LTC3115-2_15 Datasheet, PDF (23/42 Pages) Linear Technology – 40V, 2A Synchronous Buck-Boost DC/DC Converter

LTC3115-2
Applications Information
The denominator of the buck mode transfer function ex-
hibits a pair of resonant poles generated by the LC filtering
of the power stage. The resonant frequency of the power
stage, f_O, is given by the following expression where L is
the value of the inductor:
f_O
=
1
2π
LC_O^R₍^+RR+SR_C
)
≅
1
2π
1
LC_O
The quality factor, Q, has a significant impact on compen-
sation of the voltage loop since a higher Q factor produces
a sharper loss of phase near the resonant frequency. The
quality factor is inversely related to the amount of damping
in the power stage and is substantially influenced by the
average series resistance of the power stage, R_S. Lower
values of R_S will increase the Q and result in a sharper
loss of phase near the resonant frequency and will require
more phase boost or lower bandwidth to maintain an
adequate phase margin.
Q
=
LC_{O (}R+R_{C )(}R+R_{S )}
( ^RR_C^C_O ^{+L +C}_O^R_S ^R+R_C
)
≅
L
R
LC_O
+ C_OR_S
Boost Mode Small-Signal Model
When stepping up from a lower input voltage to a higher
output voltage, the buck-boost converter will operate in
boost mode where the small-signal transfer function from
control voltage, V_C, to the output voltage is given by the
following expression.
V_O
V_C
BOOST
MODE
= G_BOOST
⎛
⎝⎜
1+
s
2π f_Z
⎞
⎠⎟
⎛
⎝⎜
1–
1+
s
2π f_O Q
+
⎛
⎝⎜
s
2π f_RHPZ
s ^⎞2
2πf_O ⎠⎟
⎞
⎠⎟
The boost mode gain, G_BOOST , is comprised of three
components: the analog divider, the pulse width modulator
and the power stage. The gain of the analog divider and
PWM remain the same as in buck mode operation, but
the gain of the power stage in boost mode is given by the
following equation:
( ) ^{GPOWER â‰}
V_OUT2
1– t_LOW f
V_IN
By combining the individual terms, the total gain in boost
mode can be reduced to the following expression. Notice
that unlike in buck mode, the gain in boost mode is a
function of both the input and output voltage.
G_BOOST
≅
29.7V_OUT2
V_IN2
In boost mode operation, the frequency of the right half
plane zero, f_RHPZ, is given by the following expression.
The frequency of the right half plane zero decreases at
higher loads and with larger inductors.
( ) _fRHPZ
R
=
1– t_LOW f ² V_IN2
2πL V_OUT2
In boost mode, the resonant frequency of the power stage
has a dependence on the input and output voltage as shown
by the following equation.
f_O
=
1
2π
( ) ^RS
+
RV_IN2
V_OUT2
≅
1_•
V_IN
LC_O R+R_C 2π V_OUT
1
LC
Finally, the magnitude of the quality factor of the power
stage in boost mode operation is given by the following
expression.
In boost mode operation, the transfer function is character-
ized by a pair of resonant poles and a zero generated by
the ESR of the output capacitor as in buck mode. However,
in addition there is a right half plane zero which generates
increasing gain and decreasing phase at higher frequen-
cies. As a result, the crossover frequency in boost mode
operation generally must be set lower than in buck mode
in order to maintain sufficient phase margin.
Q=
⎛
LC_OR _⎝⎜
R_S
+
RV_IN2
V_OUT2
⎞
⎠⎟
L +C_OR_SR
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