design tool help

8/10/2019 Design Tool Help

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Inductance factor calculation

Inductance Calculations - Ferroxcube Standard Cores

L

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 dene your own material by directlyentering its features.

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

F When a bias current is entered, the resulting magnetic bias eld (H bias) is calculated. The bias ux density (B bias) in the core is estimated with the help of the effective permeability ( e).

G Click Go to start a new calculation,

A B C D

E

GHI

F

H Click Datasheet to view the data sheet ofthe selected core type.

I Click User dened cores if you want todesign your own core.

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

K Effective core parameters for the selectedcore are shown.

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

M Inductance value (L) is caculated based on A L and number of turns. These parameters can

be changed freely to optimize the design. If required the number of turns can be xed.

JK

M


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Using the program Inductance Factor Calculation

Functional description

Ferroxcube Standard Cores

The program consists of two parts. The rst screen appears when you click the button Inductance factorCalculations and can be used for calculations on standard Ferroxcube Cores.The second program can be activated by clicking the button User Dened in the lower right corner of thewindow.

User Dened CoresThis program calculates effective magnetic dimensions of user dened, magnetically closed core shapes.With these data, the properties of the chosen ferrite material and airgap length, the inductance factor A L of the core is derived. With the number of turns the inductance value of an inductor based on this coreis calculated. For magnetically open circuits like rods the so-called rod permeability is caculated. Withthe 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, Ecore, P core and rod. Several other core types are variations on the principal shapes so that the sameformulas can be used. The core dimensions are entered with the help of a dimensional drawing in thecore type window. In the case of cores with known effective dimensions, these dimensions can be entereddirectly 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, ux level, eld strength or inversecalculations. Every core shape, including ring cores, can be gapped. The inuence of fringing ux and parasiticgaps is accounted for. In case of a rod the calculation can be inductance, or an inverse calculation. An ALvalue is not appropriate here, because the inductance also depends on coil length and not only on the rodand the number of turns.

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


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Inductance factor calculations for User Dened Cores

L

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

B Enter its main dimensions using the outlinedrawing as a guide.

C Use direct input when effective coreparameters l e 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 theestimation of the fringing ux.

D Available materials for the chosen shapeare displayed. When you select one, itspermeability and the usual surface qualityare shown.

E You can also dene your own material by

directly entering its features.

F Flux density (Bpeak ) is calculated based on,

core parameters, a given voltage and thenumber of turns.

A

E

HI

F

calculated. The bias ux density (B bias)in the core is estimated with the helpof the effective permeability ( e).

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 itsposition.

K Effective core parameters for the selectedcore are shown.

L Inductance factor (A L) is calculated basedon material permeability, total airgap and itsposition.

M Inductance value (L) is calculated based onAL and number of turns. These parameterscan be changed freely to optimize thedesign. If required the number of turns canbe xed.

JK

M

B C D

G


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Denition of terms

Permeability

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

B 0 H J or B+ 0 H M+( )= = [1]

where 0 = 4 .10-7 H/m, J is the magnetic polarization and M is the magnetization.

The ratio of ux density and applied eld is called absolute permeability.

[2]BH---- 0 1

MH-----+

absolute= =

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

[3]BH---- 0r=

Since there are several versions of r depending on conditions the index r is generally removed andreplaced by the applicable symbol e.g. i, a, etc.

Initial permeability

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

[4]i10------ BH-------- 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 difcult. As a result, theux density for a given magnetic eld strength is lower.Effective permeability is dependent on the initial permeability of the soft magnetic material and thedimensions of airgaps and circuit.

[5]e

i

1G i

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

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

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

Amplitude permeability

The relationship between higher eld strength and ux densities without the presence of a bias eld, isgiven by the amplitude permeability ( a).

[6]a10------ B

H----=

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

Incremental permeability

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

[7] 10------ BH-------- HDC=

If the amplitude of the alternating eld is negligibly small, the permeability is called the reversiblepermeability ( rev).


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Effective core dimensions

To facilitate calculations on a non-uniform soft magnetic core, the effective dimensions are given on eachdata sheet. These dimensions, effective area (A e), effective length (l e) and effective volume (V e) dene ahypothetical 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

Ae------------------

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

[10]1e------ lA----

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

[11]L0 N

2

1e------ lA--------------------------- 1.257 10

9 N2

1e------ lA------------------------------------------------------ (in H)= =

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

[12]B U 2 109

Ae N------------------------------ 2.25U 10

8

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

for square wave:

[13]B 0.25U 109

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


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where:Ae is the effective area in mm

2 .U is the voltage in Vf is the frequency in HzN is the number of turns.

The magnetic eld strength (H) is calculated using the effective length (I e):

[14]H IN 2le

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

If the cross-sectional area of a core is non-uniform, there will always be a point where the real cross-sectionis minimal. This value is known as A min and is used to calculate the maximum ux density in a core. In welldesigned ferrite core a large difference between A e and A min is avoided. Narrow parts of the core couldsaturate or cause much higher hysteresis losses.

Inductance factor (A L)To facilitate inductance calculations, the inductance factor, known as the A L value (nH), is given in each datasheet. The inductance factor of a core is dened as:

[15]L N2 AL (nH)=

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

[16]AL =0 e 10

6

l A ( )------------------------------ =1.257 e 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 andtherefore the mechanical dimensions have more inuence on the inductance than the ferrites 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(rod ).


