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AN2644 Datasheet(PDF) 51 Page - STMicroelectronics

Part # AN2644
Description  An introduction to LLC resonant
PDF  64 Pages
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Manufacturer  STMICROELECTRONICS [STMicroelectronics]
Direct Link  http://www.st.com
Logo STMICROELECTRONICS - STMicroelectronics

AN2644 Datasheet(HTML) 51 Page - STMicroelectronics

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AN2644
Resonant transitions of half-bridge midpoint
51/64
Note that the integral in Equation 26 is the total charge provided to Cr during the conduction
time of Q1, when the converter draws current from the input source. This charge can be
expressed also as the product of the DC input current Iin times the switching period, then:
Equation 27
On the other hand, remembering that VC has a DC value equal to Vin/2, for symmetry the
following identity holds true:
Equation 28
By combination of (Equation 39) and (28) we find:
Equation 29
After some algebraic manipulations, it is possible to find that the equations governing the
operation of the circuits in Figure 30 are:
Equation 30
with the following initial conditions:
Equation 31
The solutions of these equations are:
Equation 32
with
Equation 33
V
C
Ts
2
-------
⎝⎠
⎛⎞
V
C 0
()
Iin
Cr
-------Ts
+
=
V
C
Ts
2
-------
⎝⎠
⎛⎞ Vin
2
--------
V
in
2
--------
V
C 0
()
–
=
–
V
C 0
()
1
2
--- V
in
Iin
Cr
-------Ts
–
⎝⎠
⎛⎞
=
V
C
Ts
2
-------
⎝⎠
⎛⎞
1
2
--- V
in
Iin
Cr
-------Ts
+
⎝⎠
⎛⎞
=
d
2
V
HB
dt
-----------------
1
C
HB Ls
Lp
+
()
------------------------------------V
HB
V
C 0
()
C
HB Ls
Lp
+
()
------------------------------------
=
+
d
2
V
HB
dt
-----------------
1
C
HBLs
------------------V
HB
V
C 0
() aV
out
V
F
+
()
⋅
–
C
HBLs
-----------------------------------------------------------
=
+
⎩
⎪
⎪
⎨
⎪
⎪
⎧
DCM
CCM
V
HB 0
()
0
dV
HB
dt
--------------
t0
=
I
R 0
()
C
HB
-------------
=
,
=
V
HB t
()
V
DD
ω
DDt
ϕ
DD
–
() V
C 0
()
+
sin
V
CC
ω
CCt
ϕ
CC
–
() V
C 0
() aV
out
V
F
+
()
⋅
–
+
sin
⎩
⎨
⎧
=
DCM
CCM
ω
DD
1
Ls
Lp
+
()C
HB
----------------------------------------
ϕ
DD
tan
1
–
V
C 0
()
I
R 0
()
----------------
C
HB
Ls
Lp
+
--------------------
⎝⎠
⎜⎟
⎛⎞
V
DD
V
C 0
()
ϕ
DD
sin
-------------------
ω
CC
1
LsC
HB
----------------------
,
=
,
=
,
=
,
=
ϕ
CC
1
–
V
C 0
() aV
out
V
F
+
()
⋅
–
I
R 0
()
-----------------------------------------------------------
C
HB
Ls
-----------
⎝⎠
⎜⎟
⎛⎞
V
CC
V
C 0
() aV
out
V
F
+
()
⋅
–
ϕ
CC
sin
-----------------------------------------------------------
=
,
tan
=



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