Embed Size (px)
Citation preview
So
lid
Sta
te R
ela
ys &
Its
Ap
pli
cati
on
sA
pp
lica
tio
ns
Pre
sen
ted
By
Dr.
Mo
sta
faA
bd
el-
Ge
lie
l
Co
urs
e O
bje
ctiv
es
•K
no
w n
ew
te
chn
iqu
es
in r
ela
y i
nd
ust
rie
s.
•U
nd
ers
tan
d t
he
ty
pe
s o
f st
ati
c re
lays
an
d i
ts
com
po
ne
nts
.
•U
nd
ers
tan
d D
igit
al
rela
y f
un
da
me
nta
ls•
Un
de
rsta
nd
Dig
ita
l re
lay
fu
nd
am
en
tals
•a
pp
ly t
he
dif
fere
nt
typ
es
of
dig
ita
l re
lays
Co
urs
e c
on
ten
ts
•Ty
pe
of
rela
y s
yste
m
•G
en
era
l o
verv
iew
of
sta
tic
rela
ys.
•S
tati
c R
ela
y e
lem
en
ts a
nd
de
sig
n
•C
om
pu
ter
ap
pli
cati
on
s to
pro
tect
ive
re
lay
ing
.•
Co
mp
ute
r a
pp
lica
tio
ns
to p
rote
ctiv
e r
ela
yin
g.
•D
igit
al
rela
y f
un
da
me
nta
ls
•M
ath
em
ati
cal
Ba
ckg
rou
nd
of
dig
ita
l re
lay.
•A
pp
lica
tio
ns
of
Dig
ita
l re
lay
•A
dva
nce
d t
op
ics
in r
ela
yin
g s
yste
m
Co
urs
e o
utc
om
es
•B
ase
d o
n N
AR
S (
Na
tio
na
l A
ccre
dit
ati
on
refe
ren
ce S
tan
da
rd)
the
co
urs
e s
ho
uld
co
ver
som
e in
ten
de
d l
ea
rnin
g o
utc
om
es
1-
Kn
ow
led
ge
an
d U
nd
ers
tan
din
g1
-K
no
wle
dg
e a
nd
Un
de
rsta
nd
ing
2-
skil
ls
–1
-In
tell
ect
ua
l S
kil
ls
–2
-P
ract
ica
l S
kil
ls
–3
-G
en
era
l S
kil
ls
Kn
ow
led
ge
an
d u
nd
ers
tan
din
g
•U
nd
ers
tan
d e
lect
rom
ech
an
ica
l re
lays
ve
rse
s st
ati
c a
nd
dig
ita
l re
lay.
•K
no
w r
ece
nt
de
velo
pm
en
ts in
th
e a
rea
of
mic
rop
roce
sso
r re
lays
an
d p
rote
ctio
n s
yste
ms.
•D
em
on
stra
te t
he
uti
liza
tio
n a
nd
va
rio
us
•D
em
on
stra
te t
he
uti
liza
tio
n a
nd
va
rio
us
ap
pli
cati
on
s o
f d
igit
al
rela
ys i
n in
teg
rate
d
ap
pa
ratu
s p
rote
ctio
n.
•K
no
w h
ow
to
in
teg
rate
pro
tect
ion
an
d c
on
tro
l in
su
bst
ati
on
s.
•D
isti
ng
uis
h b
etw
ee
n d
iffe
ren
t m
ath
em
ati
cal t
oo
ls
use
d in
dig
ita
l re
lay
Inte
lle
ctu
al sk
ills
•D
ete
rmin
e t
he
re
lay
fu
nct
ion
an
d c
lass
ify
th
e r
ela
y
typ
es
•D
ete
rmin
e t
he
re
lay
co
mp
on
en
ts
•S
ele
ct t
he
su
ita
ble
ma
the
ma
tica
l to
ols
to
bu
ild
dig
ita
l re
lay
alg
ori
thm
rela
y a
lgo
rith
m
•D
em
on
stra
te t
he
ha
rdw
are
an
d s
oft
wa
re o
f d
igit
al
rela
y
•R
ep
ort
th
e m
ain
dif
fere
nce
s b
etw
ee
n d
igit
al
an
d s
tati
c re
lay
•A
pp
ly d
igit
al
rela
y f
or
dif
fere
nt
ele
ctri
c e
qu
ipm
en
ts
Pra
ctic
al
(Pro
fess
ion
al)
Sk
ills
•D
esi
gn
a c
om
ple
te s
tati
c re
lay
•S
ele
ct t
he
ha
rdw
are
th
at
is s
uit
ab
le f
or
use
in
d
igit
al r
ela
ys e
ffe
ctiv
ely
.