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

[17]L 0 ro dN

2A

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

where:N = number of turnsA = cross sectional area of rod (mm 2)I = length of coil. (mm)

Fig.1 Rod permeability ( rod ) as a function of length todiameter ratio with material permeability as a parameter.

10 3

10 2

10

Length / diameter ratio

rod

i = 10.000

5000

20001000

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 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.

EClick 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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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 curreApplying this to the pulse shaped input voltage Vp and current Ip with duty cycle (active part of period gives :

Pthr (Vp Ip)dt

0

T

= = Vp Ip [1]1TThe relation of voltage Vp and ux density B is given by Faradays law of induction. Using this for the signals and keeping in mind that 1/T equals the frequency f, and Np stands for the number of primary tthis results in:

Vp = Np Ae = Np Ae [2]dBdt

B = Np Ae

fB

Note that B in eq.[2] is referring to the total ux excursion, So, from its minimum to its maximum valueSubstituting 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 toconsider the effective (or rms) current. This current (Ieffective) is related to power losses in a resistance R byP = I 2R. The effective current for a time varying current I(t) with period T is dened in general as:

I2effective

I2 (t) dt

0

T

= [3A]1T


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

Ipeffective = Ip [4]

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

Ipeffective = [5]

Ip =1

1 Pw prRac pr

The a.c. resistance of the windings is related to the d.c. resistance by a constant factor FR which is equalto 1 for a d.c. current. The increase of the a.c. resistance value is caused by the skin and proximity effect

Normally these effects strongly depend on the arrangements of the windings in the available winding sand on the frequency of the signals. In literature, methods to calculate the FR value are published (e.g. basedon the theory of Dowell). For these calculations computer programs can be helpfully.

The d.c. resistance Rdc is equal to: Rdc = .Np.Lav/Acopper. is the copper resistivity inm 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 ll factor Fw, which is dened 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]Rac prim = F R Rdc =FR Np lav F R Np lav

Acopper=

Fw Awprim

2

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

[7]

Ip =1 1

Pw prim Fw Awprim

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 Pw, where Pw is the total winding losses.

Fw.Aw.pr Fw.Aw.s Fw.AwApplying this to [7] gives the primary current in terms of the total winding losses and total area:

[8]

Ip =1 1

Pw Fw Aw

lav FRNp2

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

Ae (fB) [9]

Pthr =1

Pw Fw Aw

lav FR2

The same formula is valid for a yback transformer, however the derivation is somewhat more complicdue to the triangular current shapes. Due to the inuence of the airgap (fringing ux) the throughput pois 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 highercompared to the forward transformer. For transformers used in other converter topologies only the facto might change. So in general the througput power factor is equal to:

Ae (fB) [10]Pthr Pw Fw Aw

lav FR


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2. Determination of the optimum ux density.

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

P trafo = [11]TR th

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

Rth = 1/ (C th .Ve 0.54 ).

The constant Cth depends on core shape and winding arrangement. For standard wound cores like RM oETD and EFD it is about 17. For planar cores with windings integrated in a PCB (epoxy) it is about 24,for 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 riseT andeffective volume Ve:

[12]P trafo = T C th Ve 0.54

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

Pcore C m fx

By

ac ct ct 1 T Tct 2+2

C m C T( ) fx

By

ac= = [13]

Note that Bac is here the peak ux density, so half the total ux excursion .(Bmax Bmin).The parameters Cm, x, y, ct, ct1 and ct2 are specic for each power ferrite and often only applicable in adened frequency range. Total transformer loss consists of winding loss plus core loss. So, winding lossbe written as:

Pw = Ptrafo

P core = Cm

C T( ) fx

B Vey

[14]T C th Ve 0.54

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


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[15]

Ae f

C1 =2

C3 = C m C T( ) fx

Ve Fw AwFR lav

C2 = T C th Ve 0.54

Fw AwFR lav

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

C1 Bac [16]Pthr (B ac ) = (C2 C3 Bac ) y

The throughput power is now expressed as a function of Bac only because the other terms are consideredas constants.Remark:Bac is the peak 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 ux density, can be calculated by solvin

d[Pthr(B ac )]/dB ac = 0.

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

= Cm C T( ) fx

Bac Vey

[17]T C th Ve 0.542

2 + y

From expression [17] it can be seen that:

[18]22 + y

P core =y

2 + yP w =P trafo P trafoand/or


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

fx/y

[19]Ve 0.46/y2

2 + yBopt = 2 ( ( )T C th

Cm C(T)

1/y1/y

)

Bopt is expressed as a function of : 6 t parameters for the ferrite Cm, x, y, ct, ct1 and ct2 Allowed temperature riseT 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 theselection on core family(ies) and type of ferrite(s).For each core the required information as e.g. A

e, V

e, l

av, A

w is read from a data base, which is also done for

each ferrite with the constants Cm, x, y, ct, ct1 and ct2.

The rst step is the calculation of the optimum ux density Bopt with [19] for all the core / ferritecombinations. The required parameters can be read from the data bases or are known because of the inpon 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 throughputpower factor.The copper resistivity is used as temperature dependent(T) in the following way:

[20]T = 393 . 10 -5 T =1 + T ( T-20)

1 + T 76(T) = (T=100) T

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

Dependent on whether it is a forward or a push-pull transformer a multiplying factor of 2/ or 22/ is used respectively to calculate the throughput power. If the value for a ferrite core is outside the range the inserted values for minimum or maximum throughput power it will not be mentioned in the range othe determined available cores.

