Texas Instruments SLVU013 - Manual (Page 39)

External Component Selection
2-19 Design Procedure
Peak to peak value of the inductor current
∆
I is given by the following equation:
D I +
V
I
– Io
ǒ
R
DS(on)
) R
L
Ǔ
–V
O
L
D Ts
(1)
Where:
V
I
=the input voltage
V
O
= the output voltage
Ts = the switching period
D
+
V
O
)
Io
ǒ
R
DS(on)
)
R
L
Ǔ
V
I
is the duty cycle which is defined as :
D +
t
ON
Ts
and t
ON
is the on time of the high-side MOSFET.
Referring to Figure 2–6 (e), the output voltage ripple, V
p-p
, is higher than the
hysteresis band, Hyst, because of the delays, t
del
. Assume that delays for both
switching instants are equal for simplicity. The ideal capacitor voltage
component has the same initial value during switching instants
t
ON
and
( Ts–t
ON
) (see Figure 2–6 (b)). In this case the voltage V
p-p
is:
V
p-p
+
ESL
L
V
I
) D
I ESR
(2)
On the other hand, the hysteresis band is equal to the difference between the
peak-peak values of the V
O
ripple, v
ripple,
at the moments t
ON
– t
del
and
t
OFF
– t
del
Hyst + V
ripple
ǒ
t
ON
* t
del
Ǔ
* V
ripple
ǒ
t
OFF
* t
del
Ǔ
(3)
After substituting equation (1) into equations for v
C
, v
ESR
and v
ESL
(Figure 2–6) and using equations (2) and (3), the following equation for the
switching frequency,
f
S
, can be derived:
f
s
+
V
O
ǒ
V
I
– V
O
Ǔ ǒ
ESR–t
del
ń Co
Ǔ
V
I
ǒ
V
I
ESR t
del
) Hyst L–ESL V
I
Ǔ
(4)
Equation (4) shows that the switching frequency strongly depends on ESR and
ESL. It is important that ESL meet the following condition:
ESL
t
ǒ
ESR t
del
)
Hyst L D
ń
Vout
Ǔ
If it does not, the voltage step across the ESL during switching exceeds the
hysteresis window, and the switching frequency becomes too high and
uncontrollable.
The switching frequency does not depend on the load current in equation (4),
because a synchronous regulator has the same two stages of operation during
the switching period over the load current range, including no load condition.
There is no discontinuous mode operation at very light loads. In reality there
is a weak dependence of the switching frequency on the load current because