transformadores e inductores

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Chapter 1

Transformer and Inductor Design Philosophy

opyright 2002 by Marcel Dekker, Inc. All Rights Reserved.



Table of Conten ts

1. I n t r o d u c t io n

2. Pow er-Handl ing Abi l i ty3. Trans fo rmer Des ign

4. Tr a n s f o r me r s w i t h M u l t ip le O u t pu t s

5. Regula t ion

6. Rela t ionship K g to Power Trans fo rmer Regula t ion Capab i l i ty

7. Rela t ionship A p to Tr a n s f o r me r P o w e r H a n d l in g C a pa b i l i t y

8. Inducto r Des ign

9 . F u n d a m e n t a l C o n d i t io n s i n D e s ig n in g I n d u c t o r s

10. Fr ing ing F lux

11. To r o id a l P o w d e r Core Select ion

12. Rela t ionship Kg t o P o w e r I n d u c t o r R e g u la t io n C a pa b i l it

13. Rela t ionship A p to I n d u c t o r E n e r g y H a n d l in g C a pab i l i t y

14. Trans fo rmer Losses

15. Inducto r Losses

16. Eddy Current Losses , Skin Effect

17. Eddy Curren t Losses , Proximity Effect

18. Tempera ture Rise and Surface Area




Introduct ion

The conversion process in power electronics requires the use of transformers, and components which

f requently are the heaviest an d bulkiest i tems in the conversion circuits. They also have a significant effect

upon the overall performance and efficiency of the system. Ac cord ing ly, the design of such transform ers

has an important in f luence on overall system weight, on the power conv ersion efficiency and cost. Because

of the interdependence and interaction of parameters, judicious tradeoffs are necessary to achieve design

optimization.

Po we r-Ha ndl ing Abi l i ty

For years, manufacturers have assigned numeric codes to their cores; these codes represent the power-

handl ing abi l i ty . This method assigns to each core, a num ber, which is the product of its window area,

(W a), and core cross-section area, (A c), and is called, "Area Product," A p .

These numbers are used by core suppliers to summarize dimensional and electrical properties in their

catalogs . They are available for laminations, C-cores, pot cores, powd er cores, ferrite toroids, and toroidal

tape-wound cores.

The reg ulation and po wer-han dling ability of a core is related to the core geometry, K g . Every core has its

own inherent, K g. The core geometry, Kg, is a relatively new for magnetic cores. M anuf acturers do not l istthis coefficient .

Because of their significanc e, the area produc t, A p, and core geometry, K g , are treated extensively in this

book. A g reat deal of other information is also presented for the convenience of the designer. Mu ch of the

material is in tabular form to assist th e designer in making th e tradeoffs, best-suited for his particular

application, in a min imum amoun t of t ime.

These relationships can now be used as new tools to simplify and standardize th e process of transformer

design. They make it possible to design transformers of a l ighter weight and smaller volum e, or to optimize

eff iciency without going through a cut and try, design procedure. While developed specifically fo r

aerospace appl icat ions , th e information has a wider uti l i ty , and can be used for the design of non-aerospace

transformers, as well .




Tra nsf o rme r Design

The designer is faced with a set of constraints which must be observed in the design of any transformer.

One of these is the output power, PQ, (operat ing voltage m ult ipl ied by maximum current demand) , which

the secondary w inding m ust be capable of deliverin g to the load within specified reg ulation limits. Ano ther

relates to minimum eff iciency of operat ion which is dependent upon th e maximum power loss , which can be

allow ed in the transformer. Still another defines the maxim um permissible temperature rise for the

t ransformer, when used in a specif ied temperature environm ent .

Other constraints relate to volume occupied by the t ransformer, part icularly , in aerospace applications, and

weight, since weight minimization is an important goal in the design of space f l ight electronics. Lastly, cost

effectiveness is always an important consideration.