•D
esi
gn
a d
igit
al re
lay
fo
r d
iffe
ren
t e
lect
ric
eq
uip
me
nt.
eq
uip
me
nt.
•D
iffe
ren
tia
te b
etw
ee
n t
he
su
ita
ble
re
lay
a
lgo
rith
m f
or
a s
pe
cifi
ed
re
lay
fu
nct
ion
•M
an
ipu
late
th
e m
ea
sure
d s
ign
al
nu
me
rica
lly
•P
erf
orm
ing
la
bo
rato
ry e
xpe
rim
en
ts t
o g
et
fam
ilia
r w
ith
dif
fere
nt
typ
es
of
pro
tect
ive
re
lays
Ge
ne
ral s
kil
ls
•C
om
mu
nic
ate
eff
ect
ive
ly w
ith
co
lle
ag
ue
s a
nd
p
rofe
ssio
na
l e
ng
ine
ers
to
inte
rch
an
ge
kn
ow
led
ge
a
nd
in
form
ati
on
in
th
e p
ow
er
en
gin
ee
rin
g f
ield
of
pra
ctic
e.
•P
rep
are
eff
ect
ive
an
d i
nfo
rma
tive
te
chn
ica
l •
Pre
pa
re e
ffe
ctiv
e a
nd
in
form
ati
ve t
ech
nic
al
rep
ort
s a
nd
pre
sen
t re
sult
s o
n e
lect
rica
l po
we
r e
ng
ine
eri
ng
.
•U
sin
g in
form
ati
on
te
chn
olo
gy
(IT
) to
fo
llo
w t
he
co
nti
nu
ou
s a
dva
nce
me
nt
in e
lect
rica
l po
we
r in
no
vati
on
s a
nd
te
chn
olo
gic
al a
pp
lica
tio
ns.
•P
ract
ice
dig
ita
l re
lay
op
era
tio
n a
nd
tu
nin
g
Lect
ure
1
: In
tro
du
ctio
n
•In
tro
du
ctio
n t
o r
ela
yin
g s
yste
m
•M
ult
i-la
yer
stru
ctu
re o
f p
ow
er
syst
em
•E
lem
en
ts o
f p
rote
ctio
n s
yste
m
•Ty
pe
of
pro
tect
ion
de
vic
es
•Ty
pe
of
pro
tect
ion
de
vic
es
•Ty
pe
of
rela
yin
g s
yste
m
Pro
tect
ive
Re
lay
ing
•P
ow
er
syst
em
is
an
ess
en
tia
l a
nd
th
e m
ain
ste
m o
f m
od
ern
ind
ust
ria
l so
cie
ty,
the
refo
re i
ncr
ea
sin
g a
vail
ab
ilit
y, s
afe
ty a
nd
qu
ali
ty o
f p
ow
er
syst
em
is m
an
da
tory
.
•T
he
po
we
r sy
ste
m is
alw
ays
su
bje
ct t
o a
lmo
st u
n-m
ea
sura
ble
an
d u
np
red
icta
ble
dis
turb
an
ces
cre
ate
d b
y d
iffe
ren
t a
bn
orm
al
an
d u
np
red
icta
ble
dis
turb
an
ces
cre
ate
d b
y d
iffe
ren
t a
bn
orm
al
rea
son
s su
ch a
s:
–ra
nd
om
loa
d c
ha
ng
es
–E
qu
ipm
en
t a
nd
op
era
tor
fail
ure
–Fa
ult
s d
ue
to
na
tura
l ca
use
s
Pro
tect
ive
Re
lay
ing
–T
he
fu
nct
ion
of
pro
tect
ive
re
lay
ing
is
to p
rom
ptl
y r
em
ove
fro
m s
erv
ice
an
y e
lem
en
t fr
om
th
e p
ow
er
sys
tem
th
at
sta
ts t
o
op
era
te in
an
ab
no
rma
l m
an
ne
r.
