Current Wave

If the spacing between the lines is large compared with the radius of the conductor r
the flux within the conductors can be neglected.
With this assumptions the expressions for inductance and capacitance can be given
as
L= µo/π ln(d/r) H/ m
L = 2×10-7ln(d/r) H/ m
C = πεo / ln (d/r) F/m
Substitute Land C in equation (3)
1
𝑣=
µ𝒐 ln d x πεo / ln (d/r)
π r
1
𝑣=
µ𝐨εo
The equation of current can be alternatively stated as
CV
I=CVv=CV/I = CVv = 𝐿𝐶

𝑉/𝐼 = √(𝐿/𝐶) = 𝑍𝑂
The above expression is the ratio of voltage and current having the dimension of
impedance. Therefore it is called Surge Impedance. Note that Surge Impedance is
the square root of ratio of series inductance L per unit length of line and shunt
capacitance C per unit length of line. This simply means that this value will remain
constant for a given transmission line.
This value will not change due to change in length of line.
The value of surge impedance for a typical transmission line is around 400
Ohm and that for a cable is around 40 ohm. Notice that the value of surge
impedance for cable is less than that of transmission line. This is due to the higher
value of capacitance of cable compared to the transmission line.
A travelling wave of voltage passes along the line at a velocity approaching the
speed of light, establishing an electric field between the conductors. The voltage
V
wave is accompanied by a current wave of amplitude 𝑍𝑂
which in turn creates a
magnetic field in the surrounding space. The energy is supplied to the line at a rate
of VI watts. In the absence of losses this energy must be stored in the
electromagnetic field.
Relationship between Voltage and Current Wave
 Transient Response of Systems with Series and Shunt
Lumped Parameters and Distributed lines

In practice, lumped elements are connected to lines. In this section we

will consider the development of equations suitable for solution based on transform
methods or finite difference methods using the Digital Computer.

Figure shows a general single-line diagram of a source E(s) energizing the

line with distributed-parameters (r, l, g, c per unit length). There is an impedance Zs
in between the source and line which normally is composed of the transient
reactance x' of the synchronous machine and any resistance that can be included in
the circuit breaker during the switching operation. A shunt impedance Zsh is also in
the circuit which can represent the shunt compensating reactor. The line is
terminated with an impedance Zt which consists of a transformer, shunt reactor, or
an entire substation.
Let z = r + ls and y = g + cs as before
Theory of Travelling Waves and Standing Waves
Then, at any point x from the termination Zt, the equations are
𝝏𝑉
= zI (1)
𝝏𝑥

𝝏𝐼
= yV (2)
𝝏𝑥

with solutions

V = AepX + Be−pX (3)

And
I = plz ( AepX − Be−pX ) (4)
 Transient Response of Systems with Series and Shunt
Lumped Parameters and Distributed lines

The boundary conditions are at x = L, V = Ve, the voltage at the line

entrance at x = 0, V(0) = I(0).Zt
Using Z0 = z/y = z/p,
we obtained the solutions, equations (3) and (4) for the voltage and
current at any point on the line as follows;

