Inductance Calculations - Ferroxcube Standard … Inductance factor calculation Inductance...

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Inductance factor calculation

Inductance Calculations - Ferroxcube Standard Cores

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A Select a Core Family from our standard range.

B Available sizes are displayed, make your selection.

C Available materials for the chosen size are displayed. When you select one, its permeability and the usual surface quality are shown.

D You can also defi ne your own material by directly entering its features.

E Flux density (Bpeak) is calculated based on a given voltage and number of turns.

F When a bias current is entered, the resulting magnetic bias fi eld (Hbias) is calculated. The bias fl ux density (Bbias) in the core is estimated with the help of the effective permeability (µe).

G Click “Go” to start a new calculation,

A B C D

E

GHI

F

H Click “Datasheet” to view the data sheet of the selected core type.

I Click “User defi ned cores” if you want to design your own core.

J Effective permeability is caculated based on material permeability, total airgap and its position.

K Effective core parameters for the selected core are shown.

L Inductance factor (AL) is caculated based on material permeability, total airgap and its position.

M Inductance value (L) is caculated based on AL and number of turns. These parameters can be changed freely to optimize the design.

If required the number of turns can be fi xed.

JK

M

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Using the program “Inductance Factor Calculation”

Functional description

Ferroxcube Standard CoresThe program consists of two parts. The fi rst screen appears when you click the button “Inductance factor Calculations” and can be used for calculations on standard Ferroxcube Cores. The second program can be activated by clicking the button “User Defi ned” in the lower right corner of the window.

User Defi ned CoresThis program calculates effective magnetic dimensions of user defi ned, magnetically closed core shapes. With these data, the properties of the chosen ferrite material and airgap length, the inductance factor AL of the core is derived. With the number of turns the inductance value of an inductor based on this core is calculated. For magnetically open circuits like rods the so-called rod permeability is caculated. With the number of turns and the position of the winding with respect to the rod the inductance of the coil is predicted.

The program can handle a variety of core shapes. The principal shapes are : Toroid (ring core), U core, E core, P core and rod. Several other core types are variations on the principal shapes so that the same formulas can be used. The core dimensions are entered with the help of a dimensional drawing in the core type window. In the case of cores with known effective dimensions, these dimensions can be entered directly without the need to specify the whole geometry.

After the calculation of magnetic dimensions, a menu appears for the choice of the inductor calculation. For a magnetically closed core type this can be AL value, inductance, fl ux level, fi eld strength or inverse calculations. Every core shape, including ring cores, can be gapped. The infl uence of fringing fl ux and parasitic gaps is accounted for. In case of a rod the calculation can be inductance, or an inverse calculation. An AL value is not appropriate here, because the inductance also depends on coil length and not only on the rod and the number of turns.

Further remarks:If entered values are out of range, a warning message box will displayed. After changing a value, a recalculation can be triggered by clicking the Go button or by pressing <ENTER> in the updated text fi eld.Calculations can be saved in a fi le and later be opened again with the Save en Open options in the File menu. Results can be printed with the Print option in File menu. Core sizes are always listed in ascending order of effective volume Ve. In cases where the calculated gap would become smaller than the parasitic gap, the parasitic gap length is taken as default.

Inductance factor calculations for User Defi ned Cores

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A Select a Core Shape from the list according to the design you want to make.

B Enter its main dimensions using the outline drawing as a guide.

C Use “direct input” when effective core parameters le and Ae are known. Choose between a closed core or 2 halves. Enter the type of airgap and its position. The core window height is required for the estimation of the fringing fl ux.

D Available materials for the chosen shape are displayed. When you select one, its permeability and the usual surface quality are shown. E You can also defi ne your own material by directly entering its features.

F Flux density (Bpeak) is calculated based on, core parameters, a given voltage and the number of turns.

G When a bias current is entered, the resulting magnetic bias fi eld (Hbias) is

A

E

HI

F

calculated. The bias fl ux density (Bbias) in the core is estimated with the help of the effective permeability (µe).

H Click “Go” to start a new calculation,

I Click “Ferroxcube Standard Core” to switch back to the standard core ranges.

J Effective permeability is caculated based on material permeability, total airgap and its position.

K Effective core parameters for the selected core are shown.

L Inductance factor (AL) is calculated based on material permeability, total airgap and its position.

M Inductance value (L) is calculated based on AL and number of turns. These parameters can be changed freely to optimize the design. If required the number of turns can be fi xed.

JK

M

B C D

G

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Defi nition of terms

Permeability

When a magnetic fi eld is applied to a soft magnetic material, the resulting fl ux density is composed of that of free space plus the contribution of the aligned domains.

B µ0H J or B+ µ0 H M+( )= = [1]

where µ0 = 4π.10 -7 H/m, J is the magnetic polarization and M is the magnetization.The ratio of fl ux density and applied fi eld is called absolute permeability.

[2]BH---- µ0 1 M

H-----+

µabsolute= =

It is usual to express this absolute permeability as the product of the magnetic constant of free space and the relative permeability (µr).

[3]BH---- µ0µr=

Since there are several versions of µr depending on conditions the index ‘r’ is generally removed and replaced by the applicable symbol e.g. µi, µa, µ∆ etc.

Initial permeability

The initial permeability is measured in a closed magnetic circuit (ring core) using a very low fi eld strength.

[4]µi

1µ0------ × B∆

H∆--------H∆ 0→( )

=

Initial permeability is dependent on temperature and frequency.

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Effective permeability

If an airgap is introduced in a closed magnetic circuit, magnetization becomes more diffi cult. As a result, the fl ux density for a given magnetic fi eld strength is lower.Effective permeability is dependent on the initial permeability of the soft magnetic material and the dimensions of airgaps and circuit.

[5]µe

µi

1G × µi

le-----------------+

---------------------------=

where G is the gap length and le is the effective length of magnetic circuit. This simple formula is a good approximation only for small airgaps. For longer airgaps some fl ux will cross the gap outside its normal area (fringing fl ux) causing an increase of the effective permeability.

Amplitude permeability

The relationship between higher fi eld strength and fl ux densities without the presence of a bias fi eld, is given by the amplitude permeability (µa).

[6]µa1µ0------ × B

H----=

Since the BH loop is far from linear, values depend on the applied fi eld strength.

Incremental permeability

The permeability observed when an alternating magnetic fi eld is superimposed on a static bias fi eld, is called the incremental permeability.

[7]µ∆1µ0------

B∆H∆-------- HDC

=

If the amplitude of the alternating fi eld is negligibly small, the permeability is called the reversible permeability (µrev).

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Effective core dimensions

To facilitate calculations on a non-uniform soft magnetic core, the effective dimensions are given on each data sheet. These dimensions, effective area (Ae), effective length (le) and effective volume (Ve) defi ne a hypothetical ring core which would have the same magnetic properties as the non-uniform core.The reluctance of the ideal ring core would be:

[9]le

µ × Ae------------------

For the non-uniform core shapes, this is usually written as:

[10]1µe------ × Σ l

A----

the core factor divided by the permeability. The inductance of the core can now be calculated using this core factor:

[11]Lµ0 × N

2

1µe------ × Σ l

A----

-----------------------1.257 × 10

9– × N2

1µe------ × Σ l

A----

-------------------------------------------------- (in H)= =

The effective area is used to calculate the fl ux density in a core, for sine wave:

[12]BU 2 × 10

9

ωAeN------------------------------

2.25U × 108

fNAe----------------------------------(in mT)= =

for square wave:

[13]B0.25U × 10

9

fNAe----------------------------------(in mT)=

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where:Ae is the effective area in mm2 .U is the voltage in Vf is the frequency in HzN is the number of turns.

The magnetic fi eld strength (H) is calculated using the effective length (Ie):

[14]HIN 2

le--------------(A/m)=

If the cross-sectional area of a core is non-uniform, there will always be a point where the real cross-section is minimal. This value is known as Amin and is used to calculate the maximum fl ux density in a core. In well designed ferrite core a large difference between Ae and Amin is avoided. Narrow parts of the core could saturate or cause much higher hysteresis losses.

Inductance factor (AL)To facilitate inductance calculations, the inductance factor, known as the AL value (nH), is given in each data sheet. The inductance factor of a core is defi ned as:

[15]L N2 × AL (nH)=

The value of AL is calculated from the core factor and the effective permeability:

[16]AL =µ0µe × 10

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Σ l A⁄( )------------------------------ =1.257µe

Σ l A⁄( )--------------------- (nH)

Inductance calculations on rods and tubesRods and tubes are generally used to increase the inductance of a coil. The magnetic circuit is very open and therefore the mechanical dimensions have more infl uence on the inductance than the ferrite’s permeability (see Fig.1) unless the rod is very slender. In order to establish the effect of a rod on the inductance of a coil, the following procedure should be carried out:

• Calculate the length to diameter ratio of the rod (l/d).

• Find this value on the horizontal axis and draw a vertical line.The intersection of this line with the curve of the material permeability gives the effective rod permeability (µrod).

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The inductance of the coil, provided the winding covers the whole length of the rod is given by:

[17]L µ0µrodN

2A

l----------- H( )=

where:N = number of turnsA = cross sectional area of rod (mm2)I = length of coil. (mm)

Fig.1 Rod permeability (µrod) as a function of length to diameter ratio with material permeability as a parameter.

103

102

10

Length / diameter ratio

µrod

µi = 10.000

5000

2000

1000

700500

400

300

200

150

100

70

40

20

10

1 10 100

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Transformer Core Selection

A B D

E

G

H

F

C


A Enter the design parameters. Take a reasonable value for the maximum throughput power, for instance 2 times the minimum required power, to limit the choice of suitable core sizes.

B Make a choice of converter type. Enter the required creepage distance between windings, expected copper fi ll factor and duty cycle, or accept the default values.

C Check the core families you wish to consider for your design.

D Select the material you wish to apply or “all” to leave the choice up to the program.

E Click “Go” to start a new calculation.

F The core types capable of handling the required throughput power are displayed in this window.

G Click “Graph” to view a graph of throughput power versus frequency for the selected core type.

H Click “Datasheet” to view a data sheet of the selected core type.

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IntroductionThis program can be used to select ferrite cores from the actual product range of FERROXCUBE for the design of a forward or fl yback transformer for use in a switched mode power converter. The user is asked to fi ll in the required power level, expected operating conditions and a number of other variables. Then he chooses the core families and the ferrite materials he wants to consider. The program calculates the expected throughput power for all chosen cores and displays the cores that fulfi l the criteria in the result window. The background of these calculations and the formulas used are explained in the following paragraphs.For the core set(s) a graph showing the losses as a function of frequency can be generated. This can be useful for the optimization of the converter design. When the choice is made, the data sheet of the product is displayed on request.

Input dataThe requested input data, grouped in 3 sets, can be inserted arbritrary order.

1. Application conditions

Description Symbols (used in calculations)

• Minimum throughput power Pthrmin • Maximum throughput power Pthrmax • Converter operating frequency f

• Ambient temperature Tamb • Allowed temperature rise ∆T

• Converter type Forward, Flyback or Push-Pull

• Creepage distance Cr *)

• Copper fi ll factor Fw • Effective duty factor δ

• Preferred airgap lg

*) For all core families, except toroids (T, TN, TX, TL, TC), the total creepage distance is the sum of the creepage distances between winding and the ferrite core. If one choses e.g. a value of 8 mm it is assumed that the distance between the primary and secundary winding package and the ferrite are 4 mm each. For toroids the creepage dis-tance is the sum of both insulation distances between the primary and secondary winding.

2. Selection of core family(ies)

The search should be limited to those families one wants to consider to speed up the proces.

3. Selection of power ferrite(s)

The search can be limited to those materials one wants to consider to speed up the proces or the option “all” can be checked to leave it up to the program.

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Background of the calculations

1.Calculation of througput power (example forward converter)

In general the throughput power can be calculated as a time average of the product of voltage and current. Applying this to the pulse shaped input voltage Vp and current Ip with duty cycle (active part of period) δ gives :

Pthr (Vp × Ip)dt

0

T

∫= = Vp × Ip × δ [1]1T

The relation of voltage Vp and fl ux density B is given by Faraday’s law of induction. Using this for the applied signals and keeping in mind that 1/T equals the frequency f, and Np stands for the number of primary turns, this results in:

Vp = Np × Ae × = Np × Ae × [2]dBdt

BδΤ

= Np × Ae ×fBδ

Note that B in eq.[2] is referring to the total fl ux excursion, So, from its minimum to its maximum value. Substituting eq.[2] for Vp in formula [1] shows that throughput power is proportional to the product of (f.B), which is interpreted as the material performance factor.

Pthr = Np × Ae × fB × Ip [3]

Next the current Ip will be worked out in terms of winding losses. For this purpose it is necessary to consider the effective (or rms) current. This current (Ieffective) is related to power losses in a resistance R by P = I2R. The effective current for a time varying current I(t) with period T is defi ned in general as:

I2effective

I2(t) dt

0

T

∫= [3A]1T

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Applying this defi nition to the appropriate current signals we get the following results:

Ipeffective = Ip × [4]δ

Relating the current Ip to the primary winding losses Pw.pr and primary resistance Racpr gives:

× Ipeffective = [5]δ

Ip =1

δ

1 ×Pwpr

Racpr

The a.c. resistance of the windings is related to the d.c. resistance by a constant factor FR which is equal to 1 for a d.c. current. The increase of the a.c. resistance value is caused by the skin and proximity effects. Normally these effects strongly depend on the arrangements of the windings in the available winding space and on the frequency of the signals. In literature, methods to calculate the FR value are published (e.g. based on the theory of Dowell). For these calculations computer programs can be helpfully.

• The d.c. resistance Rdc is equal to: Rdc = ρ.Np.Lav/Acopper. • ρ is the copper resistivity in Ωm • Lav is average winding turn length • Acopper is the cross-sectional area of the copper wire • Acopper can be expressed in terms of a copper fi ll factor Fw, which is defi ned as total area of copper divided by total winding area, or in symbols : Fw = N.Acopper / Aw • This gives : Acopper = Fw.Aw / N. • Using the quantity FR and the expression for Acopper one can write for the the a.c. resistance in:

[6]Racprim = FR × Rdc = FR × ρ × Np × lav FR × ρ × Np × lav

Acopper=

Fw × Awprim

2

Substituting formula [6] in formula [5] for the current Ip gives:

× [7]δ

Ip =1 1 ×

Pwprim × Fw × Awprim

ρ × lav × FRNp

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In transformers the following rules apply : Np.Ip = Ns.Is Pw.pr ∝ (Np.Ip)2 and Pw.s ∝ (Ns.Is)2.

Also it is assumed that :

• Pw.pr ≈ Pw.s ≈ ½Pw, where Pw is the total winding losses.

• Fw.Aw.pr ≈ Fw.Aw.s ≈ ½ Fw.Aw

Applying this to [7] gives the primary current in terms of the total winding losses and total area:

× [8]δ

Ip =1 1 ×

Pw × Fw × Aw

ρ × lav × FRNp2

If equation [8] for Ip is substituted in [3] the following expression for the throughput power is obtained:

× Ae × (fB) [9]δ

Pthr =1 ×

Pw × Fw × Aw

ρ × lav × FR2

The same formula is valid for a fl yback transformer, however the derivation is somewhat more complicated due to the triangular current shapes. Due to the infl uence of the airgap (fringing fl ux) the throughput power is about 90 % compared to a forward concept.For a standard push-pull transformer it can be derived that the throughput power is a factor 2 higher compared to the forward transformer. For transformers used in other converter topologies only the factor ½δ might change. So in general the througput power factor is equal to:

Ae × (fB) [10]Pthr ∝ ×Pw × Fw × Aw

ρ × lav × FR

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2. Determination of the optimum fl ux density.

The total loss in a transformer and its temperature rise are related by an analogon of Ohm’s law:

Ptrafo = [11]∆TRth

Measurements on wire wound transformers resulted in an empirical formula for the thermal resistance

Rth = 1/ (Cth.Ve 0.54 ).

The constant Cth depends on core shape and winding arrangement. For standard wound cores like RM or ETD and EFD it is about 17. For planar cores with windings integrated in a PCB (epoxy) it is about 24, and for fl at frame and bar cores used for LCD backlighting the value is about 20. The total transformer losses (in mW) can be related directly to its allowed temperature rise ∆T and effective volume Ve:

[12]Ptrafo = ∆T × Cth × Ve 0.54

The power loss density (mW/cm3) in a core can be determined with the following empirical fi t formula:

Pcore Cm fx

By

ac ct ct1 T Tct2+2

– Cm C T( ) f

xB

y

ac= =× × × ×× × × × [13]

Note that Bac is here the peak fl ux density, so half the total fl ux excursion ½.(Bmax − Bmin). The parameters Cm, x, y, ct, ct1 and ct2 are specifi c for each power ferrite and often only applicable in a defi ned frequency range. Total transformer loss consists of winding loss plus core loss. So, winding loss can be written as:

Pw = Ptrafo − Pcore = − Cm C T( ) fx

B × Vey

× × × [14]∆T × Cth × Ve 0.54

If eq. [14] is substituted in equation [10] the result is a long formula with many terms. Denoting some terms in advance makes it easier to calculate:

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[15]

× Ae × fδ

C1 =2

C3 = Cm C T( ) fx

× × ×Ve × Fw × AwFR × lav × ρ

C2 = ∆T × Cth × Ve 0.54

×Fw × Aw

FR × lav × ρ

Substituting formula [14] in [10] and using the denoted terms [15] gives:

C1 × Bac [16]Pthr (Bac) = × (C2 − C3 × Bac) y

The throughput power is now expressed as a function of Bac only because the other terms are considered as constants. Remark:Bac is the peak fl ux density which is half the value of B as used in equation [10].The value of Bac for which Pthr is maximum, called the optimum fl ux density, can be calculated by solving:

d[Pthr(Bac)]/dBac = 0.

Using formula [16], the result for which this differential quotient equals zero is given by:

= Cm C T( ) fx

Bac × Vey

× × × [17]× ∆T × Cth × Ve 0.542

2 + y

From expression [17] it can be seen that:

[18]22 + y

Pcore =y

2 + yPw = × Ptrafo × Ptrafoand/or

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By isolating Bac from [17] and noting that B = 2.Bac we fi nd for the optimum B:

f−x/y

× [19]× Ve − 0.46/y2

2 + yBopt = 2 × ( × ( )∆T × Cth

Cm × C(T)

1/y1/y)

Bopt is expressed as a function of :

• 6 fi t parameters for the ferrite Cm, x, y, ct, ct1 and ct2 • Allowed temperature rise ∆T

• Constant Cth (17 for wire-wound and 24 for planar transformers)

• The effective core volume Ve • The switching frequency f

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3. Calculation of throughput power for forward and push-pull transformers

The calculations are done for all the combinations of cores and ferrites which are given as input by the selection on core family(ies) and type of ferrite(s).For each core the required information as e.g. Ae, Ve, lav, Aw is read from a data base, which is also done for each ferrite with the constants Cm, x, y, ct, ct1 and ct2.

The fi rst step is the calculation of the optimum fl ux density Bopt with [19] for all the core / ferrite combinations. The required parameters can be read from the data bases or are known because of the input on the application condition.

The second step is the calculation of Ptrafo with [12] followed by the calculation of Pw with equation [18].

The third step is the inserting of the calculated values of Bopt and Pw in [10] to calculate the throughput power factor.The copper resistivity ρ is used as temperature dependent ρ(T) in the following way:

[20]αT = 393 . 10-5 ρT =1 + αT × ( T-20)1 + αT × 76

ρ(T) = ρ(T=100) × ρT

For ρ(T=100) the value of 2.24 .10-8 Ωm is used

Dependent on whether it is a forward or a push-pull transformer a multiplying factor of 2/√δ or 2√2/√δ is used respectively to calculate the throughput power. If the value for a ferrite core is outside the range of the inserted values for minimum or maximum throughput power it will not be mentioned in the range of the determined available cores.

4. Calculation of throughput power for fl yback transformers

The operating principle of a fl yback transformer differs from that of a forward or push-pull type. In a fl yback transformer magnetic energy is stored during the active part of the duty cycle. The energy is transferred to the secondary side during the off period of the duty cycle.

The energy density (J/m3) of a magnetic fi eld is: E = ½.B.H

The amount of energy Eind ( J ) stored in an inductor is:

[21]Eind = ½ B × H × Ve = × VeB2

2µ

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Writing out eq.[21] for the ferrite part (Ve = Ae.le ) and the airgap (VLg = Ae.lg) gives:

[22]Eind = 2 × µ0 × µa 2 × µ0

B2 × Ae × le B2 × Ae × lg2 × µ0

B2 × Ve + = × ( )+1µa

lgle

The throughput power Pthr = Eind/T = Eind . f becomes with [22]:

[23]2 × µ0

B2× f × Ve × ( )+1µa

lgle

Pthr(lg) =

The fringing fl ux has the effect of increasing the airgap area. The reluctance Rg, which is equal to lg/µ.Ag, decreases by the fringing fl ux factor F. Then Rg becomes lg/(µ.Ag.F). For the calculations it is more convenient to keep a constant cross-section Ae for the total circuit. Then the effect of the fringing fl ux on the effective airgap lg is accounted for by deviding lg by a factor F.

A formula for the fringing fl ux factor F can be derived with conformal transformations and is equal to:

× In ( [24]Ae

F = 1 +lg

lg

2 × Wh )

where Wh is the height of the winding window.

The throughput power for a fl yback inductor choke with fringing is equal to:

[25]2 × µ0

B2× f × Ve × ( )+1µa

lg/Fle

Pthr(lg) =

The throughput power capability of a ferrite core is (for a fl yback) equal to:

× Ae × (fB) [9]δ

Pthr =1 ×

Pw × Fw × Aw

ρ × lav × FR2

Expression [9] can be written out in three steps by: • Substituting for Pwinding in eq.[18] : Pwinding = [y/(2+y)].Ptrafo • Substituting for Ptrafo in [18] : Ptrafo = ∆T/ Rth • Substituting for B in [9] the optimum value from eq. [19]

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The result of these substitutions gives for the throughput power capability:

[26]f(1−x/y) (0.27−0.46/y)

× y½

× Ae × Ve 22+y( ) ( )Cth

CmC(T)

1/y (1/y+½) (1/y+½) ×δ

Pthr =1

× ×× ∆TAw( )Lav

½×

Fw( )ρFR

½×

1 2 3 4

Remark :For a better understanding of this long formula it is useful to have a closer look at its different parts. 1. Circuit dependent part 2. Ferrite dependent part, important is to see how Pthr depends on: - frequency : Pthr ∝ f (1 − x/y)

- temperature : Pthr ∝ ∆T(1/y + ½)

3 Core dependent part, known as the core performance factor 4 Winding dependent partThe throughput power in eq. [9] or [26] can, in theory, be equal to the value from eq.[25] when the airgap lg is given its maximum size. The maximum airgap can be calculated by reworking [25] and using Pthr from eq. [9] or [26] denoted now as Pthrmax:

[27]2 × µ0

Ae× B × f ×Pthrmax − (( )) le

µalgmax =

Remarks :1. Fringing factor F is ignored2. The maximum airgap is in principle strongly dominated by the allowed temperature rise ∆T, and is almost lineair with ∆T.3. If a larger airgap than lgmax is used the allowed temp. rise ∆T will be exceeded. Magnetically this means that the transformer is not operating with the optimum fl ux density as stated in [19]. Larger airgaps lead to a higher fi eld strength H, which means higher currents and winding losses. In this case the optimum equilibrium between core and winding losses is disturbed.4. Even if lgmax is used for the airgap, Pthr will always be somewhat lower (10 to 15 %) than calculated with eq. [26], due to fringing effects, and additional losses in the windings due to fringing fl ux.

The sequence to calculate Pthr for a fl yback transformer is:

• The steps as mentioned in chapter 3; The calculation of Pthr for forward and push-pull transformers. The values obtained from eq. [10] are interpreted as Pthrmax.

• The calculated values for Bopt are used to calculate values for µa. For this purpose the graphs µa(B) from our Data Handbook, are approximated by 4 straight lines for each power ferrite.

• For each selected ferrite core Pthrmax and other data is used in eq.[27] to calculate lgmax. These are the maximum values that can be used for the airgap.

