PSMA...pi Select A pi +1 Calculate Turns Calculate J o Select Wires Calculate Copper Loss Calculate...
Transcript of PSMA...pi Select A pi +1 Calculate Turns Calculate J o Select Wires Calculate Copper Loss Calculate...
PSMA
1Power Electronics Research Centre, NUI Galway
High Frequency Effects in the Core
Losses in Magnetic Components
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Copper losses
Core losses
Hysteresis loss
Eddy current loss
Skin effect loss
Proximity effect loss
Ferromagnetic Materials
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(a) Hard magnetic materials (b) Soft magnetic materials
B
Br
HHc-Hc
-Br
(a)
B
H
(b)
Br
Hc-Hc
-Br
Core Loss
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Hysteresis loss in a ferromagnetic material Eddy current loss in a ferromagnetic material
dB
0
a
H
B
bc
die ie/n
t
ˆfe cP K f Bα β=
Hysteresis loss is the area inside the B-H loop Eddy current loss is reduced by laminations in steel Eddy current loss is reduced by higher resistivity in ferrites
Ferromagnetic Materials
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Soft magnetic materials are classified as:
Ferrites
Laminated iron alloys
Powered iron
Amorphous alloys
Nanocrystalline materials
Core Shapes
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Toroid core PQ core Pot core RS/DS core RM core
EP core
EE core EI core ER core EFD core ETD core
U coreUR core C core Planar core
Core Loss Density vs Frequency
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B=0.1 T
Performance Factor
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Fringing (Flux)
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Gap in the centre leg Gap in the outer leg
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Transformer Design
Basic Equations
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Kv =4.44 for a sinewave=4.00 for a squarewave
Voltage equation
Power equation
rmsˆ v cV K f NBA=
Vrms: the rms value of the applied voltageKv: the voltage waveform factorf: the frequency of the applied voltageJo: the current density in each winding
: the maximum flux density in the coreIi: the current in winding iNi: the number of turns in winding iAwi: the conductor area in winding iku: the window utilisation factor
ˆVA v i i c
i o wi
K fB N I AI J A
= ⋅ ⋅∑ ∑=
Window utilisation factor
ˆVA v o u a cK fBJ k W A=∑
wi1
n
ii
u
a
N Ak
W=∑
=
secp a cA W A
Windowarea cross tional area= ×
× −
B
ˆv u p oVA K fBk A J=∑
Transformer Losses
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Winding losses
Total resistive losses
is window utilization factor
is volume of the windings
Core losses
Typical layout of a transformer
22 wi
cu1
wi
( )ni o
wi
N MLT J AP RIA
ρ=
= =∑ ∑
2
cu w w u oP V k Jρ=
wi1
n
ii
u
a
N Ak
W=∑
=
w aV MLT W= ×
feˆ=Vc cP K f Bα β
Volume ofcore,VC
Cross-sectionalarea, AC
Mean Length of a Turn, MLT
Volume ofwindings,VW
Window area,Wa
2ro
Heat loss by convection
total cu fe= c tP P P h A T+ = ∆
Dimensional Analysis
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kw=10, kc =5.6 and kt=40 are typical values
Typical layout of a transformer
3/4 2
cu w w p u oP k A k Jρ=3/4
feˆ=k c p cP A K f Bα β
Volume ofcore,VC
Cross-sectionalarea, AC
Mean Length of a Turn, MLT
Volume ofwindings,VW
Window area,Wa
2ro
3/4w w pV k A=
3/4c c pV k A=
1/2t t pA k A=
(14)
(15)
1/2
total =h c t pP k A T∆ˆ
v u p oVA K fBk A J=∑
Losses Optimization
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Winding losses
Total losses
Core losses
At a given operation frequency,
The minimum losses occur when
2
cu 2 2
VAˆ ˆw w u
v f u p
aP V kK fBk k A f B
ρ ∑= =
feˆ ˆ= c cP V K f B bf Bα β α β=
2 2ˆ
ˆaP bf B
f Bα β= +
1
2 3
2 ˆ 0ˆ ˆP a bf BB f B
α β∂ β∂
−= − + =
cu fe2P Pβ
= total cu
2P Pββ+
=
Losses Optimization
15Power Electronics Research Centre, NUI Galway
Winding, core and total losses at different frequencies
The first step in the design is to establish whether the optimum flux density given by the optimization criterion is greater or less than the saturation flux density.
