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Power Magnetic Devices: A Multi-Objective Design Approach
Chapter 6: Magnetic Core Loss
S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.1 Eddy Current Losses
• Eddy current lost is a resistive power loss associated with induced currents
• Let’s consider a rectangular conductor
2S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.1 Eddy Current Losses
• Defining
• We can show
1 min( , )k w d=
2k w d= −
214 3 2 2 3 4
1 2 1 2 1 2 1 22 0
2 114 8 2 2 ln128
Tkl dBP k k k k k k k k dtk T dt
σ += + + − +
3S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.1 Eddy Current Losses
• Derivation
4S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.1 Eddy Current Losses
• Derivation
5S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.1 Eddy Current Losses
• Derivation
6S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.1 Eddy Current Losses
• Derivation
7S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.1 Eddy Current Losses
• Derivation
8S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.1 Eddy Current Losses
• Example 6.1A. Eddy current loss in a toroid
Table 6.1A-1 Aluminum alloy test samples.
Material
σ(MS/m)
rμ ir
(mm) or
(mm)
w (mm)
d (mm)
6061T6 25.3 1.23 140 150 10 12.7 6013 23.3 1.20 140 150 10 27.2 9S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.1 Eddy Current Losses
• Example 6.1A. Normalized power loss
• Thus, with our expression for eddy current loss, we have
2
0
1n TPpdBV dt
T dt
=
4 3 2 2 3 4 11 2 1 2 1 2 1 2
2
214 8 2 2 ln
128n
kk k k k k k k kk
pwd
σ ++ + − + =
10S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.1 Eddy Current Losses
• Example 6.1A. Measured power loss. We can show
0
1 Tps p
s
NP v i dt
N T=
11S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.1 Eddy Current Losses
• Example 6.1A. Derivation of average power.
12S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.1 Eddy Current Losses
• Example 6.1A. Derivation of average power.
13S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.1 Eddy Current Losses
• Example 6.1A. Now we measured power, we can find the measured normalized power loss density
• Clearly – we need to know B. There are two approaches.
2
0
1n TPpdBV dt
T dt
=
14S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.1 Eddy Current Losses
• Example 6.1A. Approach 1 to finding B (predicted)• We use
0
2r piBr
μ μπ
=
15S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.1 Eddy Current Losses
• Example 6.1A. Approach 1 to finding B (measured)• We use
s
s
dB vdt wdN
=
16S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.1 Eddy Current Losses
• Example 6.1A. Results:
17S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.1 Eddy Current Losses
• Sinusoidal excitation
• Periodic excitation
( )cospk eB B tω=2
2 2
0
1 12
T
e pkdB dt B
T dtω =
, ,1
cos( ) sin( )K
a k e b k ek
B B k t B k tω ω=
= +
( )2
2 2 2 2, ,
10
1 12
T K
e a k b kk
dB dt k B BT dt
ω=
= +
18S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.1 Eddy Current Losses
• Thin laminations. Suppose k2 >> k1• Then we can show
• Can be approximated as
4 3 2 2 3 4 11 2 1 2 1 2 1 2
2
214 8 2 2 ln
128n
kk k k k k k k kk
pwd
σ ++ + − + =
23
2 10
112
Tl dBP k k dtT dt
σ =
19S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.1 Eddy Current Losses
• To show this
20S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.1 Eddy Current Losses
21S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.1 Eddy Current Losses
• It follows that
• To show this, we start with
22
0
112
Tw dBp dtT dt
σ =
23
2 10
112
Tl dBP k k dtT dt
σ =
22S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.1 Eddy Current Losses
• Continuing …
23S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.2 Hysteresis Loss and the B-H Loop
• Domain wall motion
24S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.2 Hysteresis Loss and the B-H Loop
• Sample B-H characteristics
25S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.2 Hysteresis Loss and the B-H Loop
• The area of a B-H trajectory represent energy loss.