• The program uses eq.[25] to calculate Pthr for a certain airgap lg. If the desired value for the airgap is too large for certain cores, automatically the value lgmax is used.

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5. Option of Graph Pthr(f) for forward and push-pull transformers

Select one of the displayed cores and click on the button Pthr(f) for a graph of througput power as a function of frequency.

The graph is generated by stepping through the following procedure at several frequencies in the operating range of the applied ferrite.

• Optimum fl ux density Bopt with eq. [19]

• Calculation of Ptrafo with eq.[12]

• Calculation of Pwinding with eq.[18]

• Inserting the calculated values of Bopt and Pwinding in eq.[10] to calculate the throughput power and fi nally multiplying it with 2/√δ for forward or 2√2/√δ for push-pull transformers.

6. Option of Graph Pthr(f) for fl yback transformers

For this calculation it is assumed that for the selected core:

• Airgap lg is constant

• Number of turns is fi xed

• f and δ can change (to f ’ and δ’) but, they have to be balanced that new values B’ and I’ are ensuring still Pcore + Pwinding = Ptrafo. So the control loop has to ensure that f and δ are kept in balance.

Suppose a transformer was optimized for nominal frequency fn which fi xed Np, Lg and δn (usually 0.5).This can be written as:

[28]Pcore (fn, δn) + Pw (fn, δn) = Ptrafo

Consider the following period T for the fl ux density:

• From t = 0 to δprim the fl ux increases from 0 to Bmax.

• From t = δprim to δsec the fl ux decreases from Bmax to 0.

• From t = (δprim + δsec) to T, the fl ux remains 0.

The following assumptions are than usually justifi ed:

[29]δprim ≈ δsec ≈ δ and 1−δ ≈ 2δ

[30]feff = = 1

2δTf

2δ

13


[31]at fn: 2δn ≈ 1

Using equations [29], [30] and [31] leads to:

[32]Pcore (f, δ) = const. × feff × By (1-d) = const. × ( ) × By × 2δx x

2δf

[33]Pw (f, δ) = const. × I2 × (1-d) = const. × I2 × 2δ

Airgap lg and Np are already fi xed. And because V is fi xed and since V = LdI/dt ~ dB/dt the slopes I(t) and B(t) are fi xed as well:

• For f, δ the slope B(t) = B(f,δ)/δT

• For fn, δn the slope B(t) =B(fn,δn)/δnTn

[34]B (f, δ) = B (fn, δn) × ( ) = B (fn, δn) × ( )δnTnδT

fδ

δnfn×

Using expression [34] for B(f,δ) in eq.[32] to fi nd the core losses Pcore(f,δ) gives:

[35]Pcore (f, δ) = const.× ( )x × [B (fn, δn) × ( )]

y× 2δ

2δf

fδ × δn

fn

Equation [35] can be expanded to eq.[36]:

[36]Pcore(f,δ) = const.×( )

x× B (fn,δn)

y× 2δn ×( )

x×( )

x×( )

y×( ) 2δn

fn δ.fnδn.f

2δ2δn

2δn2δfn

f

Pcore (fn, δn)

By using the substitution for Pcore(fn ,δn) in [36] it can be written as:

14


Pcore(f,δ) = Pcore (fn, δn) × ( )y-x

× ( )1-x+y

ffn δ

δn [37]

The winding losses can be expressed in the same way. Since dI/dt ≈ dB/dt the function for I(f ,δ) is similar to B(f,δ) in equation [34]:

I (f,δ) = I (fn, δn) × ( × ) δnfn

fδ [38]

Using expression [38] for I(f,δ) in eq. [33] to fi nd the winding losses Pw (f,δ) gives:

Pwe(f,δ) = const.[ I (fn, δn) × ( × )]2

× 2δ δnfn

fδ

[39]

Equation [39] can be reworked to eq. [40] and [41]:

Pwe(f,δ) = const. I (fn, δn)2 × 2δn × ( )3 × ( )2 δn

δffn

[40]

Pwe(f,δ) = Pw (fn, δn) × ( )3 × ( )2 δnδ

ffn

[41]

For the nominal case (fn ,δn) as well as for another (f ,δ) this yields:

Pcore(fn,δn) + Pw(fn,δn) = ( ) Ptrafo + ( ) Ptrafo = Ptrafo2+y2

2+yy

[42]

Pcore(f,δ) + Pw(f,δ) = Ptrafo [43]

15


Substituting equations [41] and [37] in eq. [43] and comparing with eq. [42] gives:

Pcore (fn, δn) × ( )y-x

× ( )1-x+y

ffn δ

δnPw (fn, δn) × ( )3 × ( )2

= δnδ

ffn

+

( ) × ( )y-x

× ( )1-x+y

ffn δ

δn ( ) × ( )3× ( )2× Ptrafo= Ptrafoδnδ

ffn

+2+y2

2+yy× Ptrafo

[44]

From equation [44] follows that:

( ) × ( )y-x

× ( )1-x+y

ffn δ

δn2+y2 ( ) × ( )3× ( )2= 1

δnδ

ffn

+ 2+yy

[45]

This equation can only be solved numerically. The parameters y, fn and δn are known. Frequency f has to be chosen and δ can be calculated (or approximated).Using eq. [23] and [34], the throughput power as a function of f and δ can be written as:

[46]Pthr(f,δ) = const × f × B(f,δ)

2= const × f × B(fn,δn)

2× ( )2

= Pthr(fn,δn) × ( )× ( )2

ffn

δnδ

fδ

δnfn×

The duty cycle δ is always ≤ 0.5. It is assumed that the nominal duty factor δn has the optimum and maximum value of 0.5 at the nominal frequency fn. Two situations can be distinquished for the new frequency f:

f > fnIn this case the new duty cycle δ cannot increase above δn but will stay on the maximum value of δn = 0.5 Using eq. [23] and [46] shows that in this case we can write the new fl ux density B(f,δ) and Pthr(f,δ) as:

[47]B(f,δ) = B(fn,δn) × ( )Pthr(f,δ) = Pthr(fn,δn) × ( )f

fnffn

16


f < fnFor this situation the new duty cycle δ can be lowered by the control circuit with respect to the maximum nominal value of δn = 0.5. The new duty cycle δ can be calculated by solving numerically equation [45]. By using eq. [23] and [46] again with δn = 0.5 the new fl ux density B(f,δ) and Pthr(f,δ) can be written as:

[48]B(f,δ) = B(fn,δn) × 2δ × ( )Pthr(f,δ) = Pthr(fn,δn) × (2δ)2 × ( )f

fnffn

Equations [47] and [48] are used to calculate Pthr(f) for a fl yback transfomer.A rough estimation can be made to give some insight in the curve of Pthr as a function of frequency.The left term in eq [45] is constant if:

( ) = ( ) y−x

≈ ( )− (0.45...0.60)

δnδ

ffn

ffn1−x+y [49]

The right term in eq. [45] is constant if :

( ) = ( ) 2

≈ ( ) − 0.6

δnδ

ffn

ffn3 [50]

So, the solution is:

( )

≈ ( ) − 0.6

δnδ

ffn [51]

For a given f, the δ is adapted to:

δn = δn × ( )

− 0.6

ffn

[52]

Using [52] in eq. [46] the approximation for Pthr in the situation : f > fn can be seen as follows:δ cannot increase above δn. So, Pthr which is a linear function of (f.B2) decreases. Because B decreases with 1/f also Pthr(f) will decrease with 1/f.

17


Using [52] in [48] the approximation for Pthr in the situation: f < fn is:

Pthr(fn,δn) = Pthr(fn,δn) × ( )

− 0.2

ffn

[53]

So, for frequencies lower than nominal, Pthr(f) increases with about approximately f 0.2 up to the maximum for Pthr(f) at fn. For frequencies higher than nominal, Pthr(f) decreases with 1/f.

1

A

E F

K

J

A Choose the type of inductor to design.B Choose the type of coil former to be used.C Choose the type of magnetic circuit to be used for the design.D Fill in the design parameters and limitations.E Make a choice from the available core shapes.F Choose a ferrite material from the list.G Shows the ìeffectiveî initial permeability for this core / ferrite combination.H Shows the usual surface Ý nish for this core / ferrite combination.

I Press ìGoî button to start the calculation.J Shows the minimum core size suitable for the present design complete with effective core parameters.K Overview of all relevant properties of the Ý nished design.L Press ìDatasheetî button to open PDF Ý le of the data sheet of the resulting core type.

Inductor Design

B

C

D

G

H

I

L

Inductor design

2

Inductor designIntroductionThe design of an inductor is often done by trial and error. There are many parameters to play with. Current and inductance value are usually Ý xed, but there is a choice of core types, materials and airgap lengths to realize the design with.

This program greatly simpliÝ es the design procedure by checking all cores in a range for magnetic saturation and overheating. Then it proposes the smallest possible core size Ý tting all boundary conditions for the design.

In common-mode chokes, the 2 windings generate counteracting Ð uxes of the same magnitude. The resulting Ð ux is practically zero, there is no danger of saturation so airgaps are not required.However, because of the safety isolation requirements there is a certain distance between both windings. This allows some stray Ð ux to escape from the core before passing through the other winding. This stray Ð ux is not compensated and will cause saturation at very high current levels. The program accounts for this effect and proposes the smallest core for the design.

Design procedure

Inductor designThe program starts by checking the cores from the chosen range for energy storage capability using the following relations:

[1]

---------------------------Ve min = µe × µ0 × E

Bsat2

Energy stored: E = Imax × L2

The saturation Ð ux density Bsat depends on the ferrite material and the temperature of the core. Initially, the ambient temperature is taken. The effective permeability µe is equal to the material permeability µi in case of a closed core without mating faces or airgaps. In a gapped core it is a variable, mainly controlled by the length of the airgap.The iteration start is different for closed and gapped cores.• Closed core µe = µi, so Ve.min follows from E and deÝ nes the Ý rst core of the range to check.• Gapped core The iteration is started with the smallest core of the range, so µe follows from E and the corresponding core volume Ve.

When a core with the required Ve is found, the winding is designed by Ý rst calculating the number of turns N from :

Inductor design

3

[2]L = µe × µ0 × N × Ae2

le

Then the conductor length (lcopper) and cross-section (Acopper) are derived from the available winding window (Aw), copper Ý ll factor (Fw) and the average turn length (lav).

[3]Acopper = Fw × Aw

N

lcopper = lav × N

This gives the total winding resistance :

[4]R = ρ × lcopper

Acopper

Since the resistance of copper is a function of temperature, the program takes that into account with the following formula :

[5]ρ = ρ20˚C × [ 1 + α ( Toperating − 20 )]

where α = 0.0039 / °C.The winding losses can now be calculated with :

[6]Pwinding = Ieff × R2

Usually this loss is the predominant cause of heat dissipation in the inductor. The temperature rise of the inductor design is calculated from :

[7]∆T = Toperating - Tamb = Pwinding

Rth

Rth is the thermal resistance of the inductor which is a function of Ve of the ferrite core and is calculated with the following empirical equation:

Inductor design

4

[8]Rth =Cth . Ve

0.54

1

The constant Cth depends on core shape and winding arrangement. For standard wound cores like RM or ETD and EFD it is about 17. For planar cores with windings integrated in a PCB (epoxy) it is about 24. The final operating temperature of the proposed design can be found by combining eq. [4] to [7], resulting in a linear equation for Toperating.With this new temperature, the core is checked for saturation again, using the Bsat value at this temperature. Bsat(T) is approximated by a piecewise linear function. Also the other boundary conditions like Tmax and RDCmax are checked. If one of these conditions is not fulfilled the core is found not suitable and the core with the next larger Ve is taken for a new design cycle. The first core size fulfilling all boundary conditions is proposed for the the inductor design. All necessary core parameters such as airgap length and µe are given in the output window. For the calculation of this airgap the effect of fringing flux is taken into account. Also the resulting winding design is presented together with power dissipation and temperature rise.In order to fine-tune the design, the core size can be fixed to find out what happens when a parameter is adapted manually.

Inductor design

5

Common-mode Choke

Design procedureThe Ý rst step in the design procedure is to estimate the non-compensated proportion of the Ð ux. The amount of stray Ð ux is controlled by the geometry of the windings and the number of turns only. The total generated Ð ux is controlled by the same parameters but also by the core permeability. This means that the fraction of non-compensated Ð ux only depends on the core permeability. It is approximated by a multi-step function.

This non-compensated Ð ux fraction s is used as input for the design procedure using the same steps and largely the same formulas as for the normal inductor design.• The stored energy is reduced

[1a]Energy stored: E = s × Imax × L2 2

• The winding space is divided

[3a]Acopper = Fw × Acoil

N

Acoil = (Aw − Aiso) / 2

The area Aiso follows directly from the safety isolation distance diso.• The losses double

[6a]Pcoil = Ieff × R

2

Pwinding = 2 x Pcoil

Inductor design

1

Magnetic Regulator Design

Magnetic Regulator

A B

C

AMake a choice from the available toroid sizes

BInsert design parameters

CGraph explains the meaning of the abbreviations used in the output window

DA list with results for different numbers of turns is presented in the window

D

2


The magnetic regulator calculation formulas

Hc

-Hc

∝ td

∝ tb

∝ tblBda=Vin·tb/Ae·N

Bsr=Vin·td/Ae

∆B=Vin·tbl/Ae·N

Br

Bs

Ireset=Bda⋅Hc⋅le/(BrN)

The picture above shows the origin of the terms used in the equations that describe the fi elds and delay times in the regulator. A summary of the formulas that are valid for the different parts of the BH-loop are quoted below :

( )bloninoutttVV −⋅=⋅T

[1]

blinetVNAB ⋅=⋅⋅∆

[2]

binedatVNAB ⋅=⋅⋅

[3]

dinesrtVNAB ⋅=⋅⋅

[4]

( ) 235.0235.011

eoutsrlINB ⋅⋅=

[mT] [5]

( ) 3250/T0048.013.17.1

⋅−⋅∆= fBPcore

[mW/cc], B in mT, f in kHz.[6]

( )avCublonoutwlNttfIP ⋅⋅⋅−⋅⋅⋅⋅= ρ12

108 [mW], f in kHz[7]

etotalVP ⋅⋅∆= 17T [mW] for V

e in cc, [W] for V

e in m

3

.[8]

These are the basic equations for a magnetic regulator in which we shall now step by step insert the known input variables.

List of input variables :core magnetic regulator core sizeVin Input voltage Vout Output voltageIout Output currentf , T Frequency or cycle periodton on-time of the input voltage, related to the duty factor of the input cycleTa Ambient temperature∆T Maximum temperature rise

3


Some of the input variables are related to each other:

ton = δ . T and f = 1 / T

The fi rst step in the magnetic regulator program is to calculate the blocking time tbl.From equation [1] and the relation above it can be shown that :

TV

Vt

in

out

bl⋅

−= δ

[9]

The blocking time is needed when we want to calculate the minimum number of windings Nmin.First we introduce some constants for convenience :

eV⋅=′ 17αT∆⋅′=αα

( )avCublonoutlttfI ⋅⋅−⋅⋅⋅⋅= ρβ 12

108

( )a

efV

T0048.013250

3.1

⋅−⋅⋅

=′′γ

0048.03250

3.1fV

e⋅

=′′′γ

( ) T3250/T0048.013.1

∆⋅′′′−′′=⋅−⋅⋅=′ γγγ fVe

γγ ′

⋅⋅=

7.13

10

e

blin

A

tV

We can now express the formulas for the power losses in a more compact form :

α=

totalP NP

w⋅= β 7.1

BPcore

∆⋅′= γ

from :

Ptotal = Pw + Pcore

Where Pw represents the winding losses,we fi nd for the total fl ux swing :

7.1

1

'

⋅−=∆

γβα N

B[10]

By inserting [10] in equation [2] we can derive an expression for Nmin :

4


07.2

min

7.1

min=−⋅−⋅ γβα NN [11]

The computer iterates Nmin until [11] > 0.

Inserting Nmin in [10] gives the maximum fl ux swing. The maximum fl ux swing is limited to 2 Br.

The following step is to calculate Nmax. How many turns can we wrap around the chosen core ?For the calculation of Nmax we assume that the current density J = 8 A/mm2

ππ 2

82

4

1

out

Cu

Cu

outI

dd

IJ =⇒== [12]

The effective diameter is calculated with an empirical formula for winding cores. A 5% isolation and pitch of the winding is incorporated in this equation :

921.014.105.1

Cueffdd ⋅⋅=

[13 a]

For a core with inner diameter di it can be shown that :

−⋅= 1

max

eff

i

d

dN π

[14]

Note that dCu from [12] or [13] is a minimum value. The maximum value is calculated with :

( )ππ +⋅= Nddieff max, [13b]

The maximum diameter is limited to one quarter of the inner diameter.There is one more restriction to Nmax. Increasing N means that Bsr is increasing. We are applying a larger H fi eld. Bsr is causing the delay time td. We do not want td to exceed tbl. The number of turns for which this happens is when td=tbl. With [4] and [5] we can calculate Nmax :

( )

235.11

.2350max

011.0

⋅⋅

⋅=

el

I

blin

A

tVN

e

out

[15]

For Nmax we choose the minimum of [14] and [15]

5


Finally the program iterates N from Nmin to Nmax and calculates ∆B from [2], td from [4] and [5], tb from tbl-td, Ireset and the temperature rise. Ireset is calculated by :

da

r

ec

resetB

NB

lHI ⋅

⋅

⋅=

[16]

Iteration from Nmin to Nmax gives various values for ∆B from [2] which result in a specifi c temperature rise.The temperature rise can be calculated by rewriting [10].

⇒∆⋅′′′−′′

⋅−′⋅∆=∆

T

T7.1

γγβα N

B

7.1

7.1

T

B

NB

∆⋅′′′+′

⋅+∆⋅′′=∆

γαβγ

[17]

List of input variables :Nmin minimum number of turns [from eq. 11]Nmax maximum number of turns [from eq. 14 & 15]dCu,min minimum wire diameter [from eq. 13a]∆Bmax maximum fl ux swing [from eq. 10]

Output results from iteration Nmin to Nmax∆B fl ux swing [from eq. 2]dCu,max maximum wire diameter [from eq.13b]Ireset reset current [from eq.16]td time Br to Bs [from eq. 4 & 5]tb time Bda [tb = tbl-td]tbl total blocking time [from eq. 9]∆T temperature rise [from eq. 17]

1

Power Loss Calculations

A Enter the switching frequency

BMake a choice of ferrite material

CEnter the AC peak fl ux density

DEnter the DC bias fl ux density

E Make a choice between sine wave and customized waveform

FPress to draw a graph (up to 4 per screen)

GPress to clear the graphs

AB

C

D

E

FG

Power Loss Calculation

2


Power Loss Calculation

Introduction With this program it is possible to calculate the power loss density for Ferroxcube power ferrites under various conditions. The fl ux density signal can be an arbitrary wave shape or a sine wave, with or without a bias fl ux. The calculations are based on fi t formulas describing the power loss density as a function of frequency, fl ux density and temperature. In the program the desired operating frequency (in kHz) or cycle period (in µs) and magnetic fl ux densities (in mT) can be given as input. The program calculates the losses in the temperature range from 0 to 140 °C for the selected power ferrite. The program protects the user for wrong choices. Once a frequency is given as input, only those ferrites can be selected which are intentended for use at that frequency.The fl ux density cannot be chosen higher than the saturation level of the selected ferrite. Theory behind the fi t formulas and part of the fi t parameters can be found in the references.

1.Basic formulaThe calculation is based on an empirical fi t formula for the core loss density of power ferrites:

Remarks • The constants are based on measurements with symmetric sine waveforms (no bias). • Bac is the peak fl ux density, so half of the total fl ux excursion : ½.(Bmax − Bmin) • The parameters Cm, x, y, ct, ct1 and ct2 are specifi c for each power ferrite and often applicable only in a limited frequency range. • The temperature dependent coeffi cients ct, ct1 and ct2 are dimensioned in such a way that at T = 100 °C the temperature term (ct - ct1.T + ct2.T 2) is equal to 1.

3


2. Calculation for sine waveIf there is no DC fl ux density (bias), the basic formula [1] can be applied directly to calculate the power loss density in kW/m3 which is identical to mW/cm3.With a DC fl ux density a correction is required. For this purpose measuring data from reference [2] is used. For several power ferrites the power loss density at defi ned values of Bac and Bdc is given there.The relative increase of the losses at Bac+ Bdc levels with respect to corresponding levels without bias is calculated from this data. The maxtrix obtained in this way consists of the elements: Pcore(Bac,j ,Bdc,i)/Pcore(Bac,j ,Bdc = 0).

The determination of the loss density on a arbritrary point of (Bac, Bdc) is done in four steps:

1. First the Bdc level under consideration is rounded to the nearest Bdc value available in the matrix.

2. The Bac level under consideration is rounded to the nearest lower and higher Bac value available in the matrix . A lineair interpolation between these 2 Bac values is made to arrive at the desired Bac level.

3. Finally with this rounding of Bdc and interpolation of Bac the approximated value of PV(Bac,Bdc)/PV(BacBdc = 0) for the levels of Bac, and Bdc under consideration is obtained.

4. The obtained correction factor [1 + PV(Bac,Bdc)/PV(BacBdc = 0)] is multiplied by the power loss density for the same Bac but without bias.

3. Calculation for custom waveformFor an arbritrary waveform the power loss density can be approximated as described in the following chapters:

3.1. Sine wave equivalent of an arbritrary waveformThe arbitrary waveform is approximated by a maximum of ten pieces of straight lines. Each interval is transformed to an “sinus frequency equivalent”

The method used is partly described in reference [3] and makes use of the sum of the weighted time derivative of B which is defi ned as:

In words: the sum of the time derivative × weight factor.

4


Example : For a symmetric (no bias) triangle waveform with its maximum Bm at ¼T, zero at ½T, −Bm at ¾T and zero again at T, Bw can be calculated with eq. [2] as:

------------------------------------------------------------------------------------------

For a symmetric sine wave with the same period and values for Bm the calculation of can Bw can be done by writing it in a integral representation:

Comparing the end result of equations [3] and [4] shows that the sine wave equivalent frequency of the triangle form can be given as:

A piece of line representing ¼ part of a complete triangle period T is denoted as ∆Ti:

Inserting eq. [6] in eq. [5] gives the expression for the sinus equivalent frequency of a line piece with period ∆Ti :

Example :Suppose ∆ Ti = 2.5 µs. From eq. [7] follows that fsin.eq = 81 kHz.∆ Ti = 2.5 µs corresponds to T = 10 µs, so by using eq. [5] we also fi nd fsin.eq. = 81 kHz. A sine wave with T = 10 µs corresponds with fsin = 100 kHz

5


3.2 Division of a waveform into pieces of straight linesThe division of fl ux levels into periods of ∆Ti is done in the following steps:

1.First the DC level of the fl ux waveform Bdc is determined. This is done by calculating numerically:

2.The begin and end points of the pieces of fl ux lines are now (numerically) placed at the : • Corners, where the inserted wave form changes from increasing B to decreasing B and vice versa. • Intersections of the inserted wave form with the calculated value for Bdc.

3.3. Calculation of total power lossFor all periods ∆Ti, which are constructed as explained in chapter 3.2, the equivalent sine wave frequency is calculated with eq. [7]: fsin.eq = 2/(B²∆Ti).The corresponding fi t parameters (Cm, x, y, ct, ct1, ct2) valid for the calculated frequency fsin.eq are now read from the data base.

The power loss density contribution for each period is now calculated by inserting eq. [7] in eq.[1] and multiplying it by its weighting factor ∆Ti / T

The total power loss Ptotal( J ) is calculated by taking the sum of the obtained values with formula [9]:

If there is a DC fl ux density the same correction will be made as explained for the sine waveform in chapter 3.1 . The DC fl ux density is already calculated as in chapter 3.2. As AC fl ux density the maximum value of the waveform with respect to the DC fl ux Bdc is used. The total losses are in this case:

6


References

[1] Mulder S.A., 1993, “Loss formulas for Power Ferrites and their use in transformer design”, Application note Philips Components.

[2] Brockmeyer A., 1995 “Experimental Evaluation of the infl uence of DC Premagnetization on the properties of Power electronic ferrites”, Aachen University of Technology.

[3] Durbaum, Th, Albach M, 1995, “Core losses in transformers with an arbitrary shape of the Magnetizing current”, 1995 EPE Sevilla.

