Analysis of errors in some simplified
textbook interpretations of coil
coupling coefficient
Sérgio Francisco Pichorim and Paulo José Abatti
Department of Electronics, Federal University of Technology, Paraná (UTFPR – DAELN),
Curitiba, Paraná, Brazil
E-mail: pichorim@utfpr.edu.br; abatti@utfpr.edu.br
Abstract The coupling coefficient (k) between coils is a subject that is presented in textbooks of
electricity, electric circuit analysis and electromagnetism. Some authors, however, make a partial
or incomplete interpretation of the coupling coefficient, leading students to wrong results and
misunderstandings. This paper presents the most common errors found in some textbooks, comparing
them with the correct definition for coupling coefficient. A practical example is also provided.
Keywords circuit analysis; coupled coils; coupling coefficient
There are, basically, two ways of defining the coupling coefficient (k) between coils:
a relation between self-inductances and mutual inductance; and a relation between
magnetic fluxes. The first one, always correctly presented, has the advantage of using
only common electric circuit concepts.1 This definition, in order to improve the
students’ comprehension, is frequently followed by commentaries on the relation of
k with the degree of magnetic coupling between the coils (tightly or loosely
coupled).1–8 The second definition, although it involves electromagnetism concepts,
is frequently presented in the theory of electric circuit analysis.9–22 However, some
authors carelessly present a particular case as a general definition of k.15–22 The use
of this particular case as a general rule does not correspond to the truth, leading the
students to errors and misunderstandings. In this paper the correct definition for
coupling coefficient is compared with the most common errors found in some textbooks of electric circuit analysis. A practical example is also presented.
The coupling coefficient k
An electric circuit with two coils (coil 1 and coil 2 with self-inductances L1 and L2,
respectively) magnetically coupled, with a mutual inductance (M) between them, is
shown in Fig. 1. It is easy to demonstrate that the relation between mutual inductance
(M) and geometric mean of the self-inductances involved ( L1.L2 ) has an upper
limit of unity and as a lower limit, zero.1 This relation is called the coupling coefficient (k), so that it can be written
k=
M
,
L1.L2
(1)
D
with k assuming any value between 0 and 1.1–8
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Fig. 1 Magnetically coupled coils (a transformer) with their magnetic fluxes.
Primary (coil 1) with N1 turns, voltage V1, and current i1, secondary (coil 2)
with N2 turns, voltage V2, and current i2.
Perhaps because this relation does not allow a direct intuitive physical interpretation, some authors prefer to define the coupling coefficient (k) as a relation between
the magnetic fluxes involved (see Fig. 1). This can be done knowing that
dφ1
,
di1
dφ
L2 = N 2 2 , di2
dφ21
dφ
M = N1
= N 2 12 , di2
di1
L1 = N1
(2)
(3)
(4)
where N1 and N2 are the numbers of turns, ϕ1 and ϕ2 are the total magnetic fluxes of
coils 1 and 2 generated by electrical currents i1 and i2, respectively, ϕ12 is the part
of magnetic flux of coil 1 that involves coil 2, and ϕ21 is the part of magnetic flux
of coil 2 that involves coil 1. Thus, substituting eqns (2), (3), and (4) into eqn (1)
and considering the magnetic media linear (dϕ/di = ϕ/i), it can be written
k=
D
φ12 φ21
φ12
φ21
=
.
(φ11 + φ12 ) (φ22 + φ21 )
φ1 φ2
(5)
where ϕ11 and ϕ22 (called leakage fluxes) are parts of the total magnetic fluxes
(ϕ1 and ϕ2) that do not contribute to the coupling effect.10–11
The appeal of this way of presenting k (eqn (5)) is that, by drawing the magnetic
fluxes and imagining the coil 1 and 2 modifying their relative position, it can
be ‘visualised’ that k increases (tending to unity) as coils approach to each other
(ϕ12 tending to ϕ1 and ϕ21 tending to ϕ2) or decreases (tending to zero) as the coils
are pulled apart (ϕ12 and ϕ21 tending to zero).
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Coupling coefficient textbook errors
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The definitions of k just presented are general and therefore valid whenever the
coils are constructed using linear magnetic media, being found in many textbooks
on circuit analysis (eqn (1)),1–8 and electromagnetism (eqn (5)). 9–14
However, some authors,15–22 perhaps trying to present the subject in an even more
intuitive form, use an expression for k that is valid only in a particular case. In this
paper such an equivocated presentation of k and its consequences are discussed in
detail. A practical example is also provided.
Equivocated definition of k
In some of the literature,15–22 k is defined as
k=
φ12 φ21
=
.
φ1
φ2
(6)
Note that eqn (6) is valid only whenever ϕ12/ϕ1 is equal to ϕ21/ϕ2, being a particular
case of eqn (5). Of course, to use eqn (6) instead of eqn (1) or (5), as a general
expression for k, would lead someone to wrong conclusions.