𝑍𝑂
𝐶𝑜𝑠ℎ 𝑝𝑥 + ൗ𝑍 𝑆𝑖𝑛ℎ 𝑝𝑥
𝑡
𝑉 𝑥 = 𝑉𝑒 (5)
𝑍𝑂
𝐶𝑜𝑠ℎ 𝑝𝐿 + ൗ𝑍 𝑆𝑖𝑛ℎ 𝑝𝐿
𝑡

𝑍𝑂
1 𝑆𝑖𝑛ℎ 𝑝𝑥+ ൗ𝑍 𝐶𝑜𝑠ℎ 𝑝𝑥
And 𝐼 𝑥 = 𝑍𝑂
𝑡
𝑉𝑒 (6)
𝑍 𝑂 𝐶𝑜𝑠ℎ 𝑝𝐿+ ൗ 𝑍𝑡 𝑆𝑖𝑛ℎ 𝑝𝐿

At the end of the line, x = 0, the voltage and current are

𝑉𝑒
𝑉 0 = (7)
𝑍𝑂ൗ
𝐶𝑜𝑠ℎ 𝑝𝐿 + 𝑆𝑖𝑛ℎ 𝑝𝐿
𝑡

𝑉 0
And 𝐼0 = 𝑍 (8)
𝑡

At the entrance to the line, x = L, they are

𝑉 𝐿 = 𝑉𝑒 9

𝑍𝑂
1 𝑆𝑖𝑛ℎ 𝑝𝐿 + 𝑍𝑡 𝐶𝑜𝑠ℎ 𝑝𝐿
𝐼 𝐿 = 𝑉𝑒 (10)
𝑍𝑂 𝐶𝑜𝑠ℎ 𝑝𝐿 + 𝑍𝑂
ൗ 𝑍𝑡 𝑆𝑖𝑛ℎ 𝑝𝐿
ൗ
Therefore, if the voltage Ve is found in terms of the known excitation
voltage of source (step, double exponential, or sinusoidal) then all quantities in
equations(5) to (10) are determined in operational form. By using the Fourier
Transform method, the time variation can be realized.
Transient Response of Systems with Series and Shunt
Lumped Parameters and Distributed lines

Referring to the above Fig. the following equations can be written down:
𝑉𝑒
𝐼𝑠ℎ =
𝑍𝑠ℎ

𝐼𝑠 = 𝐼𝑠ℎ + 𝐼(𝐿)

𝑉𝑒 = 𝐸 𝑆 − 𝑍𝑆𝐼𝑆 (11)

𝑉𝑒= 𝐸 𝑆 − ( 𝐼𝑠ℎ + 𝐼 𝐿 ) (12)

𝑍𝑂
𝑍𝑆 𝑍𝑆 𝑆𝑖𝑛ℎ 𝑝𝐿 + 𝑍𝑡 𝐶𝑜𝑠ℎ 𝑝𝐿
𝑉𝑒 = 𝐸 𝑆 − 𝑉 − (13)
𝑍𝑆ℎ 𝑒 𝑍𝑜 𝐶𝑜𝑠ℎ 𝑝𝐿 + 𝑍𝑂
ൗ 𝑍𝑡 𝑆𝑖𝑛ℎ 𝑝𝐿
ൗ

Solving for Ve, the voltage at the line entrance, we obtain

𝑍𝑆 𝑍𝑂 𝑍𝑆𝑍𝑂 𝑍𝑆
1+ 𝑍𝑆ℎ + 𝐶𝑜𝑠ℎ 𝑝𝐿 + ൗ𝑍 + ൗ𝑍 𝑍 + ൗ𝑍 𝑆𝑖𝑛ℎ 𝑝𝐿
𝑉𝑒 = 𝐸 𝑆 𝑍𝑆ൗ 𝑡 𝑡 𝑠ℎ 𝑡 𝑂
14
𝑍 𝑍𝑂ൗ
ൗ 𝐶𝑜𝑠ℎ 𝑝𝐿 +
𝑡
𝑆𝑖𝑛ℎ 𝑝𝐿

When Zs = 0 and Zsh = ¥ , the entrance voltage equals the source voltage E(s).
 Travelling wave- Step response

Consider a lossless two wire line.

The fig. shows a small element of a transmission lines.
If the line has an inductance of L henries per meter and a capacitance of C
farads per meter and an elementary length Δx will have inductance and
capacitance LΔx and CΔx as shown in fig.

The voltage across the element will be

ΔV = −L∆𝑥 𝜕𝐼 (1)
𝜕
ΔV = −L 𝜕𝐼
𝜕
𝜕𝑉 𝜕𝐼
𝜕𝑥
= −L (2)
Here the partial derivatives are used
𝜕 because V and I are functions of both
position and time.
The current to charge the elementary capacitance ΔC is given by
𝜕𝑉
-ΔI = 𝐶∆𝑥 (3)
𝜕