Output power, (P 0) , is of greatest interest to the user. To the transform er designer it is the apparent pow er,

(P t) , whi ch is associated w ith the gre ater important geom etry of the transform er. Assum e, for the sake of

sim plicity, the core of an isolat io n transform er has only two win din gs in the window area, (W a), a primary

and a secondary. Also , assume that the window area, (W a) , is divid ed up in proportion to the power

handling capabil i ty o f the windings, us ing equal current densi ty . This includes the primary w inding

handles , Pjn, and the secondary handles , P 0 , to the load. Since the power transform er has to be designed to

accomm odate the primary, P in and P 0 , then:

By d ef in i tion :

P> = — , [watts]

The pr imary turns can be expressed using Faraday's law:

N =—̂ — , [turns]/' A D f IS




The winding area of a transformer is fully utilized when:

By definition the wire area is:

Rearranging the equation shows:

Now, substitute in Faraday's equation:

AcBiicfKfj

Rearranging shows:

The output power, P0, is :

^=^/,, [watts]

The input power, P in , is :

Then:

[watts]




Substitute in P,

W A =

By d ef ini t ion , A p, equals:

BiJJK,Ku

Ap=W

aA

c,

[cm

4

]

Then:

., [ c m4]

T he des igner must be concerned with th e apparent power, handl ing capabi l i ty , P t, of the t ransformer core

and wind ing s . P^ may vary by a factor , ranging fro m 2 to 2 .828 t imes the input power, P^, depending upon

th e type of c i rcu i t in which th e t rans form er is used. If the cur ren t in the recti f ier t ransformer becomes

interrupted, i ts effective RMS value changes . Thus , t ransfo rm er s ize, i s not on ly determined by the load

demand, b ut a lso, by appl icat ion , because of the different copper losses incurred, due to the current

waveform.

For example , for a load of one watt , compare the power hand l ing c apabi l i t ies required for each wind ing ,

(neglect ing transform er and d iode losses , so that Pj n = Po) for the fu l l -wave bridge c irc ui t of Figure 1-1, th e

full-wave center- tapped seco ndary c ircu i t of Figure 1-2, and the push-pul l , center- tapped ful l -wave circui t

in Figure 1-3, where all the windings have the same number of turns, (N).

o < R,

F i g u r e 1 - 1 . Ful l -W ave Bridge Secondary .

T he tota l apparent power, P, , for the circui t shown in F ig ure 1-1 is 2 watts .

This i s shown in the fol low ing equat ion :

P,=Pin+Pn, [watts]




P t = 2 P i H , [watts]

Figure 1-2. Full-Wave, Center-TappedSecondary.

O

(-

Cl

s~

\ —

Q i

JE-N '

HPQ2

ii —i

N

C R l

Figure 1-3. Push-Pull Primary, Full- Wave, Center-Tapped Secondary.

The total power, Pt, for the circu it shown in Figure 1-2, increased 20.7%, due to the distorted wave form of

the interrupted current flowing in the secondary winding. This is shown in the followin g equation:

P,=Pin+P0j2, [watts]

Pf=Pin(l+j2), [watts]

The total power, P t, for the circuit is shown in Figure 1-3, which is typical of a dc to dc converter. It

increases to 2.828 times P jn , because of the interrupted current flowing in both the primary and secondary

windings.

Pr=P:nj2+Poj2, [watts]

Pt=2Pin̂ 2, [watts]opyright 2002 by Marcel Dekker, Inc. All Rights Reserved.



T ra n s f o rm e rs w i t h M u l t i p l e O u t p u t s

This i s an example of how the apparent power, P t, changes w ith a mu lt ip le output t ransformers .

Circui t

center tapped V^ = d iode drop ~ 1 V

full wave br idge V^ = diode drop = 2 V5 V(o), 1A

Eff ic iency = 0.95

o £ R,

o < R,

F i g ure 1 - 4 . Multiple Output Converter.

The output power seen by the transform er in Figure 1-4 is :

and:

^ , = ( K , , + K / ) ( / » , ) . [watts]

P0, =(5 + l ) ( l O ) , [ wa tt s ]

p g ] = 60 , [watts]

^=(^+^)(/J, [watts]

Po2 = ( 1 5 + 2)(1.0) , [watts ]

/Jo; =17, [watts ]




Because of the different winding con figurations th e apparent power, P t, th e transformer w ill have to be

summed to reflect this. When a winding has a center tap and produces a discontinuous current, then, th e

power in that wind ing, be it primary or second ary, has to be m ultiplied by the factor, U. The factor, U ,

corrects for the rm s current in that winding, if the winding has a center tap, then, the factor U is equal to

1.41. If not, the factor, U, is equa l to 1.

For an example, summing up the output power of a mu ltiple output transformer, w ould be:

then:

/ > = 60(1.41) + 17(1), [watts]/ > = 1 0 1 . 6 , [watts]

After th e secondary ha s been totaled, then th e primary power can be calculated.

