SKIN AND PROXIMITY EFFECTS IN TWO
PARALLEL PLATES
A thesis submitted in partial fulfillment
of the requirements for the degree of
Master of Science in Engineering
By
Hamdi Altayib Abdelbagi
B.S.EE, 2004
Wright State University
2007
Wright State University
WRIGHT STATE UNIVERSITY
SCHOOL OF GRADUATE STUDIES
August 14, 2007
I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY
SUPERVISION BY Hamdi Eltayib Abdelbagi ENTITLED
Skin and Proximity Effects in Two Parallel Plates BE ACCEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
Master of Science in Engineering
Marian K. Kazimierczuk, Ph.D.
Thesis Director
Fred D. Garber, Ph.D.
Department Chair
Committee on
Final Examination
Marian K. Kazimierczuk, Ph.D.
Gregory Kozlowski, Ph.D.
Ronald Riechers, Ph.D.
Dr. Josph F. Thomas, Jr, Ph.D.
Dean, School of Graduate Studies
Abstract
Abdelbagi, Hamdi. M.S., Department of Electrical Engineering, Wright State
University, 2007. SKIN AND PROXIMITY EFFECTS IN TWO PARALLEL PLATES
Time varying currents within winding and core conductors induce magnetic fields.
When more than one conductor is present the resultant magnetic field can be found
by adding the individual magnetic fields by superposition. The resultant magnetic
field in turn induces eddy currents within each electrical component within the vicinity of the resultant magnetic field. Eddy currents flow in the opposite direction of the
primary current and increase the resistance by reducing the area in which the primary
current has to travel. Eddy currents also reduce the effectiveness of the conductors
to conductor high frequency currents. Skin and proximity effects were numerically
investigated for two parallel plate conductors while a laminated core was designed to
reduce the power losses. Maxwell’s equations were solved to obtain analytical equations for magnetic fields eddy current distribution and power losses. These equations
were illustrated in MATLAB for various frequencies to validate the theoretical analysis. Results demonstrate current within an isolated conductor flows near the surface.
However, when the same conductor is placed near another conductor the flow path
is affected. For the case when the current is flowing in the opposite direction, the
magnetic fields are added in the area between the conductors and subtracted on the
outer side of the conductor. This causes an increase of the current density within the
conductor areas, where the conductors are close to each other. This is the proximity
effect. The anti-proximity effect occurs when two conductors carry current in the
same direction. In this case the magnetic fields are subtracted from each other in
the area between the conductors and are added to each other in the area outside the
conductors resulting in a higher current density in these areas. The eddy currents
iii
can be reduced in two ways. Using a highly resistant material for the core increases
the skin depth making the distribution of the magnetic flux more uniform. Laminating the core with an oxide film can be used to reduce the eddy current loss as well.
The study shows that the eddy current power loss in a sold core is greater than loss
in a laminated core by a factor of K 2 , where K is the number of the sheets in the
laminated core.
iv
Contents
1 Introduction
2
1.1
Research Background and Motivation of Study . . . . . . . . . . . . .
2
1.2
The Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3
Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2 Skin and Proximity Effects
5
2.1
Skin Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2
Proximity Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
3 Skin Effect in Single Rectangular Plate
9
4 Proximity and Skin Effects in Two Parallel Plates
23
5 Anti-proximity and Skin Effects in Two Parallel Plates
41
6 Laminated Cores
48
6.1
Low-Frequency Solution . . . . . . . . . . . . . . . . . . . . . . . . .
50
6.2
General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
7 Summary
7.1
72
Future Work: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
74
List of Figures
1
Skin depth δw as a function of frequency f . . . . . . . . . . . . . . .
6
2
Single isolated plate carrying current . . . . . . . . . . . . . . . . . .
9
3
Plot of 2a|H(x)|/I as a function of x/w . . . . . . . . . . . . . . . . .
10
4
Plot of the real part os 2a|H(x)|/I as a function of x/w . . . . . . . .
10
5
Plot of the imaginary part of 2a|H(x)|/I as a function of x/w . . . .
11
6
Plot of 2a|J(x)|/I as a function of x/w . . . . . . . . . . . . . . . . .
12
7
Plot of the real part of 2a|J(x)|/I as a function of x/w . . . . . . . .
13
8
Plot of the imaginary part of 2a|J(x)|/I as a function of x/w . . . . .
13
9
Plot of J(x)/Jdc as a function of x/w at selected values for w/δw . . .
14
10
Plot of the real part of J(x)/Jdc as a function of x/w at selected values
for w/δw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
Plot of of the imaginary part of J(x)/Jdc as a function of x/w at
selected values for w/δw . . . . . . . . . . . . . . . . . . . . . . . . .
12
15
Time-average skin-effect power loss 4aδw PD /bρw I 2 as a function of
w/δw at fixed w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
15
Time-average skin-effect power loss 4awPD /bρw I 2 as a function of w/δw
at fixed δw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
14
16
Time-average skin-effect energy stored in the plate 4aWm /bδw µw I 2 as
a function of w/δw . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
15
Ratio of Rw /Rwdc as a function of w/δw . . . . . . . . . . . . . . . . .
18
16
Ratio of XL /Rwdc as a function of w/δw . . . . . . . . . . . . . . . . .
19
17
Plot of |Z|/Rwdc as a function of w/δw . . . . . . . . . . . . . . . . .
20
18
Ratio of φZ as a function of w/δw . . . . . . . . . . . . . . . . . . . .
21
19
Ratio of XL /Rw as a function of w/δw . . . . . . . . . . . . . . . . .
21
20
Two plates carrying currents in opposite directions. . . . . . . . . . .
23
vi
21
Plot of a|H(x)|/I as a function of x/w for selected values of w/δw in
the left plate due to the proximity effect. . . . . . . . . . . . . . . . .
22
24
Plot of the real part of a|H(x)|/I as a function of x/w for selected
values of w/δw . The current density is given by in the left plate due to
the proximity effect. . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
24
Plot of the imaginary part of a|H(x)|/I as a functed of x/w for selection
values of w/δw . The current density is given by in the left plate due to
the proximity effect. . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
Plot of a|J(x)|/I as a function of x/w for selected values of w/δw in
the left plate due to the proximity effect. . . . . . . . . . . . . . . . .
25
30
Plot of the imaginary part of |J(x)|/Jdc as a function of x/w for selected
values of w/δw in the left plate due to the proximity effect. . . . . . .
30
29
Plot of the real part of |J(x)|/Jdc as a function of x/w for selected
values of w/δw in the left plate due to the proximity effect. . . . . . .
29
28
Plot of |J(x)|/Jdc as a function of x/w for selected values of w/δw in
the left plate due to the proximity effect. . . . . . . . . . . . . . . . .
28
27
Plot of the imaginary part of a|J(x)|/I as a function of x/w for selected
values of w/δw in the left plate due to the proximity effect. . . . . . .
27
27
Plot of the real part of a|J(x)|/I as a function of x/w for selected
values of w/δw in the left plate due to the proximity effect. . . . . . .
26
25
30
Plots of P (x)/ρw as a function of x/w for selected values of w/δw in
the left plate due to the proximity effect. . . . . . . . . . . . . . . . .
31
31
Plots of |Jsp (−w/2)/Js (−w/2)| as a function of w/δw . . . . . . . . .
32
32
Plot of |Jsp (w/2)/Js (w/2)| as a function of w/δw . . . . . . . . . . . .
32
33
Plot of the power loss due to skin and proximity effects awPsp /bρw I 2
as a function of w/δw at fixed w. . . . . . . . . . . . . . . . . . . . .
vii
33
34
Plot of the power loss due to skin and proximity effects aδw Psp /bρw I 2
as a function of w/δw at fixed δw . . . . . . . . . . . . . . . . . . . . .
33
35
Plot of the power loss due to skin effect awPs /bρw I 2 as a function of w . 34
36
Plot of the power loss due to skin effect aδw Ps /bρw I 2 as a function of
w/δw at fixed δw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
Plot of the power loss due to proximity effect awPp /bρw I 2 as a function
of w/δw at fixed w. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
34
35
Plot of the power loss due to proximity effect aδw Pp /bρw I 2 as a function
of w/δw at fixed δw . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
39
Plot of the ratio Pp /Psp as a function of w/δw at fixed w. . . . . . . .
36
40
Plot of the ratio Ps /Psp as a function of w/δw .
. . . . . . . . . . . .
36
41
Plot of the ratio Pp /Ps as a function of w/δw . . . . . . . . . . . . . .
37
42
Plot of the ratio Rw /Rwdc as a function of w/δw . . . . . . . . . . . .
37
43
Plot of XL /Rwdc as a function of w/δw . . . . . . . . . . . . . . . . .
38
44
Plot of |Z|/Rwdc as a function of w/δw .
. . . . . . . . . . . . . . . .
38
45
Plot of φZ as a function of w/δw . . . . . . . . . . . . . . . . . . . . .
39
46
Typical pattern of multi-layer inductor winding. . . . . . . . . . . . .
41
47
Two plates carrying currents in the same directions. . . . . . . . . . .
41
48
Plot of a|H(x)|/I as a function of x/w for selected values of w/δw in
the left plate due to the proximity effect. . . . . . . . . . . . . . . . .
49
43
Plot of the real part of a|H(x)|/I as a function of x/w for selected
values of w/δw . The current density is given by in the left plate due to
the proximity effect. . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
43
Plot of the imaginary part of a|H(x)|/I as a function of x/w for selected
values of w/δw . The current density is given by in the left plate due to
the proximity effect. . . . . . . . . . . . . . . . . . . . . . . . . . . .
viii
44
51
Plot of a|J(x)|/I as a function of x/w for selected values of w/δw in
the left plate due to the proximity effect. . . . . . . . . . . . . . . . .
52
44
Plot of the real part of a|J(x)|/I as a function of x/w for seleced values
of w/δw . The current density is given by in the left plate due to the
proximity effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
45
Plot of the imaginary part of a|J(x)|/I as a functed of x/w for selection
values of w/δw . The current density is given by in the left plate due to
the proximity effect. . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
45
Plot of P (x)/ρw as a function of x/w for selected values of w/δw in the
left plate due to the proximity effect. . . . . . . . . . . . . . . . . . .
46
55
Cross section of single lamination used to analyzing eddy-current loss.
48
56
Distribution of the envelope of the amplitude of magnetic field intensity
H(x)/Hm and the eddy density J(x). (a) For a single solid core at
w = 8δc . (b) For laminated core at w/2δc . . . . . . . . . . . . . . . .
49
57
Plot of |H(x)|/Hm as a function of x/w for selected values of w/δc . .
55
58
The real part of |H(x)|/Hm as a function of x/w for selected values of
w/δc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
56
The imaginary part of |H(x)|/Hm as a functed of x/w for selected
values of w/δc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
60
Plot of δc |J(x)|/Hm as a function of x/w for selected values of w/δc . .
57
61
2
Plot of δc2 Pe (x/ρc Hm
) as a function of x/w for selected values of w/δc .
58
62
2
Plot of δc PE ρc Hm
hlc as a function of w/δw . . . . . . . . . . . . . . . .
59
63
2
Plot of δc Pe ρc Hm
hlc as a function of w/δc . . . . . . . . . . . . . . . .
61
64
Plot of Rc /Rcdc as a function of w/δc . . . . . . . . . . . . . . . . . . .
62
65
Plot of XL /Rcdc as a function of w/δc . . . . . . . . . . . . . . . . . .
62
66
Plot of |Z|/Rcdc as a function of w/δc .
63
ix
. . . . . . . . . . . . . . . . .
