 72
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Chapter 3
Combining Lenz s law with the inductor equation V L dI/dt , we get
VN
d
dt
NA
dB
dt
L
dI
dt

f
(3-42)
From this we get the two key equations used in power conversion
B
LI
NA
(voltage independent equation)
(3-43)
B
Vt
NA
(voltage dependent equation)
(3-44)
The rst equation can be written symbolically as
B
LI
NA
(voltage independent equation)
(3-45)
And the latter equation can be written in a more “ power-conversion-friendly ” form as
follows
B
VD
NAf
AC
ON
(voltage dependent equation)
2
(3-46)
For most inductors used in power conversion, if we reduce the current to zero, the ﬁ eld
inside the core also goes to zero. Therefore, an implicit assumption of complete linearity
is also usually made—that is, B and I are considered proportional to each other as shown
in Figure 3.13 (unless of course the core starts saturating, at which point, all bets are off!).
The voltage independent equation can then be expressed as any of the equations shown in
the ﬁ gure—in other words, this proportionality applies to the peak values of current and
eld, their average values, their AC values, their DC values, and so on. The constant of
proportionality is equal to
L
NA
(3-47)
where N is the number of turns and A the actual geometrical cross-sectional area of the core
(its center limb usually, or simply the effective area A
e
given in the datasheet of the core).
3.17 Worked Example (5)—When Not to Increase
the Number of Turns
Note that the voltage-independent equation is useful if, for example, we want to do a
quick check to see if our core may be saturating. Suppose we are custom-designing our www.newnespress.com
73DC-DC Converter Design and Magnetics
inductor. We have wound 40 turns on a core with an area of A 2
cm
2
. Its measured
inductance is 200 H, and the peak inductor current in our given application is 10 A. Then
the peak ﬂ ux density can be calculated as follows:
B
L
NA
I
PK PK
teslas
200 10
40
2
10
025
6
4
.
Note that we have converted the area to m
2
in the above equation, because we are using
the MKS version of the equation.
For most ferrites, an operating ﬂ ux density of 0.25 T is acceptable, since the saturation
ux density is typically around 0.3 T.
Based on the B and I linearity, we can also linearly extrapolate and thus conclude that
the peak current in our application should under no condition be allowed to exceed
(0.3/0.25) 10 12 A, because at 12 A, the ﬁ eld will be 0.3 T, and the core will then
start to saturate.
Time
Time
B
PK
B
PP
B
PK
B
AC
B
DC
I
PP
I
PK
B
DC
I
DC
B
DC
I
PK
I
DC
B
AC
I
AC
ΔB
ΔB
NA
L
I
B
NA
L
=
====
Current
ΔI
ΔI
B
I
B-field
B I
Proportionality constant is:
NA
L
(symbolically)
Figure 3.13 : B and I can usually be considered proportional to each other

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