P + P/ > = -2! -21, [watts]

(60)P =

V/ , [watts](0.95) L J

f > n = 8 1 , [watts]

Then, th e apparent power, P t, equals:

P ^ [watts]

/ ? = ( 8 1 ) ( 1 . 4 1 ) + (101.6), [watts]

P,=215.S, [watts]

Regulat ion

The minimum size of a transformer is usually determined either by a temperature rise l imit, or by al lowable

voltage regulation, assuming that size an d weight are to be minimized. Figure 1-5 shows a circuit diagram

of a transformer with one secondary.

Note that a = regulation (%).




Primaiy Secondary

n = Ns/Np = 1

Figure 1-5 . Transformer Circui t Diagram.

The assumption is that distributed capacitance in the secondary can be neglected because the frequency and

secondary voltage are not excessiv ely high. Also , the wind ing geom etry is designed to limit the leakage

inductance to a level, low enough to be neglected under most operating condit ions.

Transformer voltage regulat ion can now be expressed as:

V (F .L.)

. . , .

- ( 1 0 0 ) , [% ]V '

in which, V O (N .L. ) , is the no load voltage and, V O (F.L.), is the ful l load voltage. For the sake of simplicity,

assume the transformer in Figure 1-5, is an isolat ion transformer, w ith a 1:1 turns rat io , and the core

impedance, Re, is infinite .

If the transformer has a 1:1 turns rat io , and the core impedance is inf ini te, then:

R,,=R5, [ohms]

With equal window areas allocated for the primary and secondary windings, and using the same current

density, J ,

A I/, = / , . „ / ? , , = A F = I 0 R f , [volts]

Regulation is then:




Mult iply th e equation by currents, I:

V I . V Ip in s o

Primary copper loss is :

Pp=&ypI.H, [watts]

Secondary copper loss is :

Ps=&VsIo, [watts]

Total copper loss is:

Then, th e regulat ion equation can be rewritten to :

a = " - ( 1 0 0 ) , [%]

Regulat ion can be expressed as the power lost in the copper. A transform er, with an output power of 100

watts an d regulation of 2%, will have a 2 watt loss in the copper:

PaPr =-2—, [watts]

c" 100

(100)(2)

Pcl =

A;

, [watts]100

Pcu =2, [watts]




R e la t i o n s h i p K p to Powe r T r a n s f o r m e r Regulat ion Capabi l i tyo

Transformers

Although most transformers are designed for a given temperature rise, they can also be designed for a given

regulation. The regulat ion an d power-handling abil i ty of a core is related to two constants:

rv -(J. —

2KgKc

a = Regulation (% )

The constant, *Kg, is determined by the core geometry, which may be related by the following equations:

W aA ;K H ,- —-, [ c m ' ]M LT

The constant, K e, is determined by the magnetic and electric operating co ndition s, which may be related by

th e fo l lowing equat ion:

K =

Where:

K f = waveform coefficient

4.0 square wave

4.44 sine wave

From the above, it can be seen that factors, such as flux density, frequency of operation, and waveform

coefficient, have an in f luence on the transformer size.

The der ivat ion fo r these equations is set forth, in detail, by the author in the book, "Transformer an d

Inductor Design Handbook," Marcel Dekker, Inc., N ew York, 1988.




Relationship Ap to Transformer Pow er H and l ing C apabil i ty

Transformers

According to the newly developed approach, the power handling capability of a core is related to its area

product, *Ap, by an equation which m ay be stated as:

A=--—, [cm4]

Where:

Kf = waveform coefficient

4.0 square wave

4.44 sine wave

From the above, it can be seen that factors, such as flux density, frequency of operation, an d window

utilization factor Ku, defines the maxim um space which may be occu pied by the copper in the win dow .

The area product, Ap, of a core is the product of the available window area, Wa, of the core in square

centimeters, (cirr), mu ltiplied by the effective, cross-sectional area, A c, in square centimeters, (cirr), which

may be stated as:

Ap=WaAc, [cm4]

Figures 1-6 through Figure 1-8 show, in outline form, three transformer core types that are typic al of those

shown in the catalogs of suppliers.