–T
he
pu
rpo
se i
s
•Li
mit
fu
rth
er
da
ma
ge
of
eq
uip
me
nt
•Li
mit
fu
rth
er
da
ma
ge
of
eq
uip
me
nt
•M
inim
ize
da
ng
er
to p
eo
ple
•R
ed
uce
str
ess
on
oth
er
eq
uip
me
nts
•M
ain
tain
inte
gri
ty a
nd
sta
bil
ity
•P
rote
ctiv
e r
ela
yin
g i
s a
de
vic
e t
ha
t d
ete
cts
ab
no
rma
l p
ow
er
sys
tem
co
nd
itio
ns
an
d i
nit
iate
s a
co
rre
ctiv
e a
ctio
n a
s fa
st a
s p
oss
ible
in
o
rde
r to
re
turn
th
e p
ow
er
syst
em
to
its
no
rma
l st
ate
. T
he
re
spo
nse
m
ust
be
au
tom
ati
c, q
uic
k a
nd
ca
use
min
imu
m a
mo
un
t o
f d
isru
pti
on
of
po
we
r sy
ste
m
•M
ain
re
qu
ire
me
nts
of
pro
tect
ive
sys
tem
–Q
uic
k r
esp
on
se–
Qu
ick
re
spo
nse
–C
orr
ect
dia
gn
osi
s o
f tr
ou
ble
–M
inim
um
dis
turb
an
ce t
o p
ow
er
syst
em
–A
bso
lute
dis
crim
ina
tio
n
–S
en
siti
ve a
nd
acc
ura
te
–R
eli
ab
le
–S
tab
le
Th
ere
fore
exp
en
sive
an
d s
op
his
tica
ted
eq
uip
me
nt
is n
ee
de
d t
o a
cco
mp
lish
th
ese
ta
sks.
Mu
lti-
laye
r st
ruct
ure
of
po
we
r sy
ste
m
•T
hre
e l
aye
red
str
uct
ure
of
po
we
r sy
ste
m
–A
pp
ara
tus
such
as
(ge
ne
rato
r, T
ran
sfo
rme
r, e
tc.)
Co
ntr
ol
Eq
uip
me
nts
Pro
tect
ive
Eq
uip
me
nt
Po
we
r A
pp
ara
tus
–A
pp
ara
tus
such
as
(ge
ne
rato
r, T
ran
sfo
rme
r, e
tc.)
–C
on
tro
l Eq
. w
hic
h c
on
tro
l vo
lta
ge
, fr
eq
ue
ncy
, p
ow
er,
op
tim
al e
con
om
ic a
nd
se
curi
ty in
in
terc
on
ne
cte
d n
etw
ork
–R
ela
y:
mo
re f
ast
th
an
co
ntr
ol
an
d i
t a
cts
to o
pe
n a
nd
clo
se
Cir
cuit
Bre
ake
r (C
B),
wh
ich
le
ad
s to
ch
an
gin
g s
tru
ctu
re o
f
po
we
r sy
ste
m.
Wh
ile
co
ntr
ol a
cts
con
tin
uo
usl
y a
dju
st
syst
em
va
ria
tio
ns.
Ele
me
nts
of
a P
rote
ctio
n s
yste
m
•Tr
an
sdu
cer
–S
uch
as
P.T
an
d C
.T.
•B
rea
ker
–C
B t
o i
sola
te t
he
fa
ult
•R
e la
y
Tra
nsd
uce
rB
rea
ker
Re
lay
•R
e la
y
–To
de
tect
fa
ult
an
d r
em
ove
it
•S
up
ply
–To
pro
du
ce t
rip
pin
g p
ow
er
of
bre
ake
r a
nd
op
era
te
rela
y
–S
ho
uld
be
un
inte
rru
pte
d s
uch
as
ba
tte
rie
s a
nd
UP
S
Su
pp
ly
Re
lay
Co
nst
ruct
ion
•B
ase
d o
n t
he
ele
me
nts
of
pro
tect
ion
sys
tem
the
re
lay
co
nst
ruct
ion
ca
n b
e d
ivid
ed
in
to
thre
e s
ub
syst
em
–S
en
sin
g–
Se
nsi
ng
–C
om
pa
riso
n
–C
on
tro
l a
ctio
n (
trip
)
Re
lay
Fu
nct
ion
•T
he
re
lay
fu
nct
ion
is
de
pe
nd
ing
on
th
e a
bn
orm
al
situ
ati
on
th
at
sho
uld
be
pro
tect
ed
so
, th
e r
ela
y
ma
y b
e–
Ove
r lo
ad
–O
ver
an
d u
nd
er
volt
ag
e–
Ove
r a
nd
un
de
r vo
lta
ge
–O
ver
an
d u
nd
er
fre
qu
en
cy
–R
eve
rse
po
we
r
–S
ho
rt c
ircu
it
–E
tc.
