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A PRIMER TO TRANSFORMERS AND COUPLED INDUCTORS MODELING
Technical Report · March 2007
DOI: 10.13140/RG.2.2.17208.57600
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APPLICATION NOTE

Preliminary
A PRIMER TO TRANSFORMERS AND
COUPLED INDUCTORS MODELING
by C. Adragna
This application note brushes up the basic concepts related to electrical modeling of transformers and
coupled inductors. Measurement techniques, which are vital for a successful implementation of these
magnetic devices, are reviewed as well.
Coupled inductors and transformers
A system of coupled inductors is a set of coils that share one or more common magnetic paths because of
their proximity. In such systems, magnetic flux changes in any one coil will not only induce a voltage across
that coil by self-induction, but also across the others by mutual induction.
Accurate descriptions of coupled inductors use the reluctance model approach and its derivations, which
closely represent the physical structure of the magnetic element. This approach is especially useful when
dealing with complex magnetic structures, which is not our case. Hence, we will follow a simpler way based
on the examination of the terminal equations describing the electrical behavior of the magnetic structure.
From the electrical standpoint, a system of m coupled inductors, will be defined by m coefficients of selfinductance, which relate the voltage across any inductor to the rate of change of current through the same
inductor, and m·(m-1) coefficients of mutual inductance, equal in two by two, relating the voltage induced
across any inductor to the rate of change of current in each other inductor. In general, a system of m coupled
inductors will be identified m·(m+1)/2 total coefficients.
Considering the important practical case of coupled inductors wound on the same core of magnetic material,
each inductor is commonly termed “winding”. Limiting our attention to the case m=2, a system of two coupled
inductors (which will be designated as the primary and the secondary winding) is a linear, time-independent
two-port circuit described by the following branch-constitutive equations, derived from Faraday’s law:
v1(t)
L
M d i1(t)
,
= 1
v 2 (t)
M L 2 dt i 2 (t)
(1)
where L1 is the self-inductance of the primary winding made up of N1 turns, L2 is the self-inductance of the
secondary winding made up of N2 turns and M is their mutual inductance. Winding resistance is assumed to
be negligible.
Unlike L1 and L2, which are inherently positive, M can be either positive or negative, depending on the
voltage polarity of the windings relative to one another: a positive rate of change of the current in one
winding can induce a voltage either positive or negative in the other winding. As shown in figure 1, this is
indicated by dot notation, which follows three important rules:
1. Voltages induced in any winding due to mutual flux changes have the same polarity at dotted terminals.
2. Positive currents flowing into the dotted terminals produce aiding magnetic fluxes.
3. If one winding is open-circuited and the current flowing into the dotted terminal of the other winding has a
positive rate of change, the voltage induced in the open winding will be positive at the dotted terminal.
Based on rule 1 and on the sign convention of the terminal voltages and currents of two-port circuits, it is
easy to see that for the coupled inductors in figure 1 on the left-and side it is M>0, while for those on the
right-and side it is M<0.
The mutual inductance M cannot be assigned arbitrarily but must fulfill the inequality:
M ≤
March 2007
L1 L 2 .
1/12
APPLICATION NOTE
Figure 1. Coupled inductors.
M
M
i1(t)
v1(t)
i1(t)
i2(t)
L1
L2
v2(t)
v1(t)
i2(t)
L1
M>0
The case M =
L2
v2(t)
M<0
L1 L 2 is that of perfectly coupled inductors, that is when the magnetic flux generated by
either winding is totally linked to the other. However, in real-world systems there is always some flux linking
one winding but not the other or not completely linking the winding itself, e.g. because it “leaks” into the
surrounding air; hence it is possible to write:
M=k
L1 L 2 ;
(2)
k, which lies in the range -1≤k≤1 is the so-called coupling coefficient. For simplicity, from now on we will
neglect winding polarity and assume M>0 (and 0≤k≤1) always, being obvious the extension to the case M<0.
The coupling coefficient is not a mere mathematical entity but is a measure of the degree of magnetic
coupling between the windings. Consistently with the definition of self-inductance Lj of the j-th winding as the
ratio of the magnetic flux Φj linked to all its Nj turns (flux linkage) to the current ij flowing in that winding:
Lj =
Nj ⋅Φ j
ij
, (j = 1,2)
(3)
the mutual inductance M of the two windings can be defined as:
M=
N 2 Φ 21 N1 Φ 12
=
,
i1
i2
(4)
where Φ21 is the magnetic flux linking the secondary winding due to the current i1 of the primary winding and
Φ12 is the magnetic flux linking the primary winding due to the current i2 of the secondary winding. These
mutual fluxes can be expressed as:
Φ 21 = k 1 Φ 1
Φ 12 = k 2 Φ 2 ;
then k1 represents the portion of the flux generated by the first winding linking the secondary winding and k2
the portion of the flux generated by the second winding linking the first winding. Obviously, both k1 and k2 are
less than unity but note that, in general, k1 ≠ k2: fluxes do not necessarily link windings symmetrically to one
another.
Based on the above definitions, it is possible to write:
M=
N1 Φ 12 N1 k 2 Φ 2 N 2 N1 k 2 Φ 2 N1
=
=
=
k2 L2
i2
i2
N2
i2
N2
(5a)
and similarly:
M=
N2
k 1 L1 ;
N1
(5b)
multiplication of these two relationships yields:
M2 = k 1 k 2 L1 L 2
 M=
k1 k 2
L1 L 2 ,
By comparing this result with (2), it is possible to state that k is the geometric mean of k1 and k2:
k=

