MagneticsMagnetics

Courtesy of Dr. Slobodan Cuk

P L = N2 P

dt

dv

Finally, 1221/

2

1

tttvdt tt

t

t

SIZE AND WEIGHT OF MAGNETIC COMPONENTS

Basic equation:

Fundamental reason for increasing switching frequency in switched mode power converters.

1. By Increase of fs smaller cores with smaller cross-section S can be used with no saturation.Also, smaller number of turns require smaller window area.

2. Better ferromagnetic materials with higher Bs lead directly to smaller, more compact designs.

For example, iron with Bs = 1.6T compared to ferrite with Bs = 0.3T would result in more compact andsmaller design.

SSSfBN

V4

I N C R E A S E O F T H E D R I V E V O L T A G E A M P L I T U D E

- flux linkages

0t

v(t)

v(t)

0 t

0

i

S y m m e tr ic a l a n d a lte r n a t iv e (b o th p o s it iv e a n d n e g a t iv e ) v o lta g e e x c ita t io n is a s su m e d a n d o n lyp o s it iv e e x c u r s io n i l lu s tr a te d h e r e . A lso fo r s im p lic ity o f d e m o n str a t io n a sq u a r e - lo o p c o r e

m a te r ia l is a s su m e d .

RmRg

NLeff

i

2

From reluctance model:

and

so

so

LgLmLeff //

Uniform Flux assumed

S

g

S

morRgRm

R 00

henceg

R

m

DC CURRENT CAPABILITY AND AIR-GAP

s

At no-gap or with smallair gap the inductor cansupport only a small DCcurrent without saturation

Small DC Current

i

Note that from s = Li

for any air-gap valueHence, higher peakcurrent are followed bylower inductance since s isconstant for given core.

(material)

221000250~

cm

A

cm

AJ

(1)

(2)

INDUCTOR DESIGN VIA CORE GEOMETRY APPROACH

Here instead of current density J, the copper losses Pcu are taken as a design parameter cThis is more realistic criterion in practice, since designer allocates total Watts lost in windings.

DERIVATION:

where t is mean length per turn of the winding and is resistivityof copper at 25C.To fully utilize flux capability: (1)

To fully utilize window area W: (2)

Substitution of (1) into (2) and rearranging results in:

(3)

PcukB

LI

t

WSKg

2

222

core geometry (m5)geometrical parametersonly

electrical and magneticparmaters

From Faraday’s Law:

From Ampere’s Law:

Assuming core with extremely steepB-H loop

EVEN MORE REALISTIC MODEL...

Note that the core function (mutual coupling ) can be represented bymagnetizing inductance either on primary or on secondary.

C O U P L E D I N D U C T O R S M O D E L

N o t e t h a t m u t u a l i n d u c t a n c e L M a n d m a g n e t i z i n g i n d u c t a n c eL m a r e d i f f e r e n t i n g e n e r a l a n d r e l a t e d b y :

GENERAL MODEL : a ANY NUMBER!

"PHYSICAL" MODEL : a = N 1 / N 2

V 1 L 1 L 2 V 2

i 1 i 2L M

L 1 a : 1 L 2 -1a

L M- aL M

V 2V 1aL M

i 1 i 2

L 1 N 1 : N 2L 2

N 1

N 2

L M

a - arbitrary number

C o u p l e d i n d u c t o r e q u a t i o n s

v Ld i

d tL

d i

d tM1 11 2

v Ld i

d tL

d i

d tM21

22

L M - M U T U A L I N D U C T A N C E

Intuitive argument:

Proportional

AREA PRODUCT ApTo fully utilize the core capability:

(square wave, V1 primary, V2 - second)

To fully utilize window area:

(1)

(2)

(3)

(4)

Substitution of (1) in (2) and rearranging

SSkJfB

IVIVWSAp

42211

OR

SfBVN SS422