67
Plot of φz as a function of w/δc . . . . . . . . . . . . . . . . . . . . . .
65
68
Plot of XL /Rc as a function of w/δc . . . . . . . . . . . . . . . . . . .
65
69
Plot of R/ωLo as a function of w/δc . . . . . . . . . . . . . . . . . . .
66
70
Plot of XL /ωLo as a function of w/δc . . . . . . . . . . . . . . . . . . .
67
71
Plot of L/Lo as a function of w/δc . . . . . . . . . . . . . . . . . . . .
67
72
Plot of |Z|/ωLo as a function of w/δc . . . . . . . . . . . . . . . . . . .
68
73
Plot of φz as a function of w/δc . . . . . . . . . . . . . . . . . . . . . .
68
74
Plot of Rp /ωLo as a function of w/δc . . . . . . . . . . . . . . . . . . .
69
75
Plot of Xp /ωLo as a function of w/δc . . . . . . . . . . . . . . . . . . .
69
x
Acknowledgements
I would like to thank my advisor, Dr. Marian K. Kazimierczuk, for his guidance and
input on the thesis development process.
I also wish to thank Dr. Gregory Kozlowski and Dr. Ronald Riechers for serving
as members of my MS thesis defense committee, giving the constructive criticism
necessary to produce a quality technical research document.
I would also like to thank the Department of Electrical Engineering and Dr. Fred D.
Garber, the Department Chair, for giving me the opportunity to obtain my MS degree
at Wright State University.
I would also like to thank my family, my friends, and Alhamduleelah Wa Alsalat
Wa Alsalam Ala Rsoul Allah.
1
2
1
1.1
Introduction
Research Background and Motivation of Study
In many electrical devises such as inductors and transformers,there are power losses
because electricity converts to heat in their wires. This is one of the main reasons
why many devices get hot when they are in use. Studying how the change from
electricity to heat occurs and how to reduce the amount of heat released is necessary
in order to conserve energy and enhance the operation of electric devices. In power
engineering, a conductor is a piece of metal used to conduct electricity, known colloquially as an electrical wire. Conductors have different shapes and sizes. In the
United States, smaller conductors are measured by American wire gauges, and large
conductors are measured by circular mils. The metal commonly used for conductors
is copper because it has high conductivity. Silver is more conductive than copper,
but due to cost, it is not practical in most cases and copper is still the most common
choice for light-gauge wire. Compared to copper, aluminium has lower conductivity
per unit volume, but better conductivity per unit wight. For this reason aluminium is
commonly used for large-scale power distribution conductors such as overhead power
lines.
However, although many metals have good conductivity, they loose power by converting energy into heat at high frequencies. A conductor and core conductor that carry
time-varying currents produce a magnetic field because of its own current, and also
it produces another magnetic field because of the currents in adjacent conductors.
According to linz’s law these magnetic fields produce eddy-current that is opposite
to the original current. There are two kinds of eddy-current effects: skin effect and
proximity effect. Both of these effects cause nonuniform current density in conductors
at high frequencies. The eddy currents flow in the center of the conductor and in the
opposite direction to the original current. As a result, the original current tends to
3
flow near the surface and its density decreases from the surface to the center and
increases the (dc) resistance at high frequencies by restricting the conducting area
of the wire to the thin skin surface. This is called the skin effect. The proximity
effect is similar to the skin effect, but it is caused by the current carried by nearby
conductors.
This thesis will focus on the influence of the skin and proximity effects on rectangular
conductors and using the lamination method to reduce these effects. The fundamental laws and units of the magnetic theory will be reviewed, magnetic relationships
will be given, the equations of the conductors will be derived, and eddy-current losses
will be studied. Solutions to reduce the power losses using certain materials on the
cores and the lamination method will be provided. Each topic will be discussed using
equations, figures, and MATLAB graphs to give a better understanding of the thesis’s
objectives.
1.2
The Objective
The objectives of this thesis are as follows.
(1) To review and study the skin and proximity effects on the conductors at high
frequencies.
(2) To analyze and study the influence of the skin and proximity effects on rectangular conductors, and to interpret high-frequency behaviors.
(4) To analyze the lamination core to in order reduce the eddy-current power loss.
(5) To gain a deep understanding of high-frequency behaviors to obtain a better design in the future to reduce the eddy-current power loss.
4
1.3
Thesis Outline
The skin and proximity effects are introduced in Chapter 2, and will be explained
when and how they occur and how they effect the power loss. Also the skin depth is
briefly discussed. In Chapter 3, a deep study of the skin effect on a single rectangular
plate is presented, and will use MATLAB simulation to get accurate values and make
a better understanding of high-frequency behaviors. The same techniques in Chapter
3 are used in Chapters 4, and 5 to analyze and study proximity and anti-proximity
and skin effects in two parallel rectangular plates. In Chapter 6, two solutions are
studied to reduce the eddy-current power loss. A high resistivity core and a laminated
core are used for this solution. A summary and future work follow in Chapter 7.
5
2
Skin and Proximity Effects
In order to provide a better understanding of this thesis, skin and proximity effects will
be discussed briefly. A winding conductor and core conductor that carry time varying
current experience the magnetic field due to its own current and also the magnetic
field due to all current carrying conductors in vicinity. In turns, these magnetic fields
induce eddy currents in conductors. They oppose the penetration of the conductor by
the magnetic field and convert energy into heat. There are two kinds of eddy current
effect: skin effect and proximity effect. Both these effects cause non-uniform current
density in conductors at high frequencies. These are high frequency phenomena and
limit the ability of conductor to conduct high frequency currents. The skin effect and
proximity effect are orthogonal to each other and can be considered separately.
2.1
Skin Effect
A time varying currents generate magnetic field that in turn generate currents. The
skin effect arises because of the conductor carrying time varying current is immersed
in its own magnetic field which, causes eddy current in conductor itself. In accordance
with lenz’s law, the eddy currents produce a secondary magnetic field that opposes
the primary magnetic field. When a time varying AC current flows in conductor,
the magnetic field is induced in the conductor by its own current, which causes extra
circulating current in the conductor. As a result, the current tends to flow near the
surface and the current density tends to decrease from the surface to the center of
the conductor. This is called skin effect. As the frequency increases, the conductor effective resistance increases and so does the power loss. At low frequencies, the
current takes the path of lowest resistance. At high frequencies, the current takes
the path of the lowest inductance. The winding power loss also called the copper
loss, is caused by the current flow through the winding resistance. At dc and low
6
4
10
3
δw(mm)
10
2
10
1
10
0
10
−3
10
−2
10
−1
0
10
1
10
10
2
10
f(Hz)
Figure 1: Skin depth δw as a function of frequency f .
frequencies, the current density J is uniformly distributed throughout the conductor
cross-sectional area. when a conducting material is subjected to an alternating magnetic field, eddy current are induced in it. At high frequencies, the current density J
becomes nonuniform due to eddy currents caused by two mechanisms: the skin effect
and proximity effect. Neither the electric field E nor the magnetic field H penetrate
far into a conductor. The point where these fields are reduced by factor 1/e is called
skin depth δw
δw =
2.2
s
2
1
=√
=
ωµσ
πµσf
s
ρw
=
πµf
s
ρw
.
πµr µof
(1)
Proximity Effect
A time varying current i in one conductor generates both external and internal time
varying magnetic field H1 . This field in turn induced a time varying current i2 known
as eddy current in nearby conductors, causing power loss. Proximity or closeness of
7
other current carrying conductor affects the ability of the conductor to carry high
frequency current. The magnetic field induced by conductors in close proximity will
add or subtract depending in their directions. The proximity effect in inductors
and transformers is caused by the time varying magnetic field arising from currents
flowing in adjacent winding layers in multiple layer winding. Currents flow in two
opposite directions in the same conductor of multiple layer winding, except for the
first layer. As a result, the amplitudes and rms values of eddy currents caused by
magnetic fields in the adjacent layers due to proximity effect increase significantly as
the number of layers Nl increases. Therefore, the power loss due to proximity effect in
multiple layer windings is much higher than the power loss in skin effect . In general,
the proximity effect occurs when the current in nearby conductors causes a time
varying magnetic field and induces a circulating current inside the conductor. The
proximity effect is similar to the skin effect, but the difference is that the proximity
effect is caused by the current carried by nearby conductors. In the other words,
the proximity effect causes magnetic fields due to high frequency currents in one
conductor to induce voltages in adjacent winding layers in multi-layer inductor and
transformers. Each conductor is subjected to its own field and the fields generated by
other conductors. Eddy currents are induced in a conductor by time varying magnetic
field whether or not the conductor carries current. If the conductor carries current,
the skin effect eddy current and the proximity effect eddy current superimpose to
form the total eddy current. If the conductor does not carry current, then only the
proximity effect affect eddy current induced. The skin effect current and the proximity
effect current are orthogonal. The current density due to the skin effect exhibits an
even symmetry and the current density due to the proximity effect exhibits an odd
symmetry. The proximity effect causes nonuniform current density in the cross section
of the conductors, increasing significantly the winding loss at high frequencies. When
8
two or more conductors are brought into close proximity, their magnetic fields may add
or subtract. The high frequency current will concentrate within a conductor, where
the magnetic fields are additive. The magnitude of the proximity effect depend on
(1)frequency,(2)conductor geometry (shape and size),(3)arrangement of conductors,
and (4)spacing. Mathematically, the proximity effect is very complex.
9
3
Skin Effect in Single Rectangular Plate
y
a
b
J
H
H
w
2
0
w
2
x
z
Figure 2: Single isolated plate carrying current
Fig 2 shows a single conducting plate. A sinusoidal current i(t) = Icoswt flows
through the plate in −y-direction, causing a magnetic field in side and outside the
plate in the y − z plate in the z direction for z > 0 and in −z-direction for x <
0. Invoking the odd symmetry of the magnetic field, H(x) = −H(−x) and using
Ampere’s circuital law
I
c
H . dI = I
(2)
for the magnetic field intensity on the plate surface, we get
w
w
aH( ) + aH(− ) = 1.
2
2
(3)
Assuming that w << a and w << b, we obtain the amplitude of the magnetic field
intensity at the surface of the plate
w
w
1
H( ) = −H(− ) =
= Hm .
2
2
2a
(4)
The magnetic field intensity H outside the conductor is assumed to be uniform in
10
1
0.9
w /δw = 1
0.8
2a |H(x)| / I
0.7
5
0.6
0.5
0.4
10
0.3
0.2
20
0.1
0
−0.5
0
x/w
0.5
Figure 3: Plot of 2a|H(x)|/I as a function of x/w
1
0.8
w /δw = 1
0.6
10
0.4
Re{(2aH(x) / I )}
5
0.2
20
0
−0.2
−0.4
−0.6
−0.8
−1
−0.5
0
x/w
0.5
Figure 4: Plot of the real part os 2a|H(x)|/I as a function of x/w
11
0.4
5
0.3
10
0.2
Im{(2aH(x) / I )}
20
0.1
w /δw = 1
0
−0.1
−0.2
−0.3
−0.4
−0.5
0
x/w
0.5
Figure 5: Plot of the imaginary part of 2a|H(x)|/I as a function of x/w
the y − z plane and parallel to the current sheet. The magnitude of this field changes
only in the x-axis. Therefore, one-dimensional solution is sufficient. The magnetic
field is described by the ordinary second-order differential Helmholtz equation,
d2 H(x)
= jwµw σw H(x) = γ 2 H(x)
dx2
(5)
whose general equation is
H(x) = H1 eγx + H2 e−γx .
