Expert Design & Empirical Test Strategies for Practical Transformer Development

Expert Design and Empirical Test
Strategies for Practical Transformer
Development

Victor W. Quinn
Director of Engineering / CTO
RAF Tabtronics LLC

1

Outline
• Units
• Design Considerations
• Coil Loss
• Magnetic Material Fundamentals
• Optimal Turns
• Empirical Shorted Load Test
• Core Loss Estimation and Recommended
Empirical Test Method (iGSE,FHD,SED)
• Example Application
• Summary
Victor Quinn
victor.quinn@raftabtronics.com
1-585-243-4331 Ext 340
2

New Product Development Premise
Component Verification !
Application Validation
• Transformer developers strive to design, develop and
consistently manufacture transformers based on in depth
understanding of application conditions and implementation of
best practice design, production and test methods.
• The initial first article transformer must be carefully validated
in the respective application circuit to ensure the scope of the
component development was sufficient for the end use.
• Based on the challenges of accurate measurement of
transformers in complex circuit applications, calorimetric
(temperature rise) measurements are often used to
characterize transformer power dissipation levels.
• The transformer developer may work closely with the power
system developer to correlate loss and thermal measurements
under specific conditions.
• Once the transformer design is validated, the transformer
developer must implement tight QA controls on component
materials and processes to ensure consistent results.

3

UNITS: Rationalized MKS+
Volt
Electric Field "
Meter
Webers Weber
B" 2 # 1 2 $ 1Tesla $ 10 Gauss
4
Meter Meter
Ampere % Turn AT
H" # 1 $ 1.257 & 10% 2 Oe
Meter M
Ampere 1 Oe = 2.02 AT/inch
J"
Meter 2
1 1 ( . 0.676)- % inch (Cu @ 20 Celsius)
' $ "
( Ohm % Meter
Henry H
)0 $ 4* & 10%7 $ 1.257 & 10%6 (permeability constant)
Meter M
H G
% 12 Farad ) 0 . 3.19 /10 %8 . 0.5 inch
+ 0 $ 8.85 & 10 (permittivity constant) inch A
Meter
( F
,$ , where f = frequency in Hertz + 0 . 0.225 / 10%12
*)0 f inch
2.6
,. inch (Cu @ 20 Celsius )
f
4

Design Considerations
Cross Sections and Operating Densities
Ip Is J

!B

!

J
Primary Secondary

!B
Core
Core cross section [Acore]
Coil cross section [Awindow] constrained by aperture
constrained by aperture of of coil.
core.
5

Faraday’s Law
• When a voltage E is applied to a E +
winding of N Turns
Np Ns
d0
E $ (% ) N -
dt (Instantaneous Volt/Turn Equivalence)
t1 2 T
Voltage

3E
t1
p dt
10 $ E
2Np 0
Time
t1 2 T
1
E $
T 3
t1
E dt (Average Absolute Voltage)
6

Flux Density Considerations
i(t)
• Voltage driven coil
t1 2 T

3t1
E p dt
E
v(t)
+

L
10 $ Np
2Np -
• Line filter inductor

Current

v(t) +

Time

i(t) L
Voltage

-
Time
(Two induced consecutive, additive voltage pulses)

7

Ampere’s Law
• When a current i is applied to a Ip Is
winding of N Turns

3 H / dl $ ienclosed Np Ns

Curve C
• For ideal core having infinite
permeability, Hc=0 Curve C

N p I p $ Ns I s
Is

Ip
Hc
(Instantaneous Amp-Turn
Equivalence)
The Amp-Turn imbalance is
precisely the excitation required to
achieve core flux density. Magnetic
material reduces the excitation
current for a given flux density.
8

Transformer VA Rating
• Faraday’s Law and Ampere’s Law imply that
instantaneous Volt/Turn and Amp-Turn are
constant.

• Therefore the winding Volt-Ampere product is
independent of turns and is a useful parameter
to rate transformers.

• Since core flux excursion is based upon
average absolute winding voltage E
and winding dissipation is based upon
RMS current i RMS ,
define:
Transformer VARating $ 5i
Ei 4 i RMSi
9

Fundamental Transformer Equation

• Derive a relationship between Volt-Ampere rating
and transformer parameters:

VA Rating 6 Acore 4 Awindow 4 freq 4 J 4 1 B 4 K Fe 4 KCu

VA Rating
VAP 7 Acore 4 Awindow 6 4
freq 4 J 4 1 B 4 K Fe 4 KCu
• Therefore, by increasing frequency, current
density, flux density, or utilization factors, the area
product can be reduced and the transformer can
be made smaller.

