ELECTRONICA DE POTENCIA - Revista de la SOBRAEP, vol. 5, núm. 1, 2000, pp.
16-27.
FLYBACK VS. FORWARD CONVERTER
TOPOLOGY COMPARISON BASED
UPON MAGNETIC DESIGN .
H. E. TACCA.
Cita: H. E. TACCA (2000). FLYBACK VS. FORWARD CONVERTER TOPOLOGY
COMPARISON BASED UPON MAGNETIC DESIGN. ELECTRONICA DE
POTENCIA - Revista de la SOBRAEP, 5 (1) 16-27.
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1
FLYBACK VS. FORWARD CONVERTER TOPOLOGY COMPARISON BASED
UPON MAGNETIC DESIGN
Hernán Emilio TACCA
Universidad de Buenos Aires , Facultad de Ingeniería
LABCATYP - Depto. de Electrónica
[email protected]
Abstract - Based on approximative equations for single
step calculation design procedures, a simple method
useful to adopt the most suitable topology in low power
DC to DC conversion is derived from the volume
calculation of the magnetic cores needed.
la
leav
air gap
average mean turn length
f fV
voltage form factor
S Cu L
mean turn length
effective magnetic length
primary turns
secondary turns
winding losses
device rated switching power
core losses
converter output power
a.c. resistance
d.c. resistance
total forward core volume to flyback core
volume ratio
thermal resistance
inductor wire cross section
FP
partition factor of the primary windings :
FP = SWP SW
S Cu P
primary wire cross section
S CuS
secondary wire cross section
power switch profit factor (or power device
S dis
S Fe
FW
I Lav
utilization factor)
total winding eddy currents factor
window factor : FW = SW S Fe
average value of the inductor current
S Fe ef
heat dissipation surface
minimum core section
effective core section
S FeFB
flyback core section required
I Lef
effective value of the inductor current
S Fe L
inductor core section required
peak value of the inductor current
S FeTFW
forward transformer core section required
I Lmax
I Pav
average value of the primary current
window area
primary window area
I Pef
effective value of the primary current
SW
SWP
SWS
secondary window area
I Pmax
peak value of the primary current
tC
kC
VFeFB
power switch conduction time
core volume of the flyback coupled inductor
VFeFW
(usually named flyback “transformer”)
total volume occupied by the cores of the
VFe L
forward converter
core volume of the smoothing inductor
kS
k uT
core sizing characteristic converter topology
coefficient
effective core shape factor
skin effect factor
proximity effect factor
geometrical core shape factor
transformer utilization factor
VFeTFW
core volume of the forward transformer
L
inductance of the smoothing inductor
VP
Δ
primary supply voltage
penetration depth
Index terms - Low power magnetic components, switching
power converters, flyback and forward magnetics design.
NOMENCLATURE
Bmax
Bmin
D
f
Fc P / S
peak value of the induction
induction minimum value
duty-cycle factor
switching frequency
window filling factor of the primary/secondary
coil (also known as copper factor) :
Fc P / S = n P / S S Cu P / S SWP / S
f pr
Fr
k ES
k rS
k rX
lem
l Fe
nP
nS
PCu
PD
PFe
PO
Rac
Rdc
ℜVFW / FB
Rθ
2
ΔB
δ iL
δ iP
η
Φ
μo
μ rs
ρ
σL
σP
maximum induction increment
normalized amplitude of the inductor ripple
current
normalized primary current variation
converter efficiency
magnetic flux
magnetic field constant
relative static permeability (derived from the
static magnetization curve)
conductor resistivity
inductor current density
primary current density
δ i P = ΔI P I Pmax
D = tC T
and from the current waveforms it follows :
⎛ δi ⎞
I Pav = D I Pmax ⎜ 1 − P ⎟
⎝
2 ⎠
(2.2)
D I Pmax
(2.5)
I Pef =
II. FLYBACK CONVERTER
A. Basic flyback circuit
Fig. 1.a depicts the basic circuit of a flyback converter and
fig. 1.b shows the corresponding waveforms. From fig. 1.b it
may be defined :
(2.1)
ΔI P = I Pmax − I Pmin
(2.4)
1
1 − δ iP + δ iP2
3
⎛ δi ⎞
PO = η PP = η V P I Pav = η D⎜ 1 − P ⎟ V P I Pmax
⎝
2 ⎠
Then, the output power is :
+ Vp
I. INTRODUCTION
In DC to DC converters and off-line switching power
supplies for low power applications, the most used
topologies are the flyback and the forward converters [1].
The flyback structure has the advantage of requiring only
a single magnetic component. This one, if designed for
minimum size, results smaller than the overall volume
occupied by both the smoothing inductor plus the power
transformer of the equivalent forward converter. However,
when the flyback coupled inductor size is minimum, the
profit factor of the power switch becomes poor [2][3]. If an
improved profit of the flyback power device is required, it
results mandatory to supersize the magnetic component . So,
a trade-off arises between the core size and the rated
switching power (maximum theoretical switching power) of
the power transistor [4]. The better the profit factor, heavier
the magnetic cores become and this apply to both topologies
here compared. Nevertheless, the required core volumes
result from different laws, so for a wanted profit factor, one
topology will be the most suitable regarding the core weight.
In this work, the total magnetic material volume required
for both alternatives are computed and related. The
comparison criterion is stated as the ratio of core sizing
approximative equations, which become justified by the
closed matching with results obtained using manufacturer
design data and recommended procedures.
Derivation of some equations involved on design
procedures are presented in appendices (including some
application examples agreeing with manufacturer data).
In order to compare topologies, first, the required core
sections are obtained and then, the core volumes are
computed assuming identical core shapes for all the magnetic
components involved.
Finally, the volume rate is plotted to bring an easy method
to know wich topology will be the lightest.
(2.3)
D
iS
nP
(2.6)
iO
nS
RL
iP
Vo
(a)
iP
Ipmax
Ipmin
0
tc = DT
T
t
T
t
iS
I S max
I S min
m. m. f .
tc
T - tc
m. m. f .max
nS i S
nP iP
magnetomotive force
m. m. f .= n P i P + nS i S
m. m. f .min
T
tc
DT
T - tc
t
(b)
Figure 1 : Flyback converter, (a) basic circuit, (b) basic
waveforms.