1

A

E F

A Choose Ferroxcube cores or user deÞ ned cores.

B Make a choice from the available core shapes.

C Choose a core size.

D Choose a ferrite material from the list.

E Choose an operating temperature

F Fill in core parameters for a customized core design

G Choose an effective permeability, inductance fator or gap length. After pressing ÒenterÓ the other windows are updated and show the calculated results.H Shows the resulting plots (after pressing ÒenterÓ).I The graph can be exported or stored.

Power Inductor Properties

B

C D G

HI

2

Power Inductor PropertiesThe power inductor properties program is a tool to explore the properties of a power inductor in more detail. One can select various materials for a core shape and it is possible to defi ne an effective permeability, inductance factor or gap length. All properties are then calculated and displayed in graphs. The plots give more insight in how the inductor will behave as a function of fi eld strength as well as the energy storage as a function of gap length.

BH-loop

The BH-loop of our power materials were measured at three temperatures at fi eld strengths from –250 A/m to 250 A/m. The measurements were carried out on cores without gap. With this design tool one can calculate the gap length required to get the given effective inductance or permeability or visa versa, using the relations:

e

e

eL

l

AA ⋅⋅= µµ

0 [1]

FAl

Al

e

eg

ie⋅⋅

⋅+=

g

11

µµ[2]

⋅+=

g

leg

e

g

l

l

A

lF ln1

[3]

2

The factor F is the contribution of the fringing fl ux to the effective permeability.When a gap is introduced, one can recalculate the BH-loop by re-scaling the applied fi eld H.Ampere’s law for a gapped core can be written as:

ggefgggefeap

ap

lHlHlHllHlHNI ⋅+⋅≈⋅+−⋅=⋅= )(

g

eg

fg

g

gg

A

AlBl

BlH

⋅

⋅⋅==⋅

00µµ

HHlA

AlBHH

l

NIf

eg

eg

ff

e

∆+=⋅⋅

⋅⋅+==

0µ

[4]

Hf = fi eld strength in the ferrite,Hg = fi eld strength in the gap,Bf = fl ux density in the ferriteBg = fl ux density in the gap∆H = extra fi eld strength to compensate for reluctance of gapHap = applied fi eld strength,

3

So, for the gapped core an extra fi eld ∆H is needed to reach the same level of B compared to a non gapped core.

original BH loop without gap

sheared BH loop with gap

-Hc

Br

Br(gap)

∆Hpoint (∆H, Br)

Using equation [2] we can rewrite equation [4] as:

−⋅+=

ie

ffBHH

µµµ

111

0

[5]ap

This enables the program to draw a so-called sheared BH-loop for any gap.

Incremental permeability versus H

The incremental permeability is defi ned as the slope of the BH-loop at given fi eld strength H.The BH-loop can be divided into an upper loop and a lower loop. For each value of H there are 2 values of B. In the calculation of H according to [5] the average of these two B values is used.

−⋅+

−⋅+=

ie

lowerf

ie

upperffBBHH

µµµµµµ

111111

0

,2

1

0

,2

1 [6]ap

The value of B of the upper and lower loop is also averaged:

)(',f,f2

1

upperlowerBBB +=

[7]f

The slope of B’f(Hap) gives the incremental permeability µ∆:

Power Inductor properties

4

H

B

∂

′∂=

∆µ [8]

ap

f

The plot produced by the program gives an indication of the fi eld strength for which the permeability of the selected ferrite material collapses. This happens when the gapped core is saturated at the given operating temperature.

AL of a gapped core versus NI

The behaviour of the AL of the gapped core set can now be calculated from the incremental permeability:

e

e

L

l

AA ⋅⋅=

∆ 0µµ

[9](gapped)

From the average fi eld strength H’ [6] the program calculates NI:

elHNI ⋅= [10]

ap

I²L versus NI

Important design parameters for a power inductor are inductance value (L) and the maximum current (I). This is related to the required energy storage per cycle in the inductor, which can be expressed as ½I²L. Following the information given in the application section of the FERROXCUBE data book, the program plots I²L as a function of NI.

=elHLI222

[11]LA (gapped)⋅

ap

Important design information can be taken from these graphs. The peak in the curve gives an indication of the maximum energy storage in the core. If this is too low for the design a larger ferrite volume or another material is obviously required. The value of NI below the peak gives the optimal number of turns.

Power Inductor properties

1

Transformer Design

In switching power supplies, problems often arise with the design of the mag-netic components, the ferrite-cored transformers and chokes. The interaction between electronic and magnetic design deserves particular attention. Firstly, in switched-mode power supplies (SMPS), the power transformer and choke, and the electronic circuitry are so interdependent that design is hardly possible without the magnetic aspects being constantly taken into account. Secondly, by combining magnetic and electronic design, a far better insight is gained into the operation of the circuit, with a consequent improvement in the design itself. In the following sections these problems are tackled. Design procedures are explained with the help of many useful formulas and graphs.

Part 1 - Magnetic components functional requirementsPart 1 covers most aspects of switching power supply design, with emphasis on the magnetic aspects. Here, the basic electrical relationships for SMPS are given for forward, push-pull and fl y back converters. Practical formulae are given for inductance and effective-current values; auxiliary outputs and other special features are also covered. Some aspects of control are treated. All treatments are related to the magnetic design.

Part 2 - Selection of suitable cores for a transformer designPart 2 deals with the design of the magnetic components themselves, especially the selection of the appropriate core.

Part 3 - Transformer winding designPart 3 deals with the design of transformer windings

Part 4 - Power inductor designPart 4 concentrates on the design of power chokes

Each Part contributes to an overall, step-by-step design procedure. Use of the pro-cedure requires only a general electronic engineering background. Before entering intothe procedure, a choice of converter type must have been made.

2

Part 1

Part 1 - Magnetic components functional requirements

1.The forward converter

1.1 Non-isolated

Principle of operation

Figure 1 shows the outline circuit of a forward converter. The basic operation of an ideal converter is described by: Vo = δVi

ViD

S L

C Vo Io

+

-

+

-

Fig.1 Outline circuit of a forward converter.

Figure 2 shows the voltage waveform across the inductance and the associated current waveforms.

Fig. 2 Voltage waveform across the power choke in a forward converter and the associated current waveform.

1/f

δ/f (1−δ)/f

IL

VL Vi - Vo

Vo

time

time

2Iac

Io

3

Part 1

Minimum choke inductance For continuous-mode operation (uninterrupted current fl ow through the choke) - otherwise regulation deteriorates-output current Io must always be greater than half the choke ripple current 2Iac. This is ensured by using a minimum value for the choke inductance

Lmin = 1 Vo ( 1- δmax Vi min ) [2]

2f Io min Vi max

An increase in δ as a result of a sudden load increase will cause a temporary increase in choke ripple current. As long as the ripple component is much smaller than the d.c. component, which is usually the case, this does not affect the design of the choke.

Choke design Output choke L carries a direct current equal to the d.c. current in the load. Thus, to avoid saturation, an air gap is required in the core. The design steps are:

- determine IM = σIo max + Iac - calculate I2MLmin - proceed to Part 4 of this series.

For core loss, see Part 2.

Deriving auxiliary power from the choke During the fl ywheel period (1-δ)/f, that is, while the power switch is not conducting, the voltage across the choke is stabilized. By adding secondary windings to the choke, auxiliary, stabilized, low voltages can be obtained as shown in Fig. 3. These auxiliary voltages are rectifi ed by diodes that conduct during the fl ywheel period. Since auxiliary loads decrease the amount of energy recovered by fl ywheel diode D, the amount of auxiliary power that can be derived is limited to 20 to 30% of the total output power.

Fig. 3 Auxilliary low-voltage outputs can be obtained by adding windings to the choke of a forward converter.

V1 aux

Vo

V2 aux

D C

0

4

Part 1

In order to store suffi cient energy in the choke to supply an auxiliary load, inductance Lmin calculated from Eq. 2 must be increased to Laux. This might lead to the use of a larger core: see Part 4. The relationship between Lmin and Laux is

Laux > Lmin [4]

1− δmax1 −

(0.3 to 0.4)

The factor (0.3 to 0.4) corresponds to an auxiliary load being 20 to 30% of the total. If output ripple is not important, a higher proportion of auxiliary load can be drawn. The turns ratio

r ≈ Vo = ND [5]

Vaux Naux

During forward conversion, during the period δ/f the input power is: Pin = δViIo Similarly, the throughput power is: Pth = δVoIoThe difference is the power stored in and then removed from the choke: PL = Pin − P = δIo ( Vi − Vo )

From Eq.1: PL = δIo ( Vi − δVi ) = δViIo ( 1 − δVi ) = Pin ( 1 − δ )

If, during fl ywheel period (1- δ )/f, the choke current is divided between output and fl ywheel diodes as indicated in Fig.4 by line a, Paux is maximum, Paux max . The current below a is that through the fl ywheel diode of the primary output. Fig.5 shows some waveforms. Due to leakage induction, fl ywheel diode current does not begin at A, but at Imax as the auxiliary output currents are then zero. This decreases the value of Paux rnax . If the current is shared according to line b in Fig.4, Paux = kPaux max, where k = 0.5. In practice, k will be somewhat higher: a value of k = 0.7 is reasonable.

Fig.4 Current through a forward converter choke with auxiliary windings.

(1−δ)/f

time

I

Imax

L = Lmin

L = Laux

A

k = 0.7

a (k=1)

a (k=0.5)

5

Part 1

If L = Lmin, no auxiliary power can be drawn. From these considerations it follows that the auxiliary power that can be obtained is limited and depends on k, δ and L, such that

Paux ≤ k ( 1 - δ ) ( 1 - Lmin

) PauxLaux

The required value of Laux is obtained from

Laux > Lmin

k Ptot (1− δmax )1 −

Paux

Substituting k = 0.7 and Paux / Ptot = 0.2 to 0.3 yields Eq.4.

Ptot = Pth - I0 ( VF + VR + Vo )

Further discussion may be found in Ref. 1.

Fig.5 Oscillogram of choke waveforms (a) L> Laux and (b) L < Laux>

a b

V

I

6

Part 1

1.2 Single forward converter with isolating transformer

Principle of operation The outline circuit of a forward converter with mains isolation is given in Fig.6. The magnetic energy stored in the transformer while S is ON must be removed while S is OFF, otherwise the energy stored and removed during a complete switching cycle would not be zero and the transformer core would rapidly saturate. A solution involving minimum power loss is to add a winding, closely coupled to the primary, and a diode D3 such that a fl ow of magnetizing current is ensured while S is OFF.

Vi

D2

D3

SD1

L

C Vo

+

-

+

-

Fig.6 Outline circuit of a mains-isolated forward-converter SMPS.

The operation of the transformer-isolated forward converter is described by the same basic expression, Eq. 1, as was used for the non-isolated version. The transformer also adds an extra degree of freedom of choice of output voltage for practical values of δ. This output voltage becomes Vo = δVi / rVoltage and current waveforms for a transformer-isolated forward converter are given in Fig. 7

1/f

δ/f (1−δ)/f

IL

VL Vi /r - Vo

Vo

time

time

2IacIo

Fig.7 Output circuit waveforms for the mains-isolated forward converter.

7

Part 1

Duty factor The maximum allowable duty factor at which the core will not saturate due to fl ux staircasing depends on r and m:

[7]δmax < 1 − mm + r

The maximum voltage across the power switch (here, a transistor) is then

VCEM = Vi max m + rm

[8]

When using a transistor with VCESM = 850 V in a forward converter with a maximum (rectifi ed) input voltage of 375 V, it is usually adequate to limit VCE to 2 x 375 = 750 V. Thus, with m = r, there is 100 V to spare for ringing and integrated supply-voltage surges. The effective duty factor depends on frequency, turns ratio r, rated load current, the leakage inductance of the transformer and the inductance of the leads to the output diodes. As a guide for mains-operated SMPS, decrease the conduction time so that

[9]δe = δ - rIo 1.2 10-9 f f

The inductance of the leads to the output diodes is refl ected into the primary as the square of the turns ratio. With large turns ratios, combined with high switching currents, the loss due to commutation delay becomes substantial. The effect is as if the available duty factor is decreased. This problem is discussed in greate detail in Ref. 3. An example is given in Fig8a. The commutation delay is: tc ≈ IorLs / Vi . Now, Ls is about 1 nH/mm of leads and, for a 220 V mains supply, Vi min ≈ 200 V. With the shortest possible leads to the output diodes, experiment shows that the commutation delay is tc ≈ 1.2Ior × 10-9

This is an important reason for not operating low-voltage high-current SMPS at high frequencies, but, rather, to use a frequency just above the audible range.

Io /r

Io

I2Ls

Ls

I2Ls

δe/f

Io

tc time0

(a) (b)

Fig.8 Effect of stray inductances in the output circuit of an isolated forward converter: (a) inductance in the secondary is reflected back into the primary as the square of the turns ratio;

(b) there is a commutation delay tc during which neither output diode conducts.

8

Part 1

Preliminary turns ratio and core selection The preliminary turns ratio is

[10]r' = δe max Vi min

Vo max + VF + VR

For transformer core selection, see Part 2.

Multiple-output transformers Additional outputs at any d.c. Ievel can be obtained simply by adding further secondaries of the appropriate number of turns to the output transformer. The regulation of the additional outputs will be better than with a fl yback converter. However, each output needs two diodes and a power choke, against the single diode needed with a fl yback converter. Warning: the fl ywheel diode must always conduct when the forward diode does not. Otherwise, peak forward conversion rectifi cation occurs and the output voltage could rise to the peak value of the forward-conversion voltage, which might be much higher than the nominal voltage, with disastrous results. So, ensure that the appropriate minimum load is always present at each output. With several different outputs, it is necessary to fi nd that value of volts-per-turn for the transformer that allows each output voltage to be obtained within the permitted tolerance with an integral number of turns. The procedure for this is described in Part 3.

Control method The function of the control circuit is to stabilize the output against variations in input voltage and load by adjusting the duty factor of the switching device. However, the effect of step load changes cannot be corrected immediately because some time is needed for the current through the choke to assume the new value of the load current. A momentary change in output voltage is thus inevitable. The time required for resumption of the desired level of output voltage after the sudden load change depends greatly on the properties of the control system. Two basic control characteristics can be distinguished: Fig.9; these are discussed in greater detail in Ref.2.

δmin

Vi min0

Vi

Vi min

Vi bo

time0

δmax Vi maxδmax

Vi maxVi bo

1.1Viδ

Vi

δVi

10% feed forward

Fig. 9 Input voltage and duty factor combinations under transient load conditions for feedback and feedforward control.

9

Part 1

Feedback: A step increase in load current causes the powerswitch duty factor to increase instantly to its maximum value δmax regardless of the level of input voltage. It is, thus, possible for δVi max = δmax Vi max.

Feedforward: A step rise in load current causes the powerswitch duty factor to increase to a value x % higher than its steady-state value for a constant load. The δVi max product in response to a step load increase will be higher with feedback control. This results in a shorter delay in adjusting to the new load because the choke current is forced to increase at a maximum rate. However, the output transformer must be so designed that it is able to cope with the product δmax Vi max without saturating.

With feed forward control,

δVi max = ( 1 + x

) δminVi max100

The corresponding value of α for the transformer design is

α = 1 + x

100

Diffi culties may be encountered with converter starting at full load with minimum mains voltage (15 % below nominal). If the duty factor is close to maximum, all the output current will fl ow into the load and little or no current will be available to charge the output capacitor. Starting will be improved if the steady-state value of the duty factor is, say, 10 % below its maximum allowable value.

One disadvantage of feedback control is that a larger transformer core is generally required to avoid saturation. The transient response time resulting from the method of control is discussed at the end of Section 1.2.

Primary inductance and de-saturation The primary inductance is given by

[11]L1 = µ0 µa n1 Ae

le

where the value of µa is obtained from the core data. The maximum, peak, primary magnetizing current is

[12]ImagM = (δ Vi) max

L1 f

The peak primary current is

[13]ImagM = 1 ( Io + Io min ) + ImagMf 2

10

Part 1

Vi

D

L1

C

+

-

Fig. 10 Forward converter transformer with demagnetizing (energy recovery) winding, and a slow rise capacitor in the primary circuit.

Primary inductance L1 together with slow-rise dV/dt capacitor C (Fig. 10) across the switch forms a resonant circuit with a natural frequency

[14]fr = 1

2π (L1 C)

The value of C used should satisfy the relationship

I1M tf = C > I1M2VCESM dV/dt

It is recommended that fr > f. At a lower value of fr, the core might fail to de-saturate and fl ux staircasing could occur, with disastrous results. A higher fr offers suffi cient safety margin under all conditions; a ratio fr/f of about 1.2 is a good compromise. An electronic solution to the problem can be found in some control ICs (e.g. TDA1060) where there is a facility for reducing the duty factor when core saturation is possible.The value and spread of L1 may be reduced by introducing an air gap in the core; the penalty is higher magnetizing current.

11

Part 1

For these small values of air gaps, the value of µe ( and, thus, L1 ) can be obtained for cores of constant cross-sectional area from

[16]s = le ( 1 - 1 )2 µe µa

If it is not possible to decrease L1 suffi ciently to make fr > f, the addition of a sensor winding on the transformer to generate a signal to prevent premature switch-on should be considered. (20)

Transformer currents The ripple in the output choke is, in general, only a few percent of the d.c. load current. For this reason, the transformer current can be regarded as a square wave for the purposes of winding loss calculation. The maximum r.m.s. current values are given, approximately, for the primary and secondary by

[17]Ie 1 = Io δo max = Ie 2r r

where

[18]δo max = δmax Vi

Vi av min

Duty factor δomax is used in these calculations for the following reason. For 220 V mains, the minimum line voltage is roughly 185 V r.m.s. Using an input fi lter capacitor of about 2 uF/W (to cope with a mains drop-out of 10 ms), the peak-to-peak ripple at 185 V mains input is about 20 V. Under these conditions the minimum average (steady-state) input voltage

[19]Vi av min = 2Vi min − ½Vir

= 252 V

However, δmax is set so that the converter can handle a 10 ms drop out. But mains drop out is not a steady-state condition, so that δomax should be used to calculate Ie and, thus, the loss and consequent temperature rise of the transformer.

Choke design To determine the minimum required choke inductance and for the choke design, see Section 1.1.

Transient response timeThe transient response time required for a forward converter to adjust to a step in the load current of ∆Io is

[20]tr = ∆Io L( Vo + VF + VR ) (δtr/δ - 1 )

12

Part 1

1.3 The double forward converter Principle of operation The equivalent circuit diagram of a double forward converter is shown in Fig. 11. It comprises two forward converters in parallel, with fl ywheel diode D and fi lter LC common to both. Switches S1 and S2 operate alternately, which doubles the ripple frequency of the choke current. Since energy is pumped twice per converter period, the output voltage is Vo = 2 δ Vi

ViD

S1

S2

L

C Vo Io

+

-

+

-

Fig. 11 Outline circuit of a double-forward SMPS converter. Switches S1 and S2 operate alternately.

Choke inductanceThe minimum choke inductance is calculated in a similar way to that for the single forward converter. However, since there are now two charges and two discharges per converter period

[22]L > Vo 1 −

2δmax Vi min

Vi max 4fIo min

Further choke design proceeds according to Section 1.1.

Preliminary transformer turns ratio The transformer is designed in a similar way to that for the single forward converter, except that now the turns ratio is twice as great. The preliminary turns ratio of a double forward converter transformer is

[23]r' = δe max Vi min

Vo max + VF + VR

Further design proceeds according to Part 2 of this series.

Transient response time The transient response time required for a double forward converter to adapt to a step in the load current of ∆Io is

[24]tr = ∆Io L2(Vo max + VF + VR) (δtr/δ -1)

13

Part 1

THE FLYBACK CONVERTER

2.1 Non-isolated inverting converter The outline circuit of an inverting fl yback converter is given in Fig. 12. The basic operation of an ideal inverting fl yback converter is described by

[25]Vo = δ Vi 1 − δ

Vi

DS

L C VoIo

+

-+

-

Fig. 12 Outline circuit of a non-isolated, inverting flyback converter.

The voltage waveform across inductance L and the associated current waveforms under steady-state conditions are given in Fig. 13.

1/f

δ/f (1−δ)/f

IL

VL

Vi

Vo

time

time

2Iac

Io

S D

Fig. 13 Voltage and current waveforms for the choke of a non-isolated, inverting flyback converter.

14

Part 1

Power inductor The minimum inductance required to ensure continuous mode operation at minimum load Po min is

[26]L > ( δmin Vi max )

2

2fPo min

Note: use of Po allows output-diode losses to be neglected. The maximum peak current through the inductor is IM = Idc max + 2 Iac, and

[27]IM = σPo max + δmax Vi min

2fLδmax Vi min

For inductor design, see Part 4.

It is evident that the value of the inductor is inversely proportional to the minimum load. On the other hand, the choke must not saturate at maximum load if the supply voltage is minimum. Thus, continuous choke current with large output-power variations is not always practical. The alternatives are:

- Pre-load the supply to decrease the ratio P o max /P o min - Change the operating frequency, as in the ’hiccup’ mode described in Ref. 2. - Accept a discontinuous choke current at the expense of higher peak current through switch and output capacitor.

In the last case, the choke current waveform will be triangular rather than trapezoidal. For a triangular current waveform. Eqs (26) and (27) become

[28]L > ( δmin Vi max )

2

2fPo min

and

[29]IM = 2 σPo max

δmax Vi min

15

Part 1

Non-inverting boost converter

Principle of operation The outline circuit of a boost converter is shown in Fig. 14. Its operation is described by

Vo = Vi

1 − δ

The voltage waveform across inductor L and the associated current waveforms under steady-state condi-tions are shown in Fig. 15

.

Vi

D

S

L

C VoIo

+

-

+

-

Fig. 14 Outline circuit of a boost flyback converter.

Fig. 15 Voltage and current waveforms for the choke of a boost flyback converter.

1/f

δ/f (1−δ)/f

IL

VL

Vi

Vo − Vi

time

time

2Iac

Io

16

Part 1

Power inductor The minimum inductance required to ensure continuous- mode operation of the choke at load Po min is

[30]L > 2 Vo

2

27 fPo min

The maximum peak inductor current is

[31]ImaxM = σPo max + δmax Vi min

2fLVi min

Proceed now with Part 4.

17

Part 1

2.2 Transformer-isolated flyback converter

Principle of operation For high-voltage low-current supplies, a useful variant of the inverting fl y back converter is obtained by adding secondary windings to the choke to form a transformer, Fig. 16. A further benefi t of this arrangement is mains isolation.

Io

Vo

+

-

Vi

+

-

r : 1S

Fig. 16 Outline circuit diagram of a transformer-isolated flyback converter.

Turns ratioIn order to protect the switching devices, the turns ratio

[32]r < VCESM - ( Vi max + Vr )

Vo + VF + VR

At high voltages, such as rectifi ed mains, a good compromise between inductor size, switching-transistor peak current, and diode peak current can usually be obtained by one of the following procedures. At moderate input-voltage range, say

Vi max < 2 Vi min

take δmin = 0.3. This yields

1 + 7/3 ( Vi min / Vi max )δ max = 1 [33]

so that

Vo + VF + VRr' =

3/7 Vi max [34]

At large input-voltage range, that is

Vi max > 2

Vi min

18

Part 1

take

Vo + VF + VRr' =

1 (Vi max Vi min ) [35]

This yields

r'(Vo + VF + VR) 1 +

Vi max [36]δmin = 1

Note: If dmin > 0.3, take dmin = 0.3 and proceed as for a small input-voltage range, otherwise

1 = 1 + (1 - δmin) Vi min δminVi maxδmax

A voltage limiting winding with turns ratio between primary and limiting winding r/m limits switching-device voltage to β Vi max The maximum duty factor must then be such that

δmax < 1 − 1β

[37]

whence

δmax Vi min 1 +

(1 - δmax) Vi max [38]δmin = 1

and

[39]r' = δmin Vi max

( 1 − δmin) (Vo + VF + VR)

with

m = r

β − 1[40]

19

Part 1

Power inductor To ensure continuous-mode operation, design the choke primary for minimum load. (See also Section 2.1.)

