NCV8851
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14
The ripple current is at a maximum when the duty cycle
is at a minimum value and vice versa, as follows:
i
L(max)
+
V
OUT
@ (1 * D
MIN
)
L @ F
SW
i
L(min)
+
V
OUT
@ (1 * D
MAX
)
L @ F
SW
Where: i
L(max)
: maximum inductor current ripple [App]
i
L(min)
: minimum inductor current ripple [App]
From this equation it is clear that the ripple current
increases as L decreases, emphasizing the trade−off between
dynamic response and ripple current. The peak and valley
values of the triangular current waveform are as follows:
I
L(pk)
+ I
OUT
)
i
L
2
I
L(vly)
+ I
OUT
*
i
L
2
Where: I
L(pk)
: peak (maximum) value of ripple current [A]
I
L(vly)
: valley (minimum) value of ripple current [A]
Saturation current is specified by inductor manufacturers
as the current at which the inductance value has dropped a
certain percentage from the nominal value, typically 10%.
For stable operation, the output inductor must be chosen so
that the inductance is close to the nominal value even at the
peak output current, I
L(pk)
. It is recommended to choose an
inductor with saturation current sufficiently higher than the
peak output current, such that the inductance is very close to
the nominal value at the peak output current. This introduces
a safety factor and allows for more optimized compensation.
Inductor efficiency is another consideration when
selecting an output inductor. Inductor losses include dc and
ac winding losses and core losses. Core losses include eddy
current losses, which are very low due to high core
resistance, and magnetic hysteresis losses, which increase
with peak−to−peak ripple current. Core losses also increase
as switching frequency increases.
Ac winding losses are based on the ac resistance of the
winding and the RMS ripple current through the inductor,
which is much lower than the dc current. The ac winding
losses are due to skin and proximity effects and are typically
much less than the dc losses, but increase with frequency. Dc
winding losses account for a large percentage of output
inductor losses and are the dominant factor at switching
frequencies at or below 500 kHz. The dc winding losses in
the inductor can be calculated with the following equation:
P
L(dc)
+ I
OUT
2
@ R
dc
Where: P
L(dc)
: dc winding losses in the output inductor
R
dc
: dc resistance of the output inductor (DCR)
As can be seen from the above equation, to minimize
inductor losses, an inductor with very low DCR should be
chosen.
(5) Output Capacitor Selection
The output capacitor is a basic component for the fast
response of the power supply. During a load step, for the first
few microseconds, it supplies the current to the load. The
controller immediately recognizes the load step and
increases the duty cycle, but the current slope is limited by
the inductor’s slew rate. During a load release, the output
voltage will overshoot. The capacitance will dampen this
undesirable response, decreasing the amount of voltage
overshoot.
In the case of stepping into a short, the inductor current
approaches zero with the worst case initial current at the
current limit and the initial voltage at the output voltage set
point, calculating the voltage overshoot as follows:
DV
OS
+
L @ I
CL
2
C
) V
OUT
2
* V
OUT
Ǹ
Accordingly, a minimum amount of capacitance can be
chosen for a maximum allowed output voltage overshoot:
C
MIN
+
L @ I
CL
2
(V
OUT
) DV
OS(max)
)
2
* V
OUT
2
Where: C
MIN
: minimum amount of capacitance to minimize
voltage overshoot to DV
OS(max)
[F]
DV
OS(max)
: maximum allowed voltage overshoot
during a short [V]
A maximum amount of capacitance can be found based on
the inrush current and current limit. To calculate the input
startup current, the following equation can be used:
I
INRUSH
+
C
OUT
@ V
OUT
t
SS
) I
OUT(i)
Where: I
INRUSH
: input current during startup
I
OUT(i)
: initial output current
If the inrush current is higher than the steady−state input
current with the maximum load, then the input fuse should
be rated accordingly, if one is used. During soft−start, the
inductor current must provide current to the load, as well as
current to charge the output capacitor. The maximum current
which the inductor is allowed to conduct is the current limit.
Setting the inrush current to the current limit, this puts a limit
on the maximum capacitor size, as follows:
C
MAX
+
(I
CL
* I
OUT(i)
) @ t
SS
V
OUT
Where: C
MAX
: maximum output capacitance [F]
Capacitors should also be chosen to provide acceptable
output voltage ripple with a dc load, in addition to limiting
voltage overshoot during a dynamic response. Key
specifications are equivalent series resistance (ESR) and
equivalent series inductance (ESL). The output capacitors
must have very low ESL for best transient response. The
PCB traces will add to the ESL, but by putting the output
capacitors close to the load, this effect can be minimized and
ESL neglected in determining output voltage ripple.
