71 CHAPTER-III MODELING OF 245kV GIS SYSTEM FOR ESTIMATION AND SUPPRESSION OF VFTOs 3.1 INTRODUCTION An accurate representation of each component of a system is essential for a reliable simulation of its transient performance. This representation must be done taking into account the frequency range of the transients to be simulated. Very fast transients (VFT) belong to the highest frequency range of the power system. The simulations are suitable for frequencies varying from 100 KHz to 100MHz [13] Due to the travelling nature of the transients the modeling of GIS makes use of electrical equivalent circuits composed of lumped elements and especially by distributed parameter lines, surge impedances and travelling times. Disconnector switches are used primarily to isolate operating sections of high voltage installations from each other as a safety measure. In addition, they must also be able to perform certain switching duties such as load transfer from one bus bar to another busbaror off load connection or disconnection of bus sections, circuit breakers etc. The connection or disconnection of energized but unloaded substation sections involve the disconnector having to switch small capacitive currents, typically few mA[21]. During closing and opening operations the voltages develop across the switching contacts which
102
Embed
CHAPTER-III MODELING OF 245kV GIS SYSTEM FOR ...shodhganga.inflibnet.ac.in/bitstream/10603/19779/12/12...Fig 3.1 Single-line diagram of 245kV GIS 3.3 REPRESENTATION OF IMPORTANT GIS
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
71
CHAPTER-III
MODELING OF 245kV GIS SYSTEM FOR
ESTIMATION AND SUPPRESSION OF VFTOs
3.1 INTRODUCTION
An accurate representation of each component of a system is
essential for a reliable simulation of its transient performance. This
representation must be done taking into account the frequency range of
the transients to be simulated. Very fast transients (VFT) belong to the
highest frequency range of the power system. The simulations are
suitable for frequencies varying from 100 KHz to 100MHz [13]
Due to the travelling nature of the transients the modeling of GIS
makes use of electrical equivalent circuits composed of lumped elements
and especially by distributed parameter lines, surge impedances and
travelling times. Disconnector switches are used primarily to isolate
operating sections of high voltage installations from each other as a
safety measure. In addition, they must also be able to perform certain
switching duties such as load transfer from one bus bar to another
busbaror off load connection or disconnection of bus sections, circuit
breakers etc. The connection or disconnection of energized but unloaded
substation sections involve the disconnector having to switch small
capacitive currents, typically few mA[21]. During closing and opening
operations the voltages develop across the switching contacts which
72
subsequently collapse in a series of spark discharges often in extended
sequences. Within just nanoseconds, the channel of such a spark
discharge rapidly establishes a conducting bridge across the contacts.
Lasting few hundreds of micro seconds, it momentarily connects the
potential equalization is accompanied by transient oscillations with very
high frequencies in the adjacent GIS elements, giving to VFTOs. The
frequency and amplitudes depends up on the layout of the GIS network.
The estimation of these transients and rise times are very important in
order to design insulation levels[14]. The fast transients over voltages are
generated during switching operation of disconnectors. VFTO generated
in a GIS should be considered as an important factor in the insulation
design. In EHV class voltages, VFTOs can reach to high amplitudes
and steepness, the insulation failures of 500kV transformers due to
effect of VFTO have happened several times [15]. Hence it is important
to estimate and suppress these over voltages for protection of internal
systems. The simulation depends on the quality of the model of each
individual GIS component. In order to achieve reasonable results in GIS
structures highly accurate models for each internal equipment and also
for components connected to the GIS are necessary.
The disconnector spark itself has to be taken into account by
transient resistance according to the toepler’s equation and subsequent
arc resistance of a few ohms. The wave shape of the over voltage surge
due to disconnector switch is affected by all GIS elements. Accordingly,
73
the simulation of transients in GIS assumes an establishment of the
models for the bus, bushing, elbow, transformers, surge arresters,
breakers, spacers, disconnectors and enclosures and so on. One of the
ways is avoiding dangerous layout of GIS and dangerous operation
procedures of the disconnectors however; this brings a big limitation to
the design and control operation of GIS. Another way is to use high
speed disconnector, but there will be a problem of high trapped charge
on the floating electrode[23]. The exisisting method of suppressing
VFTOs is resistance switching. In this method a resistor of range 400Ω to
500Ω is fitted to disconnector[29,31], in this method in the event of
restriking the resistor is inserted in the circuit, so that ,the over voltages
can be suppressed, but this method is complicated in structure and has
reduced reliability and also the probability of failure of resistors are
great. Therefore this method needs to combine practical considerations.
Another method is using R-C filter circuits, this method has been
widely used in vacuum circuit breakers to suppress over voltages of
arcing, the R-C filters can be applied to suppress the VFT, but in this R-
C filter absorbs high frequency components, consumes the energy of
VFTO, but selection of R and C for different ratings applications is
difficult[18]. Another method suggested by Hongsheng Li is use of metal
oxide arrester but this can inhibit the amplitude of the VFTO, but cannot
inhibit its steepness and high frequency oscillations.
74
In this chapter, first suppression effect of VFTOs has been verified
with switching resistor across the disconnector switch and secondly the
suppression effect of VFTOs verified with new technique of application of
ferrite rings on bus bar. The frequency spectrums are also obtained
using FFT technique. The results obtained from the above methods are
compared.
In this chapter, the single line diagram of 245kV GIS and its
description is given in section 3.2. The representations of important GIS
components are given in section 3.3. The modeling of GIS components is
given in section 3.4. Calculation of various parameters for modeling is
given in the section3.5 A 245kV GIS system considered for modeling to
estimate VFTOs is presented in section 3.6 The EMTP-RV model of the
system given in section 3.7 The Simulation of the EMTP-RV GIS model to
estimate transients due to disconnector switch 1 closing operation with
fixed arc resistance is given in section 3.7.1. The Simulation of the
EMTP-RV GIS model to estimate transients due to disconnector closing
operation with variable arc resistance given in section3.7.2. The
Simulation of the EMTP-RV GIS model to estimate transients due to
disconnector opening operation with fixed arc resistance given in section
3.7.3. The simulation of the EMTP-RV GIS model to estimate transients
due to disconnector opening operation with variable arc resistance given
in section3.7.4. The Simulation of the EMTP-RV GIS model to estimate
transients due to disconnector opening operation with variable arc
75
resistance and trapped charges is given in section 3.7.5. The GIS
simulation model with DS2 opening operation and variable arc resistance
is given in the section 3.7.6. The results of various simulations are given
in the section 3.8.1. The transients on source side and load side of the
disconnector switch with different trapped charges are given in the 3.8.2.