–T
he
se
lect
ion
of
rela
yin
g f
un
ctio
n d
ep
en
ds
ma
inly
on
th
e
eq
uip
me
nt
th
at
sho
uld
be
pro
tect
ed
Re
lay
Ty
pe
s
•T
he
re
lay
ty
pe
s ca
n b
e c
lass
ifie
d b
ase
d o
n t
he
tech
niq
ue
s u
sed
to
pe
rfo
rm i
ts f
un
ctio
n s
uch
as –
Co
nve
nti
on
al
(Ele
ctro
-me
cha
nic
al
rela
y)
–C
on
ven
tio
na
l (E
lect
ro-m
ech
an
ica
l re
lay)
–S
oli
d s
tate
re
lay
–D
igit
al re
lay
Co
nve
nti
on
al r
ela
y
•It
is
an
ele
ctro
-me
cha
nic
al d
ev
ice
, it
ha
s th
e
ad
van
tag
e t
ha
t
–R
ob
ust
–E
asy
to
ma
inte
na
nce
–E
asy
to
ma
inte
na
nce
–W
hil
e i
t h
as
dis
ad
van
tag
e
•La
rge
siz
e
•H
igh
po
we
r co
nsu
mp
tio
n
•It
de
pe
nd
s o
n m
ovin
g p
art
s w
hic
h n
ee
d m
ore
m
ain
ten
an
ce
•E
ach
re
lay
pe
rfo
rm a
sin
gle
fu
nct
ion
Exa
mp
le
•D
iscu
ss t
he
ba
sic
stru
ctu
re o
f o
ver
curr
en
t
rela
y (
OC
)
Lect
ure
2:
So
lid
sta
te r
ela
y
Fu
nd
am
en
tals
•B
asi
c fu
nd
am
en
tal
of
sta
tic
rela
y
•Lo
gic
Sig
na
l co
nd
itio
nin
g
•A
na
log
Sig
na
l C
on
dit
ion
ing
•C
om
pa
rato
r ci
rcu
its
•C
om
pa
rato
r ci
rcu
its
•T
ime
de
lay
cir
cuit
•E
xam
ple
of
sta
tic
rela
y
So
lid
Sta
te R
ela
y
•It
ha
s n
o m
ov
ing
pa
rts
mo
st o
f it
s co
mp
on
en
ts
are
sta
tic
de
vic
e b
ase
d o
n s
em
i-co
nd
uct
or
ma
teri
als
.
•T
he
ma
in p
art
s •
Th
e m
ain
pa
rts
sen
sin
gp
rote
ctio
nTr
an
sdu
cer
an
d
sig
na
l co
nve
rsio
n
Sig
na
l
con
dit
ion
ing
(Fil
tra
tio
n)
Co
mp
ara
tor
fun
ctio
n
Tim
e d
ela
y
fun
ctio
nTr
ip C
ircu
itC
ircu
it
Bre
ake
r (C
B)
Re
lay
lo
gic
Re
lay c
ircu
itP
ow
er
syst
em
So
lid
Sta
te R
ela
y
1-
Se
nsi
ng
: vo
lta
ge
an
d c
urr
en
t se
nso
rs s
uch
as
CT
a
nd
PT
2-
Pro
tect
ion
: Z
ine
rd
iod
e,
curr
en
t li
mit
er,
etc
…
3-
Tra
nsd
uce
r: s
uch
as
ha
ll t
ran
sdu
cer,
F/V
, A
C-D
C,
etc
…3
-Tr
an
sdu
cer:
su
ch a
s h
all
tra
nsd
uce
r, F
/V,
AC
-DC
, e
tc…
4-
Sig
na
l co
nd
itio
nin
g:
Low
pa
ss f
ilte
r,…
5-
Co
mp
are
r: v
olt
ag
e a
nd
cu
rre
nt
com
pa
rato
rs,
…
6-
Tim
e d
ela
y c
ircu
it:
trip
tim
e c
alc
ula
tio
n c
ircu
it
7-
Trip
cir
cuit
: S
ign
al co
nd
itio
nin
g t
o o
pe
rate
CB
So
lid
Sta
te R
ela
y
•A
dva
nta
ge
s:–
Sm
all
siz
e
–E
asy
to
mo
un
ted
–Fa
st r
esp
on
se
–Lo
w p
ow
er
con
sum
pti
on
–N
o m
ovin
g p
art
s(le
ss m
ain
ten
an
ce)
–C
an
pe
rfo
rm m
ore
so
ph
isti
cate
d f
un
ctio
n
•D
isa
dva
nta
ge
s•
Less
ro
bu
st t
o v
ibra
tio
n &
Te
mp
.
