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Magn
netic Circcuit Dot Conventi
C
ion
ed
uc
at
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co
m
When two aree more coilss are inductivvely coupledd, there will be two induuced emfs in each
W
coil. Thee first one is
i the self innduced emff which is because
b
of thhe time varriation of cuurrent
flowing through
t
it. The
T second one
o is the muutual induced emf, whicch is due to the
t time variiation
of currennt flowing in the other coil. It is necessary
n
too know wheether these induced emffs are
additive or otherwise. This mainnly dependss on the sennse in whichh the windinng is woundd, and
directionns of currentss in the coilss.
ks
hi
Let us consider the coils 1 and
a 2 whichh are wound on a magnettic core. Lett be the cuurrent
flowing through
t
the coil1. The direction
d
of magnetic
m
fluux produced by this curreent
is dirrected
upwards (this is acco
ording to rigght hand rulle). Now let us considerr flowing in the coil2. The
current produces
p
a flu
ux
directted downwarrds.
=
+M
=
+M
------------------ (1)
w
w
w
.s
a
In othher words th
he two magnnetic fluxes produced
p
byy and arre aiding one another. Hence
H
the two innduced voltaages in the tw
wo coils are having the same
s
sign i.ee. positive. Hence
H
If the flux
f
produced by andd opposes each other, then the mutual
m
inducced emf andd self
induced emf
e will hav
ve opposite sign
s
(M is neegative). Hennce
=
-M
=
-M
------------------ (2)
(
The above method
T
m
of decciding polarrity for inducced emfs reqquire the reprresentation of
o the
coils as shown in fig, indicatting the seense in whiich the coils are wouund. But suuch a
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representtation is cu
umbersome to
t be used in represennting the cooupled coils in an elecctrical
network. To simplify
fy the repressentation, we use dot coonvention too indicate thhe corresponnding
terminalss of the coilss. It facilitatees to determiine the sign of mutual innduced emf.
In casee winding diirection not given and too give the siggn of mutuaal inductancee, this conveention
is employyed.
ed
uc
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Dot Con
nvention:
me that the current enteers at the dootted end off coil,
Plaace a dot at one end off coil1; assum
determinne the directiion of flux produced duee to this currrent. Then pllace anotherr dot at one of
o the
ends of coil2
c
such th
hat the currennt entering at
a that dottedd end in coil22 produces a flux in the same
directionn. It is illustraated in fig. consider
c
two coils1 and 2 as shown in
i fig.
ks
hi
1) Place a dot at
a one end of
o coil1 and assume thatt the currentt enters at thhat dotted ennd in
cooil1.
2) Place anotherr dot at one of the ends of coil2 succh that the cuurrent enteriing at that end
e in
cooil2 establish
hes magnetic flux in thee same directtion.
.s
a
Inn order thatt the flux prroduced by flowing in
i coil2, prooduces the flux
f
in the same
upward direction
d
so that it enterrs at lower end
e of coil2. Hence placce a dot at that
t
end of coil2.
c
The dotteed ends acco
ording to the above systeems are know
wn as corresponding endds.
w
To determ
mine the pollarity of Mu
utual Induceed Voltage:
w
w
Thhe self inducced voltagess are always taken as poositive and iff the mutuall induced vooltage
aids self induced volltage, then itt is taken as positive or else
e negativee. Having pllaced the dots we
can deterrmine the sig
gn of mutuall induced voltage as folloows.
1) Place the dotss at the correesponding teerminal of cooupled coils..
2) Mark
M
the direections of cuurrent in the two
t coils.
If the currrents enter or leave at dotted ends in both thee coils, then effect of mutual
m
inductancce is additiv
ve and sign of M is takeen as positivve. The volttage expressiions are givven in
equation.. Otherwise,, if one curreent enters inn and in the other
o
currennt leaves the dotted termiinals,
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then the sign of M iss taken as neegative and the
t effect off M is oppossing. The voltage expresssions
are givenn in an equattion.
Iff more than two coils arre mutually coupled the correspondiing ends of different paiirs of
coils are indicated ussing differennt symbols. Itt is illustrateed in fig.
M
Mutual
inducctance
co
m
Coils
1 and 2
1 and 3
2 and 3
ed
uc
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Symbolss
dot
Triangle
square
w
w
w
.s
a
ks
hi
Ex: tw
wo coils A and
d B are connected in seriess. The coils ha
ave self induceed and with a mutua
al ind
ductance M. w
what is the efffective inducttance of the series circuit. The two coils A and B are conneected in seriees as shown in
i fig
The voltaage induced in the circuiit (a) =
+M
=(
Therefore,
T
=
+
+
+
+M
+ 2M)
+ 2M
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Voltage induced
i
in th
he circuit (b)) =
-M
=(
Therefore,
T
=
+
+
+
-M
- 2M)
- 2M
co
m
Positive sign
s
for aiding connectioon, negative sign for series oppositioon.