B. Core sizing
The primary current density is :
I Pef
σP =
S Cu P
(2.7)
where S Cu P is the cross section of the primary wire that
must verify :
S Cu P = FP Fc P SW n P
(2.8)
3
where, ΔB = Bmax − Bmin and
μo
Bmax =
la +
n P I Pmax ≅
μ rs
l Fe
μo
la
(2.9)
n P I Pmax
(2.10)
μo
ΔB ≅
n P ΔI P
la
Relating 2.10 and 2.11 yields :
ΔI P
ΔB
=
= δ iP
Bmax
I Pmax
where,
( p)
G δi
=
Bm
D PO
f σ P Fc P FP FW
1
1 − δ iP + δ iP2
3
⎛ δ iP ⎞
⎜1 −
⎟
⎝
2 ⎠
( p)
simplified assuming G δ i
Dm
iP
Ipmax
Ipmin
0
iS
DT
iL
T
t
Ismax = I Lmax
iL
iS
I Lmin
v CE
(2.14.b)
DT
m. m. f .
VCEmax
T
t
T
t
v CE
m. m. f .
Vp
DT
tm
A. Basic circuit and transformer core sizing
The typical circuit of a forward converter is presented in
fig. 2.a , while its waveforms are depicted in fig. 2.b . The
equation of the primary voltage becomes :
B
1
V P = n P m S Fe =
n P Bm f S Fe
(3.1)
tC
D
from which, the sizing equation results :
Bm
v CE
(2.14.a)
III. FORWARD CONVERTER
( p) η
RL
(a)
(2.12)
≅1 .
D PO
f σ P Fc P FP FW
Vo
DL
nS
(2.11)
To complete the design, it is necessary to adopt the
partition factor of the primary windings (see section IV), the
primary current density (section V) and the maximum value
of the induction (Appendix I).
S FeTFW = G δ i
iO
iP
For all the possible values 0 ≤ δ i P ≤ 1 it results
1 ≤ G δ i < 1075
.
, so the expression 2.14.a may be
( p)
L
iL
nP
using this equation (with the simplified notation
Bm = Bmax ) the expression 2.9 yields :
f
S Fe
V P = n P δ i P Bm
(2.13)
D
With eqs. 2.5 , 2.6 , 2.7 , 2.8 and 2.13 , using the window
factor definition FW = SW / S Fe , it results :
( p) η δ iP
as it was done with flyback converters.
The equations 3.1 and 3.2 are particular cases of
generalized expressions valid for many other converter
topologies (Apendix II).
+ Vp
where μ rs is the relative static permeability.
By a similar way,
S FeFB = G δ i
≅1
DFW
la >> l Fe μ rs
because usually :
( p)
G δi
In accordance with the Faraday law :
ΔB
S Fe
VP = nP
tC
(b)
Figure 2 : Forward converter, (a) basic circuit, (b) basic
waveforms.
B. Inductor core sizing
The effective value of the
expressed as :
SWL
I Lef = σ L S Cu L = σ L Fc
nL
On the other hand,
n Φ nL
L= L
=
iL
inductor current can be
= σ L Fc FW
Bm S Fe L
I Lmax
S Fe L
nL
(3.3)
(3.4)
Using eqs. 3.3 and 3.4 :
(3.2)
S Fe L =
where it may be approximated
From fig. 2.b it follows :
L I Lef I Lmax
σ L Fc FW Bm
(3.5)
4
I Lef = I Lmax
I Lav = I Lmax
1
1 − δ iL + δ iL2
3
⎛ δ iL ⎞
⎟
⎜1 −
⎝
2 ⎠
and then, the output power results :
n
⎛ δi ⎞
PO = VO I Lav = S D ⎜ 1 − L ⎟ V P I Lmax
nP ⎝
2 ⎠
During the interval 0 ≤ t ≤ D T it must be
nS
ΔI
V P − VO = L L
DT
nP
that may be expressed as
n
L f
I L δ iL
(1 − D ) S =
n P D V P max
Substituting eqs. 3.6 , 3.8 and 3.9 in 3.5 , yields :
S Fe L = G( δ i )
L
(1 − D) PO
(3.7)
and secondary coils, while FP is the partition factor of the
primary windings defined as the ratio between the window
area occupied by the primary winding and the total window
area , FP = SWP SW .
(3.8)
Substituting the eqs. 4.3 into eqs. 4.2 , then the result into
4.1, and considering both
n P I Pef = n S I Sef and SW = FW S Fe ,
(3.6)
I Pef
(3.9)
Optimal partition factor :
The optimum is defined as that value which minimizes the
winding losses, given by
(4.1)
PCu = PCu P + PCuS = I Pef 2 RCu P + I Sef 2 RCuS
where, the winding resistances are
RCuS = ρ eqS n S
and, lemP / S
S Cu P
lemS
(4.2.a)
(4.2.b)
S CuS
are the primary and secondary mean turn
lengths, while ρ eq P / S are the equivalent resistivities of the
primary and secondary conductors, taken into account skin
and proximity effects [7][8][12][13] :
(4.2.c)
ρ eq P / S = FrP / S ρ Cu
FrP / S = k rS P / S + k rX P / S
(4.2.d)
being k rS P / S the primary and secondary skin effect factors
and k rX P / S the proximity factors (Appendix III).
The conductor sections are :
S Cu P = FP Fc P
SW
nP
S CuS = (1 − FP ) FcS
(4.3.a)
SW
nS
(4.3.b)
where, Fc P / S are respectively, the fill factors of the primary
)
⎞
σ P FcS ⎛ 1
=
− 1⎟
⎜
Fc P ⎝ FP
σS
⎠
(3.10)
IV. PARTITION FACTOR OF THE WINDING AREA
lemP
2
that related give :
δ i L Bm f σ L Fc FW
where , as previously stated , it may be aproximated
G( δ i ) ≅ 1 . Also, it can be assumed that δ i L ≅ δ i P
L
provided that the magnetizing inductance be large enough.
In order to complete the design, the air-gap and the
winding turns must be determined, which may be done by
several graphical [5][6] or analytical methods [7]-[9][11].
