ISL78208_14 Datasheet, PDF (19/24 Pages) Intersil Corporation – Wide VIN Dual Standard Buck Regulator with 3A/3A Continuous Output Current

ISL78208
Theory of Compensation
The sensed current signal is injected into the voltage loop to
achieve current mode control to simplify the loop compensation
design. The inductor is not considered as a state variable for
current mode control and the system becomes a single order
system. It is much easier to design a compensator to stabilize the
voltage loop than voltage mode control. Figure 46 shows the
small signal model of the synchronous buck regulator.
^i_IN
V^_IN
+
^iL _L
ILd^ ^{1:D VINd^}
R_T
V^_O
Rc
Ro
Co
d^
T_i(S)
K
Fm
+
He(S)
T_v(S)
V^_{COMP -Av(S)}
FIGURE 46. SMALL SIGNAL MODEL OF SYNCHRONOUS BUCK
REGULATOR
PWM Comparator Gain F_m
The PWM comparator gain F_m for peak current mode control is
given by Equation 14:
F_m
=
-_vˆ---c---_o^d-ˆ--_m-----_p-
=
---------------¹----------------
S_e + S_nT_s
(EQ. 14)
Where S_e is the slew rate of the slope compensation and S_n is
given by Equation 15.
S_n
=
R_t
-^V----ⁱ-ⁿ-----–----^V----^o--
L
(EQ. 15)
Where:
R_T is trans-resistance, and is the product of the current sensing
resistance and gain of the current amplifier in current loop.
CURRENT SAMPLING TRANSFER FUNCTION H_e(S)
In current loop, the current signal is sampled every switching
cycle. Equation 16 shows the transfer function:
H_eS=
-^S----²-
_n2
+
^-----_n^-S--Q^-----_n^-
+
1
(EQ. 16)
Where Q_n and _n are given by ^Q_n ^{= –2-}_^{- = }_n^{= f}_s .
Power Stage Transfer Functions
Transfer function F₁(S) from control to output voltage is
calculated in Equation 17:
F₁S
=
v_{-ˆ-d-ˆ-o--}
=
V_i
n
---------¹-----+-----^{---------Se}------^s-------^r--------
-^S----²-
_o2
+
^-----_o^-S--Q^-----_p^-
+
1
(EQ. 17)
Where
_esr
=
------¹--------
R_cC_o
,Q_p

R_o
-^C----^o-
L
,_o=
-------¹-------
LC_o
Transfer function F₂(S) from control to inductor current is given
by Equation 18:
F₂S
= ^ˆ-Id-ˆ^o--
=
-------^V----ⁱ--ⁿ-------
R_o + R_L
------------¹-----+------^--^S------^z-------------
-^S----²-
_o2
+
------^S--------
_oQ_p
+
1
(EQ. 18)
Where
_z
=
-------¹-------
R_oC_o
.
Current loop gain T_i(S) is expressed as Equation 19:
T_iS = R_TF_mF₂SH_eS
(EQ. 19)
The voltage loop gain with open current loop is calculated in
Equation 20:
T_vS = KF_mF₁SA_vS
(EQ. 20)
The voltage loop gain with current loop closed is given by
Equation 21:
L_vS = ₁-----^T_+----^v-_T--^--^S_i--_--_S-^----_
(EQ. 21)
Where
K
=
V_-----F---B--
V_o
,
V_FB
is the feedback voltage of the voltage
error amplifier. If T_i(S)>>1, then Equation 21 can be simplified as
shown in Equation 22:
L_vS=
-^V----^F---^B--
V_o
R_{-----o-----+----}R_-----L-
R_T
-¹----+-----^{----------Se------s------r}
1
+
--^S----
_p
-^A----^v---^---^S----^-
H_eS
,
_p

-------¹-------
R_oC_o
(EQ. 22)
From Equation 22, it is shown that the system is a single order
system, which has a single pole located at _P before the half
switching frequency. Therefore, a simple type II compensator can
be easily used to stabilize the system.
Submit Document Feedback ₁₉
FN8354.1
July 29, 2014