Summary is given in the section 3.9. Various methods for suppression of
VFTOs in GIS systems are discussed in section3.10. The VFTO
suppression using resistance switching is discussed in 3.10.1. The FFT
analysis of reduced VFTOs given in section 3.10.2. Single phase
equivalent circuits of 245kV GIS system with opening and closing
resistance given in 3.10.3. Simulation results with opening and closing
resistance are presented in the section3.10.4. The VFTO suppression
using ferrite rings given in 3.11. The simulation results are given in the
section 3.14. The summary is given in the section 3.15.
76
3.2 A TYPICAL 245kV GAS INSULATED SUBSTATION
Fig 3.1 Single-line diagram of 245kV GIS
3.3 REPRESENTATION OF IMPORTANT GIS COMPONENTS
a) Bus ducts: bus duct can be represented as a loss less transmission
line for a range of frequencies lower than 100MHz. The surge impedance
is calculated from the physical dimensions of the duct (using Eq 3.4).
77
The Experimental results show that the propagation velocity in GIS duct
is close to 95% of the speed of light other devices such as closed
disconnectors can also be modeled as lossless transmission lines.
b) Surge arresters: A surge arrester model is taken into account the
steep front wave effect. The voltage developed across the arrester for a
given discharge current increases, and reaches crest prior to crest of the
discharge current. A detailed model is represent each internal shield and
block individually, and the travel times along shield sections as well as
capacitances between these sections as well as capacitances between
blocks and shields, and the blocks themselves. The detailed model is
shown in the table. Usually the switching operations do not produce
voltages high enough to cause MOAs to conduct its capacitance is taken
into account.
c) Circuit breakers: The representation of circuit breakers is very
complicated in GIS systems because of its internal irregularities. In
addition , circuit breakers with several chambers contain grading
capacitors as these components are not arranged symmetrically, a circuit
breaker has a different transient response depending up on which
terminal is connected to the surge source. A closed circuit breaker can
be represented as a lossless transmission line. The surge impedance is
calculated from the diameters of the conductor and enclosure. The effect
of grading capacitors can be ignored. The representation of a closed
circuit breaker is more complicated because the electrical length is
78
increased and the speed of progression is decreased due to the effects of
the higher dielectric constant of the grading capacitors. If the
intermediate voltages are needed, the breaker is divided into as many
sections as there are interrupters, all connected by the grading
capacitors. A simple model consists of two equal lengths of bus
connected by a capacitor with a value equivalent to the series
combination of sections are calculated from the physical dimensions of
the breaker
d) Disconnector switches: Closed disconnectors are modeled as a
transmission line with distributed parameters. Capacitance of the
switching contacts towards the ground is considered. Disconnectors in
open condition are represented with inter electrode capacitance of the
switching contacts towards the ground is considered.
e) Earth switches: Earth switches can be modeled as lumped
capacitance to ground.
79
3.4 MODELING OF GIS COMPONENTS
ELEMENT MODEL EQUIVALENT
CIRCUIT
CHARACTERISTI
CS
BUS DUCT
Transmission line with
distributed parameters.
Loss in transmission line
because of skin
effect.(Neglected)
Loss free
distributed
parameter
transmission line
SPACER Lumped Capacitance
towards the ground.
C > 20pf
ELBOW
Transmission line with
distributed parameters
and capacitance added
in between the line.
Parameters
depending on the
ratio between
conductor and
enclosure radius.
Value of the
capacitance C
depending on the
system topology.
CABLE
Transmission line with
distributed parameters.
Each end of cable is
terminating with a
lumped capacitance.
The values are
Depends up on
voltage rating of
GIS
CURRENT
TRANSFORMER
Lumped capacitance
towards the ground
The values are
Depends up on
voltage rating of
GIS
80
CAPACITIVE
VOLTAGE
TRANSFORMER
Lumped capacitance
towards the ground
BUSHING
(Capacitively
Graded Bushing)
Transmission line of
varying surge
impedances are
connected in series
Zg1, Zg2,.. are
variable surge
impedance in SF6
side. Za1, Za2, …
are variable surge
impedance in air
side.
SURGE
ARRESTER
Arrester capacitance is
considered. Protection
characteristic connected
in parallel with arrester
capacitance
In case of VFT
(0.5µs) the
protection
characteristic is
corrected in
reference to the
characteristic for
the surge 8/20µs.
Inductance of
grounding
connection is
taken into account.
POWER
TRANSFORMER
Lumped capacitance
towards the
ground.Inductive branch
toards ground is
neglected due to a very
high impedance at very
high
frequencies.Nonlinear
behavior of the core is
neglected
Value of
capacitance
depends on the
transformer type,
voltage level,
winding connection
and winding type.
81
DIS-CONNECTOR
CLOSED
Transmission line with
distributed parameters.
Capacitance of the
switching contacts
towards the ground is
considered.
Parameters
depending on the
ratio between
conductor and
enclosure radius.
Value of
capacitance C
depends on the
system topology.
DIS-CONNECTOR
OPENED
Inter electrode
capacitance of the
switching contacts
towards the ground is
considered.
C includes spacer
capacitance also.
EARTH
SWITCHING
Lumped capacitance
towards the ground.
SPARK
RESISTANCE (in
case of DS
operation)
It is a non-linear
function of time. It varies
according to the
Toepler’s Spark Law
if t < 1µs, R = 0 Ω
if t > 1µs, R varies
from 0 to 5 Ω
SPARK
(earth fault)
Spark resistance varies
according to Toepler’s
Spark Law. L is the
inductance of the spark
channel.
R is in the range of
1 to 3 Ω
CIRCUIT
BREAKER (C.B)
CLOSED
Transmission line with
distributed parameters
equivalent capacitance of
switching contacts
The surge
Impedance of C.B
bus duct is less
than 70 Ω because
82
towards the ground is
considered.
of additional
capacitance.
CIRCUIT
BREAKER (C.B)
OPENED
The capacitance between
switching contacts is
considered. C.B bus duct
is represented with
distributed parameters
on both sides of the
contacts.
The length of bus
duct on both sides
of contacts is
equal. The inter
electrode
capacitance incase
of C.B is high,
because of large
arc of the contacts.
3.5 CALCULATION OF VARIOUS PARAMETERS OF GIS
3.5.1 Calculation of Inductance
The inductance of the bus duct can be calculated by using the formula
[16] given below:
Where r1, r2, r3, r4, are the radii of the conductors in the order of decreasing
magnitude and ‘l’ is the length of the section.