10

Current Density (J ),
Window Utilization Factor (KCu )
and Coil Loss

11

Coil Loss

• At low frequencies, for uniform conductor resistivity and uniform mean
lengths of turn, minimum winding loss is achieved when selected
conductors yield uniform current density throughout the coil.

• For uniform current density J:
Low Frequency Coil Loss $ ( / Cond Vol / J 2
2
= :
or Low Frequency Coil Loss $ - / ;5 NI i 8
< i 9
where ( / 1y ( / 1y
-$ $
1x / Thicknesscu K Cu / Core Window Area

Window Utilization Factor

12

Window Utilization Factor (KCu )
Coil Insulation Penalty
Core Window
• Conductor Coatings
• Insulation Between
Windings
• Insulation Between
Layers
• Margins
• Core Coatings or
Bobbin / Tube Supports
These considerations can
total more than half of the
available core window limiting
utilization to less than 35%!

13

Example of KCu Estimate (PQ 20/20)
"x
tol
2IW
2IW
"z
support
tol
margin
tol

K Cu x $
@!x % 2>tol? % 2(margin or bobbin flange)A $ @0.55 - 2(0.003) - 2(0.036)A $ 0.858
!x 0.55
*
K Cu shape ( round versus square) $ $ 0.785
4
K Cu insul ( heavy film solid wire) $ 0.85

K Cu z $
@!z % 2>tol? % (support) - 2(2IW)A $ @0.169 % 2>0.010? % (0.035) - 2(0.01)A $ 0.556
!z 0.169

So,
KCu . K Cu x / K Cu shape / K Cu insul / K Cu z $ 0.318

14

Window Utilization Penalties
Typical Core Window Utilization

Coil Loss 100%

14

12 75%
10
Loss (W)

8

6 50%

4

2
Ideal 25%
0
0 5 10 15 20
Window Utilization
Frequency (kHz)
0%
100%
90% Insulation occupies
80%
70%
more than 50% of
core window.
Utilization

60%
50%
40%
30%
20%
Eddy current
10% losses can further
0%
0 5 10 15 20 reduce effective
Frequency (kHz) conductor area by
more than 50%. 15

Other Design Considerations Affecting
Window Utilization: KCu

Eddy
Current Leakage
Losses Inductance

Maximum
Kcu

Insulation
Requirements

Corona / DWV
Number Of Capacitance
Requirements Windings

16

Low Frequency Example Calculation of Loss

H ( z ) x 7 RH 0 RMS x
ˆ ˆ

ˆ
J ( z) y 1z
z
!
H ( z ) x 7 H 0 RMS x
ˆ ˆ Ampere ' s Law # (R - 1)H 0 RMS 1x $ iRMS
iRMS
R$ 21
H 0 RMS1x
dH 1H ( R % 1) H 0 RMS iRMS
x
! J y RMS $ $ $ $
dz 1z 1z 1 x1 z
y
! ( 1y iRMS 2
Loss $ (1x1z 3 J y RMS dy $
2

y
1 x1 z
Effective conductor thickness

17

Qualitative Example
Severe Skin / Proximity Effects
J0 2 1 J ( H0 2 1 H ) cos(Bt ) x
!
2 J0

1J
i 1z CC ,

H0 cos(Bt ) x
!

% J0 1x CC 1z
Circulating 2 2
Current J0 $ H0 , 1 J $ 1H
, ,
" " i
3 H 4 dl $ i # 1 H cos(Bt ) $ lw
curve1
18

Dissipation And Energy As Function Of B
(single layer) Hs1 $ 0, Hs2 $ Hs
2.5 2.5
Single layer with one
surface field at zero is
2 forgiving of design 2
Dissipation errors.
Dissipation
However, other
Average Energy
1.5 constructions can 1.5
yield lower loss.
1x1y( 2 1 x1 y)0,Hs 2
HS 1 1
2, 8
0.5 0.5
Average Energy

0 0
0.5 1.5 2.5 3.5 4.5 1z
7B
Minimum Dissipation @ B="/2
,

19

Loss Equivalent and Energy Equivalent
Thicknesses (Single Layer Portion)

