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AN3180 Datasheet, PDF (32/39 Pages) STMicroelectronics – A 200 W ripple-free input current
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Electrical equivalent circuit models of coupled inductors and transformers
AN3180
inductance Lb is associated to the secondary leakage flux, the flux generated by the
secondary winding and not completely linked to itself nor to the primary winding: it is called
secondary leakage inductance and is designated by Ll2.
This choice of 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 25, but it
is not the only one that makes sense.
Figure 25. Model of coupled inductors with a = n (a=n model)
L W
/O
Q
/O
L W
L0 W
Y W
/0
Y W
LGHDO
!-V
Another possible choice, which leads to always positive values for La and Lb, is a = L2 ⁄ L1 .
This quantity, usually indicated with ne, is termed “equivalent turn ratio” and is conceptually
and numerically different from the physical turn ratio n:
Equation 27
ne =
L2 = Ll2 + n2LM
L1
Ll1 + LM
which coincides with n if and only if Ll2 = n2 Ll1, that is in case of symmetrical magnetic
coupling (k1 = k2 = k).
This choice 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, Equation 26, considering Equation 23, takes the form:
Equation 28
⎧L
⎪⎨L
a
μ
=
=
(1− k) L1
k L1
⎪⎩Lb = (1− k) L2 = ne2 La
It is worth noting that Lb, reflected back to the primary side, equals La, as pointed out by the
third Equation 28. 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, and so Ll2 ≠ n2 Ll1. It is 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 = k).
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