RCu P = ρ eq P n P
(
n P 2 ⎡ lem
lemS FrS FrP ⎤
P
⎢
⎥
(4.4)
PCu = ρ eq P
+
FW S Fe ⎢ Fc P FP
FcS (1 − FP ) ⎥
⎣
⎦
On the other hand, the current densities in each winding
are :
σ P = I Pef S Cu P and σ S = I Sef S CuS ,
yields :
(4.5)
Considering the common winding techniques, two
alternatives will be studied.
1. Shared coil-former windings
In this case : lemP = lemS = lem so , the eq. 4.4 becomes :
(
)
⎡
FrS FrP ⎤
lem
⎥
⎢ 1
+
FW S Fe ⎢ Fc P FP Fc S (1 − FP ) ⎥
⎦
⎣
(4.6)
The optimal partition factor will be the one which minimizes
the copper losses given by eq. 4.6 . That is :
1
FPopt =
(4.7)
Fc P FrS
1+
Fc S FrP
PCu = ρ eq P I Pef
2
nP2
and from eq. 4.5 , the current densities become related by :
Fc S Fr
σP
S
=
(4.8)
Fc P FrP
σS
For the particular case when Fc P = Fc S and FrP = FrS
it results :
and
FPopt =
1
2
σP =σS
In such a case, the Joule losses result :
4 lem
PCu = ρ eq P I Pef 2 n P 2
Fc P FW S Fe
Defining the equivalent turn factor :
Fc
FrS ⎤
FP
1⎡
⎥
Fle = ⎢1 + P
Fc S (1 − FP ) FrP ⎥
2⎢
⎦
⎣
the eq. 4.6 may be rewritten as :
Fle lem
PCu = 2 ρ eq P I Pef 2 n P 2
FP Fc P FW S Fe
(4.9.a)
(4.9.b)
(4.10)
(4.11)
(4.12)
5
(
)
ρ eq P = k rS P + k rX P ρ Cu
2. Stacked windings
In case of superposed winding sections (for example
secondary over primary) the mean turn lengths become
different and will depend on the the primary partition factor
adopted. For the core of fig. 3.a , according to fig. 3.b , the
sectional mean turn lengths will be :
(4.13.a)
lemP = 2( A + B ) + π C P
C ⎞
⎛
lemS = 2( A + B) + 2π ⎜ C − S ⎟
⎝
2 ⎠
(4.13.b)
C = C P + CS
where :
(4.13.c)
A/2
H
C
A/2
(4.15)
( C P − CS )
2
For C P = CS the average mean turn length coincides
with the single section mean turn length, defined by :
leav = 2( A + B ) + π C +
lem = leav
]C = C
P
S
π
= 2( A + B ) + π C
C
lemP
FW
FP
Fh
(4.20.b)
S Fe
Substituting 4.20.a and b into 4.4 yields :
l em
PCu = ρ eq P I Pef 2 n P 2
⋅
FW S Fe
(primary)
lemS
⎡ 1 − M (1 − F P )
1 + M FP
⋅⎢
+
Fc S (1 − FP )
⎢ Fc P FP
⎣
where :
FW
π
M=
S Fe
lem Fh
(secondary)
B
CP
C
CS
A
(b)
SdisFe
Sdis
Cu
Sdis
Fe
[
= 2 (π C + A)( H + C ) + C A
{
]
}
= 2 ( A + B )[2 ( A + C ) + H ] + A B
(c)
Figure 3 : E-type core, (a) dimensions, (b) mean turn lenghts,
(c) dissipation surfaces.
Defining the average mean turn length as :
(4.16)
and from fig. 3.a , the following geometrical coefficients are
defined :
Fh = H C
(4.17.a)
and
Fs = B A
(4.17.b)
The primary partition factor is :
SW
C
FP = P = P
(4.18)
SW
C
where C may be expressed by ,
FW
C=
(4.19)
S Fe
Fh
lemS = lem + π
(a)
SdisCu
)
1
lemP + lemS
(4.14)
2
and using the eq. 4.13 , the definition 4.14 may be expressed
as :
Using eqs. 4.13.c , 4.16 , 4.18 and 4.19 , the eqs. 4.13.a
and 4.13.b may be expressed as :
F
lemP = lem − π W (1 − FP ) S Fe
(4.20.a)
Fh
B
A
(
leav =
where, according to eqs. 4.2.c and 4.2.d , it is
⎛ FrS
⎜
⎜F
⎝ rP
⎞⎤
⎟⎥
⎟⎥
⎠⎦
The expression 4.21.a has a minimum for :
1
FPopt =
Fc P (1 + M ) ⎛ FrS ⎞
⎟
⎜
1+
Fc S (1 − M ) ⎜⎝ FrP ⎟⎠
Fc S (1 + M ) ⎛ Fr
⎜ S
Fc P (1 − M ) ⎜⎝ FrP
(4.21.a)
(4.21.b)
(4.22)
therefore , from eq. 4.5 , the optimal current density ratio is :
σP
=
σS
⎞
⎟
⎟
⎠
(4.23)
For square section scrapless lamination cores, with
Fc S = Fc P and FrS = FrP the optimal values are :
FPopt = 0.43
and
.
.
σ P σ S = 134
Adopting these values PCu becomes 28 % lower with
respect to the case corresponding to stacked windings
Nevertheless,
adopting
with FP = 0.5 and σ P = σ S .
6
FP for minimal losses implies , as disadvantage, one
higher current density in the inner winding , which has the
worst heat transfer capability. By this reason , equal current
densities are often adopted , even if this selection leads away
of the optimum value from the winding losses point of view.
In a similar fashion as done with the shared coil-former
case an equivalent turn factor ( Fle ) may be defined
keeping the Joule losses still given by eq. 4.12 . For this
purpose it must be :
Fc P Fr FP (1 + M FP ) ⎤
1⎡
S
⎥ (4.24)
Fle = ⎢1 − M (1 − FP ) +
2⎢
1 − FP ) ⎥
Fc S FrP
(
⎦
⎣
(where M is defined by 4.21.b).