∆𝐼 𝜕𝑉
= −C
∆𝑥 𝜕𝑡

𝜕𝐼
= −C 𝜕𝑉 (4)
𝜕𝑥 𝜕𝑡
 Travelling wave- Step response

Differentiate equation (2) w.r.t x and equation (4) w.r.t t

2
𝜕2𝑉
= −𝐿. 𝜕 𝐼 (5)
𝜕𝑥 2 𝜕 X𝜕 t

𝜕2 𝐼 = −𝐶. 𝜕2 𝑉
(6)
𝜕x𝜕t 𝜕𝑡2
Eliminating and rearranging the terms,
Voltage wave equation
𝜕2 𝑉 𝜕2 𝑉
= 𝐿𝐶. 2 (7)
𝜕𝑥 2 𝜕𝑡
Solving equation (2) and (4) for I
𝜕2 𝐼 𝜕2 𝐼
= 𝐿𝐶. 2 (8)
𝜕𝑥 2 𝜕𝑡
Equations (7) and (8) constitute the transmission line wave equations.
The wave equations (7) and (8) can be transformed into ordinary differential
equations and then these equations can be transformed into algebraic equations.
The voltage equation can be satisfied by a solution
1
𝑉 = 𝑓 𝑡 ± 𝐿𝐶 2 𝑥 (9)
𝑡
Or 𝑉 = 𝑓 𝑥 ± 1 (10)
𝐿𝐶 2
1
where LC 2 is the velocity. Therefore equation (9) states V is the function of time
and equation (10) states V is a function of distance.
Rewriting the equation in terms of v(velocity of electromagnetic waves in freespace)

𝑉 = 𝑓1 𝑥+𝑣𝑡 +𝑓2(𝑥−𝑣𝑡) (11)

Differentiate the above equation w.r.to x

𝜕𝑉
= 𝑓′1 𝑥+𝑣𝑡 +𝑓′ 2(𝑥−𝑣𝑡) (12)
𝜕𝑥
𝜕2 𝑉
= 𝑓′′1 𝑥+𝑣𝑡 +𝑓′′ 2(𝑥−𝑣𝑡) (13)
𝜕𝑥 2
 Travelling wave- Step response

Differentiate equation (11) twice w.r.to t

𝜕𝑉
= 𝑣 𝑓′1 𝑥+𝑣𝑡 −𝑓′ (14)
2(𝑥−𝑣𝑡)
𝜕𝑡

𝜕2 𝑉
2
= 𝑣 2 𝑓′′1 𝑥+𝑣𝑡 +𝑓′′ 2(𝑥 −𝑣 𝑡 (15)
𝜕𝑡 )
Equation (15) becomes
𝜕2 𝑉 1
= 𝑓′′1 𝑥+𝑣𝑡 +𝑓′′ 2(𝑥−𝑣𝑡) (16)
𝜕𝑡2 𝐿𝐶
1 1
Where 𝑣 = , therefore 𝑣 2 =
𝐿𝐶 𝐿𝐶

Consider the function f1(x+vt),

At time t=0,the spatial distribution f1(x) becomes f1(a),since x=a.
 Travelling wave- Step response

At any subsequent time t=𝑟,x=(a-v𝑟)

This implies that the voltage distribution has moved intact a distance v𝑟 in the
direction of minus x.
Similarly for the function f2(x- v𝑟) a voltage distribution moves with a velocity V in
the direction of plus x.
The current wave accompany the voltage waves,which can be derived from equation
(2).
𝜕I 1 𝜕v
=−
𝜕𝑡 𝐿 𝜕𝑡
𝜕v
Substitite 𝜕
from equation (14) in the above equation
𝜕𝐼 1
= − 𝑣 𝑓 ′ 1 𝑥+𝑣𝑡 +𝑓 𝘍2 𝑥−𝑣𝑡
(12)
𝜕𝑡 𝐿
Integrate the above expression w.r.to t on both sides

1
𝐼= 𝑓1𝑥+𝑣𝑡 −𝑓2(𝑥−𝑣𝑡)
𝐿𝑉
1
𝐼= 𝑓1𝑥+𝑣𝑡 −𝑓2(𝑥−𝑣𝑡)
1
𝐿
𝐿𝐶

𝐶
= 𝑓
𝐿 1𝑥+𝑣𝑡 −𝑓2(𝑥−𝑣𝑡)

𝑉
𝐼= (13)
𝑍𝑜

𝐿
𝑍𝑜 = = 𝐶ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟𝑖𝑠𝑡𝑖𝑐𝑠 𝑖𝑚𝑝𝑒𝑑𝑎𝑛𝑐𝑒
𝐶
The equation (13) states that the current waves are directly proportional to the
voltage waves with the proportionality factor of 𝑍𝑜.
The current waves travelling in the direction of minus x which is opposite sign to the
voltage wave.
 Travelling wave- Step response

The various combination of current and voltage waves are shown in fig.