• The first step to do this is to show
• To do end, consider the toroid0
TB
B
e HdB=
26S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.2 Hysteresis Loss and the B-H Loop
• Proceeding …
27S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.2 Hysteresis Loss and the B-H Loop
• Now let us consider a closed path
1 2
0 1
3 0
2 3
Term 2Term 1
Term 3 Term 4
B B
B B
B B
B B
e HdB HdB
HdB HdB
= +
+ +
28S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.2 Hysteresis Loss and the B-H Loop
• Minor loop loss
01
0 1
Term 2Term 1
BB
B B
e HdB HdB= +
29S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.2 Hysteresis Loss and the B-H Loop
• Now that we have established the energy loss per cycle, the average power loss may be expressed
h BHp fe=
30S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.3 Empirical Modeling of Core Loss
• The Steinmetz Equation1
pkhBH
b bb
Bfkef Bf
βα − =
pkh h
b b
Bfp k
f B
βα =
31S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.3 Empirical Modeling of Core Loss
• The Modified Steinmetz Equation (MSE)– Motivated by observation that localized eddy current due
to domain movement being tied rate of change• MSE equivalent frequency
mx mnB B BΔ = −2
0
1 T dBB dtB dt
= Δ
2
2 20
2 Teq
dBf dtB dtπ
= Δ 32S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.3 Empirical Modeling of Core Loss
• MSE loss equation1
2eqh
BHbbb
fkeBff
α β− ΔΒ =
1
2eq
h hbb b
f fp kBf f
α β− ΔΒ =
33S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.3 Empirical Modeling of Core Loss
• Note
34S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.3 Empirical Modeling of Core Loss
• MSE with sinusoidal flux density
35S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.3 Empirical Modeling of Core Loss
• MSE with sinusoidal flux density
36S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.3 Empirical Modeling of Core Loss
• Example 6.3A. MSE with triangular excitation
37S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.3 Empirical Modeling of Core Loss
• Example 6.3A. MSE with triangular excitation
38S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.3 Empirical Modeling of Core Loss
• Generalized Steinmetz Equation (GSE)– Assumes instantaneous loss is a function of rate of
change and flux density value
– A suggested choice is
,idBp f Bdt
=
1i
b b b
dB Bp k
f B dt B
α β α−
=
39S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.3 Empirical Modeling of Core Loss
• The GSE then becomes
where we choose
0
11 Th
b b b
dB Bp k dt
f B dt BT
α β α−
=
( )2
1
02 cos sin
hkkd
πα α β α θπ θ θ− −
=
40S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.3 Empirical Modeling of Core Loss
• Example 6.3B. Consider MN60LL ferrite with α=1.034, β=2.312, kh = 40.8 W/m3
• Let us compare MSE and GSE models for
• Cases– Case 1: B1=0.5 T, B3=-0.05T– Case 2: B1= 0.45T, B3=0 T– Case 3: B1= 0.409 T, B3=0.0409 T
1 3cos(2 ) cos(6 )B B ft B ftπ π= +
41S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.3 Empirical Modeling of Core Loss
• Example 6.3 flux density waveforms
42S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.3 Empirical Modeling of Core Loss
• Results– MSE model: 87.4, 86.5, 86.2 kW/m3
– GSE model: 86.9, 86.5, 86.8 kW/m3
• Comments on MSE and GSE models
43S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.3 Empirical Modeling of Core Loss
• Combined loss modeling– Eddy current loss
– Hysteresis loss (MSE)
– Combined loss model
2
0
1 Te e
dBp k dtT dt
=
1
2eq
h hbb b
f fp kBf f
α β− ΔΒ =
1 2
0
12
Teq
t h ebb b
f f dBp k k dtBf f T dt
− ΔΒ = + α β
44S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.3 Empirical Modeling of Core Loss
• Example 6.3C. Lets look at losses in M19 steel with kh=50.7 W/m3, α=1.34, β=1.82, and ke=27.5·10-3Am/V. We will assume
• Since the flux density is sinusoidal
cos(2 )pkB B ftπ=
pkh h
b b
Bfp k
f B
βα =
2 2 22e e pkp k B fπ=
45S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.3 Empirical Modeling of Core Loss
• Example 6.3C results
46S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.4 Time Domain Modeling of Core Loss
• These models can capture effects such as – waveforms with dc offset– aperiodic excitation– waveforms with minor loops
• Two approaches– Jiles-Atherton– Praisach
47S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.4 Time Domain Modeling of Core Loss
• Jiles-Atherton model0 ( )B H Mμ= +
irr revM M M= +
M ( )(M ( ) )an irrirr
an irr
H MdM dHk H Mdt dtδ α
− = − − 1 0
0 0
1 0
dHdt
dHdtdHdt
δ
>= =−
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6.4 Time Domain Modeling of Core Loss
• Preisach model is based on behavior of a hysteron
• Total magnetization based on
Q( , ) P( , )U
M U V U V dVdU∞
−∞ −∞
= 49S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.4 Time Domain Modeling of Core Loss
• Saturated magnetization
• Normalized magnetization
• Normalized hysteron density
P( , )U
sM U V dVdU∞
−∞ −∞
=
s
MmM
=
1p( , ) P( , )s
U V U VM
=
50S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.4 Time Domain Modeling of Core Loss
51S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.4 Time Domain Modeling of Core Loss• Lets consider behavior of model• Start in State 1 – all hysterons neg• State 1 to State 2
• State 2 to State 3
1
2 1 2 p( , )H U
s sm m U V dVdU−∞ −∞
= +
1 1
2
3 2 2 p( , )H H
s sH V
m m U V dUdV= −
52S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.4 Time Domain Modeling of Core Loss• State 3 to State 4
3
2 2
4 3 2 p( , )H U
s sH H
m m U V dVdU= +
53S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.4 Time Domain Modeling of Core Loss• Reconsider State 3 to State 4
• Suppose field decreases after State 4
2 2
3 2 p( , )H U
sH H
m m U V dVdU= +
2
2 p( , )H
H
dm H V dVdH
=
3 3
4 2 p( , )H H
sH V
m m U V dUdV= − 3
2 p( , )H
H
dm U H dUdH
= 54S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.4 Time Domain Modeling of Core Loss
• Completing the modeldm dm dHdt dH dt
=
sM M m=
0B H Mμ= +
55S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach
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6.4 Time Domain Modeling of Core Loss
• Relative advantages and disadvantages of approaches
56S.D. Sudhoff, Power Magnetic Devices: A Multi-Objective Design Approach