4. Calculation of throughput power for yback transformers

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

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

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

[21]E ind = B H Ve = VeB2

2


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

[22]E ind = 2 0 a 2 0B2 Ae le B2 Ae lg

2 0B2 Ve + = ( )+1a

lgle

The throughput powerPthr = E ind /T = E ind . f becomes with [22]:

[23]2 0

B2 f Ve ( )+1algle

Pthr(l g) =

The fringing ux has the effect of increasing the airgap area. The reluctance Rg, which is equal tolg/.A g, decreases by the fringing ux factor F. Then Rg becomeslg/(.A g.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 ux onthe effective airgap lg is accounted for by deviding lg by a factor F.

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

In ( [24]Ae

F = 1 +lg

lg2 Wh )

where Wh is the height of the winding window.

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

[25]2 0

B2 f Ve ( )+1alg /Fle

Pthr(l g) =

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

Ae (fB) [9]

Pthr =1

Pw Fw Aw

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(1x/y) (0.27 0.46/y)

y Ae Ve 22+y( ) ( )C thCmC(T)

1/y (1/y+) (1/y+)

Pthr =1 T Aw( )Lav

Fw( )FR

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 + ) 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 airgag is given its maximum size. The maximum airgap can be calculated by reworking [25] and using Pthr fro[9] or [26] denoted now as Pthrmax:

[27]2 0Ae B f

Pthr max (( )) lealgmax =

Remarks :1. Fringing factor F is ignored2. The maximum airgap is in principle strongly dominated by the allowed temperature riseT, and is

almost lineair withT.3. If a larger airgap than lgmax is used the allowed temp. riseT will be exceeded. Magnetically thismeans that the transformer is not operating with the optimum ux density as stated in [19]. Largeairgaps lead to a higher eld strength H, which means higher currents and winding losses. In this 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 calculawith eq. [26], due to fringing effects, and additional losses in the windings due to fringing ux.

The sequence to calculate Pthr for a 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 fora. For this purpose the graphsa(B) fromour 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 themaximum 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 toolarge 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 afunction of frequency.

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

Optimum 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 nally multiplying it with 2/ for forward or 22/ for push-pull transformers.

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

For this calculation it is assumed that for the selected core: Airgap lg is constant Number of turns is xed f and can change (to f and) 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 arekept in balance.

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

[28]P core (fn, n) + P w (fn, n) = P trafo

Consider the following period T for the ux density: From t = 0 toprim the ux increases from 0 to Bmax. From t =prim to sec the ux decreases from Bmax to 0. From t = (prim + sec) to T, the ux remains 0.

The following assumptions are than usually justied:

[29]prim sec and 1 2

[30]feff = =1

2Tf

2


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[31]at fn: 2 n 1

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

[32]P core (f, ) = const. feff By (1-d) = const. ( ) By 2xx

2f

[33]P w (f, ) = const. I2 (1-d) = const. I2 2

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

For f, the slope B(t) = B(f,)/T

For fn,n the slope B(t) =B(fn,n)/nTn

[34]B (f, ) = B (fn, n) ( ) = B (fn, n) ( )nTnT f nfn

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

[35]P core (f, ) = const. ( )x [B (fn, n) ( )]

y2

2f

f n

fn

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

[36]Pcore (f,) = const. ( )

xB (fn,n)

y2n ( )

x( )

x( )

y( ) 2nfn .fnn.f 22n2n2fnf

P core (fn, n)

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


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

P thr (fn, n) = P thr(fn, n) ( ) 0.2

ffn [53]

So, for frequencies lower than nominal, Pthr(f) increases with about approximately f0.2 up to the maximumfor Pthr(f) at fn. For frequencies higher than nominal, Pthr(f) decreases with 1/f.


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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 effective initial permeability for this core / ferrite combination.H

Shows the usual surface nish for this core / ferrite combination.

I Press Go 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 nished design.

L Press Datasheet button to open PDF le of the data sheet of the resulting core type.

Inductor Design

B

C

D

GH

IL

Inductor design


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[2]L = e 0 N Ae2le

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

[3]Acopper =

Fw AwN

lcopper = lav N

This gives the total winding resistance :

[4]R = lcopper

Acopper

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

[5] = 20C [ 1 + ( Toperating 20 )]

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

[6]P winding = Ieff R2

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

[7]T = T operating - Tamb =P winding

R th

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

Inductor design


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[8]R th =C th . Ve

0.54

1

The constant Cth depends on core shape and winding arrangement. For standard wound cores like RM oETD and EFD it is about 17. For planar cores with windings integrated in a PCB (epoxy) it is about 24The final operating temperature of the proposed design can be found by combining eq. [4] to [7], resultin a linear equation for Toperating.With this new temperature, the core is checked for saturation again, using the Bsat value at thistemperature. Bsat(T) is approximated by a piecewise linear function. Also the other boundary conditionslike Tmax and RDCmax are checked. If one of these conditions is not fulfilled the core is found not suitableand the core with the next larger Ve is taken for a new design cycle. The first core size fulfilling all boundconditions is proposed for the the inductor design. All necessary core parameters such as airgap length e are given in the output window. For the calculation of this airgap the effect of fringing flux is taken i

account. Also the resulting winding design is presented together with power dissipation and temperaturrise.In order to fine-tune the design, the core size can be fixed to find out what happens when a parameter iadapted manually.