For the particular case with Fc S = Fc P and FrS = FrP ,
adopting FP = 0.5 yields :
Fle = 1
For square section scrapless
FP = FPopt it results : Fle = 0.84 .
(4.25)
laminations with
FrS = FrP ).
Popt
V. CURRENT DENSITY ADOPTION
A. Thermal considerations. Dissipation of losses
For temperature rises ranging about 50 ºC , assuming an
ambient temperature of 40 ºC , the power dissipated by
radiation may be estimated as :
Pr[ W]
= 7.08 Δθ [ º C]
(5.1)
S
[ ]
dis m2
(linear approximation obtained from the Stefan-Boltzmann
[15] equation assuming an emissivity coefficient of 0.8).
On the other hand , the power dissipated by natural
convection may be approximated through [15] :
Pc[ W]
(5.2)
= 2.17 Δθ [ º C] 1.25
S
[ ]
dis m2
[
]
(valid for bodies with dimensions smaller than 0.5 m).
Therefore , the total dissipated power is :
Pt [ W]
= 7.08 + 2.17 Δθ [ º C] 0.25 Δθ [ º C ]
S
[ ]
dis m2
and always assuming Δθ = 50 º C , the above equation leads
to the estimative expression :
Pt [ W]
Pt [ W]
≅ 780
Δθ [ º C] = 0.0778
(5.3)
S
S
[ ]
dis m2
example, for square section scrapless laminated cores , from
fig. 3.a and c , one obtains :
S disCu = ( 5 + 2π ) A 2 , S disFe = 14.5 A 2
and the total dissipation surface becomes :
. A2
S distot = S disCu + S disFe = (19.5 + 2π ) A 2 = 258
Defining the thermal resistance as :
Δθ
Rθ =
Pt
from eqs. 5.3 and 5.4 it results :
30
Rθ
≅
⎡ º C/W ⎤
A[cm ]
⎢⎣
⎥⎦
(
tot
For E ferrite cores with the following typical dimensions :
A=C
; B = 15
. A ;
H = 15
. A
one obtains ,
FPopt = 0.4 and Fle ] F
= 0.77
(always with Fc S = Fc P and
composed by one part corresponding to the coil windings
S disCu and other one S disFe concerning to the core. For
[ ]
dis cm2
(linear equation valid only in the near range of
Δθ = 50 º C , rising over 40 ºC ambient temperature).
In case of E type cores the surface of heat dissipation is
(5.4)
(5.5)
)2
(5.6)
This expression may be used to estimate the transformer
temperature rise when the copper and iron losses cause
similar rises. Notice that S disCu and S disFe are similar areas.
Therefore, when the core and winding losses are similar, the
temperature rise will be comparable , and the approximation
made will be acceptable.
In other circumstances , it will be suitable to verify the
coil and core temperature rise separately, assuming isolated
dissipations paths and using each kind of loss with its own
dissipation area. It should be ensured that the highest
temperature rise computed be lower than the maximum
specified.
The thermal conduction resistance between coil and core
is usually high because the coil former is plastic made and an
air filled gap lies between the coil former and the core central
leg.
As usually PCu and PFe are similar, so they are S disCu
and S disFe , the core and coil temperature rise become
similar. Therefore, stated the thermal resistance is high, the
heat conduction exchange between coil and core must be
neglectible in a first trial approximation.
Example :
From the transformer thermal model [16] depicted in fig.
4 , one concludes :
PCu Rθ Cu − PFe Rθ Fe
(5.7)
PCu Fe =
Rθ Cu / Fe + Rθ Cu + Rθ Fe
Utilizing a core E42-15 , with 230 turns of 0.60 mm wire
the D.C. resistance results : R DC = 13
. Ω .
Applying a direct current of 1.13 A the temperature rise
measured was Δθ = 26 º C when vertical mounted, and
33 ºC if horizontal mounted. Therefore, the average
value Δθ = 29.5 º C
will be adopted for calculation
purposes.
From equations deduced for S disCu and S disFe :
Rθ Cu = 24.2 º C W
and
Rθ Fe = 17.6 º C W .
From the model shown in fig. 4 (with PFe = 0 ) the total
equivalent thermal resistance becomes :
7
Rθ eq
tot
=
1
1
Rθ Cu
1
+
Rθ Cu / Fe + Rθ Fe
from which :
Rθ Cu / Fe =
1
I DC 2 R DC
1
−
Δθ
Rθ Cu
=
Δθ
(5.8)
I DC 2 R DC
− Rθ Fe
(5.9)
and using the above estimated values for Rθ Cu and Rθ Fe ,
this gives Rθ Cu / Fe = 49.7 º C W .
Assuming a typical case where PCu = PFe =
yields :
Ptot
, eq. 5.7
2
Rθ Cu − Rθ Fe
1⎛
⎜
⎜
2 ⎝ Rθ Cu / Fe + Rθ Cu + Rθ Fe
⎞
⎟
(5.10)
⎟
Ptot
⎠
thus, for the above experimental values, eq. 5.10 gives :
PCu Fe = 0.036 Ptot which allows, in first instance, to
PCu Fe
=
neglect the thermal exchange between the core and the coil.
PCu / Fe
Rθ
PCu
Rθ
Cu / Fe
Rθ
Cu
PFe
Fe
Figure 4 : Thermal model of an open-core transformer
(C , E , EC or RM type cores).
B. Loss balance between core and coil [3] [7] [8] [14]
The iron losses may be estimated by [6][12][18] :
(5.11)
PFe = k Fe f ξ Bm β VFe
where for most ferrites : ξ = 13
. and 2 ≤ β ≤ 2.7 being
β ≅ 2 for high permeability materials aimed for switching
frequencies ranging from 20 to 40 kHz , and β ≅ 2.5 for low
permeability ferrites suitable for higher frequencies.
Depending on the switching frequency the material choice
should be done using the loss charts available from ferrite
manufacturers [6][9][10].
The copper losses may be expressed by eq. 4.12
(with Fle defined by eq. 4.11 or 4.24 , depending on the
winding structure).