−∗
∗+
+
+
××= 1
r
rln
r
r-1
r
r
2 r
rln
r
rln
r
rln 0.001 L
2
1
2
1
2
2
1
2
3
4
1
2
3
1l
3.1
83
Fig 3.2 Cross section of typical GIS System
3.5.2 Calculation of Capacitance in micro farads
The Capacitance is calculated with the assumption that the conductors
are Cylindrical. Capacitance is calculated by using the standard formulae
given below:
0
1 0
2 * * *
2 .3 * ln
r lC
b
a
π ε ε∗=
3.2
Where εo = 8.854 * 10-12, εr = 1
b = Outer Cylinder Radius
a = Inner Cylinder Radius
l = Length of the Section
3.5.3 Calculation of Capacitance due to Spacer
Spacers are used for supporting the inner conductor with reference
to the outer enclosure. They are made with Alumina filled epoxy
material whose relative permittivity (εr) is 4. The thickness of the
spacer is assumed to be the length of the capacitance for calculation.
84
3.5.4 Calculation of Short Circuit Inductance(mH) &
Resistance
Assuming a short circuit fault level of 2000MVA for 245kV system
voltage, Inductance and Resistance are calculated as follows: In the
derivation of short circuit inductance and resistance, the GIS
considered for the study is 200MVA,22.8/220Kv with leakage
reactance of 10%. The symmetrical short circuit MVA will be about 8
to 12 times the rated MVA capacity of the ttransformer.
S V * ph =phI
⇒ phV
S =phI
And V
I * X Z% =
⇒ I
V * %Z X =
But L * f * * 2 X Π=
⇒ f * * 2
X
Π=L
And it is assumed that R = XL
3.5.5 Calculation of Inductance due to Load
For 200MVA, 245KV transformer with 10% impedance and 0.8 power
factor the inductance is calculated as follows:
85
P Cos * I * V * 3 =φ
⇒ φ
=Cos * V * 3
P I
And V
I * X Z% =
⇒ I
V * %Z X =
But L * f * * 2 X Π=
⇒ f * * 2
X L
Π=
3.5.6. Calculation of Variable Arc Resistance
Based on earlier studies in SF6 gas, Toepler’s Spark Law is valid for
calculation of Variable Arc Resistance. The Variable Arc Resistance due to
Toepler’s formulae [5] is given below
= ∗ +
3.3
Where KT = Toepler’s Constant
= 0.005 volt.sec/m for SF 6 under Uniform Field conditions
L = Spark Length in meters
qo = Initial Charge or Charge at the instant of breakdown
t = Spark collapse time in sec.
The value of time varying spark resistance R (t), is calculated until
it reaches a value of 1 to 5 ohms. The integral in the denominator sums
86
up the absolute value of current ‘i’ through the resistance R (t) over the
time beginning at breakdown inception. Thus, it corresponds to the
charge conducted through the spark channel up to time‘t’.
Initial charge qo is an important parameter while considering the non-
uniform fields. But the field between the disconnector contacts is almost
uniform. Therefore qo is very small.
3.5.7 Surge Impedance
A typical 245kV gas insulated bus duct in the substation
considered has an inner conductor of 3.6 inches (8.9cm) and its
grounded outer sheath has a diameter of about 12inches (30.5cm). The
surge impedance of the 245kV SF6 bus can be calculated from the
following formula [6]
= 138√
3.4
Where Z= surge impedance in Ω
= inner radius of outer sheath
= radius of inner conductor
K= permittivity of dielectric (unity for SF6)
Hence, for a typical 245kV Gas insulated bus, the surge impedance is
about 75Ω
Ω== 8.739.8
5.30log
1
138Z
87
3.6 TYPICAL SECTION OF SEGREGATED-PHASE 245kV GIS SYSTEM
Fig.3.3 Single line diagram of typical section of segregated-phase
245kV GIS system
T - Generator transformer E.S - Earthing Switch B1 = Air –to- SF6 Gas Bushing C.B- Circuit Breaker L.A - Lightning arrester C.T- current Transformer P.T - Potential Transformer B2 = SF6 Gas - to – XLPE cable
Table.3.1 Dimensions of 245kV GIS system
Name of the GIS component Distance
in meters
Air-to-SF6bushing(From DS1)
12.6
Lightning arrester(From DS1)
11.3
Potential transformer(From DS1)
9.5
Earth switch(From DS1)
1.5
Current transformer(From DS2) 1.5
Earth switch(From DS2) 2.0
Potential transformer(From DS2) 9.5
SF6-to-XLPE cable termination
13.5
88
3.6.1 Description of the circuit
A typical section of 245kV GIS substation has been considered
for VFTO study. Its single line diagram is shown in Fig. 3.3; a complete
EMTP-RV electrical equivalent models have been developed for the GIS
system.
A substation has 200MVA, 22.8kV/220kV transformer and the
over head line from transformer is connected to 245kV GIS system
through an Air-to-SF6 gas bushing. The length of the over head line is
about 20meters. The typical 245kV GIS system consisting of lightning
arrester, instrument transformers, and high speed earthing switch and
SF6 gas to air bushing and SF6 gas to XLPE cable termination have
shown in the diagram. The surge impedance of the over head line is
considered as 350Ω with wave velocity of 300m/µs.
Due to the travelling wave nature of the VFT the modeling of
GIS makes use of electrical equivalent circuits composed by lumped
elements and especially by distributed parameter lines, defined by surge
impedances and travelling times. The power transformer is assumed as a
source side and load side of the disconnector switch being operated.
It is assumed that, initially the load side circuit breaker is
operated. The fast transient over voltages results at both load side and
source side of the disconnector switch. The switching of capacitive
current is difficult; under these conditions restrikes occurs and cause
large number of transients on the supply and load side. The variations of
89
transient voltages at Air-to-SF6 gas bushing, load side & source side are
calculated during disconnector Switch1 operation and transient voltages
at XLPE cable termination has been estimated during Disconnector
switch2 opening and closing operations using EMTP-RV. The step size
for the analysis taken as 0.1ns and stop time is selected between 5 to
8µsec. In the circuit a great deal of restrikes occurs across the switching
contacts when disconnectors are operated. These restrikes lead to
generation of VFTO; Consequently VFTO appears in the circuit. Due to
random nature of trapped charge at the commencement of disconnector
opening or closing two important parameters variable arc resistance and
trapped charge on the floating bus bar are considered during estimation
of VFTO.
The transients generated due to the operation of a disconnector switch1
have been simulated by the injection of a unit step voltage source. The
patterns of transient voltages at air to gas bushing are estimated during
Disconnector switch 1 closing and opening operations with fixed and
variable arc resistance. The patterns of transient voltages at SF6-to-XLPE
cable termination is estimated during Disconnector switch2 opening
operation the results are presented. The patterns of transient voltage at
source side, load side have been analyzed. The trapped charge is varied
from -0.1p.u to -1p.u. insteps of 0.1p.u. (1 p.u = Vm (ph))
The electrical equivalent models with trapped charge are given
in the Fig. 3.5.9 to Fig 3.5.18. The simulation waveforms are given in
90
section 3.8.1. The patterns of transient voltages at SF6-to-XLPE cable
termination during Disconnector switch 2 opening is presented in the
Fig.3.42. The VFTOs on source side with different trapped charges are
presented in the Table 3.1. The VFTOs on load side with different trapped
charges are presented in the table 3.2. The VFTOs at air to SF6 gas bushing
during opening and closing operation of DS1 are given in the table 3.3.
