Prof. S. Ben-Yaakov , DC-DC Converters
[3- 1]
Magnetics Design
3.1 Important magnetic equations
3.2 Magnetic losses
3.3 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]
Faraday’s law
BDC
B
H
Ae
Φ
HDC
∆B
dΦ
dB
= nA e
; µ=
∆H
dt
dt
Φ - magnetic flux Weber [ Wb ]
V - voltage [ V ]
Wb
B - flux density 2 = Tesla [ T ]
m
Also : Gauss [ G ]
1T = 10,000 G
V =n
1
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 3]
Ampere’s law
le
I
n
H - magnetic field [ A/m ]
∫ Hdl = n ⋅ I
n ⋅I = H ⋅ le
H=
n ⋅I
[ A/m ]
le
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 4]
Magnetic losses
BDC
mW
P
cm3
B
H
HDC
∆B
Magnetic losses ~∆B
“Good number” = 100mW/cm3 =100KW/m3
2
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 5]
Magnetic Losses
z
“Good number”=100mW/cm3=100 kW/m3
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 6]
Magnetic losses
BDC
mW
P
cm3
B
H
HDC
∆B
Magnetic losses ~∆B
“Good number” = 100mW/cm3 =100KW/m3
3
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 7]
Magnetic Losses
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 8]
Magnetic Losses
z
z
z
4
Curves for constant loss: 500mW/cm3
Figure of merit B*f
Each material has optimum operating temperature
(minimum loss)
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 9]
Transformer currents
I2
I1
I1
n2
n1
I2
n1
For ideal transformer
n2
I1 n2
=
I2 n1
n1I1 = n2I2
At any given moment n1I1 = n2I2
I1,I2 opposite direction.
No magnetic energy stored due to useful
currents I1, I2 (they cancel each other)
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 10]
Transformer voltages
V1
φ
V1 = n1
dφ1
dt
n1
n2
V2 = n2
dφ 2
dt
Assu min g φ1 = φ2
dφ1 dφ2
=
dt
dt
V1 n1
=
V2 n2
5
V2
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 11]
Voltages
Since each winding also represents an inductance,
therefore for any winding
Vn = 0
Permissible voltages: AC only on any winding
V
A
S1
S1
t
S2
S2
V
S1
B
S1
S2
t
S2
V
C
S1
S2
S1
t
S2
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 12]
Equivalent circuit (preliminary)
Ideal
Ideal
Llkg1
Llkg1
Lm2
Lm1
1:n
Ideal
Llkg2
Lm1
Lm2 = Lm1n2
Ideal transformer L → ∞
1:n
6
1:n
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 13]
Leakage
z
Leakage inductance
I1
V1
z
Leakage inductance is the
uncoupled magnetic flux
Lkg1
I2
n1
n2
V2
1:n
Lkg2
Lm1
ideal
z
Relationship between Llkg, M and k (coupling coefficient).
M = k L1 ⋅L 2
L lkg1 ≅ L m1(1 − k )
L lkg2 ≅ L lkg1 ⋅ n2
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 14]
Leakage
Llkg1
1:n
Llkg2
Vo
Lm1
ideal
Llkg1
Lm1
7
Vo′ =
L'lkg2
V'o
Vo
n2
L′lkg2 =
L lkg2
n2
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 15]
Magnetization Current
I1
Ideal
Im
Vin
Vo
Lm1
Vin
Llkg2
I2
R
t
Vo
t
I2
1:n
t
Im
t
I1
t
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 16]
Transformer
I1
V1
n1
n2
B sat+
B
I2
B max+
V2
∆Β
B sat-
H
Bmax-
1. Bmax ( could be symmetrical or asymmetrical )
2. Bmax < Bsat
3. In most case ( high frequency ) Bmax limit by magnetic
losses.
V1 = n1
8
dΦ
dB
= n1A e
dt
dt
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 17]
Symmetrical operation
V1
B=
Vm
1
∫ Vdt
n1A e
Bmax + = Bmax −
t on
Ts
B
∆B = 2Bmax =
B max+
n1 =
Bmax-
n1A e ~ t on t
on →
TS
2
~
Vm t on
n1A e
{Vm ,ton }
2Bmax A e
1
fs
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 18]
Skin effect
High Frequency
DC
δ
R AC
>1
RDC
δ − skin depth
72
δ (mm ) =
f
f in Hz
9
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 19]
Skin Effect Solutions
Litz wire
Tape
Prof. S. Ben-Yaakov , DC-DC Converters
Proximity effect
I
I
Current crowding due to magnetic fields
10
[3- 20]
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 21]
Aw
Aw =
[w
A1
⋅ n1 + w A 2 ⋅ n2 ]
k
k - filling factor k<1
w A1 =
Aw
wA
I1 rms
J
J - current density A/m2
J ≅ 4.5 A/mm2
Aw - winding area
I2 =
Prof. S. Ben-Yaakov , DC-DC Converters
n1
I1
n2
[3- 22]
Ap
n1I1rms
⋅2
Jk
{V1, t on }
n1 =
2Bmax A e
Aw =
n1 =
JkA w
2 ⋅ I1rms
Ap = A w Ae =
Ap =
Ap =
11
{V1, t on }
JkA w
=
I1rms ⋅ 2 2Bmax A e
{V1, t on }2 ⋅ I1rms
{2Bmax }Jk
{V1, t on }2 ⋅ I1
rms
∆B ⋅ Jk
{V1,Don }2 ⋅ I1
rms
fs ⋅ ∆B ⋅ Jk
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 23]
Transformer design stages
Ap =
1. Calculate Ap
{V1,Don }2 ⋅ I1
rms
fs ⋅ ∆B ⋅ Jk
In symmetrical operation
In asymmetrical operation
∆B = Bmax+ - Bmax-
∆B = Bmax - 0
2. Look for core
3. Calculate n1 by:
4. Calculate n2
n1 =
{Vm ,ton }
2Bmax A e
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 24]
Inductor design
Need to store energy
( in transformer n1·I1= n2·I2 )
I
L
B
µ
12
H
µ = µ oµr
µo - air (vacuum) permeability
µr - relative permeability
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 25]
Permeability
Henry
m
µo = 1.26·10-6
µr of ferrites ∼ 2000 - 4000
B = µH
Bo
B
µ r1
µr 2
H
Ho
If µ is high B will reach
quickly Bsat
Need to slower µ
µ r 2 < µ r1
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 26]
Gaps
Discrete
air gap
µo
µr
Φ
Distributed
air gap
Same Φ magnetic lines in ferromagnetic material and in air.
l g << l e
lg + le ≅ le
13
le
lg
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 27]
Current Crowding
Current crowding due to magnetic fields
RAC high around gap
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 28]
Inductance with Gap
l g << l e
le
lg
lg + le ≅ le
Φ = constant
Hg =
B
µo
Hm =
B
µm
nI = l eH = Hml e + Hgl g
14
B ≅ constant
Hl e =
Bl e Bl g
+
µm
µo
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 29]
Inductance with Gap
Hl e =
zDividing
H=
Bl e Bl g
+
µm
µo
out le and defining
µe =
B
H
Ba
B Bm
=
+
l
µ e µm
µo e
lg
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 30]
Gap Calculation
1
1
=
+
µ e µm
1
1
=
+
µreµo µrmµo
1
l 
µo  e 
 