Obtaining n P from eq. AII.2 (see Appendix II) and
substituting into eq. 4.12 , yields:
PCu
⎛ V Pef I Pef
= 2 ρ eq P ⎜⎜
⎝ k C f fV f
⎞
⎟
⎟
⎠
2
Fle l em
1
FP Fc P FW S Fe 3 Bm 2
=
K Cu
Bm 2
(5.12)
The total losses are the sum of the core losses plus the
windings ones , becoming a function of Bm which should be
adopted looking for minimal total losses [14].
According to eqs. 5.11 and 5.12 :
Ptot = PFe + PCu = K Fe Bm β +
KCu
Bm 2
expression that has a minimum for :
2 K Cu Bm 2 = β K Fe Bm β
which leads to the optimal condition :
PCu =
(5.13)
(5.14)
β
PFe
(5.15)
2
Notice that only if β = 2 then PCu = PFe which is the
optimal condition. However, even if β = 2.6 , the optimal
values of PCu and PFe are not quite different.
C. Current density adoption as function of temperature rise
1. Transformers: General expression
Assuming winding and core losses of the same order , and
provided similar dissipation surfaces , the final temperatures
will resemble.
Therefore, given a high winding-core thermal resistance ,
the heat exchange will be small enough to assume that the
windings will dissipate only through the air-exposed coil
surface , this should be the area used for calculations
( S disCu ). According to the nomenclature of fig. 3.a :
⎞
⎛π
(5.16)
S disCu = 2( π C + A) H + 4⎜ C 2 + A C⎟
⎠
⎝2
With definitions 4.17 and eq. 4.19 :
⎡
⎤
S disCu
2 + Fh
1
(5.17)
= 2 π FW ⎢1 +
+
⎥
S Fe
⎢⎣ Fh π FW Fs Fh ⎥⎦
The effective value of the primary current may be
expressed as function of the current density by eqs. 2.7 and
2.8 as :
S
I Pef = σ P S Cu P = σ P Fc P FP FW Fe
(5.18)
nP
that substituted into eq. 4.12 gives :
(5.19)
PCu = 2 ρ eq P σ P 2 FcP FP FW Fle lem S Fe
where Fle depends on the the winding structure (given by
eqs. 4.11 or 4.24) , but in most cases adopting Fle ≅ 1 may
be an acceptable first trial approximation.
From fig. 3.b , using eq. 4.16 yields lem , and utilizing the
definitions 4.17 and eq. 4.19 it results :
⎡ (1 + Fs )
FW ⎤
(5.20)
+π
lem = ⎢2
⎥ S Fe
Fh ⎥⎦
Fs
⎢⎣
Substituting eq. 5.20 into 5.19 :
PCu = 2 ρ eq P σ P 2 Fc P FP FW Fle ⋅
⎡ (1 + Fs )
FW
⋅ ⎢2
+π
Fh
Fs
⎢⎣
Substituting eqs. 5.21 and 5.17
Δθ = 0,078ρ eq P σ P 2 Fc P
⎤
⎥ S Fe 3/ 2
⎥⎦
into 5.3 yields :
⎡ (1 + Fs ) 1 FW
+
⎢
2 Fh
⎢ π Fs
FP Fle ⎢
2 + Fh
⎢1 + 1 +
Fh π FW Fs Fh
⎢
⎣
(5.21)
⎤
⎥
⎥
⎥ S Fe
⎥
⎥
⎦
8
(5.22)
from which one obtains :
Δθ ºC
[ ]
,
=
0
358
σ ⎡
⎤
A
P⎢
ρ eq
⎣ mm2 ⎥⎦
P [ μΩ cm]
Fgm
1
FP Fc P Fle 4 S ⎡ 2 ⎤
Fe cm
⎣⎢
⎦⎥
(5.23.a)
where Fgm is a core geometry dependent factor :
Fgm =
1+
1
+
Fh π
2 + Fh
(1 + Fs ) + 1
π Fs
FW Fs Fh
2
(5.23.b)
FW
Fh
For example, for square section scrapless lamination
cores, assuming a temperature rise of 50 ºC and
ρ eq P = 20 μΩ cm , the following estimative expression is
obtained :
σ
⎤
⎡
P ⎢A
⎣ mm2 ⎦⎥
≅
5
A[ cm]
classical empirical formula well known by craftsmen, where
A is the central leg width.
2. Ferrite made chokes
In this particular case, it is possible to find from
manufacturer tables AR such as :
RCu = AR n 2
(5.24)
Usually AR ]0.5 is specified as the
corresponding to Fc = 0.5 [6]. Therefore :
AR =
AR ] 0.5
2 Fc
The copper losses may be estimated by :
PCu = I ef 2 RCu = σ 2 S Cu 2 AR n 2
where :
AR
value
(5.25)
(5.26)
S Cu = Fc
S
SW
= Fc FW Fe
(5.27)
n
n
Substituting 5.25 and 5.27 into 5.26 yields :
F
PCu = σ 2 S Fe 2 FW 2 c AR ]0.5
(5.28)
2
Neglecting the iron losses (respect to the copper ones) it
may be assumed that :
(5.29)
PCu = Δθ Rθ tot
Equating expressions 5.28 and 5.29 yields :
2 Δθ ºC
10
[ ]
σ⎡ A ⎤=
F
S
⎤
⎥
⎢
W
Rθ ºC W Fc AR ⎥
Fe ⎡cm2 ⎤
⎢⎣ mm2 ⎥⎦
⎣⎢
⎦⎥
]
tot [
⎦ 0.5 [ μ Ω]
(5.30)
Notice that eq. 5.30 do not consider the resistance rise due
to both the skin and proximity effects. In most choke
application cases, this is not important because the D.C.
component is the main harmonic current component.
VI. VOLUMETRIC CORE COMPARISON BETWEEN
FLYBACK AND FORWARD CONVERTERS
A. Volumetric ratio
The volume of a magnetic core is related to the effective
length by ,
⎛ S Feef ⎞
⎟ S Fe l Fe = k E S Fe l Fe
(6.1)
VFe = S Feef l Fe = ⎜⎜
⎟
⎝ S Fe ⎠
and the magnetic length is related to the geometric minimal
core section by,
l Fe = k S S Fe
(6.2)
which yields :
VFe = k E k S S Fe 3 2 = k SE S Fe 3 2
where :
k SE = k E k S = S Feef l Fe
S Fe
(6.3)
3
For scrapless laminations k E = 1 and k S = 6 so
k SE = 6 , while for ferrite E cores k SE usually lies
between 5 and 7 [6][10] (as it may be calculated from core
manufacturer data).