k 1 k 2 . (5c)
2/12
APPLICATION NOTE
Additionally, equating (5a) and (5b):
M=
N1
N
k 2 L 2 = 2 k 1 L1
N2
N1
2

k 1  N1  L 2

. (6)
=
k 2  N 2  L 1
In case of symmetrical magnetic coupling k1 = k2 = k, then k1/k2 = 1. Thereby, (6) can be used to determine
experimentally the degree of symmetry of magnetic coupling in a system of coupled inductors.
Transformers are composed of two (or more) windings that share a common magnetic path just like coupled
inductors and, as physical objects, they look very much like coupled inductors. Additionally, since the same
induction phenomena governed by Faraday’s law seen in coupled inductors take place in transformers as
well, it is expected that they can be described from the electrical point of view by the same type of terminal
equations (1). Despite this analogy, however, transformers and coupled inductors work in a substantially
different way.
To better understand differences and similarities, it is useful to start from the concept of “ideal transformer”.
From the electrical standpoint it is a lossless device “transparent to power”, which means that the power it
outputs equals the power entering its input port; just the output power parameters, voltage and current, may
be different from those of the input. If v1(t) i1(t) are the input power parameters and v2(t) i2(t) the output power
parameters, consistently with the sign convention of the terminal voltages and currents of two-port circuits,
such a device can then be described by the fundamental relationship:
v1(t) i1(t) = -v2(t) i2(t).
(7)
Equation (7), however, does not uniquely identify the device: in fact it does not provide information on how
the parameters of power are changed when going from the input to the output. Introducing a real positive
number n, equation (7) can be expanded in two equivalent equations:
v 2 (t)
=n
v1(t)
;
i 2 (t) 1
= ,
i1(t) n
(8)
which now completely identify the device (it would make no difference defining n = v1(t) / v2(t)). Assuming
that n = N2/N1, the device described by (8) can be represented by the two coupled inductors shown in figure
2. Examining the properties of the ideal transformer it is possible to derive the characteristics of these
coupled inductors that, as such, should be described by equations like (1).
In such a device, if a voltage v1(t) is applied to the primary winding, a voltage v2(t) = N2/N1 v1(t) will appear on
the secondary winding. If this is open, i.e. i2(t) = 0, the primary current i1(t) will be zero as well. Since the
terminal equations hold because of the magnetic flux linking the two windings, this means that there must be
flux in the magnetic core even with no circulating current. It is possible to deduce from (3) and (4) that L1, L2
and M must tend to infinity to allow that. Therefore, it is possible to say that the two coupled inductors of
figure are still represented by equations (1), although in a degenerate form.
For L1, L2 and M to be infinite, the permeability of the magnetic material of the core must tend to infinity as
well, so that the reluctance of the magnetic path is zero. Such a material is lossless and does not require
energy to be magnetized; this is consistent with (7). Note also that the first of (8) requires that flux linkage be
the same for both windings, i.e. that there is no leakage flux (it could not be otherwise with a material having
infinite permeability). Then there is no energy stored in an ideal transformer.
Figure 2. Ideal transformer.
i1(t)
N1
N2
i2(t)
i1(t)⋅ N1 + i2(t)⋅ N2 = 0
v1(t)
v2(t)
v1(t)
N1