Inductor design


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Common-mode Choke

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

This non-compensated ux fraction s is used as input for the design procedure using the same steps anlargely 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 AcoilN

Acoil

= (Aw

Aiso

) / 2

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

[6a]P coil = Ieff R

2

P winding = 2 x P coil

Inductor design


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Magnetic Regulator Design

Magnetic Regulator

A B

C

AMake a choice from the available toroid sizes

BInsert design parameters

CGraph explains the meaning of theabbreviations used in the output window

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

D


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The magnetic regulator calculation formulas

Hc-Hc

td

t b t blBda=Vint b/AeN

Bsr =Vintd/Ae

B=Vint bl/AeN

Br Bs

Ireset=Bda Hc le/(Br N)

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

( )bl oninout t t V V =T [1]bl ine t V N A B = [2]bineda t V N A B = [3]d ine sr t V N A B = [4]

( ) 235.0235.011 eout sr l I N B = [mT] [5]( ) 3250/T0048.013.17.1 = f B P core [mW/cc], B in mT, f in kHz. [6]

( ) avCubl onout w l N t t f I P = 12108 [mW], f in kHz [7]etotal V P = 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 knowninput variables.

List of input variables :core magnetic regulator core sizeV in Input voltageV out Output voltageIout Output currentf , T Frequency or cycle periodt on on-time of the input voltage, related to the duty factor of the input cycleTa Ambient temperature T Maximum temperature rise


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Some of the input variables are related to each other:

ton = . T and f = 1 / T

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

T V V

t in

out bl

= [9]

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

eV = 17

T= ( ) avCubl onout l t t f I = 12108( )ae

f V T0048.01

3250

3.1

=

0048.03250

3.1 f V e =

( ) T3250/T0048.013.1 == f V e

=

7.1310

e

bl in

At V

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

=total P N P w = 7.1 B P core =

from :

Ptotal = P w + Pcore

Where P w represents the winding losses,we nd for the total ux swing :

7.11

'

=

N

B [10]

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


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37/152

Power Loss Calculations

Power Loss Calculation

IntroductionWith this program it is possible to calculate the power loss density for Ferroxcube power ferrites undervarious conditions. The ux density signal can be an arbitrary wave shape or a sine wave, with or witho

a bias ux. The calculations are based on t formulas describing the power loss density as a function offrequency, ux density and temperature. In the program the desired operating frequency (in kHz) or cycperiod (ins) and magnetic ux densities (in mT) can be given as input. The program calculates the lossthe 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 ferritescan be selected which are intentended for use at that frequency.The ux density cannot be chosen higher than the saturation level of the selected ferrite. Theory behind the t formulas and part of the t parameters can be found in the references.

1.Basic formulaThe calculation is based on an empirical 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 ux density, so half of the total ux excursion : .(Bmax Bmin) The parameters Cm, x, y, ct, ct1 and ct2 are specic for each power ferrite and often applicable only in a limited frequency range. The temperature dependent coefcients 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.


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2. Calculation for sine waveIf there is no DC ux density (bias), the basic formula [1] can be applied directly to calculate the powedensity in kW/m3 which is identical to mW/cm3.With a DC ux density a correction is required. For this purpose measuring data from reference [2] is uFor several power ferrites the power loss density at dened values of B

ac and B

dcis given there.

The relative increase of the losses at Bac+ Bdc levels with respect to corresponding levels without bias iscalculated 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 availablein 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 ofPV(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 lossdensity 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 followingchapters:

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

The method used is partly described in reference [3] and makes use of the sum of the weighted timederivative of B which is dened as:

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


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Example : For a symmetric (no bias) triangle waveform with its maximum Bm at T, zero at T,Bm at T and zeroagain 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 bywriting it in a integral representation:

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

A piece of line representing part of a complete triangle period T is denoted asTi:

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

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


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

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

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

3.3. Calculation of total power lossFor all periodsTi, which are constructed as explained in chapter 3.2, the equivalent sine wave frequenccalculated with eq. [7]:f sin.eq = 2/(B T i).The corresponding t parameters (Cm, x, y, ct, ct1, ct2) valid for the calculated frequency f sin.eq are now readfrom the data base.

The power loss density contribution for each period is now calculated by inserting eq. [7] in eq.[1] andmultiplying it by its weighting factorTi / 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 ux density the same correction will be made as explained for the sine waveform in ch3.1 . The DC ux density is already calculated as in chapter 3.2. As AC ux density the maximum valuethe waveform with respect to the DC ux Bdc is used. The total losses are in this case:


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References

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

[2] Brockmeyer A., 1995 Experimental Evaluation of the inuence of DC Premagnetization on theproperties of Power electronic ferrites,Aachen University of Technology.

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


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A

E F

A Choose Ferroxcube cores or user

dened cores.

B Make a choice from the available coreshapes.

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 enter the other

windows are updated and show the calculated results.H Shows the resulting plots

(after pressing enter).I The graph can be exported or stored.

Power Inductor Properties

B

C D G

HI


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

original BH loop without gap

sheared BH loop with gap

-Hc

Br

Br(gap)

H point ( H, Br)

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

+=

ie f f B H H

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 dened as the slope of the BH-loop at given 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 valueB. In the calculation of H according to [5] the average of these two B values is used.

