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PAPER PREPARATION GUIDELINES
UDC 621.3
ELECTROMAGNETIC FIELD ENERGY FLUX IN TRANSFORMER
Špaldonová D., research, doc., RNDr, PhD, assistant professor
Technical University of Košice
Park Komenského 3, 04200 Košice, Slovak Republic, E-mail: [email protected]
This paper deals with the electromagnetic field energy flux in the basic electric components and engines by the
means of Maxwell’s theory of the electromagnetic field. The shell transformer, as one of the often used electric engine,
was selected to illustrate the application of this theory on the computing the electromagnetic field energy flux. At the
first, electric intensity E(t) and magnetic intensity H(t) of the electromagnetic field in the transformer core and in its
winding was expressed. At the second, the density of the transmitted electromagnetic field energy per unit time i.e. the
density of power, known as Poynting’s vector, was calculated for the transformer winding as well as its core. Next, the
transmitted electromagnetic field energy per unit time, i.e. the transmitted power was calculated for once again for the
transformer core and for its winding too.
Keywords: Theory of electrical engineering, Theory of electromagnetic field, Electromagnetic field energy,
Electromagnetic field energy flux in the basic electric components, Electromagnetic field energy flux.
ДОСЛІДЖЕННЯ ПОТОКУ ЕЛЕКТРОМАГНІТНОЇ ЕНЕРГІЇ
У ТРАНСФОРМАТОРАХ
Спальданова Д., д.прир.н., PhD, доц.
Технічний університет Кошице
вул. Парк Коменскего, 3, м. Кошице, 04200, Словаччина. Е-mail: [email protected]
У статі розглянуто питання дослідження потоку енергії в електромагнітному полі на основі аналізу
елементарних електричних компонентів та електромагнітних перетворювачів на основі електромагнітної теорії
Максвела. Для ілюстрації можливостей застосування даної теорії для розрахунку потоку електромагнітної
енергії було обрано броньовий трансформатор, як один з найчастіше використовуваних електромагнітних
перетворювачів. По-перше, було отримано вирази для опису напруженості електричного поля E(t) та
напруженості магнітного поля H(t) осердя трансформатора та його обмоток. По-друге, було розраховано
щільність переданої електромагнітним полем енергії за одиницю часу, тобто щільність потужності, відому як
вектор Пойнтінга, для обмоток і осердя трансформатора. На наступному кроці було повторно розраховано
передану електромагнітним полем енергію за одиницю часу, тобто передану потужність, для осердя та обмоток
трансформатора.
Ключові слова: теорія електротехніки, теорія електромагнітного поля, енергія електромагнітного поля,
потік енергії електромагнітного поля.
Introduction. The flux of the electromagnetic field
energy in the electric circuits, basic electric components and engines is uncommon studied and it paid
only minimal attention, so that ideas of this process are
often incorrect. However, the knowledge of this process is very important for precisely understanding of its
operation.
The best way to describe this process is by the
means of Maxwell’s theory of the electromagnetic
field, because the differential values describe electromagnetic field in each point of space where it exist and
not only in the points of electric circuit as integral
values do.
The shell transformer, as one of the often used electric engine, was selected to illustrate the application of
this theory on the computing the energy flux.
The description of the processes running in the
transformer is usually concentrated on the specifying of
magnetic flux (t) enclosing in its ferromagnetic core.
The electromagnetic field is understood as constant in
the space and changing in the time only. This implies
the incorrect idea that the electromagnetic field energy
spread from the transformer primary winding through
the ferromagnetic core to the secondary winding while
energy of the electric field in primary winding transform to the energy of magnetic field in the core and
then transform to the energy of the electric field in
secondary winding. This idea is wide extended up to
this day nevertheless that it has been adverted on the
inaccuracy of such an idea several times.
Basic relationships. The base for the expression of
electromagnetic field energy flux is the integral form of
the first two of Maxwell’s equations. The first of them
is Faraday law
 E(t ).dl E  ui (t )
(1)
lE
where
is the induced voltage, which is
B(t )
d(t )
.dS  
dt
S t
ui (t )   
and the second one is Ampere law
 H(t ).dl H  iC (t )
(2)
(3)
lH
where
iC (t )  iv (t )  ip (t )
and there
(4)
PAPER PREPARATION GUIDELINES
iv(t) is the conduction current, which is expressed as
iv (t )   J(t ).dS ;
(5)
S
ip(t) is the displacement current, which is expressed as
D(t )
.dS .
S t
ip (t )  
 the points of the outer primary winding surface as
A1 and the points on its inner surface as A2,
 the points of the core outer surface as B (Fig. 2).
Electromagnetic field in the transformer is best described in the cylindrical coordinate system [r, α, z]
illustrated on the Fig. 2.
z
The density of the transmitted electromagnetic field
energy per unit time i.e. the density of power is defined
by Poynting’s vector as
N(t) = E(t)  H(t)
(6)
and the transmitted electromagnetic field energy per
unit time, i.e. the transmitted power is
P(t )   N(t ).dS 
S
 E(t )  H(t ) .dS .
B
(7)
lH
lB
B
S
So we need to know both the electric intensity E(t)
and the magnetic intensity H(t) to be able to calculate
process of electromagnetic field energy flux.
SHELL TRANSFORMER. The shell transformer is
one of the often used electric engines, so it was selected to illustrate the application of Maxwell’s theory of
electromagnetic field to compute its electromagnetic
field energy flux.
Let us consider a shell transformer in the idle state
(Fig. 1) whose primary winding radius is R0, its height
is l0, its winding thickness is  and it is negligible in
comparison to other sizes of the primary winding. The
transformer core is made from the ferromagnetic material whose permeability is  and cross section of its
middle column is S0. The primary winding is connected
to the voltage u(t).
z
S0/2
S0/2
A1
A2
lEA2
A2
A1
lEA1
r
Figure 2 – Points of transformer winding
surfaces and core surface
TRANSFORMER WINDING. The electric intensity
E(t) in the transformer primary winding may be determined by the means of Faraday law. Under this law

