ISL6253 Datasheet, PDF (18/22 Pages) Intersil Corporation – Highly Integrated Battery Charger for Notebook Computers

ISL6253
PWM Comparator Gain F_m
The PWM comparator gain Fm for peak current mode
control is given by:
F_m
=
-_vˆ---c---_o^d-ˆ--_m-----_p-
=
--------¹---------
V_PWM
(EQ. 18)
where V_PWM is the peak-peak voltage of the PWM ramp
signal.
Current Sampling Transfer Function H_e(S)
In current loop, the current signal is sampled every switching
cycle. It has the following transfer function:
H_e(S)
=
-^S----²- + ------^S-------- + 1
ω_n2 ^ωn^Qn
(EQ. 19)
where Q_n and ω_n are given by
Q_n
=
– ²-- ,
π
ω_n
=
πf_s
,
respectively.
Power Stage Transfer Functions
Transfer function F₁(S) from control to output voltage is:
F₁(S)
=
-^vˆ-_d-ˆ-^o-
=
1 + -----^S------
V_i
n
-------------------^ω----^e----^s---^r--------
-^S----²- + ------^S-------- + 1
ω_o2 ^ωo^Qp
(EQ. 20)
Where
ω_esr
=
------¹--------,
R_cC_o
Q_p
≈
R_o
-^C----^o- ,
L
ω_o
=
-------¹-------
LC_o
Transfer function F₂(S) from control to inductor current is,
F₂(S) = ^ˆi-_d-ˆ^o- = _R-----_o--^V--+--ⁱ--ⁿ-_R-----_L-
1 + --^S----
----------------------^ω----^z------------
-^S----²- + ------^S-------- + 1
where
ω_z
≈
-------¹-------
R_oC_o
(EQ. 21)
ω_o2 ^ωo^Qp
Current loop gain T_i(S) is expressed as the following
equation:
T_i(S) = R_TF_mF₂(S)H_e(S)
(EQ. 22)
where R_T is the trans-resistance in current loop. R_T is
usually equal to the product of the current sensing resistance
of the current amplifier. For ISL6253, R_T = 24R₁.
The voltage gain with open current loop is:
T_v(S) = KF_mF₁(S)A_v(S)
(EQ. 23)
where
K
=
V_-----F---B--
V_o
,
V_FB
is
the
feedback
voltage
of
the
voltage
error amplifier.
The Voltage loop gain with current loop closed is given by:
L_v(S) = ₁-----^T_+----^v-_T--⁽--^S_i--₍--_S⁾-----₎
(EQ. 24)
If T_i(S)>>1, then the above equation can be simplified as
follows:
L_v(S)
=
-^V----^F---^B--
V_o
R_{-----o-----+----}R_-----L-
R_TV
-¹----+-----^{ω----------Se------s------r}
1 + --^S----
H^{-A----ve---((---SS----)-}),
ω_p
≈
-------¹-------
R_oC_o
ω_p
(EQ. 25)
Where RTV is the trans-resistance due to the current
information fed into the voltage loop. From the above
equation, it is shown that the system is a single order
system, which has a single pole located at ω_p before the
half switching frequency. Therefore, a simple type II
compensator can be easily used to stabilize the system.
Figure 20 shows the type II compensator, and its transfer
function is expressed as follows:
i^_in
v^_in
+
i^_L ^L
+
I_Ld^ 1:D
V_ind^
R_T
V^_o
Rc
Ro
Co
T_i(S)
d^
K
Fm
+
He(S)
T_v(S)
V^_{comp -Av(S)}
FIGURE 19. SMALL SIGNAL MODEL OF SYNCHRONOUS
BUCK REGULATOR
Vo
V_FB
V_REF
-
g_m
+
V_COMP
R1
C2
C1
FIGURE 20. TYPE II COMPENSATOR
18