The derivation fo r these equations is set forth in detail by the author in the book "Transformer an d

Inductor Design Handbook," Marcel Dekker, Inc., New York, 1988.




W ,

G

D

Figure 1-6 . C, Core.

G D

Ar

F i g u r e 1-7. EE Core.

OD

F i g u r e 1 - 8 . Toroidal Core.




Indu ctor Des ign

The designer is faced with a set of constraints which m ust be observed in the design of any inductor. One

of these constraints is copper loss. The w inding must be capable of delivering current to the load within

specified regulation limits. Another cons traint relates to minimum efficiency of operation, which is

dependent upon th e maximum power loss that can be al lowed in the indu ctor. Still another defin es th e

maximum permissible temperature rise for the inductor, when used in a specified temperature environment.

T he gapped inductor has three loss compo nents. They are copper loss, P cu , core loss P fe , and gap loss, Pg.

Maximum efficiency is reached in an inductor , as in a transformer, when the copper loss, P c u , and the iron

loss, P f e , are equal, but only , when the core gap is zero. The loss does not occur in the air gap itself, but is

caused by magnetic flux fringing around the gap, and re-entering the core in a direction of high loss. As the

ai r gap increases, the fringing flux increases more and mo re. Some of the fringing flux strikes the core

perpend icular to the lamination, and sets up eddy currents, which cause additional loss. Designing w ithmolypermalloy powder core, the gap loss is minimized because th e powder is insulated with a ceramic

ma terial, which provides an uniformly distributed air gap. Also design ing with ferrites, the gap loss is

minimized because ferrite m aterials have such high resistivity.

Other constraints relate to volume o ccupied by the ind uctor, and weight, since weight minim ization is an

important goal in the design of space fl ight electronics. Lastly, cost effectiveness is always an important

consideration.

Fundamental Condit ions in Designing Inductors

T he design of a linear reactor depends upon four related factors:

1. Desired inductance, L.

2. Direct current, Idc .

3. Alternating current, AI.

4. Power loss and temperature rise, Tr.

With these requirements established, th e designer must determine th e maximum values for , Bdc, an d for, Bac,

which will not produce magnetic saturation. The designer must make tradeoffs, which w ill yield the highest

inductance for a given volume. T he core material, which is chosen, dictates th e maximum flux density

which can be tolerated for a given design. The basic equation fo r maximum flux is :




0.471 N(I, +A /

( l ( T4)

- [ t e s l a ]

/'

T he inductance of an iron-core inductor, carrying dc and having an air gap, may be expressed as :

0.47t/V2/ f (10

8)

L- •

[henrysl

Inductance is dependent on the effective length of the magnetic path, which is the sum of the air gap length,

(lg), and the ratio of the core mean length to relat ive permeability, (lm/u r).

When th e core air gap, (l g), is larger, compared to relat ive permeability, (lm/u r), because of the high relat ive

permeability (u r) , variat ions in ( u r ) do not sub stant ial ly effect the total effectiv e magnetic path length, or the

inductance. The in ductance equat ion , then reduces to :

0.471 N2

A ( l 08)

L = , [henrys]

Final determination of the air gap size requires consideration of the effect of fr inging flux, which is a

func t ion of gap dimension, the shape of the pole faces, and the shape, size and location of the winding. Its

net effect is the short ing of the air gap.

F r i n g i n g F l u x

Fringing f lux decreases the total relucta nce of the magn etic path and,therefore, increases the inductance by

a factor, F, to a value greater than that calculated from in ductance equat ion . Fringing flux is a larger

percentage of the total for larger gaps. T he f r inging f lux factor is :

/ 2 G

I nduc tance , L, computed, does not include the effect of fr inging f lux. The value of inductance, L, corrected

fo r f r ing ing flux is :




2A F ( l O 8

)

M P L[henrys]

Distribution of fringing flux is also affected by another aspect of core geometry, the proximity of coil turns

to the core, an d whether, there are turns on both legs.

OAnNFU, + — l O ~4

The fringing flux is around the gap and re-entering the core in a direction of high loss as shown in Figure 1 -

9.

Eddy currents

Fringing Flux

Magnetic Path

N©0©©©©©©©©

Figure 1-9. Fringing Flux Around the Gap of an Inductor Design with a C Core.