A
B
C
D
Losses
Flux densityBoptD BoptBBsat
P
Pfe
Pcu
P
Pfe
Pcu
50Hz50kHz
Losses Optimization
Power Electronics Research Centre, NUI Galway 16
( )8 27
7 2 7( )8 7 2
[ ]2( 2) [ ][ ] VA
t v uo o o
w w c c
hk T K kf B fk k K
β α β ββ ρ
− − ∆
= + ∑
8/74/7 VA2 1ˆ
w wp
t u v o
kAhk k T K fBρ β
β +
= ∆
∑
1/41
2t
ow u p
hk TJk k A
ββ ρ
∆=
+
3/4 2
cu w w p u oP k A k Jρ=3/4
feˆ=k c p cP A K f Bα β
1/2
total =h c t pP k A T∆
ˆv u p oVA K fBk A J=∑
Core size
Current density
Optimum flux density
total cu
2P Pββ+
=
total fe
22
P Pβ +=
Core Losses Correction
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ˆ( ) 1cr
V ccr
fP K fB ff
αβ α β−
= +
feˆ=Vc cP K f Bα β
Core size v Frequency
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27 127
47 2 8( )22 2 2
7 22( 2) ( 2)7
VA( ) ( 2)( )
2
c w w cp
t v u
k k KA fhk K k T
ββ
βα βββ ββ
β β β
ρ β
β+
−−+−
+ +
+ = ∆
∑
Optimum Core Size
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27 127
48
7 2 8( ) 7 222 2 27 2
2( 2) ( 2)7
VA( ) ( 2) 1( )
2
cr
c w w cp o
t crv u
k k K fA fhk fK k T
ββ
β αα β βββ ββ
β β β
ρ β
β+
−− −+−
+ +
+ = + ∆
∑
Design Methodology
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Specifications : ∑VA,K,f,ku,ΔT
Select Material : Bsat,ρc,Kc,α,β
Calculate Bo
AcWaMLTm
Bo < Bsat
Yes No
Calculate Ap
Select Ap
Calculate Api
Select Api+1
Calculate Turns
Calculate Jo
Select Wires
Calculate Copper Loss
Calculate Core Loss
AcWa
MLTm
Calculate High Frequency LossesSelect Ap
Bmax ≤ Bsat
Calculate Efficiency, η
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Push-pull Converter TransformerCircuit Waveforms
+
_
vp2
is1
+
_
vs1
Lo
Co
+
_
Vo
is2
+
_
vs2
+
_
vp1
D1
D2
Np : Ns
Vs
+
_
S2 S1
t0
Vp,Vs
t0
ΦVs
Io
ip1 ip2
t0
Io
2oI
t0
Io
2oI
is1
is2
TDT’ T’
τ
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Push-pull Converter: Specifications
Input 36 → 72 V
Output 24 V, 300 W
Frequency, f 50 kHz
Temperature Rise, ΔT 35 ºC
Ambient Temperature, Ta 45 ºC
Kc 9.12
α 1.24
β 2.0
Bsat 0.4 T
Design specifications Core data: EPCOS N67 Mn-Zn
fe c mP K f Bα β=
Core loss
Push-pull Converter: Voltage factor
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Calculations:
(1) Voltage waveform factor KvVp,Vs
t0
ΦVs
τTDT’ T’
Push-pull converter voltage and flux waveforms
4 4.88vKD
= =
24 0.6736
D = =
max maxs max'
4 4V = = = = / 2p p c p c p c p c
B Bd dBN N A N A N A fN A Bdt dt DT DT Dφ
=
rms max max4V = = K s p c v p cDV fN B A fN B AD
=
Push-pull Converter: Power factor
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Calculations:
(2) Power factor kpp, kps
( )rms rms; 12
o os s s
V IV DV I DD
= = = +
'
0
1( ) ( ) '2
DT
s s s s
Dp v t i t dt V I DT V IT
< >= = =∫
rms rms rms rms
1 ; 12pp ps
p p s s
p p Dk kV I V I D< > < >
= = = =+
rms rms
1 1 12 2
os s o o
D D PV I V ID D+ +
= =
rms rms; ( / 2) ; p s p sV DV I D I= =
t0
Vp,Vs
t0
ΦVs
Io
ip1 ip2
t0
Io
2oI
t0
Io
2oI
is1
is2
TDT’ T’
τ
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Push-pull Converter: Core sizeCalculations:
(4) Optimum Ap
The optimum flux density is less than Bsat
(3) VA ratings of the windings
1 1 122 2 2 2
1 0.672 (300) 898.6 VA0.67
o o o oo
pp ps
P P P P DVA Pk k D
+ = + + + = +∑
+= + =
[ ][ ]
18 7 2.07 2 (7 1.24 2)7 2
7 2.0 278 28
(10)(40)(35)2 2.0 4.899 (0.4)ˆ 50000(2.0 2) 898.6[(1.72 10 )(10)] (5.6)(9.12)
0.127T
oB× − − × −
× −−
× = • + × =
4/7 8/784(1.72 10 )(10) 2.0 2 1 898.6 2.54cm
(10)(40) 2.0 0.4 35 (4.899)(50000)(0.127)pA− × +
= = ×
ETD44 Core Data
Power Electronics Research Centre, NUI Galway 26
Ac 1.73 cm2
Wa 2.10 cm2
Ap 3.63 cm4
Vc 17.70 cm3
kf 1.0
ku 0.4
MLT 7.77 cm
ρ20 1.72 µΩ-cm
α20 0.00393
Fringing (Flux)
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Gap in the centre leg Gap in the outer leg
Fringing (Different Frequencies)
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Frequency 1kHz Frequency 100kHz
Width of conductor: 0.2mm Core: Magnetics® port core
Magnetic Field Intensity
Fringing (Different Frequencies)
29Power Electronics Research Centre, NUI Galway
Frequency 1kHz Frequency 100kHz
Magnetic Flux
Fringing (Different Frequencies)
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Frequency 1kHz Frequency 100kHz
Current Density
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High Frequency Effects in the Windings
High Frequency Effects
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High frequency
effects
Windings optimization
Skin effect
Proximity effect
Core optimization Eddy current
Windings arrangement
Thickness optimization
Design Issues for High Frequency
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High frequency winding loss
Core loss: Steinmetz equation, iGSE.