the voltage inducted in one of the primary winding turns is
ui (t ) 
 E(t ).dl E ,
(8)
lE
 the total voltage inducted in all N turns of the primary winding is
S0
S0/2
uic (t )  N  E(t ).dl E  u (t ) .
S0/2
(9)
lE
l0
If the primary winding resistivity as well as its thickness δ be neglected, then the electric intensity EA1(t) in
the points A1 of the outer primary winding surface and
electric intensity EA2(t) in the points A2 of the inner
primary winding surface (Fig. 2) will be the same:
δ
δ
r
R0
u(t)
Figure 1 – Shell transformer
We will examine the electromagnetic field energy
flux in both basic parts of the transformer – in its winding and in its ferromagnetic core.
Denote:
EA1(t) = EA2(t) = E(t)
and the length of one winding turn lE1 for the points A1
of the outer primary winding surface and lE2 for the
points A2 of the inner primary winding surface (Fig.2)
will be the same too:
lE1 = lE2 = lE = 2R0 .
Each of the primary winding turn is an electrical
line of force, so that electric intensity E(t) is tangent to
the lE and have a constant size E(t) at any point of lE.
Then it will be
PAPER PREPARATION GUIDELINES
ui (t )   N  E(t ).dl E   N  E (t )dlE   NE (t )  dlE 
lE
lE
lE
N A2 (t )  E(t )  H A1 (t ) 
  NE (t )lE   NE(t )2R0 ,

so we get
E (t ) 
u (t )
N 2R0
E(t ) 
and
u (t )
α0 .
N 2R0
(11)
The magnetic intensity H(t) in the transformer primary
winding may be determined by the Ampere law, i.e.
 H(t ).dl H  iv (t )  Ni(t ) .
lH
Elected closed integration curve lH closely surrounds winding turns and its area is vertical to the
winding turns area (Fig. 2). If the primary winding
thickness δ is neglected, then the length of the curve lH
will be
1 u (t )i (t )
(r0 ) .
( r  1) 2R0l0
P(t )   N(t ).dS
The magnetic intensity H(t) is tangent to the lH at any
point too, but its size is HA1(t) in the points A1 of outer
surface and HA2(t) in the points A2 of inner surface of
the primary winding and they are
S
z
EA1
H A1 (t )  0 B(t ) and H A2 (t )  0 r B(t ) ,
A1
so we have
NA1
HA1(t) = r HA2(t) .
Then
A1
NA1
lEA
EA1
HA1
lH
HA1
 H(t ).dl H   H A1 (t ).dl H   H A2 (t ).dl
l0
H
r