Effective permeability may be calculated from the following expression:

Urn

1+AM P L

After establishing th e required inductance and the dc bias current which will be encountered, d imensions

can be determined. This requires consideration of the energy handling capabil i ty, which is controlled by the

size of the inductor.

The energy handling capability of a core is:

LI2

Energy = , [watt-second]




T o r o i d a l P o w d e r Core Select ion

The design of an inductor , a l so, f requent ly involves cons iderat ion of the effect of its magnet ic field on other

devices near where it i s p laced. This is especially true in the desig n of high-current ind uctors for converters

and sw itching regulators used in spacecraft, which may also em ploy sen sitive m agnetic f ield detectors. For

this type of design problem, it is frequen tly imperative that a toroidal core be used. The magnetic flux in apowder toroid, (core), can be con tained inside the core more read ily, than in a lam ination, or C type core, as

the winding covers the core alon g the whole magnetic path length.

The author has developed a simp lif ied method of designing optimum, dc carrying inductors with powder

cores. This method allow s the correct core permeabil i ty to be determined, withou t relying on trial and

error.

With these requirements established , the designer must determine the maximum values for, Ba c, and for B ac ,

which wil l not produce mag netic saturation, and m ust make tradeoffs that will yield the highest ind uctance

fo r a given volume. The chosen core permeabil i ty dictates the max imu m dc flux density, which can be

tolerated for a given design. Po wder cores come in a range of permeabi l i ty . Figure 1-10 shows how the

required permeability ch anges with dc bias. As the m agnetizin g fo rce (dc b ias ) is increased, the

permeabil i ty wil l start to drop. W hen the perm eability has reached 90% of its initial valu e, as shown in

Figure 1-10, the permeabil i ty falls off, relatively fast.

1.0 10 100

D C M a g n e t i z i n g F o r c e , ( O e r s t ed s )

F igure 1 - 1 0 . Typical Perm eabi l ity Versus dc Bias fo r Powder Cores.

1 000




If an ind uctan ce is to be co nstant with an increasing direct current, there must be a negligib le drop in

inductance over the operating current range. Then the maximum, H, is an indicat ion of a core's capability.

The magnetization force, H, is in oersteds:

,, 0.471 N I rH = , oersteds

M P L

Inductance decreases with increasing f lux density and magnetizing force for various m aterials of different

values of permeabil i ty. The selection of the correct permeabil i ty for a given design is made after solving

fo r the energy handling capabil i ty:

=̂ gm(MPL)(l04)

JK.

It should be remembered that maximum flux density depends upon, Id c + AI/2, in the manner shown in

Figure 1-11. Different powder cores materials , with different permeabi l i ty , will operate with a high or

lower dc bias. Table 1-1 shows the different types of powder cores that are offered to the design engineer.

0.471 / / C+4)

M P L- , [tesla]

H (oersteds)

Figure 1- 11 . Flux Density Versus 1̂ + AI.




Table 1-1

Powder Cores

Materia ls

Initial Permeability

Flux Density

Density

M i

B m

5

M P P

14 - 550

0.7T

8.5

Hugh F lux

1 4 - 160

1.4T

8

Kool M u

60- 125

LOT

6.15

I ron Powder

10- 100

1.03- 1.4T

6 . 5 - 7

Re l a t i o nsh i p KK to P o w e r I n d u c t o r R e g u l a ti o n C a p a b i l it y

Inductors

Inductors, like transform ers, are designed for a given temperature rise. They can also be designed for a

given regulat ion . The regulat ion and energy hand ling abi l i ty of a core is related to two constants:

a = ( E n e r g y ) '

KK.,

a = Regula t ion (% )

The constant , *Kg, is determined by the core geometry:

,M L T

T he constant , Ke, is determined by the magnet ic and electric operating condit ions:

Where:

Po = output power

AD

B = B, +—, [tesla]ma x (tc " L J

From the above, it can be seen that f lux dens ity is the predom inant factor, governing the size.

*The der ivatio n for these equatio ns are set forth in detail by the author in the book, "Transformer and

Inductor Design Hand book," M arcel Dekker, Inc. , N ew York, 1988.