•D
iffe
ren
t ci
rcu
it f
or
dif
fere
nt
fun
ctio
n
•Le
ss f
lexi
ble
Sig
na
l C
on
dit
ion
ing
•It
is t
he
Ad
just
me
nt
of
sig
na
l to
be
su
ita
ble
to
th
e n
ext
sta
ge
(co
ntr
oll
er,
d
isp
lay,
etc
…)
for
exa
mp
le–
Am
pli
fica
tio
n
–A
tte
nu
ati
on
–F
iltr
ati
on
–Is
ola
tio
n
–Li
ne
ari
zati
on
–Li
ne
ari
zati
on
–E
xcit
ati
on
–C
on
vers
ion
–E
tc.
•A
na
log
sig
na
l co
nd
itio
nin
g
•B
ina
ry s
ign
al
Co
nd
itio
nin
g
•D
igit
al S
ign
al C
on
dit
ion
ing
Log
ic S
ign
al
Co
nd
itio
nin
g
•Tw
o s
tate
sig
na
l (h
igh
-lo
w)
–H
igh
: 5
v, 1
2V
, 2
4v (
dc)
–2
4,
11
0,
22
0 v
(a
c)
•B
uff
er
(Iso
lati
on
)•
Bu
ffe
r (I
sola
tio
n)
–Tr
i-st
ate
bu
ffe
r
–U
sin
g T
ran
sist
or
en
ab
le
Inp
ut
Ou
tpu
t
en
ab
le
inp
ut
ou
tpu
t
Log
ic S
ign
al
Co
nd
itio
nin
g (
con
t.)
•O
pt.
iso
lati
on
–P
ho
to D
iod
e
–P
ho
to t
ran
sist
or
–P
ho
to t
ria
c
•R
ela
yV
cc
VC
C
ou
tpu
tin
pu
t
•R
ela
y
•2
. In
vert
er
•3
. A
mp
lifi
er
Inp
ut
ou
tpu
t
Inp
ut
VC
C
ou
tpu
t
An
alo
g s
ign
al
Co
nd
itio
nig
•It
De
pe
nd
s m
ain
ly o
n O
P-A
mp
An
alo
g s
ign
al
Co
nd
itio
nig
•It
De
pe
nd
s m
ain
ly o
n O
P-A
mp
•U
nit
y G
ain
•U
nit
y G
ain
An
alo
g S
ign
al
Co
nd
itio
nin
g (
Co
nt.
)
•A
dd
itio
n a
nd
su
btr
act
ion
•In
teg
rato
r
•Li
ne
ari
zati
on
•C
om
pa
rato
r•
Co
mp
ara
tor
–N
orm
al
com
pa
rato
r
–H
yste
resi
s O
utp
ut
Inp
ut
Vl
Vh
An
alo
g s
ign
al
con
dit
ion
ing
An
alo
g s
ign
al
Co
nd
itio
nig
•U
nit
y G
ain
•A
mp
lifi
er
An
alo
g S
ign
al
Co
nd
itio
nin
g (
Co
nt.
