AN966 Datasheet, PDF (10/21 Pages) STMicroelectronics – The front-end stage of conventional off-line converters

AN966 APPLICATION NOTE
being Ton and Toff the ON-time and the OFF-time of the power MOSFET respectively, ILpk the maxi-
mum peak inductor current in a line cycle and θ the instantaneous line phase (θ∈(0, π)). Note that the
ON-time is constant over a line cycle.
As previously said, ILpk is twice the line-frequency peak current, which is related to the input power and
the line voltage:
ILpk
=2
⋅
√2
⋅
P_i
V_irms
.
Substituting this relationship in the expressions of Ton and Toff, after some algebra it is possible to find
the instantaneous switching frequency along a line cycle:
f_sw (θ) =
1
Ton + Toff
=
2⋅
1
L⋅
P_i
⋅
V^2irms
⋅
(V_O − √2 ⋅ V_irms
V_O
⋅ sin(θ))_.
The switching frequency will be minimum at the top of the sinusoid (θ = π/2 ⇒ sin(θ) =1 ), maximum at
the zero crossings of the line voltage (θ = 0 or π ⇒ sin(θ) = 0) where Toff = 0.
The absolute minimum frequency ƒsw(min) can occur at either the maximum or the minimum mains volt-
age, thus the inductor value is defined by:
L
=
V^2irms ⋅ (V_O −
2 ⋅ ƒ_{sw (min)}
√2 ⋅ V_irms
⋅ P_i ⋅ V_O
)
,
where Virms can be either Virms(min) or Virms(max), whichever gives the lower value for L.
The minimum suggested value for ƒsw(min) is 15 kHz, not to interfere with the internal starter (see ZCD
and triggering block description) .
Once defined the value of L, the real design of the inductor can start. As to the magnetic material and
the geometry, the need of isolation due to the high voltage, and the operating frequency range make the
standard high frequency ferrite (gapped core-set with bobbin) the usual choice in PFC applications.
Among the various types offered by manufacturers the most suitable one will be selected with technical
and economic considerations.
The next step is to estimate the core size. To get the approximated value of the minimum core size, it is
possible to use the following practical formula:
Volume ≥ 4K ⋅ L ⋅ I_r2ms,
where Volume is expressed in cm³, L in mH and the specific energy constant K depends on the ratio of
the gap length (lgap) and the effective magnetic length (le) of the ferrite core:
K
≅
14
⋅
10^−3
⋅
l_e
l_gap
.
The ratio le/lgap is fixed by the designer.
Then the winding must be specified. The turn number and the wire cross-section are the quantities to be
defined.
The (maximum) instantaneous energy inside the boost inductor (1/2 ⋅ L ⋅ ILpk2) can be expressed in
terms of energy stored in the magnetic field, given by the maximum energy density times the effective
core volume Ve:
1
2
⋅
L
⋅
I_L2pk
=
1
2
⋅
∆H
⋅
∆B
⋅
V_e
≈
1
2
⋅
∆H
⋅
∆B
⋅
A_e
⋅
I_e,
where: Ae is the effective area of the core cross-section, ∆H is the swing of the magnetic field strength
and ∆B is the swing of the magnetic flux density.
To prevent the core from saturating because of its high permeability and allow an adequate ∆H, it is nec-
essary to introduce an air gap.
Despite the gap length lgap is few per cent of le, the permeability of ferrite is so high (for power ferrites,
typically µr = 2500) that it is possible to assume all the magnetic field concentrated in the air gap with good
approximation (∆H ≈ ∆Hgap). For instance, with 1% of lgap/le (which is the minimum suggested value) the er-
ror caused by this assumption is about 4%. The error will be smaller if the lgap/le ratio is larger.
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