[41]L < ( δ min Vi max )

2

2 f Po min

The maximum peak current through the primary is

ImaxM = σPo max + δmax Vi min

2fLδmax Vi min

Proceed with Part 4. The number of turns on the choke secondary, together with r, are found in Part 3.

Multiple output Additional output voltages at any d.c. level can be obtained by simply adding additional secondaries of the appropriate numbers of turns. Note that, if the range of d.c. output voltages is large, leakage inductance will increase and regulation deteriorate. The design procedure is given in Part 3.

Effective currents The maximum value of the r.m.s. current through the primary winding is

Ie1 = Po max δ0 max

δ0 max Vi min

1 + 13

2

Po max

Po min( ) [42]

and through secundary x,

Iex = rxPox max (1 − δ0 max)

δ0 max Vi min

1 + 13

2

Po max

Po min( ) [43]

where

r (Vo + VF + VR) 1 +

Vi av min δ0 max = 1

20

Part 1

Fig. 18 Fig. 17 The basic configurations of push-pull converters. (a) Two-winding (1 + 1) transformer in a single-ended push-pull converter with a bridge-rectified output. (b) Two-winding (1 + 1) transformer in a bridge converter with a bridge-rectified output. (c) Three- winding (1 + 2) transformer in a single-ended push-pull converter with a bi-phase output. (d) Three-winding (1 + 2) transformer in a bridge converter with a bi-phase output. (e) Three-winding (1 + 2) transformer in a push-pull converter with a bridge- rectified output. (f)

T1

½Nt½Nt

C1

D2D1

D3 D4

L1S1

S2

C2 C3

T1

½Nt½Nt D2D1

D3 D4

L1S3

S4

S1

S2

C3

T1

1/3Nt

1/3Nt1/3Nt

C1

L1S1

S2

C2 C3

T1

½Nt

S3

S4

S1

S2

1/3Nt

1/3Nt

L1

C3

T1

S1

S2

D2D1

D3 D4

L1

C3

1/3Nt1/3Nt

1/3Nt

T1

1/4Nt

1/4Nt

L1

C3

S1

S2

1/4Nt

1/4Nt

21

Part 1

3 THE PUSH-PULL CONVERTER

3.1 Principle of operation Outline circuits of the various basic confi gurations of push-pull converters are shown in Fig. 17. Since energy is pumped twice per converter period, the basic operation of a push-pull converter is described by

Vo = z δ Vi

2[44]

For full bridge and conventional push-pull converters, z = 4; for half bridge push-pull converters z = 2. The voltage waveform across the choke and the associated current waveforms are shown in Fig. 18.

1/f

δ/f1/2f - δf

IL

VL

Vo

time

time

2IacIo

Vi - Vor

Fig. 18 Voltage waveform across the choke of a push-pull converter together with the associated current waveforms.

Duty factor With the equal conduction times per converter cycle, the maximum allowable duty factor is 0.5, but a more practical value is 0.45. In practice, wiring and transformer stray inductances result in a fi nite commutation time between output diodes. As a result, the interval during which energy is supplied to the output is shorter, and the effective duty factor smaller. This effective duty factor depends on operating frequency, transformer ratio, load current, and the stray inductance of the leads to the rectifi er diodes. To compensate for the increased commutation time, the energy transfer period δ/f should be decreased to about

[45]δe = δ - rIo 1.2 10-9 f f

Further discussion will be found in Section 1.2.

22

Part 1

Transformer turns ratio The preliminary turns ratio is

[46]r' = 2δe max Vi min

Vo + VF + VR

For further transformer design, proceed to Part 2.

3.2 Power choke inductance The minimum choke inductance that ensures continuous choke current, and, thus, continuous-mode operation is

[47]L > Vo

4fIac Vi min

Vi max( 1 - 2δmax )

here,

Iac = Io min − ImagM [48]

and

ImaxM = r δ max Vi min le

2n1 µ0 µe Ae f2

[49]

During transistor conduction, Fig.19, the magnetizing current changes from + I magM to - I magM. While both transistors are off, during the interval (l/2f - δ/f) the transformer primary is open circuit. This forces the magnetizing current to fl ow through the output diodes in series. Thus, the load and magnetizing currents reinforce in one diode and cancel in the other.

23

Part 1

1/f

δ/f 1/2f−δf

1/2f

IL

Im1

ID1

ID2

2Iac

Io

Im

S1 ON

D1 ON

D2 ON

S2 ON

Fig. 19 Waveform for calculating the peak choke current of a push-pull converter.

If, at low output current, one diode ceases to conduct, there is no path for the magnetizing current, which is then diverted through the conducting diode and the output choke to the output capacitor. This causes the output voltage to rise. From this, it follows that the minimum load current that can be drawn from the converter without one diode ceasing to conduct in this way is

Io min = ImagM + Iac

The magnetizing current fl owing through the output diodes is

ImaxM = δ max Vi min r

2 L1 f

where

ImaxM = 2n1 µ0 µe le

Ae

2

These two expressions together yield Eq. 49. From Fig.19, the peak current through the power choke is

IM = σ (Io max + Io min + ImagM) [50]

Further choke design proceeds with Part 4.

24

Part 1

3.3 Transformer currents In a push-pull transformer, only one half of the double winding conducts at a time. The peak current through each half of the double winding is

I1M = IM [51]r

The effective primary current in each winding half is

Ie1 ≈ Io [52]r

δo max

and, in the secundary,

Ie2 ≈ Io [53]δo max

where

δo max = δmax Vi min

Vi av min

Transient response time The transient response time required for a push-pull converter to adjust to a step in the load current of ∆ Io

[54]tr = ∆Io L2 ( Vo max + VF + VR ) (δtr/δ - 1 )

1

Part 2

Part 2 - Selection of suitable cores for a trans-former design

Available ferrite core types cater for a wide range of SMPS application requirements.If the proper core for a given application is to be selected, a number of factors should be taken into account. The type of converter circuit used, for example, determines to a large extent the throughput power capacity of transformer wound on a given core type.

The discussion here assumes that no premagnetization - magnetic bias by a permanent magnet - is applied to the core.

2

Part 2

1.1. Core selection for forward and push-pull transformers

The practical throughput power of a transformer is represented as a shaded area, as shown in Fig. 1.

10 100f(kHz)

range of Pthfor good, practicaldesigns

Pth

Fig. 1 The powerhandling capability of a core is plotted as a shaded area extending from 10 kHz to 100 kHz. The vertical boundaries of this area represent the upper and lower limits of throughput power

capacity achievable by good design, but depending on conductor type and mains insulation requirements.

The actual throughput power obtainable with a given core depends to a large extent on the following characteristics:

• fl ux density sweep (Section 2)

• the winding confi guration (simple or split/sandwiched windings, sensor or demagnetization windings)

• conductor type (solid, strip, Litz)

• single or multiple output

• mains insulation requirements.

3

Part 2

The upper limit of the shaded area for each core refers to a transformer design with optimized fl ux-density sweep, maximum use of the winding window, and Litz wire for minimum a.c. resistance. The lower limit refers to a design with 8 mm creepage distance for IEC 435 mains insulation, optimized fl ux-density sweep, a (1 + 2) winding confi guration (Part 1), and optimized but solid-wire windings. In addition, the following general conditions were assumed in the calculation of both boundaries :

• the hot-spot (peak) temperature of the core is 100oC; the temperature rise is 40 K

• the maximum fl ux-density sweep is limited to 1/1.72 of the maximum permissible fl ux den sity for the

core material (0.32 T for 3C94) to cope with transient conditions.

• the thermal behaviour of wound cores, but without potting or additional heatsinking was assumed.

• core fl ux densities are calculated assuming minimum cross- section areas.

Where the ambient temperature is lower than 60oC, when feedforward is used to ease the restriction on maximum fl ux density, or when heatsinking or potting are employed to improve heat transfer, throughput power capacity will be increased.

It may happen that, at a given power level, more than one type of the core may be used. The following criteria may be used to make a choice :

• copper foil secondary windings are preferable for low-voltage, high-current supplies; thus, for ease of

winding, a core with a round centre pole should be chosen.

• coil formers and mounting hardware are not available as standard for all core types.

• production logistics may be improved if one core type is used for both transformer and choke.

Where these considerations do not apply, the choice of the core should be guided by the discussion of Section 2.

1.2. Core selection for fl yback converters and chokesThe magnetic design of fl yback transformers and output chokes for forward push-pull converters is essentially the same: the main design parameter is the energy to be stored. Core selection, therefore, is made on the basis of energy stored, ½ I2ML. . This leads directly to airgap length and number of turns.

4

Part 2

2. Operating flux densityWhen determining operating fl ux density, a distinction must be made between transformers and chokes. For chokes, and fl yback-converter transformers (which also function as chokes), the most important parameter is the maximum peak fl ux density. The fl ux density sweep follows from the inductance value required.

For push-pull and forward-converter transformers, both a.c. and d.c. components of fl ux den sity must be taken into account from the start of the design process.

2.1. Flux-density levels for forward and push-pull transformersThe operating fl ux density of a transformer can seldom approach the maximum permissible fl ux density in practice since an allowance must be made for transient conditions, such as sudden load increases.Unless special electronic measures (Part 1) are taken, a transient factor is required to cope with sudden load changes. This factor is related to the ability of the power supply to accept a range of input voltages. The input voltage range of mains-fed power supplies may be 215 V to 370 V, or 200 V to 340 V; for telephone supplies, 40 V to 70V; and for mobile supplies, 9V to 15.5 V. The usual transient factor is 1.72.

For symmetrical excitation of the core (push-pull), the maximum possible fl ux-density sweep (Fig.2) is, in principle, twice that for asymmetrical excitation. In practice, however, allowance has to be made for unbalance when determing the operating fl ux density.

B

2Bac

(a)

H

B

2Bac

(b)

H

B

2Bac

(c)

H

Fig. 2 Flux density excursions and the corresponding flux-density sweeps for (a) push-pull converter transformer; (b) forward-converter transformer

(with slow-rise capacitor) or ringing-choke flyback converter; and (c) flyback converter choke.

The maximum operating fl ux density depends on the protection circuitry. One source of unbalance is unequal fl ux linkage between two halve of a centre-tapped winding. For this reason, bifi lar windings are to be preferred. However, this is not possible in mains-fed supplies since the voltage across the winding might be greater than the maximum voltage between adjacent turns.The major reason for asymmetry is unequal conduction times or saturation voltages of the power switches in a push-pull converter. Storage effects can result in different switch-off delays. Core saturation occurring due to a delay decreases the primary inductance so that the magnetizing current rises steeply. This may lead to the destruction of the power switches. As a further safeguard, the maximum operating fl ux density should be decreased by an additional factor that depends on the effi ciency of the protection circuit. A pratical guide is a 15% allowance for a fully protected converter (unbalance factor ε = 1.15), but a 100% allowance unbalanced push-pull converter (ε = 2).

In forward converters, remanence should, in theory, also be allowed for. However, to obtain the correct primary inductance, some airgap is often useful. This airgap, together with the slow-rise capacitor (Part 1), results in the whole fi rst quadrant of the BH loop being useful in practice, as is shown in Fig. 2.

5

Part 2

1009080706050403020151030

40

50

60

708090

100

150

200

300

400

saturationlimit

push-pull

forward

Bac(mT)

f (kHz)

1

2

34

1. UU15/22/6. EE20/20/5 2. UU20/32/7. EE25/25/7. EE30/30/73. UU25/40/13 4. UU30/50/16

7Z90894

1009080706050403020151030

40

50

60

708090

100

150

200

300

400

saturationlimit

push-pull

forward

Bac(mT)

f (kHz)

1

2345678

1. EC 35 2. EC41, EC52 3. EC70 4. EE42/42/155. EE42/42/20. EE42/54/20. EE42/66/20 6. EE55/55/217. EE55/55/25. UU64/79/20 8. EE65/66/27

7Z90895

100 1509080706050403020151030

40

50

60

708090

100

150

200

300

400

saturationlimit

push-pull

forward

Bac(mT)

f (kHz)

1234

Curve 1 ETD34Curve 2 ETD39Curve 3 ETD44Curve 4 ETD49

7Z90893

Fig. 3 Optimum peak centre-pole flux density sweeps Bac opt for a variety of cores for SMPS

applications. Horizontal lines indicate the limits for various converter types with α = 1.72.

The curves are calculated for a transformer temperature rise of 40 K.

Pth/Pth max

B/Bopt

1

0,9

0,8

0,8 0,9 1 1,1 1,2

Fig. 4 The effect on throughput powercapacityof a deviation in Bac from the optimum value indicated in

Fig. 3.

6

Part 2

Where feedforward control is used, the transient factor can be reduced considerably and a higher fl ux-density sweep can often be applied. The actual transient factor is determined by the feedforward percentage used. Note, however, that application of feedforward reduces the transient response of the power supply. TABLE 2 Maximum Values of flux-density sweep for various converter types and control circuitsboundary conditions

flux-density sweep Bac cp (T) forward push-pull

maximum flux sweep (100oC) 0.16 0.32

0.32 0.32at transient factor α 2α α

0.32with unbalance factor ε - εα

0.32 0.32with x% feedforward 2(1 + x/100) (1 + x/100)

with unbalance factor ε and x% - 0.32 feedforward ε(1 + x/100)

Practical fl ux density and sweep limits are summarized for various converter types in Table 2. The curves of Fig. 3, show the optimum fl ux density sweep, where throughput power is maximum, for a range of cores in the frequency range 10 kHz to 100 kHz. Horizontal lines indicate the maximum allowable sweep for various converter types. Further lines can be added for other boundary conditions with the aid of Table2. The given curves are calculated for a transformer temperature rise of 40 K.

The converter operating frequency is set by the required output voltage and current, and by the type of switch to be used. One this frequency is known, the optimum fl ux-density sweep can be found for any core type.

Where the frequency is more or less fi xed, and two core types could be used (Section 1), preference should be given to that core type for which the intersection of the optimum-sweep curve with the set frequency is closest to the maximum fl ux density sweep.

Where frequency can be chosen freely, the frequency corresponding to the intersection of the optimum-sweep curve with the line for the maximum fl ux-density sweep represents the optimum use of the core material.Operation at the optimum sweeps represented by the curves of Fig. 3 means that core loss and permissible winding loss are in optimum proportion of core and winding loss, results in a lower throughput power.When the design is limited by saturation fl ux density, deviation is inevitable. The effect of deviation from the optimum fl ux density is plotted in Fig. 4. This plot, which applies to any frequency, gives a rough indication for the reduction in throughput power from the optimum value. Once the optimum fl ux density sweep has been determined for the core and converter combination, the number of turns can be calculated as shown in Section 3.

2.2. Flux densities for chokes and fl yback transformersThe maximum fl ux density is taken into account in the design procedures given in Parts 1 and 4. The design centres around the maximum stored energy ½I2

M L in the choke, which, in turn, is related to the inductance

value from which follows the number of turns.The fl ux density sweep in chokes is generally relatively low, and so, in consequence, is the core loss.

7

Part 2

It may happen, however, especially in ringing choke fl yback converters, that the fl ux-density sweep is comparable to that in forward converters. Core loss is not then negligible and should be calculated as shown in Section 4.Once core type, spacer thickness, and number of turns have been established, the peak fl ux density sweep can be calculated:

L IacBac max =

Nprim Amin

where Iac is found during the design process (Part1). In case of a fl yback transformer, all quantities refer to the primary. If Iac is relatively high, core loss will be signifi cant.

3. Number of turnsOnce the fl ux-density sweep is known, the number of turns can be determined.

3.1. Forward and push-pull converter transformers.The minimum number of turns in the secondary

δVin2 min = r'Amin Bac max f z

z = 2 for forward and half-bridge push-pull convertersz = 4 for full-bridge push-pull converters.

The values of δ and Vi are obtained from δVi = δ maxVi min = δ minVi max (this and r’ are found in Part 1).

n2 min will generally not be an integer and must be rounded. If the value of n2 min is small, rounding will change the fl ux density sweep signifi cantly. To prevent saturation, rounding should be to the next higher integer. The effect of rounding can be counteracted by changing the operating frequency.

Where operating frequency is not fi xed, iteration will result in a satisfactory design. When the actual fl ux-density, sweep is within 10% of the optimum, throughput power will be within 5% of its maximum (Section 2.1). Use the fl ux density sweep obtained after rounding to determine core loss (Section 4).

3.2 Flyback converters and chokesFlyback converters and output chokes: the number of turns is derived and rounded in the choke design procedure of Part 4.

Flyback transformers: the rounded number of primary turns is found in the choke design procedure of Part 4. The value of r' is calculated in Part 1.

8

Part 2

4. Core lossCore loss has two main components: hysteresis loss and eddy-current loss. The hysteresis loss in a typical low frequency power ferrite is for instance:

Ph ≈ 8 f1.3 B2.5ac Ve

Both coeffi cients of f and B are frequency dependent. In the design tools software this type of formula is used to predict core losses during transformer design.The eddy-current loss for a ferrite, under the same conditions, is:

Pe ≈ 0,8 f2 B2ac Ae Ve /ρ.

Where ρ is a function of frequency and temperature.

The contribution of eddy-current loss in the core is greatly dependent on core size, frequency and fl ux density sweep. Below 100 kHz, eddy-current loss may be neglected for small cores. However, for the larger cores, such as the E65 and EC70, eddy-current loss becomes important at much lower frequencies.For very large cores, with centre-pole diameters exceeding 35 mm, the expressions for both Ph and Pe do not hold, even below 100 kHz. In the above expressions, Bac is the peak effective fl ux density sweep. It is related to Bac max by

Bac max Acp = Bac Ae

For cores of about constant cross sectional area, such as ETD, and most E and U cores, Bac max and Bac are similar in value, but they differ signifi cantly for EC cores.

9

Part 2

5. Thermal resistanceIn order to determine the maximum permissible dissipation of a transformer or choke, its thermal behaviour must be know. This depends on core size, conductor form, and insulation requirements. For standard, more or less cubical transformers based on cores like E or ETD the constant Cth is about 50. Very fl at designs like Planar E cores have much better thermal properties, the constant is around 25. (Ve is in cm3).

Rth = Cth Ve-0.54 K/W

These thermal resistances were measured with the transformer mounted on a printed circuit board with no local heat sources. They are referred to the hot-spot temperature in the middle of the centre leg of the core (as deducted from the measured value on the ferrite surface).For hot-spot temperatures up to 100oC, the maximum permissible winding dissipation may be obtained from :

∆TP w = - Pc

Rth

This expression may be used for any ratio of core to winding dissipation.In a choke with a large d.c. component of current, where core loss is negligible, P w = ∆T / Rth.The temperature rise should be checked again once the winding design is complete. Winding design is treated in part 3 for transformers and Part 4 for chokes. If the actual temperature rise is too high, the estimated throughput capacity was optimistic and a larger core should be used.

10

Part 2

6. Data necessary for winding designBefore it is possible to move to the design of the windings, the following data should be available.

Forward and push-pull converters • core type(s) (Section 1.1)• coil former dimensions (Ref. 1)• mains insulation requirements (Section 1.)• preliminary turns ratio (Part 1)• rounded number of secondary turns (Section 3.1)• operating frequency• secondary current and waveform• winding dissipation permitted (Section 5)• other windings - auxiliary outputs or sensor windings

Flyback transformers• core types (Part 4)• coil former dimensions• number of primary turns (Part 4)• preliminary turns ratio (Part 1)• operating frequency (Part 3)• primary and secondary currents and wave forms• other winding - sensor or auxiliary outputs• winding dissipation permitted (Section 5)• core loss (Section 4)• mains insulation requirements (Section 1.1.)

Output and fl yback chokes• core loss (Section 4)

Proceed to part 4 for remaining design.

1

Part 3

Part 3 - Transformer winding design

1. Introduction At ultrasonic frequencies, the simple design rules used for low-frequency transformer windings are no longer valid. Winding resistance and thus winding loss are increased through eddy-current effects, the proximity effect. At mains frequencies these effects can be ignored. The design methods and aids presented here aim at fi nding the optimum winding design: that is, the winding geometry that yields lowest loss.In drafting these design aids, the number of decisions made for the designer has been restricted to a minimum. He thus obtains a maximum of freedom of choice. The essential background information will be given after presenting practical design procedures. A certain amount of background information is required to use that freedom properly.

2. Definition of the problem: boundary conditions

2.1. Given boundary conditionsThe following boundary conditions are set once a core is selected:

• frequency• primary and secondary r.m.s. currents• primary and secondary turns numbers• winding breadth• available winding height• permissible winding loss

Coil former and winding dimensions are defi ned in Fig. 1.These parameters can only marginally - if at all - be varied in winding design, therefore, they are termed given boundary conditions.

Winding breadth bw follows from coil former breadth BCF and the required creepage allowance c as set by the insulation standard. Available winding height is the height of the winding window HCF in the coil former

BCF

HCF

CL

c/2 c/2

screen

7Z80234

H

bw

Fig. 1 Definitions of coil former and winding dimensions.

2

Part 3

less the height occupied by the screens with the insulation, and the auxiliary windings (control windings not taking part in power transfer.) The available height should accommodate the power-transfer windings. It is not possible to determine beforehand the maximum height of individual windings.

2.2. Chosen boundary conditions

In the winding design process the designer can make his own choice of • winding confi guration (Section 4.1.1.) • conductor form (wire, strip, bunched wire, etc.)

At the moment of choice, it is not always clear which choice is best. When a cheap design has an acceptable loss it may be preferable to a more expensive design with even lower loss. It may therefore be desirable to work out more than one design by making preliminary choices for these boundary conditions.

2.3. The design problem

The actual winding design is now to determine the most suitable winding geometry.

Solving this problem requires (preferably convenient) means and methods to determine. • for strip winding (strip width equal to winding breath) - strip thickness • for wire windings - number of layers - wire size.Having chosen the interleaving thickness, all further desired parameters can be calculated. Such means and methods are presented here. For the prevailing set of (given and chosen) boundary conditions they are directed to fi nding the geometry giving minimum loss.

3

Part 3

3. Design procedures

The design process is organized in the following fi ve phases.

I Collecting the boundary conditions. II Determining the ideal power windings. III Their evaluation for winding loss and required height. (According to the result, the design process then branches to Phase IV of Phase V, or a new start is required.) IV Determination of the optimum combination of non-ideal designs that will fi t into the available height.

V Design fi nalization.

Note: here, the term ideal is used to describe windings designed for minimum loss regardless of the height of coil former required. Non-ideal windings have a height less than that required for ideal windings and the lowest possible loss that this height permits.

3.1. Phase I :Collect the boundary conditions

3.1.1. Given boundary conditions

Collect the given boundary conditions as set by circuit requirements and the core selected.

1. Core related :• operating frequency• maximum permissible winding loss Pw max• preliminary primary to secondary turns ratio r’.

For forward and push-pull transformers:• number of turns in secondary Nsec• full load r.m.s. secondary current Ie 2 .

For fl yback transformers:• number of turns in primary N1• full-load r.m.s. primary current Ie 1• full-load r.m.s. secondary current Ie 2.

2. Circuit related :• in the case of mains isolation, creepage distance c• data on windings other than power windings, such as demagnetizing and sensor windings• single or multiple secondaries.

4

Part 3

3.1.2. Chosen boundary conditions

1. Winding confi guration: Choose between the simple confi guration (where C = 1) or the split/sandwiched confi guration (where C = 2), see Sects 4.1.1 and 5.1. In the latter case, decide which winding will be split, and which sandwiched. (It is usual to split the winding with the larger number of turns.) Sketch the complete winding arrangement (not to scale), including any auxiliary or multiple-output windings, as in Fig. 4, 19, and 20. Draw also the leakage fl ux (or NI) diagram and determine for each of the power windings the value of ϑ, the ratio of minimum to maximum leakage fl ux density over the height of the winding. Normally ϑ = 0, but in multiple output transformers it occurs that ϑ ≠ 0, see Sect. 4.5.1.In the case of a fl yback transformer, draw the leakage-fl ux diagram for both the period of primary conduction and the period of secondary conduction, see Sect. 4.5.3. For push-pull transformers see Sect. 4.5.2.

2. Conductor form : Assume single round wire initially. The results thus obtained will guide the fi nal choice of conductor.

3.1.3. Derived boundary conditions.

1. Effective frequency: If the waveform of the current through the windings is known from previous designs, the effective frequency fe can be obtained by analysis of the current waveform , as given in Sect. 6. If the waveform is unknown, assume that fe is the switching frequency in kHz.