(6)
Form the boundary conditions,
w
w
w
H( ) = H1 eγ 2 + H2 e−γ 2
2
(7)
w
w
w
H(− ) = H1 e−γ 2 + H2 eγ 2 .
2
(8)
and
Adding these two equations, we obtain
w
w
H( ) + H(− ) = (H1 + H2 )(eγx + e−γx ) = 0
2
2
(9)
12
2.5
w /δ = 1
2aδw|J(x)|/ I
2
w
1.5
2
1
3
0.5
5
0
−0.5
10
0
x/w
0.5
Figure 6: Plot of 2a|J(x)|/I as a function of x/w
which produces
H1 = −H2 =
Hm
.
2
(10)
Substitute (10) into (7), we get
w
w
w
w
w
(e−γ 2 − e−γ 2 )
w
H( ) = H1 (e−γ 2 − e−γ 2 ) = 2H1
= 2H1 sinh(γ )
2
2
2
(11)
13
0.5
10
0
5
w
Re{2aδ J(x)} / I
−0.5
3
2
−1
−1.5
−2
−2.5
−0.5
w /δw = 1
0
x/w
0.5
Figure 7: Plot of the real part of 2a|J(x)|/I as a function of x/w
0.4
2
3
0.2
w /δw = 1
5
10
w
Im{2aδ J(x)} / I
0
−0.2
−0.4
−0.6
−0.8
−1
−1.2
−0.5
0
x/w
0.5
Figure 8: Plot of the imaginary part of 2a|J(x)|/I as a function of x/w
14
8
7
w /δ = 10
w
6
|J(x)| / Jdc
5
4
5
3
2
3
1
1
0
−0.5
0
x/w
0.5
Figure 9: Plot of J(x)/Jdc as a function of x/w at selected values for w/δw
1
10
0
5
3
Re{J(x) / Jdc}
−1
w /δw = 1
−2
−3
−4
−5
−0.5
0
x/w
0.5
Figure 10: Plot of the real part of J(x)/Jdc as a function of x/w at selected values
for w/δw
15
2
10
1
5
w /δw = 1
0
Im{J(x) / Jdc}
3
−1
−2
−3
−4
−5
−0.5
0
x/w
0.5
Figure 11: Plot of of the imaginary part of J(x)/Jdc as a function of x/w at selected
values for w/δw
5
10
4
10
3
4a w PD /bρw I2
10
2
10
1
10
0
10
−1
10
−2
10
−2
10
−1
10
0
10
w/δw
1
10
2
10
Figure 12: Time-average skin-effect power loss 4awPD /bρw I 2 as a function of w/δw
at fixed δw
16
3
10
2
4aδ w PD /bρw I2
10
1
10
0
10
−1
10
−2
10
−1
10
0
10
w/δw
1
10
2
10
Figure 13: Time-average skin-effect power loss 4aδw PD /bρw I 2 as a function of w/δw
at fixed w
1
4a Wm /bµwδwI2
10
0
10
−1
10
−1
10
0
10
w/δw
1
10
Figure 14: Time-average skin-effect energy stored in the plate 4aWm /bδw µw I 2 as a
function of w/δw
17
from which
H( w2 )
I
1
H1 = −H2 =
.
w =
2sinh(γ 2 )
2a 2sinh(γ w2 )
(12)
Hence, the magnetic field intensity distribution inside the plate is
w
w
x
sinh(γx)
(eγ 2 − e−γ 2 )
I sinh[(1 + j)( δw )]
H(x) = 2H1
= Hm
=
2
sinh(γ w2 )
2a sinh[ (1+j)
( δww )]
2
w
x
I sinh[(1 + j)( δw )( w )]
=
2a sinh[ (1+j)
( δw )]
2
(13)
w
Fig. 3, 4, and 5 show plots of 2aδw |H(x)|/I and its real and imaginary parts as
function of x/w for selected values of w/δw .
The current density in the plate is
J(x) = Jy (x) = −
dH
cosh(γx)
(1 + j)Hm cosh(γx)
= −γHm
w = −
dx
sinh(γ 2 )
δw
sinh(γ w2 )
w
x
(1 + j)I cosh[(1 + j)( δw )( w )]
.
=−
2aδw
sinh[ (1+j) ( w )]
(14)
w
w
I cosh( δw ) + cos( δw )
.
2aδw cosh( δww ) − cos( δww )
(15)
2
δw
Hence,
|J(x)| =
Fig. 6, 7, and 8 show plots of 2aδw |J(x)|/I and its real and imaginary parts as
function of x/w for selected values of w/δw .
The dc current density in the plate is
Jdc =
I
.
aw
(16)
Hence, the normalized current density is
w
x
J(x)
(1 + j) w cosh[(1 + j)( δw )( w )]
=−
( )
.
Jdc
2
δw
sinh[ (1+j)
( δw )]
2
(17)
w
Fig. 9, 10, and 11 show the normalized current density |J(x)|/Jdc and its real and
imaginary parts as a function of x/w for selected values of w/δc . That shows the Ac
current almost equal to DC current at low frequencies, and the ratio between the Ac
18
2
Rw / Rwdc
10
1
10
0
10
−1
10
0
1
10
w/δ
10
2
10
c
Figure 15: Ratio of Rw /Rwdc as a function of w/δw
current and DC current increases by increasing the frequency.
The time-average skin-effect power loss at a constant frequency is
w
w
w
1Z aZ bZ 2
ρw bI 2 sinh( δw ) + sin( δw )
2
.
PD =
ρw |J(x)| dxdydz =
2 0 0 − w2
4aδw cosh( δww ) − cos( δww )
(18)
and for varying frequencies is
ρw bI 2 sinh( δww ) + sin( δww )
PD =
.
4aδw cosh( δww ) − cos( δww )
(19)
Fig. 12 shows the time-average skin-effect power loss 4aδw PD /ρw bI 2 as a function of
w/δw .
Fig 13 shows the time-average skin-effect power loss 4awPD /ρw bI 2 as a function of
w/δw .
For high frequencies, w/δw >>1 and values of the hyperbolic are much larger than
the values of the trigonometric functions
ρw bI 2
ρw bI 2
PD ≈
=
4aδw
4a
s
πµw f
bI 2 q
πµw f ρw .
=
ρw
4a
(20)
19
2
10
1
10
XL / Rwdc
0
10
−1
10
−2
10
−3
10
−1
0
10
10
1
w/δ
2
10
10
w
Figure 16: Ratio of XL /Rwdc as a function of w/δw
The time-average magnetic energy stored in the plate is
1
Wm =
2
=
abµw
2
Z
µw
µw |H| dV =
2
v
Z Z Z
w
2
−w
2
|
2
Z
a
0
Z
0
b
Z
w
2
−w
2
ρw |H|2 dxdydz
w
w
I sinh(γx) 2
µw bδw I 2 sinh( δw ) − sin( δw )
|
dx
=
.
2a sinh(γ w2 )
4a cosh( δww ) − cos( δww )
(21)
Fig. 14 shows the normalized magnetic energy as a function ofw/δw .
The power loss in the plate at any frequency can be expressed as
w
w
1 2
1 2 ρw b sinh( δw ) + sin( δw )
PD = I Rw = I
2
2 2aδw cosh( δww ) − cos( δww )
(22)
yielding the resistance at any frequency
Rw =
w
w
ρw b w sinh( δw ) + sin( δw )
2awδw δw cosh( δww ) − cos( δww )
(23)
The dc resistance of the plate is
Rwdc = ρw
b
.
aw
(24)
The ratio of the resistance at any frequency to the dc resistance is
FR =
w
w
Rw
1 w sinh( δw ) + sin( δw )
=
Rwdc
2 δw cosh( δww ) − cos( δww )
(25)
20
2
| Z | / Rwdc
10
1
10
0
10
−1
10
0
10
1
w/δ
10
2
10
w
Figure 17: Plot of |Z|/Rwdc as a function of w/δw
Fig. 15 shows the ratio of Rw /Rwdc as a Function of w/δw .
Assuming that the current flows uniformly over the skin depth δw only near both
surfaces of the effective area Ae = 2δw a, the plate ac resistance at hight frequencies
is
Rw ≈
ρw b
bq
≈
πµw ρw f
2aδ
2a
Rw
w
b
FR =
=
=
Rwdc
2δw
2
s
πµ0 f
.
ρw
(26)
(27)
The magnetic energy stored inside the plate can be expressed as
w
w
1 2 µw bδw I 2 sinh( δw ) − sin( δw )
LI =
2
4a cosh( δww ) − cos( δww )
(28)
yielding the internal inductance
w
w
XL
µw bδw sinh( δw ) − sin( δw )
L=
=
w
2a cosh( δww ) − cos( δww )
(29)
The reactance XL due to the internal inductance L normalized with respect to the
21
50
45
40
35
φZ (°)
30
25
20
15
10
5
0
−1
10
0
1
10
w/δ
2
10
10
w
Figure 18: Ratio of φZ as a function of w/δw
1
10
0
XL/Rw
10
−1
10
−2
10
−3
10
−1
0
10
10
1
w/δ
10
2
10
w
Figure 19: Ratio of XL /Rw as a function of w/δw
dc resistance Rwdc is given by
w
w
XL
1 w sinh( δw ) − sin( δw )
FX =
=
.
Rwdc
2 δw cosh( δww ) − cos( δww )
(30)
22
Fig. 16 shows the ratio XL /Rwdc as a function of w/δw . The total internal impedance
is
w
w
sinh( δww ) − sin( δww )
1
w sinh( δw ) − sin( δw )
Z = Rwdc ( )
+j
2
δw cosh( δww ) − cos( δww )
cosh( δww ) − cos( δww )
"
#
= Rwdc + jXL = |Z|ejφz
(31)
Figs. 17 and 18 show |Z|/Rwdc and φz as function of w/δw .
Fig. 19 shows the ratio of XL /Rw . An alternative method of deriving an expression
for the plate impedance is given below. The electric field intensity is given by
E = ρw J = ρw
dH
ρw γI cosh(γx)
.
=−
dx
2a sin(γ w2 )
(32)
The plate impedance is
bE w2
V
bρw γ
w
bρw
w
Z=
=
=
coth(γ ) =
(1 + j)coth(γ )
I
I
2a
2
2aδw
2
1
w
w
= Rwdc ( )(1 + j)coth(γ ) = R + jXL .
2
δw
2
(33)
Using the relation
coth(1 + j)x =
sinh2x − jsin2x
cosh2x − cos2x
we obtain the same result in equation (30)
(34)
23
4
Proximity and Skin Effects in Two Parallel Plates
y
a
a
b
b
J
J
H
H
w
2
0
w
2
w
x
z
Figure 20: Two plates carrying currents in opposite directions.
Fig. 20 shows two parallel rectangular conducting plates. Considering the case in
which the currents i(t) = Icoswt flow in the plates in opposite directions. The current
in the left plate flows in the −y-directions and in the right plate in the y−direction.
The currents cause the magnetic field inside and outside the plates. The magnetic
fields generated by both plates add up between the plates, causing larger current
density j(x) in the plate areas, where the plates are close to each others. Using
Ampere’s circuital law,
I
c
H . dI = I
(35)
we have
w
w
w
aH( ) − aH(− ) = aH( ) = 1
2
2
2
(36)
w
H(− ) = 0.
2
(37)
w
1
Hm = H( ) = .
