# Power Electronics Notes 30H Power Electronics Notes 30H Magnetic Fields from Power Cables (Case Studies) (Case Studies)

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Oct 18, 2013 (4 years and 8 months ago)

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PowerElectronicsNotes30H
Power

Electronics

Notes

30H
Magnetic Fields from Power Cables
(CaseStudies)
(Case

Studies)
Marc T. Thompson, Ph.D.
Thompson Consulting, Inc.
9 Jacob Gates Road
Harvard, MA 01451
Phone: (978) 456-7722
Fax: (240) 414-2655
Email: marctt@thompsonrd.com
Web: http://www.thompsonrd.com
Magnetic Fields from Power Cables1©M. T. Thompson, 2009
© Marc Thompson, 2009
Ampere’s Law

Flowingcurrentcreatesamagnetic
Flowing

current

creates

a

magnetic

field
dt
d
o
v
r
v
v
v
r
⋅+⋅=⋅

ε
•In magnetic systems, generally there
ishighcurrentandlowvoltage(and
dt
SSC

is

high

current

and

low

voltage

(and

hence low electric field) and we can
approximate for low d/dt:
∫∫
⋅≈⋅
SC
v
v
v
r
•In words: the magnetic flux density
integrated around any
closed contour
e
q
uals the net current flowin
g
throu
g
h
Magnetic Fields from Power Cables2©M. T. Thompson, 2009
qgg
the surface bounded by the contour
André-Marie Ampère

Achangingmagneticfluximpingingon
A

changing

magnetic

flux

impinging

on

a conductor creates an electric field
and hence a current (eddy current)
d

Theelectricfieldintegratedarounda
∫∫
⋅−=⋅
SC
dt
d
ldE
v
r
v
r

The

electric

field

integrated

around

a

closed contour equals the net time-
varying magnetic flux density flowing
through the surface bound by the
contour

Inaconductorthiselectricfieldcreates

In

a

conductor
,
this

electric

field

creates

a current by:
EJ
r
r
σ
=
Magnetic Fields from Power Cables3©M. T. Thompson, 2009•Induction motors, brakes, etc.
Gauss’ Magnetic Law
G'tilthtth
B
1
,
A
1
B
2
,
A
2

G
auss
'
magne
ti
c
l
aw says
th
a
t

th
e
integral of the magnetic flux density over
an
y
closed surface is zero, or:

1
,
1
2
,
2
y
T桩猠污眠業灬楥猠瑨慴⁭慧湥瑩挠晩敬摳⁡牥f

=

S
dAB0
B3, A3
d
ue to e
l
ectr
i
c currents an
d
t
h
at
magnetic charges (“monopoles”) do not
exist.
221133
A
B
A
B
A
B
+
=
•Note: similar form to KCL in circuits !
(We’ll use this analogy later…)
Magnetic Fields from Power Cables4©M. T. Thompson, 2009
Carl Friedrich Gauss
Gauss’ Law ---Continuity of Flux Lines
123
0
φ
φφ
+
+=
Magnetic Fields from Power Cables5©M. T. Thompson, 2009
Finite-Element Analysis (FEA)
•Very useful tool for visualizing magnetic fields
•FEA is often used to simulate and predict the performance
ofmotors,etc.
of

motors,

etc.
•Following we’ll see some 2-dimensional (2D) FEA results to
help explain Maxwell’s equations
Magnetic Fields from Power Cables6©M. T. Thompson, 2009
Example 1: Circular Current Loop (Dipole)
CillidilithdiltIA(I

Ci
rcu
l
ar
l
oop
i
s a
di
po
l
e w
ith

di
po
l
e momen
t

IA

(I
=
current, A = loop area)
Magnetic Fields from Power Cables7©M. T. Thompson, 2009
Example 1: Field From Current Loop, NI = 500 A-
Turns
Turns

•Coil radius R = 1”; plot from 2D finite-element analysis
Magnetic Fields from Power Cables8©M. T. Thompson, 2009
Example 2: Isolated Wire, Return Far Away
Thiiilld

Thi
s one
i
s eas
il
y so
l
ve
d
Magnetic Fields from Power Cables9©M. T. Thompson, 2009
Example 3: Magnetic Dipole, 2D
Wilit

Wi
res are
l
ong
i
n
t
o page
•What’s the magnetic flux density Bfar away?

A
ssume I = 15A; wire-wire s
p
acin
g
d = 10 mm;
r
wire
=2 mm
pg
wire
Magnetic Fields from Power Cables10©M. T. Thompson, 2009
Example 3: Magnetic Dipole, 2D
Magnetic Fields from Power Cables11©M. T. Thompson, 2009
Example 3: Magnetic Dipole, 2D
Fild1t(d)dilt

Fi
e
ld
r =
1
me
t
er away
(
r >>
d)
, ra
di
a
l
componen
t
Magnetic Fields from Power Cables12©M. T. Thompson, 2009
Example 3: Magnetic Dipole, 2D
Fild1tthtt

Fi
e
ld
r =
1
me
t
er away,
th
e
t
a componen
t
Magnetic Fields from Power Cables13©M. T. Thompson, 2009
Example 4: Magnetic Quadrupole, 2D
2

A
ssume
I
=
15A
;
d
=
10
mm;
r
wire=
2
mm
•Note orientation of + and -currents
Magnetic Fields from Power Cables14©M. T. Thompson, 2009
Example 4: Magnetic Quadrupole, 2D
Magnetic Fields from Power Cables15©M. T. Thompson, 2009
Example 4: Magnetic Quadrupole, 2D
Fild1tdiltFildd

Fi
e
ld
r =
1
me
t
er away, ra
di
a
l
componen
t
.
Fi
e
ld
s
d
ecay
as 1/r3
Magnetic Fields from Power Cables16©M. T. Thompson, 2009
Example 4: Magnetic Quadrupole, 2D
Fild1tthttFildd

Fi
e
ld
r =
1
me
t
er away,
th
e
t
a componen
t
.
Fi
e
ld
s
d
ecays
as 1/r3
Magnetic Fields from Power Cables17©M. T. Thompson, 2009
Example 5: Twisted Pair
Fild1tthttFildd

Fi
e
ld
r =
1
me
t
er away,
th
e
t
a componen
t
.
Fi
e
ld
s
d
ecays
as e-kr
Magnetic Fields from Power Cables18©M. T. Thompson, 2009
AC Magnetic Shielding
•Flux densit
y

p
lots at DC and 60 Hz
yp
•At 60 Hz, currents induced in plate via magnetic induction
create lift force
DC
60H
DC
60

H
z
Magnetic Fields from Power Cables19©M. T. Thompson, 2009
Good Reference
Magnetic Fields from Power Cables20©M. T. Thompson, 2009