The disconnector switches DS1, DS2 and circuit breaker arrangement as
shown in the diagram.
The disconnectors are of motor driven, with rated voltage is
245kV and rated short-time current 40/50kA 3sec.
Earth switches are located on either side of the disconnector
switches. The rated voltage is 245kV, rated short time current 40/50kA
3sec, Method of operation is motor driven, Bus bar type is segregated
phase, Rated voltage is 245kV,Rated current is 2500/3150A, Rated short
time current rating is 40/50kA,The rated gas pressure is 0.6Mpa.
The ratings of the circuit breaker are, Rated voltage is 245kV, Rated
current is 1250/2500/3150A rated breaking current 40/50A, rated gas
pressure is about 0.6MPa and method of operation is motor driven with
spring. The instrument transformers are located with in the bay. Their
secondary connections are routed through a gas-tight bushing plate to a
terminal box.
The pressurized SF6 gas in the module serves as the primary
insulation. The high voltage connection to the switchgear is established
91
by means of conductor, which is supported by means of a gas-tight
bushing plate to the terminal box.
The surge arrester consists of metal oxide resistors with a non-
linear current/voltage characteristic. The arrester is flange-joined to the
switchgear via gas tight bushing. In the tank of the arrester module,
there is an inspection hole, through which the internal conductor can be
inspected. At the bottom there are the connections for gas monitoring,
arrester testing, and an operation counter. The SF6gas/air termination is
a combination of an angle type module and an outdoor/ SF6 bushing.
The surge impedance of the 245kV XLPE-600 Cable is taken as 30Ω and
wave velocity of 103.8m/µs.
3.7 EMTP-RV MODEL OF THE SECTION OF 245KV GIS SYSTEM
Fig. 3.4 EMTP-RV model of the section of 245kV GIS system
92
The EMTP-RV equivalent circuit representation of typical
245kV GIS system is shown in Fig.3.4. According to their internal design
all parts of the GIS have been represented thoroughly by line sections
with the corresponding surge impedance and travelling times. The
VFTOs during opening and closing operation of the disconnector switch1
are simulated and estimated for various conditions. The variable arc
resistance with respect to time is given in the software according to
customised equation.
3.7.1 GIS (EMTP-RV) model of DS1 closing operation with fixed arc
resistance (Rarc=0.5Ω)
Fig3.5 The Electrical equivalent network of the GIS system during
disconnector switch 1 closing with fixed arc resistance.
The trapped charge equivalent is considered by assigning different
voltage values to VL i .e from -0.1 to -1 pu. This can be considered as
equivalent magnitude of trapped charge.
VS =VFTO at Source side
VL = VFTO at Load side
Vagb = VFTO at Air-to- SF6 bushing
Z Z Z Z
ZZZZ ZZ
250 75 75 75
75 75 757575 30
+2nF
C1 +
0.003nF
C2 +
0.003nF
C3 +
0.2nF
C4 +
0.1nF
C5 +
0.0045nF
C6 +
0.003nF
C8
+
0.003nF
C10
+
0.003nF
C11 +
0.0045nF
C12 +
0.005nF
C13 +
0.003nF
C14 +
0.003nF
C15 +
0.0045nF
C16 +
0.1nF
C17
+
0.5
r
+
0.003nF
C7
+
1 /_0
+ C9
0.0
05
nF
+0.4nF
C18
VM+
VS
?v
VM+
VL
?v
VM+
?v
VM+
?v
Vagb+
0|2.5ms|0
DS1
93
VS =VFTO at Source side
VL = VFTO at Load side
Vagb = VFTO at Air-to- SF6 bushing
Z Z Z Z
ZZZZ ZZ
250 75 75 75
75 75 757575 30
+2nF
C1 +
0.003nF
C2 +
0.003nF
C3 +
0.2nF
C4 +
0.1nF
C5 +
0.0045nF
C6 +
0.003nF
C8
+
0.003nF
C10
+
0.003nF
C11 +
0.0045nF
C12 +
0.005nF
C13 +
0.003nF
C14 +
0.003nF
C15 +
0.0045nF
C16 +
0.1nF
C17
+
0.5
r
+
0.003nF
C7
+
1 /_0
+ C9
0.0
05
nF
+
0.4nF
C18
VM+
VS
?v
VM+
VL
?v
VM+
?v
VM+
?v
Vagb+
-1|2.5ms|0
DS1
3.7.2 GIS model of DS1 closing operation with variable arc resistance
Fig.3.6 The Electrical equivalent network of the GIS system during
disconnector switch 1 closing with variable arc resistance
3.7.3 GIS simulation model of DS1 opening operation with fixed arc
resistance
Fig. 3.7 The Electrical equivalent network of the GIS system during
disconnector switch 1 opening with fixed arc resistance.