When two waves travelling in opposite directions will meet at one

point,addedalgrbraically and then pass away from each other.this is illustrated in fig.
 Transient Response Of Systems With Distributed-
Parameter

In the transmission lines, the line parameters are distributed throughout the
Length. The properties and behavior of transmission line can be analyzed under any
type of excitation. These can be applied specifically for lightning impulses and
switching surges.
The distributed parameters are series resistance (r) ,series inductance (l),shunt
capacitance(c),shunt conductance (g).In the case of overhead transmission lines the
shunt conductance(g) is omitted when the corona losses are neglected.
By considering all four quantities (r, l, g, c) the differential equations for voltage
and current can be developed through the method of Laplace Transforms.

General Method of Laplace Transforms

According to the method of Laplace Transform, the general series impedance
operator per unit

Length of line is z(s) = r + ls, and the shunt admittance operator per unit length
is y(s) = g + cs,
where s = the Laplace-Transform operator.
Consider a line of length L energized by a source whose time function is e(t) and
Laplace Transform E(s), as shown in Fig.

Let the line be terminated in a general impedance Zt(s).

We will neglect any lumped series impedance of the source at the terminal end at
x=0
 Transient Response Of Systems With Distributed-
Parameter

Let the Laplace Transforms of voltage and current at any point x be V(x, s) and
I(x, s).

The two basic differential equations governing the steady-state excitation can
be given as,
𝜕𝑉(𝑥, 𝑠)
= 𝑧 𝑠 . 𝐼 𝑥, 𝑠 (1)
𝜕x

𝜕I(x, s)
= 𝑦 𝑠 . 𝑉 𝑥, 𝑠 (2)
𝜕x

The solutions for voltage and current in equation will be

𝑉(𝑥) = 𝐴𝑒 𝑝𝑥 + 𝐵𝑒 −𝑝𝑥 (3)
𝐼(𝑥) = 𝑝𝑙𝑧 (𝐴𝑒 𝑝𝑥 − 𝐵𝑒 −𝑝𝑥 ) (4)

where p = the propagation constant = 𝑧𝑦= (𝑟 + 𝑙𝑠)(𝑔 + 𝑐𝑠)

𝑝 𝑧𝑦 𝑌
= = = 𝑌𝑜 (5)
𝑧 𝑧 𝑧
and

𝑧 𝑧 𝑧
= = = 𝑍𝑜 (6)
𝑝 𝑧𝑦 𝑦

Zo-Characteristics or surge impedance

 Transient Response Of Systems With Distributed-
Parameter

For this problem, the boundary conditions are:

At x = L, V(L) = source voltage = E(s); and
At x = 0, V(0) = Zt. I(0).
Using the boundary conditions in (3) and (4) yields

(𝑍𝑡 + 𝑍0 )𝐸(𝑆)
𝐴= 7
𝑍𝑡 + 𝑍0 𝑒 𝑝𝐿 + 𝑍𝑡 − 𝑍0 𝑒 −𝑝𝐿
𝑍𝑡 − 𝑍0
𝐵= (8)
(𝑍𝑡 + 𝑍0 )
Therefore

𝑍
𝐶𝑜𝑠ℎ𝑝𝑥 + 𝑍𝑜𝑡 𝑆𝑖𝑛ℎ𝑝𝑥
𝑉 𝑥 = 𝐸(𝑆) (9)
𝑍𝑜
𝐶𝑜𝑠ℎ𝑝𝐿 + 𝑆𝑖𝑛ℎ𝑝𝐿
𝑍𝑡

𝑍𝑂ൗ
1 𝜕𝑣 𝑦 𝑆𝑖𝑛ℎ 𝑝𝑥 + 𝑡
𝐶𝑜𝑠ℎ 𝑝𝑥
𝐼 𝑥 = = . 𝐸𝑆 (10)
𝑧 𝜕𝑥 𝑧 𝐶𝑜𝑠ℎ 𝑝𝐿 + 𝑍𝑂ൗ 𝑆𝑖𝑛ℎ 𝑝𝐿
𝑡

The equation (9) and (10) states the voltage and current equation at point x on the
transmission line.
Three cases of Line Terminations:
Case (1) Open Circuit Zt = ¥