2. Effective skin thickness in copper at 100oC is

∆e = √(5,62 / fe) (mm)(with fe in kHz)

3. Number of turnsForward and push-pull transformers :

secondary, N2 = Nsec/Cprimary, n1 = Nsec r’.

Initially, round n1 to the next even integer Nprim. It is essential that Nprim is even if the winding is to be split, and it avoids a further rounding if there are two layers in a simple winding. Then,

N1 = Nprim/ C.

Flyback transformers :A fl yback transformer may use a split/sandwiched winding confi guration to reduce leakage inductance, but this will not, however, reduce winding loss. The design procedures are always those for simple windings.But, of course, the number of turns of the split winding must be rounded to an even number.

N2 = N1 /r’

rounded to an integer.

5

Part 3

Multiple-output windings (see alsoSect. 4.5.1) • Calculate for the output with the lowest voltage, that is, with the smallest number of turns, and round up to the next integer. • To obtain the number of turns for the other outputs divide the required output voltage by the volts-per-turn value for the lowest-voltage winding. Round the numbers of turns of all output to the nearest integer. • Check whether the various output voltages are within the required limits with these numbers of turns. • If not, increase the number of turns of the lowest-voltage winding by one, and repeat the calculations, starting with the volts per turn. • When the numbers of turns for the correct output voltage have been established, correct the number of primary turns using the volts-per-turn value last found.

4. Winding window

Winding breadth bw = BCF - c, see Fig.1.

Available height: the height available for the complete power windings is HCF less- an allowance for imperfect contact between layers, interleavings and screens;- the height of screens and their insulation (C times), see Sec. 4.5.3;- the height of auxiliary windings handling little or no power, such as : • a sensor winding on the primary side of the screen that may also supply a little power for the control circuitry. • a demagnetizing winding.Such windings are not designed for minimum loss. To save height, single layers are assumed of winding pitch

t = bw/(N + 1).

Choose a wire size such that tmin ≤ t and d ≤ ∆e. Then with an interleaving of thickness i suitable for the winding pitch, the required height is

H = do + i.

The height that remains after all deductions divided by C is Ha, the height available for one primary and one secondary winding portion.

6

Part 3

3.2. Phase II:Determine the ideal power windings

• For push-pull multiple output and fl yback transformers, fi rst see Sect. 4.5.

Not yet knowing the value of p, multilayer windings are assumed initially If, during the design procedure , this assumption proves incorrect, the design procedure switches to that for single-layer windings. Follow the procedure below for all power windings.

3.2.1. Ideal multilayer windings of solid, round wire

1.Calculate

d∞ = 17.1 bw N fe

3

2. If ϑ = 0, read Cp for the table below. If p is not known, p = 1.5 for a sandwiched winding or p = 2 in other cases. p 1.5 2 2.5 3 to 4.5 >4.5 if p > 3check Cp 1.06 1.03 1.02 1.01 1 for errors

-If ϑ ≠ 0, calculate

Cp = 1 - ϑ

(1 - ϑ3)1/3

Then calculate did = d∞ Cp.

3.Select the nearest standard wire size from a wire table (see page 49) and note d, do, tmin and rdc.

4.Number of layers

a

Pid = Nbw / tmin - 1

Note: this expression is valid only for tmin from Step 3. If Pid ≤ 1.5 a strip or foil alternative may be preferable, see Sect. 3.2.23.If ϑ = 0 and pid ≤ 1, the expression for d∞ in Step 1 is not valid. Go to the single-layer winding procedure, Sect. 3.2.2.

b. Find p by rounding up Pid to the next suitable value: to a multiple of 0.5 for a sandwiched winding, and to an integer in other cases.

c. Calculate Nl = N / p. If that value is not an integer, consider an adaptation of N to facilitate manufacture. Start again with the new value of N.

7

Part 3

d. If ϑ = 0, check the value of p used equals that assumed in Step 1. If not, repeat from Step 2 using the correct value of p.

5. Determine the winding pitcht = bw /( Nl + 1 ) = pbw / ( Nl + p ). Nl may not be the same in all layers (Step 3c); this will result in different values of t. Remember that all layers should occupy the breath bw. do not allow a difference greater than one turn in Nl.

6. Select a suitable interleaving, thickness i, for the winding pitch and calculate the required height Hid = p ( do + i )

7. Resistance factor FR = 1 + 1/2 (d /did)6 when d/did < 1.25. But when p = 1.5, this expression holds only for d/did < 1.15.

8. AC resistance per metre length of wire rac = FR . rdc.

9. Winding loss of a complete winding ( of C portions ) Pw = C I2e N lav rac, taking for the average turn length lav the value for a fully wound coil former in m.

Having found the ideal designs for all power windings using this procedure, proceed to Sect. 3.3 (Phase III).

8

Part 3

3.2.2. Single and half-layer windings of solid round wire

This procedure applies only when ϑ = 0.Arriving form Step 4a of the previous procedure, take p = 1.

1. Winding pitch t = pbw /( N+p).

2. Select from a wire table the thickest standard wire for which tmin ≤ t and note d, do and rdc. Also choose a suitable interleaving, thickness i.

3. H = p (do + i).

4. ϕ = √(0.124 fe d3/t),

(fe in kHz, d and t in mm).

Note : If fe differs from the switching frequency, use the lower value of frequency.

5. Read FR from Fig.2. Beyond the chart, FR = pϕ.

6. rac = FRrdc.

7. Pw = C I2e N lav rac, where lav is in m.

In the case of a sandwiched winding, consider a strip or foil winding, Sect. 3.2.3. If a wire winding is preferred, and if ϕ from Step 4 is about 4 or greater, a half-layer winding will probably have about equal rac, about half the wire size and about one quarter of the copper content; follow the above procedure using p = 0.5.

Having found the ideal designs for all power windings, go to Sect. 3.3 (Phase III).

8

7Z898434FR

3

2

1,5

1,2

1,1

1,05

1,02

1,01

p = 1 p = 0,5

0,5 1 2 3 4 5 10ϕ

Fig. 2 FR versus ϕ for single and half-layer windings. Above FR = 4, take FR = pϕ.

9

Part 3

3.2.3. Ideal strip or foil windings

1. If ϑ = 0, read CN from the table below.

N 0.5 1 1.5 2 2.5 3 to 4.5 ≥ 5 CN 1.69 1.19 1.06 1.03 1.02 1.01 1

If ϑ ≠ 0, calculate

CN =

(1 - ϑ)3

1 - ϑ3

4

Then determine fe (kHz).

hid = CN√ 9.74 /N fe mm, using fe in kHz.

2. Select the nearest available strip or foil thickness h.

3. FR = 1 +(1/3)(h/hid)4 when h/hid < 1.4.

4. rac = FR/(45bwh) Ω/m (bw and h in mm).

5. Pw = CI2eNlavrac.

6. Select a suitable interleaving, thickness i, then Hid = N(h = i).

Note: this procedure is for copper at 100oC, resistivity.1/45 Ω mm2/m. It can be adapted to aluminium conductor (62% higher resistivity at 100oC). In the expression for hid (Step 1) re-place 9.74 by 15.8; in Step 4, replace 45 by 28. Compare Pw from Step 5 with that of the wire version of the winding (Sect. 3.2.1, Step 9 or Sect. 3.2.2, Step 7) and choose the ideal solution for lowest rac. Having houdn the ideal design for all power windings go to Sect. 3.3 (Phase III).

10

Part 3

3.3. Phase IIIEvaluate the desigh with respect to winding loss and height.

The success of a design attempt is judged on the basis of winding-loss margin Pm and remaining free height Hr. The winding-loss margin is calculated from Pw max and the total loss of the power windings as found in Sect. 3.2.1. (Step 9), Sect. 3.2.2 (Step 7) and Sect. 3.2. (Step 5). The remaining free height is calculated from Ha as found in Sect. 3.1.3 (Step 4b), and the heights of the power windings found in Sect. 3.2.1 (Step 6), Sect. 3.2.2 (Step 3) and Sect. 3.2.3 (Step 6).

There are four possible results of the evaluation:

- Pm and Hr both positive : design attempt successful, proceed to Sect. 3.6 (Phase V), - Pm and Hr both negative : boundary conditions do not permit a successful design. Make a fresh start with a higher operating frequency or a larger core, for example. - Pm positive and Hr negative : a frequent occurrence, especially at lower frequencies. Try non-ideal winding designs requiring less height, Sect. 3.5 (Phase IV). - Pm negative and Hr positive : reconsider the chosen boundary conditions and start again. The background of Sect. 5 may be useful; some basic considerations are : • Split/sandwiched designs of offer lower loss than simple ones, specially at higher frequencies. • Strip or foil windings tend to have lower losses than (single layer) wire windings. • If Hr is large, try replacing a single-wire winding by a multiple-wire or bunched-wire winding. Design procedures are given ins Sect. 3.4. A Litz-wire winding may also be possible If Hr is particularly large. Evaluate the result again.

If adoption of one of these measures results in both Hr and Pm being positive, the design is successful; if not, try a higher operating frequency or a larger core in a new design.

3.4. Alternative designs: multiple, bunched and Litz-wire windings

3.4.1. Ideal multiple-wire winding

For background and possible loss reduction, see Sect. 5.2.2.

1. Choose the number of parallel strands ns : usually 2 or 3.

2. Following the design procedure for ideal multilayer windings of solid, round wire, Sect. 3.2.1, as if the winding had nsN turns. If step 4a results in pid ≤ 1, follow the single-layer design procedure, Sect. 3.2.2, again replacing N by nsN.

3. Divide the value of rac by ns to fi nd the a.c. resistance of the ns parallel strands.

11

Part 3

3.4.2. Ideal bunched-wire windings

Bunched wire consists of a few strands of insulated wire twisted together to form a bunch. The strand diameter is not so mall that eddy-current loss is negligible. For background see Sect. 5.2.3.The minimum winding pitch of a bunched wire of ns strands is nttmin, and the layer height without interleaving is nhtmin, where tmin is the minimum winding pitch given in the wire tables (page 27) for a single strand. Values of nt and nh for crude experimental bunches are given for guidance in the table below.

nS 4 5 6 7 8 9 10

nt 2,45 2,94 2,98 3,11 3,61 3,89 4,34 nh 2,31 2,69 2,93 2,93 3,16 3,16 3,26

3.4.2.1. Multilayer bunched-wire windings

1.Choose ns. A bunch of 7 strands gives the best space (copper) factor.

2.If ϑ = 0 read Cp from the table below. If p is not known, take p = 1.5 for a sandwiched winding, and p = 2 in other cases.

p 1.5 2 2.5 3 to 4.5 >4.5 if p > 3check Cp 1.06 1.03 1.02 1.01 1 for errors

If ϑ ≠ 0, calculate

Cp = 1 - ϑ

(1 - ϑ3)1/3

Then

did = 17.1 bw ns N fe

3

3. Select the nearest standard wire size from a wire table (page 27) and not d, do, tmin and rdc.

4. Number of layers

a.

Pid = N

bw / (nt tmin) - 1

Note: the expression holds only for tmin from the previous steps. If Pid ≤1, the expression in Step 2 does not apply: go to Sect. 3.4.2.2.

b. Find P by rounding Pid up to the next suitable value. For sandwiched windings, round to multiples of 0.5 ; in other cases, round to integers.

12

Part 3

c. Calculate.Nl = N / p If this does not yield an integer, consider adapting N to facilitate manufacture. Start again using the new value of N.

d. If ϑ = 0 check, that the value of p found is that assumed in Step 2. If not, repeat from Step 2, using the correct value of p.

5. Winding pitch t = bw / (Nl + 1) = pbw / (N + p).Nl may not be the same in all the layers. This will result in different values of t, since all layers should occupy the full winding breath. Do not permit differences of more than one turn in Nl.

6. Select a suitable interleaving, thickness i, for the winding pitch. The required height H = p(nhtmin + i) where nh is given in table in Sect. 3.4.2.

7. Resistance factor FR = 1+1/2(d/did)6.

8. Resistance per metre length of bunch rac = FRrdc / ns where rdc is the resistance per metre length of strand.

9. Pw = ci2enlavrac, where lav in m.

3.4.2.2. Single-layer and half-layer bunched-wire windings

This procedure applies only if ϑ = 0Arriving from Sect. 3.4.2.1, Step 41, take p = 1.

1. Winding pitch t = pbw / (N + p)

2. From a wire table, select the thickest strand for which tmin ≤ t/nt and note d, tmin and rdc.Also choose a suitable interleaving, thickness i, for the winding pitch.

3. Required height H = p(nhtmin + i).

4. ϕ = √(0.124nsfed3/t).

Note : if fe differs from the switching frequency, use the lower of the two values (in kHz).

5. Read FR from Fig.2. Beyond the range of Fig.2, FR = pϕ

6. The a.c. resistance per metre length of bunch Rac = FRrdc/ns, Where rdc is the d.c. resistance of a metre length of strand

7.Winding loss Pw = CI2eNlavrac, Where lav is in m.

If ϕ in Step 4 is greater than or equal to about 4, a half-layerdesign might be preferable for a sandwiched winding.

12

13

Part 3

3.4.3. Litz-wire windings

Litz wire is here taken to be a kind of unched wire in which the strands are so thin (d < ∆e) that the eddy-current effects can be neglected. It mains drawback is its low space (copper) factor, often only 25% to 30%.Litz wire is standardized and commercially available. The Standards specify numbers of strands, strand diameter, overall diameter and d.c. resistance (see also Sect. 5.2.4).Mechanical stress deforms the funch so that the breath in the layer and, thus the minimum winding pitch are greater than the overall diameter of the bunch, but the height of the layer is smaller.

1.Knowing the winding current IX, select a bunch of strand diameter d and number of strands ns such that πd2ns ≈ IX. The resulting current density will be about 4 A/mm2.

2. Due to deformation, the winding pitch t ≈ 1.2do, where do is the overall diameter of the bunch.

3. Provisional number of layers p' = Nt/(bw - t).

4. If p' is within 10% of the next lower integer, the winding may be feasible with care with this lower value. Otherwise, take for p the next higher integer. Then calculate n's = ns(p/p')2.

5. Select a standard Litz wire with the same strand diameter, and a number of strands as large as possible, But lower than n's.

6. The required height of the winding will be less than H = p(do + i), where i is the thickness of the interleaving.

7. The d.c. resistance per metre length is usually given for 20oC, multiply that value by 1.3 to get the resistance at 100oC. Then calculate Pw = CI2eNlavrdc, using lav in m. Note: If all windings in the transformer are of Litz wire, the split/sandwiched winding confi guration does not lead to lower losses.If a suitable Litz wire is not readily available, the above procedure can be adapted to use multiple Litz wire using similar methods to those for multiple wire, Sect. 3.4.1.

14

Part 3

3.5. Phase IVDetermine the optimum combination of non-ideal designs

When ideal designs overfl ow the winding space, non-ideal designs must be used. First, collect a few non-ideal versions of, if possible, primary and secondary. The accommodation procedure will then show which combinations fi t into the available height. The one having the lowest loss can then be used as the fi nal design.

3.5.1. Non-ideal design versions in solid round wire

Non-ideal design versions of wire windings are generated simply by successively reducing the number of layers to reduce winding height.The procedure given in this Section is not valid when at the start p = 1: in that case use the procedure of Sect. 3.2.2. However, it remains valid if step 2 yields values of p ≤ 1.

1. Note, for the foil former, bw and lav (is in m); and for the ideal design did (before rounding), N, Ie, p, t, d, H, rac and Pw. A table such as that shown below will found a convenient design route.

2. Reduce p by one step • of 0.5 for a sandwiched winding • of 2 in other cases.

3. If Nl = N/p is not integer, consider adapting N to obtain an integral number of turns per layer. If possible, let this adaptation cancel a previous one. Calculate the winding pitch t = pbw(N + p).

4. Select (from a wire table) The thickest standard wire for which tmin ≤ t and note d, do and rdc. Select a suitable interleaving thickness i.

5. Required height H = p(do + i).

6. Resistance factor FR = 1 + 1/2(d/did)6.If pid > 1, this expression holds for p ≤ 1, but this is not the case of pid ≤ 1.

7. Resistance per metre length of wire rac = FRrdc.

8. Winding loss Pw = CI2eNlavrac.

For each design version, repeat this procdure from Step 2.

15

Part 3

Having collected several versions of, as far aspossible, all power windings, follow Procedure 3,5.4. Example: In this table of design versions, the 3-layer version is the ideal design, the others are derived using the procedure above.

Primary: split, C = 2, bw =13.4 mm, did = 0.437 mm, Ie = 0.565 A, lav =0.053 m.

P 3 2 1 N 57 58 57 t 0.670 0.447 0.231 d 0.450 0.355 0.180 do - 0.414 0.222 rdc - 0.225 0.873 FR - 1.144 1.002 rac 0.223 0.257 0.875 pw 0.430 0.505 1.688 i - 0.06 0.06 H 1.728 0.948 0.282

16

Part 3

3.5.2. Non-ideal design versions of strip or foil windings

Here, the required reduction in height is obtai ned by using thinner conductor.

1. Note, for the coil former, bw and lav, and for the ideal design N, Ie, hid (before rounding ) h, i, rac, H and Pw. Set out a table in the form shown below.

2. For h, take the next available size below that used in the last design, and select a suitable interleaving thickness in i.

3. Resistance factor FR = 1 + (1/3) (h/hid)4.

4. Resistance per metre length of conductor rac = FR/(45bwh) .

5. Winding power loss Pw = CI2eNlavrac .

6. Required height H = N(h + i) .

For each successive version, repeat from Step 2 onward.

The table below shows an example of design versions of a strip conductor.

Secondary: sandwiched, C = 2, N = 4, bw =13 mm, hid = 0.222 mm, Ie = 8.05 A, lav =0.053 m.

h 0.2 0.15 0.1 0.073 i 0.1 0.06 0.06 0.04 FR - 1.069 1.014 1.004 rac 0.0104 0.0122 0.0173 0.0235 pw 0.286 0.335 0.475 0.646 H 1.20 0.84 0.64 0.452

Once several versions of, preferably all power windings have been collected, follow the procedure in Sect. 3.5.4.

16

17

Part 3

3.5.3. Alternative winding designs with reduced height

Multiple wire:To change ns follow the procedure in Sect. 3.4.1. It is diffi cult to predict the value of ns that is the best starting point for the following procedure when the loss margin is positive.

- If pid > 1, use the procedure in Sect. 3.5.1, but divide t (Step 3) and rdc (Step 4) by ns.

- For a sandwiched winding, if pid ≤ 1, use procedure 3.2.2 to design a half-layer winding, again dividing t (Step 1) rdc (Step 2) by ns.

Bunched wire:Use the procedure in Sect. 3.4.2.1 as follows.

- If pid > 1, commence with Step 4b and continue with values of p < pid.

- For a sandwiched winding, if pid ≤ 1 and p = 1, use the procedure in SEct. 3.4.2.2 with p = 0.5.

Litz wire:Use the procedure in Sect. 3.4.3 with lower values of p.

Having collected several versions of, preferably, all power windings, follow the procedure in Sect. 3.5.4.

3.5.4. Accommodation procedure

This procedure seeks combinations of design versions that fi t into the available winding height, together with their total winding loss.

1. Calculate, for all versions of the primary, the maximum permissible secondary height, as shown in the table below H2 max = Ha -H1 Note, also primary dissipation Pw1 of all versions.

2. Select, for each primary version, the secondary that most nearly fi lls H2 max and note its dissipation Pw 2.

3. Calculate the remaining ree height Hr = H2 max - H2 and total winding loss Pw = Pw 1 + Pw 2.

4. Select the combination having the lowest value of Pw.

The following table illustrates the accommodation procedure for a transformer with two power windings. The same principles apply for transformers with more power windings, even though the number of combinations is greater. Much labour might be saved if design versions with excessive loss per mm of winding height are not considered.

18

Part 3

Table illustrating the accommodation procedure for Ha = 1.778 mm and Pw max = 1.23 W.

primary versions p1 3 2 1 (split. single wire) pw 1 (W) 0.430 0.505 1.688 see Sect. 3.5.1 H1 (mm). 1.728 0.948 0.282

Height available For secondary H2max 0.050 0.830 1.496

Secondary versions H2 (mm) - 0.84 0.64 1.20 (Sandwiched. strip) h (mm) - 0.15 0.1 0.20 see Sect. 3.5.2 pw 2 (W) - 0.335 0.475 0.286

Remaining height Hr (mm) - -0.01 0.19 0.296 Total winding loss pw (W) - 0.840 0.980 1.974

Since the maximum permissible winding loss is 1.23 W, the 2-layer primary combined with the secondary using 0.1 mm strip results in a satisfactory design. There is no need to consider the secondary using 0.15 mm strip, although the excess height of 0.2 mm (C = 2) could, perhaps, be accommodated by careful manufacture.Note: The example is for a split/sandwiched winding confi guration (C = 2), so that winding heights are for one portion of each winding.

3.6. Phase V:Finalizing the design

In order to prepare the transformer design for production, four further steps are necessary.

1. Check that the design route followed was the correct one, and that the correct value of winding-confi guration factor C was used throughout.

2. Make a dimensioned sketch of a cross-section through the windings in the winding window. Calculate the conductor lengths required for each winding, using the actual average turn lengths. For wire windings, check that Nl turns of the wire selected will fi t into the winding breadth, that is, that bw ≥ (Nl + 1) tmin.

3. Prior to prototype evaluation, make a fi nal estimate of the transformer temperature rise using winding losses recalculated from the actual winding lengths obtained in the previous step.

4. In preparation for manufacture, collect all required information about core and coil former, windings and interleavings, screens and their interleaving, interwinding insulation, and terminations. Specify winding pitches for wire windings: windings are generally not close-packed since all layers should be of equal breadth.

This concludes the practical design procedures. The following sections contain supplementary information only.

19

Part 3

4. Background

The essential difference between a designer and a computer is creative thinking. Thus, a true designer will fi nd no lasting satisfaction in following design procedures that resemble computer fl ow charts. He cannot help asking why the procedures are as they are, and how they can be extended to solve related problems. This Section seeks to answer these and other questions with a view to avoiding mistakes due to misinterpretation of the instructions. It is mainly qualitative in nature : mathematics have been reduced to a minimum, and higher-order effects, although included in the formulas behind the design aids, are generally not discussed.

4.1. Eddy-current effects in windings

4.1.1. Transformer magnetics

The main fl ux that couples primary and secondary windings is core bound. Since primary and secondary ampere-turns oppose, the instantaneous value of the mutual fl ux is determined by the instantaneous difference between them. The magnetizing fl ux is proportional to the induced m.m.f. per turn. In an ideal transformer (zero core reluctance and zero winding resistance) with a short-circuited secondary, the induced m.m.f. and, thus, the magnetizing fl ux are both zero. Flux also crosses the winding space. this fl ux is not common to all windings, and not even to all turns in the same winding, and is, therefore, known as leakage fl ux.

Figure 3a shows the leakage fl ux paths in a simple, ideal, short-circuited transformer (with no main fl ux). Within the winding space, primary and secondary leakage fl ux lie in the same direction. Since they mutually repel, they are in parallel with the interface between primary and secondary: that is, in parallel with the layers if the windings are on top of each other. Thus, in order not to jeopardize this parallelism, only complete layers of equal breadth will be considered.The leakage fl ux density is maximum at the interface between primary and secondary, Fig.3b. On both sides of this maximum, the fl ux density falls roughly linearly to zero over the height of the winding.