For instance, combining the equivocated eqn (6) with eqns (2), (3), and (4)
someone would conclude that
L1
L
= 2 ,
N12 N 2 2
(7)
which, evidently, it is not correct in general, otherwise any two coils with the same
number of turns would have the same self-inductances (independently of geometric
parameters such as shape, diameter, length, number of layers, wire diameter, and
other factors related to the magnetic core, such as, size, position, and magnetic
permeability). Unfortunately, such a generalisation is found in the literature.20
As a second example, we can examine the coils voltage ratio (V2/V1) with coil 2
open (i2 = 0) to facilitate the computations. Remembering that, under the above
condition (i2 = 0), V1 = L1.di1/dt and V2 = M.di1/dt, it is easy to demonstrate, using
the correct definition of k (eqn (1)), that the voltage ratio can be given by
V2
L
=k 2
V1
L1
for i2 = 0. (8)
Clearly, whenever eqn (6) is valid, eqn (8) can be written as
V2
N
=k 2
V1
N1
for i2 = 0 and φ12 φ1 = φ21 φ2. (9)
Eqn (8) is valid in general, whenever i2 = 0, while eqn (9) is valid only whenever
both i2 = 0 and ϕ12/ϕ1 = ϕ21/ϕ2. Unfortunately, again in the literature, it is possible to
find eqn (9) presented as if it would be valid requiring only that coil 2 is open.17,19
Beyond all the above arguments, it is important to emphasise that, to those who
do not have a good background in circuit analysis or electromagnetism, it is not easy
to identify, at first glance, that the condition ϕ12/ϕ1 = ϕ21/ϕ2 can only be attained in
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S. F. Pichorim and P. J. Abatti
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Fig. 2 A practical didactic transformer used to measure coupling coefficient
(dimensions in mm). Primary (multi-layer of 90 turns) and secondary
(single-layer of 45 turns) coils are wound on a ferrite core.
For measured results, see text.
a very particular case. In fact, even more expert people sometimes can make a wrong
judgment about this subject.15–22 Anyhow, at this point it is interesting to show to
students the results of some practical experiments (ideally they should be invited to
do it in the classroom).
A practical experiment
D
In order to clarify to the students the concepts of coupled coefficient of coils, and
also to confirm the errors generated by the misinterpretation (the use of a particular
case as a general definition) of coupling coefficient (k), a didactic set of coupled
coils has been constructed (Fig. 2). Other configurations of the didactic transformer
can be implemented using simple materials and easy construction. Primary and
secondary (coils 1 and 2) have been wound on a ferrite core, with measured inductances of 549 μH for L1 (multi-layer of 90 turns) and 105.4 μH for L2 (single-layer
of 45 turns). The measured value of mutual inductance (M) is 145.8 μH, yielding a
coupling coefficient (k) of 0.6061. It is important to observe that coils do not have
the same ratio L/N2. At this point it is important show to students that the equality
present in eqn (7) is not valid in general.
A sinusoidal voltage (17 V and 30 kHz) is applied in the primary coil (L1), and
an induced voltage (V2) of 4.54 V is measured in the secondary. Using eqn (8) the
value of 4.51 V is obtained for the induced voltage in the secondary coil. Discounting the errors of measurement, this can confirm the validity of eqn (8). On the other
hand, if eqn (9) is applied a different value of the induced voltage (5.15 V) is found.
So this simple experiment shows the problem of some equations presented in the
literature17,19 caused by an equivocated definition of k.15–22
Finally, dividing eqn (4) by eqn (2) a practical flux ratio ϕ12/ϕ1 of 0.5311 is
obtained and dividing eqn (4) by eqn (3) the practical ratio ϕ21/ϕ2 of 0.6916 is
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Coupling coefficient textbook errors
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obtained. Knowing that the measured k is 0.6061, we can confirm, with this example,
that
k≠
φ12 φ21
≠
φ1
φ2
(10)
and eqn (6) is not valid as a coupling coefficient definition.
Thus, the problem caused by a misinterpretation of coupling coefficient in circuit
analysis presented by some authors is obvious.15–22
Discussion and conclusion
The coupling coefficient between coils can be defined as a relation of inductances,
always correctly presented in textbooks, or as a relation of magnetic fluxes. This
second definition is presented by some authors15–22 in a simplified form valid only
in a particular case where both coils have the same ratio L/N2. Of course, this may
lead the students to wrong conclusions, including numerical errors in calculations,
as presented here. Authors of electric circuit analysis or electricity theory books
should avoid using the simplified equation for coupling coefficient, since it causes
misinterpretations and errors, as shown in this paper, especially for beginning students who are being introduced to coupled coils or transformer theory. The authors
of this paper believe that this analysis can be an alert to those that are dealing with
inductive coupling to use the correct basic definitions and formulas about this
subject.
Acknowledgment
The authors would like to thank to CNPq (Brazilian Council for Scientific and
Technological Development) for its financial support.
References
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