=
v2(t)
N2
3/12
APPLICATION NOTE
Before going any further, it is worthwhile reminding another interesting property of ideal transformers that will
be useful for future considerations. Dividing side by side the two equations (8) it is possible to find:
v1(t)
1 v 2 (t)
=
i1(t) n 2 i 2 (t)
Then, if an impedance Z2 is connected across the secondary winding (which defines the ratio v2(t) / i2(t)), the
impedance Z2 as seen from the primary side becomes Z1:
Z1 =
v 1( t )
1
=
Z 2 . (9)
i1( t ) n 2
Of course, this property is symmetrical, i.e. (9) still applies and provides Z2 if Z1 is assigned. Impedances can
be transferred from one side to the other of an ideal transformer multiplying or dividing their value by n2.
Going to real-world transformers, the magnetic material used for the core has a finite permeability, the
magnetic path has non-zero reluctance and some energy is required to magnetize the material.
Consequently, (7) and the second of (8) are no longer true and some of the input current i1(t) is used to
provide the energy necessary to magnetize the magnetic material. This component is called magnetizing
current and flows on the primary side even with the secondary side open. To account for that, in the
schematic of figure 2 we need to consider an inductor LM, called magnetizing inductance, in parallel to the
primary winding of the ideal transformer.
Additionally, the first of (8) is no longer true either: the finite permeability causes a non-zero leakage flux that
develops in air and makes flux linkage different for the two windings. To account for that, in the schematic of
figure 2 we need to consider two extra inductors Ll1, Ll2, called leakage inductances, connected in series to
the ideal transformer. Additional energy is stored in these inductances. Note that, on the primary side, the
leakage inductance Ll1 must be placed upstream the magnetizing inductance LM for consistency with the
physical models. The resulting schematic of the “real transformer”, shown in figure 3, is often referred to as
π-model. Its terminal equations will be again (1), where L1, L2 and M now have finite values; the relationships
(2) to (6) as well as the related concepts will obviously apply too.
Figure 3. Real transformer (losses are not considered).
i1(t)
Ll1
N1
N2
Ll2
i2(t)
iM(t)
v1(t)
LM
v2(t)
ideal
Finally, note that relationships (7) and (8) hold however vj (t) and ij(t) (j=1,2) change in time, then even if they
do not change at all, i.e. they are dc quantities. An ideal transformer can then work also in dc, whereas a real
transformer cannot. The magnetizing inductance of figure 3 accounts for that as well: as the frequency of
v1(t) is reduced, the impedance of LM becomes lower and lower and tends to short out the ideal transformer
and cut off the primary-to-secondary energy transfer.
From the above discussion it is possible to state the fundamental difference between coupled inductors and
transformers: while coupled inductors are energy storage devices (under magnetic field form), just like simple
inductors or capacitors (under electrical field form), transformers are coupling devices assigned to transfer
power instantaneously from a source to a load where energy storage is an unwanted side-effect, a nonideality that normally should be kept as limited as possible. In terms of parameter values, for the same
winding current capability, in transformers the values of L1, L2 and M (infinite if the transformer was ideal) will
typically be much higher than in coupled inductors.
As previously said, coupled inductors and transformers look very much like to one another, to the point that it
is sometimes impossible to understand by simple external inspection if a device is a coupled inductor or a
transformer; however, their different functions and parameter values have a major impact on they way the
two types of device are built.