Fig. 3 (a) Leakage flux paths in an idealized,short-circuited (to main flux) transformer. (b) Flux-density distribution.

secondary

primary

CLB

7Z80235

(b)

H1x

(a)

µ0N2I2/b

µ0N1I1/b

Consider a fl ux path crossing the winding window at a distance x from the wall of the coil former. Fig. 3a. Assuming that the current density over the height of the primary is constant,

20

Part 3

B ds = NI xH1µ0∫

where B is the fl ux density, s is the distance along the fl ux path, and NIx / H1 is the number of ampere-turns enclosed. The fi eld strength is negligible in the (high-permeability) core, and is assumed constant over the winding breadth, so that the leakage fl ux density

B = µ0 × N × I × x

b H1

where b is the fl ux path length outside core material.Figure 3b shows how B varies over the height of the wound area. There is a good case for arguing that, if bw is smaller than the width of the winding window (perhaps due to creepage allowance), b should be taken equal to bw (Ref. 2, page 355). Note that leakage fl ux is not due to imperfect core material, but is an intrinsic property of a winding.The leakage fl ux through the windings gives rise to eddy currents in the conductors. In a transformer of normal construction, the leakage fl ux lies parallel to the layers of the winding and roughly normal to the turns.Note that the leakage fl ux density varies from layer to layer and the leakage fl ux density is proportional to the sum of the ampere-turns in that layer and the ampere-turns in the layer between that layer and the nearest point of zero fl ux density. The average fl ux density and, thus, the eddy-current losses, can be reduced by suitably mixing primary and secondary windings. One result of this process is the split/sandwiched confi guration of fi g. 4b.

primary

secondary

(a)

(b)

B

B

7Z80236

primary/2

primary/2

secondary

Fig. 4 The effect of splitting a transformer primary into two portions either side of the secondary is to halve the peak

leakage fulx and, consequently, the eddy currents.

Figure 4a again shows the leakage fl ux distribution of a basic transformer winding arrangement. In Fig. 4b, the primary winding is split into halves, with the secondary sandwiched between them. This halves the peak fl ux density.To make this construction possible, the split winding must have an even number of turns, and an even total number of layers. The sandwiched winding, although physically one unit, can also be regarded as consisting

21

Part 3

of two portions; its leakage fl ux distribution is shown in more detail so that it can be seen that it may have an odd number of layers. Then, each portion contains a “half” layer ; a layer having half the height of a normal layer. Similarly, a portion of a sandwiched winding may contain a “half” turn if the “half” layer has an odd number of turns.

Due to the symmetry of the split/sandwiched confi guration, only one portion of each winding need be considered in calculating the eddy-current effects.

The splitting and sandwiching process could, of course, be repeated to further reduce eddy-current loss. However, this quickly becomes unpractical: each interface between primary and secondary portions requires extra insulation, usually including screens to reduce radio-frequency interference. Their presence reduces the space (copper) factor attainable in the winding window, eventually to such a degree that an improvement is either lost or not worth the extra complication.

A distinction can thus be made between• simple windings as in fi g. 4a.• split windings, such as the primary in fi g 4b• sandwiched windings, such as the secondary in fi g.4b.

The design process of Sect. 3 deals exclusively with winding portions:• a complete simple winding• one half of a split winding• one half of a sandwiched winding.

In the split/sandwiched winding confi guration, the number of turns in a winding portion is half that in the complete primary or secondary.

22

Part 3

4.1.2. Penetration of an electromagnetic wave into a conductor

Eddy currents are induced in a conductor exposed to an elctro-magnetic wave. They oppose the penetra-tion of the wave and, in resistive conductors, transform electromagnetic energy into heat.Let plane x = 0 be the surface of a conductor of infi nite depth. The electric and magnetic fi elds of a plane electro-magnetic wave propagating in the non-conducting semi-space above the conductor, and incident perpendicularly upon it, are tangential to the surface and mutually perpendicular. If the positive x axis is into the conductor, the penetrating wave can be described by

A - x + j ( ωt - x + φ )∆ ∆

Constant A and φ (part of the incident energy is refl ected from the surface) are not of interest here. The amplitude of the wave decays within the conductor as e-x/∆, where ∆ depends on the properties of the conductor and the frequency of the wave.At a depth x = ∆, the amplitude of the wave has decreased to 1/e of its value at the surface and the phase is delayed by one radian. At a depth equal to a small multiple of ∆, there is almost no fi eld in the bulk of the conductor and, consequently, no induced current.

At x = 2π∆ where the phase delay is just 2π radian, the fi eld strengths decay to the negligible value e-2π = 0,0019. This is at a depth one wavelength of the penetrating wave.

4.1.3. Skin effect

A straight, isolated, round wire carrying alternating current generates a concentric, circular magnetic fi eld, both in the wire itself and in the surrounding space.Isolated in this context means that there are neither other conductors nor magnetic fi elds in the vicinity of the wire. The low frequency fi eld distribution under these circumstance is shown in Fig. 5.

Fig.5 Low-frequency distribution of flux density B about a round current-carrying conductor. Here, x is the distance from the centre of

the conductor, and r is the conductor radius.

7Z80237

x

BB=

2πr2µ0Ix

23

Part 3

The fi eld is tangential to the surface of the wire. This fi eld induces eddy currents that oppose its penetration, enhacing the current fl ow near the surface and reducing it near the centre of the wire as shown in Fig.6.

Fig. 6 The magnetic field generated by an a.c. current in a wire induces currents in the wire that oppose its own penetration.

eddycurrents

magneticflux

I

I

7Z80238

As was the case with the penetrating electromagnetic wave, there is a tendency for current to fl ow only near to the surface of the wire: the skin effect. Figure 7 shows the current distribution carrying the same alternating current at various frequencies. As frequency increases, ∆ reduces and d/∆ increases.This current redistribution results in the a.c. resistance of the wire being greater than its d.c. resistance.The voltage is constant over any cross-section of the wire normal to its axis since there can be no radial current fl ow. In general, the voltage across a length of the wire is the sum of resistive and induced voltage drops. With a relatively thick wire (d/∆ >> 1), the voltage drop near the centre is mainly induced by the eddy currents, whereas, near the surface, where the current density is high, the voltage drop is mainly resistive.If the wire is replaced by a tube of the same material and diameter, such that the tube has the same d.c. resistance as the wire for a.c., its wall thickness will be equal to ∆, provided that the curvature of the surface is negligible (d >> ∆ ). For this reason, ∆ is known as the (equivalent) skin thickness.The term penetration dept is often used for ∆, but some authorities prefer this term for the wavelength (2π∆) of the penetrating wave discussed above. The (uniform) current density in the equivalent tube is equal to that at the surface of the wire it replaces. The ratio of a.c. resistance to d.c. resistance due to skin effect can be deduced from the relative cross-sectional areas of wire and tube :

FR ≈ d , ( d >> 1, in practice FR ≈ 1 ( d + 1 ) if d ≥ 5 )∆ ∆4

The a.c. resistance per meter length of wire is proportional to d-1, although the d.c resistance is proportional to d-2.

24

Part 3

d0

1

2

3

4

5

7Z89855

5

7

10

d/∆ =20

43

Fig. 7 Current distribution in a wire carrying constant a.c. at various frequencies.

Skin thickness depends on conductor material and frequency.

∆ = ρ π µ0 µr f

The current redistribution establishes an equilibrium between inductive and resistive voltage drops. Increas-ing the frequency does not alter the fl ux density at the surface because the current remains constant, but does cause an increase in the induced voltage which results in a smaller skin thickness and, thus, a higher current density. This, in turn results in a higher resistive voltage drop which opposes the concentration of current at the surface. A new equilibrium is achieved at a skin thickness proportional to 1/√ f. Similar reasoning can be established for the effect of µr and ρ.

The skin thickness itself is independent of the current carried by the wire. Because the wire is immersed in its own fi eld, there is a fi xed relationship between average current density and magnetic fl ux density.

25

Part 3

4.1.4. Proximity effect

A different class of eddy currents is found in wire exposed to an external alternating magnetic fi eld normal to the axis of the wire. In that case, eddy-current fl ow in opposite sides of the wire is in opposite directions, Fig. 8.

eddycurrents

external field

field due toeddy currents

7Z80239

Fig. 8 Eddy currents induced in a conductor exposed to an alternating magnetic field normal to its length. A rectangular

conductor is shown for clarity.

Eddy-current fl ow is confi ned to a skin below the conductor surfaces that are tangential to the external fi eld. No net current fl ow is assumed in the wire. To avoid the complications of a curved surface, a rectangular conductor is shown in Fig. 8.The situation resembles that of a turn in a winding exposed to leakage fl ux of the type described in Sect. 4.1.1. The essential difference is that the turn carries an alternating current, and that the leakage fl ux density is proportional to this current. The relationship between current and leakage fl ux is not the same all over the winding, but depends on the position of the layer which contains the turn under consideration.Because the leakage fl ux to which the turn is exposed originates from other turns in proximity to it, the eddy-current phenomenon is known as the proximity effect in this case.

(a)

(b)7Z80240

Fig. 9 Four turns from a layer of a winding: the concentric flux due to the winding current (a) cancels between turns so that the

resultant flux lines are parallel to the layer (b).

26

Part 3

Figure 9a represents a section through a few turns in a layer with the concentric fl ux paths associated with an isolated wire. Between the turns, however, individual fl ux lines, which are roughly perpendicular to the layer, oppose and so tend to cancel. The result is the fl ux pattern of Fig.9b. Such fl ux patterns from a number of layers result in the fl ux distribution of Figs. 3 and 4.

The difference between the two types of eddy-current effect is now evident.

Skin effect: tendency for the current to fl ow near the conductor surface, no current reversal and thus no increase in current fl ow.

Proximity effect: there are two skin regions below the surfaces tangential to the magnetic fi eld: besides the tendency for the main current to fl ow in the skin region where the magnetic fi eld is highest, the skin regions carry opposite eddy currents that increase the effective current fl ow.

The leakage fl ux density varies from layer to layer, but is constant over the breadth of each layer (Figs 3 and 4). The proximity effect is greatest in the layers at both sides of the interface between primary and secondary, where the leakage fl ux density is greatest. The simplest case is that of a strip or foil winding, where each layer has only one turn. The number of layers (and turns) is thus known at the start of the design process. Figure 10, which is based on normalised conductor height, shows the relationship between proximity effect and leakage fl ux density.

The proximity effect in a wire winding depends not only on wire diameter, but also on the layer space (copper) factor. The main geometrical parameters of a turn in a wire winding are wire diameter d and winding pitch t. Of course, t ≥ do . A useful simplifi cation is to regard a round wire as a square wire of equal cross section and, thus, equal current density, Fig. 11. Then, h = d√ (π/4) ≈ 0.886d. Extending this model allows a layer to be regarded as a strip of thickness h and layer space (copper) factor Fl = h / t carrying a current Nl times greater than that in the wire. The leakage fl ux is the same as that of the wire-wound layer, but the current density is 1/Fl times greater

(a)

(b)

(c)

1stlayer 2ndlayer 3rdlayer 4thlayer 5thlayer

ϕ=0,59

7Z89852

2

1

0

4

3

2

1

0

543210-1-2-3

1,33


ϕ=2

7Z89853

20

10

0

300

200

100

0

543210-1-2-3

27,9


ϕ=8

7Z89856

50

25

0

3000

2000

1000

0

543210-1-2-3

137

1,01 1,09 1,251,49

1,82

1,9 8,4 21,440,9

66,9

8 40 105 202 331

Fig. 10 The proximity effect in a winding of 5 layers. Effective conductor heights are ϕ = 0.59 (left), ϕ = 2 (centre) and ϕ = 8 (right). Sections through the ends of the layers of the winding are shown, hatched. The layers are of strip conductor, but could also be Nl turns of rectangular wire. The current density distribution is plotted in (a). The full line is the amplitude, the broken line the real part, and the dotted line the imaginary part. The losses are plotted in (b). The loss distribution (solid line) is proportional to the square of the current density amplitude. The broken line shows the average loss in each layer, and the dotted line the average loss in the complete winding. Since the scale calibration is multiples of the d.c. loss, the dotted line shows the value of FR. (c) is the leakage flux diagram, obtained by integration of the current density. Scale divisions are layer current; line types are as in (a).

27

Part 3

The equivalent conductor height ϕ = h/∆ for strip or foil windings, and ϕ = (h/∆)√Fl = (d/∆)√(Fl4/π) ≈ 1.128 (d/∆)√Fl for windings of solid round wire

Comparison of h/∆ for strip with (h/∆)√ Fl for wire indicates that the skin dept in wire seems 1/√Fl times greater than in strip. This can be understood by remembering that ∆ is proportional to √ρ and that increasing ρ or increasing the current density has the same effect on the resistive voltage drop on which the equilibrium depends.

do d t

th

h

(a)

(b) 7Z80241

Fig. 11 (a) Principal geometrical parameters describing a turn in a given layer. (b) Replacing a round wire with a square one of equal cross sectional area simplifies the discussion since now a

layer resembles a turn of strip conductor.

4.2. A.C. resistance

4.2.1. Resistance factor FR

The increase in conductor resistance due to proximity effects can be expressed in terms of resistance factor FR: the ratio of a.c. resistance to d.c. resistance.The chart of Fig. 12, based on Dowell’s work, Ref.1, gives FR as a function of equivalent conductor height ϕ with the number of layers p as a parameter. Inspection of these curves shows that there is a region where FR is proportional to ϕ. In multi-layer windings, this region commences when ϕ is greater than about 3, but for single layer windings, when ϕ ≥ 1. This is the region of skin conduction, where no current fl ows in the bulk of the conductor. The situation is similar to that for skin effect when d/∆ > 1, where FR is proportional to d.

The curve for p = 1 (eddy-current loss negligible when ϕ ≤ 1) also represents the behaviour of the fi rst layer of a multilayer winding. The curve for p = 0.5 shows similar behaviour, but at double the values of ϕ. In all layers of a multilayer winding except the fi rst, eddy-current loss (proportional to FR - 1) increases as ϕ4 up to ϕ ≈ 2, and in proportion to ϕ above ϕ ≈ 3. This most clearly demonstrated by the curve for p = 1.5. Eddy-current losses in the higher layers evidently dominate the a.c. losses in the fi rst layer, unless ϕ < 1. This explains the difference that will become apparent later between single and half-layer windings, and multi-layer windings.

It should be noted that eddy-current loss remains proportional to ϕ4 in the low bend of the curves, below FR ≈ 2 The bend is due to d.c. loss not being negligible compared to eddy-current loss in that region (see also Fig. 2, mathematically a plot of FR − 1, scale calibrated for FR).

28

Part 3

Figure 12 is most useful for determing the a.c. resistance of a winding of known geometry. However, it is not a convenient basis for optimizing winding geometry itself.

Fig. 12 Resistance factor FRversusequivalen conductor height ϕ, with number of layers p as a parameter, after Dowell (Ref.1).

7Z89845

p =10

6

4

3

2,5

2

1,5

1

0,5

0,1 0,2 0,5 1 2 5 10ϕ

1

2

5

10

20

50

100

200

500

1000FR

4.2.2. A.C. resistance per metre conductor length rac

The A.C. resistance per metre conductor length is a useful quantity for comparing various versions of a given winding. Winding loss is proportional to winding Rac = Nl av rac. In a given design problem, the number of turns and coil former dimensions are known. Conductor length Nl av is only slightly infl uenced by the number of layers or the conductor size. Thus, achievement of the design goal - the winding geometry for minimum loss - can be reduced to the achievement of minimum rac.

29

Part 3

4.2.3. Windings of strip of foil conductor

The geometry of windings of full-width strip or foil conductors is completely estblished once strip thickness h is known. Strip breadth is bw, and the number of layers is equal to the number of turns in the winding portion. These are all set by the boundary conditions, so that the only parameter remaining to the determined is thickness. Since rac is proportional to FR/ϕ, its value can be derived from Dowell’s chart, Fig. 12.

rac = rdc FR = ρ FR = ρ . FR (since ϕ = h/∆)ϕbw∆bwh

thus

rac bw∆ = FR ϕρ

Therefore, it is suffi cient not plot FR/ϕ against ϕ as in Fig. 13. There, the straight, dash-dot line labelledFR = 1 represents rdc. Since rdc is proportional to ϕ−1 its slope is -1.Each curve in Fig. 13 has a minim, marked by a dot. Each minimum occurs at the normalized, ideal strip thickness hid/∆ and the lowest possible value of racbw∆/ρ from which the value of rac id can be calculated.The term 'ideal' is used in preference to 'optimum' since the former implies desirability but not necessarily feasibility. A thickness other than hid may have to be used due either to conductor or space availability limitations.

20

10

5

2

1

0,5

0,20,1 0,2 0,5 1 2 5 10

bW∆ρrAC

7Z89844

h/∆

FR= 1FR= 4/3

0,5

1

1,5

2

2,5

3

3,5

4

5681015N = 20

Fig. 13 Plot of FR/ϕ versus ϕ which virtually solves the design problem for strip windings. Given the number of turns N, the

minumum in the appropriate curves gives the ideal (lowest loss) strip thickness as a multiple of the skin thickness. The resistance

per metre length of strip can also be determined.

30

Part 3

As, for a given ∆, strip thickness increases beyond hid eddy-current loss increases more quickly than d.c. loss decreases. Note that hid is independent of winding current because leakage fl ux density is proportional to current density. All minima in Fig. 13 lie close to the straight line FR = 4/3, although some deviation can be seen at low values of N. In the practical design procedure, in Sect. 3.2.3, hid is found assuming this straight line, but a correction factor CN has been added for the deviation that occurs when N < 5.since, usually h ≠ hid, the approximation

FR = 1 + (1/3) (h/hid)4

is included in the design procedure to give the a.c. resistance.Foil or strip conductor is only recommended for sandwiched windings, not for split or even simple windings. There is no guarantee that current distribution is uniform over the breath of a strip. In fact, there is a tendency for current density to be higher near the edges of a strip when ∆ << √ (bwh), which is invariably the case (Ref. 3). Non-uniform current density cannot occur when the leakage fl ux is truly parallel to the layers, so advantage should be taken of the repulsive forces between the leakage fl ux of primary and secondary (Sec. 4.1.1). Sandwiching a strip winding between two portions of a wire winding, whose current density must be uniform over the layer breadth, ensures that there are strong repulsive forces on both sides of the strip winding, where leakage fl ux density is maximum.

4.2.4. Wire windings : winding breath-to-turns ratio T.

The a.c. resistance per metre length of wire rac can be derived from a plot of FR/ϕ 2 against ϕ , Fig. 14.For rectangular wire of breadth b (in the layer) and height h (normal to the layer)

ϕ = h b∆ t

and FR = rac ∆2 t = rac ∆2 t

rdc h2 b ρhϕ2

For round wire of diameter d

ϕ = h b = ( π ) d d = t ( d π )∆ t t t∆ ∆4 4

3/4 3/2

so that, for agiven ratio t/d

FR = rac ∆2 t = rac ∆

2 t 4πρh ρdϕ2

is proportional to rac.Here, rdc ∝ ϕ-2 so that the line FR = 1 has a slope of - 2. Whereas Fig. 13 gave the solution of the design of a strip winding, where the number of layers and conductor breadth were known, the plot of Fig. 14 does not contain the solution for wire windings. The number of layers is still not known.The winding breadth is introduced in the form of the winding breadth-to-turns ration T = bw/N. Physically, T is the winding pitch in a single-layer winding containing all N turns. Note that at the interface between primary and secondary, the leakage fl ux density B = µ0 I/T, Sect. 4.1.1 and also that T is known at the start of the design process.

31

Part 3

Since t = pT, for a rectangular wire

ϕ = h b ∆ pT

and for round wire

ϕ = ( π ) d d ∆ pT4

3/4

FR= 1

7Z89846FR/ϕ2

ϕ

100

50

20

10

5

2

1

0,5

0,2

0,10,1 0,2 0,5 1 2 5 10

2,5

p =10

6

4

3

2

1,5

10,5

Fig. 14 Plot of FR/ϕ2 versus ϕ, a first step towards the solution of

the design problem for wire windings .

32

Part 3

4.2.4.1. Close-packed windings of round wire

For close-packed windings, rac∆2/ρ can be plotted against T/∆, Fig. 15; this gives better access to the design problem. In close-packed wire windings, t = do, so the layer space (copper) factor depends on the ratio d/do for the wire. This ratio is not constant : standard-wire tables show that d/do is smallest for fi ne wire, but the rate of change is rather low. If. Fig. 14 were replotted to give rac∆2/ρ against, t, the calibration for the axes would , of course, change, but the shape of the curves would hardly be affected because the rate of change of d/t is so low. In order to replace t by T, the values for each of the curves must be divided by p, since T = t/p. This shifts every curve horizontally a distance -log p.The introduction of T is an important step forward: knowing T/∆, it is possible to read the lowest possible value of rac∆2/ρ, the optimum number of layers, and, fi nally, from do = t = pT, the optimum wire size can be determined.In order to plot Fig. 15, values of d/do and its variation with wire size were estimated. So, although the plot of Fig. 15 is adequate for illustrating the theory, its accuracy is not suffi cient for practical design purposes.

rAC∆2

ρ

7Z89847

T/∆

5

2

1

0,5

0,2

0,1

0,05

simple, split or sandwichedsandwiched only

8 76

5

4 3 2 1 p=0,5

0,1 0,2 0,5 1 2 5 10 20

Fig. 15 Plot of rac∆2/ρ versus T/∆ with p as a parameter for close-packed wire windings.

33

Part 3

4.2.4.2. Spaced windings of round wire

Windings are also possible with a pitch greater than the overall wire diameter: spaced windings. This introduces an additional degree of freedom that can be exploited to achieve a further reduction in a.c. resistance. This is illustrated by the plof of Fig. 16.

The solid curves in Fig. 16 are for close-packed windings (t = do), as in Fig. 15. The dash-dot curves are for windings of two layers, in which t > do. These show that there is a range of T/∆ where spacing results in lower values of rac. However, spacing should not be excessive : better performance is always possible if a smaller number of layers can be used.The straight dashed lines in Fig. 16 are in further addition : they show that spacing displaces the curves so that similar points lie on a line of slope -2/3. They also show that the curves for p < 1 are so steep that spacing brings no improvement.In effect, spacing allows designs intermediate between close-packed versions.

rAC∆2

ρ

7Z898485

2

1

0,5

0,2

0,10,2 0,5 1 2 5 10

T/∆

p=3 p=2 p=1

Fig 16 Plot of Rac∆2/ ρ versus T/∆ extended for spaced windings

4.2.5. The basic solution of the design problem

Suffi cient information is now available to make a plot that effectively solves the design problem. This is given as Fig. 17. Figure 17a differs from Fig. 14 in that spaced windings are included where they give lower loss. Furthermore, those parts of the curves for close-packed windings that are not useful for design purposes have been suppressed.The full lines apply to optimum designs with an integral number of layers. The broken lines apply to optimum designs with a half layer (except in the plot of winging height, these often coincide with the full-line curves). The dotted lines are for non-optimum, close-packed windings.There it can be seen that, if T/∆ is known, the optimum design is completely determined.

34

Part 3

Further inspection reveals that : - for T/∆ greater than about 2, the optimum design is a single-layer winding but, in sandwiched designs, when T/∆ is greater than about 6, the optimum is a half-layer winding- for T/∆ less than about 2, the optimum design has more than one layer.

Both the full and the broken lines for p > 1 show that the resistance minima occur generally for spaced windings (Sect. 4.2.4), so that it is important to include in the manufacturing instructions a specifi c statement that windings should not be close packed. The curves for p ≤ 1 have a slope of about -1, indicating that FR∝ ϕ−1, so that they belong to the region of sin conduction (Sect. 4.2.1.). Moreover, they are, in principle, close-packed windings (Sect. 4.2.4.1). It is clear that multilayer (p > 1) windings and those with p ≤ 1 will require different design methods.

rAC∆2

ρ

7Z89849

T/∆7Z89850

T/∆

7Z89851

T/∆

5

2

1

0,5

0,2

0,1

0,050,1 0,2 0,5 1 2 5 10 20

0,5

1

2

5

10

0,1 0,2 0,5 1 2 5 10 20

1

2

5

10

0,1 0,2 0,5 1 2 5 10 20

86

5

8

8 7 6 54,543,5 3

2,5 21,5

1

p=0,5

p-0,511,557

4 3 2 1,5 1 p=0,57

d/∆

He/∆

46

3 2

Fig. 17 Basic design charts for wire windings: (a)Rac∆2/ ρ, (b) (a) d/∆, and (c) He /∆ versus T/∆ with p as a parameter. He is the required height excluding interteaving. Knowing T/∆ , the ideal winding geometry and the resistance per meter length of wire

can be determined.