Therefore the above two equations can be written as,

𝐶𝑜𝑠ℎ𝑝𝑥
𝑉𝑜𝑐 𝑥, 𝑠 = 𝐸(𝑆) (11)
𝐶𝑜𝑠ℎ𝑝𝐿

𝑦 𝑆𝑖𝑛ℎ 𝑝𝑥
𝐼𝑜𝑐 𝑥, 𝑠 = . 𝐸𝑆 12)
𝑧 𝐶𝑜𝑠ℎ 𝑝𝐿
 Transient Response Of Systems With Distributed-
Parameter

Case (2) Short Circuit Zt = 0

𝑆𝑖𝑛ℎ𝑝𝑥
𝑉𝑠𝑐 𝑥, 𝑠 = 𝐸 𝑆 13
𝑆𝑖𝑛ℎ𝑝𝐿
𝑦 𝐶𝑜𝑠ℎ 𝑝𝑥
𝐼𝑠𝑐 𝑥, 𝑠 = . 𝐸 𝑆 (14)
𝑧 𝑆𝑖𝑛ℎ 𝑝𝐿
Case (3) Matched Line
In this case, the impedance at the termination of the transmission line is
equal to the surge impedance or characteristics impedance (ie Zt=Zo).Therefore the
equations (9) and (10) can be written as

𝐶𝑜𝑠ℎ𝑝𝑥 + 𝑆𝑖𝑛ℎ𝑝𝑥
𝑉𝑚 𝑥, 𝑠 = 𝐸(𝑆) (15)
𝐶𝑜𝑠ℎ𝑝𝐿 + 𝑆𝑖𝑛ℎ𝑝𝐿

𝑦 𝑆𝑖𝑛ℎ 𝑝𝑥 + 𝐶𝑜𝑠ℎ 𝑝𝑥
𝐼𝑚 𝑥, 𝑠 = . 𝐸𝑆 (16)
𝑧 𝐶𝑜𝑠ℎ 𝑝𝐿 + 𝑆𝑖𝑛ℎ 𝑝𝐿
The ratio of voltage to current at every point on the line is
𝑉
𝐼
= 𝑍𝑜
𝑧
= 𝑍𝑜
𝑦

1 𝑦
𝑌𝑜 = 𝑍𝑜
= 𝑧

𝑔 𝑠 +𝑐
𝑌𝑜 = 𝑟 +𝑙

By taking inverse Laplace transform, time domain solution of above

expression can be obtained.
Three cases of Line Terminations:
Case (1) Open Circuit Zt = ¥
Therefore the above two equations can be written as,

𝐶𝑜𝑠ℎ𝑝𝑥
𝑉𝑜𝑐 𝑥, 𝑠 = 𝐸𝑆 11
𝐶𝑜𝑠ℎ𝑝𝐿
𝑦 𝑆𝑖𝑛ℎ 𝑝𝑥
𝐼𝑜𝑐 𝑥, 𝑠 = . 𝐸𝑆 (12)
𝑧 𝐶𝑜𝑠ℎ 𝑝𝐿
 Transient Response Of Systems With Distributed-
Parameter

Case (2) Short Circuit Zt = 0

𝑆𝑖𝑛ℎ𝑝𝑥
𝑉𝑠𝑐 𝑥, 𝑠 = 𝐸(𝑆) (13)
𝑆𝑖𝑛ℎ𝑝𝐿

𝑦 𝐶𝑜𝑠ℎ 𝑝𝑥
𝐼𝑠𝑐 𝑥, 𝑠 = . 𝐸𝑆 (14)
𝑧 𝑆𝑖𝑛ℎ 𝑝𝐿

Case (3) Matched Line

In this case,the impedance at the termination of the transmission line is
equal to the surge impedance or characteristics impedance (ie Zt=Zo).Therefore the
equations (9) and (10) can be written as