+

+=

ielower f

ieupper f f B B H H

111111

0,2

1

0,2

1 [6]ap

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

)(' ,f ,f 21

upper lower B B B += [7] f

The slope of Bf(Hap) gives the incremental permeability :

Power Inductor properties


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H B

= [8]ap

f

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

A L of a gapped core versus NI

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

e

e L l

A A = 0 [9](gapped)

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

el H NI = [10]ap

IL versus NI

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

= el H L I 222 [11] L A (gapped)ap

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

Power Inductor properties


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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 interactionbetween 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, bycombining magnetic and electronic design, a far better insight is gained into theoperation of the circuit, with a consequent improvement in the design itself.In the following sections these problems are tackled. Design procedures areexplained 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 y back converters. Practicalformulae are given for inductance and effective-current values; auxiliary outputs and other special featuresare 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 theappropriate 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 enteringintothe procedure, a choice of converter type must have been made.


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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 isdescribed by: V o = Vi

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 forwardconverter and the associated current waveform.

1/f

/f (1) /f

IL

VL Vi - Vo

Vo

time

time

2Iac

Io


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3

Part 1

Minimum choke inductanceFor continuous-mode operation (uninterrupted current ow through the choke) - otherwise regulationdeteriorates-output current I o must always be greater than half the choke ripple current 2 I ac. This is ensuredby using a minimum value for the choke inductance

Lmin =1 Vo

( 1- max Vi min

) [2]2f Io min Vi maxAn increase in as a result of a sudden load increase will cause a temporary increase in choke ripplecurrent. 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 designOutput choke L carries a direct current equal to the d.c. current in the load. Thus, to avoid saturation, an airgap is required in the core. The design steps are:

- determine I M = I o max + I ac

- calculate I 2MLmin - proceed to Part 4 of this series.

For core loss, see Part 2.

Deriving auxiliary power from the chokeDuring the ywheel period (1- )/f, that is, while the power switch is not conducting, the voltage acrossthe choke is stabilized. By adding secondary windings to the choke, auxiliary, stabilized, low voltages can beobtained as shown in Fig. 3. These auxiliary voltages are recti ed by diodes that conduct during the ywheelperiod. Since auxiliary loads decrease the amount of energy recovered by ywheel diode D, the amount ofauxiliary 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 addingwindings to the choke of a forward converter.

V1 aux

Vo

V2 aux

D C

0


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4

Part 1

In order to store suf cient energy in the choke to supply an auxiliary load, inductance L min calculated fromEq. 2 must be increased to L aux. This might lead to the use of a larger core: see Part 4. The relationshipbetween L min and Laux is

Laux > Lmin [4]

1 max1 (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 notimportant, a higher proportion of auxiliary load can be drawn.The turns ratio

r Vo = ND [5]Vaux Naux

During forward conversion, during the period /f the input power is: P in = ViI o Similarly, the throughput power is: P th = VoI oThe difference is the power stored in and then removed from the choke: P L = P in P = I o ( Vi Vo )

From Eq.1: P L = I o ( Vi Vi ) = ViI o ( 1 Vi ) = P in ( 1 )

If, during ywheel period (1- )/f, the choke current is divided between output and ywheel diodes asindicated in Fig.4 by line a, P aux is maximum, P aux max . The current below a is that through the ywheel diodeof the primary output. Fig.5 shows some waveforms. Due to leakage induction, ywheel diode current doesnot begin at A, but at I max as the auxiliary output currents are then zero. This decreases the value of Pauxrnax . If the current is shared according to line b in Fig.4, P aux = kPaux max, where k = 0.5. In practice, k will besomewhat higher: a value of k = 0.7 is reasonable.

Fig.4 Current through a forward converter choke withauxiliary windings.

(1 ) /f

time

I

Imax

L = L min

L = L aux

A

k = 0.7

a (k=1)

a (k=0.5)


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5

Part 1

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

P aux k ( 1 - ) ( 1 -Lmin

) P auxLaux

The required value of L aux is obtained from

Laux > Lmin

k P tot (1 max )1

P aux

Substituting k = 0.7 and P aux / P tot = 0.2 to 0.3 yields Eq.4.

Ptot = P th - I0 ( VF + VR + Vo )

Further discussion may be found in Ref. 1.

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

a b

V

I


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6

Part 1

1.2 Single forward converter with isolating transformer

Principle of operationThe outline circuit of a forward converter with mains isolation is given in Fig.6. The magnetic energy storedin the transformer while S is ON must be removed while S is OFF, otherwise the energy stored andremoved during a complete switching cycle would not be zero and the transformer core would rapidlysaturate. A solution involving minimum power loss is to add a winding, closely coupled to the primary, and adiode D3 such that a 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-converterSMPS.

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 choiceof output voltage for practical values of . This output voltage becomes V o = Vi / rVoltage and current waveforms for a transformer-isolated forward converter are given in Fig. 7

1/f

/f (1) /f

IL

VL Vi /r - Vo

Vo

time

time

2IacIo

Fig.7 Output circuit waveforms for the mains-isolated forwardconverter.