1z
(HS1 7 0 and HS2 7 HS with B 7 )
,

e2 B % e %2 B 2 2 sin(2B)
[ 2 B %2 B ]
(1y e 2 e % 2 cos(2B) (1y
Power $ (NIRMS ) 2 / $ (NIRMS ) 2 / Thkequivloss
1x , 1x

)0 1y 3 e2 B % e%2 B % 2 sin(2B) ) 1y
Energy $ (NIRMS ) / [ 2 B %2 B
2
], $ 0 (NIRMS ) 2 / Thkequivmagnetic
61x 2 e 2 e % 2 cos(2B) 61x

20

Define Normalized Densities
for General Sine Wave Boundary Conditions
1 x1 y( Dimensionless densities (p0 , e0)

Power $ H 2
p 0 (R, B , 0 )
,
0RMS

Magnetic Energy $ 1x1y) 0, H 0RMSe 0 (R, B, 0 )
2

• !x !y represents the surface area of the conductor layer

• H0RMS=RMS value of sinusoidal magnetic field intensity at one boundary

• R=Ratio of magnetic surface field Intensities

• B=Ratio of conductor thickness to " (conductor skin depth)

• "=Shift of magnetic surface field phases

21

Complex Winding Currents
Dissipation For Frequency Dependent Resistor

V $ IR
V
_
P$ I R 2
+
2
Pave $ 1
T 3I Rdt
T I Power
1 2
Pave $ R 3 I dt
TT For complex waveshape and frequency
Pave $ R * Irms 2 dependent resistor, Harmonic
Orthogonality =>
1 2
Irms $ 3 I dt
TT
Pave
total
$ 5 n
R( f n ) * Irmsn 2

22

Nonsinusoidal Excitation
I dc $ D 0 I rms
I rms1 $ D1 I rms

I rmsk $ D k I rms
• I rmsk is the RMS current value at the kth
.
harmonic
• I rms is the RMS value of the complex
waveshape
• D k is the ratio of the RMS value of the
kth harmonic to the RMS value of the
complex waveshape

23

Nonsinusoidal Currents and
Multi-Layer Winding Portion
Derive equivalent power density function

p0 equiv >R, B1 ,0 ? $ D 0
2 >1 % R ?2 2 E i D 2 p >R, i / B1 ,0 ?
i 0
B1 i

Where B1 is the ratio of conductor thickness to skin depth
for the first harmonic
Sum results to derive equivalent winding thickness and
effective window utilization
,
Loss Equivalent Winding Portion Thickness $ 2
nl
Kn H
F p0 equiv K1 % , B1 ,0 H
1
5 I
In F I
J n
F
G
n $1 J l G
1
K Cu $ 2
nl
Kn H
F p0 equiv K1 % , B1 ,0 H
1
nl B1 5 I
In F I F
n $1 J l G J n G
24

Portion Utilization Sum of 10 Harmonics

Sine Wave, No Insulation
2

Amplitude (normalized)
1.5
1
0.5
0

Conductor Utilization In Portion -0.5 0
-1
2 4 6 8 10 12 14

-1.5
No Insulation -2
Phase (radian)
100.0%

90.0%
1
For portions less than
80.0% 2 3 skin depths thick,
70.0% 3 utilization increases
4 with increasing layers.
60.0%
KCuz 5
50.0%
6 Layers
40.0% 7
30.0%

20.0%

10.0%

0.0%
0.00 1.00 2.00 3.00 4.00 5.00 6.00
Portion Thickness (in skin depths)

25

Sum of 10 Harmonics

Portion Utilization 2

Sine Wave, Insulation Penalty
1.5
1
0.5
0
-0.5 0 2 4 6 8 10 12 14
-1

Conductor Utilization In Portion -1.5
-2
Phase (radian)
IL=0.5 ! IW =0.5 !
50.0%

Maximum conductor
40.0%
utilization is achieved
1 with single layer
portions, but a large
2
30.0%
number of portions is
KCuz 3
required.
4
20.0%
5
6
10.0%
7
Layers
0.0%
0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00

26

Loss Equivalent Thickness
Sine Wave, No Insulation
Loss Equivalent Portion Thickness
Sum of 10 Harmonics
IL=0.0 ! IW =0.0 !
2
3.0

1.5
1

Layers 0.5
0

2.5 -0.5 0 2 4 6 8 10 12 14
-1
-1.5
Equivalent Thickness (in skin depths)