𝐶𝑜𝑠ℎ𝑝𝑥 + 𝑆𝑖𝑛ℎ𝑝𝑥
𝑉𝑚 𝑥, 𝑠 = 𝐸𝑆 15
𝐶𝑜𝑠ℎ𝑝𝐿 + 𝑆𝑖𝑛ℎ𝑝𝐿

𝑦 𝑆𝑖𝑛ℎ 𝑝𝑥 + 𝐶𝑜𝑠ℎ 𝑝𝑥
𝐼𝑚 𝑥, 𝑠 = . 𝐸𝑆 (16)
𝑧 𝐶𝑜𝑠ℎ 𝑝𝐿 + 𝑆𝑖𝑛ℎ 𝑝𝐿

The ratio of voltage to current at every point on the line is

𝑉
= 𝑍𝑜
𝐼

𝑧
= 𝑍𝑜
𝑦

1 𝑦
& 𝑌𝑜 = 𝑍𝑜
= 𝑧

𝑔𝑠 + 𝑐𝑠
𝑌𝑜 =
𝑟 + 𝑙𝑠

By taking inverse Laplace transform, time domain solution of above

expression can be obtained.
Bewley’s lattice diagram
Bewley’s lattice diagram
Bewley’s lattice diagram
 Bewley’s lattice diagram

At the receiving end At the sending end Time unit

1 0 0
1 1
2
1 3
4
1 5
6
Bewley’s lattice diagram
 Standing waves and natural frequencies

A standing wave, also known as stationery waves, is a wave that remains

constant position. This phenomenon can occur because the medium is moving in the
opposite direction to the wave, or it can arise in a stationery medium as a result of
interference between two waves travelling in opposite direction. Standing waves are
confined to a given space in a medium and still produce a regular wave pattern which is
readily descramble amidst the motion of the medium. Any wave travelling along the
medium will reflect back when they reach the end. This effect is most noticeable in musical
instruments where at various multiples of a vibrating string or air columns natural
frequency, a standing wave is created, allowing harmonics to be identified.

Traveling waves have high points called crests and low points called
troughs (in the transverse case) or compressed points called compressions and
stretched points called rarefactions (in the longitudinal case) that travel through the
medium. Standing waves don't go anywhere, but they do have regions where the
disturbance of the wave is quite small, almost zero. These locations are
called nodes. There are also regions where the disturbance is quite intense, greater
than anywhere else in the medium, called antinodes. Nodes occur at fixed ends and
anti-nodes at open ends. If fixed at only one end, only odd-numbered harmonics are
available. At the open end of a pipe the anti-node will not be exactly at the end as it
is altered by its contact with the air and so end correction is used to place it exactly.
 Standing waves and natural frequency
Analysis of natural frequency with an example

Where

r---resistance of transmission line

g—conductance of transmission line

l---inductance of transmission line

c—capacitance of transmission line

l∆x-Inductance of infinitesimal area

c∆x-capacitance of infinitesimal area

Standing waves and natural frequencies
Standing waves and natural frequencies
Standing waves and natural frequencies
Standing waves and natural frequencies
Reflection and refraction of travelling waves
Reflection and refraction of travelling waves
Reflection and refraction of travelling waves
Reflection and refraction of travelling waves
Reflection and refraction of travelling waves
Reflection and refraction of travelling waves
 Assignment

1.A transmission line is 300 km long and open at the far end. The attenuation of
surgeis 0.9 over one length of travel at light velocity. It is energized by (a) a step of
1000kV, and (b) a sine wave of 325 kV peak when the wave is passing through its
peak.Calculate and plot the open-end voltage up to 20 ms.

2. For the step input in problem 1, draw the Bewley Lattice Diagram. Calculate the
final value of open-end voltage.
 Part-A Questions & Answers
Q.No Question & Answer K Level CO
Level
Define lumped parameters.
The lumped element (also called lumped parameters
(or) lumped components) simplifies the description of the

1. behavior of spatially distributed physical system in to a K1 CO4

topology Consisting of discrete entities that a approximate
the behavior of the distributed system under certain
assumptions.
What are the specifications of travelling wave?
A travelling wave is characterized by the four
2. specifications Crest of a wave, Front of a wave, Tail of a K1 CO4

wave and polarity.

What is the importance of Bewley’s Lattice diagram?

In a complex electrical network with number of
interconnections and with various terminations, the
travelling wave initiated by single incident wave will upstart
with a considerable rate as the wave splits. Due to this
3. K1 CO4
multiple reflection occur. It is possible for the voltage to
build up certain points by the reinforcing action of several
waves. In order to study such effects, Bewley proposed
transient.