Loop equ
uations:
ed
uc
at
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n.
Itt is easier to solve the coouple circuitt using loop or mesh meethod. This is
i illustratedd with
below exxample.
ks
hi
Ex: writee the loop eq
quations for the
t circuit shhown in fig.
.s
a
In coils1, we will hav
ve self inducced emf due to ( - ) floowing througgh it and muutual inducedd emf
due to ( - ) flowing
g through cooil2 and flowing throuugh coil3.
For looop1,
+
w
w
w
+
( - )-
+
( - )=
+
-
For looop2,
( - )+
-
( - )+
( - )+
+
+
( - )=0
For looop3,
( - )+
-
( - )-
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( - )-
( - )+
dt = 0
Conducttively coupled equivalent circuits:
(
+j
+
)
-j M + (
ed
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co
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Itt is possible in analysis to
t replace a mutual couppled circuit in
i the fig witth a conducttively
coupled circuit
c
of fig
g. Let us connsider a mutuually coupleed circuit as shown in figg. let and be
the loop currents. Th
he loop equattions in matrrix form for the coupled circuit are:
-j M
+j
=
)
=-
----------- (1)
(
w
w
w
.s
a
ks
hi
The loopp equations for
fo the netwoork shown inn below fig,
For loop1,
(
+j
(
+j
-M))
+
)- j
(
- )j
+ j
+j
M=
-
)-
j
j
=
=
----------- (2)
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For loop22,
-
)j M+
j
M-
j
(
+
+j
M+
(
-
-M)) = +j
)-
j M+
(
j M=-
+j
)=-
------- (3)
hi
ed
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The equations for the networks show
T
wn in 2 and 3 are exactly identical. Hence
H
the couupled
circuits shown
s
in fiig are mutuually coupled circuits and
a can be replaced byy a conducttively
coupled circuit
c
as giv
ven as below
w in fig.
w
w
w
.s
a
ks
i below
Ex: obtaiin the effectiive inductance of the circuit shown in
Writing the
t loop equ
uation valuess and we get,,
( - )+M
=
--------- (1)
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(
- )-M
+
-M
(
- ) = 0 ------ (2)
Under steady state,
)j
j
+
-
-
)j
-
j
j
-
j M=
(
=
j M+
---------------- (3)
j
-j M
j
+
- )=0
- 2M ) =0 --------- (4)
The loop equations in matrix form,
j
j
ed
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at
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j
j
The effective impedance =
M
=
co
m
-
2M
0
M
=
M
ks
But
M
=
hi
Solve this we get
=
.s
a
M
Therefore
=
w
Similarly for opposing,
M
=
(this is for aiding)
M
w
w
Ex: Two coupled coils with = 0.03H, = 0.02H and K= 0.6 are connected in four different
ways Series, aiding, series opposing and parallel with both arrangement of the winding sense.
What are four equivalent inductances?
Given data,
= 0.03H
= 0.02H
K= 0.6
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= 0.6 √0.03 0.02 = 0.0146H
From the formula, M = K
=
+
+ 2M
ed
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= 0.03 + 0.02 + 2 (0.0146)
= 0.0792H
Series opposing:
+
- 2M
hi
=
ks
= 0.03 + 0.02 - 2 (0.0146)
.s
a
= 0.0208H
w
w
w
Parallel aiding:
=
co
m
Series adding:
M
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=
.
.
.
.
.
.
.
= 0.018598H
=
ed
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Parallel opposing:
M
=
.
.
.
.
.
.
.
= 0.004884H
hi
Ex: two coils with inductances in the ratio of 5:1 having coefficient of coupling, k = 0.5. When
these coils are connected in series aiding, the equivalent inductance is 44.4mH. Find , , M
= 5:1Æ 5 =
.s
a
:
ks
Given data,
K = 0.5
= 0.5
5
M = 1.118
w
w
w
M=K
In series aiding,
=
+
0.0444 =
+ 2M
+5
+ 2(1.118
= 0.00539H
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5 =
= 5 *0.00539 = 0.02695
M = 1.118
= 1.118 * 0.00539 = 0.006026H
Given data,
When coils connected in aiding,
+
When coils connected in opposing,
=
+
Performing (1) – (2) we get
+ 2M = 0.5H --------- (1)
ed
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=
co
m
Ex: Two coils connected in series having an equivalent inductance of 0.5H, when connected in
aiding and equivalent inductance 0.3H, when connected in opposing. Calculate the mutual
inductance of the coil.
- 2M = 0.3H ---------- (2)
w
w
w
.s
a
ks
hi
4M= 0.2 Æ M = 0.05H
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