-2
7 Phase (radian)
2.0 6
5
4
1.5
3

2
1.0
1

0.5

0.0
0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00

27

Fourier Decomposition Of Theoretical Waveshapes

@
f (t ) $ 5 a cos(nBt ) 2 b sin( nBt )
n n
A
n
Amplitude an $ 0
8Ip n*
Ip bn $ sin( DC)
2 * Ip DC* n 2 2
2
n $ 1, 3, 5, ...
radian
4 Ip n*
an $ sin( DC)
Ip * n 2
2 * Ip bn $ 0
n $ 1, 3, 5, ...
- DC / * / 2 DC / * / 2 (1 - DC/2) / *
2 /*

28

Fourier Decomposition Of Theoretical Waveshapes

@
f (t ) $ 5 a cos(nBt ) 2 b sin( nBt )
n n
A
n
Amplitude
an $ 0
2 Ip
Ip bn $ sin(n* DC)
2 * Ip DC (1 - DC) * n2 2

n $ 1, 2, 3, ...
radian
2 Ip
an $ sin( n * DC)
Ip * (1- DC)
* n
Ip bn $ 0
- Ip * (DC) n $ 1, 2, 3, ...
- DC / * DC / * (2 - DC) / *
2 /*

29

Complex Wave, No Insulation
IL=0.0 ! IW =0.0 !
3.5

7 Layers Sum of 10 Harm onics

3.0 2

6 1.5

1

0.5

2.5 5 0
-0.5 0 2 4 6 8
-1

4 -1.5

2.0 -2
Phase (radian)

3
1.5
2
1.0
1

0.5

0.0
0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00

30

Complex Wave, No Insulation
IL=0.0 ! IW =0.0 !

7 Layers Sum of 10 Harmonics

Amplitude
2
6
3.5 0
0 2 4 6 8
5 -2
3.0

Phase
4
2.5
3
2.0

2
1.5

1.0 1

0.5

0.0
0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00

31

Core Loss Introduction

32

Magnetic Material
• One Oersted is the value of applied field strength
Iex
which causes precisely one Gauss in free space.
• Permeability is the relative gain in magnetic
induction provided by magnetic material.
! • Magnetic Material consists of matter
spontaneously magnetized.
– Magnetic Material in demagnetized state is
divided into smaller regions called domains.
– Each domain spontaneously magnetized to
saturation level.
– Directions of magnetization are distributed so
there is no net magnetization.
– Crystal anisotropy and stress anisotropy
create preferential orientation of spontaneous
magnetization within each domain.
– Magnetostatic energy is associated with
induced monopoles on surface.
– Exchange energy resides in domain wall as a
result of nonparallel adjacent atomic spin
vectors.
– Domain structure evolves to achieve relative
local minimum of total stored energy.
33

Magnetization
Microscopic Effects
• Soft magnetic material
provides increased magnetic
induction from induced
alignment of dipoles in
domain structure.
B • Magnetization process
converts material to single
domain state magnetized in
Br same direction as applied
field.
• Magnetization process
involves domain wall motion
Hc then domain magnetization
rotation.
H • Low frequency magnetization
characteristics for magnetic
material display energy
storage and dissipation
effects.
• Material voids and inclusions
=>irreversible magnetization
(Br, Hc)

34

Optimal Transformer Turns

35

Design Considerations Affecting
Current Density: J

Coil Loss D ( J )
Customer
Limit 2

Considering insulation and
eddy current penalties
Maximum J

Electrical
Requirements
(regulation,
efficiency)
Maximum
Winding
Loss
Maximum
Temperature
Rise

36

Design Considerations Affecting
Flux Density: !B
Saturation
Bpeak Bpeak OCL
L
Core Loss D (1B )
Steady State Transient Linearity

Where exponent # is determined
through empirical core loss
testing
Maximum 1 B

Electrical
Requirements
(efficiency)

Maximum
Core Loss
Maximum
Temperature
Rise

37

Selection Of Transformer Parameters:
J , !B , KCu , KFe

J

!B

• For given core and winding regions, J and !B function in a
competing relationship.
– Decreasing !B is equivalent to increasing turns of all
windings by the same proportion.
– Increasing turns of all windings by a given proportion
generates proportionately higher current density.
• For a linear transformer, selection of turns on any one
winding, uniquely determines J and !B densities and
therefore total loss for a specific application.