 lg 
1
1
1
=
+
µre µrm  l e 
 
 
 lg 
15
 le 
  + µrm
1  l g 
=
µre
l 
µrm  e 
l 
 g
1
l 
µo  e 
 
 lg 
µre
If
l 
µrm  e 
l 
 g
=
 le 
  + µrm
 
 lg 
le
< µrm
lg
l 
µre ≈  e 
 
 lg 
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 31]
Inductance
V =L
L
dI
dt
V =n
dΦ
dt
dI
dΦ
=n
dt
dt
L-?
n
dΦ
dB
dH
n dI
= nA e
= nA eµ
= nA eµ
dt
dt
dt
l e dt
L
dI n2 A eµ dI
=
dt
l e dt
L=
n2 A eµ
le
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 32]
Two windings on same core
L1  n1 
= 
L 2  n2 
2
Inductor design
B
Bmax
H
16
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 33]
Saturation Limits
L
dI
dΦ
=n
dt
dt
Ipk  dI 
Bmax  dB 
L ∫   dt = nA e ∫ 
 dt
0
0
 dt 
 dt 
L Ipk = nAeBmax
n=
LIpk
quick design and check
A eBmax
Ae =
LIpk
n=
nBmax
JkA w
Irms
Prof. S. Ben-Yaakov , DC-DC Converters
Ap = A eA w =
LIpkIrms
[3- 34]
Ap
Bmax Jk
LIpkIrms ≈ LI2
Energy stored =
LI2
2
Air gapped core Design
1. Calculate Ap
2. Choose a core
3. Iterate
4. Calculate l g ( or increase gap until L is as required )
17
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 35]
Cores
z
Transformer core
z
Inductor core
air gap
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 36]
Cores
1. E - core
2. TOROID
3. ARENCO
18
4. POT
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 37]
Commercial cores
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 38]
Distributed gap core
zThe
AL =
Hy
turn
concept of AL
( sometime
Hy
1000 turns
)
L for n turns: L = n2 ⋅ A L
Distributed
air gap
19
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 39]
AL
Prof. S. Ben-Yaakov , DC-DC Converters
Toroid Data
20
[3- 40]
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 41]
Permeability change
1 Amp/m =79.5 Oe
L decreases with DC current !
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 42]
Losses
Misleading notations !
∆B NOT B
“Good number”=100mW/cm3
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
21
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 43]
Temp. Rize
z
“Hot Spot” - Critical parameter
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 44]
Hanna Curve
n=
LI pk
Ae Bmax
Hn =
Hn =
22
H LI pk
Ae Bmax
nI LI
le Ae Bmax
H=
LI2
Ve
HBmax =
1
Bmax
LI2
Ve
Bmax
= µ = µ rµ o
H
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 45]
Hanna Curve
Prof. S. Ben-Yaakov , DC-DC Converters
Core Size Selection
23
[3- 46]
Prof. S. Ben-Yaakov , DC-DC Converters
[3- 47]
Basic Design of Distributed Gap Core
1. Calculate LI2
2. Look up manufacturer data
3. Select Core
4. Calculate n = 1000
5. Check Lmin
L
AL(1000)
6. Calculate losses. Temp rise and and µ = f (
7. Iterate
24
nI
)
le