The required core volumes will be compared assuming
equal shapes and proportions, so with the same k SE .
The total volume occupied by the cores of the magnetic
components of a forward converter is :
VFeFW = VFeTFW + VFe L
Then, the volumetric ratio for converter comparison may
be defined as :
VFeFW
VFeTFW + VFe L
(6.4)
ℜVFW / FB =
=
VFeFB
VFeFB
Substituting the aproximative form of the sizing equations
2.14 , 3.2 and 3.10 in the expression 6.3 and replacing the
results in 6.4 yields :
⎡ (1 − D )
σ IP ⎤
(6.5)
FP
ℜVFW / FB =
= δ i P + ⎢η
⎥
VFeFB
σ I L ⎥⎦
D
⎢⎣
The most commonly partition factor adopted is
FP = 0.5 . Assuming this partition value and σ P = σ L ,
the volumetric ratio is shown in figs. 5.a and 5.b with
parameters D and δ i P for η = 1 . Obviously, when
VFeFW
34
34
ℜVFW / FB < 1 the forward topology must be preferred.
B. Power switch sizing considerations
The profit factor (or power utilization ratio) of the power
device is defined as :
P
f pr = O
PD
where PO is the maximum available output power of the
converter and PD is the rated switching power for the
power device. The profit factor becomes better in continuous
operation mode and it may be easily demonstrated that for
both topologies it results :
⎛ δi ⎞
f pr = η D (1 − D) ⎜ 1 − P ⎟
(6.6)
⎝
2 ⎠
9
The profit factor becomes maximum when D = 0.5 . In
this case, if η = 1 the former expression yields :
f pr =
1 ⎛ δ iP ⎞
⎟
⎜1 −
4⎝
2 ⎠
(6.7)
Substituting 6.7 in 6.5 , for FP = 0.5 :
[(
ℜVFW / FB = 2 1 − 4 f pr
)]
34
+ 2−9 8
In accordance with 6.6 it must be :
1
1
≤ f pr ≤
8
4
so from 6.8 it results :
0.458 < ℜVFW / FB < 146
.
The boundary value for the profit factor is
f pr lim = 0195
.
≅ 0.2 .
(6.8)
(6.9)
When the selected operation duty cycle is near the optimal
value 0.5 , adopting σ L higher than σ P does not change
the boundary value obtained.
Assuming D as the maximum duty cycle for nominal
output power, the maximum voltage over the power switch
will be : VCEmax = V P (1 − D) expression valable for both
topologies here involved. Using this equations with 6.5 and
6.6 , ℜVFW / FB is plotted in fig. 5.c , as function of
f pr with the parameter VCEmax V P . There, the most
suitable operation area for each topology are marked.
ℜVFW / FB
2,5
(6.10)
δ iP = 1
1,5
1
δ i P = 05
.
δ i P = 01
.
0,5
0
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
D
1
(a)
η=1
ℜVFW / FB
σ P =σ L
Fp = 0.5
2,5
D = 01
.
2
D = 0.4
1,5
D = 0.8
1
D=1
0,5
aproximative value 0.2 is again valid.
0
0
VII. CONCLUSIONS
core volume than the flyback one and vice versa, so for
f pr < f pr lim a flyback implementation should be preferred.
σ P =σ L
Fp = 0.5
2
may be too
Actually, the adoption σ P = σ L
conservative, because the core for the inductor usually results
smaller than the one required for the transformer. Therefore,
the inductor can dissipate heat better than the transformer
due to its higher surface/volume ratio. Also, the inductor core
power losses are lower, since the hysteresis loop there is
smaller than the one performed in the transformer.
Consequently, higher copper losses (per volume) should be
admissible, allowing higher current density on inductor
windings (according to eq. 5.30).
The inductor current density is usually adopted between
. σ P . A higher current density reduces the
σ P ≤ σ L ≤ 15
inductor volume but degrades the efficiency and complicates
the close loop converter operation making the voltage
transfer ratio a function of the output load.
. σ P and recalculating the boundary value
Using σ L = 15
for the profit factor yields,
f pr lim = 0.209 , so the
The widely used stacked winding technique allows
partition factors others than 1/2 if required , but usually , the
inner winding density current has to be limited to values such
as the optimal efficiency cannot be achieved. However, these
kind of windings have smaller leakage inductance than
shared coil-former made, but they present a bigger
interwinding capacity and poorer isolation features.
Therefore, even if the secondary was allocated in a
separated coil-section, the demagnetizing coil should be
placed over the primary winding to ensure a good magnetic
coupling. In battery powered converters, bifilar winding will
improve the magnetic coupling. In off-line SMPS bifilar
windings are not reliable enough [7] and a good practice
should be to interpose the primary between both the halved
demagnetizing winding sections . This increases the parasite
interwinding capacity [12] but an appropriate connection of
the demagnetizing diode may overcome this drawback [7].
For f pr > f pr lim the forward topology requires small
η=1
0,1
0,2
ℜVFW / FB
2,5
2
0,3
0,4
0,5
0,6
1
0,9
1
δ iP
η=1
σ P =σ L
VCEmax V P = 11
.
Fp = 0.5
VCEmax V P = 167
.
VCEmax V P = 2
forward VCE
VP = 8
max
area
0,5
0,8
(b)
flyback area
1,5
0,7
0
0
0,05
0,1
0,15
0,2
0,25
fpr
(c)
Figure 5 : Volume ratio as function of (a) duty-cycle,
(b) primary current increment and (c) device profit factor.