VS =VFTO at Source side
VL = VFTO at Load side
Vagb = VFTO at Air-to- SF6 bushing
Z Z Z Z
ZZZZ ZZ
250 75 75 75
75 75 757575 30+2nF
C1 +
0.003nF
C2 +
0.003nF
C3 +
0.2nF
C4 +
0.1nF
C5 +
0.0045nF
C6 +
0.003nF
C8
+
0.003nF
C10
+
0.003nF
C11 +
0.0045nF
C12 +
0.005nF
C13 +
0.003nF
C14 +
0.003nF
C15 +
0.0045nF
C16 +
0.1nF
C17
+
0.5
r
+
0.003nF
C7
+
1 /_0
+ C9
0.0
05
nF
+
0.4nF
C18
VM+
VS
?v
VM+
VL
?v
VM+
?v
VM+
?v
Vagb+
0|2.5ms|0
DS1
R(t)+0
Rt1
94
3.7.4 GIS simulation model of DS1 opening operation with variable
arc resistance
Fig3.8 The Electrical equivalent networks of the GIS system during
disconnector switch 1 opening with variable arc resistance
3.7.5 GIS simulation models of DS1 opening operation with variable
arc resistance and with trapped charge of -0.1p.u to -1p.u
Fig.3.9 The Electrical equivalent network of the GIS system during
opening operation of disconnector switch 1 with -0.1 p.u. trapped
charge
VS =VFTO at Source side
VL = VFTO at Load side
Vagb = VFTO at Air-to- SF6 bushing
Z Z Z Z
ZZZZ ZZ
250 75 75 75
75 75 757575 30
+2nF
C1 +0.003nF
C2 +
0.003nF
C3 +
0.2nF
C4 +
0.1nF
C5 +
0.0045nF
C6 +
0.003nF
C8
+
0.003nF
C10
+
0.003nF
C11 +
0.0045nF
C12 +
0.005nF
C13 +
0.003nF
C14 +
0.003nF
C15 +
0.0045nF
C16 +
0.1nF
C17
+
0.5
r
+
0.003nF
C7
+
1 /_0
+ C9
0.0
05
nF
+
0.4nF
C18
VM+
VS
?v
VM+
VL
?v
VM+
?v
VM+
?v
Vagb+
-1|2.5ms|0
DS1
R(t)+0
Rt1
VS =VFTO at Source side
VL = VFTO at Load side
VB = VFTO at SF6-to- XLPE cable termination
Z Z Z Z
ZZZZ ZZ
250 75 75 75
7575 7575
75 30
+
C1
2nF +C2
0.003nF
+C3
0.003nF
+C4
0.2nF
+C5
0.1nF
+C6
0.0045nF
+C7
0.003nF
+ C8
0.003nF
+ C9
0.003nF
+ C10
0.0045nF
+ C11
0.005nF
+ C12
0.003nF
+ C13
0.003nF
+ C14
0.0045nF
+ C15
0.1nF
+R1
0.5
+ C16
0.003nF
+
1 /_0
+
0.0
05
nF
C
17
+C18
0.4nF
VM+
?v
VSVM+
?v
VL+
DS1
?v
-1|1E15|0
VM+
m4
?v
R(t)+0
Rt1
95
VS =VFTO at Source side
VL = VFTO at Load side
VB = VFTO at SF6-to- XLPE cable termination
Z Z Z Z
ZZZZ ZZ
250 75 75 75
7575 7575
75 30
+
C1
2nF +C2
0.003nF
+C3
0.003nF
+C4
0.2nF
+C5
0.1nF
+C6
0.0045nF
+C7
0.003nF
+ C8
0.003nF
+ C9
0.003nF
+ C10
0.0045nF
+ C11
0.005nF
+ C12
0.003nF
+ C13
0.003nF
+ C14
0.0045nF
+ C15
0.1nF
+R1
0.5
+ C16
0.003nF
+
1 /_0
+
0.0
05
nF
C
17
+C18
0.4nF
VM+
?v
VSVM+
?v
VL+
DS1
?v
-1|1E15|0
VM+
m4
?v
R(t)+0
Rt1
Fig3.10 The Electrical equivalent network of the GIS system during
opening operation of disconnector switch 1 with- 0.2 p.u. trapped
charge
Fig3.11 The Electrical equivalent network of the GIS system during opening
operation of disconnector switch 1 with -0.3 p.u. trapped charge
VS =VFTO at Source side
VL = VFTO at Load side
VB = VFTO at SF6-to- XLPE cable termination
Z Z Z Z
ZZZZ ZZ
250 75 75 75
7575 7575
75 30
+
C1
2nF +C2
0.003nF
+C3
0.003nF
+C4
0.2nF
+C5
0.1nF
+C6
0.0045nF
+C7
0.003nF
+ C8
0.003nF
+ C9
0.003nF
+ C10
0.0045nF
+ C11
0.005nF
+ C12
0.003nF
+ C13
0.003nF
+ C14
0.0045nF
+ C15
0.1nF
+R1
0.5
+ C16
0.003nF
+
1 /_0
+
0.0
05
nF
C
17
+C18
0.4nF
VM+
?v
VSVM+
?v
VL+
DS1
?v
-1|1E15|0
VM+
m4
?v
R(t)+0
Rt1
96
Fig. 3.12 The Electrical equivalent network of the GIS system during
opening operation of disconnector switch 1 with -0.4 p.u. trapped
charge
Fig.3.13. The Electrical equivalent network of the GIS system during opening
operation of disconnector switch 1 with -0.5 p.u. trapped charge.
VS =VFTO at Source side
VL = VFTO at Load side
VB = VFTO at SF6-to- XLPE cable termination
Z Z Z Z
ZZZZ ZZ
250 75 75 75
7575 7575
75 30+
C1
2nF +C2
0.003nF
+C3
0.003nF
+C4
0.2nF
+C5
0.1nF
+C6
0.0045nF
+C7
0.003nF
+ C8
0.003nF
+ C9
0.003nF
+ C10
0.0045nF
+ C11
0.005nF
+ C12
0.003nF
+ C13
0.003nF
+ C14
0.0045nF
+ C15
0.1nF
+R1
0.5
+ C16
0.003nF+
1 /_0
+
0.0
05
nF
C
17
+C18
0.4nF
VM+
?v
VSVM+
?v
VL+
DS1
?v
-1|1E15|0
VM+
m4
?v
R(t)+0
Rt1
VS =VFTO at Source side
VL = VFTO at Load side
VB = VFTO at SF6-to- XLPE cable termination
Z Z Z Z
ZZZZ ZZ
250 75 75 75
7575 7575
75 30
+
C1
2nF +C2
0.003nF
+C3
0.003nF
+C4
0.2nF
+C5
0.1nF
+C6
0.0045nF
+C7
0.003nF
+ C8
0.003nF
+ C9
0.003nF
+ C10
0.0045nF
+ C11
0.005nF
+ C12
0.003nF
+ C13
0.003nF
+ C14
0.0045nF
+ C15
0.1nF
+R1
0.5
+ C16
0.003nF
+
1 /_0
+
0.0
05
nF
C
17
+C18
0.4nF
VM+
?v
VSVM+
?v
VL+
DS1
?v
-1|1E15|0
VM+
m4
?v
R(t)+0
Rt1
97
Fig. 3.14 The Electrical equivalent network of the GIS system during
opening operation of disconnector switch 1 with -0.6 p.u. trapped
charge
Fig. 3.15 The Electrical equivalent network of the GIS system during
opening operation of disconnector switch 1 with -0.7 p.u. trapped
charge