2
a
(38)
because
thus,
24
1
0.9
0.8
w /δ = 1
w
0.7
a |H(x)| / I
2
0.6
3
0.5
0.4
5
0.3
0.2
10
0.1
0
−0.5
0
x/w
0.5
Figure 21: Plot of a|H(x)|/I as a function of x/w for selected values of w/δw in the
left plate due to the proximity effect.
1.2
1
w /δw = 1
Re{ a |H(x)| / I }
0.8
2
0.6
3
0.4
0.2
5
0
−0.2
−0.5
10
0
x/w
0.5
Figure 22: Plot of the real part of a|H(x)|/I as a function of x/w for selected values
of w/δw . The current density is given by in the left plate due to the proximity effect.
The magnetic field inside the plates is described by the Helmholtz equation
25
0.05
10
0
−0.05
5
Im{ a |H(x)| / I}
w /δ = 1
w
−0.1
−0.15
−0.2
3
−0.25
2
−0.3
−0.35
−0.5
0
x/w
0.5
Figure 23: Plot of the imaginary part of a|H(x)|/I as a functed of x/w for selection
values of w/δw . The current density is given by in the left plate due to the proximity
effect.
d2 H(x)
= jwµw σw H(x) = γ 2 H(x).
dx2
(39)
H(x) = H1 eγx + H2 e−γx .
(40)
whose general solution is
From the boundary conditions,
w
w
w
1
H( ) = H1 eγ 2 + H2 e−γ 2 =
2
a
(41)
w
w
w
H(− ) = H1 e−γ 2 + H2 eγ 2 = 0.
2
(42)
and
hence,
H2 = −H1 e−γw
(43)
producing
w
1
eγ 2
H1 =
a eγw − e−γw
(44)
26
and
w
1 e−γ 2
H2 = − γw
a e − e−γw
(45)
The magnetic field is given by
H(x) =
I sinh[γ(x + w2 )]
.
a sinh(γw)
(46)
Fig. 21, 22 and 23 shows plots of a|H(x)|/I and its real and imaginary parts as a
function of x/w for selection values of w/δw .
The current density is given by
J(x) = −
dH(x)
γI cosh[γ(x + w2 )]
=−
.
dx
a
sinh(γw)
(47)
Fig. 24, 25 and 26 show plots of a|(Jx)|/I and its real and imaginary parts as a
function of x/w for selection values of w/δw .
The dc current density in each plate is
Jdc =
I
.
aw
(48)
Hence, the normalized current density in the plate is
cosh[γ(x + w2 )]
w cosh[γ(x + w2 )]
J(x)
= wγ −
= −(1 + j)( )
.
Jdc
sinh(γw)
δw
sinh(γw)
(49)
Fig. 27, 28 and 29 show normalized current density of |J(x)|/Jdc and its real and
imaginary parts as a function of x/w for selection values of w/δw . The current density
due to skin and proximity effects at the surfaces of the conductor for x = w/2 and
x = −w/2 is given by
Fig. 30 shows plot of P (x)/ρw for selection values of w/δw in the left plate due to the
proximity effect.
w
γI cosh(γw)
Jsp ( ) = −
.
2
a sinh(γw)
(50)
27
1.5
w /δw = 1
|J(x)| / I
1
w
2
aδ
3
0.5
5
10
0
−0.5
0
x/w
0.5
Figure 24: Plot of a|J(x)|/I as a function of x/w for selected values of w/δw in the
left plate due to the proximity effect.
1.2
w /δ = 1
w
1
2
3
0.6
Re{aδ
w
|J(x)| / I }
0.8
5
0.4
0.2
10
0
−0.2
−0.5
0
x/w
0.5
Figure 25: Plot of the real part of a|J(x)|/I as a function of x/w for selected values
of w/δw in the left plate due to the proximity effect.
28
1.2
1
Im{aδ
w
|J(x)| / I }
0.8
0.6
2
0.4
3
0.2
w /δ = 1
w
0
5
10
−0.2
−0.4
−0.5
0
x/w
0.5
Figure 26: Plot of the imaginary part of a|J(x)|/I as a function of x/w for selected
values of w/δw in the left plate due to the proximity effect.
and
γI
1
w
.
Jsp (− ) = −
2
a sinh(γw)
(51)
The current density due to the skin effect only for x = w/2 and x = 0 is
w
γI cosh(γ w2 )
Js ( ) = −
2
2a sinh(γ w2 )
(52)
and
Js (0) = Jsp (0) = −
γI
1
.
2a sinh(γ w2 )
(53)
Hence,
Jsp (− w2 )
1
=
w
Js (− 2 )
1 + sinh2 (γ w2 )
(54)
Jsp ( w2 )
w
= 1 + tanh2 (γ )
w
Js ( 2 )
2
(55)
Jsp (0)
= 1.
Js (0)
(56)
and
29
8
7
6
|J(x)| / Jdc
5
w /δ = 5
w
4
3
3
2
2
1
1
0
−0.5
0
x/w
0.5
Figure 27: Plot of |J(x)|/Jdc as a function of x/w for selected values of w/δw in the
left plate due to the proximity effect.
Fig. 31 and 32 show plots of|Jsp (−w/2)|/Js(−w/2) and |Jsp (w/2)|/Js(w/2) as function of w/δw . The power density distribution is given by
1
1 ρw I 2
P (x) = ρw |J(x)|2 =
2
2 a2
γcosh[γ(x + w )] 2
2 .
sinh(γw)
(57)
The total power loss in each conductor due to the proximity and skin effects at a
constant frequency is
Psp
1Z Z Z
ρw Z a Z a Z f racw2 I 2 cosh[γ(x + w2 )] 2
=
ρw |J(x)| dV =
γ
dxdydz
2
2 0 0 −f racw2 a2 sinh(γw) v
=
2w
2w
ρw bI 2 sinh( δw ) + sin( δw )
.
2aδw cosh( 2w
) − cos( 2w
)
δw
δw
(58)
and at varying frequencies is
Psp
δw
δw
) + sin( 2w
)
ρw bI 2 sinh( 2w
=
δw
δw .
2aδw cosh( 2w ) − cos( 2w )
(59)
30
1
5
0
3
w /δw = 1
2
Re{|J(x)| / J
dc
}
−1
−2
−3
−4
−5
−0.5
0
x/w
0.5
Figure 28: Plot of the real part of |J(x)|/Jdc as a function of x/w for selected values
of w/δw in the left plate due to the proximity effect.
2
1
Im{|J(x)| / Jdc }
0
w /δw = 1
−1
2
−2
3
−3
5
−4
−5
−0.5
0
x/w
0.5
Figure 29: Plot of the imaginary part of |J(x)|/Jdc as a function of x/w for selected
values of w/δw in the left plate due to the proximity effect.
31
Fig. 33 shows the power loss due to skin and proximity effect awPsp /bρw I 2 as a
function of w/δw .The plot shows that when the frequency increases, the power loss
due to skin and proximity effects also increases.
Fig. 34 shows the power loss due to skin and proximity effect aδw Psp /bρw I 2 as a
function of w/δw . The plot shows that when the width of the plate increases, the
power loss due to skin and proximity effects decreases.
The power due to the skin effect at a constant frequency is the same at that for a
single plate and it is given by
Ps =
2w
2w
ρw bI 2 sinh( δw ) + sin( δw )
2w .
4aδw cosh( 2w
)
−
cos(
)
δw
δw
(60)
δw
δw
) + sin( 2w
)
ρw bI 2 sinh( 2w
δw
δw .
4aδw cosh( 2w ) − cos( 2w )
(61)
and for varying frequencies is
Ps =
1
0.9
0.8
w /δw = 1
0.7
0.6
P(x) /ρ
w
2
0.5
0.4
0.3
5
0.2
10
0.1
0
−0.5
0
x/w
0.5
Figure 30: Plots of P (x)/ρw as a function of x/w for selected values of w/δw in the
left plate due to the proximity effect.
32
1
0.9
0.7
0.6
s
|J (−w / 2) /J (−w / 2)|
0.8
0.5
sp
0.4
0.3
0.2
0.1
0
0
1
2
3
4
w/δw
5
6
7
8
Figure 31: Plots of |Jsp (−w/2)/Js (−w/2)| as a function of w/δw .
2
sp
s
|J (w / 2) /J (w / 2)|
2.5
1.5
1
0
1
2
3
4
w/δ
5
6
7
8
w
Figure 32: Plot of |Jsp (w/2)/Js(w/2)| as a function of w/δw .
The proximity loss is:
Pp = Pps − Ps =
δw
δw
) − sin( 2w
)
ρw bI 2 sinh( 2w
δw
δw .
4aδw cosh( 2w ) + cos( 2w )
(62)
33
5
10
4
10
3
10
a w Psp / ρw b I
2
2
10
1
10
0
10
−1
10
−2
10
−3
10
−2
10
−1
10
0
10
w/δ
1
10
2
10
w
Figure 33: Plot of the power loss due to skin and proximity effects awPsp /bρw I 2 as a
function of w/δw at fixed w.
6
5
aδw Psp / ρw b I
2
4
3
2
1
0
−1
10
0
10
w/δ
1
10
w
Figure 34: Plot of the power loss due to skin and proximity effects aδw Psp /bρw I 2 as
a function of w/δw at fixed δw .
34
4
10
3
10
2
1
10
s
w
awP /ρ bI
2
10
0
10
−1
10
−2
10
−3
10
−2
10
−1
10
0
10
w/δ
1
10
2
10
w
Figure 35: Plot of the power loss due to skin effect awPs /bρw I 2 as a function of w .
6
5
3
w
s
w
aδ P / ρ b I2
4
2
1
0
−1
10
0
10
w/δ
1
10
w
Figure 36: Plot of the power loss due to skin effect aδw Ps /bρw I 2 as a function of w/δw
at fixed δw .
35
5
10
4
10
3
10
a w Pp / ρw b I
2
2
10
1
10
0
10
−1
10
−2
10
−3
10
−2
10
−1
10
0
10
w/δ
1
10
2
10
w
Figure 37: Plot of the power loss due to proximity effect awPp /bρw I 2 as a function
of w/δw at fixed w.
0.35
0.3
0.2
w
p
w
aδ P / ρ b I2
0.25
0.15
0.1
0.05
0
−1
10
0
10
w/δ
1
10
w
Figure 38: Plot of the power loss due to proximity effect aδw Pp /bρw I 2 as a function
of w/δw at fixed δw .
36
0.38
0.36
0.34
Pp/Psp
0.32
0.3
0.28
0.26
0.24
0.22
−2
10
−1
10
0
10
w/δ
1
10
2
10
w
Figure 39: Plot of the ratio Pp /Psp as a function of w/δw at fixed w.
−0.3
10
−0.4
Ps/Psp
10
−0.5
10
−0.6
10
−2
10
−1
10
0
10
w/δ
1
10
2
10
w
Figure 40: Plot of the ratio Ps /Psp as a function of w/δw .
Fig. 35 shows a plot of awPs /ρw bI 2 as a function of w/δw .The plot shows that
when the frequency increases, the power loss due to the skin effect also increases.
37
3
2.5
p
P /P
s
2
1.5
1
0.5
−2
10
−1
10
0
10
w/δ
1
10
2
10
w
Figure 41: Plot of the ratio Pp /Ps as a function of w/δw .
1
R
w
/R
wdc
10
0
10
−1
10
0
10
w/δ
1
10
w
Figure 42: Plot of the ratio Rw /Rwdc as a function of w/δw .
Fig. 36 shows a plot of aδw Ps /ρw bI 2 as a function of w/δw .The plot shows that when
the width of the plate increases, the power loss due to the skin effect decreases.
