Coils Coupling Coefficient
Sérgio Francisco Pichorim
In an electric circuit with two coils (primary
and secondary with self-inductance of L1
and
L2,
respectively)
magnetically
coupled, the mutual inductance (M) has
the maximum value when all energy is
transferred from primary to secondary coil.
That means, all the magnetic flux lines
generated by primary coil involve the
secondary one. In this situation mutual
inductance M is defined as the geometric
mean of self-inductances L1 and L2 (i.e.,
M = L1 .L2 ). If primary and secondary are
completely decoupled (apart to the
infinitum) mutual inductance is zero. For
all real cases between the above limits,
the coupling coefficient (k) can be defined
as
(1)
M
k=
.
L1 .L2
Observe that k can assume any
value between 0 and 1, being 1 for tightly
coupled coils and 0 for completely
decoupled coils.
If secondary coil is open (I2=0),
knowing
that
V1 = L1 .di1 / dt
and
V 2 = M .di1 / dt , it is easy to demonstrate,
using equation (1), that the induced
voltage in secondary coil for generic coils
can be given by
(2)
V2
L
=k 2 .
V1
L1
However, in an IDEAL transformer,
where k = 1, the ratio of secondary (V2) to
primary (V1) coil voltages could be written:
(3)
V2 N 2
=
V1 N 1
Also, for an IDEAL transformer, it is
assumed that there is any power loss
(secondary power P2 is equal to primary
power P1). So, the ratio of secondary (I2)
to primary (I1) coil currents could be
written:
(4)
V2 N 2 I 1
=
=
V1 N 1 I 2
Using equations (3) and (4) it is
possible to show that the resistance in
secondary R can be reflected to the
primary circuit by
N
Rrefle = R. 1
 N2
Fig. 1 Magnetically coupled coils: A Transformer.



2
(5)