38

Selection Of Transformer Parameters
• For given conductor and core regions, the normalized expression for low
frequency winding loss can be extended and combined with an empirically
Empirical tests
derived core loss result to yield: characterize
core material in
Insulation and eddy current impacts
Fcore application

Total transformer loss $ Fcoil / N 2 L 2 conditions
p
Np
• Fcoil and Fcore relate exponential functions of primary turns (Np) to coil and core
losses respectively.

• n is the empirically derived exponential variation of core loss with magnetic flux
density.

• Without a constraint of core saturation, minimum transformer loss is achieved
with primary turns:

L / Fcore Core Loss 2
Np $ L 22 $
2 / Fcoil implying => Winding Loss L

39

Determination Of Fcoil

2
= :
Low Frequency Coil Loss $ - / ;5 NI i 8
< i 9
= VAsec :
5 NI i $ ; I p 2 V 8 N p
i ; 8
< p 9

2
(1y = VAsec :
Coil Loss $ ;I p 2 8 N p2
K Cu Awindow ; Vp 8
< 9
2
(1y = VAsec :
Fcoil $ ;I p 2 8
K Cu Awindow ; Vp 8
< 9

Reflects insulation and eddy current loss penalties

40

Determination Of Fcore

K0 Vp
B$
fN p Acore
If Core Loss $ VolumeFe K core @B A ,
L

L
= K0 Vp : 1
then Core Loss $ VolumeFe K core ; 8
; fAcore 8 N pL
< 9
L
= K0 Vp :
and Fcore $ VolumeFe K core ; 8
; fAcore 8
< 9
Reflects empirical characterization of core loss for specific
operating conditions and environment

41

Empirical Test: Verify

42

Transformer Realities
Winding loss
Ip Is

!leakage !leakage
!

Core Loss and Leakage
Magnetic Energy magnetic
applied to excite core flux
43

Coupled Coils
M,k

ip(t) is(t)

+ +
vp(t) Lp Ls RL vs(t)
- -

Zi
k is coupling coefficient
M is mutual inductance

di p dis dip
dis
v p $ Lp %M vs $ M % Ls
dt dt dt dt
44

Finding k and M Empirically

ip(t) is(t)

+
vp(t) Lp Ls
-

Lsc
k $ 1% whe re L sc $ primary inductance with secondary shorted
Lp

M $ k Lp Ls

45

Leakage Inductance
M,k

ip(t) is(t)

+ +
vp(t) Lp Ls RL vs(t)
- -

Zi
L p = RL 2 jBLs (1 % k 2 ) : RL
Z i ( jB ) $ ; 8 where Q p 7
Ls ;< 1 % jQ p 8
9 B Ls

• Leakage inductance represented if Qp <<1

46

Lumped Series Equivalent Test Method

Rp Rs
Primary Lp Ls Secondary

If BLs CC Rs ,
Primary Equivalent Leakage Inductance $ ( 1 % k 2 )Lp
Lp
Primary Equivalent Resistance $ R p 2 k 2
Rs
Ls

47

Empirical Shorted Load Test
ip(t) is(t)

+
5 I n (nf ) Lp
n -

Zi

AC Winding Loss . 5n Measured Winding Loss[ I n (nf )]
Average Energy . 5n Measured Average Energy[ I n (nf )]

Coil Loss . AC Winding Loss 2 DC Winding Loss

48

Core Loss Test Method Challenges
• Core magnetization is nonlinear process with
nonlinear loss effects so Fourier Decomposition
may be problematic
• Application waveshapes are difficult to generate
– Rectangular voltage
– Triangular current
– Commercial impedance test equipment is
based on sinusoidal testing at low amplitude
• Accurate loss measurement for complex
waveshapes is challenging

49

Sine Voltage Loss Measurement Challenge
Measured Loss Error from Uncompensated Phase Shift
vs Inductor Q
Sine Wave
180

160

Power Measurement Error Multiplier (x % phase error)

140

120

Q
of
10 rror
100

0c ~
e
80

au 158
se x
60

s p ph
ow ase
40

er
me error
20

as
ure
0
0 20 40 60 80 100 120

me
Q

nt
50

51
or
ci t
pa s
ca los
Empirical Open Load Test

nt te
n a ra t
so ccu en
l re a em
lle es ur
ra itat as
Pa acil me
f
ip(t)

Lp
ir(t)

