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Compute leakage inductance

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Hindawi Publishing CorporationAdvances in Power ElectronicsVolume 2012, Article ID 635715, 6 pagesdoi:10.1155/2012/635715
Research Article
ImprovedExpressionforEstimationofLeakageInductancein
E
CoreTransformerUsingEnergyMethod
SivanandaReddyThondapu,
1
MangeshB.Borage,
2
YashwantD.Wanmode,
1
andPurushottamShrivastava
1
1
Pulse High Power Microwave Section, Raja Ramanna Centre for Advanced Technology, Indore 452013, India
2
Power Supplies and Industrial Accelerator Division, Raja Ramanna Centre for Advanced Technology, Indore 452013, India
Correspondence should be addressed to Sivananda Reddy Thondapu, sivananda@rrcat.gov.inReceived 31 December 2011; Revised 13 April 2012; Accepted 29 April 2012Academic Editor: Pavol BauerCopyright © 2012 Sivananda Reddy Thondapu et al. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the srcinal work isproperly cited.This paper proposes a simpler and more accurate expression for estimation of leakage inductance in
E
core transformer, which isthe most widely used transformer structure. The derived expression for leakage inductance accounts for the ﬂux extending intoair. The ﬁnite element method (FEM) analysis is made on the secondary shorted transformer to observe the
H
-ﬁeld pattern. TheresultsobtainedfromFEManalysisareusedforapproximatingtheﬁeldthatisextendingintoairtoderiveanexpressionforleakageinductance. This expression is experimentally validated on prototype transformers of di
ﬀ
erent core dimensions.
1.Introduction
Transformer is one of the basic building blocks of many power converters. The following are some of the cases whereaccurate estimation of leakage inductance is required.(i) Di
ﬀ
erent resonant converter topologies, discussed in[1–5], use parasitics of transformer as a part of reso-
nant tank network. For designing power converterwith such topologies, one requires accurate estima-tion of leakage inductance.(ii) In hard switched converters, in every cycle the energy stored in the parasitics appears as loss in converter. Inestimation of e
ﬃ
ciency of such converters, one needsto estimate leakage inductance before hand.(iii) For designing snubber circuits to limit device voltageduring turn-o
ﬀ
transients [6–8], one needs to esti-
mate leakage inductance. These turn-o
ﬀ
transientsmainly occur due to energy stored in the leakageinductance of the transformer.Methods that are usually employed for estimation of leakage inductance are (i) energy method [8–13] and (ii)
method of mutual ﬂuxes.In energy method, the energy stored in magnetic ﬁeld of the secondary shorted transformer is calculated and equatedto (1
/
2)
L
leak
I
2
p
where
L
leak
is the leakage inductance of thetransformer when referred to primary, and
I
p
is currentﬂowing through primary.The
H
-proﬁle inside the coil is calculated using Ampere’slaw. The energy stored in magnetic ﬁeld is calculated by evaluating the volume integral in (1):
E
stored
=
µ
2
H
2
dv
=
12
L
leak
I
2
p
.
(1)The expression derived for leakage inductance usingenergy method is independent of frequency. Hence, it doesnot consider any frequency-dependent e
ﬀ
ects on leakageinductance. The energy method is used for comparing leak-age inductance, in di
ﬀ
erent winding conﬁgurations.On the other hand, method of mutual ﬂuxes uses Max-well’s equations to predict the leakage inductance moreaccurately at high frequencies. As this method accounts forfrequency-dependent e
ﬀ
ects like eddy current losses andaltered ﬂux pattern due to eddy currents, it gives moreaccurate results, particularly at high frequencies. In [14], afrequency-dependent formula is presented to ﬁnd leakageinductance in a toroidal core transformers.
2 Advances in Power Electronics
x y z
(a) Perspective
x y z
(b) Front view
z x y
(c) Top view
z x y
(d) Side view
Figure
1: Simulation result of FEM analysis.
In this paper, an expression for leakage inductance isderived using energy method. Expression derived accountsfortheﬂuxthatextendsintoairinasecondaryshortedtrans-former, which is not considered earlier [8, 10, 11]. Therefore,
the estimated leakage inductance using this expression hasbetter accuracy.