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7

Part 1

Duty factorThe maximum allowable duty factor at which the core will not saturate due to ux staircasing dependson r and m:

[7]max < 1 m

m + r

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

VCEM = V i max m + r

m[8]

When using a transistor with V CESM = 850 V in a forward converter with a maximum (recti ed) input voltageof 375 V, it is usually adequate to limit V CE to 2 x 375 = 750 V. Thus, with m = r, there is 100 V to spare forringing and integrated supply-voltage surges.The effective duty factor depends on frequency, turns ratio r, rated load current, the leakage inductance ofthe 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]e = - rIo 1.2 10 -9 f f

The inductance of the leads to the output diodes is re ected into the primary as the square of the turnsratio. With large turns ratios, combined with high switching currents, the loss due to commutation delaybecomes substantial. The effect is as if the available duty factor is decreased. This problem is discussedin greate detail in Ref. 3.An example is given in Fig8a. The commutation delay is: t c I orLs / Vi .Now, L s is about 1 nH/mm of leads and, for a 220 V mains supply, V i min 200 V.With the shortest possible leads to the output diodes, experiment shows that the commutation delay istc 1.2 I or 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

e /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 t c during which neither output diode conducts.


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8

Part 1

Preliminary turns ratio and core selectionThe preliminary turns ratio is

[10]r' =e max Vi min

Vo max + VF + VR

For transformer core selection, see Part 2.

Multiple-output transformersAdditional outputs at any d.c. Ievel can be obtained simply by adding further secondaries of the appropriatenumber of turns to the output transformer. The regulation of the additional outputs will be better thanwith a yback converter. However, each output needs two diodes and a power choke, against the singlediode needed with a yback converter.Warning: the ywheel diode must always conduct when the forward diode does not. Otherwise, peakforward conversion recti 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, ensurethat the appropriate minimum load is always present at each output.With several different outputs, it is necessary to nd that value of volts-per-turn for the transformer thatallows 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 methodThe function of the control circuit is to stabilize the output against variations in input voltage and loadby adjusting the duty factor of the switching device. However, the effect of step load changes cannot becorrected immediately because some time is needed for the current through the choke to assume the newvalue of the load current. A momentary change in output voltage is thus inevitable. The time required forresumption of the desired level of output voltage after the sudden load change depends greatly on theproperties of the control system. Two basic control characteristics can be distinguished: Fig.9; these arediscussed in greater detail in Ref.2.

min

V i min0

V i

V i min

V i bo

time0

max V i max max

V i maxV i bo

1.1V i

V i

V i

10% feedforward

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


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9

Part 1

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

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

With feed forward control,

Vi max = ( 1 +x

) minVi max100

The corresponding value of for the transformer design is

= 1 +x

100

Dif 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 ow into the loadand little or no current will be available to charge the output capacitor. Starting will be improved if thesteady-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 avoidsaturation. The transient response time resulting from the method of control is discussed at the end ofSection 1.2.

Primary inductance and de-saturationThe primary inductance is given by

[11]L1 = 0 a n1 Aele

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

[12]I magM = ( Vi) maxL1 f

The peak primary current is

[13]I magM = 1 ( I o + I o min ) + I magMf 2


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10

Part 1

Vi

D

L1

C

+

-

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

Primary inductance L 1 together with slow-rise dV/dt capacitor C (Fig. 10) across the switch forms aresonant circuit with a natural frequency

[14]fr = 12 (L1 C)

The value of C used should satisfy the relationship

I 1M tf = C > I 1M

2VCESM dV/dt

It is recommended that f r > f. At a lower value of f r, the core might fail to de-saturate and ux staircasingcould occur, with disastrous results. A higher f r offers suf cient safety margin under all conditions; a ratio f r/fof 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 L 1 may be reduced by introducing an air gap in the core; the penalty is highermagnetizing current.


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11

Part 1

For these small values of air gaps, the value of e ( and, thus, L 1 ) can be obtained for cores of constantcross-sectional area from

[16]s = le ( 1 - 1 )2 e a

If it is not possible to decrease L 1 suf ciently to make f r > f, the addition of a sensor winding on thetransformer to generate a signal to prevent premature switch-on should be considered. (20)

Transformer currentsThe ripple in the output choke is, in general, only a few percent of the d.c. load current. For this reason, thetransformer 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]I e 1 =I o o max =

I e 2r r

where

[18]o max = maxVi

Vi av min

Duty factor omax 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 lter capacitor of about2 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

Vir

= 252 V

However, max is set so that the converter can handle a 10 ms drop out. But mains drop out is not asteady-state condition, so that omax should be used to calculate I e and, thus, the loss and consequenttemperature rise of the transformer.

Choke designTo 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 currentof I o is

[20]tr =I o L

( Vo + VF + V R ) (tr / - 1 )


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12

Part 1

1.3 The double forward converterPrinciple of operationThe equivalent circuit diagram of a double forward converter is shown in Fig. 11. It comprises two forwardconverters in parallel, with ywheel diode D and lter LC common to both. Switches S 1 and S 2 operatealternately, which doubles the ripple frequency of the choke current. Since energy is pumped twice perconverter period, the output voltage is V o = 2 Vi

V iD

S 1

S 2

L

C V o Io

+

-

+

-

Fig. 11 Outline circuit of a double-forward SMPS converter.

Switches S 1 and S 2 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

2max Vi minVi max 4fIo min

Further choke design proceeds according to Section 1.1.

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

[23]r' =e max Vi min

Vo max + VF + VR

Further design proceeds according to Part 2 of this series.