Zi
+

-

Rectangular Voltage Loss Measurement Challenge

Rectangular Voltage Loss Measurement
Potential Time Delay
1.5

Time Delay
1

Voltage
Loss Impact
0.5 Current
Normalized Amplitude

0

-0.5

-1

-1.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time (Period)

52

Rectangular Voltage Loss Measurement Challenge
Loss Measurement Error vs Uncompensated Time Delay
0.5 Duty Cycle (Q ~ 31.83)
100

Rectangular Voltage Loss Measurement
80 Potential Time Delay
1.5

60

1
Loss Measurement Error (%)

40
Voltage
0.5 Current

20

0

0

0. Time Delay
05 -0.5

20 %
-20

% pe
lo r io
-1
-40

ss d
m tim
ea e
-60 -1.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

su de
Time (Period)

-80 re la
m y
en ca
-100
t e us
rro es
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

Time Delay (% of Period)
r

53

Linear System Assumption i(t)

Illustrates Loss Variation
(Function of Duty Cycle) +

Normalized Rectangular Wave Loss vs Duty Cycle v(t) R L
calculated from fixed core loss resistance
("=#=2) -
10

9

8

No es
nl pi t
d
in e
7 Reverse Voltage

ea lin
r l ea
os r
Increase
Normalized Loss

6 Fixed Voltage

s sy
Fixed ET Fixed ET

va st
ria em
5

tio
n
4 Duty Cycle
Increase
3

2

1

0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Duty Cycle

54

Modified Core Loss Estimation

55

Notions for Modified Core Loss
Calculation Method
• Characterize a given piecewise Sinusoidal Equivalent Derivative
flux excursion using a sinusoidal SED

equivalent derivative which
1

provides same peak flux and 0.8
Sinusoidal Equivalent Derivative
SED
average slope over the interval 0.6
Linear Flux Trajectory

• The resultant induced loss by this 0.4

sinusoidal equivalent derivative will

0.2

be weighted by the effective duty 0

cycle of the induced loss -0.2

• Leverage benefits of sinusoidal -0.4

empirical test methods -0.6

• Facilitate consistent and
application relevant transformer
-0.8
[1]
core loss test results
-1
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Time (Period)

– Correct peak flux density
– Consistent piecewise flux
derivative

56

Further Loss Weighting

If instantaneous loss is presumed to
Ratio of Sine Wave Loss to Square Wave Loss vs Alpha "
vary with the power of the flux
derivative [1]:
1.5

D 1.45

dB
P(t ) fixed peak flux density 6 1.4

dt 1.35

Normalized Ratio
then a further weighting factor can be 1.3

determined to scale the 1.25

measured sinusoidal loss 1.2

1.15
calorimetric

Note that for an exponent greater 1.1 calorimetric

than 1, sine wave voltage 1.05

generated loss is greater than
1

square wave voltage generated 1 1.2 1.4 1.6 1.8 2 2.2 2.4

Alpha "
loss as demonstrated historically

57

Compare Three Core Loss Estimation
Methods
• iGSE
– Improved Generalized Steinmetz Equation [1]
• FHD
– Fourier Harmonic Decomposition
• SED
– Sinusoidal Equivalent Derivative

Initial comparisons will be based on Steinmetz power
law representation as established for specific
domains of sinusoidal excitation.

58

Comparison of Core Loss Calculation Methods
Triangular Flux Shape
Fixed Equivalent Core Loss Resistance
$=#=2
Methods consistently calculate loss for linear system
Calculated Core Loss: Triangle Wave Calculated Core Loss: Quasi Triangle Wave
(normalized to peak flux equivalent sine wave at fundamental frequency) (normalized to peak flux equivalent sine wave at fundamental frequency)
("=#=2) ("=#=2)
4.5 6

4

5

3.5

3 4
Normalized Loss

Normalized Loss
2.5 iGSE iGSE
3
FHD FHD
2
SED SED

1.5 2

1

1

0.5

0 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Duty Cycle Duty Cycle

Duty Cycle
Duty Cycle

Duty Cycle = 1.0 Duty Cycle = 1.0

59

Eddy Current Shielded Lamination
$=1.5 , #=2
Calculated Core Loss: Quasi Triangle Wave
Calculated Core Loss: Triangle Wave
te
stima
("=1.5 #=2) ("=1.5 #=2)
ndere
ds U
2.5 8