38
1
10
0
X /R
wdc
10
−1
L
10
−2
10
−3
10
−1
10
0
10
w/δ
1
10
w
Figure 43: Plot of XL /Rwdc as a function of w/δw .
2
| Z | /R
wdc
10
1
10
0
10
−1
10
0
10
w/δ
1
10
w
Figure 44: Plot of |Z|/Rwdc as a function of w/δw .
Fig. 37 shows the power loss due to proximity effect awPp /ρw bI 2 as a function of
39
50
45
40
35
φZ (°)
30
25
20
15
10
5
0
−1
10
0
10
w/δ
1
10
w
Figure 45: Plot of φZ as a function of w/δw .
w/δw .The plot shows that when the frequency increases, the power loss due to the
proximity effect also increases.
Fig. 38 shows the power loss due to proximity effect aδw Pp /ρw bI 2 as a function of
w/δw .The plot shows that when the width of the plate increases, the power loss due
to the proximity effect decreases.
Fig. 39 shows the ratio of Pp /Psp as a function of w/δw .
Fig. 40 shows the ratio of Ps /Psp as a function of w/δw .
Fig. 41 The ratio of Pp /Ps as a function of w/δw .
The power loss in each plate can be expressed as
Psp
2w
2w
1 2
ρw bI 2 sinh( δw ) + sin( δw )
= I Rw =
.
2
2aδw cosh( 2w
) − cos( 2w
)
δw
δw
(63)
Thus, the normalized resistance of each plate is
FR =
2w
2w
Rw
w sinh( δw ) + sin( δw )
=( )
.
Rwdc
δw cosh( 2w
) − cos( 2w
)
δw
δw
(64)
40
The skin and proximity effects are orthogonal. It can be shown that
Z Z
s
Js Jp = 0.
(65)
The electric field intensity is
dH
ρw γI cosh[γ(x + w2 ]
E = ρw J = −ρw
=−
.
dx
a
sinh(γw)
(66)
The impedance of each plate is
Z=
bE( w2 )
V
w
=
cotrh(γw) = Rwdc ( )(1 + j)coth(γw) = FR + jFX
I
a
δw
(67)
where
2w
2w
w sinh( δw ) + sin( δw )
Rw
=( )
2w .
Rwdc
δw cosh( 2w
)
−
cos(
)
δw
δw
(68)
2w
2w
XL
w sinh( δw ) − sin( δw )
=( )
.
FX =
Rwdc
δw cosh( 2w
) − cos( 2w
)
δw
δw
(69)
FR =
and
At high frequencies, the resistance of each plate is
FR = FX ≈
w
.
δw
Figs. 42 and 43 show plots of Rw /Rwdc and XL /Rwdc as a function of w/δw .
figs. 44 and 45 show plots of |Z|/Rwdc and φz as a function of w/δw .
(70)
41
5
Anti-proximity and Skin Effects in Two Parallel
Plates
A B
Figure 46: Typical pattern of multi-layer inductor winding.
y
a
a
b
b
J
J
H
H
w
2
0
w
2
w
x
z
Figure 47: Two plates carrying currents in the same directions.
A typical pattern of inductor winding is shown in Fig.46. It can be seen that
the current flows in the same direction in the adjacent layers. Fig. 47 shows two
rectangular parallel conducting plates. Consider the case in which the current i(t) =
Icoswt flow in the plate in the same direction. Both currents flow in the −y direction.
42
The current induce the magnetic field in side and out side the plates. the magnetic
fields generated by both plates subtract from each other between the plates and add
to each other out side the plates, causing larger current density J(x) in the plate
areas, where the plates are far from each other. The magnetic field is described by
the Helmholtz equation
d2 H(x)
= γ 2 H(x).
2
dx
(71)
A general solution of this equation is
H(x) = H1 eγx + H2 e−γx .
(72)
where
w
I
eγ 2
H1 =
.
a eγw + e−γw
(73)
and
w
I
e−γ 2
.
H2 = − γw
a e + e−γw
(74)
I sinh[γ(x − w2 )]
.
H(x) =
a sinh(γw)
(75)
The magnetic field is given by
Fig. 48, 49 and 50 show plots of a|(Hx)|/I and its real and imaginary parts as a
function of x/w for selection values of w/δw .
The current density is
dH(x)
γI cosh[γ(x − w2 )]
J(x) = −
=−
.
dx
a
sinh(γw)
(76)
Fig. 51, 52 and 53 show plots of a|(Jx)|/I and its real and imaginary parts as a
function of x/w for selection values of w/δw .
The current density due to the skin and proximity on effects on the surfaces of the
conductor for x = w/2 and x = −w/2 is given by
w
γI cosh(γ w2 )
.
Jsp ( ) = −
2
a sinh(γw)
(77)
43
1
0.9
w /δ = 1
w
0.8
0.7
a |H(x)| / I
2
0.6
0.5
3
0.4
0.3
5
0.2
0.1
10
0
−0.5
0
x/w
0.5
Figure 48: Plot of a|H(x)|/I as a function of x/w for selected values of w/δw in the
left plate due to the proximity effect.
0.4
0.2
Real ( a |H(x)| /I)
0
10
−0.2
5
−0.4
3
−0.6
2
−0.8
w /δ = 1
w
−1
−0.5
0
x/w
0.5
Figure 49: Plot of the real part of a|H(x)|/I as a function of x/w for selected values
of w/δw . The current density is given by in the left plate due to the proximity effect.
44
0.35
2
0.3
3
Imag( a |H(x)| /I )
0.25
0.2
0.15
0.1
5
0.05
0
w /δ = 1
w
10
−0.05
−0.5
0
x/w
0.5
Figure 50: Plot of the imaginary part of a|H(x)|/I as a function of x/w for selected
values of w/δw . The current density is given by in the left plate due to the proximity
effect.
1.5
w /δ = 1
w
|J(x)| / I
1
w
2
aδ
3
0.5
5
10
0
−0.5
0
x/w
0.5
Figure 51: Plot of a|J(x)|/I as a function of x/w for selected values of w/δw in the
left plate due to the proximity effect.
45
1.2
w /δw = 1
1
2
3
0.6
Re{aδ
w
J(x) / I }
0.8
0.4
5
0.2
10
0
−0.2
−0.5
0
x/w
0.5
Figure 52: Plot of the real part of a|J(x)|/I as a function of x/w for seleced values
of w/δw . The current density is given by in the left plate due to the proximity effect.
1.2
1
Im{aδ
w
J(x) / I }
0.8
0.6
2
0.4
3
0.2
5
w /δ = 1
w
0
−0.2
−0.4
−0.5
10
0
x/w
0.5
Figure 53: Plot of the imaginary part of a|J(x)|/I as a functed of x/w for selection
values of w/δw . The current density is given by in the left plate due to the proximity
effect.
46
1
0.9
0.8
0.7
w /δw = 1
P(x) /ρ
w
0.6
2
0.5
0.4
0.3
5
0.2
0.1
10
0
−0.5
0
x/w
0.5
Figure 54: Plot of P (x)/ρw as a function of x/w for selected values of w/δw in the
left plate due to the proximity effect.
and
w
γI
1
Jsp (− ) = −
.
2
a sinh(γw)
(78)
Jsp (− w2 )
1
=
w
Js (− 2 )
1 + sinh2 (γ w2 )
(79)
Jsp ( w2 )
w
= 1 + tanh2 (γ )
w
Js ( 2 )
2
(80)
Jsp (0)
= 1.
Js (0)
(81)
1
P (x) = ρw |J(x)|2 .
2
(82)
hence,
and
The local power density is
Fig. 54 shows plots of P (x)/ρw as a function of x/w for selection values of w/δw in
the left plate due to the proximity effect.
47
The total power loss in each conductor due to the proximity and skin effects is
Psp
2w
2w
ρw bI 2 sinh( δw ) + sin( δw )
=
.
) − cos( 2w
)
2aδw cosh( 2w
δw
δw
(83)
Comparing the power loss in a single conductor and two conductors, we can see that
the presence of second conductor has increased the loss in the first. this increase is
simply the proximity loss set up in the first conductor by the magnetic field of the
second.
The proximity loss is
Pp =
2w
2w
ρw bI 2 sinh( δw ) − sin( δw )
2w .
)
+
cos(
)
4aδw cosh( 2w
δw
δw
(84)
hence
Ps = Psp − Pp =
2w
2w
ρw bI 2 sinh( δw ) + sin( δw )
.
4aδw cosh( 2w
) − cos( 2w
)
δw
δw
(85)
48
6
Laminated Cores
y
lc
i(t)
h
B(t)
J
φ (t)
w
2
w
2
z
dx
x
Figure 55: Cross section of single lamination used to analyzing eddy-current loss.
As previously discussed in Chapter 2, there are two kinds of eddy-currents. One
occurs due to the skin effect and the other occurs due to the proximity effect. In this
chapter, the method for the reduction of these two kinds of eddy-current loss will
be discussed. The first method is by using a high-resistivity material for the cores.
This high-resistivity increases the skin depth δc , and , thus reduces w/δc , that makes
the distribution of the magnetic flux density B more uniform. If this happens, the
condition w/δc < 1 is satisfied over a wider frequency range. In order to increase
the core resistivity ρc and therefore reduce the eddy-current amplitude, the iron is
used with a small amount of silicon or chrome (1 to 5%), producing a magnetic steel.
Ferrite and powder cores are also example of material with high resistivity. The second
method to reduce the eddy-current that occurs due to proximity effect is to divide
the core in to large number of thin slices that are insulated from each other using
49
H(x)
1
Hm
0
w = 8δc
J(x)
0
(a)
H(x)
Hm
1
0.78
0
w = 2δc
J(x)
0
(b)
Figure 56: Distribution of the envelope of the amplitude of magnetic field intensity
H(x)/Hm and the eddy density J(x). (a) For a single solid core at w = 8δc . (b) For
laminated core at w/2δc .
50
oxide film. This thin insulated sheet called lamination, are oriented parallel to the
magnetic flux φ. Thin laminations have little flux in individual layers and therefore,
the induced voltage decreases. Also the resistance of a single lamination is k times
higher than that of the corresponding solid core, where k is the number of sheets.
Laminations are insulated from each other by an oxide film or insulated varnish and
then stacked together to form the magnetic core. If thin sheets are used then, more
of core volume is needed. The lamination thickness w ranges from 0.01 to .5 mm.
The typical thickness usually is 0.3 mm for frequencies up to 200 Hz, 0.1 mm for
frequencies between 200 Hz and 2 kHz, and 0.05 mm for higher frequencies and pules
applications. The typical thickness of insulation is 0.015 mm. The most common
core use 97 % iron and 3 % silicon. They have µr = 10,000 and lamination thickness
equal to 0.3 mm. In powder core, eddy-current loss is low because individual particles
are insulated from each other. Laminations do not affect the magnetic performance
of the core at low frequencies. Line frequency transformers and electric machines use
laminated cores.
The other reason for using lamination is the skin effect. Otherwise, a solid core would
contain flux only in a shell with thickness equal to the core skin depth δc . As w/δc is
increased from 1 to 10, H(0)/H(w/2) and B(0)/B(w/2) decrease from approximately
1 to 10. For example.B(0)/B(w/2) = 0.98 at w/δc = 1, B(0)/B(w/2) = 0.78 at w/δc
= 2, B(0)/B(w/2) = 0.46 at w/δc = 3, and B(0)/B(w/2) = 0.1 at w/δc = 8. When
the core resistivity ρc is increased, the core skin depth δc also increased. Therefore,
there will be low frequency range, in which δc > w, the distribution of H is nearly
uniform, and the skin effect in the core can be neglected.
