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1
Prof. S. Ben-Yaakov , DC-DC Converters [3- 1]
Magnetics Design
3.1 Important magnetic equations3.2 Magnetic losses3.3 Transformer
3.3.1 Ideal transformer (voltages and currents)3.3.2 Equivalent circuit of transformer
(coupling, magnetization current)3.3.3 Design of transformer
3.4 Inductor design
Prof. S. Ben-Yaakov , DC-DC Converters [3- 2]
;dtdBnA
dtdnV e=Φ=
HB
∆∆=µ
Φ - magnetic flux Weber [ Wb ]
B - flux density ]T[TeslamWb
2 =
V - voltage [ V ]
Also : Gauss [ G ] 1T = 10,000 G
B
H
BDC
HDC
ΦAe
Faraday’s law
2
Prof. S. Ben-Yaakov , DC-DC Converters [3- 3]
Ampere’s law
H - magnetic field [ A/m ]
∫ ⋅= InHdl
eHIn l⋅=⋅
e
InHl
⋅= [ A/m ]
In
el
Prof. S. Ben-Yaakov , DC-DC Converters [3- 4]
Magnetic losses ~∆B
“Good number” = 100mW/cm3 =100KW/m3
P3cmmW
B∆
Magnetic losses
B
H
BDC
HDC
3
Prof. S. Ben-Yaakov , DC-DC Converters [3- 5]
“Good number”=100mW/cm3=100 kW/m3
Magnetic Losses
Prof. S. Ben-Yaakov , DC-DC Converters [3- 6]
Magnetic losses ~∆B
“Good number” = 100mW/cm3 =100KW/m3
P3cmmW
B∆
Magnetic losses
B
H
BDC
HDC
4
Prof. S. Ben-Yaakov , DC-DC Converters [3- 7]
Magnetic Losses
Prof. S. Ben-Yaakov , DC-DC Converters [3- 8]
Curves for constant loss: 500mW/cm3
Figure of merit B*fEach material has optimum operating temperature (minimum loss)
Magnetic Losses
5
Prof. S. Ben-Yaakov , DC-DC Converters [3- 9]
Transformer currents
n1 n2
I1I2
For ideal transformer 2211 InIn =1
2
2
1nn
II=
At any given moment 2211 InIn =
I1,I2 opposite direction.
No magnetic energy stored due to useful currents I1, I2 (they cancel each other)
n1 n2
I1 I2
Prof. S. Ben-Yaakov , DC-DC Converters [3- 10]
Transformer voltages
φ
dtdnV 1
11φ=
dtdnV 2
22φ=
21gminAssu φ=φ
dtd
dtd 21 φ=φ
2
1
2
1nn
VV
=
n1 n2V1 V2
6
Prof. S. Ben-Yaakov , DC-DC Converters [3- 11]
Since each winding also represents an inductance, therefore for any winding 0=nVPermissible voltages: AC only on any winding
S1
V
S1
S2 S2
S1
V
S1
V
S2S2
S1S2
S1S2
t
t
t
A
B
C
Voltages
Prof. S. Ben-Yaakov , DC-DC Converters [3- 12]
Equivalent circuit (preliminary)
2mm nLL
12=
∞→LIdeal transformer
Ideal
1:n
Lm1
Llkg1
Ideal
1:n
Lm2
Llkg1
Ideal
1:n
Lm1
Llkg2
7
Prof. S. Ben-Yaakov , DC-DC Converters [3- 13]
Leakage inductance
I2V2
I1V1 n2
n1
Leakage inductance is theuncoupled magnetic flux
1:nLkg1
Lm1
ideal
Lkg2
Relationship between Llkg, M and k (coupling coefficient).
