J. Basic. Appl. Sci. Res., 1(12)2655-2662, 2011
© 2011, TextRoad Publication
ISSN 2090-4304
Journal of Basic and Applied
Scientific Research
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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
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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 j1   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.
j1
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)
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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


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(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  Lm     


  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.
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J. Basic. Appl. Sci. Res., 1(12)2655-2662, 2011
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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
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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
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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.
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