6.1
Low-Frequency Solution
The derivation of the eddy-current power loss in lamination iron for low frequencies
(where δc >> w) is as follows. The eddy-current density is assumed to be uniform.
51
Consider an iron lamination without an air gap, shown in Fig. 7. Assumed that
h >> w and δc > w, let us assume that the distribution of H along the x−axis is
uniform. Also, assume that a sinusoidal current flows in the inductor winding
i(t) = Im sinωt.
(86)
resulting in the magnetic flux density in the core
B(t) = Bm sinωt.
(87)
The distribution of the magnetic flux B = Bz in the core is uniform for w < δc . The
area through which the magnetic flux is passing for x > 0 is given by
Aφ = hx for x > 0.
(88)
and the magnetic flux passing through half of the area encircled by eddy-current loop
is
φt = Aφ Bm sinωt = hxBM sinωt for x > 0.
(89)
By Faraday’s low, the voltage induced between the bottom and top terminals is equal
to the derivative of the magnetic flux passing through half of the area enclosed by
the eddy-current loop
v(t) =
dφ(t)
dB(x)
= Aφ
= hxwBm cosωt = Aφ wBm cosωt = Vm coswt
dt
dt
(90)
where
Vm = ωBm hx.
(91)
Thus, Vm is proportional to f, Bm , h and x. By Lenz’s law, the voltage v(t) induces the
eddy-currents J, which, in turn, generate flux φ′ (t) that tend to oppose the imposed
flux φ(t). Since Vm increases with x, the maximum current density in the middle of
the lamination.
The lamination resistance of thickness x and areaAex = xlc is
Rcx = ρC
h
h
= ρc
Aex
lc x
(92)
52
resulting in the amplitude of the eddy-current
Iem =
Vm
ωBm lc x2
=
Rcx
ρc
(93)
and in the eddy-current density
|Jm (x)| =
Iem
ωBm x
=
.
Aex
ρc
(94)
It can be seen that the amplitude of the eddy current density change linearly with the
distance x. It is zero in the middle of the lamination and increases with the distance x
from the middle of the lamination. The maximum amplitude of eddy-current density
is at the surface
w
ωBm w
Jm(max) = Jm ( ) =
.
2
2ρc
(95)
The time-average eddy-current power loss in the lamination at low frequencies is
PE =
Z Z Z
= ρc
vc
Z
0
lc
Z
0
h
2
ρc Jm
(x)
ρc
dxdydz =
2
2
Z
0
w
2
lc
Z
0
Z
0
h
Z
w
2
−w
2
2 2
2
ω 2Bm
x
π 2 f 2 Bm
hlc w 3
dx
=
ρ2c
6ρc
2 2
ω 2Bm
x
dxdydz
ρ2c
for w < ρc .
(96)
yielding the time-averaged eddy-current power loss per unit volume (or the specific
eddy-current power loss) at sinusoidal waveform of B(t) = Bm sinwt
2
2
PE
π 2 f 2 w 2 Bm
ω 2w 2 Bm
Pe =
=
=
Vc
6ρc
24ρc
for w < δc =
s
ρc
.
πµrc µ0 f
(97)
where Vc = whlc is the single lamination volume and w the single lamination thickness.
Note that PE is proportional to w 3 and Pe is proportional to w 2 . It is evident from
the above expression that the eddy-current power loss density at low frequencies is
reduced by using thinner laminations and core material with high resistivity.
An alternative method of deriving an expression for PE is as follows. The resistance
of a thin strip with eddy-current is
R = ρc
h
.
lc dx
(98)
53
Hence, the eddy-current power dissipated in the thin lamination is
dPE =
2
Vm2
hlc ω 2 Bm
=
x2 dx
2R
2ρc
(99)
resulting in the eddy-current power dissipated in lamination at low frequencies
PE =
Z
w
2
−w
2
dPE = 2
=
Z
0
w
2
dPE =
2
hlc ω 2 Bm
ρc
2
2
w 2 ω 2Bm
π 2 f 2 w 2 Bm
Vc =
Vc
24ρc
6ρc
Z
w
2
0
x2 dx =
2
hlc w 3ω 2 Bm
24ρc
for w < δc .
(100)
From the equation, the loss due to eddy-current is proportional to the square of the
frequency and the cube of the thickness.
Let’s assume that a solid core width is ws = kw, the ratio of the eddy-current power
loss density in sold core Pes to that in the single lamination Pe is
ws
Pes
= ( )2 = k 2
Pe
w
for w < δc =
s
ρc
.
πµrc µ0 f
(101)
The ratio the eddy-current power loss in a sold core PEs to the eddy-current power
loss that in the lamination core consisting of k laminations PEl is
PEs
ws
= ( )3 = k 3
PE
w
for w < δc =
s
ρc
.
πµrc µ0 f
(102)
Now, the ratio of the eddy-power loss in a solid core PEs that in the single lamination
Pe is
PEs
PEs
ws
=
= ( )2 = k 2
PEl
kPE
w
for w < δc =
s
ρc
.
πµrc µ0 f
(103)
Therefore, the eddy-current power loss in the sold core k 2 time higher than in that
in the lamination core, assuming that both core have the same conducting volume.
The solid core resistance at low frequencies is
h
h
= ρc
.
Acs
ws lc
(104)
h
h
= ρc
= kRcsd .
Acs
ws lc
(105)
Rcsd = ρc
The resistance of a single lamination is
Rcsl = ρc
54
The core resistance of the laminated core with k laminations at low frequencies is
Rcl = ρc
6.2
h
= Rcsd .
kwlc
(106)
General Solution
At any frequency, the magnetic field intensity is described by the Helmholtz equation
d2 Hz (x)
= γ 2 Hz (x) = jµc ωHz (x)
dx2
(107)
where
γ=
q
q
jµc σc ω = (1 + j) µc σc f =
1+j
δc
(108)
with the depth into the conductor sheet given by
1
δc = √
=
πµc σc f
s
ρc
.
πµc f
(109)
H(x) = H1 eγx + H2 e−γx .
(110)
w
w
w
H( ) = H1 eγ 2 + H2 e−γ 2
2
(111)
w
w
w
H(− ) = H1 e−γ 2 + H2 eγ 2 .
2
(112)
A general solution of (107) is
Hence,
and
The condition of even symmetry requires that H(x) = H(−x), from which H(w/2) =
H(w/2).
Subtract (112) from (111), we get
w
w
w
w
w
w
w
w
H( ) − H(− ) = H1 eγ 2 + H2 e−γ 2 − H1 e−γ 2 − H2 eγ 2 = (eγ 2 + e−γ 2 )(H1 − H2 ) = 0
2
2
(113)
from
H1 = H2 .
(114)
55
1
w /δ = 1
c
0.9
2
0.8
|H(x)| / H
m
0.7
0.6
3
0.5
0.4
0.3
5
0.2
0.1
10
0
−0.5
0
x/w
0.5
Figure 57: Plot of |H(x)|/Hm as a function of x/w for selected values of w/δc .
Substituting for H2 in (110), we get
H(x) = H1 (eγx + e−γx ) = 2H1
eγx + e−γx
= 2H1 cosh(γx).
2
(115)
Using the boundary condition,
γw
w
H( ) = 2H1 cosh( )
2
2
(116)
we obtain
H1 =
H( w2 )
2cosh( γw
)
2
(117)
the amplitude of the magnetic field intensity at x = w/2 is
w
NIm
H( ) = Hm =
.
2
lc
(118)
Substitute (117) and (118) into (115), we get the solution of the Helmholtz equation
cosh( δxc + j δxc )
cosh[(1 + j)( δwc )( wx )]
cosh(γx)
H(x) = Hm
= Hm
= Hm
.
cosh(γ w2 )
cosh( 2δxc + j 2δxc )
cosh[ (1+j)
( δw )]
2
c
(119)
56
1.2
1
w /δ = 1
c
2
m
Re{H(x) / H }
0.8
3
0.6
5
0.4
0.2
10
0
−0.2
−0.5
0
x/w
0.5
Figure 58: The real part of |H(x)|/Hm as a function of x/w for selected values of
w/δc .
0.1
10
0
m
Im{H(x) / H }
−0.1
5
−0.2
w /δ = 1
−0.3
c
−0.4
3
−0.5
2
−0.6
−0.5
0
x/w
0.5
Figure 59: The imaginary part of |H(x)|/Hm as a functed of x/w for selected values
of w/δc .
57
1.6
1.4
δ c |J(x)| / Hm
1.2
w /δc = 3
5
1
0.8
0.6
1
0.4
10
0.2
0
−0.5
0
x/w
0.5
Figure 60: Plot of δc |J(x)|/Hm as a function of x/w for selected values of w/δc .
The magnitude of the magnetic field intensity is
|H(x)| = Hm
v
u
u
t
v
u
u cosh 2x
cosh2 δxc − sin2 δxc
+ cos 2x
δc
δc
t
= Hm
.
cosh2 2δwc − sin2 2δwc
cosh δwc + sin δwc
(120)
Fig. 57 ,58 and 59 show plots of H(x)/Hm and its real and imaginary parts as a
function of x/w at different values of w/δc .
The current density is
sinh( δxc + j δxc )
dHz (x)
sinh(γx)
1+j
J(x) = Jy (x) = −
= −γHm
=−
Hm
. (121)
dx
cosh(γ w2 )
δc
cosh( 2δxc + j 2δxc )
The magnitude of the current density is
v
u
2[cosh 2x
− cos 2x
]
Hm u
δc
δc
t
|J(x)| =
w
w .
δc
cosh δc + cos δc
(122)
Figs. 60 show plot of δc |J(x)|/Hm as a function of x/w at different values of w/δc .
At high frequencies, δc << w, the distribution of the envelope of magnetic field
intensity H(x) is not uniform and the amplitude of eddy-current density J(x) does
58
1.2
w /δc = 3
1
δ 2c Pe(x) / ρcH2m
0.8
5
0.6
0.4
0.2
1
10
0
−0.5
0
x/w
0.5
2
Figure 61: Plot of δc2 Pe (x/ρc Hm
) as a function of x/w for selected values of w/δc .
not vary linearly with distance x, as illustrated in Fig. 56 for a solid core at w = 8δc .
In this case, J(w/2) is very high, resulting in high core loss. the distribution of
H(x)/Ho and J(x) in the laminated core at w = 2δc are depicted in Fig.56 (b).
Silicon is often added to steel to reduce σc , and reduce w/δc , making H(x) more
uniform.
Using Ohm’s law E = σc J, we obtain the time-average eddy-current power loss density
2 cosh 2x − cos 2x
1 . ∗ ρc . ∗ ρc |J(x)|2
ρc Hm
δc
δc
Pe (x) = E J = J J =
=
2
2
2
δc2 cosh δwc + cos δwc
(123)
2
Fig. 61 shows plot of δc2 Pe (x/ρc Hm
) as a function of x/w at different values of w/δw .
Since sinh(x/δc + jx/δc ) ≈ (1 + j)x/δc and cosh(w/δc + jw/δc ) ≈ 1 for w/deltac << 1
and x/δc << 1, the current density at law frequencies is approximated by
J(x) = Jy (x) ≈ Hm
1+j
δc
2
Bm
x=
µc
1+j
δc
2
x=
2Bm
x.
µc δc2
(124)
The complex power for sinusoidal waveforms in phasor form is
SE(complex) =
1
2
Z Z Z
v
E. J∗ dV.
(125)
59
1
10
0
δcPE / ρ cbHm2 h lc
10
−1
10
−2
10
−3
10
−4
10
−1
0
10
1
10
2
10
w/δc
10
2
Figure 62: Plot of δc PE ρc Hm
hlc as a function of w/δw .