21 LLkM ⋅=)k1(LL 1m1lkg −≅
21lkg2lkg nLL ⋅≅
Leakage
Prof. S. Ben-Yaakov , DC-DC Converters [3- 14]
1:nLlkg1
Lm1
ideal
Llkg2
Vo
Llkg1
Lm1
L'lkg2
V'o 22lkg
2lkg nL
L =′
2o
o nVV =′
Leakage
8
Prof. S. Ben-Yaakov , DC-DC Converters [3- 15]
Magnetization Current
Vint
Vo
t
I2t
Imt
I1t
Ideal
1:n
Lm1
Llkg2
RI2
Vo
I1
ImVin
Prof. S. Ben-Yaakov , DC-DC Converters [3- 16]
Transformer
1. Bmax ( could be symmetrical or asymmetrical )2. Bmax < Bsat
3. In most case ( high frequency ) Bmax limit by magnetic losses.
dtdBAn
dtdnV e111 =Φ
=
I2V2
I1V1 n2
n1
H
B
B sat-
Bsat+
B max+
∆ΒBmax-
9
Prof. S. Ben-Yaakov , DC-DC Converters [3- 17]
Symmetrical operation
∫= VdtAn1B
e1
−+ = maxmax BB
e1
onmmax An
tVB2B ==∆
{ }emax
onm1 AB2
t,Vn =
B
Vm
Ts
Bmax-
Bmax+
ton
V1
s2Ttone1 f
1~t~An Son→
Prof. S. Ben-Yaakov , DC-DC Converters [3- 18]
Skin effect
depthskin−δ
f72)mm( =δ
Hzinf
δ 1RR
DC
AC >
DC FrequencyHigh
10
Prof. S. Ben-Yaakov , DC-DC Converters [3- 19]
Skin Effect Solutions
Litz wire
Tape
Prof. S. Ben-Yaakov , DC-DC Converters [3- 20]
Proximity effect
I
I
Current crowding due to magnetic fields
11
Prof. S. Ben-Yaakov , DC-DC Converters [3- 21]
[ ]k
nwnwA 22A1A
w1
⋅+⋅=
k - filling factor k<1
JI
w rms1A1=
J - current density A/m2
J ≅ 4.5 A/mm2
12
12 I
nnI =
Aw - winding area
AwwA
Aw
Prof. S. Ben-Yaakov , DC-DC Converters [3- 22]
2JkIn
A rms11w ⋅=
rms1
w1 I2
JkAn⋅
=
{ }emax
on11 AB2
t,Vn =
{ }emax
on1
rms1
w
AB2t,V
2IJkA
=⋅
{ }{ }JkB2
I2t,VAAAmax
rms1on1ewp
⋅==
{ }JkB
I2t,VA rms1on1
p ⋅∆⋅
=
{ }JkBfI2D,V
As
1on1p
rms
⋅∆⋅⋅
=
Ap
12
Prof. S. Ben-Yaakov , DC-DC Converters [3- 23]
Transformer design stages
1. Calculate Ap
2. Look for core3. Calculate n1 by:4. Calculate n2
{ }JkBfI2D,V
As
1on1p
rms
⋅∆⋅⋅
=
In symmetrical operation∆B = Bmax
+ - Bmax-
In asymmetrical operation∆B = Bmax - 0
{ }emax
onm1 AB2
t,Vn =
Prof. S. Ben-Yaakov , DC-DC Converters [3- 24]
Inductor design
Need to store energy( in transformer n1·I1= n2·I2 )
roµµ=µµo - air (vacuum) permeabilityµr - relative permeability
I
L
B
Hµ
13
Prof. S. Ben-Yaakov , DC-DC Converters [3- 25]
µo = 1.26·10-6m
Henry
If µ is high B will reach quickly Bsat
Need to slower µ
B
HHo
<2rµ
2rµ
1rµ
1rµBo
µr of ferrites ∼ 2000 - 4000
B = µH
Permeability
Prof. S. Ben-Yaakov , DC-DC Converters [3- 26]
Same Φ magnetic lines in ferromagnetic material and in air.