What are the standing waves?

A standing wave, also known as stationary wave, is a
wave that remains in a constant position. This
phenomenon can occur because the medium is moving in
4. K1 CO4
the opposite direction to the wave, or it can arise in a
stationary medium as a result of interference between two
waves travelling in opposite directions.
 Part-A Questions & Answers
Q.No Question & Answer K Level CO
Level
What is attenuation? How they are caused?
The decrease in the magnitude of the wave as a
5. K1 CO4
propagates along the line is called attenuation. It is caused
due to the energy loss in the line.
What are the principles observed in lattice diagram?
All waves travel down hill in to the positive time. The
6. position of the wave at any instant is given by the means K1 CO4

of the time scale at the left of the lattice diagram.

What are the damages caused by the travelling

waves.
The high peak (or) crest voltage of the surge may cause
7. K1 CO4
Flashover in the internal winding their by spoil the windings
insulation. The steep wave front of the surge may cause
internal flashover between their turns of the transformer.
Define front and crest of a travelling wave
Front: The front of the wave is the portion of the wave
before crest and is expressed in time from beginning of the
8. wave to the crest value in ms (or) µs. K1 CO4
Crest: The crest of the wave is maximum amplitude of the
wave and is usually expressed in KV (or) KA.

What is travelling wave? What is the role of

distributed parameters (R,L ,C) in it.
Any disturbance on a transmission line (or) system such as
sudden opening or closing of the line, short circuit or s

9. fault results in the development of over voltages or over K1 CO4

current at that point. The disturbance propagate as a
travelling wave to the ends of a line or transmission , such
as a substation.
 Part-A Questions & Answers
Q.No Question & Answer K CO
Level Level
Define attenuation and distortion.
The decrease in the magnitude of The wave at a propagates
10. along the line is called attenuation. The elongation or change of K1 CO4
wave shape that occurs called distortion.

Distinguish between reflection and refraction of

travelling waves with expressions.
a= [ZB – ZA / ZA+ZB ]and is called the reflection
11. K1 CO4
coefficient 1≤ a ≤ +1

b= [2ZB / ZB + ZA] and is called the refraction coefficient

Define coefficient of reflection

The coefficient of reflection (a) is given by the ratio of
reflected wave to the voltage of incident wave of a transmission

12. line due to the travelling waves caused by switching surges. K1 CO4
Coefficient of reflection a =Vr / Vi Where V r- is the reflected
wave ,Vi – is the incident wave.

Define reflection and refraction.

Whenever there is an abrupt change in the parameters of a
transmission line, such as an open circuit or a termination, the
travelling wave undergoes a transition, part of the wave is
reflected or sent back only a portion is transmitted forward. At
13. the transition point (or) junction, the voltage or current wave K1 CO4

may attain a value which can vary from zero to twice its initial
value. The incoming wave is called incident wave and the other
waves are called reflected and transmitted (or) refracted waves
at the transition point.
 Part-A Questions & Answers
Q.No Question & Answer K CO
Level Level
Define tail and polarity of a wave.
Tail: tail of the wave is a portion beyond the crest, It is
expressed in time µs from beginning of the wave to the point.

14. Where the wave has to reduced to 50% of its value at crest. K1 CO4
Polarity: It is polarity of crest voltage or current a positive wave
of 500Kvcrest, 1µs from time and 25 µs tail time will be
represented as +500/1.0/25.0.
What is the effect of shunt capacitance at the terminal of
a transmission lines?
The effect of shunt capacitance at the terminal of a
15. transmission line is to cause the voltage at the terminal is to rise K1 CO4
to full value gradually instead of abruptly. i.e, to cause flattening
of the wave front which reduces the stress on the line end
windings of transformer connected to the lines.
What are the design principles observed in lattice
diagram?
16. All waves travel downhill in to the positive time. The position of K1 CO4
the wave at any instant is given by the means of the time scale
at the left of the lattice diagram.
Define SWR.
Standing wave ratio: is the ratio of the amplitude of a
17. K1 CO4
partial standing waves at an antinode to the amplitude at an
adjacent node is an electrical transmission line.
What is surge impedance of a line and why is it also
called the natural impedance?
The ratio of voltage to current which has the dimension of
18 K1 CO4
impedance is called as surge impedance of the line.1.