2.EstimationofLeakageInductanceon
E
CoreTransformer
FEM analysis on secondary shorted transformer is madeusing CST Ver 2008.5 (with magneto-static solver). Thesimulation result of FEM analysis is shown in Figure 1. Thesimulation result is the
H
-ﬁeld pattern of secondary shorted
E
core transformer. This gives an idea of proﬁle of
H
-ﬁeldfrom the surface of the core to the outer surface of the wholewinding. The proﬁle of
H
-ﬁeld of a
E
core transformer anditswindingconﬁgurationisshowninFigures2and3,respec-
tively.Energy stored in the magnetic ﬁeld in the secondary shorted transformer is equated to energy stored in leakageinductor, which is given in (1).The volume represented with purple color in Figure 4 isconsidered for volume integration. This constitutes (i) thevolume occupied by the coils and (ii) the volume formedby extrusion of the coils along the centre limb of the core,excluding the volume of the core. The ﬂux extends intoair shown in Figure 5. The volume shown in Figure 4 is
divided into four parts, whose top view is shown in Figure 3.The proﬁle of
H
-ﬁeld is calculated using Ampere’s law, thesame proﬁle is observed in FEM analysis. The
H
-proﬁle isexpressed mathematically in (2):
H
(
z
)
=
H
m
z h
1
0
< z < h
1
,
H
m
h
1
< z < h
1
+
t
,
H
m
h
1
+
h
2
+
t
−
z h
2
h
1
+
t < z < h
1
+
h
2
+
t.
(2)Using
H
-proﬁle given in (2) to evaluate volume integralgiven in (1), we get expression of the leakage inductance:
L
leak
=
13
µ
o
N
21
F
2
(
h
+ 2
t
)[
FC
+
B
(
E
+ 2
h
)], (3)where
h
=
h
1
+
h
2
+
t
is total thickness of the windingmeasured from the surface of the core to the outer surface of the outer winding,
t
is total thickness of the insulation usedbetween the layers.
B
,
C
,
E
,
F
is are the dimensions of the
E
core shown in Figure 6.
Advances in Power Electronics 3
t h
1
h
2
H
(
z
)
z H
m
Figure
2: Proﬁle of
H
(
z
) from the surface of the core.
Part 3Part 1Part 2
Ex z C
Part 4Center limbof the coreSecondary winding (S)Primary winding (P)Interwindinginsulation
Figure
3: Top view of
E
core transformer with windings.
y x z
Figure
4: Volume around the core considered for integration.
( A / m )
15.413.511.59.617.695.773.841.920
z x y
Figure
5: Scaled side view to show the ﬂux that extends into air.
4 Advances in Power Electronics
Table
1: EE core transformer with interwinding insulation.Sample no. 1 2Core type EE42/21/15 EE65/38/13
F
14.45mm 22.65mm
C
15.20mm 13.45mm
B
21.10mm 32.59mm
E
12.05mm 19.77mm
h
1
thickness of primary 3.20mm 3.81mm
h
2
thickness of secondary 1.90mm 1.55mm
t
thickness of insulation 1.27mm 2.00mm
N
1
34 48
N
2
17 24Photograph
h
1
h
2
PS
t
1
Windingconﬁguration
Leakage inductance
L
leak,earlier
from (5) 11.91
µ
H 22.52
µ
HLeakage inductance
L
leak
from (3) 15.32
µ
H 28.21
µ
HLeakage inductance
L
exp
at 10kHz measured 14.13
µ
H 26.76
µ
H% deviation of
L
exp
from calculated
L
leak,earlier
15.71% 15.84%% deviation of
L
exp
from calculated
L
leak
8.38% 5.12%
EF BD AC
Figure
6:
E
core dimensions given in data sheets.
It is seen in [8, 11–13], by sandwiching the windings,
leakage inductance reduces by
p
2
times, where
p
is the num-ber of interfaces between primary and secondary [8]:
L
leak
=
13
p
2
µ
o
N
21
F
2
(
h
+ 2
t
)[
FC
+
B
(
E
+ 2
h
)]
.
(4)
3.ExperimentalResults
Three di
ﬀ
erent samples are made to validate the derivedexpression. The results are compared with the expressionsavailable in literature. The expression given in [8, 10, 11] is
replaced with the dimensions of the core (shown in Figure 6)and the dimensions of windings will give (5):
L
leak,earlier
=
13
p
2
µ
o
N
21
F
2
(
h
+ 2
t
)
F
(
C
+
E
+ 2
h
)
.
(5)Sample no. 1 is made with
E
core (EE42/21/15) woundwith stranded conductor made with three conductors of 24SWG. With interwinding insulation of thickness 1.27mm isused. Standard winding conﬁguration as shown in Table 1 isused to wind primary and secondary.Sample no. 2 is made with
E
core (EE65/38/13) woundwith stranded conductor made with three conductors of 24SWG. With interwinding insulation of thickness 2.00mm.Standard winding conﬁguration as shown in Table 1 is usedto wind primary and secondary.Sample no. 3 is made with
E
core (EE42/21/15) woundwith stranded conductor made with two conductors of 24 SWG. With total interwinding insulation of thickness0.46mm + 0.26mm
=
0.72mm is used. Sandwiched winding

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