The time-average power dissipated at low frequencies (w << δc ) is given by
1
Pe = Re
2
Z Z Z
v
≈
2
4ρc Bm
=
2µ2c δc4
Z
0
E. J∗ dV
2
ρc Bm
2µ2c
lc
Z
h
Z
0
Z
w
2
−w
2
0
lc
Z
0
1
2
=
h
Z
Z Z Z
w
2
−w
2
ρc |J|2 dV =
v
ρc
2
Z Z Z
v
|J|2 dV
2
1+j
x dxdydz
δc
2
µc lc hw3 Bc2
lc hw 3 ω 2 Bm
=
.
x dxdydz =
6µ2c δc2
24ρc
2
(126)
this expression is identical to that given by (97).
using 119, the time-average eddy-current power loss dissipated in the lamination is
PE =
=
2
ρc 2Hm
2
2
2δc
Z
0
h
Z
0
1
2
w
2
Z Z Z
v
ρc |J(x)|2 dV =
1
2
Z
0
lc
Z
0
h
Z
w
2
−w
2
ρc |J(x)|2 dV
2x
2x
2
cosh( 2x
) − cos( 2x
)
ρc Hm
lc h sinh( δc ) − sin( δc )
δc
δc
dV
=
.
cosh( δwc ) + cos( δwc )
δc
cosh( δwc ) + cos( δwc )
(127)
2
Fig. 62 shows δc PE ρc Hm
hlc as a function of w/δc . As w/δc increases from 0, PE
increases from 0, reaches the maximum values, decreases to minimum value, increase
60
again to a maximum, and so on.
Taking the derivative of PE with respect to w/δc and setting it to zero
w
w
2
dPE
2ρc Hm
hlc sinh( δc ) − sin( δc )
=
=0
d( δwc )
δc
[cosh( δwc ) + cos( δwc )]2
(128)
We get
w
sinh
δc
= 0.
(129)
Hence, the maximum and minimum values of PE occur at
w
δc
= nπ.
(130)
where n is the integer.
The largest maximum value of PE occurs for n =1, resulting in the most lossy lamination thickness
wpE(max) = πδc =
s
πρc
.
µc f
(131)
Substitution of (131) into (127) gives the maximum time-average eddy-current total
power loss
PE(max)
2
2
2
2
1.09Hm
hlc
1.09Bm
hlc
1.09Bm
hlc
ρc Hm
hlc sinh(π)
=
=
=
=
2
δc
cosh(π) − 1
δc
µc δc
µc
s
πρc
.
µc f
(132)
Using (124)
2
d 2 PE
2ρc Hm
hlc
=
w 2
w
d( δc )
δc [cosh( δc ) + cos( δwc )]3
w
w
w
w
w
w
sinh( )sin2 ( ) − sinh2 ( )sin( ) + 1 + cosh( )cos( )
δc
δc
δc
δc
δc
δc
w
w
sinh( ) + sin( ) .
δc
δc
(133)
Hence,
2
d2 PE sinh(nπ)
n 2ρc H + m hlc
=
(−1)
.
w 2 w
d( δc ) ( δc )=nπ
δc
[cosh(nπ) + (−1)n ]2
(134)
For odd values of n(n = 1, 3, 5, ....),
d 2 PE
<0
d( δwc )2
(135)
61
0
10
−1
δc2 Pe / ρ cHm2
10
−2
10
−3
10
−1
0
10
1
10
2
10
w/δc
10
2
Figure 63: Plot of δc Pe ρc Hm
hlc as a function of w/δc .
resulting in maximum values of PE . For even values of n(n = 2, 4, 6, ....),
d 2 PE
>0
d( δwc )2
(136)
yielding the minimum values of PE . For n = 2,
wPE (min) = 2πδc = 2
s
πρc
µc f
(137)
and
PE(min) =
ρc Hc2 hlc
ρc Hc2 hlc sinh(2π)
= 0.996
.
δc
cosh(2π) + 1
δc
(138)
The time-average eddy-current power loss per unit volume at any frequency is
PE
ρc
Pe =
=
Vc
2Vc
Z
w
2
−w
2
ρc
|J| dx =
lc hw
2
Z
0
w
2
|J|2 dx
2 sinh( w ) − sin( w )
2 sinh( w ) − sin( w )
ρc Hm
ρc Hm
δc
δc
δc
δc
=
w
w = 2 w
w
wδc cosh( δc ) + cos( δc )
δc ( δc ) cosh( δc ) + cos( δwc )
2
Fig. 63 shows δc Pe ρc Hm
hlc as a function of w/δc .
W
.
m3
(139)
62
2
10
1
10
0
10
−1
Rc / Rcdc
10
−2
10
−3
10
−4
10
−5
10
−6
10
−1
10
0
10
1
w/δc
10
2
10
Figure 64: Plot of Rc /Rcdc as a function of w/δc .
2
10
1
10
0
XL / Rcdc
10
−1
10
−2
10
−3
10
−1
10
0
10
1
w/δc
10
Figure 65: Plot of XL /Rcdc as a function of w/δc .
2
10
63
2
10
1
10
| Z | / Rcdc
0
10
−1
10
−2
10
−3
10
−1
0
10
10
1
w/δc
2
10
10
Figure 66: Plot of |Z|/Rcdc as a function of w/δc .
For low frequencies or a very thin plate(w << δc )
sinh( δwc ) − sin( δwc )
≈
cosh( δwc ) + cos( δwc )
w
δc
+ 16 ( δwc )3 − δwc + 16 ( δwc )3
1 w 3
≈
.
1+1
6 δc
(140)
Hence, the time-average eddy-current power loss per unit volume of the core at low
frequencies is
Pe ≈
2 ( w )3
2 2
2
ρc Hm
ρc Hm
w
π 2 Bm
w2f 2
δc
=
=
wδc 6
6δc4
6ρc
(141)
and PE is approximated by by (127). the eddy-current loss is low when the plates is
very thin compared to δc . in this case, the magnetic field produced by eddy current
is very low. The eddy currents are restricted by lack of space or high core resistivity
and are resistance limited.
At high frequencies or very thick plate (w >> δc ), ew/δc ≈ 0, sinh(w/δc ) = cosh(w/δc ) ≈
ew/δc /2, resulting in
sinh( δwc ) − sin( δwc )
≈
cosh( δwc ) + cos( δwc )
ew/δc
2
ew/δc
2
= 1.
(142)
64
Therefore, the time-average eddy current power loss per unit volume of the core at
high frequencies is given by
2
2
2 q
2
ρc Hm
ρc Bm
Hm
Bm
Pe ≈
=
=
πρ
µ
f
=
c c
wδc
wµ2c δc
w
wµc
s
πρc f
µc
W
m3
(143)
and the total time average eddy-current power loss at high frequencies is
q
2
ρc Hm
hlc
B 2 hlc
2
PE ≈
= Hm
hlc πρc µc f = m
δc
µc
s
πρc µc f
.
µc
(144)
At the high frequencies, the eddy currents are limited by their own magnetic field
and are said to be inductance-limited. At low frequencies, Pe is proportional to w 2 .
in contrast, at high frequencies, Pe is inversely proportional to w. thus, there is the
worst case of w at which Pe reaches a maximum value [32]. the lamination thickness
for the maximum power loss per unit volume occurs at frequencies is
wpe(max) ≈ 2.252δc
(145)
resulting in maximum time-average eddy-current power loss per unit volume frequencies is
pe(max) ≈ 0.4172
2
2
ρc Bm
ρc Hm
=
0.4172
.
δc2
µ2c δc2
(146)
The dc resistance of single lamination is
Rcdc =
ρc h
.
wlc
(147)
the impedance of a lamination is given by
hρc γ
w
Z=
coth γ
2lc
2
hρ(1 + j)
w
=
coth γ
2lc δc
2
1
w
w
= Rcdc
(1 + j)coth γ
2
δc
2
= Rc + jXL .
(148)
The normalized resistance of lamination is
FR =
w
w
Rc
1 w sinh( δc ) − sin( δc )
=
.
Rcdc
2 δc cosh( δwc ) + cos( δwc )
(149)
65
90
85
80
75
φZ (°)
70
65
60
55
50
45
40
−1
10
0
10
1
w/δc
10
2
10
Figure 67: Plot of φz as a function of w/δc .
3
10
2
XL/Rc
10
1
10
0
10
−1
10
−1
10
0
10
1
w/δc
10
Figure 68: Plot of XL /Rc as a function of w/δc .
2
10
66
0
10
−1
R /ω Lo
10
−2
10
−3
10
−1
0
10
10
1
w/δc
10
2
10
Figure 69: Plot of R/ωLo as a function of w/δc .
Fig. 64, and 65 show a plot of Rc /Rcdc and XL /Rcdc as a function of w/δc . The
lamination reactance XL normalized with respect to the core dc resistance Rcdc is
given by
FX =
w
w
XL
1 w sinh( δc ) + sin( δc )
=
Rcdc
2 δc cosh( δwc ) + cos( δwc )
(150)
yielding the core inductance
w
w
µc hδc sinh( δc ) + sin( δc )
L=
4lc cosh( δwc ) + cos( δwc )
(151)
the core impedance is
1
w
Z = Rcdc
2
δc
sinh( δwc ) − sin( δwc )
sinh( δwc ) + sin( δwc ) +j
cosh( δwc ) + cos( δwc )
cosh( δwc ) + cos( δwc )
= Rw + jXL = |Z|ejφz .
(152)
Fig. 66 and 67 show plots of |Z|/Rcdc and φz . Fig. 68 shows the ratio XL /Rc as
a function of w/δc . The magnetic flux flowing through the cross section of a single
67
1
0.9
0.8
0.7
XL /ω Lo
0.6
0.5
0.4
0.3
0.2
0.1
0
−1
10
0
10
1
w/δc
10
2
10
Figure 70: Plot of XL /ωLo as a function of w/δc .
1
0.9
0.8
0.7
L /Lo
0.6
0.5
0.4
0.3
0.2
0.1
0
−1
10
0
10
1
w/δc
10
Figure 71: Plot of L/Lo as a function of w/δc .
2
10
68
1
0.9
0.8
0.7
| Z | /ω Lo
0.6
0.5
0.4
0.3
0.2
0.1
0
−1
10
0
10
1
w/δc
10
2
10
Figure 72: Plot of |Z|/ωLo as a function of w/δc .
90
85
80
75
φZ (°)
70
65
60
55
50
45
40
−1
10
0
10
1
w/δc
10
Figure 73: Plot of φz as a function of w/δc .
2
10
69
3
10
2
10
1
Rp /ω Lo
10
0
10
−1
10
−2
10
−1
10
0
10
1
w/δc
10
2
10
Figure 74: Plot of Rp /ωLo as a function of w/δc .
1
0.9
0.8
0.7
Xp /ω Lo
0.6
0.5
0.4
0.3
0.2
0.1
0
−1
10
0
10
1
w/δc
10
Figure 75: Plot of Xp /ωLo as a function of w/δc .
2
10
70
laminations given by
φc1 =
Z Z
s
.
B(x) ds =
Z
h
0
dy
Z
w
2
−w
2
hµc H − m
µc H(x)dx =
cosh(γ w2 )
w
2hµc Hm
tanh γ
=
γ
2
w
2
Z
2hµc NIm
w
=
tanh γ
γlc
2
−w
2
cosh(γx)dx
(153)
where
Hm =
NIm
lc
(154)
and N is the number of turns. Since the laminations are close to each other, it can
be assumed that the total magnetic flux through n lamination only
φc = nφcl .