E /I=√(L/C) = Zc = Zn (natural impedance)

 Part-B Questions & Answers
Q.No Question & Answer K CO
Level Level
Explore the steps involved in Bewely’s lattice diagram K3 CO4
1.
construction with an example
Evaluate the value of current in a transmission line K2
2. CO4
considering its series and shunt lumped parameters.
Draw the step response of a travelling wave. Explain it by
3. K2 CO4
using Bewely’s lattice diagram

4. Discuss elaborately on reflection and refraction travelling K3 CO4

Examine multi-velocity waves of travelling waves in

5 K3 CO4
transmission lines
Explain multi-conductor system of travelling waves in
6. K2 CO4
transmission lines
Develop wave equation of travelling waves in
7. K2 CO4
transmission lines
Describe the transient response of systems with series K3
8. CO4
and shunt distributed parameters
Examine the behavior of travelling waves at open
9. K2 CO4
circuited transmission line
Describe briefly about standing waves and Standing Wave K2
10. CO4
Ratio (SWR) and natural frequency
Derive the reflection and refraction co efficient of a
11. K3 CO4
travelling wave with diagrams
Analyze the phenomenon of current interruption in a
12. lumped capacitive circuit and a distributed constant K3 CO4
transmission lines
Analyze the phenomenon of current interruption in a
13. lumped capacitive circuit and a distributed constant K2 CO4
transmission lines
 Part-C Questions & Answers

Q.No Question & Answer K CO

Level Level

With neat diagrams discuss the behaviour of a travelling

1. wave when it reaches the end of i)open circuited K1 CO4
transmission line ii)Short circuited transmission line

Explore the steps involved in Bewely’s lattice diagram

2. K3 CO4
construction with an example
Discuss and drive transient response of systems with
3. series and shunt lumped parameters and Distributed K2 CO4
lines
Obtain the value of current in a transmission line
4. K2 CO4
considering its series and shunt lumped
 Supportive Online Certification Courses

NPTEL
Power System Engineering
Electrical Distribution System Analysis

UDEMY
Introduction to Power System Harmonics
Short Circuit Analysis for HV three Phase systems
Real time Applications in day to day life and to Industry

1. Application of Bewely’s lattice diagram for 400KV Extra high voltage Transmission
line in Kosice region, Slovakia.

2. Travelling wave concept applicable to locate the fault area i in Korean power
transmission system.
 Contents beyond the Syllabus

Termination of line with different type of condition- open circuited line, short
circuit line, T-Junction.

http://web.cecs.pdx.edu/~greenwd/xmsnLine_notes.pdf

Travelling of propagation of surges- Attenuation, Distortion.

https://core.ac.uk/download/pdf/35466135.pdf
 Assessment Schedule

Sl Examination Proposed Date Actual Date

No

1 First Internal Assessment Test 09.9.2023

to
15.9.2023

2 Second Internal Assessment 26.10.2023

Test to
01.11.2023
3 Model Exam 15.11.2023
to
25.11.2023
 Books

Prescribed Text Books

Allan Greenwood, Electrical Transients in Power Systems, Wiley Inter Science,
New York, 2ndEdition, 1991.
Pritindra Chowdhari, Electromagnetic transients in Power System, John Wiley and
Sons Inc., Second Edition, 2009.
C.S. Indulkar, D.P.Kothari, K. Ramalingam, Power System Transients – A statistical
approach, PHI Learning Private Limited, Second Edition, 2010.

Reference Books
M.S.Naidu and V.Kamaraju, High Voltage Engineering, McGraw Hill, Fifth Edition,
2013.
R.D. Begamudre, Extra High Voltage AC Transmission Engineering, Wiley Eastern
Limited, 1986.
Y.Hase, Handbook of Power System Engineering, Wiley India, 2012.
J.L.Kirtley, Electric Power Principles, Sources, Conversion, Distribution and use,
Wiley, 2012.
Akihiro ametani, Power System Transient theory and applications, CRC press,
2013.
 Mini Project suggestions

1.Bewleys lattice diagram implementation by using power system

tool.
2. Finding abnormal voltage disturbances in a transmission line.
 Thank you

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