Relat ionship Ap t o Induct o r E n e rg y Ha n dl ing C a pa b i li ty

Inductors

Acc ord ing to the newly developed approach, the energy han dlin g capability of a core is related to its area

product, *Ap, by an equation, which may be stated as follows:

2 ( E n e rg y ) ( l Q4)

A = , [cm 1P

K..BJ

From the above, it can be seen that factors, such as flux density, Bmax, window utilization factor, K u,

(which defines the maximum space, which may be occupied by the copper in the window), and the current

density, J. All have an inf luence on the inductor area prod uct.

The area produc t Ap of a core is the product of the available window area, W a, of the core in square

centimeters, ( cm r ) , multiplied by the effective, cross-sectional area, A c, in square centimeters, (cn~r), which

may be stated as :

Ap=WaAc, [cm 4]

Figures 1-6 through 1-8 show, in outline form , three transformer core types that are typical of those shown

in the catalogs of the suppliers.

*The derivation fo r these equations are set forth in detail by the author in the book, "Transformer an d

Inductor Design Handbook," Marcel Dekker, Inc., New Y ork, 1988.




Transformer Losses

Transformer efficiency, regulation, and temperature rise are all interrelated. Not all of the input power to

the transformer is delivered to the load. The difference between th e input power an d output power is

converted into heat. This power loss can be broken down into tw o components: core loss an d copper loss.

The core loss is a fixed loss, and the copper loss is a variable loss that is related to the current deman d of

th e load. Copper loss increases by the square of the current and is also termed quadratic loss. Maxim um

efficiency is achieved when the fixed loss is equal to the quadratic loss at the rated load. Transformer

regulation is the copper loss, Pcu, divided by the output power, Po:

P * = P c u + P f c , [watts]

Inductor Losses

T he losses in an inductor are made up of three components: ( 1 ) copper loss, Pcu; (2 ) iron loss, P f e; and (3)

gap loss, Pp. The copper loss an d iron loss have been previo usly discussed. G ap loss is independent of

core material thickness an d permeabil i ty. Maximum efficiency is reached in an inductor , as in a

transformer, when th e copper loss, P cu , and the iron loss, P te , ar e equal, but only when th e core gap is zero.

The loss does not occur in the air gap itself, but is caused by magnetic f lux fringing around the gap and re-

entering the core in a direction of high loss. As the air gap increases , the flux across the gap fringes more

an d more. Some of the fringing flux strikes th e core, perpend icular to the strip or tape, an d sets up eddy

currents, which cause addi tional losses in the core. If the gap dim ensio n gets too large the fringing flux will

strike th e copper winding an d produce eddy currents, generatin g heat, just l ike an induction heater. Thefringing flux will j ump the gap and produce eddy currents, in both the core an d wind ing , as shown in Figure

1-12.

pv =

Pc u + Pe +

p»' [watts]

Eddy cu rrents

Core

W i n d i n g

Fringing Flux

Magnetic Path

L/2

F igure 1 - 12 . Fringing Flux Around the Gap of an Inductor Design with a C Core.




Eddy C u r r e n t Losses, Skin Effec t

As the f requency increases, there ar e additional losses that occur in the winding, due to eddy currents

induced in the con duc tors by the mag netic f ields within the windin g. Skin effect is caused by edd y currents,

induced in a wire by the magnetic field of the curren t carried by the wire itself. Skin depth is defined as the

distance belo w the surface, where the current density has fallen to , 1/e, or 37 percent of its value at the

surface. Skin depth of copper at 20°C is :

6.62e = - 7 = r > [ c m ]

Skin effect is i l lustrated in Figure 1-13. The required wire size is a num ber 17, m agnet wire. The operating

f requency is 100 kHz. The skin depth is 0.0209 centimeters. A number 17 magnet wire, operating at 100

kHz, wil l yield an unused area of 0.00422 cm2, as shown in Figure 1-13. If you take th e skin depth s, and,

assume it to be the radius of the wire, then, you can calculate th e minimum wire area. Take this area an d

match i t with the closest, AWG. Then, take the area of the AW G, and d ivide i t into the required area, and

that wil l be the number of strands.

Frequency = 100 kHz

#17 Required, area = 0.010398 cm2

Area not used = 0.00422 cm2

8 #26, area = 0.00128 cm2

x 8 = .01024 cm2

Skin depth

Figure 1-13. Skin Depth Illustration.