VS =VFTO at Source side
VL = VFTO at Load side
VB = VFTO at SF6-to- XLPE cable termination
Z Z Z Z
ZZZZ ZZ
250 75 75 75
7575 7575
75 30+
C1
2nF +C2
0.003nF
+C3
0.003nF
+C4
0.2nF
+C5
0.1nF
+C6
0.0045nF
+C7
0.003nF
+ C8
0.003nF
+ C9
0.003nF
+ C10
0.0045nF
+ C11
0.005nF
+ C12
0.003nF
+ C13
0.003nF
+ C14
0.0045nF
+ C15
0.1nF
+R1
0.5
+ C16
0.003nF+
1 /_0
+
0.0
05
nF
C
17
+C18
0.4nF
VM+
?v
VSVM+
?v
VL+
DS1
?v
-1|1E15|0
VM+
m4
?v
R(t)+0
Rt1
VS =VFTO at Source side
VL = VFTO at Load side
VB = VFTO at SF6-to- XLPE cable termination
Z Z Z Z
ZZZZ ZZ
250 75 75 75
7575 7575
75 30
+
C1
2nF +C2
0.003nF
+C3
0.003nF
+C4
0.2nF
+C5
0.1nF
+C6
0.0045nF
+C7
0.003nF
+ C8
0.003nF
+ C9
0.003nF
+ C10
0.0045nF
+ C11
0.005nF
+ C12
0.003nF
+ C13
0.003nF
+ C14
0.0045nF
+ C15
0.1nF
+R1
0.5
+ C16
0.003nF
+
1 /_0
+
0.0
05
nF
C
17
+C18
0.4nF
VM+
?v
VSVM+
?v
VL+
DS1
?v
-1|1E15|0
VM+
m4
?v
R(t)+0
Rt1
98
VS =VFTO at Source side
VL = VFTO at Load side
VB = VFTO at SF6-to- XLPE cable termination
Z Z Z Z
ZZZZ ZZ
250 75 75 75
7575 7575
75 30
+C1
2nF +C2
0.003nF
+C3
0.003nF
+C4
0.2nF
+C5
0.1nF
+C6
0.0045nF
+C7
0.003nF
+ C8
0.003nF
+ C9
0.003nF
+ C10
0.0045nF
+ C11
0.005nF
+ C12
0.003nF
+ C13
0.003nF
+ C14
0.0045nF
+ C15
0.1nF
+R1
0.5
+ C16
0.003nF
+
1 /_0
+
0.0
05
nF
C
17
+C18
0.4nF
VM+
?v
VSVM+
?v
VL+
DS1
?v
-1|1E15|0
VM+
m4
?v
R(t)+0
Rt1
Fig.3.16 The Electrical equivalent network of the GIS system during opening
operation of disconnector switch 1 with -0.8 p.u. trapped charge
Fig.3.17 The Electrical equivalent network of the GIS system during opening
operation of disconnector switch 1 with -0.9 p.u. trapped charge
VS =VFTO at Source side
VL = VFTO at Load side
VB = VFTO at SF6-to- XLPE cable termination
Z Z Z Z
ZZZZ ZZ
250 75 75 75
7575 7575
75 30
+
C1
2nF +C2
0.003nF
+C3
0.003nF
+C4
0.2nF
+C5
0.1nF
+C6
0.0045nF
+C7
0.003nF
+ C8
0.003nF
+ C9
0.003nF
+ C10
0.0045nF
+ C11
0.005nF
+ C12
0.003nF
+ C13
0.003nF
+ C14
0.0045nF
+ C15
0.1nF
+R1
0.5
+ C16
0.003nF
+
1 /_0
+
0.0
05
nF
C
17
+C18
0.4nF
VM+
?v
VSVM+
?v
VL+
DS1
?v
-1|1E15|0
VM+
m4
?v
R(t)+0
Rt1
99
Fig.3.18 The Electrical equivalent network of the GIS system during
opening operation of disconnector switch 1 with -1 p.u. trapped
charge.
3.7.6 GIS simulation model of DS2 opening operation with variable
arc resistance
Fig.3.19 The EMTP-RV equivalent network of the GIS system during closing
operation of disconnector switch 2 with -1pu trapped charge.
VS =VFTO at Source side
VL = VFTO at Load side
VB = VFTO at SF6-to- XLPE cable termination
Z Z Z Z
ZZZZ ZZ
250 75 75 75
7575 7575
75 30+
C1
2nF +C2
0.003nF
+C3
0.003nF
+C4
0.2nF
+C5
0.1nF
+C6
0.0045nF
+C7
0.003nF
+ C8
0.003nF
+ C9
0.003nF
+ C10
0.0045nF
+ C11
0.005nF
+ C12
0.003nF
+ C13
0.003nF
+ C14
0.0045nF
+ C15
0.1nF
+R1
0.5
+ C16
0.003nF+
1 /_0
+
0.0
05
nF
C
17
+C18
0.4nF
VM+
?v
VSVM+
?v
VL+
DS1
?v
-1|1E15|0
VM+
m4
?v
R(t)+0
Rt1
250 75 75 75 75
75 75 75 75 30
z z z z z
z z z z z
VS = VFTO at source side
VL = VFTO at load side
Vxlpe = XLPE cable termination
VS VL
+
2nF
C1 +
0.2nF
C4 +
0.1nF
C5 +
0.0045nF
C6 +
0.003nF
C7 +
0.003nF
C8 +0.003nF
C9
+
0.003nF
C10
+
0.003nF
C11
+
0.0045nF
C12 +
0.005nF
C13 +
0.003nF
C14 +
0.003nF
C15 +
0.0045nF
C16 +
0.1nF
C17 +
0.4nF
C18
+
0.0
05
nF
C
19
+
1 /_0
AC1 +
0.003nF
C3+
0.003nF
C2
+1ms|2.5ms|0
DS2
R(t)+0
Rt1VM+
?v
Vxlpe
3.8 RESULTS OF THE SIMULATION
Fig. 3.20 VFTO at Air
DS1 with fixed arc resistance.
Fig. 3.21 VFTO at Air
the DS1 with variable arc resistance.
100
RESULTS OF THE SIMULATIONS
VFTO at Air-to-SF6 bushing during the opening operation of the
DS1 with fixed arc resistance.
VFTO at Air-to-SF6 bushing during the opening operation of
the DS1 with variable arc resistance.
SF6 bushing during the opening operation of the
operation of
101
Fig. 3.22 VFTO at Air-to-SF6 bushing during the closing operation of the
DS1 with fixed arc resistance.
Fig. 3.23 VFTO at Air-to-SF6 bushing during the closing operation of the
DS1 with variable arc resistance.
Fig. 3.24 VFTO at source side of DS1 with variable arc resistance and
trapped charge of
Fig. 3.25 VFTO at load side of DS1 with variable arc resistance and trapped
charge of -0.1p.u.
102
at source side of DS1 with variable arc resistance and
trapped charge of- 0.1p.u.
at load side of DS1 with variable arc resistance and trapped
0.1p.u.
at source side of DS1 with variable arc resistance and
at load side of DS1 with variable arc resistance and trapped
Fig. 3.26 VFTO at sou
charge of -0.2p.u.
