Copyright 20 13 Dorling Kindersley (India) Pvt. Ltd

14.1.2 
dot Polarity convention
One does not want to physically examine every coupled coil-set to find out the relative winding 
directions in order to decide the sign connecting the self-induced voltage drop and mutually induced 
voltage drop in each coil. Therefore, the manufacturer marks this information in the form of two dots – 
one each on each coil in a coupled set of two coils. He/She has a choice of two terminals in the first 
coil to put a dot on. He/She chooses that arbitrarily. That choice will fix the terminal of the second 
coil at which the second coil dot has to be marked. The following rule helps us to write the voltage 
equation in a set of coupled coils.
Increasing current entering a 
dot
point in one coil will produce a mutual emf with its 
positive polarity at the 
dot
point in the other coil.
Increasing current leaving a 
dot
point in one coil will produce a mutual emf with its 
negative polarity at the 
dot
point in the other coil.
With this interpretation for dot polarity assignment, the mutual inductance between a pair 
of coils is always a positive quantity. 
Self-induced voltage is always positive at the current entry point for both coils.
Various possibilities for mutual emf polarity in a two-coil coupled system are shown in Fig. 14.1-4. 
A double-headed arrow indicates magnetic coupling with M value marked by the side.

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14.6
Magnetically Coupled Circuits
+
–
+
–
+
–
+
–
i
i
i
i
M
M
M
M
=
M
di
dt
=
M
di
dt
=
M
di
dt
=
M
di
dt
Fig. 14.1-4 
Dot polarity convention and sign of mutually induced voltage in a two-coil 
system
But markings can fade with time. And people can forget to do things they are supposed to do. 
Therefore, the following experimental method can help us to put the dots if they are not known already.
Let A and B be the two terminals of the first coil and C and D be the two terminals of the second 
coil. Join B and D. Apply a sinusoidal source of suitable amplitude and frequency between A and B as 
shown in Fig. 14.1-5. Use three rms reading voltmeters to record the readings V
1
, V
2
and V
3
. Then, V
2
will be close to either V
1
+ 
V
3
or | V
1
-
V
3
|. The dot point assignment for these two cases are shown in 
Fig. 14.1-5.
A
C
B
D
VM
VM
VM
+
–
V
1
V
2
V
3
A
C
M
If
B
D
A
C
M
B
D
+
=
V
1
V
2
V
3
A
C
M
B
D
A
C
M
B
D
If
|
|
=
V
2
–
V
1
V
3
Fig. 14.1-5 
Experimental determination of dot point assignment in a two-coil system
Dot polarity convention can be applied in the case of more than two coupled coils too. Consider 
three mutually coupled coils with self-inductance L
1
, L
2
and L
3. 
Let M
1
be the mutual inductance 
between coil-1 and coil-2, M
2
be the mutual inductance between coil-2 and coil-3 and M
3
be the 
mutual inductance between coil-1 and coil-3. We need three pairs of dots to indicate the relative 
polarity of mutually induced emf in the three coils. Each coil has a self-induced emf and two mutual 
emf from the remaining two coils. A possible dot polarity assignment is shown in Fig. 14.1-6. 
+
–
L
1
M
1
M
2
M
3
i
1
v
1
+
–
L
2
i
2
v
2
+
–
L
3
i
3
v
3
Fig. 14.1-6 
Dot polarity assignment in a three coil system

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The Two-Winding Transformer 
14.7
The KVL equations (assuming that there is no resistance in the coils) can be obtained as follows:
v t
L
di t
dt
M
di t
dt
M
di t
dt
v t
M
di t
dt
L
d
1
1
1
1
2
3
3
2
1
1
2
( )
( )
( )
( )
( )
( )
=
+
+
=
+
ii t
dt
M
di t
dt
v t
M
di t
dt
M
di t
dt
L
di t
dt
2
2
3
3
3
1
2
2
3
3
( )
( )
( )
( )
( )
( )
−
=
−
+