eg ll <<
eeg lll ≅+
Discreteair gap
Φµo
µr
el
gl
Distributedair gap
Gaps
14
Prof. S. Ben-Yaakov , DC-DC Converters [3- 27]
Current crowding due to magnetic fields
RAC high around gap
Current Crowding
Prof. S. Ben-Yaakov , DC-DC Converters [3- 28]
Φ = constant B ≅ constant
og
BHµ
=m
mBHµ
=
ggeme HHHnI lll +==ogB
meB
eHµ
+µ
=ll
l
eg ll <<
eeg lll ≅+
el
gl
Inductance with Gap
15
Prof. S. Ben-Yaakov , DC-DC Converters [3- 29]
g
eo
a
m
m
e
BBBH
l
lµ
+µ
=µ
=
o
g
m
ee
BBHµ
+µ
=ll
l
Dividing out le and defining HB
e =µ
Inductance with Gap
Prof. S. Ben-Yaakov , DC-DC Converters [3- 30]
µ
+µ
=µ
g
eo
me
111
l
l
µ
+µµ
=µµ
g
eo
ormore
111
l
l
+
µ=
µ
g
ermre
111
l
l
µ
µ+
=µ
g
erm
rmg
e
re
1
l
l
l
l
rmg
e
g
erm
re
µ+
µ
=µ
l
l
l
l
rmg
eIf µ<l
l
≈µ
g
ere
l
l
Gap Calculation
16
Prof. S. Ben-Yaakov , DC-DC Converters [3- 31]
dtdILV =
dtdnV Φ=
dtdInnA
dtdHnA
dtdBnA
dtdn
eeeel
µ=µ==Φ
dtdn
dtdIL Φ
=L-?
dtdIAn
dtdIL
e
e2
l
µ=
e
e2AnLl
µ=
Inductance
Prof. S. Ben-Yaakov , DC-DC Converters [3- 32]
2
2
1
2
1nn
LL
=
Inductor design
B
H
Bmax
Two windings on same core
17
Prof. S. Ben-Yaakov , DC-DC Converters [3- 33]
dtdn
dtdIL Φ
=
∫∫
=
maxpk B
0e
I
0dt
dtdBnAdt
dtdIL
L Ipk = nAeBmax
maxe
pk
BALI
n =
max
pke nB
LIA =
rms
wI
JkAn =
quick design and check
Saturation Limits
Prof. S. Ben-Yaakov , DC-DC Converters [3- 34]
JkBILI
AAAmax
rmspkwep ==
2rmspk LIILI ≈
2LIstoredEnergy
2=
Air gapped core Design1. Calculate Ap
2. Choose a core3. Iterate4. Calculate ( or increase gap until L is as required ) gl
Ap
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Prof. S. Ben-Yaakov , DC-DC Converters [3- 35]
Cores
Transformer core Inductor core
air gap
Prof. S. Ben-Yaakov , DC-DC Converters [3- 36]
1. E - core
2. TOROID
3. ARENCO 4. POT
Cores
19
Prof. S. Ben-Yaakov , DC-DC Converters [3- 37]
Commercial cores
Prof. S. Ben-Yaakov , DC-DC Converters [3- 38]
Distributed gap core
turnH
A yL = )
turns1000H
sometime( y
L for n turns: L2 AnL ⋅=
The concept of AL
Distributedair gap
20
Prof. S. Ben-Yaakov , DC-DC Converters [3- 39]
AL
Prof. S. Ben-Yaakov , DC-DC Converters [3- 40]
Toroid Data
21
Prof. S. Ben-Yaakov , DC-DC Converters [3- 41]
1 Amp/m =79.5 Oe
L decreases with DC current !
Permeability change
Prof. S. Ben-Yaakov , DC-DC Converters [3- 42]
These curves are measured by feeding ac signals.If the current is composed of DC + ripple, core loss is due only to ripple component !
DC bias tend to increase loss
“Good number”=100mW/cm3
Misleading notations !∆B NOT B
Losses
22
Prof. S. Ben-Yaakov , DC-DC Converters [3- 43]
“Hot Spot” - Critical parameter
Temp. Rize
Prof. S. Ben-Yaakov , DC-DC Converters [3- 44]
Hanna Curve
orHmaxB
eV
2LImaxHB
maxB1
eV
2LIH
µµ=µ=
=
=
maxBeAelLInI
Hn
maxBeApkLIH
Hn
maxBeApkLI
n
=
=
=
23
Prof. S. Ben-Yaakov , DC-DC Converters [3- 45]
Hanna Curve
Prof. S. Ben-Yaakov , DC-DC Converters [3- 46]
Core Size Selection
24
Prof. S. Ben-Yaakov , DC-DC Converters [3- 47]
Basic Design of Distributed Gap Core
)nI(fel
=µ
1. Calculate LI2
2. Look up manufacturer data 3. Select Core
4. Calculate
5. Check Lmin
6. Calculate losses. Temp rise and and
7. Iterate
)(LALn1000
1000=