Eddy C u r r e n t Losses, Proximity Effec t

Proximity effect is caused by eddy currents induced in a wire, due to the alternating magnetic field of other

conductors in the vicinity. Proximity effect is more serious than skin effect because skin effect can be

overcome by going to a smaller diameter wire. In the proximity effect, eddy currents, caused by adjacent

layers, increase expon ential ly in amplitude, as the num ber of layers increases. Proxim ity effect, skin effect

and high f requency together w il l cause the transform er, with m ultiple layers, to have losses that are

excessive, due to current crowding from the skin effect. With each additional layer, the I2R losses in that

layer, increase by the square of the current of the previous layer. Selecting th e correct AWG, as well as the

wind ing geom etry, is very important in keeping the losses down. The proximity effect can be reduced,

s ign i f icant ly , by in terleavin g the primary and secon dary, as shown in Figures 1-14 and Figure 1-15.

pyright 2002 by Marcel Dekker, Inc. All Rights Reserved.



Secondary-layer 2

Secondary- layer 1

Primary- layer 2 —

Primary- layer 1

Insula t ion

Core

OOO0OOO0O0000000

Bobb in Magnet wire

Center leg

Spatial I l lustrat ion

0, mmf

F i g u r e 1 - 14 . Pr imary and Secondary are Separated.

Spatial I llustrat ion

— * •

— * •

— *•

-y

Core

00000000

®00@®0O®00000000

©0®<S©®©®

/ /Bobb in Magnet wire

Center leg

(), mmf

\̂^̂̂Secondary-layer 2 -

Primaiy- layer 2 —

Secondary- layer 1 -

Primary-layer 1 —

Insulat ion

Figure 1-15. Primary and Secondary are Interleaved.

Temperature Rise and Surface Area

The heat, gen erated by the core loss, copper loss, gap loss, and the losses due to the skin effect and

proximity effect , produces a temperature rise, which must be controlled to prevent damage to, or the failure

of the wind ings by the breakdo wn of the ins ulat ion at e levated temperatures . This heat is dissipated fromth e exposed surfaces of the t ransformer or inductor by a combinat ion of radiat ion and convect ion .

Therefore , the diss ipat ion is dependent upon the total, exposed surface area of the core and windings.

Temperature rise in a t ransformer winding cannot be predicted with complete precis ion , despite th e fact that

many techniques ar e described in the l i terature for its calculat ion . One, reasonably accurate method fo r

open core and win din g con struc tion is a hom ogen eous m ethod. It is also based up on the assum ption that

th e core and windin g losses may be lumped together as :

Transformer:

/ > = pcu + / ? / 5 [watts]

Inductor :

p- =

Pc u + P f c +

P*' [watts]

Also, The assumption is made that the thermal energy is dissipated uniformly throughout the surface area of

th e core an d winding assembly . T he effective surface area, A,, requ i red to dissipate heat, (expressed as

watts diss ipated per u n i t area) , is :




4=^, [c m2]

"y" is the power density of the average power, dissipated per unit area from th e surface of the t ransformer

and, P v, is the total power lost or dissipated.

The tem perature rise that can be expected for various levels of power loss is shown in Figure 1-16. It is

based on data obtained from Blume, (1938 ), for heat transfer, effected by a com binat ion of 55% radiation

and 45% convect ion, from surfaces having an emissivity of 0.95, in an ambient temperature of 25°C, at sea

level. Power loss, (heat dissipation), is expressed in watts per square centimeter of the total surface area.

Heat dissipation, by convection from the upper side of a horizontal flat surface, is on the order of 15-20%

more than from a vertical surface. Heat dissipation, from th e unders ide of a hor izonta l flat surface, depends

upon surface area and conductivity. Below are two, prominently used power loss factors, expressed in watts

per square centimeter of the total surface area.

y =0.03^/cm2@25°C rise

y/ = 0 . 0 7 H// c m

2@ 2 5 ° C rise

1.0

I00

0.1

0.01

0.001

Ambient Temperature

Emiss ivi ty 0.95

45% Convection

55% Radiat ion

10° C 100° C

AT = Temperature Rise, °C

Tr =450

F igure 1 -1 6 . Temperature Rise Versus S urface Diss ipat ion .

transformadores e inductores

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