Figure 3 – Energy density NA1
l0
  H A1 (t )dlH   H A2 (t )dlH  H A1 (t )l0  H A2 (t )l0 
l0
(15)
Therefore it is obvious then the transmitted electromagnetic field energy coming out through the lateral
primary winding surfaces 2πR0l0 perpendicular to these
surfaces. Energy density transmitted through the outer
primary winding surface NA1(t) is directed to the secondary winding (Fig. 3) and energy density transmitted
through the inner primary winding surface NA2(t) is
directed to the transformer core (Fig. 4). It is important
to remember that the energy density NA1(t) directed to
the secondary winding is μr times greater than the energy density NA2(t) directed to the core.
The transmitted electromagnetic field energy per
unit time, i.e. the transmitted power is
lH = 2l0 .
lH
u (t )
Ni(t )
α0 
k
N 2R0
(r  1)l0
.
z
l0
 r HA2 (t )  HA2 (t )  l0  (r  1) HA2 (t )l0
HA2
so we get
A2
( r  1) H A2 (t )l0  Ni (t ) ,
HA2
EA2
NA2 NA2
lEA
A2
EA2
and then
lH
Ni(t )
H A2 (t ) 
( r  1)l0
and
Ni(t )
k,
(r  1)l0
(12)
r Ni(t )
 k  .
(r  1)l0
(13)
H A2 (t ) 
H A1 (t ) 
 Ni(t )
H A1 (t )  r
,
( r  1)l0
Subsequently the density of the transmitted electromagnetic field energy per unit time i.e. the density of
transmitted power consists of two parts:
N A1 (t )  E(t )  H A1 (t ) 

 Ni(t )
u (t )
α0  r
 k  
N 2R0
(r  1)l0
r u (t )i (t ) ,
r0
(r  1) 2R0l0
(14)
r
Figure 4 – Energy density NA2
We select the area of cylinder S as the closed integration
surface which its bases area is S1, S2 and its shell area is S3.
This area closely surround the outer surface of primary
winding for the vector NA1(t) (Fig. 5) and the inner surface of
primary winding for the vector NA2(t) (Fig. 6).
For the first part of the transmitted electromagnetic
field energy per unit time we get
PA1 (t ) 
 NA1 (t ).dS 
S

 N A1 (t ).dS1  S N A1 (t ).dS2  S N A1 (t ).dS3 
S1


3
 N A1 (t ).dS3  S NA1 (t )r0 .dS3r0  S N A1 (t )dS3 
S3

2
S3
3
3
r
u (t )i(t ) r
dS3 
u (t )i(t )
S3 (r  1)

 r  1
(16)
PAPER PREPARATION GUIDELINES
and for the second part of the transmitted electromagnetic field energy per unit time we have
PA2 (t )   NA2 (t ).dS 
S