Metho mpacts
ased ing I
tive B x Crowd
7

Deri va
Flu
2
6

iGSE
5

Normalized Loss
Normalized Loss

1.5 FHD iGSE

SED 4
FHD

SED
1
3

2

0.5

1

0 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Duty Cycle
Duty Cycle


60

ct
0.001 Permalloy 80 Im
pa
ing
$=1.71 , #=1.96 x Cr
ow
d

Flu
al
tenti
Calculated Core Loss: Triangle Wave Po Calculated Core Loss: Quasi Triangle Wave
("=1.71 #=1.95) ("=1.71 #=1.95)
3.5 10

9
3

8

2.5
7
iGSE
Normalized Loss

Normalized Loss
FHD 6 iGSE
2

SED 5
FHD

1.5 SED
4

3
1

2

0.5
1

0 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Duty Cycle
Duty Cycle


61

3C85 Ferrite
$=1.5 , #=2.6
Calculated Core Loss: Triangle Wave Calculated Core Loss: Quasi Triangle Wave
("=1.5 #=2.6) ("=1.5 #=2.6)
2 8

1.8
7

1.6

6
1.4
iGSE
Normalized Loss

5

Normalized Loss
1.2 FHD iGSE

1
SED FHD
4

SED
0.8
3

0.6

FH 2
0.4
D
un
de
1
0.2

res
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 tim
1
0

ate
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
s fer
Duty Cycle
rite Duty Cycle
tria
ng
ula
r lo
ss

62

Harmonic Decomposition Problem
$=1.5 , #=2.6
Sum of 1 Harmonic

Calculated Normalized Harmonic Loss: FHD 2
1.5

1
1
0.9
0.5
0.8 0
-0.5 0 2 4 6 8 10 12 14
Normalizeed to Total

0.7
-1
0.6 -1.5
-2
0.5 Presumed Significant Waveshaping Phase (radian)
0.4 with Minimal Incremental Loss
Sum of 10 Harmonics
0.3
2
0.2 1.5

1
0.1
0.5
0 0
1 2 3 4 5 6 7 8 9 10
-0.5 0 2 4 6 8 10 12 14
Harmonic Order
-1
-1.5
-2
Phase (radian)

63

Observations of Loss Methods
• iGSE and SED deliver consistent loss estimates from
Steinmetz parameters since both methods presume
instantaneous loss to be proportional to the power of the flux
derivative
• SED facilitates meaningful empirical test implementation
beyond mathematical application of Steinmetz equation:
– Evaluates core capacity for application driven flux
density
– Generates application rated average winding voltages
– Generates rated instantaneous loss over flux excursion
interval
– Facilitates resonant test methods for accurate and
consistent results in production environment

64

Evaluate Loss using Harmonic Coil
Loss Analysis and SED Core Loss
Estimating Method

65

Example Of Coil Harmonic Loss Testing
Sum of 10 Harmonics

2

1.5

1 168 W Forward Transformer 300 kHz
0.5 3 Watt Maximum Winding Dissipation
Losses By Harmonic: 16 V in
0
0 2 4 6 3.08 10 12 Losses By Layer
14
-0.5 Tabtronics: 568-6078, 16 V in,
Radian Phase (radian) 53% Duty Cycle
2.5
0.6

Loss (Watt)
0.5

2.0
0.4
AC
Loss (Watt)

0.3
0.2 DC
0.1

1.5 0.0
1 2 3 4 5 6 7

Layer Number
1.0
Calculated
0.5 Actual (5 harmonics)

0.0
DC 2 4 6 8 10
Harmonic Order
Total

66

Loss Summary
168 W Forward Converter 8 V, 21 A, 300 kHz

3.5 Total
3
AC
Power Loss (W)

2.5
Coil
2
AC
1.5 DC
1 Core
Coil
0.5
0
Loss Summary

67

Other Observed Core Loss Factors
• Fundamental variables (peak field density, frequency,
temperature)
• Biased operating application waveshape [2]
• Flux density relaxation during circuit commutation
intervals
• Flux crowding from eddy current shielding
• Designed air gap => fringing effects
• Nonuniform core cross sections => nonuniform
magnetic field densities => nonuniform loss density
• Unintended micro-crack(s)
• Mechanical Stress (magnetic anisotropy)
• Irregular or mismatched core segment interfaces
• Material manufacturing lot variances (unintended
substitutions !)

68

Expert Design & Empirical Test Strategies for Practical Transformer Development

Expert Design & Empirical Test Strategies for Practical Transformer Development

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