Transient response timeThe transient response time required for a double forward converter to adapt to a step in the loadcurrent of I o is

[24]tr =I o L

2(Vo max + V F + VR) (tr / -1)


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Part 1

THE FLYBACK CONVERTER

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

[25]Vo = Vi 1

Vi

DS

L C V o Io

+

-+

-

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

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

1/f

/f (1 ) /f

IL

VL

Vi

Vo

time

time

2I ac

Io

S D

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


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Part 1

Non-inverting boost converter

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

Vo =Vi

1

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 Vo Io

+

-

+

-

Fig. 14 Outline circuit of a boost flyback converter.

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

1/f

/f (1) /f

IL

VL

Vi

Vo Vi

time

time

2Iac

Io


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Part 1

2.2 Transformer-isolated flyback converter

Principle of operationFor high-voltage low-current supplies, a useful variant of the inverting y back converter is obtained byadding secondary windings to the choke to form a transformer, Fig. 16. A further bene t of this arrangementis mains isolation.

Io

V o

+

-

V i

+

-

r : 1S

Fig. 16 Outline circuit diagram of a transformer-isolated flybackconverter.

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

Vi max< 2Vi min

take min = 0.3. This yields

1 + 7 / 3 ( V i min / Vi max ) max =

1 [33]

so that

Vo + VF + VRr' =

3 / 7 Vi max [34]

At large input-voltage range, that is

Vi max> 2Vi min


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Part 1

take

Vo + VF + VRr' =

1(Vi max Vi min ) [35]

This yields

r'(Vo + VF + VR) 1 +

Vi max[36]min =

1

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

1 = 1 + (1 - min )Vi min

min Vi maxmax

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

max < 1 1[37]

whence

max Vi min1 +

(1 - max ) Vi max[38]min =

1

and

[39]r' =min Vi max

( 1 min ) (Vo + VF + VR)

with

m =r

1[40]


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Part 1

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

[41]L < ( min Vi max )2

2 f P o min

The maximum peak current through the primary is

I maxM =P o max +

max Vi min2fLmax Vi min

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

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

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

I e1 =P o max 0 max

0 max Vi min1 + 1

3

2

P o max

P o min( ) [42]

and through secundary x,

I ex =rxP ox max (1 0 max )

0 max Vi min1 + 1

3

2

P o max

P o min( ) [43]

where

r (Vo + VF + VR) 1 + Vi av min

0 max =1


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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-pullconverter with a bridge-rectified output. (b) Two-winding (1 + 1) transformer in a bridge converterwith a bridge-rectified output. (c) Three- winding(1 + 2) transformer in a single-ended push-pullconverter with a bi-phase output. (d) Three-winding (1 + 2) transformer in a bridge converterwith a bi-phase output. (e) Three-winding (1 +2) transformer in a push-pull converter with abridge- rectified output. (f)

T1

Nt Nt

C1

D2D1

D3 D4

L1S 1

S 2

C2 C3

T1

NtNt D2D1

D3 D4

L1S 3

S 4

S 1

S 2

C3

T1

1 / 3Nt

1 / 3Nt1 / 3Nt

C1L1

S 1

S 2

C2 C3

T1

Nt

S 3

S 4

S 1

S 2

1 / 3Nt

1 / 3Nt

L1

C3

T1

S 1

S 2

D2D1

D3 D4

L1

C3

1 / 3Nt1 / 3Nt

1 / 3Nt

T1

1 / 4Nt

1 / 4Nt

L1

C3

S 1

S 2

1 / 4Nt

1 / 4Nt


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21

Part 1

3 THE PUSH-PULL CONVERTER

3.1 Principle of operationOutline circuits of the various basic con 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 isdescribed by

Vo =z 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

/f1/2f - f

IL

V L

V o

time

time

2I acIo

V i - V or

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

Duty factorWith the equal conduction times per converter cycle, the maximum allowable duty factor is 0.5, but a morepractical value is 0.45. In practice, wiring and transformer stray inductances result in a nite commutationtime between output diodes. As a result, the interval during which energy is supplied to the output isshorter, 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 recti er diodes.To compensate for the increased commutation time, the energy transfer period /f should be decreasedto about

[45]e = - rIo 1.2 10 -9 f f

Further discussion will be found in Section 1.2.


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Part 1

Transformer turns ratioThe preliminary turns ratio is

[46]r' =2e max Vi minVo + VF + VR

For further transformer design, proceed to Part 2.

3.2 Power choke inductanceThe minimum choke inductance that ensures continuous choke current, and, thus, continuous-modeoperation is

[47]L > Vo4fI ac Vi min

Vi max( 1 - 2 max )

here,

I ac = I o min I magM [48]

and

I maxM =r max Vi min le2n 1 0 e Ae f

2[49]

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


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Part 1

1/f

/f 1/2f f

1/2f

IL

Im1

ID1

ID2

2I ac

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, whichis then diverted through the conducting diode and the output choke to the output capacitor. This causesthe output voltage to rise. From this, it follows that the minimum load current that can be drawn from theconverter without one diode ceasing to conduct in this way is

I o min = I magM + I ac

The magnetizing current owing through the output diodes is

I maxM = max Vi min r

2 L1 f

where

I maxM =2n 1 0 e le

Ae

2

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

I M = (I o max + I o min + I magM ) [50]

Further choke design proceeds with Part 4.


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Part 1

3.3 Transformer currentsIn a push-pull transformer, only one half of the double winding conducts at a time. The peak current througheach half of the double winding is

I 1 M =I M [51]r

The effective primary current in each winding half is

I e1 I o [52]r

o max

and, in the secundary,

I e2 I o [53]o max

where

o max = maxVi min

Vi av min

Transient response timeThe transient response time required for a push-pull converter to adjust to a step in the load currentof I o

[54]tr =I o L

2 ( V o max + V F + V R ) (tr / - 1 )


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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 intoaccount. 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 appliedto the core.