APPENDIX I : Magnetic flux density adoption
A. Maximum efficiency selection
For the optimal condition stated by eq. 5.15 the total
losses become :
10
β⎞
⎛
Ptot = PCu + PFe = ⎜ 1 + ⎟ PFe = k Fe f
⎝
2⎠
β⎞
⎛
Bm β ⎜ 1 + ⎟ VFe
⎝
2⎠
(AI.1)
and the temperature rise may be estimated through :
(AI.2)
Δθ = Ptot Rθ tot
ξ
From eqs. AI.1 and AI.2 it results :
1
Δθ
⎤ β
⎡
V
Fe
⎥
⎢ R
θ tot
⎥
(AI.3)
Bm ] opt = ⎢
⎢
β⎞ ⎥
ξ ⎛
⎢ k Fe f ⎜⎝ 1 + ⎟⎠ ⎥
2 ⎦
⎣
For most ferrite cores , this value is smaller than the
maximum allowable (which depends on the switching
frequency adopted). Therefore, a higher flux density might
be adopted to increase the available output power at price of
an efficiency degradation.
B. Maximum output power selection
In this case, the current density will be adopted as function
of the temperature rise (see section V - C ).
For a given core, once the current density was
adopted, PCu becomes determinated from eq. 5.19 . For
these copper losses , it should be avoided that the core
temperature rise surpasses the maximum specified for the
coil , in order to prevent an additional heat transfer to the
windings. Consequently , it must be :
PCu
P
= Fe
(AI.4)
S disCu
S disFe
Being both temperature rises identical :
Δθ = ( PCu + PFe ) Rθ tot
From eqs. AI.4 and AI.5 it results :
Δθ ⎛ S disFe ⎞
⎟
⎜
PFe =
Rθ tot ⎜⎝ S distot ⎟⎠
Using eqs. 5.11 and AI.6 , one obtains :
1
⎡ Δθ ⎛ S dis ⎞ ⎤ β
Fe
⎟⎥
⎜
⎢
⎢ Rθ tot ⎜⎝ S distot ⎟⎠ ⎥
=⎢
Bm ] Po
⎥
ξ
max
⎢ k Fe f VFe ⎥
⎥
⎢
⎥⎦
⎢⎣
Relating the expressions AI.3 and AI.7 :
1
Bm ] Po
⎡⎛ S dis ⎞ ⎛
β⎞⎤ β
Fe
max
⎟ ⎜1 + ⎟ ⎥
= ⎢⎜⎜
2⎠⎥
Bm ] opt
⎢⎣⎝ S distot ⎟⎠ ⎝
⎦
(AI.5)
(AI.6)
(AI.7)
(AI.8)
For example, for a core E55-21 made with a material such
becomes approximately 14 %
as β = 2.5 , Bm ] Po
greater than Bm ] opt .
max
Using the Bm adoption criteria stated by eqs. AI.3 or
AI.7 requires a previously made core selection, but it is just
for selecting the core (as output power function) that Bm is
first needed. To overcome this obstacle , one estimation a
priori of Bm can be made based upon known reference
values , through the equation deduced keeping constant the
core density losses :
ξ
⎛ f ref ⎞ β
Bref
Bm ( f ) = ⎜
(AI.9)
⎟
⎝ f ⎠
For high permeability ferrites used from 10 to 50 kHz , the
references may be Bref = Bsat and f ref = f min , where
f min is the minimum recommendable operating frequency,
that is, the maximal switching frequency which allows using
Bm = Bsat . Under f min it should be preferable to adopt
other material , while for f > f min it is necessary to reduce
Bm according to eq. AI.9 .
Usually ξ β ≅ 0.5 so, the eq. AI.9 may be approximated
as :
Bm ( f ) =
f ref
Bref
(AI.10)
f
For low permeability ferrites suitable for high frequency
switching , the reference values may be determined as
function of temperature rise from manufacturer issued curves
(or tables).
For example, suitable reference values for N27 and N47
materials are :
N27 :
; Bref = 0.2 T
f ref = 20 kHz
N47 :
f ref = 100 kHz
;
Bref = 0.1 T .
APPENDIX II : Transformer core sizing equation
The Faraday law applied to a transformer magnetic circuit
yields :
ΔB
V P = n P S Fe
(AII.1)
Δt
For
symmetrical
converters
ΔB = 2 Bmax and
Δt = D T 2 , while for continuous operating mode forward
converters ΔB = Bmax − Br ≅ Bmax and Δt = D T , where
Br is the residual flux density (assumed that
Br << Bmax in soft magnetic materials).
On the other hand , in flyback converters :
ΔB = Bmax − Bmin = Bmax δ i P and Δt = DT .
In symmetrical converters operating in continuous mode :
V Pef = D V P and V Pav = D V P ,
so the voltage form factor results :
f fV = 1 D .
For asymmetrical converters operating in continuous
mode :
D
V Pef =
V P and V Pav = 2 D V P
1− D
therefore ,
1
f fV =
2 D (1 − D )
which allows expressing the eq. AII.1 as :
11
V Pef = k C f f V n P S Fe Bm f
(AII.2)
where :
kC = 4
for symmetrical converters
kC = 2
for forward converters
kC = 2 δ i P
for flyback converters
and the notation was simplified writing Bm = Bmax .
Using eq. 5.18 the primary current may be expressed as
density current function. Then , using eqs. AII.2 and 5.18 the
primary apparent power is obtained :
(AII.3)
V Pef I Pef = k C f f V Bm f σ P Fc P FP FW S Fe 2
from which the minimal required core section ( S Fe ) may
be found.
The output power may be expressed as function of the
apparent primary power, yielding :
(AII.4)
PO = η k uT V Pef I Pef
where k uT is the transformer utilization factor , defined as
the ratio between the active primary power (or D.C. primary
supply power) and the apparent primary power adopted for
transformer design. Suitables design values are :
k uT = 1 − D for asymmetrical converters and
k uT = 1
for symmetrical ones.
Substituting AII.4 into AII.3 and rearranging , the required
minimal core section is obtained as output power function :
PO
S Fe =
(AII.5)
η k uT k C f f V Bm f σ P Fc P FP FW
Examples:
For a ferrite core E42-15 made with material N27 ,
assuming Δθ = 30 º C yields
σ P = 31
. A / mm 2
(see
section 4). Adopting the typical values : D = 0.4 ,
Fb = 0.36 , FP = 0.5 , η = 0.9 , f = 20 kHz and
Bm = 0.2 T , for the forward topology , from eqs. AII.4
and AII.5 , it results : PO = 100 W , while the empirical
manufacturer graphic [6] yields PO = 110 W .
