J. Basic. Appl. Sci. Res., 1(12)2655-2662, 2011 © 2011, TextRoad Publication ISSN 2090-4304 Journal of Basic and Applied Scientific Research www.textroad.com Presentation of an Algorithm to Optimal Design of High Frequency Transformers H. Feshki Farahani1,* 1 Department of Electrical Engineering, Ashtian Branch, Islamic Azad University, Ashtian, Iran ABSTRACT High frequency transformers are used in different applications such as switching power supply. So, their optimal designing are of great importance. In all transformer applications, their sizes should be minimized. Transformer size depends on different parameters such as flux density, operational frequency and core type. Therefore, by optimizing these parameters, transformer size can be minimized. As well as size, losses are also important in these transformers. Because switching losses are high in these transformers, so, this parameter should be optimally chosen. The objective of this paper is to present an algorithm to optimal design of high frequency transformers based on losses minimization. Transformer losses include copper and core losses which depends on flux density. As a result, designing based on flux density optimization is recommended. This optimization is done by optimally choosing of window factor, core type and switching frequency. Furthermore, a sample transformer is designed, constructed and tested for switching power supply applications using recommended algorithm. Experimental results verify this algorithm efficiency.. KEY WORDS: High frequency Transformer, Copper Loss, Core loss, Optimal Frequency and Flux Density I. NOMENCLATURE ρ D Itot k n1 ,n 2 λ1 ku β ffe kgfe Kgfe,desired AC WA MLT lm Bmax Aw1, Aw2, … α1, α2, … no_string1 no_string1 Ptot,desired Pcu,tot Ptot Pfe Bmax Wire Effective Resistivity Duty Ratio Total RMS Winding Current Referred to Primary Turn Ratio Primary and Secondary Turn Primary Volt-Seconds Wining Fill Factor Core Loss Exponent Core Loss Coefficient Core Geometrical Constant Core Geometrical Constant desired Core Cross Sectional Area Core Area Winding Mean Length per Turn Magnetic Path Length Peak ac Flux Density Wire Areas Window Factor Number of String Primary and Secondary Desired Total Loss Desired Total Copper Loss Total Loss Core Loss Maximum Flux Density of Core II. INTRODUCTION One of the key issues which should be taken into account is losses (includes core and switching losses). Total transformer losses depend on core type, frequency, winding factor, wire diameter and flux density. Hence, by choosing optimal values for these parameters, core loss and consequently core size can be minimized. Winding losses includes skin effect and proximity effect. To decrease former loss, some conductors with wire area approximately equal to skin depth have been used. Many papers have been introduced designing of switching power supply transformer [1-3]. To decrease latter effect loss, sandwich winding way has been used. Winding losses are a function of winding factor which by choosing optimum of this parameter, loss can be reduced. Reference [4] presents some methods to optimize winding losses which are used to design optimal high-frequency transformer. In [5], Core losses depend on flux density and frequency. Therefore, optimization of these two parameters leads to optimal core selection [6-8]. *Corresponding Author: H. Feshki Farahani, Department of Electrical Engineering, Ashtian Branch, Islamic Azad University, Ashtian, Iran. Tel.: +98 (912) 2549149; fax: +98 862 7222627. E-mail: hfeshki@yahoo.com 2655 Feshki Farahani, 2011 In [9], a designing method with optimization of both copper and core losses has been proposed. In reference [10], optimal transformer considering thermal and electrical characteristics has been designed by selecting apparent endurable power for Ferrite core, flux density and current density. Reference [12], presents a method to construct a high frequency planar transformer model with improved accuracy which is suitable for design, optimization and circuit simulations. A method of optimizing planar transformers’ design based on the optimization tool iSIGHT is also presented. Some earlier studies have indicated that there was an optimum spacing between primary and secondary layers that minimized losses [12-15]. The trend in SMPS is towards miniaturization, and transformers are the largest objects in SMPS circuits. It also became evident that the ease with which interleaving can be implemented in planar structures allows the minimization and control of leakage inductance within the winding [12]. Furthermore, an analytic model has been developed for the parasitic capacitances of planar transformers [16]. It is obvious that windings can be close together, thus, reducing leakage inductance has usual unwanted effects of increasing AC resistance and parasitic capacitance. In this