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 ﬁ 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

ﬁ eld, their average values, their AC values, their DC values, and so on. The constant of

proportionality is equal to

L

NA

BI(proportionality constant linking and )

(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

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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

ﬂ 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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