Fig. 3.27 VFTO at load side of DS1 with variable arc resistance and trapped
charge of -0.2p.u
103
at source side of DS1 with variable arc resistance and trapped
0.2p.u.
at load side of DS1 with variable arc resistance and trapped
0.2p.u.
rce side of DS1 with variable arc resistance and trapped
at load side of DS1 with variable arc resistance and trapped
Fig. 3.28 VFTO at source side of DS1 with variable arc resistance and
trapped charge of
Fig. 3.29 VFTO at load side of DS1 with variable arc resistance and trapped
charge of -0.3p.u.
104
at source side of DS1 with variable arc resistance and
trapped charge of -0.3p.u.
at load side of DS1 with variable arc resistance and trapped
0.3p.u.
at source side of DS1 with variable arc resistance and
at load side of DS1 with variable arc resistance and trapped
Fig. 3.30 VFTO at source side of DS1 with variable arc resistance
trapped charge of
Fig. 3.31 VFTO at load side of DS1 with variable arc resistance and trapped
charge of -0.4p.u.
105
at source side of DS1 with variable arc resistance
trapped charge of -0.4p.u.
at load side of DS1 with variable arc resistance and trapped
0.4p.u.
at source side of DS1 with variable arc resistance and
at load side of DS1 with variable arc resistance and trapped
106
Fig. 3.32 VFTO at source side of DS1 with variable arc resistance and
trapped charge of -0.5p.u.
Fig. 3.33 VFTO at source side of DS1 with variable arc resistance and
trapped charge of -0.5p.u.
Fig. 3.34 VFTO at source side of DS1 with variable arc resistance and
trapped charge of
Fig. 3.35 VFTO at load side of DS1 wi
charge of -0.6p.u.
107
at source side of DS1 with variable arc resistance and
trapped charge of -0.6p.u.
at load side of DS1 with variable arc resistance and trapped
0.6p.u.
at source side of DS1 with variable arc resistance and
th variable arc resistance and trapped
Fig. 3.36 VFTO at source side of DS1 with variable arc resistance and
trapped charge of
Fig. 3.37 VFTO at load side of DS1 with variable arc resistance and trapped
charge of -0.7p.u.
108
at source side of DS1 with variable arc resistance and
trapped charge of -0.7p.u.
at load side of DS1 with variable arc resistance and trapped
0.7p.u.
at source side of DS1 with variable arc resistance and
at load side of DS1 with variable arc resistance and trapped
109
Fig. 3.38 VFTO at source side of DS1 with variable arc resistance and
trapped charge of -0.8p.u.
Fig. 3.39 VFTO at load side of DS1 with variable arc resistance and trapped
charge of -0.8p.u.
110
Fig. 3.40 VFTO at source side of DS1 with variable arc resistance and
trapped charge of -0.9 p.u.
Fig. 3.41 VFTO at load side of DS1 with variable arc resistance and trapped
charge of -0.9p.u.
111
Fig. 3.42 VFTO at source side of DS1 with variable arc resistance and
trapped charge of -1 p.u.
Fig. 3.43 VFTO at load side of DS1 with variable arc resistance and trapped
charge of -1 p.u.
112
Fig. 3.44 VFTO at SF6 – to – XLPE cable termination during opening of DS2
With variable arc resistance and trapped charge of -1 p.u.
113
3.9 FAST FOURIER TRANSFORM (FFT) ANALYSIS OF VERY FAST
TRANSIENT OVER VOLTAGES
In this section, the frequency spectrums of voltage transients are
obtained using FFT algorithm. The Fourier transform has long been a
principle analytical tool in such diverse fields as linear systems, Optics,
Probability theory, Quantum physics, Antennas and Signal analysis.
However, a similar statement is not true for the Discrete Fourier
transform (DFT). But with the development of the Fast Fourier transform
(an algorithm that efficiently computes the DFT) many facets of scientific
analysis have been completely revolutionized. The Fourier Integral is
defined by the expression:
( ) ∫+∞
∞−
Π−= dtetsfS ftj 2)( (3.5)
Where s (t) is the waveform to be decomposed into the sum of
sinusoids, S (f) is the Fourier Transform of set, and 1−=j . Typically, s
(t) is a function of the variable time and S (f) is a function of the variable
frequency. The Fourier Transform identifies or distinguishes the different
frequency sinusoids and their respective amplitudes, which combine to
form an arbitrary waveform. The Inverse Fourier Transform is defined as
∫+∞
∞−
Π= dtefSts ftj 2)()( (3.6)
Inversion transformation allows the determination of a function of
time from its Fourier transform. The validity of equations (3.5) and (3.6)
114
depend upon certain conditions as if s(t) is integral in the sense
∫+∞
∞−
∞<)(ts (3.7)
Then its Fourier transform S(f) exists and satisfies the inverse
Fourier transform. It is to be noted that the above condition is a
sufficient but not a necessary for the existence of a Fourier transform.
If s(t) = β(t) sin(2πft+α), where f and α rare arbitrary constants. If
β(t+k) < β(t) for | t | > λ > 0, then function s(t)/t is absolutely integrable
in the sense of equation (3.7) then S(f) exists and satisfies the Inverse
Fourier Transform equation (3.6).
3.9.1: Discrete Fourier transforms
1,.....,2,1,0,)()/(1
0
/2 −== ∑−
=
−NnekTgkTnG
N
K
Nnkj π (3.8)
The above expression relates N samples of time and N samples of
frequency by means of the continuous Fourier Transform. “The Discrete
Fourier Transform (DFT) is then a special case for the Continuous
Fourier Transform (CFT)”. If it is assumed that the N samples of the
original function g (t) are one period of a periodic waveform, the Fourier
Transform of this periodic function is given by the N samples computed
by equation (3.8).
115
3.9.2: Inverse Discrete Fourier Transforms
∑−
=
Π −==1
0
/2 1,...,2,1,0,)/(/1)(N
n
NnkjNnekTnGNkTg (3.9)
The above discrete inversion formula exhibits periodicity in the
same manner as the Discrete Transform (DT); the period is defined by N
samples of g(kT). This property results from the periodic nature of
ej2πnk/N.
Properties:
(l) Linearity x (t) + h (t) ↔ X (f) + H (f)
(2) Symmetry H(t) ↔ h(-f)
(3) Time scaling h (kt) ↔ 1/k H (f/k)
(4) Frequency scaling1/k h (t/k) ↔ H(kf)
(5) Time shifting h (t-to) ↔ H (f) ej2πft0
(6) Frequency shifting h (t) e j2πft0 ↔ H (f-fo)
(7) Convolution x(t) h(t) ↔ x(T)h(t- T)dT
The above properties for the Continuous Fourier Transform can be
simply restated with the appropriate notation for the discrete Fourier
transform, as the latter is a special case of the former.