35

Part 3

4.2.5.1. Multilayer windings (p > 1)

The full lines in the chart for rac∆2/ρ and d/∆. Fig. 17, consists of several sections, each for a particular value of p. Together for p > 1, they form nearly straight lines. The ideal values of d an rac are, thus, generally independent of p. There is only a slight deviation for the lower values of p: it can be seen that the lines for p = 1.5 do not coincide with those p = 2. The resistance factor for optimum or ideal design FR id ≈1.5, which means that the eddy current loss is half the frequency-independent resistive loss, also called d.c. loss.The slope of the straight liens indicates that rac id ∝ (T /∆)

-2/3 and did ∝ (T / ∆)1/3.

Since rdc ∝ d-2, FR = rac/rdc must be constant.

If the actual wire diameter d deviates from the ideal diameter did, the estimated resistance factor becomes

FR = 1 + ½ ( d )6

did

This approximation, derived form the fi rst two terms of the series expansion of Dowell’s expression, Ref. 3 is suffi ciently accurate for values of FR ≤ 2. In an ideal winding, where d = did, FR < 1.5. The expression is not valid in the region of skin conduction. To fi nd the ideal wire diameter for windings having a large number of layers, the expression d∞/∆ = 1,45 (T / ∆)1/3can be used. In a more practical form this expression reads

d∞ = 17.1 bw N fe

3

The ideal wire diameter for a practical winding, did, is then obtained using the correction factor Cp.This factor corrects for the effect of the small steps between the line sections for practical numbers of layers. Using these expressions, ideal designs can be obtained by simple calculation.Once knowing did the winding geometry can be determined. After rounding did to the nearest standard wire size, do is known. Following the design rules in the practical sections, the number of layers and the required height are easily found. Knowing rdc of the selected wire, rac can be estimated after calculation of FR.

4.2.5.2. Single-layer and half-layer windings

This is the region where the procedures of the previous section lead to a number of layers p ≤ 1.This defi nition is given because the criterion T/∆ > 2 is only an indication. First is necessary to decide between strip and wire windings. Strip of foil windings are often preferable, see Section 5.2.1. For simple and split designs, it is known that p = 1. This is also the case in sandwiched designs if T/∆ is below about 6. Since windings should be close packed, geometry is fi xed. the end of the curve for p = 1 at T/∆ = 6 is not a theoretical limit, the curve can be extended. But d may be so great that a different form of conductor will probably be chosen (Section 5.2). For sandwiched windings, a half-layer winding may be preferable, provided T/∆ is greater than about 6. That saves winding height without a loss penalty. In case of doubt (T/∆ ≈ 6), a choice can be made after having worked out both alternatives.

36

Part 3

The remaining problem is the estimation of rac. To solve that, calculate

ϕ = ( π ) d d ∆ pT4

3/4

or, alternatively,

ϕ = 0.124 fe d3 t

and read FR from the plot of FR versus ϕ, Fig.2.Figure 2, gives the FR plot in a more suitable form than Fig. 12. Do not use the expression for FR given in the previous section; it is not valid in the region of skin conduction.

4.3. Winding height

4.3.1. Ideal designs

In previous sections, methods were developed for designing minimum-loss windings of both strip and wire conductors. In developing these methods, possible height restrictions have so far been ignored.By doing this, each winding could be treated separately, the design of a primary was not infl uenced by the secondary, and vice versa. Such designs, affected by a height restriction, are called ideal designs.Similarly, terms such as ideal windings, ideal number of ayers, ideal conductor size, etc., are also used.It is, of course, necessary to check whether the situation is ideal : that the height of the winding window does not impose a height restriction on the windings.The height of a winding (portion) is calculated from:

for strip windings, H = N (h + i)for wire windings, H = p (do + i)

For the available winding height (height available for all power-transferring windings) see Section 2.1.

4.3.2. Non-ideal design versions

In the strip winding the only way to reduce winding height is to use thinner strip. Various non-ideal design versions are found by repeatedly stepping to a smaller available strip thickness. The a.c. resistance is found from the expression for FR in Sect. 4.2.3. In wire windings a smaller winding height is obtained by reducing p. This leads to reduced pitch (t = pT) and thus, also, to reduced wire size (do ≤ t), so that d < did. To reduce the difference between d and did as far as possible, a non-ideal wire winding should be close-packed using the thickest possible wire for the number of layers and winding breadth. The a.c. resistance can be estimated by means of the expression for FR in Section 4.2.5.1. Several non-ideal design versions are found by repeatedly stepping to a lower number of layers. Thus, non-ideal as well as ideal design versions can be found. Although these non-ideal design versions have smaller winding height than the ideal design, they also have higher loss.

IMPORTANT: There is no justifi cation for using conductor sizes in excess of the ideal size (apart from rounding a calculated ideal size to the available size). Reduction of current density is a false motive. The extra conductor material has only and adverse effect.

37

Part 3

4.3.3. Accommodation

Once a few design version (the ideal design and one or more non-ideal versions) of the primary and secondary have been collected, selection of the most suitable combination is made by an accommodation procedure.The accommodation procedure is given in detail in Sect. 3.5.4. It comprises the following basic steps: - list the design versions obtained, the height they require and their losses. - make combinations of primary and secondary designs having a total height not exceeding the available height, and note the winding loss of each. - fi nally, select the lowest-loss combination.Demagnetizing and control windings are usually designed for minimum height rather than minimum loss, since they do not take part in power transfer. Their height is taken into account in determing the height for the power windings.

4.4. Design evaluation

A design attempt is successful if the transformer does not run hot. Final evaluation is, of course, experimental. Unnecessary experimentation can however, be avoided if the estimated losses are fi rst compared with the estimated permissible loss. The permissible loss is estimated from the permissible temperature rise and the thermal resistance given in the data of the core selected. The thermal resistance in the published data can be reduced by potting the transformers. If this is to be done, it might be possible to eliminate the creepage distance, so increasing winding breadth. Core loss was estimated in the process of core selection and determination of the number of primary and secondary turns. The loss in each winding can be estimated as

Pw = I2nlavrac

Current densities much higher than are usual in low-frequency transformers may well be acceptable. As has been mentioned, conductors in excess of the ideal size make matters even worse.If a design attempt fails, it is advisable to fi rst review the chosen boundary conditions. This would not invalidate the results of the core selection procedure. An improvement may be obtained by using spit/sandwiched rather than simple windings. These tend to lower rac, thicker conductors and increased height. A different conductor form might also be considered.A new start, including the core selection procedure, is required if the given boundary conditions have to altered. It may be possible to use the same core size if switching frequency is increased. This leads to a fewer turns, of thicker conductors and, thus lower winding loss.To avoid fruitless effort, the practical design process of Sect. 3 is so organized that an initial design evaluation is made as soon as suitable ideal designs and their losses have been determined. Winding height can be reduced with the penalty of increased loss (Sect. 4.3.2.) and vice versa (by using bunched wire, for example, Sect. 5.2), but this will not help if the ideal design has neither loss nor height reserve.

38

Part 3

4.5. Some notes

4.5.1. Multiple-output transformers

Figure 18 shows a possible winding arrangement and leakage fl ux density distribution for a multiple output transformer. In both primary and secondary 2, the fl ux density varies from zero to maximum, so these can be designed using the present methods. That is not possible for secondary 1 because its leakage-fl ux distribution does not agree with the assumptions. The ratio for leakage-fl ux density to number of turns is much higher than has been assumed so far.

Fig. 18 An example of a multiple-output transformer with corresponding leakage-flux diagram.

secondary 2

secondary 1

primary

sreens

b

7Z80242

µ0Nsec2 Isec2bw

µ0Nsec1 Isec1bw

µ0Nprim Iprimbw

The ideal wire diameter for an infi nite number of layers d∞ in the winding exposed to the higher leakage fl ux calculated as normal, but the layer correction factor becomes

Cp = 1 - ϑ (1 - ϑ3)1/3

where υ is the ration of minimum to maximum leakage fi eld over the height of the winding. In secondary 1 of Fig. 18, ϑ = (Nsec 2Isec 2) / (NprimIprim). The factor Cp is always less than unity and decreases with increasing ϑ . The higher leakage fi eld is compensated for by using thinner wire. As is usual (ϑ = 0) for wire windings, FRid ≈ 1.5; the formula for FR still applies. Similarly, for strip conductor,

CN = (1 - ϑ)3

1 - ϑ3

4

As normal for strip windings FRid ≈ 4/3. To obtain lowest total winding loss, the layout ofthe secondary windings ( 2 or more) in the winding window must be considered carefully. - Secondaries of the same conductor type, either wire or strip: exposing the winding carrying the smallest current, that is, the one using the thinnest conductor, to the highest leakage fi eld will result in lowest loss. Remember that the eddy current loss increases in proportion to ϕ4

, Sect. 4.2.1. - Secondaries of different conductor type (wire or strip): the optimum arrangement is not easy to predict. Work out the possible layouts and select for lowest total loss.

39

Part 3

4.5.2. Push-pull transformers

In push-pull transformers, Fig. 19, not all windings carry current at the same time. The operation of push-pull converter is explained in part 1 of this series of publications. Here it is suffi cient to split up one converter operating period into - interval a : N1A and N2A are conducting, other windings carry no current. During this interval the current waveforms in N1A and N2A are the same. - interval b: N1B and N2B are conducting. - interval c, occurring twice per period: N1A and N1B are nonconducting, N2A and N2B carry equal and opposite fl ywheel currents about half as great as the secondary currents during intervals a and b (magnetizing current neglected). The closer intervals a and b approach a half period (at full load) the shorter this interval will be. Whether or not the fl ywheel current pules have an infl uence on winding, design depends on the winding arrangement.

Fig. 19 Push-pull transformer windings divided to define conduction periods.

7Z80245

N1A

N1B

N2B

N2A

Bilifar windings are often used to achieve close coupling between winding halves. Figure 20 shows a cross-section through the windings and the leakage fl ux diagrams during the three intervals. If wire windings are used the fl ywheel currents during interval cancel (higher order effects neglected). That is not quite so if “bifi lar” strip windings are used (two full-breadth strips and interleaving wound simultaneously). The peak fl ux density is then about 1/(2N2) times that during interval a or b. Since the eddy current loss in interval c may be negligible if the half-secondaries have at least a few turns. It might not be possible to ignore the contribution of the fl ywheel currents to that d.c. loss unless interval c is very short.

Fig. 20 Cross-section through the windings of a bifilar-wound push-pull transformer together with the leakage-flux diagrams

for the conduction intervals described in the text.

N2A+N2Bbifilar

N1A+N1Bbifilar

screen

7Z80246

B B B B

interval a interval b interval c(wire windings)

interval c(strip windings)(N2A=N2B=3)

During intervals a and b, eddy currents are generated in both the conducting and the non-conducting windings. Both windings of a pair are exposed to the same leakage fl ux and have about equal eddy-current loss. Only one of these has d.c. loss at a time. For minim loss (ideal designs), the ratio of eddy-current

40

Part 3

loss to d.c. loss remains one half for wire windings and one third for strip windings. That requires thinner conductors than in non-bifi lar windings have equal numbers of turns, and winding pitch or strip breath.For multilayer wire windings, the reduction is 2-1/6 = 0.89 (one step in the wire table), and for strip windings, 2-1/4 = 0.84The reader can adapt the design procedures of Section 3.2 to cope with bifi lar windings.Multilayer wire windings, procedure 3.2.1: - in step 2 multiply d∞ by 0.89 - in step 4 use 2tmin in the expression for Pid - in step 9 calculate the loss in a winding pair. Single-layer wire windings, procedure 3.2.2.: - in step 2, use 2tmin rather than tmin - in step 6, calculate rac = (2FR - 1) rdc - in step 7, estimate the loss of a winding pair.Strip or foil windings, procedure 3.2.3: - in step 1, multiply CN by 0.84 - in step 5, estimate the loss of a winding pair.Bifi lar windings are not possible if the wire insulation cannot safely withstand at least twice the peak voltage across a winding half. The transformer of Fig. 21 differs from the previous one in that the primary winding halves do not share the same space. The leakage fl ux diagrams show an asymmetry. During the interval a winding N1B, although not conducting is subjected to a high leakage fl ux and has about three times the eddy current loss of the conducting winding N1A, because the average of the r.m.s. fl ux density squared in N1A is one third of that in N1B. In interval b, however, N1A has no eddy current loss. The primary winding halves thus have different a.c. resistance due to a difference in eddy-current loss by a factor of four. Moreover, the leakage inductance is greater in N1A. The energy stored in the leakage fi eld can be expressed as ½LI2or as ½VB2/µO, where L is the leakage inductance and V the fi led volume (here proportional to the area of the fl ux diagram times the average turn length).

Fig. 21 Cross-section through the windings of a push-pull transformer with a bufilar secondary, together with the leakage flux

diagrams for two conduction intervals. Note the asymmetry.

screen

B Binterval a interval b

7Z80246

N2A=N2Bbifilar

N1B

N1A

By arranging the windings as in Fig. 22, only a minor asymmetry due to different turn lengths remains and eddy current loss in the non-conduction primary is eliminated. In this arrangement the coupling between the primary winding halves will be less close and the stray capacitance across the secondaries, now situated between two screens, will be increased.

41

Part 3

Fig. 22 Rearrangement of the windings of fig.21 showing the reduced asymmetry.

screen

B Binterval a interval b

7Z80248

N2A=N2Bbifilar

N1B

N1A

In Fig. 23, the secondaries are also divided: non-conducting windings are not exposed t leakage fl ux. Now eddy current loss during interval c occurs in the secondaries. This indicates the effect of different current waveforms in primaries and secondaries, which means hat their effective frequencies are different (see Section 6).For split/sandwiched confi gurations, similar considerations apply as indicated above for simple confi gura-tions. Here, also, leakage fl ux diagrams will be a great help in the analysis. The discussion in this Section clearly illustrates the general risk of overlooking important parameters in striving for an optimum in a particular respect.

Fig. 23 Cross-section through the windings of a push-pull transformer showing that, with divided primary and secondary

windings, the non-conducting windings are not exposed to leakage flux.

N1B

N2B

N2A

N1A

screens

7Z80249

B B B

interval a interval b interval c

42

Part 3

4.5.3. Flyback transformers

In a fl yback transformer, current conduction in primary and secondary occurs alternately. When the primary conducts, there is no current in the secondary, and vice versa. Two different leakage-fl ux density diagrams are, thus, required, Fig. 24.

Fig. 24 In a flyback converter transformer, primary and secondary conduct alternately. Thus, two leakage flux diagrams are required for loss analysis: one for the inner winding conducting (left) and

one for the outer winding conducting (right).

outer winding

inner winding

7Z80243B

B B

The fl ux diagrams shown do not pretend to be accurate. In drawing the full-line portions tight coupling to the core and a leakage fl ux parallel to the layers were assumed. However, since the repulsion between primary and secondary fl ux does not exist, the latter assumption is an idealization. The broken lines are speculative. It is therefore recommended that the outer winding has the thinner wire. The winding arrangement is recommended for the reasons given in Sect. 4.5.2. The leakage fl ux generated by the current in the inner winding also induces eddy-currents in the outer winding.As a practical guide, it is recommended that the outer winding be wound with a wire one size smaller than that calculated using the procedures given in this article. The estimated winding loss will then probably prove slightly optimistic.Although the present design methods are not as accurate for fl yback transformers as they are for forward and push-pull types, their use is helpful in preventing excessive eddy-current loss.Strip or foil windings: since the leakage fl ux might not be truly parallel to the layers we should not be surprised if the losses are higher than calculated. The current density near the edges of the conductor might be considerably higher than that near the middle of the breadth. That effect is nearly independent of the conductor thickness. The use of multiple wire, bunched or Litz wire might be a better solution for low-voltage windings.

4.5.4. Screens

Eddy current losses will also occur in the screens between primary and secondary. Screens are always situated in positions of maximum fl ux density Screens should not be thicker than necessary; there should be the minimum of overlap of the ends. copper is not the most suitable material : a (non-magnetic) conductor of higher resistivity is preferable. Eddy-current loss is directly proportional to conductivity, and proportional to the thickness cubed. Thus, screens, should be as thin as possible (UL-1244 specifi es a minimum of 0.15 mm).Although its low resistivity makes copper the natural choice for the winding conductor, it is not the fi rst choice for screens. Phosphor bronze CuSn8 has a conductivity 10.9% of that of copper at 20 oC and 13.8% at 100 oC. The use of such material reduces losses in proportion.Additional loss can be caused by the capacitance of the overlap at the ends of the screen, which causes the screen to act as a shorted turn. This overlap should be as small as possible.

43

Part 3

5. Chosen boundary conditions

Once the given boundary conditions have been established from the choice of core and numbers of primary and secondary turns, the designer can choose winding confi guration and conductor type.

5.1. Winding confi guration

The designer can choose between - simple windings - split / sandwiched windings

The lower losses of split/sandwiched windings arising from the halved peak leakage-fl ux density will be evaluated further, as will also the required winding height. In the split/sandwiched confi guration, a winding portion has half the number of turns of the complete primary or secondary winding. Hence the value of T = bw/N is doubled.

5.1.1. Strip or foil windings

The use of strip or foil windings is only recommended for sandwiched windings (Sect. 4.2.3). The following discussion is given for completeness, because the route of a design process cannot always be predicted.Analysis of the design formula in Sect. 3.2.3 shows that for moderate values of N (Where CN ≈ 1), halving N increasing hid by a factor of about √2. Since the total number of turns in the complete winding remains the same, sandwiching increases the height of the ideal winding by 41%. The infl uence of the interleaving is neglected. Besides the increase in required height, the extra winding interface with its insulation and screening reduces the available height. In the extreme case, where N changes from 1 to 0,5, hid doubles because CN also increases by a factor of √ 2.In Section 4.2.3 FRid was shown to be about constant for all values of N. So the ideal winding resistance tends to decreae as 1/√2, a reduction of 29%. For N = 0.5, rac id is half that for N = 1A smaller improvement can be obtained where the sandwiched version has to be non-ideal.Even if the simple version has to be non-ideal it may occur that its sandwiched counterpart has a lower a.c. resistance.

5.1.2 Wire windings

If, for a simple wire winding having at least 2 layers, the ideal design is replaced by an, also ideal, split or sandwiched winding rac will be improved by about 37%. The penalty is a considerable increase in required height. The actual increase in height is not readily predictable, since the number of layers may change ( the effect of doubling T/∆ is shown in Fig. 17) In many cases the split or sandwiched version will be non-ideal, so that the improvement in resistance is smaller than 37%. The improvement of a (non-ideal) split or sandwiched winding replacing a non-ideal simple winding also varies from case to case.The replacement of an ideal simple, single-layer winding by an ideal, split or sandwiched winding must be considered separately. Doubling T/∆ (Fig. 17) results in halving rac id. If the result is two single-layer portions, wire diameter will be doubled, and required height quadrupled. But, if the result is two half-layer portions, wire diameter and winding height remain the same.This will not often occur in practice, because strip or foil windings are generally preferable to single-layer windings (Sect. 4.2).

44

Part 3

5.2. Conductor form

The designer can choose between solid round wire, strip or foil conductor, multiple wire, bunched wire and Litz wire.

5.2.1. Strip or foil versus solid round wire

Strip or foil is only recommended for sandwhiched windings. (Sect. 4.2.3). There will seldom be more than 10 turns in a winding portion. Usually, the total winding may have up to 20 turns with as many interleaving in the (radial) heat-fl ow path. Depending on current density, internal overheating may occur if the number of turns and interleaving is unduly high.Strip or foil windings can be considered if T/∆ ≥ 2, as an alternative for single layer of half-layer wire windings, for example. If the winding has only a few turns, they are a must. In practical transformers , bw / ∆ may be assumed to be > 20. With N ≤ 10, then T/∆ ≥ 2. The basic design chart for wire windings, Fig. 17, shows that that is the region for single-layer or half-layer designs, for which rac id.∆2/ρ ≈ 1.1(T/∆)-1.

Choosing strip rather than wire reduces the winding resistance by a factor of about 0.9 / √N if ideal designs can be used. If N > 1, the expression for the ideal strip thickness can be written as hid /∆ ≈ 31/4N−1/2 . Then rac id = FR idρ / (bwhid) ≈ ( FR id /1.32) ρ N 1/2(bw ∆)−1

. Since FR id ≈ 1.33, dividing this result by that for wire windings obtained above yields the factor 0.9/√N.With a non-ideal strip design, the improvement of an ideal solid-wire version is, of course, smaller, but the strip version remains preferable as long as the thickness of all turns in the portion together exceeds ∆.For a non-ideal strip version which FR ≈ 1 (that is, if h < 0.5 hid), rac ≈ rdc = ρ/(bwh).To make this smaller than the rac id of the wire version, Nh > ∆/1.1 or, as a rule of thumb Nh > ∆.

5.2.2. Multiple wire

In a multiple-wire winding, each turn consists of ns strands lying side-by-side in a layer, Fig. 25.

Multiple-wire turns have the height of a single strand, but an area ns times that of strand, and a layer space (copper) factor the same as that in a layer wound conventionally with wire of the size of each strand.

Fig. 25 A multiple-wire winding of 4 turns, each of 3strands. The strands of one turn are indicated by black dots.

7Z80244

The design procedure for solid round wire windings can easily be adapted for multiple-wire.In the basic design chart of Fig. 17, after dividing T/∆ by ns, read did /∆, pid and He id /∆ as usual. The value of rac id ∆2/ρ for the single strand must be divided by ns to fi nd the resistance of the multiple wire.In effect, a single-wire winding is designed with a layer breadth bw /ns or alternatively, the winding geometry is determined as if there were nsN turns. T/∆ and rac are thus 1/ns times those for a normal wire winding.Compared with normal, single-strand, multilayer windings (more than one layer per portion, T/∆< 2), rac id and did decrease about ns

-1/3 and He in-crease about ns-1/3.

45

Part 3

With 2 parallel strands, Fig.26, the reduction in rac id and did is about 20%: the height increase is roughly 25%. With 3 parallel strands, the reduction in rac id and did is about 30%: the height increase is roughly 45%.If the increased height does not permit the use of the ideal design of a multiple-wire version, the single-strand winding will often have a lower resistance. More than 3 parallel strands are not often used due to winding diffi culty, or inadequate height.

Fig. 26 Design chart for multiple-wire windings showing the improvement in rac id obtainable for ns = 1, 2 and 3 strands. Note

that the improvement quickly vanishes between T/∆ = 2 and T/∆ = 2ns . But if T/∆ exceeds 2ns, winding height and copper

content are reduced by the use of thonner wire.

7Z80250

T/∆

5

2

1

0,5

0,2

0,1

0,050,1

1,1T ,-1

0,2 0,5 1 2 5 10 20

rac idρ ∆

ns=123

In the region of T/∆ > 2 it is usual to try to use strip or foil conductor. If that is not possible, multiple-wire may be considered. Strip conductor, although promising lower resistance, cannot always be used: because of uneven current distribution over the layer breadth, for example or due to constructional problems.If T/∆ > 2ns lower resistance cannot be achieved, Fig. 26, but a multiple-wire winding design might be justifi ed by reduced winding height, the use of thinner, more fl exible wire, and saving in copper.The curve for p = 1 Fig. 17 ends at T/∆ = 6 only for practical reasons. There is no theoretical limit: the resistance remains inversely proportional to T/∆. If p = 1 and T/∆ > 2ns, or if p = 0.5 and T/∆ > 6ns, the overall wire diameter and, thus, the winding height, will be found to reduce in inverse proportion to ns. In order to understand how less copper can have equal resistance, it is necessary to realize that this is in the region of pure skin conduction (ϕ >> 1). The breadth of a turn (either a single wire or ns strands) and, thus the skin area remains the same. Stranding reduces the amount of useless conductor in the winding.

46

Part 3

5.2.3. Bunched wire

Bunched wire comprises three or more enamelled wires, twisted around each of other to form a bunch.Here a distinction is made between bunched wire and Litz wire, which latter is discussed in the next Section. The difference is in the strand diameter: in bunched wire it is of the order of magnitude of ∆, whereas in Litz wire the strands are much fi ner. In the literature, the term bunched wire is sometimes used for what is here called Litz wire.In both bunched and Litz wire the strands undulate within the height of the layer. This ensures equal current sharing between the strands, because, on average, the position of each strand is in the middle of the layer height. Bunched-wire windings are designed in much the same way as single wire windings have Nns. turns, but adaptations of the procedures are required for the determination of winding pitch and layer height.

Practical design guidance is given in Sect. 3.42.