Parasitic parameters: leakage inductance, stray capacitance
Proximity effect
I I I I
H0
Primary SecondaryH1
Core Eddy currents
Skin effect
2r
Jz
Eddy current
r
Fringing effect
Gap
Core
Ohm loss
Eddy Current in the Core
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Eddy current losses in a toroidal core Equivalent core inductance versus frequency
The inductance terms of the core impendence is
coileddy
current
ac flux
magnetic material with ferrite conductivityand relativepermeability
σ
rμ- +0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
SLL
The inductance of the toroid under the lower frequency 2
00
r c
c
N AL µ µ=
4
0 4
0 3 4
1 2.112 1.43
1 1 1 2.116 16
sL L
L
∆ = − ∆ < + ∆ = + + ∆ > ∆ ∆ ∆
core radiusskin depth
bδ
∆ = =
Eddy Current in the Core
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Fig. 7.7 Resistivity of P type ferrite Inductance and rel. permeability v frequency and
Complex permeability 4
4
3 4
' 1 2.112 1.43
1 1 1 2.116 16
rs r
r
µ µ
µ
∆= − ∆ < + ∆ = + + ∆ > ∆ ∆ ∆
core radiusskin depth
bδ
∆ = =
Eddy Current Core Losses
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The equivalent core resistance:
The core losses may be reduced by increasing the electrical resistivity or reducing the electrical conductivity of the core material.
The use of a smaller core cross-section to reduce eddy current losses suggests the use of laminations.
The average power loss in the core due to eddy currents is
22 2
2 00 4 2
r cs
c
N l bR L fl
µ µ πω π σ ∆
= =
2 2 2
max
4f B bp π σ π
=
Eddy Current Core Losses
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Performance Factor:
Flux Distribution in the Core
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Assumptions:Homogeneous coreConstant resistivityConstant permeabilityNo dielectric effects
Axial Flux Distribution in the Core
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1. Flux bunches to the surface2. Flux is higher at the surface
Axial Flux Distribution in the Core
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Core Losses (GSE, iGSE)
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Steinmetz equation:
The time-average power loss with non-sinusoidal excitation using the iGSE
fe maxcP K f Bα β=
0 0
1 ( ) 1 ( )
( )
T T
v i i
i
dB t dB tP k B dt k B dtT dt T dt
dB tk Bdt
α αβ α β α
αβ α
− −
−
= ∆ = ∆∫ ∫
= ∆
21 1
02 cosc
i
Kkdαπβ απ θ θ− −
=∫
where
A useful approximation is
1 1 6.82442 1.10441.354
ci
Kkβ απ
α− −
= + +
Push-pull Converter Transformer
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Core data: ETD44
Kc 9.12
α 1.24
β 2.0
Bsat 0.4 T
Core data: EPCOS N67 Mn-Zn
Ac 1.73 cm2
Wa 2.78 cm2
Ap 4.81 cm4
Vc 17.70 cm3
kf 1.0
ku 0.4
MLT 7.77 cm
ρ20 1.72 µΩ-cm
α20 0.00393
Push-pull Converter Transformer
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Flux waveform for the push-pull converter
Calculations:
(1) power loss per unit volume
-Bmax
t
v
Bmax
0 DT/2 T/2 (1+D)T/2 T
ΔB
p
/2 (1 ) /2 1
0 /2
1 1 2 ( )/ 2 / 2
DT D T
Tv i i
B BP k B dt dt k B B DTT DT DT T
α αβ α β α α α− −+ − ∆ ∆ = ∆ + ≈ ∆ ∆∫ ∫
1 1
2.0 1 1.24 1
6.82442 1.10441.354
9.12 0.92756.82442 1.1044
1.24 1.354
ci
Kkβ απ
α
π
− −
− −
= + +
= = + +
Compare iGSE and GSE
44Power Electronics Research Centre, NUI Galway
Calculations:
(2) ΔB
(3) The core loss per unit volume
(5) The total core loss by GSE = 1.466 W
[ ]
1
2.0 1.24 1.24 1 1.24
5 3
1 2 ( )
(0.9275)(0.232) (50000) (2 0.232) (0.67 / (50000))0.871 10 W/m
v iP k B B DTT
β α α α− −
− −
= ∆ ∆
= × ×
= ×
(4) The total core loss
max 4
0.67(36) 0.116 T(4.88)(50000)(6)(1.73 10 )
d
p cv
cDVBK fN A −
= = =×
max2 0.232 TB B∆ = =
6 517.71 10 0.871 10 1.543 W−× × × =