4/12
APPLICATION NOTE
The essential point is that in all magnetic materials the magnetic flux density, or induction, (B) increases
proportionally to the magneto-motive force (i.e. to the magnetizing current) until it reaches a value called
saturation flux density (Bsat); from that point on, any further increase in magnetizing current will not increase
significantly the flux density any more. In the linear region, the ratio of flux density to the magnetizing current
is proportional to the permeability µ of the material. This means that in high permeability materials the
magnetizing current needed to reach the saturation flux density can be even very low. In terms of energy, it
is possible to say there is a maximum energy level that can be stored in the magnetic core before the
material saturates. Cores made of high permeability materials then have a limited energy storage capability.
While this is not usually a problem in transformers, where energy storage is incidental, to make a coupled
inductor capable of storing energy core’s storage capability must be considerably increased. To do so it is
customary to add a small region of non-magnetic material, like air, in the magnetic path; non-magnetic
materials do not saturate, have low permeability and then high energy storage capability. This region of nonmagnetic material is essentially the air gap that is usually introduced in all inductors to make them able to
carry the desired current without saturating.
The above mentioned differences between coupled inductors and transformers are reflected also in the core
saturation mechanism: in both cases saturation will occur when the magnetic core is not able to store any
more energy, however, while inductors saturate because their winding currents are too large, transformer
saturate when the magnetizing current only (related to the volt·second across LM) is too large, and this is
totally unrelated to the winding currents. Paradoxically, a transformer can saturate with the secondary open
and be away from saturation with the secondary winding shorted.
The model of figure 3 can give useful information on what happens in case the magnetic core saturates. The
leakage inductances Ll1 and Ll2 are relevant to magnetic flux that is located mostly in air, hence they do not
saturate and their value can be considered constant; the non linear element of the circuit is LM, which is
associated to the magnetic flux that flows mostly through the magnetic core and thus is affected by the nonlinearity typical of magnetic materials. If the flux density exceeds the saturation level Bsat, LM will tend to a
much lower value and shunt the ideal transformer, so that, essentially, only Ll1 and Ll2 will be visible from the
terminals. Moreover, the input to the ideal transformer will be very small, thus the windings will be as nearly
completely decoupled and energy transfer will be drastically limited.
Electrical equivalent circuit models of two-winding coupled inductors and transformers
We have seen that coupled inductors and transformers can be represented by the same branch-constitutive
equations (1), although with parameter values that are considerably different.
It is often useful to represent a system of coupled inductors with the same equivalent circuit seen for a real
transformer. Considering a two-winding coupled inductor or transformer, this equivalent circuit is shown in
figure 4 on the right-hand side. It is not difficult to show that the branch-constitutive equations of the circuit
are the following:
v1(t)
Lµ + La
a Lµ
a Lµ
a 2L µ + L b
=
v 2 (t)
d
dt
i1(t)
.
(10)
i 2 (t)
Figure 4. Electrical equivalent circuit models of coupled inductors and transformers.
M
i1(t)
i2(t)
i1(t)
La
i1'(t)
1: a
i2'(t)
Lb
i2(t)
iµ (t)
v1(t)
L1
L2
v2(t)
v1(t)
Lµ
v1'(t)
v2'(t)
v2(t)
ideal
Note that in figure 4 the ideal transformer has a turn ratio 1:a. It might be a:1 indifferently, in which case in all
the following formulae the turn ratio should be replaced by its reciprocal, leaving everything else unchanged.

5/12
APPLICATION NOTE
By comparing (10) to (1) it is possible to find the following relationships:
 L1 = L µ + L a

 M = a Lµ
 L = a2L +L
µ
b
 2


M

 L a = L1 − L µ
 L a = L1 − a


M
M

  Lµ =
 Lµ =
a
a


2
 Lb = L 2 − a Lµ
 Lb = L 2 − a M


. (11)
It is important to notice that the model (10) and the resulting relationships (11) use four parameters (Lµ, La,
Lb, a), but equations (1) show that three parameters only (L1, L2, M) are needed to completely define the
circuit. This means that one of the four parameters in (10) – a is the obvious choice - can be arbitrarily fixed,
thus leading to an infinite number of models (10) equivalent to (1).
A good criterion for choosing a is that both La and Lb have a positive value: should they result otherwise, the
terminal equations would still be represented correctly but a negative inductance does not make physical
sense and leads to wrong results as far as energy considerations are concerned. Using the relationships (5)
that relate M to the coupling coefficients k1 and k2, it is possible to express La and Lb as:


1 N2 

 L a = L 1  1 −k 1
a
N1 



 L = L  1 −k a N1 
2 
2
 b
N 2 


(12)
It is apparent that, if a equals the secondary-to-primary turn ratio n=N2/N1, La and Lb will be both positive and
the resulting circuit will be equal to that in figure 3. Moreover, it is possible to prove that this choice leads to
the same model that can be obtained with the reluctance model approach; hence the model with a = n is the
physical model of a coupled inductor or transformer.
Lµ is associated to the mutual flux that links the primary and secondary winding mostly through the magnetic
core, is called primary magnetizing inductance and is designated with LM. La is associated to the flux
generated by the primary winding and not totally linked to itself or to the secondary winding, that is, the
primary leakage flux: it is called therefore primary leakage inductance and is designated with Ll1:
L l1 = L 1 ( 1 − k 1 ) . (12a)
Similarly, on the secondary side the inductance Lb is associated to the secondary leakage flux, the flux
generated by the secondary winding and not totally linked to itself or to the primary winding: it is called
secondary leakage inductance and is designated with Ll2:
L l 2 = L 2 ( 1− k 2 ) . (12b)
Figure 5. Model of coupled inductors (transformers) with a = n (a=n model, the same as figure 3).
i1(t)
Ll1
Ll2
1: n
i2(t)
iM(t)
v1(t)
LM
v2(t)
ideal
This choice for a is perhaps the most logical one, and is also very useful because it provides a clear physical
meaning to each element of the equivalent circuit, shown in figure 5, but it is not the only one that makes
sense. Another possible choice that leads to always positive values for La and Lb is:
a=

L2
.
L1
(13)
6/12
APPLICATION NOTE
This quantity, usually indicated with ne, is termed “equivalent turn ratio” and is conceptually and numerically
different from the physical turn ratio n:
ne =
L l2 + n 2L M N 2
L2
=
=
L1
L l1 + L M
N1
k1
k1
.
=n
k2
k2
(14)
ne coincides with n if Ll2 = n2 Ll1 or, equivalently, if k1 = k2 (symmetrical magnetic structure); clearly, it is also
ne = n if Ll1 = Ll2 = 0, that is, in case of perfect coupling (k = k1 = k2 = 1).
The choice a = ne is particularly useful when the physical turn ratio n is unknown, so that it is possible to
complete the model computing ne from L1 and L2, which are easily measurable. In this case, equations (11),
considering (2), take the form:
L a = ( 1 − k ) L 1


.
L µ = k L 1

L = ( 1 − k ) L = n 2 L
2
e a
 b
(15)
The resulting model is shown in figure 6. It is noteworthy that Lb, reflected back to the primary side, equals
La, as pointed out by the third one of (15). Note that this property is often erroneously attributed to the
primary and secondary leakage inductances Ll1 and Ll2. It is worth stating once more that k represents an
average coupling between the windings and that the coupling k1 of the primary winding to the secondary one
is generally different from the coupling k2 of the secondary winding to the primary one, hence Ll2 ≠ n2 Ll1. It
will be Ll2 = n2 Ll1 only in the special case of a winding geometry symmetrical in such a way that the
reluctance of the leakage flux path is the same for both windings, resulting in symmetrical coupling between
windings (k1 = k2).
Figure 6. Model of coupled inductors (transformers) with a = ne (a = ne model).
i1(t)
v1(t)
(1-k) L1
ne2 (1-k) L1 i2(t)
1: ne
Lµ
v2(t)
ideal
Figure 7. Model of coupled inductors (transformers) with a = 1 (a =1 or T model).
L1 - M
L2 - M
i1(t)
v1(t)
i2(t)
M
v2(t)
Another interesting and quite frequently used choice is a = 1. It results in a very simple model, illustrated in
figure 7, useful for analyzing loosely coupled inductors, where L1-M and L2-M are both positive, which
happens if:

7/12
APPLICATION NOTE
Figure 8. Model of coupled inductors (transformers) with a = k ne (a = kne or ASR model).
i1(t)
L2 (1-k2 )
1: k ne
v1(t)
L1
i2(t)
v2(t)
ideal
k1
N2
N
< 1 and k 2 1 < 1 .
N1
N2
(16)
This model is then definitely applicable when N1 = N2. When N1 ≠ N2, the larger their ratio is, the lower the
magnetic coupling needs to be. Again, if both conditions (16) are not met and either L1-M or L2-M is negative
(or both are), there is no issue for calculations and terminal relationships are still correctly maintained.
Other possible choices of interest for a are a = aASR = M/L1 = k·ne = k1·n, which nulls La and puts all the
elements related (logically, not physically!) to the leakage flux on the secondary side (see figure 8), thus
originating the so-called ASR (All-Secondary-Referred) model, and a = aAPR = L2 /M = ne/k, = n/k2, which nulls
Lb and puts all the elements related to the leakage flux on the primary side (see figure 9), thus originating the
so-called APR (All-Primary-Referred) model.
From inspection of figures 8 and 9 it is apparent that the quantity L2 (1-k2) is the inductance of the secondary
winding measured with a shorted primary winding and that L1 (1-k2) is the inductance of the primary winding
with the secondary shorted out.
Figure 9. Model of coupled inductors (transformers) with a = ne/k (a = ne/k or APR model).
2
i1(t)
v1(t)
1: ne
k
Ls = L1 (1-k )
i2(t)
2
v2(t)
Lp = k L1
ideal
The APR model is particularly useful for the analysis of some common switch-mode topologies, such as the
flyback converter and the LLC resonant converter. It is convenient to provide another relationship between
the physical turn ratio n and aAPR that involves the primary side quantities:
a APR = n
k1 L1
=n
k2 L p
k1
k2


1 + Ls  = n
 Lp 


k1 L1
,
k2 L1 − Ls
which becomes:
a APR = n
L1
=n
Lp
1+
Ls
=n
Lp
L1
L1 − Ls
in case of symmetrical coupling (k1 = k2). This expression can be found starting from aAPR = L2 /M and using
(5c), (11), (12a), (12b).
The usefulness of having different models available, which depend on the value attributed to a, is that it is
possible to choose the one that best fits the specific problem under consideration, leading to a simpler
solution or minimum mathematical manipulations.

8/12
APPLICATION NOTE
An important assumption underlying the modeling techniques so far described is that the behavior of the
magnetic circuit, represented by an equivalent linear electric circuit, is linear as well, that is the magnetic flux
is proportional to the magneto-motive forces that excite the windings. This is true only if the flux density
inside the core is well below the saturation limit of the material.
Electrical equivalent circuit models of three-winding coupled inductors and transformers
It has been stated that an m-winding coupled inductor or transformer is completely identified by m·(m+1)/2
coefficients. A three-winding coupled inductor or transformer will be then identified by six coefficients.
Figure 10 shows the equivalent electrical circuit, derived as an extension of the two-winding equivalent
model shown in figure 4 by adding a tertiary winding.
Note that this model has six parameters as well: Lσ1, LM, n12, n13, Lσ2, Lσ3, which, therefore, will be uniquely
related to the six coefficients of the branch constitutive equations expressed in terms of self- and mutual
inductance coefficients:
v1(t)
L1 M12 M13
i1(t)
d
(17)
v 2 (t) = M12 L 2 M23
i 2 (t) .
dt
v 3 (t)
M13 M23 L 3
i 3 (t)
Once more, it is possible to define coupling coefficients kij < 1 (i = 1 to 2, j = i+1 to 3) between the windings
that relate the mutual inductance coefficients to the self-inductance of the windings:
M12 = k 12 L1 L 2
;
M13 = k 13 L1 L 3
;
M23 = k 23 L 2 L 3
(18).
The branch-constitutive equations of the model in figure 10 are the following:
v1(t)
v 2 (t) =
v 3 (t)
L µ + Lσ1
n12 L µ
n13 L µ
n12 L µ
2
n12
L µ + Lσ 2
n12 n 23 L µ
n13 L µ
n12 n 23 L µ
2
n13
Lµ
+ Lσ 3
i1(t)
d
i 2 (t) ,
dt
i 3 (t)
(19)
and by comparing these to (17) it is possible to find the relationship between the two models:
 L 1 = L µ + Lσ 1