5.2.4. Litz wire

Litz wire is a form of bunched wire in which the strand diameter is small compared to ∆. Eddy-current effects are thus virtually eliminated, and the a.c. resistance is equal to the d.c. resistance. Thus Litz wire windings require no special design procedures: the design methods for low-frequency windings apply.Litz wire is standardized and commercially available.For instance, IEC publication 317 -11 (Ref.4) covers Litz wire of 3 to 400 strands of 0.025 mm to 0.071 mm diameter. Standards and manufacturer’s catalogues should be consulted for engineering data.Litz wire is generally only useful where ample winding space is available. The main drawback of Litz wire is its low space (copper) factor. Typically, only one quarter to one third of the winding space will be conductor.

Design guidance is given in Sect. 3.4.3.

47

Part 3

6. Non-sinusoidal current, effective frequency

So far, sinusoidal current waveform of frequency equal to the switching frequency has been assumed.When considering proximity-effect loss compared to d.c (i.e. frequency-independent) loss, it was found that the total loss is minimum if the ratio of eddy-current loss to d.c loss is 1:3 in strip or foil windings and 1:2 in multilayer wire windings. In single-layer windings there was no such optimum ratio. The current waveforms associated with SMPS transformers are generally far from sinusoidal.A non-sinusoidal repetitive waveform can be represented by a d.c. component, a component at the fundamental frequency (the switching frequency) and several components at harmonic frequencies. The d.c component will not, of course, cause eddy-current loss: its current density is uniform over the conductor cross-section. The fundamental component has eddy-current loss as well as d.c. loss associated with it. This is also the case for the harmonic components, but the ratio of eddy-current to d.c loss will be higher than for the fundamental. Neglecting the non-sinusoidal character of the current wave-form introduces an inaccuracy. The sum of all eddy-current losses should be determined together with the total d.c loss to ensure that they have the desired optimum ratio. One problem is that, when designing the windings, it is not possible to make a reliable estimate of the current waveform. Sometimes, it is possible to make use of experence gained in previous designs.

Many designs have, in fact, been made as if the current were sinusoidal and the results have usually been satisfactory. The errors due to calculation only at the switching frequency oppose. The component of loss occurring at d.c is over-estimated, whereas the components due to harmonics are underestimated.The inaccuracy can be avoided by using an effective frequency in winding design.The effective frequency is, of course, notional. It should be used only in connection with eddy-current effects. In so far as eddy-current loss is proportional to ϕ4 (that is to f2), the effective frequency can be found from an analysis of the current waveform:

fe = 1 i2π I

where I is the r.m.s. winding current; i is the r.m.s. value of dI/dt, the fi rst derivative of the current waveform. This expression does not apply in the region of skin conduction, where the eddy-current loss is proportional to ϕ or √ f.It is apparent that i contains no contribution from the d.c. component and a more pronounced contribution from the harmonics, since d (sin nωt) /dt = nω cos nωt. The rise and fall items of the current pulses have a considerable effect on the value of i. Moreover, some (if not all) of the higher harmonics lie in the region of the skin conduction.A simple method of determining fe has not yet been found: fe can be expected to be lower than the switching frequency f, but higher than f Ie ac/Ie Where Ie ac = √ (I2e- I

20) is the r.m.s. value of the a.c.

component and I0 is the d.c. component. The use of this effective frequency, despite its not being valid in the region of skin conduction, will yield the optimum winding geometry.Single-layer windings are chose if pid ≤ 1 since fe is valid for determing pid the choice between multilayer and single-layer windings is clear. Both single and half-layer windings are close-packed, inc principle, so their layer geometry is independent of frequency.Since the effective frequency for the skin-conduction region is not known, winding loss cannot be accurately estimated, and, consequently, a reliable choice between a single and a half-layer winding cannot always be made. In most practical cases, however, these problems do not occur because, in any case, strip windings are preferable in this region.For use in choke design an expression for fe has been derived and simplifi ed, but due to the large variation

48

Part 3

in conditions it is impossible to give a typical example for transformer design.

For the waveform of Fig. 27, provided that the rise and fall times are between 15% and 85% of the repetition period,

fe ≈ 1.3 f

[ 1 + 3 ( Io / Iac )2 ]

and the effective current

Ie = ( I2o + I2

ac / 3 )

For a sinusoidal rather than a triangular waveform super-imposed on a d.c. current,

fe = f

[ 1 + 2 ( Io / Iac )2 ]

and

Ie = ( I2o + I2

ac / 2 )

Fig. 27 Current waveform in a smoothing choke idealized for the calculation of effective frequency.

7Z89485

I

1/f

Iac

IacI0

IM

References1. Dowell, P.L. 1966, Effects of eddy currents in transformer windings. Proc. IEE, 113 No. 8 (August)2. Snelling, E.C. 1969. Soft Ferrites, properties and applications. Iliffe Books Limited, London3. Casimir, H.B.G. and Ubbink, J. 1967. Skin effect -I.Philips’ Technical Review 28 No.6.4. IEC Publication 317-11 Bunched enamelled copper wire with silk covering.

49

Part 3

Appendix: Exaple of tables of standart wires sizesEnamelled dround coper winding wire, IEC-grade 2

max. overall nominal minimum nom.resistancenominal diameter diameter cross-sect. winding pitch at 100oCd (mm) do (mm) area (mm2) tmin (mm) rdc (Ω/m)

0.040 0.054 0.00126 0.059 17.68 0.045 0.061 0.00159 0.066 13.970.050 0.068 0.00196 0.073 11.32 0.056 0.076 0.00246 0.082 9.0220.063 0.085 0.00312 0.091 7.129

0.071 0.095 0.00396 0.102 5.6130.080 0.105 0.00503 0.112 4.421 0.090 0.117 0.00636 0.125 3.4930.100 0.129 0.00785 0.137 2.829 0.112 0.143 0.00985 0.152 2.256

0.125 0.159 0.0123 0.169 1.811 0.140 0.176 0.0154 0.187 1.4440.160 0.199 0.0201 0.210 1.1052 0.180 0.222 0.0254 0.234 0.87330.200 0.245 0.0314 0.257 0.7074

0.224 0.272 0.0394 0.284 0.56390.250 0.301 0.0491 0.315 0.4527 0.280 0.334 0.0616 0.348 0.36090.315 0.371 0.0779 0.387 0.2852 0.355 0.414 0.0990 0.431 0.2245

0.400 0.462 0.126 0.481 0.1768 0.450 0.516 0.159 0.538 0.13970.500 0.569 0.196 0.593 0.11318 0.560 0.632 0.246 0.659 0.090220.630 0.706 0.312 0.736 0.07129

0.710 0.790 0.396 0.823 0.056130.800 0.885 0.503 0.922 0.04421 0.900 0.990 0.636 1.032 0.034931.000 1.093 0.785 1.139 0.02829 1.120 1.217 0.985 1.268 0.02256

1.250 1.351 1.227 1.408 0.01811 1.400 1.506 1.539 1.569 0.014441.600 1.711 2.011 1.783 0.011052 1.800 1.916 2.545 1.996 0.0087332.000 2.120 3.142 2.209 0.007074

2.240 2.366 3.941 2.465 0.0056392.500 2.631 4.909 2.742 0.004527

Remark: Values of tmin are based on recommendations for mass production of one particular manufacturer.Other manufacturers may use different values.

50

Part 3

Medium enamelled copper wires - AWG (B. & S.)(diameters based on BS 1844: 1952 - mediun covering)

max. overall nominal nominal minimumAWG nominal copper diameter cross-sect. resistance at winding pitch(B. & S.) diameter (d) (do) area at 100oC (rdc ) (tmin) inch mm mm mm2 Ω/m mm

44 0.00198 0.0503 0.06604 0.00199 11.190 0.07143 0.00222 0.0564 0.07366 0.00250 8.899 0.07942 0.00249 0.0633 0.08128 0.00314 7.073 0.08741 0.00280 0.0711 0.09144 0.00397 5.594 0.09840 0.00314 0.0798 0.1041 0.00500 4.448 0.111

39 0.00353 0.0897 0.1143 0.00631 3.519 0.12238 0.00397 0.1008 0.1295 0.00799 2.783 0.13837 0.00445 0.1130 0.1448 0.01003 2.215 0.15436 0.00500 0.1270 0.1626 0.0127 1.754 0.17235 0.0056 0.1422 0.1778 0.0159 1.398 0.188

34 0.0063 0.1600 0.1981 0.0201 1.105 0.20933 0.0071 0.1803 0.2235 0.0255 0.8700 0.23632 0.0080 0.2032 0.2489 0.0324 0.6853 0.26131 0.0089 0.2261 0.2743 0.0401 0.5527 0.28730 0.0100 0.2540 0.3048 0.0507 0.4386 0.319

29 0.0113 0.2870 0.3404 0.0647 0.3435 0.35628 0.0126 0.3200 0.3759 0.0804 0.2762 0.39327 0.0142 0.3607 0.4191 0.1022 0.2175 0.43826 0.0159 0.4039 0.4699 0.128 0.1735 0.49125 0.0179 0.4547 0.5232 0.162 0.1369 0.547

24 0.0201 0.5105 0.5817 0.205 0.10860 0.60823 0.0226 0.5740 0.6502 0.259 0.08586 0.67922 0.0253 0.6426 0.7214 0.324 0.06852 0.75421 0.0285 0.7239 0.8052 0.412 0.05399 0.84120 0.0320 0.8128 0.8966 0.519 0.04283 0.937

19 0.0359 0.9119 1.003 0.653 0.03403 1.04818 0.0403 1.024 1.118 0.823 0.02700 1.16817 0.0453 1.151 1.247 1.040 0.02137 1.30316 0.0508 1.290 1.389 1.308 0.01699 1.45215 0.0571 1.450 1.557 1.652 0.01345 1.627

14 0.0641 1.628 1.737 2.082 0.010670 1.81513 0.0720 1.829 1.943 2.627 0.008460 2.03012 0.0808 2.052 2.172 3.308 0.006717 2.27011 0.0907 2.304 2.431 4.168 0.005331 2.54010 0.1019 2.588 2.720 5.261 0.004224 2.842

Remark: Values of tmin are based on recommendations for mass production of one particular manufacturer. Other manufactur-ers may use different values.

1

Part 4

A power smoothing choke which is to carry a signifi cant direct current component or to have a well-defi ned inductance is usually wound on a core whose magnetic circuit includes an air gap. The reluctance (magnetic resistance) of the air gap reduces the effective permeability of the core: either to increase the ampere-turns at which saturation occurs, or to reduce the effect of variations in the permeability of the core material on the inductance of the choke. The traditional rout to the design of a choke with a gapped core involves the use of Hanna curves, or some derivative of them. (Ref. 1, 2, 3). This design route has a number of disadvantages and limitations. Initial core selection is uncertain and designs may have to be made using a number of cores before the optimal solution is found. The design procedure involves considerable calculation and iteration, and the effects of changes in core operating conditions and mechanical tolerances, especially on the airgap, are not readily predicted. To simplify the design of power chokes we have devised a method based on computer-generated charts. The fi rst step in the design is the selection of a suitable core: this selection usually proves to be fi nal.

Design method

Starting with the peak current IM that the choke is required to pass without saturating the core, and the minimum inductance required Lmin, the designer obtains directly all the information necessary for the construction of the choke. Core size, airgap, number of turns, and winding geometry are derived by straightforward procedures. Of especial interest to those engineers to whom the subject is a black art: the magnetic properties of the core do not enter into to process at all.

Power-inductor design

2

Part 4

Core operating conditionsThe selection and design charts are constructed for cores in the power ferrite materials 3C90 or 3C94 operating at a temperature of 100oC. Operation at lower temperatures leads neither to core saturation nor to inductances lower than Lmin. The design peak fl ux density BM is 0.32 T, however, the charts can be used for a lower value by designing for a peak current 0.32I/M/BM. Symbols used in this text are listed and defi ned in the Table, symbols for currents are illustrated by Fig. 1. Note that in the equations the unit of frequency is kHz, not Hz, and the unit of length mm. Some constants in the equations are based on these units.

7Z89485A

I

1/f

Iac

Iac

I0IM

Fig. 1 Symbols for choke current used in the text.

(See also the Table)

Definition of symbols usedSymbol unit definition

AL H induction factor L/N2

bw mm winding (layer) breadth

BM T peak flux density

d mm nominal wire diameter

d o mm overall wire diameter

f kHz frequency

fe kHz effective frequency (see text)

FR - a.c. resistance factor Rac / Rdc

h mm thickness of foil conductor

H mm winding height

Ha mm available winding height

i mm thickness of interleaving

Ie A r.m.s. current at full load

IO A d.c. component of current at full load

Iac A a.c. component of current at full load

IM A peak value of current at full load

L H inductance

N - number of turns in a winding

p - number of layers

PW W winding loss

Rac Ω a.c. resistance

Rdc Ω d.c. resistance

s mm spacer thickness

Note: subscript id means 'ideal' value.

3

Part 4

101

1

10−1

10−2

10−3

10−4

10−1

air-gap (mm)

I2L(J)

E42/21/15

E41/17/12

E36/21/15

E30&31&32&34

E30/15/7

E25/13/7

E42/20

E46/23/30

E47&50

E80/38/20

E25/6

E20/10/6

E20/10/5

E19/8/9

E16/8/5

E13/6/6

E13/7/4

E19/8/5

E65/32/27

E56/24/19

E55/28/25

E55/28/21

E71/33/32

Fig. 2 Selection chart for E cores

Selection procedure The selection graphs included at the end of this part are used to select a suitable core size for the intended application. An example of such a graph is given above. The selection charts are used as follows:

• Knowing the value of peak choke current IM and the minimum inductance required Lmin, calculate the value of I2L. • Choose the shape of core based on application considerations. Draw on the appropriate selection chart a horizontal line at the level of the calculated I2L. • A core whose curve intersects this horizontal line can be used for the application. The air-gap length corresponding the intersection is, however, only an indication of the fi nal value.

Effect of core sizeWhere, as is usual, more than one core could be used, the fi nal choice may be governed by the considera-tion that operation near the right-hand end of the curves carries the risk of overheating. Moreover, selection of a larger core will generally result in a more conservative effi cient design than one based on a core that is only marginally large enough.

4

Part 4Air-gap length and number of turns

Once an air-gap length is chosen the effective core permeability µe and the inductance factor AL can be calculated with help of the program module " Inductance factor calculation".The number of turns for the choke design can now be derived as follows.

To avoid saturation the maximum number of turns allowed is given by:

Nmax = I2L

I2AL

[1]

The minimum number of turns required to achieve Lmin is:

Nmin = Lmin

AL

[2]

Now select an integral number of turns N between Nmin and Nmax.

Establish the winding geometry using the winding-design procedure in the next setion.

Winding design

The loss due to eddy-currents in a winding carrying a.c. increase rapidly with conductor size (ad d4 for wire), but resistive losses in a conductor decrease with increasing size (as d-2 for wire). It follows, therefore, that there must be a frequency-dependent “ideal” conductor size at which losses are minimum. (This is discussed fully in Ref. 5). This sets the upper limit to conductor size; there is no reason to increase losses by using a thicker conductor. The use of a thinner conductor is sometimes tolerable (low current density) or necessary (inadequate space).The procures that follow allow the ideal number of layers and wire size, or the thickness of strip, to be determined for chokes with an operating-current waveform similar to that shown in Fig.1. They also indicate the course of action in the event of the available winding window being insuffi cient to accommodate the ideal winding.Copper conductors are assumed here and the operating temperature is taken to be 100oC, so that conductor resistivity is 1/45 Ω mm2/m (30% higher than that at 20oC). Symbols used are defi ned in the table and Fig. 1.

Effective frequency and effective current

To allow for the effect of waveform on eddy-current losses in the choke windings, it is necessary to convert actual frequencies and currents to effective values. For sinusoidal currents, the effective frequency fe is equal to the actual frequency f. For small amounts of waveform distortion, and small d.c. components fe can still be taken as equal to f. For the waveform of Fig.1, and provided that the rise and fall times are between 15% and 85% of the repetition period,

fe ≈ 1.3 f

[ 1 + 3 ( Io / Iac )2 ][3]

5

Part 4

In design for class I applications fe may be only a few kilohertz. Eddy-current effects are then negligible so that windings can be designed as if they are to carry d.c. only. Remember to use the correct value for d.c. resistivity. For the waveform of Fig.1, the effective current Ie is given by

Ie = ( I2o + I2

ac / 3 ) [4]

For sinusoidal currents with a signifi cant d.c. component, however,

fe = f

[ 1 + 2 ( Io / Iac )2 ][5]

and

Ie = ( I2o + I2

ac / 2 )[6]

where Iac is the amplitude of the a.c. component.

Multi-layer wire windings of solid, round wire

In the following design procedure, it is assumed that all layers have the same breadth. However, where the number of turns in the winding cannot be divided into the ideal number of layers, a difference of one turn per layer is permissible.1. The ideal wire diameter is

did = 2.6 ( bw )1/3

Nfe[7]

2. Select the nearest standard wire size (for d and do) from a wire table such as that for IEC grade 1 winding wires.3. The ideal number of layers is now

pid = N bw/do - 1

[8]

Note: this expression is valid only for do from step 2.- If pid ≥ 1.5 and the current density in wire did is excessive, make a new design using a larger core.

6

Part 4

- If Pid ≤ 1, the expression for did in step 1 is not valid: go to the single-layer winding procedure.

Find p by rounding pid to the next highest integer. This rounding increases the spacing between turns.

4. The required winding height is H = p(do + i)

5. If H exceeds the available height Ha, or if current density is low:- reduce p by one layer- select the thickest wire for which do ≤ pbw/(N + p) repeat from step 4, if p = 1.

6. FR = 1 + 1/2(did)6

Check : if FR > 2 an error has occurred.Note : FR = 1.5 for did when d < 0.7 did, FR ² 1

7. Pw = I2e Rac = I2e FRRdc.

Single-layer windings of solid, round wire

The design procedure is to be used only when pid calculated in step 3 of the previous section comes out as equal to or less than unity.

1. Select the thickest wire for which do ≤ bw/(N + 1).

2. FR = 0.33 d fe 1/2N(N + 1), only if pid ≤ 1 in step 3 of the last section.FR has no upper limit here.

3. Pw = I2e Rac = I2e FRRdc.*

Bunched (Litz) wire windings

Eddy-current effects in bunched-conductor (or Litz-wire) windings are negligible and, thus, no special design procedure is required. Con-ductors of this type are not, however, necessarily the complete solution: their packing factor, and, consequently, winding thermal conductivity are both low. They might be an attractive alternative where the ideal solid conductor winding fi lls less than half the available height. The resistance of bunched conductors, like that of solid conductors, is 30% higher at 100oC than at 20oC.

7

Part 4

Foil or strip windings

Chokes for high-current, low-voltage SMPS often use windings of strip or foil conductor. The width bw of the strip is equal to the available winding width.

hid = 3.1 Nfe

[9]

hmin = 0.8 hid [10]N

hmax = Ha - i [11]N

Choose a value for i that suits a strip thickness about Ha/N. if hmax < hmin, try a wire winding4. Select a strip thickness h such that hmin ≤ h < hmax. Aim for h = hid.

FR = 1 + 1/3 ( h )4

hid[12]

Check that FR < 1,8; if not, reduce h when h = hid, FR = 1.33; when h <0,6 hid,FR ² 1.

6. Pw = I2e Rac = I22e FRRdc.

Note : d.c. resistance of copper strip, is 1/(45bwh)Ω/m at 100oC.

8

Part 4

Design example

A choke of 30mH minimum inductance is required for a peak current of 1A at 100 kHz with a waveform as shown in Fig.1 IacIo = 0.1.

Core selection

The value of I2MLmin = 3 x 10-2 J. A Horizontal line of this value drawn on the E core selection chart intersects the curve or the E55/28/21 core at an air-gap length of about 1.5 mm.

Number of turns and spacer thickness

Thus, from Eq.(1),

Nmax = 0.036 = 311.5 turns

371.10-9[13]

The minimum number of turns is, from Eq.(2),

Nmin = 0.03 = 284.4 turns

371.10-9[14]

Since Nmax is, at it should be, greater than Nmin, the design is successful so far and a number of turns can be selected between these limits.

Winding design

The effective operating frequency for the core is given by Eq.(3),

fe ≈ 1.3 × 100 = 7.5 kHZ

[ 1 + 3 ( 10 )2 ]

[15]

At this effective frequency, eddy-current effects can be neglected, and the winding can be designed to fi t the space available, allowing for the coil former wall thicknesses.

9

Part 4

101

CBW3191

10−1

10−2

10−3

10−4

10−1

air-gap (mm)

I2L(J)

E42/21/15

E41/17/12

E36/21/15

E30&31&32&34

E30/15/7

E25/13/7

E42/20

E46/23/30

E47&50

E80/38/20

E25/6

E20/10/6

E20/10/5

E19/8/9

E16/8/5

E13/6/6

E13/7/4

E19/8/5

E65/32/27

E56/24/19

E55/28/25

E55/28/21

E71/33/32

10

Part 4

101

CBW3201

10−1

10−2

10−3

10−4

10−1

air-gap (mm)

I2L(J)

E38/8/25

E32/6/20

E22/6/16

E18/4/10

E14/3.5/5

E43/10/28

E58/11/38

E64/10/50

11

Part 4

101

CBW3211

10−1

10−2

10−3

10−4

10−1

air-gap (mm)

I2L(J)

EC70

EC52

EC41

EC35

12

Part 4

101

CBW322

10−2

10−1

10−3

10−4

10−5

10−1

air-gap (mm)

I2L(J)

EFD25

EFD20

EFD15

EFD12

EFD10

EFD30

13

Part 4

101

CBW323

10−2

10−1

10−3

10−4

10−5

10−1

air-gap (mm)

I2L(J)

EP20

EP17

EP13

EP7 & EP10

14

Part 4

101

1

10−1

10−2

10−3

10−4

10−1

air-gap (mm)

I2L(J)

ER42

ER14.5

ER11

ER9.5

ER35

ER48 & 54 & 54S

ER28 & 28L

ER42A

ER40

15

Part 4

101

1

10−1

10−2

10−3

10−4

10−1air-gap (mm)

I2L(J)

ETD49

ETD44

ETD39

ETD34

ETD29

ETD54

ETD59

16

Part 4

101

CBW326

10−2

10−1

10−3

10−4

10−5

10−1

air-gap (mm)

I2L(J)

P66/56

P42/29

P36/22

P30/19

P26/16

P22/13

P18/11

P14/8

P11/7

P9/5

P7/4

17

Part 4

101

CBW327

10−2

10−1

10−3

10−4

10−5

10−1

air-gap (mm)

I2L(J)

P26/16/I

P22/13/I

P18/11/I

P14/8/I

P11/7/I

18

Part 4

101

CBW328

10−2

10−1

10−3

10−4

10−5

10−1

air-gap (mm)

I2L(J)

PT26/16

PT23/11

PT18/11

PT14/8

PT30/19

19

Part 4

101

CBW3301

10−1

10−2

10−3

10−4

10−1

air-gap (mm)

I2L(J)

PQ35/35

PQ20/16 & 20/20

PQ32/20 & 32/30

PQ26/20 & 26/25

20

Part 4

101

CBW331

10−2

10−1

10−3

10−4

10−5

10−1

air-gap (mm)

I2L(J)

RM8

RM6S&R

RM5

RM4

RM10

21

Part 4

101

CBW332

10−2

10−1

10−3

10−4

10−5

10−1

air-gap (mm)

I2L(J)

RM14/I

RM12/I

RM10/I

RM8/I

RM6S/I

RM5/I

RM4/I

RM7/I

22

Part 4

101

CBW333

10−2

10−1

10−3

10−4

10−5

10−1

air-gap (mm)

I2L(J)

RM14/ILP

RM12/ILP

RM10/ILP

RM8/ILP

RM6S/ILP

RM5/ILP

RM4/ILP

RM7/ILP

23

Part 4

101

CBW3341

10−1

10−2

10−3

10−4

10−1

spacer thickness (mm)

I2L(J)

U10/8/3

U93/76/16

U67/27/14

U33/22/9

U30/25/16

U25/20/13

U25/16/6

U20/16/7

U15/11/6

U100/25

U93/30

Inductance Calculations - Ferroxcube Standard … Inductance factor calculation Inductance...

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