AN3142 Datasheet, PDF (30/44 Pages) STMicroelectronics – Solution for designing a 400 W fixed-off-time controlled

Designing a fixed-off-time PFC
AN3142
where 0 < K1 < 1. Through the definition of the k2 parameter the expected time constant
τ =(R//R0)C is underlined, this is necessary to achieve the desired T_OFF@90 VAC.
Equation 63
K₂
=
T_OFF (VAC_min )
τ
Finding a way to obtain K1 and K2 means to gain the values of R and R0 and the
discharging time constant of the C capacitor.
The following part describes the mathematical way to obtain the two parameters K1 and K2.
Combining Equation 61, Equation 62, Equation 63 with the expression of the off-time
(Equation 5) the following expressions are obtained:
Equation 64
[ ^[ [ ^] ]] ^{ρ(Vmultmin,k1) =}
⎢⎢⎢⎢⎢⎢⎣⎡^{ln⎢⎢⎢⎢⎢⎣⎡}
⎡
⎢^Vmult min
⎣
⋅
VAC_max
VAC_min
⎤
+ V_{F ⎥} ⋅
⎦
⎡
⎢⎢⎣^VZCDclamp
−
⎡
⎢^Vmult min
⎣
⋅
VAC_max
VAC_min
(1− k₁)
+
V_F
⎤
⎥
⎦
⎤
⎥
⎥⎦
⎤^1−k1
⎥
⎥
⎥
⋅
(k₁)
⎥
⎥⎦
+
⎜⎛
ln_⎜⎜
⎜
⎝
V_ZCDtrigger
V_mult
min
⋅
VAC_max
VAC_min
⎢⎢⎣⎡^ln_{⎢⎢⎣⎡}
V_mult
V_ZCDclamp
min ^{+ V}F ^{⋅ (1− k}1⁾
− V_{mult min} + V_F
⋅
(k₁)
⎤
⎥
⎥⎦
1−k₁
+
ln_{⎜⎜⎝⎛}
V_ZCDtrigger
V_{mult min} + V_F
⎟⎟⎠⎞⎥⎥⎦⎤
+ V_F
⎤
⎟⎟⎟⎟⎠⎞⎥⎥⎥⎥⎥⎥⎦
Equation 65
[ _[ ] _{] k2(Vmultmin,k1)}
=
⎡
⎢
−1
⎢₁
⎣
−
k₁
⎡
⋅ ln⎢
V_{mult min}
⎢⎣ ^VZCDclamp ^−
+ V_F ⋅ (1− k₁)
V_{mult min} + V_F
⎤^1−k1
⎥
⋅ k_{1 ⎥⎦}
+ ln_{⎜⎜⎝⎛}
V_ZCDtrigger
V_{mult min} + V_F
⎟⎟⎠⎞⎥⎥⎦⎤
From Equation 61 and Equation 64, solving the following equation:
Equation 66
ρ(V_multmin,k₁) − ρ_x = 0 K₁ = 0.903
And then substituting the K1 value into the Equation 65 expression, the k2 parameter is
obtained:
Equation 67
K₂ = k ₂ (V_multmin,k₁) K₂ = 11.17
From the values of K1 and K2 it is possible to calculate the time constant τ =(R1//R2) C
necessary to achieve the desired T_OFF@90 VAC:
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Doc ID 17005 Rev 3