2
 L 2 = n12 L µ + Lσ 2
 L = n 2 L + Lσ
3
3
13 µ

 M12 = n12 L µ
M = n L
13 µ
 13
 M23 = n12 n13 L µ

M 23

 n12 = M
13

 n = M 23
 13 M
12

 L µ = M12 M13

M23

M M
 Lσ1 = L1 − 12 13

M23

M12 M23
 Lσ 2 = L 2 −
M13


M13 M23
 Lσ 3 = L 3 −
M12

.
(20)
Figure 10. Model of a three-winding coupled inductor or transformer.
Lσ2
i1(t)
Lσ1
i2(t)
1: n 12
v2(t)
iM(t)
v1(t)
Lσ3
Lµ
i3(t)
v3(t)
1: n 13
ideal

9/12
APPLICATION NOTE
It is possible to express the leakage elements Lσj (j=1 to 3) as a function of the coupling coefficients defined
by (18). After some algebraic manipulations we arrive at:

 k 12 k 13 

 Lσ1 = L1 1 −
k 23 



 k 12 k 23 
.
 Lσ 2 = L 2 1 −
k 13 



 k 13 k 23 

 Lσ 3 = L 3 1 −
k 12 


(21)
It is interesting to notice that trying to improve the coupling between any pair of windings inevitably leads to
increasing the leakage associated to the remaining winding. For example, if one wants to improve the
coupling between the secondary and the tertiary winding (k23 increases), the primary leakage inductance Lσ1
will become larger.
Coupled inductors or transformers with more than three windings cannot be modeled with the equivalent
circuit considered so far. In fact, for example, a four-winding transformer is identified by ten coefficients while
adding another winding to the model of figure 10 would result in a circuit with just eight parameter (n14 and
Lσ4 would be added). The resulting system of equations relating the two models would be overdetermined,
then with no solution. For higher number of windings different types of equivalent circuits need to be
considered [1].
Measuring parameters of two-winding coupled inductors and transformers.
From the practical point of view there is the need for measuring the parameters of coupled inductors
(transformers) L1, L2 and M or, equivalently, k and derive those of the equivalent circuits. Needless to say
that it is of particular interest to obtain the parameters of the “physical model” with a = n, LM, Ll1 and Ll2.
L1 and L2 are directly measurable, while to determine M or k appropriate measurements need to be done.
The best way to proceed is to use a Z-meter, taking the measurements at a low enough frequency that the
parasitic capacitance of the windings can be neglected. There are two basic methods to completely
characterize a coupled inductor or a transformer from the electrical point of view: the open/short-circuit
inductance (OS) method and the series-aiding/opposing inductance (AO) method.
According to the OS method, three measurements will be taken:
1.
The primary inductance with the secondary open (L1).
2.
The primary inductance with the secondary shorted (L1s).
3.
The secondary inductance with the primary open (L2).
From the inspection of figure 9 it is very easy to derive the coupling coefficient from the L1s measurement:
(
L1s = L1 1 − k 2
)

k=
1−
L1s
.
L1
(22)
M is determined using (2):
M=k
L 1L 2 .
Measuring L1s accurately might be an issue, thus this method is suitable when k is close to unity. It is not
recommended when the winding resistance is not negligible.
The AO method is based on the fact that connecting in series the two windings their combined inductance is
given by L1 +L2 ± 2M, where the sign given to 2M depends on the way the windings are connected, as shown
in figure 11. The difference of the two values is thereby equal to 4M.
According to the AO method, four measurements will be taken:
1.
The primary inductance with the secondary open (L1).
2.
The secondary inductance with the primary open (L2).
3.
The combined inductance with series-aiding connection (LA).
4.
The combined inductance with series-opposing connection (LO).