(155)
The magnetic flux linkage is given by
w
2nN 2 hµc
.
tanh γ
λc = Nφ = nNφcl =
γlc
2
(156)
The voltage across the coil is
v=
dλc
dt
(157)
which in the phasor form becomes
V = jωλc .
(158)
hence, the coil impedance is given by
V
V
jωλc
2nN 2 hµc
w
Z=
=
=
= jω
tanh γ
I
Im
Im
γlc
2
2ωLo
w
j
tanh γ
wγ
2
= ωLo
sinh( δwc ) + jsin( δwc )
δc
(1 + j)
= R + jXL
w
cosh( δwc ) + cos( δwc )
(159)
where the low frequency inductance is
µc N 2 nwh
Lo =
lc
(160)
71
and
tanh(1 + j)x =
sinh(2x)jsin(2x)
= R + jXL .
cosh(2x) + cos(2x)
(161)
The series resistance is
R=
w
w
ωLo sinh( δc ) − sin( δc )
w
cosh( δwc ) + cos( δwc )
δc
(162)
and the series reactance is
w
w
ωLo sinh( δc ) + sin( δc )
XL = w
cosh( δwc ) + cos( δwc )
δc
(163)
and the inductance at any frequency is
w
w
XL
Lo sinh( δc ) + sin( δc )
L=
= w
.
ω
cosh( δwc ) + cos( δwc )
δc
(164)
Figs. 69 through 73 show plots R/ωLo , XL /ωLo , L/Lo , |Z|/ωLo and φz as functions of
w/δc . The series equivalent circuit of the coil can be converted into parallel equivalent
circuit
q=
XL
R
(165)
Rp = R(1 + q 2 )
(166)
and
Xp = XL
1
1+ 2 .
q
Figs. 74 and 76 shows plots of Rp /ωLo and Rp /ωo as function of w/δc
(167)
72
7
Summary
The power losses in inductors and transformers consist of winding core losses. The
core losses consist of hysteresis loss and eddy-current loss. In accordance with Lenz’s
law, eddy-currents produce their own magnetic field to oppose the original field.
There are two kinds if eddy-current: skin effect and proximity effect. Both of these
effects cause current crowding. The skin effect is the tendency of (ac) currents to
flow near the surface of a conductor, thereby restricting the current to flow to a
small part of the total cross-sectional area and increasing the conductor effective
resistance. The skin effect takes place when the time-varying magnetic field induces
eddy-currents in the conductor itself. The proximity effect is the tendency to change
the current distribution in a conductor by magnetic flux produced by current in
adjacent conductors. The proximity effect takes place when a second conductor is
influenced by a nearby conductor which is carrying a time-varying current.
Eddy-currents are induced whether or not the first conductor carries current. If the
second conductor does not carry current, only the proximity effect is present in the
second conductor. If the second conductor carries current, then the total eddy current
consists of both the proximity eddy current and the skin effect eddy current.
When two conductors carry currents in opposite directions, their magnetic fields are
added in the area between them, and subtracted from each other on the outer sides
of the wires. This causes an increase of the current density in the conductor areas
where the conductors are close to each other, making the currents flow more in these
areas. This is called the proximity effect. The anti-proximity effect occurs when
two conductors carry currents in the same directions. In this case, the magnetic
fields of the conductors are subtracted from each other in the conductor areas where
they are close to each other, and added to each others on the outer sides. That
makes more current density in the outer areas. For this reason, more currents flow
73
in these areas. The skin effect eddy-current and the proximity effect eddy-current
are orthogonal. The eddy-currents cause a non-uniform distribution of the current
density, an increase in the (ac) resistance, an increase the power loss, and a reduction
in inductance. According to Lenz’s law, the direction of the eddy-currents are such
that they oppose the change that causes them. The eddy-current limits the effective
capability of a conductor to conduct high-frequency currents. The skin depth has
physical meaning. At x = δw the current density in a conductor is reduced to 1/e
= 0.37 of its value on the the surface. As the resistivity ρ decreases and the relative
permeability increases, the skin depth δ decreases. In this case, the skin and proximity
effect can be neglected, and for δw < d the skin effect increases the (ac) winding
resistance over the (dc) winding resistance, increasing the power loss. The proximity
effect loss in multiple layer winding is much higher than the skin effect loss. Increasing
the distance between conductors in the same layer reduces the proximity-effect power
loss. In leaved winding transformers reduced copper loss is caused by the proximity
effect at high frequencies, if the current through the primary and secondary windings
are in phase. Each lager operates as a single-lager winding and the proximity effect
is nearly eliminated. There is an optimum conductor thickness to the conductor or
skin depth that leads to the minimum copper loss. To reduce the copper loss due
to the proximity effect, it is highly advantageous to reduce the number of winding
lagers, increase the winding width, and interleave the winding. Litz wire increases the
effective conduction area at high frequencies and thereby reduces the eddy-current
loss. Also the eddy-current loss can be reduced by using a laminated iron core with
high resistivity. The eddy current loss in a laminated iron core is proportional to
2
f 2 , Bm
, and w 2 , and inversely proportional to ρc at w < δw for sinusoidal waveform.
74
7.1
Future Work:
The future work will be investigating the proximity effect when a conductor is surrounded by two or more conductors. Also interpreting the behaviors of the imaginary
parts of the eddy currents.
75
References
1. H. A. Wheeler, ”Simple inductance formulas for radio coils ,” Proc. IRE, vol. 16,
no 10, pp. 1398-1400, October 1928.
2. H. A. Wheeler, ”Formulas for the skin effect,” Proc. IRE, vol. 30, pp. 412-424,
September 1942.
3. C.W.T.McLyman, Transformer and Inductor Design Handbook, 2nd Ed. New
York: Marcel Dekker, 1988.
4. A van den Bossche and V .C. Valchev, Inductors and Transformers for Power
Electronics, Boca Raton: Taylor and Francis, 2005,
5. F . W Grover, Inductance Calculations, New York, NY: Van Nostrand, 1946,
reprinted by New York, NY: Dover Publications, 1962
6. M.J Hole and L.C. Apple, ”stray capacitance of a tow-layer air-cord inductor,”
IEEE Proc, part G, Circuits, Devices and Systems, Vol 125, no. 6,
PP. 565-572, December 2005.
7. E.B. Rosa, ”calculation of the self-inductance of single-layer coils,” Bull. Bureau
standards, vol. 2, no2, PP. 161-187, 1906
8. H.M Greenhouse,”Design iof Planar rectangular microelectronic inductors,”IEEE
Trans. Parts. Hybrids, packaging, vol. PHP-10, PP. 109, June 1974.
9. P.R. Gray and R. G. Mayer, ”Future directions in silicon IC’s for RF Personal
communications,” Proc IEEE 1995 custom Integrated Circuits Conf.
May 1995, PP 83-90
10. C P. Yue, C. Ray, J. Lau, T.H. Lee and S.S. Wong, ”A Physical model for Planar
spiral inductors in silicon,” International Electron Devices Meeting
Techical Digest, December 1996, PP. 155-185.
11. C.P Yue and S.S Wang, ”On-chip spiral inductors with Patterned ground shields
for Si-bases RF ICs,” IEEE j. sold-state Circuits, Vol. 33,no. 3,
76
PP. 743-752, May 1998.
12. F Mernyei, F Darrer M. Pardeon, and Sibrai ”Reducing the substrate losses of
RF integrated inductors,” IEEE Microwave and Guided Wave letters, vol. 8.
no. 9, PP. 300-3001, September 1998
13. G. Grandi, M. K. Kazimierczuk, Massarini, U. Reggiani,and G. sancineto, ”Model
of laminated iron-core inductors,” IEEE Transactions on Magnetics, vol. 40,,
no. 4 PP. 1839-1845, July 2004.
14. K.Howard and M. K. Kazimierczuk, ”Eddy-current Power loss in laminated Power
cores,” Proceeding of the IEEE International Symposium on Circuits and systems,
Sydney, Australia, May 7-9,2000, paper III-668, PP. 668-672.
15. A. Reatti and M. K. Kazimierczuk, ”comparison of various methods for
calculating the ac resistance of inductors,” IEEE Transactions on Magnetics,
vol 37, PP. 1512-1518, May 2002.
16. C. R. Sullivan ”Optimal choice for the number of strands in a litz-wire
transformer winding” IEEE transactions on power Electronics Vol. 14, no. 2,
PP. 283-291, March 1999.
17. T. L. Simpson, Effect of a conducting shield on the inductance of the an air-core
solenoid,” IEEE transactions on Magnetics, vol 35, on. 1, PP. 508-515,
January1999.
18. H. A. Wheeler, ”simple inductance formulas for radio coils”Proc. IRE, vol 16,
no. 10, PP 1398-1400, October 1928.
19. H.A. Wheeler,”formulas for the skin effect” Proc IRE, vol 30, PP. 412-424,
September 1942.
20. R.G.Medhurst, ”HF resistance and self-capacitance of single layer solenoids,”
Wireless Engineers,pp.35-34, February 1947, and PP. 80-92, March 1947.
21. R. W. Erickson and D. Maksimovic, Fundamentals of power Electronics,
77
Norwell, MA: Kluwer Academic Publishers, 2001.
22. C. P. Steinmetz, ”On the low of hysteresis,” AIEE, vol. 9, PP. 3-64, 1992. Also,
”A steinmetz contribution to the ac power revolution, ”Proc IEEE, vol 72,
PP. 196-221, 1984.
23. B. Carsten, ”High frequency conductor losses in switch mode magnetics,” Proc.
of PCI, Munich, Germany, 1986, PP. 161-182.
24. J. P. vandalec and P. D. Ziogos,”A novel approach for the minimizing high
frequency transformer copper loss,”IEEE Transactions on Power Electronics.
VOL. 3, pp. 266-276, July 1988.
25. A. M. Urling, V. A.Niemela, G. R. skutt, and T. G. Wilson, ”characterizing high
frequency effects in transformer winding: A guide to several significant papers,”
IEEE Transactions on Power Electronics Specialists Conference, 1989,
PP. 373-385.
26. P. J. Dowell, ”Effects of eddy currents in transformer winding,” Proc. IEEE, vol
. 113, no. 8, pp. 1387-1394, August 1966.
27. A. Kennelly, F. laws, and P. Pierce, ”Experimental research in skin effect in
conductors,” Trans. AIEE, vol. 34, p. 1915, 1915.
28. J. Lammeraner and M. Stafl, Essy Current, Cleveland: CRS Press, 1966.
29. J. Ebert, ”Four terminal parameters of HF inductors,” Bull. Acad. Polan.
Sci. Ser. Sci. Techn. no. 5, 1968.
30. E. C. Snellling, Soft Ferrite: Properties and Applications, London: Iliffe Books
Ltd, 1969.
31. R. L. Stall, The Analysis of Eddy Currents, Oxford: Clarendon Press, 1974,
pp. 21-27.
32. W. T. McLyman, transformer and Inductor Design Handbook, 3rd Ed.
New York: Marcel Dekker, 2000.
78
33. J. K. Watson, Applications of Magnetism, Ganainesville, 1985.
34. J. C. Maxwell, A Treatise of Electricity and Magnetism, 3rd. Ed. New York,
NY: Dover Publishing, 1997
35. J. A. Ferreia, Electromagnetic Modeling of Power Electronic Converters, Boston:
Kluwer Acaemic Publisher, 1989.