3.9.3: Fast Fourier Transform (FFT):
Consider the discrete Fourier transform
∑ −== − 1,....,2,1,0,)()( /2
0 NnekxnXNnkj π (3.10)
Where we have replaced kT by 'k' and n/NT by 'n' for convenience
of notation. We note that equation (3.10) describes the computation of N
116
equations. For example, if N = 4 and if we let
NjeW
/2π−= (3.11)
Then equation (3.10) can be written as
0
0
0
0
0
0
0
0 )3()2()1()0()0( WxWxWxWxX +=== (3.12)
3
0
2
0
1
0
0
0 )3()2()1()0()1( WxWxWxWxX +===
6
0
4
0
2
0
0
0 )3()2()1()0()2( WxWxWxWxX +===
9
0
6
0
3
0
0
0 )3()2()1()0()3( WxWxWxWxX +===
Equation (3.12) can be more easily represented in matrix form
=
)3(
)2(
)1(
)0(
)3(
)2(
)1(
)0(
0
0
0
0
9630
6420
3210
0000
X
X
X
X
WWWW
WWWW
WWWW
WWWW
X
X
X
X
(3.13)
or more compactly as
X(n) = Wnk Xo(k) (3.14)
Examination of equation (3.13) reveals that since W and possibly
Xo(k) are complex, then N2 complex multiplications and N(n-l) complex
additions are necessary to perform the required matrix computation.
“The FFT owes its success to the fact that the algorithm reduces the
number of multiplications and additions required in the computation of
equation”. To illustrate the FFT algorithm, it is convenient to choose the
number of sample points of Xo (k) according to the relation N= 2r where
'r' is an integer. From the choice of N= 4 = 2r = 22, we can apply the FFT
to the computation of equation (3.13). The first step in developing the
117
FFT algorithm for this example is to rewrite equation (3.13) as
=
)3(
)2(
)1(
)0(
1
1
1
1111
)3(
)2(
)1(
)0(
0
0
0
0
123
202
321
X
X
X
X
WWW
WWW
WWW
X
X
X
X
(3.15)
Matrix equation (3.15) was derived from (3.13) by using the
relationship Wnk = WnkmodN. Recall that [nk mod (N)] is the remainder
upon division of nk by N;
hence if N = 4, n = 2 and k = 3 then
W6 = W2 (3.16)
Since, Wnk+W6 = exp[(-jπ/4)(6)] = exp[-j3π]
exp[-jπ] = exp[(-jπ/4)(6)] = W2 = WnkmodN (3.17)
The second step in the development is to factor the square matrix
in the equation (3.15) as follows:
=
)3(
)2(
)1(
)0(
010
001
010
001
110
100
001
001
)3(
)1(
)2(
)0(
0
0
0
0
2
2
0
0
3
1
2
0
X
X
X
X
W
W
W
W
W
W
W
W
X
X
X
X
(3.18)
For the present, it suffices to show that multiplication of the two
square matrices of equation (3.18) yields the square matrix of equation
(3.15) with the exception that rows 1 and 2 have been interchanged. It is
to be noted that this interchange has been taken into account in
equation (3.18) by rewriting the column vector X(n); let the row
118
interchanged vector be denoted by
=
)3(
)1(
)2(
)0(
)(1
X
X
X
X
nX (3.19)
One should verify that equation (3.18) yields equation (3.15) with
the interchanged rows as noted above. This factorization is the key to the
efficiency of the FFT algorithm. Having accepted the fact that equation
(3.18) is correct, although the results are scrambled, one should then
examine the number of multiplications necessary to compute the
equation. The final equation can be re written as:
=
)0(
)0(
)0(
)0(
010
001
010
001
)3(
)2(
)1(
)0(
0
0
0
0
2
2
0
0
1
1
1
1
X
X
X
X
W
W
W
W
X
X
X
X
(3.20)
i.e. column vector x1(k) is equal to the product of the two matrices
on the right in equation (3.18). Element x1(0) is computed by one
complex multiplication and one complex addition
3.9.4: WEIGHTING FUNCTIONS
A weighting function, w (n), is a sequence of numbers that is
multiplied by input data prior to performing a Discrete Fourier
Transform (DFT) on that data. Weighting (also called window) functions
reduce sidelines of DFT filters and widen main lobes while, fortunately,
not altering the locations of the centers of the filters.
Weighting function selection can be made early in the design process
119
because the choice of FFT algorithm and their functions are independent
of each other. Choice of a weighting function to provide the specified side
lobe level is done without concern for the FFT algorithm that will be used
because they work for any length FFT and they work the same for any
FFT algorithm. They do not alter the FFTs ability to distinguish two
frequencies. The performance measures of weighting functions and
comparison are given in [44].
The different types of weighting functions used are:
(1) Rectangular: for n = 0 to N-l, ω(n) = 1
The rectangular weighting function is just the plain FFT without
modifying the input data samples. The peak of the highest side lobe is
only 13 dB below the main-lobe response, and the side lobe peaks do not
drop off rapidly. This makes it poor for signals with multiple frequency
components that have amplitudes that are more than 6 dB different from
each other.
(2) Triangular: for n = 0 to N/2, w (n) = 2*n/N
n = N/2+1 to N-1, w(n) = 2*(N-n)/N
The triangular window function is used to provide side lobes and
straddle loss lower than the rectangular function. The outstanding
characteristic of this window function is the smaller number of side lobes
than the others.
(3) Sine-lobe: for n = 0 to N-1, w (n) = sin(nπ/N)
The sine-lobe window function provides improved side lobe performance.
120
For power of two FFTs, this window function has a computational
advantage over triangular window function because the coefficients are
the same one as used to compute the FFT.
(4) Hanning: for n = 0 to N-l, w(n) = 0.5*[1-cos(nπ/N)]
The Hanning window function is slightly more complicated to compute
than the sine-lobe. The peaks of its side lobes fall off 50% faster than the
triangular and sine-lobe functions.
(5) Sine-cubed: for n = 0 to N-l, w(n) = sin3(nπ/N)
The sine-cubed function is a natural extension to the side-lobe window
function, but with values that are not used for the complex
multiplications between powers of two building blocks.
(6) Sine to the fourth: for n = 0 to N-l, w(n) = sin4 (nπ/N)
(7) Hamming: for n = 0 to N-l, w(n) = 0.54 – 0.46.cos(2nπ/N)
(8) Blackman:
for n = 0 to N-l , w(n) = 0.42 – 0.5*cos(2nπ/N) = 0.08* 0.08* cos(4nπ/N)
(9) Three-sample Blackman-harries:
(a) For n = 0 to N-l, w (n) = 0.449595 – 0.49364* cos(2nπ/N) = 0.05677*
cos(4nπ/N)
(b) For n = 0 to N-l, w (n) = 0.42323- 0.49775* cos(2nπ/N) = 0.07992*