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AN3180 Datasheet, PDF (31/39 Pages) STMicroelectronics – A 200 W ripple-free input current
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AN3180
Electrical equivalent circuit models of coupled inductors and transformers
Figure 24. Electrical equivalent circuit of coupled inductors
0
L W
L W
Y W
/
/
Y W
L W
/D
L W
L— W
D
L W /E
L W
Y W
/—
Y W
Y W
Y W
LGHDO
The branch-constitutive equations of the circuit are the following:
!-V
Equation 25
v1(t) = Lμ + La
v 2 (t)
aLμ
a Lμ d i1(t)
a2Lμ + Lb dt i2(t)
By comparing Equation 25 to 21 it is possible to find the following relationships:
Equation 26
⎧
⎪
⎨
L1
M
=
=
Lμ +
aLμ
L
a
⎪
⎩
L
2
=
a2Lμ
+ Lb
⇒
⎧
⎪
⎪⎪
⎨
La
Lμ
⎪
⎪ Lb
=
=
=
L1 − Lμ
M
a
L2 − a2Lμ
⇒
⎧
⎪
L
a
⎪⎪
⎨
L
μ
⎪
⎪
L
b
=
=
=
L1
M
a
L2
−M
a
− aM
⎪⎩
⎪⎩
It is important to notice that the model (Equation 25) and the resulting relationships
(Equation 26) use four parameters (Lµ, La, Lb, a), but equations (Equation 21) show that
three parameters only (L1, L2, M) are needed to completely define the two-port circuit. This
means that one of the four parameters in (Equation 25) - a is the obvious choice - can be
arbitrarily fixed, therefore leading to an infinite number of models (Equation 25) equivalent to
(Equation 21).
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.
It is possible to show that, if a equals the secondary-to-primary turns ratio n=N2/N1, La and
Lb are always positive. Moreover, it is possible to prove that this choice leads to the same
model as the reluctance model approach; and so the model with a = n is the physical model
of a coupled inductor.
Lµ is associated to the mutual flux that links the primary and secondary winding mostly
through the magnetic core, it is called primary magnetizing inductance and is designated by
LM; La is associated to the flux generated by the primary winding and not completely linked
to itself or to the secondary winding, that is the primary leakage flux: La is therefore called
primary leakage inductance and is designated by Ll1. Similarly, on the secondary side the
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