10/12
APPLICATION NOTE
Figure 11. Winding connections: aiding flux (left), opposing flux (right).
L1
L1
M
M
L2
L2
LA= L1 + L 2 + 2M
L O= L1 + L 2 - 2M
It is always LA > LO, thus, as said before:
M=
L A − LO
4
k=

L A − LO
4
L1 L 2
. (23)
The advantage of this method is its low sensitivity to winding resistance and to the impedance of the wire
used for connecting the windings. It is not recommended for low values of k because in that case it would be
given by the difference of two similar quantities, and the error might be high.
Whichever method has been used, the parameters of the equivalent circuit of figure 5 (a = n, assuming n is
known) can be readily calculated from the third group of (11) with obvious symbolism change:
M

 LM = n

M
 L l1 = L1 − L M = L1 −
n

 L l2 = L 2 − n2 LM = L 2 − n M

. (24)
The parameters of the equivalent circuit of figure 6 (a = ne, useful in case n is not known) are given by (15),
here re-written for reader’s convenience:
 L a = ( 1 − k ) L1

,
 L µ = k L1
 L = (1− k ) L
2
 b
with ne =
L2
.
L1
It is sometimes required to measure the degree of symmetry of the magnetic coupling between the windings
of a transformer or coupled inductors, in other words to find the values of k1 and k2. From (22) and (5),
reminding that k =
k 1 k 2 , it is possible to derive:
k1 =
N1 1
N2 L1
L 2 ( L1 − L s )
k2 =
N2 1
N1 L 2
L 2 ( L1 − L s ) .
(25)
Measuring parameters of three-winding coupled inductors and transformers.
In case of three-winding transformers or coupled inductors, the same methodologies seen in the two-winding
case can be applied to derive either the values of the inductance matrix in (17) or, equivalently, the
parameters in (19) of the equivalent circuit of figure 10.
Using the OS method, the following measurements will be taken:
1.
The primary inductance with the other windings open (L1).
2.
The secondary inductance with the other windings open (L2).
3.
The tertiary inductance with the other windings open (L3).
4.
The primary inductance with the secondary winding shorted and the tertiary winding open (L12).
5.
The primary inductance with the tertiary winding shorted and the secondary winding open (L13)
6.
The secondary inductance with the tertiary winding shorted and the primary winding open (L23).

11/12
APPLICATION NOTE
By inspection of figure 10 and after some algebraic manipulations it is possible to find the following
relationships:

n12 =


n =
 13

L 2 − L 23
L1 − L13
L 3 L 2 − L 23
L 2 L1 − L12

L − L13
L µ = L 2 (L1 − L12 ) 1
L

2 − L 23
Lσ1 = L1 − L µ
.

2
Lσ 2 = L 23 − n12 (L13 − Lσ1 )

L − Lσ 1
2
Lσ 3 = n13 L µ 13
L1 − L13

(26)
Alternatively, applying the AO method, the following measurements will be taken:
1.
The primary inductance with the other windings open (L1).
2.
The secondary inductance with the other windings open (L2).
3.
The tertiary inductance with the other windings open (L3)
4.
The combined inductance of the primary and the secondary windings with series-aiding connection
(L12A).
5.
The combined inductance of the primary and the secondary windings with series-opposing connection
(L12O).
6.
The combined inductance of the primary and the tertiary windings with series-aiding connection (L13A).
7.
The combined inductance of the primary and the tertiary windings with series-opposing connection
(L13O).
8.
The combined inductance of the secondary and the tertiary windings with series-aiding connection
(L23A).
9.
The combined inductance of the secondary and the tertiary windings with series-opposing connection
(L23O).
Then, the mutual inductance coefficients Mij and the coupling coefficient kij (i = 1 to 2, j = i+1 to 3) can be
calculated from:
Mij =
L ijA − L ijO
4

k ij =
L ijA − L ijO
4 Li L j
i = 1, 2; j = i + 1L 3 .
(27)
and the parameters of the model of figure 10 will be derived from (20).
Reference
[1] R. Erickson and D. Maksimovic, "A Multiple-Winding Magnetics Model Having Directly Measurable
Parameters," IEEE Power Electronics Specialists Conference, May 1998, pp. 1472- 1478. PESC98

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