)
•A
dd
itio
n a
nd
su
btr
act
ion
Dif
fere
nti
al a
mp
An
alo
g s
ign
al
con
dit
ion
ing
An
alo
g f
ilte
rs
1in
outV
V
ffc
0.7
07
1in
outV
V
ffc
0.7
07
An
alo
g s
ign
al
con
dit
ion
ing
•S
ign
al I
nte
gra
tio
n
∫−
=t
io
dt
tV
V)(
1
•Li
ne
ari
zati
on
?
∫−
=i
FG
odt
tV
CR
V0
)(
An
alo
g s
ign
al
con
dit
ion
ing
•S
ign
al
Co
nve
rsio
n
–C
urr
en
t to
vo
lta
ge
–C
urr
en
t to
vo
lta
ge
–V
olt
ag
e t
o c
urr
en
t
•A
ssig
nm
en
t
–W
rite
a r
ep
ort
ab
ou
t vo
lta
ge
-to
-cu
rre
nt
an
d
curr
en
t-to
-vo
lta
ge
m
eth
od
s
Co
mp
ara
tor
Cir
cuit
•A
mp
litu
de
(D
ire
ct o
r In
teg
rati
on
)
•P
ha
se (
Dir
ect
or
Inte
gra
tio
n)
•H
yb
rid
(D
ire
ct o
r In
teg
rati
on
)
Tim
e d
ela
y C
ircu
it
•T
he
sim
ple
cir
cuit
to
ge
ne
rate
tim
e d
ela
y i
s
RC
cir
cuit
•It
ca
n b
e b
uil
t u
sin
g d
igit
al
tim
er
Exa
mp
le:
ove
r cu
rre
nt
rela
y
Dig
ita
l R
ela
y F
un
da
me
nta
ls
•In
tro
du
ctio
n
•D
igit
al re
lay c
om
po
ne
nts
•M
ath
em
ati
cal
ba
ckg
rou
nd
–H
ow
to
im
ple
me
nt
dis
cre
te d
iffe
ren
tia
tio
n a
nd
–
Ho
w t
o im
ple
me
nt
dis
cre
te d
iffe
ren
tia
tio
n a
nd
inte
gra
tio
n
–H
ow
to
est
ima
te a
va
lue
be
twe
en
tw
o s
am
ple
o
r
pre
dic
t a
va
lue
in
th
e f
utu
re
Intr
od
uct
ion
•B
ack
gro
un
d
•A
dva
nta
ge
s o
f d
igit
al
rela
y
•O
verv
iew
of
Dig
ita
l re
lay
str
uct
ure
Dif
fere
nce
eq
ua
tio
n
1.
Ba
ckw
ard
dif
fere
nce
s1
.Fo
r co
nti
nu
ou
s fu
nct
ion
f(t
)
2.
f k=
f
(to+
kh
)
3.
∇f k
=
f k-
f k-1
4.
∇n
f k=
∇n
-1f k
-∇
n-1
f k-1
Wh
ere
to
is t
he
in
itia
l va
lue
, h
th
e s
am
pli
ng
in
terv
al,
k t
he
sa
mp
lin
g n
um
be
rW
he
re t
o is
th
e i
nit
ial
valu
e, h
th
e s
am
pli
ng
in
terv
al,
k t
he
sa
mp
lin
g n
um
be
r
•Fo
rwa
rd d
iffe
ren
ces
1.
∆f k
= f k
+1-
f k
2.
∆nf k
= ∆
n-1
f k+
1-∆
n-1
f k
1.
Sh
ift
op
era
tor
1.
f k+
1=
f
(tk+
h)
= E
fk
2.
f k+
r=
f (t
k+
rh)
= E
r fk
3.
E=
1+∆
Forw
ard
in
terp
ola
tio
n
Ba
ckw
ard
inte
rpo
lati
on
Est
ima
tio
n a
nd
da
ta P
red
icti
on
f k+
r=
(tk+
rh)
= E
r fk
;0<
r<1
–u
sin
g b
ack
wa
rd d
iffe
ren
ces
()
()(
)(
)()(
)k
n
kk
kk
rk
fn
nr
rrr
fr
rr
frr
fr
ff
∇−
++
++
+∇
++
+∇
++
∇+
=+
!
1...
21
...