paper, firstly, copper loss equation is obtained and then it is optimized respect to window factor using Lagrange method. Optimal window factor is chosen for each winding. In another section, total loss equation (copper and core losses) is obtained. This equation is a function of frequency and flux density which by optimization respect to these parameters, their optimal values are determined. In this design, skin effect (due to parallel wire series instead of one single wire) and proximity effect (due to few winding layers) are ignored. After that, an algorithm to design transformer is presented in which an acceptable loss (Ptot,desired) is defined. Total transformer loss (Ptot) should be less than this loss value. III. TRANSFORMER DESIGN To design a transformer with k outputs, leakage inductance is assumed very high. Copper and core losses are also considered in designing. Design process is based on optimization of sum of core and switching losses (Ptot) and choosing optimal flux density [6, 11, and 12]. Designing is based on (kgfe) which is explained in following parts. A. Core Windowing Factor Optimization The RMS of Transformer output current is determined by load. Winding factor (α) can be defined as: n k 1 i 1 , (1) k 1 Ptot Pcu1 Pcuk 2 2 MLT k n j I j W A k u j1 j (2) To optimize α1, α 2, … , α k values, Lagrange multiplier method is used: k g ( 1 , 2 ,, k ) 1 j (3) j 1 f (1 , 2 , , k , ) Pcu (1 , 2 , , k ) .g (1 , 2 , , k ) (4) Where f is objective function, g is considered constraint and ε is Lagrange multiplier. By deriving of (4) respect to all of variables, following equations can be obtained: 2 MLT k n j I j Pcu ,tot W A k u j 1 m nm I m nm Vm nj Vj , (5) njI j (6) j 1 m Vm I m (7) V j I j j 1 Where VmIm is apparent power of mth winding and VjI j is sum of switching apparent powers. j1 B. Copper Loss Copper loss can be obtained by [11, 18, and 19]: 2 k n MLT n12 I tot j Pcu , I tot I j WAku j 1 n1 ( A) ( B) (8) (C ) 2 I 2 MLT 1 2 Pcu 1 tot 2 ku W A Ac Bmax Equation (9) is consists of three components: (9) 2656 J. Basic. Appl. Sci. Res., 1(12)2655-2662, 2011 (A): Electrical characteristics (B): Core characteristics (C): Core loss characteristics C. Core Loss In core selection, by combination of different allays, there has been always a tradeoff between saturation magnetic flux density and core loss [9, 10]. Choosing materials with higher Bsat lead to size and price reduction. Unfortunately, materials with higher Bsat have bigger losses. Among different materials, Ferrites have relatively lower Bsat (0.2T < Bsat < 0.5T). Therefore, their core reluctance is more than those in other materials and consequently has lower eddy losses. These materials in frequency of several ten kHz to 1 MHz operate so well. Loss power density is a function of Bmax,ac for different frequency values to sinusoidal excitation which can be calculated. For a specified frequency value, core loss Pfe is estimated as following function: Pfe k fe Bmax Acl m (10) V Typical value of B for ferrite core is 2.6 < B < 2.8. Geometric coefficient Kfe depends on core temperature and Bmax which is significantly increased by increasing frequency excitation. Dependency of Kfe on frequency can be generally shown as a function like: k fe ( f max ) k fe0 [1 a1 f max f f a2 ( max ) 2 ... an ( max ) n ] f0 f0 f0 (11) D. Optimal Flux Density Maximum flux density is optimal when total core loss (Ptot) is minimized. So, optimal flux density according to Fig. 1 is a point where sum of copper loss and core loss (Ptot) is equal. At the point that Bmax is optimal, following equations can be written: Ptot Pfe Pcu (12) Pfe Ptot Pcu 0 Bmax Bmax Bmax (13) 1 Bmax 2 I 2 MLT 1 2 1 tot 3 2ku W A Ac Lm k fe (14) Fig. 1: Selecting optimal flux density from core and copper losses [11] By combination of 12 and 14, core loss can be calculated as: Ptot Ac Lm k 2 fe 2 2 12 I tot MLT 2 2 2 2 WA Ac3 2 4ku 2657 (15) Feshki Farahani, 2011 From (15), two equations can be obtained as: k gfe 2 ( 1) c WA A 2 2 2 2 2 MLT Lm 2 (16) 2 k gfe,desired 2 12 I tot k fe (17) 2 u tot 4k P To select core, following constraint should be met: (18) k gfe k gfe ,desired As it is obvious from (16), β can affect Kgfe calculation to some extent. But due to little range of β variation (2.6 < β < 2.8), tolerance value of Kgfe will be subtle. E. Optimal Frequency to Core Size Reduction Kfe is a function of frequency which has considerable impact on Kgfe value and core size as well. To obtain optimal frequency, equation (12) is derived respect to frequency and flux density. Then, Bmax and fmax are calculated: Ptot 0 (19) f max Ptot 0 Bmax (20) If Kfe is the same as (11), optimal frequency values will be roots of F function as follow: 2 1 f max 2 f max n f max ... n F 1 1 2 f0 f 0 f 0 n (21) One of the main losses in switching power supplies are switching losses which is highly