For an E55-21 core with the same material , adopting the
same topology and working conditions, for equal
temperature rise it should be σ P = 2.7 A / mm 2 , which
yields PO = 272 W , while from manufacturer data ,
PO = 275 W .
APPENDIX III : Skin and proximity effects
For a single foil conductor subjected only to the skin
effect, the increase of resistance is given by [20] :
R
ξ sinh ξ + sin ξ
k rS = ac =
(AIII.1)
Rdc 2 cosh ξ − cos ξ
where:
ξ=h Δ
Δ
: penetration depth , Δ = ρ π f μ o
h
: foil thickness
If the foil conductor is immersed into the magnetic field
due to other conductors (proximity field) the resistance will
rise even more [12][13].
Assuming the proximity field uniform over the conductor
cross section, orthogonality appears between skin and
proximity effects [20]. This decouples both effects and
simplifies calculations [21].
For the foil conductor of the m layer the increase of
resistance is given by :
Rac ξ ⎡ sinh ξ + sin ξ
2 sinh ξ − sin ξ ⎤
= ⎢
+ ( 2m − 1)
⎥ (AIII.2)
cosh ξ + cos ξ ⎦
Rdc 2 ⎣ cosh ξ − cos ξ
The first term in eq. AIII.2 is identical to eq. AIII.1 and
describes the skin effect.
For multilayer windings :
R
ξ ⎡ sinh ξ + sin ξ ⎛ 4 p 2 − 1⎞ sinh ξ − sin ξ ⎤
⎟
Fr = ac = ⎢
+⎜
⎥
3 ⎟⎠ cosh ξ + cos ξ ⎥⎦
Rdc 2 ⎢⎣ cosh ξ − cos ξ ⎜⎝
(AIII.3)
where p is the number of layers.
Therefore Fr may be expressed as :
Fr = k rS + k rX
(AIII.4)
where k rS is given by eq. AIII.1 and k rX results :
ξ ⎛ 4 p 2 − 1⎞ sinh ξ − sin ξ
⎟
⎜
(AIII.5)
2 ⎜⎝
3 ⎟⎠ cosh ξ + cos ξ
For ξ ≤ 2 the following approximations apply [12] :
k rS ≅ 1
(AIII.6.a)
and ,
5 p2 − 1 4
k rX ≅
ξ
(AIII.6.b)
k rX =
45
In order to extend this one-dimensional approach to round
wire windings, Dowell [13] introduces an equivalent square
conductor thickness from :
⎛d⎞
S Cu = h 2 = π ⎜ ⎟
⎝ 2⎠
2
therefore , h =
π
(AIII.7)
d
2
Each layer is supposed formed by nl turns of square
section equivalent conductors (fig. A3).
Since the square section conductors are separated by a gap
s , a one-dimensional layer copper factor has to be defined
as:
n
(AIII.8)
Fl = l h
bW
where,
bW : overall winding breadth
h : equivalent conductor thickness
nl : number of turns per layer
In order to adjust this model to the one-dimensional
approach of the single-foil winding , the penetration depth
must be modified due to the porosity of the winding layer
which increases the effective resistivity.
Therefore, defining :
(AIII.9)
ρ ef = ρ Fl
it yields,
12
Δ ef = ρ ef π f μ o =
Δ
(AIII.10)
Fl
which leads to the effective value of ξ to be applied in
Fr estimations in case of round wire windings :
d
π
(AIII.11)
=
Fl
Δ
2
Δ
If the increase in resistance due to eddy currents is
excessive, one alternative is to use bunched conductors or litz
wires. Then :
ξ=
Fl h
h=
π
Dst
2
(where Dst is the strand diameter)
and Fl ≅ FW .
However, when the number of turns is small, the adoption of
foil windings usually gives lower ac resistances. The use of
litz wire may be considered for multilayer winding
applications ranging over 500 kHz .
For inductors carrying DC (choke applications)
adopting d Δ ≤ 2 is suitable enough in discontinuous mode
operating converters, while d Δ ≤ 4 is acceptable in
continuous mode operation.
d
1
2
3
4
nl
layer m
h
bW
Figure A3 : A layer of square section conductors equivalent to
one of round section conductors.
ACKNOWLEDGEMENT
The author would like to thank Prof. Carlos Christiansen
for his interest and support that constantly encouraged this
work.
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de l’électronique de puissance. Tome 3: La conversion
continu-continu, Paris Fr. : Techn. et Doc. Lavoisier , 1987,
Chap. 6 .
[2]
[3]
[4]
[5]
[7]
[8]
[9]
Siemens, Ferrites and Accesories. Data Book 90/91 , Berlin,
Germany : Siemens Matsushita Components, 1990.
K. H. Billings, Switchmode Power Supply Handbook , N. Y. :
McGraw-Hill, 1989.
A. I. Pressman, Switching Power Supply Design , N. Y.:
McGraw-Hill, 1991.
C. W. T. McLyman, Transformer and Inductor Design
Handbook , N. Y. U.S.A. : M. Dekker, 1988.
[10] C. W.
T. McLyman, Magnetic Core Selection for
Transformer and Inductors , N. Y. U.S.A. : M. Dekker, 1997.
[11] H. E. Tacca, Amélioration des caractéristiques des
convertisseurs quasi-résonnants à transfert direct d’énergie ,
Thèse doctorale, Univ. des Sciences et Technologies de Lille,
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Hernán Emilio Tacca was born in Argentina on December
15, 1954. He received the B.E. degree in electrical
engineering from the University of Buenos Aires
(Argentina), in 1981 and the M.S. and Ph.D. degrees from
the University of Sciences and Technologies of Lille
(France) in 1988 and 1993 , respectively. In 1998 he
received the Doctorate degree from the University of Buenos
Aires. Since 1984 , he has been with the Faculty of
Engineering of the University of Buenos Aires, where he is
currently Assistant Professor and engaged in teaching and
research in the areas of industrial electronics. His research
interest are in the field of SMPS, UPS, battery chargers, softswitching techniques and microcontroller control of power
converters.