µ r of the ferrite amounts to It should be noted that the magnetic length of the + δ

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1 Page 9 Design of Inducors and High Frequency Transformers Inducors sore energy, ransformers ransfer energy. This is he prime difference. The magneic cores are significanly differen for inducors and high frequency ransformers: Inducors need an air gap for soring energy, ransformers do no. Transformers for flyback converers have o sore energy which means hey are no a high frequency ransformer bu hey are in fac an inducor wih and windings. The maerial of he cores is normally ferrie. In addiion o his oher marerials wih high permeabiliy and wih a high sauraion poin are used. Calculaion of Inducors: An inducor wih cerain inducance L and cerain peak curren I can be deermined by he following calculaion: Inducors should sore energy. The sored energy of an inducor is: W = 1 LI. This energy is sored as magneic field energy, wihin he ferrie core and wihin he air gap (see Fig.5.1.1). The higher he required sored he energy he larger he required core. The size of a inducor is approximaely proporional o he sored energy. I N A l fe δ Φ=ΒΑ H fe H δ I : inducor curren N: number of urns A: cross-secion area of he core l fe: magneic lengh of he core δ : air gap Φ: magneic flux : magneic flux densiy H fe: magneic field srengh wihin he ferri H : magneic field srengh wihin he air gap δ Fig : inducor wih is magneic and mechanical sizes The field energy in he inducor is: W = 1 (1) H dv 1 H Fe Fe V Fe + 1 H δ δ V δ energy in he ferrie energy in he air gap The magneic field densiy is coninuous and wihin he air gap and he ferrie is approximaely equal, i.e. Fe δ. The magneic field srengh H is no coninuous, wihin he air gap i is increased by a facor µ r compared o ha han wihin he ferrie. If his is subsiued ino equaion (1) and considering = µ 0 µ r H, V Fe = l Fe A and V δ = δ A his leads o:

2 Page 30 W 1 l Fe µ 0 µ + δ r A µ r of he ferrie amouns o I should be noed ha he magneic lengh of he ferrie is reduced by µ r in he above equaion. Therefore i can be seen ha he energy is mainly sored wihin he air gap. This leads o: W 1 A δ µ 0 Inducors require an air gap o sore energy. ecause he energy is sored wihin he air gap, an inducor requires a cerain volume for he 1 air gap o sore a cerain amoun of energy. The energy is given by LI. The core maerial has a limi for he maximum magneic flux densiy, his limi is abou max = 0, 3 T for usual ferrie maerials. This leads o a minimum required volume of he air gap: V δ V δ = A δ L I µ 0 max where max = 0, 3 T Knowing he required volume of he air gap, a core can be seleced from a daabook of ferrie cores. The number of urns N can be calculaed wih help of he magneic conducance A L (ofen simply called he -value): A L N = L A L A L :magneic conducance The -value can be verified from he daabook of he ferrie cores. A L The maximum flux densiy should no be higher han 0.3 Tesla. The maximum flux densiy wihin he ferrie can be calculaed using he daa of he core daashee. = L I = N A L I N A min A min! 0, 3 T A min : Minimum cross-cu of he core. The flux densiy has is maximum. a A min. A min can be verified from he daashee.

3 Page 31 Calculaion of he wire: The curren densiy S of he wire can be chosen beween und 5 A/mm² (depending on he size and he isolaion, which deermines he hea ranspor ou of he inducor). This leads o he diameer of he wire d: d = 4 I RMS π S wih S = 3 5 A mm Calculaion of High Frequency Transformers A high frequency ransformers ransfer elecric power. Is mechanical size depends on he power o be ransfered and on he operaing frequency. The higher he frequency he smaller he mechanical size. Usually frequencies are from 0 o 100kHz. The maerial of he core is ferrie. Daabooks for appropriae cores provide informaion abou he possible ranfer power for various cores. The firs sep o calculae a high frequency ransformer is o choose an appropriae core wih he help of he daabook, he size of he core is dependen on he ransfer power and he frequency. The second sep is o calculae he number of urns. This number deermines he magneic flux densiy wihin he core. The number of urns is he raio of o volage. Following his he diameers of he and conducors can be calculaed depending on he RMS-values of he currens. Calculaion of he minimum number of urns: I T/ T V R V N N1 N1 N simple equivalen circui: I' I N R N1 L1 I M V' R N1 N R N N1 I M I M I M : magnifiing curren Figure 5..1: Volages and currens a a ransformer

4 Page 3 The volage V 1 a he side of he ransformers has a recangle shape. This causes an inpu curren I 1, which is he addiion of he back ransformed curren I and he magneising curren I M (see figure 5..1). To keep he magneising curren I M low, a magneic core wihou an air gap is used. The recangle volage V 1 causes a riangle shape for he magneising curren I M. The magneising curren is approximaely independen of he curren I (see he simple equivalen circui in figure 5..1). The magnifiing curren is approximaely proporional o he magneic flux or flux densiy. The inpu volage V 1 deermines he magneic flux. The physical correlaions are given by Faradays law of inducion: V = N dφ. d T/ T Figure 5..: inpu volage and magneic flux densiy a he ransformer For he ransformer in figure 5..1 follows: = V 1 T/ N 1 A The change of flux densiy depends on he frequency f = 1/T and he number of urns N 1. The higher he frequency and he number of urns he lower he change of flux densiy. The minimum number of urns N 1 min can be calculaed o ensure ha a cerain change of flux densiy is no exceeded. The sauraion flux densiy of abou 0, 3 T (which means 0, 6 T) canno be used in high frequency ransformers. In push-pull converers going around he hyseresis loop wih every clock would cause unaccepable losses i.e. hea generaion. If no furher informaion on core losses and ermal resisance are available, should be limied o 0, 3 0, T for operaing frequencies from 0 o 100 khz. The lower he lower he core losses. This leads o a minimum number of urns for N 1 : N 1 min V 1 T/ A min where 0, 0, 3 T A min : minimum cross-secion area of he core. This is where he flux densiy is a a maximum. can be checked from he daashee. A min In single ended forward converers he core is magneised ino one polariy only. Iin push-pull converers he core is magneised alernaing ino boh polariies.

5 Page 33 single ended forward converers push-pull converers The calculaion of he minimum number of urns ypes of swich mode power supplies. N 1 min is equal for hese differen Calculaion of he winding conducors: The diameer of he conducors depends on he RMS-value of he curren. The curren can be calculaed wih he power. For he push-pull converer follows: I 1RMS P ou V in and I RMS = P ou V ou +Pou/Vin -Pou/Vin I +Pou/Vou -Pou/Vou For he single ended forward converer follows: I 1RMS P ou V in and I RMS = P ou V ou I Pou/Vin Pou/Vou The magneising curren can be negleced in his calculaion. The curren densiy can be chosen in a range of o 5 A/mm, depending on he ermal resisance of he choke. The cross-secion and he diameer can be calculaed as follows: A wire d wire A wire = I S and d wire = I 4 S π where S = 3 5 A mm If good coupling is imporan, he and winding should be placed on op of each oher. Improved coupling is achieved if he windings are inerlocked. The coupling is bad in a) good in b) and in c) abou four imes beer han in b). second. second. a) b) c)

6 Page 34 The number of urns should no be chosen significanly higher han N 1 min, oherwise he copper losses of he wire would increase needlessly due o he longer conducor. For high frequencies and large diameer of he wire he skin effec should be considered. For operaing frequencies of more han 0kHz and diameers of more han 1mm liz wire or copper foil should be used.

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