Transformer and Inductor Design Handbook Trade Offs Chapter_5

7/28/2019 Transformer and Inductor Design Handbook Trade Offs Chapter_5

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Chapter 5

Transformer Design Trade-Offs

yright 2004 by Marcel Dekker, Inc. All Rights Reserved.


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IntroductionThe conversion process in power electronics requires the use of transformer components that are frequentlythe heaviest and bulkiest item in the conversion circuit. They also have a significant effect upon the overallperformance an d efficiency of the system. Accordingly, th e design of such transformers has an importantinfluence on the overall system weight, power conversion efficiency, an d cost. Because of theinterdependence and interaction of these parameters, judicious trade-offs are necessary to achieve designoptimization.

The Design Problem Gen erallyThe d esigner is faced with a set of constraints that must be observed in the design on any transformer. Oneof these con straints is the output power, P 0 (operating voltage multiplied by maximum current demand) inthat the secondary winding must be capable of delivering to the load within specified regulation limits.Another constraint relates to minimum efficiency of operation, which is dependent upon the maximumpower loss that can be allowed in the transformer. Still another constraint defines the maximumpermissible temperature rise for the transformer when it is used in a specified temperature environment.

One of the basic steps in transformer design is the selection of proper core m aterial. Mag netic materialsused to design low and high frequency transformers are shown in Table 5-1. Each one of these materialshas its own optimum point in the cost, size, frequency an d efficiency spectrum. The designer should beaware of the cost difference between silicon-iron, nickel-iron, amorphous and ferrite materials. Otherconstraints relate to volume occupied by the transform er and, particularly in aerospace applications, weightminimization is an important goal. Finally, cost effectiveness is always an important consideration.

Depending upo n the application, some of these constraints will dominate. Parameters affecting others maythen be traded of f, as necessary, to achieve the most desirable design. It is not possible to optimize al lparameters in a single design because of their interaction and interdependence. For example, if volume andweight are of great significance, reductions in both can often be affected, by operating the transformer at ahigher frequency, but with the penalty being in efficiency. When, the frequency cannot be increased,reduction in weight and volume may still be possible by selecting a more efficient core material, but withth e penalty of increased cost. Thus, judicious trade-offs must be affected to achieve the design goals.



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Transformer designers have used various approaches in arriving at suitable designs. Fo r example, in manycases, a rule of thumb is used fo r dealing with current dens ity. Ty pically, an assumption is made that agood working level is 200 amps-per-cm (1000 circular mils-per-ampere). This rule of thumb will work inmany instances, but the wire size needed to meet this requirement m ay produce a heavier an d bulkiertransformer than desired or required. The info rma tion presented in this Chapter makes it possible to avoidthe assumption use of this and other rules of thumb, and to develop a more economical design with greataccuracy.

Table 5-1. Magnetic Materials and Their CharacteristicsMag netic Core M aterial Characteristics

MaterialName

InitialPermeability

H i

Flux DensityTesla

B sCurie

TemperatureC

dc , CoerciveForce, HeOersteds

OperatingFrequency

fIron Alloys

MagnesilSupermendur*

OrthonolSq.PermalloySupermalloy

1.5 K0.8 K2K

1 2 K - 1 0 0 K1 0 K - 5 0 K

1.5-1.81.9-2.2

1.42-1.580.66-0.820.65-0.82

75 094050046 046 0

0.4-0.60.15-0.35

0.1-0.20.02-0.04

0.003-0.008

< 2 k H z200>200

0.03-0.080.008-0.02


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Transformer Area Product, A pThe author ha s developed additional relationships between, Ap, numbers an d current density, J, for givenregulation an d temperature rise. The area product, A p, is a length dimension to the fourth power, ( I4 ), asshown in Figure 5-1.

Wa=FG, [c m 2]Ae=DE, [c m 2] [5-6]Ap=WaAc, [c m 4]

W,,A

*-F-H E -* * DFigure 5-1.C Core O utline Showing the Window A rea, W a and Iron Area, A c.

It should be noted. The constants fo r tape-wound cores, such as : KVO |, K w , Ks, Kj and Kp will have atendency to jum p around and not be consistent. This inconsistency has to do with the core being in ahousing, without true proportions.

Transformer Volume and the Area Product, ApThe volume of a transformer can be related to the area product, A p of a transform er, treating the volume, asshown in Figures 5-2 to 5-4, as a solid quantity without any subtraction for the core windo w. Therelationship is derived according to the following reasoning: Volume varies in accordance with the cube ofan y linear dimension, (1 ), whereas area product, Ap, varies as the fourth power:

Volume = Kf, [cm 3] [5-7]Ap=K2l\ [c m 4] [5-8]

I4=^r [5-9]\ (0.25)

7 = 1^-1 [5-10]

[5-11]

http://0.0.0.0/http://0.0.0.0/


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Volume =/d-f t5-12]K -,K ,.(0.75)

[5-13]

The volume-area product, Ap, relationship is therefore:Volume =KvolA(7i\ [cm3] [5-14]

in which, K vo l, is a constant related to core configuration whose values are given in Table 5-2. Thesevalues were obtained by averaging the values from the data taken from Tables 3-1 through Tables 3-64 inChapte r3 .

Table 5-2. Volume-Area Product Relationship.Volume -Area Product Relationship

Core TypePot CorePowder CoreLaminationsCCoreSingle-coil C CoreTape-wound Core

KVoi14.513.119.717.925.625.0

Volume Height

Figure 5-2.Toroidal Transformer Outline, Showing the Volume.