increased by frequency increase. Hence, in these devices, optimal frequency may not be used (which is determined based on loss and core size reduction). The relationship between flux density and leakage flux and turn number is as follow: B max 1 2n1A c (22) Bmax may contain DC value which has no effect on core losses. DC value just changes the point at which smaller loop, related to core loss is composed. Wire area value can be obtained from: A W1 k u WA 1 n1 (23) To decrease skin losses, several parallel wire series have been usually used. Each series diameter is chosen by considering skin effect characteristic. IV. EFFECTS OF FREQUENCY INCREASING ON THE CORE SIZE Core size and flux density variation characteristics versus frequency are shown in Fig. 2 and Fig. 3 respectively [11, 20, and 21]. If Kfe is assumed independent on frequency, by increasing frequency, core size will be smaller. As it is clear in Fig. 2, from core-size-selection point of view, (bigger core is determined by its number), both frequency values of 250 kHz and 400 kHz can be chosen. However, if optimization of switching losses is more important than core size, lower frequency is more favorable. 2658 J. Basic. Appl. Sci. Res., 1(12)2655-2662, 2011 4226 Core Size 3622 2616 2616 2213 2213 1811 25kHz 50kHz 100kHz 200kHz 250kHz Frequency 1811 400kHz 500kHz 1000kHz Fig. 2: Core size versus frequency [11] 110 100 Flux Density, B m ax [mT] 90 80 70 60 50 40 30 20 10 0 100 200 300 400 500 600 Frequency [kHz] 700 800 900 1000 Fig. 3: Flux density versus frequency [11] V. ALGORITHM OF OPTIMAL TRANSFORMER DESIGN The objective is to find optimal flux density, layer numbers, Litz wire diameters and the smallest core size that can tolerate considerable power. According to equations from (1) to (23), optimal transformer designing algorithm for switching power supply is recommended according to Fig. 4. VI. IMPLEMENTATION OF ALGORITHM TO DESIGN A TYPICAL SWITCHING POWER SUPPLY TRANSFORMER The algorithm of transformer designing is shown in Fig. 4. Using this algorithm, a sample transformer for a full wave converter is designed which its results are listed in Table. 1. Core is selected from EE57475 type. Skin depth in 25 kHz frequency and 25oC is equal to δ = 0.04cm. According to Fig. 5, wire strings of #AWG27 with 0.0409 cm value of diameter are used. Primary voltage waveform in to case with snubber and without snubber is depicted in Fig. 5. This figure shows that, transformer have been designed suitably. Secondary voltage and primary current are shown in Fig. 6 and Fig. 7 respectively. Table. 1: Input and output parameters of optimal design of transformer Input fs = 25 k Hz D = 0.35 IO=25 A Output 2.7 Itot = 17.5350 Kgfe,desired = 0.08564 Kgfe= .1191 Bmax=127.7 mT 1.724 10 6 n1 = 32.1429 k = 0.400 kfe =6.55 Ptot = 7 Watt Vg=311 V PO=1200 Watt VO=48 V ku = 0.3000 n2=18 Turn a1=0.3374 a2=0.3313 AW1=0.0241cm2 AW2=0.0423 no_string1=4.28 no_string2=5.67 2659 Feshki Farahani, 2011 Start Input parameters fs , Io , D, Pdesired , kfe , ku No Frequency Limitation? Calculation of foptimal , kgfe,optimal from (11) and (20) Yes Calculation of kgfe,desired from (17) Refer to Core Database Selection the smallest core considering constrain (18) Input the Kg , WA ,AC ,Brms selected core from database Selecting the core with high kgfe Bmax Calculation Yes Bmax ≥ Brms No Primary and secondary Turn number Calculation from (22) Rounding of n1 and n2 Recalculating of Bmax using rounded n1 and n2 Calculation of Pcu , Pfe , Ptotal Yes Pdesired ≥ Ptotal No Wire area calculation from (23) End Fig. 4: Optimal transformer design flowchart (a) (b) Fig. 5: Primary voltage of designed transformer a) without snubber b) with snubber 2660 J. Basic. Appl. Sci. Res., 1(12)2655-2662, 2011 Fig. 6: Secondary voltage of designed transformer Fig. 7: Primary Current of designed transformer VII. CONCLUSION In this paper, an algorithm has been presented to design optimal transformers of switching power supplies considering switching, copper and core losses. This has been done by optimal choosing of frequency, flux density and minimization of ferromagnetic materials volume. Copper loss equation has been optimized respect to window factor, frequency and flux density using Lagrange method and their optimal values has been determined. To decrease skin effect losses, parallel wire series (litz) has been used and to decrease proximity effect loss, sandwich winding method for wires has been used. Besides, in this algorithm, dependency of core loss factor on frequency and its effects on core size have been considered. Finally, a sample transformer 25 kHz has been designed and constructed using recommended algorithm and t has been used in a switching power supply. Experimental results have verified this algorithm practicality. VIII. 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