!3
21
!2
13
2
–U
sin
g f
orw
ard
dif
fere
nce
s
–B
y r
ep
laci
ng
th
e d
iffe
ren
ces
op
era
tor
by t
he
eq
uiv
ale
nt
form
th
e n
ew
eq
ua
tio
n is
call
ed
eq
ua
tio
n
n!!3
!2 ()
()(
)(
)()(
)k
n
kk
kk
rk
fn
nr
rr
rf
rr
rf
rr
fr
ff
∆−
−−
−+
+∆
−−
+∆
−+
∆+
=+
!
1...
21
...
!3
21
!2
13
2
•E
xam
ple
: fi
nd
th
e r
ecu
rsiv
e e
qu
ati
on
of
the
ba
ckw
ard
pre
dic
tio
n f
un
ctio
n
by a
pp
roxi
ma
tin
g t
he
re
sult
to
th
e
seco
nd
te
rmse
con
d t
erm
()
()
() (
)
()
() (
)
21
21
1
11
2
2!2
1
!2
1
!2
1
−−
+
−−
−+
−−
++
++
=
−−
−+
−+
=
∆−
∆−
+−
+=
∆−
+∆
+=
kk
kr
k
kk
kk
kk
rk
kk
kk
kr
k
kk
kr
k
cfbf
af
f
ff
fr
rf
fr
ff
ff
rr
ff
rf
f
fr
rf
rf
f(
)
()
()
!2
1
,!2
12
,!2
11
−−
=
−−
=
−+
+=
rr
c
rrr
b
rr
ra
Nu
me
rica
l d
iffe
ren
tia
tio
n
()
ft
ff
+∇
+∇
+∇
==
...
11
13
2'
'B
ack
wa
rd
()
()
rht
fr
fr
f
rht
fr
rr
kr
f
kr
rk
rk
k
r
klim
fr
+∂∂
=∂∂
=
+∂∂
→+
→=
→+
→
→
=∂∂
22
022
0
''
0
'
lim
lim
lim
00 ()
()
kk
k
kk
k
fh
tf
f
fh
tf
f
+∇
+∇
+∇
+∇
==
+∇
+∇
+∇
==
...
65
12
11
1
...
32
54
32
2
''''
32
''
()
()
kk
k
kk
k
fh
tf
f
fh
tf
f
−∆
+∆
−∆
==
−∆
+∆
−∆
==
...
12
11
1
...
31
211
43
2
2
''''
32
''
forw
rad
Nu
me
rica
l d
iffe
ren
tia
tio
n
•A
pp
roxi
ma
te d
iffe
ren
tia
tio
n
–F
irst
te
rm
()
()
''
1∇
≅=
kk
kf
ht
ff
–S
eco
nd
te
rm
()
()
()
()1
''
1−
−≅
=
∇≅
=
kk
kk
kk
k
ff
ht
ff
fh
tf
f
()
()
()
+−
≅
+−
+−
≅=
∇+
∇≅
=
−−
−−
−
21
'
21
1
''
2'
'
212
5.1
1
221
1
211
kk
kk
kk
kk
kk
k
kk
k
ff
fh
f
ff
ff
fh
tf
f
fh
tf
f
Nu
me
rica
l In
teg
rati
on
•R
ect
an
gle
ap
pro
xim
ati
on
0 1
k
0 1
k
Sin
uso
ida
l-w
ave
ba
sed
alg
ori
thm
Ch
ap
ter
4:
Pu
re s
ine
wa
ve a
lgo
rith
m
Bu
ild
a p
rog
ram
to
de
du
ce t
he
fo
llo
win
g p
ara
me
ters
1-
volt
ag
e m
ag
nit
ud
e
2-c
ure
en
t m
ag
nit
ud
e2
-cu
ree
nt
ma
gn
itu
de
3-
ph
ase
sh
ift
4-
circ
uit
imp
ed
an
ce (
Z,R
,x)
For
a g
ive
n d
ata
fil
e (
da
ta1
) w
he
re s
am
pli
ng
tim
e 1
ms
an
d t
he
sig
na
l fr
eq
ue
ncy
50
hz
ea
ch r
aw
of
the
da
ta1
fil
e
is [
sam
ple
#,
v(k
),i(
k)]
![Technical & Applications Information [Relays]](https://img.pdfslide.us/doc/110x75/61c0c3e0df57ad1a1c5b9d57/technical-amp-applications-information-relays.jpg)