Volume

Figure 5-3. El Core Transformer Outline, Showing the Volume.



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Volume

Figure 5-4. C Core Transform er Outline, Showing the Volume.The relationship between volume an d area product, Ap, for various core types is graphed in Figures 5-5through 5-7. The data fo r these Fig ures ha s been taken from Tables in Chapter 3.

1000

luOQCJofo> 10

1.00.x /x

///

Laminations

01 0.1 1.0 10 , 100 100(Area Product, (Ap, cnv)Figure 5-5. Volume Versus Area Produ ct, A p fo r El Laminations.

1000

luoooa

"o> 10

1.00._x Xx

///

cc

:

D r e s

01 0.1 1.0 10 100 100(Area Product, (A_, cm4)Figure 5-6. Volume V ersus Area Product, A p for C Cores.



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1000

100

1101.0

Toroidal Cores

0.01 0.1 1.0 10 100 1000Area Product, (Ap, cm4)Figure 5-7. Volume Versus Area Product, Ap, for Toroidal MPP C ores.

Transformer Weight and the Area Product, A pThe total weight of a transformer can also be related to the area product, Ap, of a transformer. Therelationship is derived according to the following reasoning: weight, W,, varies, in accordance with th ecube of any linear dimension 1, whereas area product, A p, varies, as the fourth power:

W,=K,f, [grains] [5-15]Ap=K2l\ [c m 4] [5-16]

/4=^ [5-17]s(0.25)

. IJ I'=!TH [5-18]A,

075- I [5-20]

(0.75)KThe weight-area product, Ap, relationship is therefore:

[5-21]

[5-22]



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l O O O F

g 100260GOI 10

1.0Toroidal Cores

0.01 0.1 1.0 10 100 1000Area Product, (Ap, cnr)Figure 5-10. Total Weight V ersus Area Product, A p, for Toroidal MPP Cores.

Transformer Surface Area and the Area Product, ApThe surface area of a transformer can be related to the area product, A p, of a transformer, treating thesurface area, as shown in Figure 5-11 through 5-13. The relationship is derived in accordance with thefollowing reasoning: the surface area varies with the square of any linear dimension ( 1 ) , whereas the areaproduct, Ap, varies as the fourth power.

A,=KJ\ [c m 2] [5-23]A =K2l\ [c m 4] [5-24]

A ",(0.25)

/ =|-M [5-26]A ,

[5-27]

,=K


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The relationship between current density, J, and area product, A p, can, therefore, be expressed as:

[5-54]

The constant, Kj, is related to the core configuration, whose values are given in Table 5-5. These valueshave been derived by averaging the values from the data taken from Tables 3-1 through Tables 3-64 inChapter3 .

Table 5-5. Constant, Kj, for Temperature Increase of 25C an d 50C.

Temperature C onstant, K jCore Type

Pot CorePowder CoreLaminationsCCoreSingle-coil C CoreTape-wound Core

Kj (A25)43 340336632 239525 0

Kj (A50)63259053 446 8569365

The relationship between current density, J, and area product, A p, fo r temperature increases of 25C and50C is graphed in Figures 5-17 through 5-19 from data calculated of Tables 3-1 through 3-64 in Chapter 3.I100080060040 0

>GOa< u2200Iu

100

P 0 = Output PowerPcu,Copper Loss = P fe , Iron Loss Laminations

0.01 0.1 1.0 10 100Area Product, A (crrr) 1000

Figure 5-17. Current Density, J, Versus Area Product, Ap, fo r El Laminations.



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A =K2l4 [5-57]From Equation 5-56,

Then,[5-58]

Substituting Equation 5-59 into Equation 5-57,

A=K

= lH [5-59]

P ~ 2

(0.8)[5-60]

Let:

Then,A [5-61]

[5-62]

The constant, Kp, is related to the core configuration, whose values are given in Table 5-6. These valueshave been derived by averaging the values from the data taken from Tables 3-1 through Tables 3-64 inChapter3.

Table 5-6. Configuration Constant, Kp, for Area Product, Ap, and Core geometry, Kg.

Constant, KpCore Type

Pot CorePowder CoreLaminationsCCoreTape-wound Core

K P8.9

11.88.312.514.0

The relationship between area product, Ap, and core geometry, Kg, is graphed inFigures 5-20through5-22,from the data taken from Tables 3-1 through Tables 3-64 in Chapter 3.



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1000

o- o100

10

1.00.01

Laminations

0.1 10 10 100 1000Core Geometry, (Kg cm3)

Figure 5-20. Area Product, A p, Versus Core Geometry, Kg, fo r El Laminations.

100

10

eProduc

o

0.1C Cores

0.001 0.01 0.1 1.0 10Core Geometry, ( K air) 100Figure 5-21. Area Product, Ap, Versus Core Geometry, K g, for C Cores.

10

gocu 1.0

rsoca*-4)10

Figure 5-22. Area Product, A p, Versus Core Geometry, K g, for MPP Powder Cores.



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4. F. F. Judd and D. R. Kessler, Design Optimization of Power Transformers, Bell Laboratories,Whippany, N ew Jersey IEEE Applied Magnetics Workshop, June 5-6, 1975

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Transformer and Inductor Design Handbook Trade Offs Chapter_5