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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 P thfor good, practicaldesigns

P th

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

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

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

ux density sweep (Section 2)

the winding conguration (simple or split/sandwiched windings, sensor or demagnetization windings)

conductor type (solid, strip, Litz)

single or multiple output

mains insulation requirements.


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Part 2

The upper limit of the shaded area for each core refers to a transformer design with optimized ux-densitysweep, 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 ux-density sweep, a (1 + 2) winding con 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 100 oC; the temperature rise is 40 K

the maximum ux-density sweep is limited to 1/1.72 of the maximum permissible 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 ux densities are calculated assuming minimum cross- section areas.

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

It may happen that, at a given power level, more than one type of the core may be used. The followingcriteria 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 discussionof Section 2.

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


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Part 2

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

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

2.1. Flux-density levels for forward and push-pull transformersThe operating ux density of a transformer can seldom approach the maximum permissible ux density inpractice 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 suddenload changes. This factor is related to the ability of the power supply to accept a range of input voltages. Theinput voltage range of mains-fed power supplies may be 215 V to 370 V, or 200 V to 340 V; for telephonesupplies, 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 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 ux density.

B

2B ac

(a)

H

B

2B ac

(b)

H

B

2B ac

(c)

H

Fig. 2 Flux density excursions and the corresponding flux-density sweepsfor (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 ux density depends on the protection circuitry. One source of unbalance isunequal ux linkage between two halve of a centre-tapped winding. For this reason, bi lar windings are tobe preferred. However, this is not possible in mains-fed supplies since the voltage across the winding mightbe greater than the maximum voltage between adjacent turns.The major reason for asymmetry is unequal conduction times or saturation voltages of the power switchesin 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 maylead to the destruction of the power switches. As a further safeguard, the maximum operating ux densityshould be decreased by an additional factor that depends on the ef ciency of the protection circuit.A pratical guide is a 15% allowance for a fully protected converter (unbalance factor = 1.15), but a 100%allowance unbalanced push-pull converter ( = 2).

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


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Part 2

Where feedforward control is used, the transient factor can be reduced considerably and a higher ux-density sweep can often be applied. The actual transient factor is determined by the feedforward percentageused. 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 circuits boundary conditions

flux-density sweep Bac cp (T) forward push-pullmaximum flux sweep (100 oC) 0.16 0.32

0.32 0.32at transient factor 2

0.32with unbalance factor -

0.32 0.32with x% feedforward

2(1 + x/100) (1 + x/100)

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

Practical ux density and sweep limits are summarized for various converter types in Table 2. The curves ofFig. 3, show the optimum ux density sweep, where throughput power is maximum, for a range of cores inthe frequency range 10 kHz to 100 kHz. Horizontal lines indicate the maximum allowable sweep for variousconverter types. Further lines can be added for other boundary conditions with the aid of Table2. The givencurves 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 typeof switch to be used. One this frequency is known, the optimum ux-density sweep can be found forany core type.

Where the frequency is more or less xed, and two core types could be used (Section 1), preference shouldbe given to that core type for which the intersection of the optimum-sweep curve with the set frequency isclosest to the maximum 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 ux-density sweep represents the optimum use of the corematerial.Operation at the optimum sweeps represented by the curves of Fig. 3 means that core loss andpermissible winding loss are in optimum proportion of core and winding loss, results in a lower throughputpower.When the design is limited by saturation ux density, deviation is inevitable. The effect of deviationfrom the optimum ux density is plotted in Fig. 4.

This plot, which applies to any frequency, gives a rough indication for the reduction in throughput powerfrom the optimum value. Once the optimum ux density sweep has been determined for the core andconverter combination, the number of turns can be calculated as shown in Section 3.

2.2. Flux densities for chokes and yback transformersThe maximum ux density is taken into account in the design procedures given in Parts 1 and 4. The designcentres around the maximum stored energy I 2M

L in the choke, which, in turn, is related to the inductancevalue from which follows the number of turns.The ux density sweep in chokes is generally relatively low, and so, in consequence, is the core loss.


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Part 2

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

L I acB

ac max=

N prim Amin

where I ac is found during the design process (Part1). In case of a yback transformer, all quantities refer tothe primary. If I ac is relatively high, core loss will be signi cant.

3. Number of turnsOnce the 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

Vin2 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 and V i are obtained fromVi = maxVi min = 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 n 2 min is small, rounding willchange the ux density sweep signi cantly. To prevent saturation, rounding should be to the next higherinteger. The effect of rounding can be counteracted by changing the operating frequency.

Where operating frequency is not xed, iteration will result in a satisfactory design. When the actual ux-density, sweep is within 10% of the optimum, throughput power will be within 5% of its maximum (Section2.1). Use the 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 ofr' is calculated in Part 1.


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Part 2

5. Thermal resistanceIn order to determine the maximum permissible dissipation of a transformer or choke, its thermal behaviourmust be know. This depends on core size, conductor form, and insulation requirements. For standard, moreor less cubical transformers based on cores like E or ETD the constant C th is about 50. Very at designs likePlanar E cores have much better thermal properties, the constant is around 25. (V e is in cm

3).

Rth = C th Ve-0.54 K/W

These thermal resistances were m

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