 N A2 (t ).dS1  S N A2 (t ).dS2  S N A2 (t ).dS3 
S1

2
3
 NA2 (t ).dS3  S NA2 (t )(r0 ).dS3r0  S NA2 (t )dS3 
S3
3
3
1 u (t )i(t )
1

dS3  
u (t )i(t ) .
r  1
S  r  1 S3
(17)
3
The sign (−) implies that energy is transmitted into
the volume V of integration surface S.
dS1
S1
Therefore the energy supplied to the primary winding per time unit coming out through the lateral primary winding surfaces S3 perpendicular to these surfaces.
The size of the energy PA1(t), which get to the space
between the primary and secondary transformer winding per time unit is r/(r + 1)-part of the energy supplied to the primary winding per time unit and size of
the energy PA2(t), which get to the space of the transformer core is 1/(r + 1)-part of the energy supplied to
the primary winding per time unit.
TRANSFORMER CORE. Let us identify the points
B of the transformer core surface with the points A2 of
the inner surface of the transformer primary winding.
Then the electric intensity EB(t) in the points B of
transformer core surface and the electric intensity
EA2(t) in the points of the transformer primary winding
inner surface will be the same (Fig. 7):
u (t )
,
N 2R0
EB (t )  EA2 (t )  E (t ) 
(18)
u (t )
α0
(19)
N 2R0
and the magnetic intensity HB(t) in the points B of
transformer core surface will be the same as the magnetic intensity HA2(t) in the points of the transformer
primary winding inner surface of the (Fig. 7):
EB (t ) 
A1
A1
NA1
NA1
S3
dS3
dS3
S1
H B (t )  H A2 (t ) 
H B (t ) 
dS2
Ni(t )
,
(r  1)l0
(20)
Ni(t )
k.
( r  1)l0
(21)
z
HB
Figure 5 – Integration area
for the vector NA1
B
lEB
HB
EB
NB
NB
B
EB
lH
dS1
r
S1
Figure 7 – Energy density NB
A2
NA2
NA2
A2
Then the density of the transmitted electromagnetic
field energy per unit time i.e. the density of transmitted
power NB(t) and NA2(t) will be the same too:
N B (t )  N A2 (t ) 
S3
dS3
dS3
S2
dS2
1 u (t )i(t )
(r0 ) .
(r  1) 2R0l0
(22)
The transmitted electromagnetic field energy per
unit time, i.e. the transmitted power PB(t) will be
PB (t ) 
 N B (t ).dS .
S
Figure 6 – Integration area
for the vector NA2
The area of cuboid S was selected as the closed integration area now which its bases area is S1, S2 and its
shell area is S3 (Fig. 8). The energy PB(t) for this elected integration area S will be the same as PA2(t):
PAPER PREPARATION GUIDELINES
PB (t ) 
 NB (t ).dS 
and the whole magnetic flux ΦC(t) closed by the all N
primary winding turns is
S

 NB (t ).dS1  S NB (t ).dS2  S N B (t ).dS3 
S1

2
3
 NB (t ).dS3  S NB (t )(r0 ).dS3r0  S N B (t )dS3 
S3
3
3
1 u (t )i(t )
1
dS3  
u (t )i(t ) ,
r  1 S3
r  1

S3
so it really is
1
PB (t )  PA2 (t )  
u (t )i(t ) .
r  1
(23)
NB
dS3
NB B
S3
dS3
S2
dS2
Figure 8 – Integration area
for the vector NB
This means that size of the energy PB(t), which get
to the transformer core is 1/(r + 1)-part of the energy
supplied to the primary winding per time unit.
The magnetic intensity H(t) inside the transformer
core is constant and such as that one on its surface
HB(t):
H (t )  H B (t ) 
H(t ) 
Ni(t )
,
(r  1)l0
Ni (t )
k.
( r  1)l0
(24)
(25)
The magnetic induction B(t) in the transformer core
then will be
B(t) =  H(t) = 0r H(t) ,
the magnetic flux Φ(t) closed by one of the primary
winding turns is
(t ) 
 B(t ).dS0  S B(t )kdS0k  S B(t )dS0 
S
0


S0
0  r Ni  t 
0
0
  Ni (t ) S0
,
dS0  0 r
( r  1)l0
( r  1) l0
(26)
0r N 2i (t ) S0
.
 r  1 l0
(27)
This expression is identical with the expression of
the whole magnetic flux ΦC expressed in determining
of the coil inductance L in the stationary electromagnetic field:
C 
dS1
S1
B
 C (t )  N (t ) 
0r N 2 IS0
,
l0
(28)
in which the electromagnetic induction does not exist,
so that the whole energy supplied to the primary winding per time unit is transmitted into the coil core and
rise the magnetic flux ΦC.
Therefore it is evident that 1/(r + 1)-part of the electromagnetic field energy which get into the transformer
core rise the magnetic flux ΦC(t).
Conclusions. The incorrect ideas of the electromagnetic field energy flux in the electric circuits, its
basic components and electric engines still appears in
the print and electronic media. Mainly it is the idea that
the electromagnetic field is constant in the space and
changing in the time only. This implies such an incorrect idea as the electromagnetic field energy spread
through conductive parts of the electric and magnetic
circuits. So in this work was demonstrated how the
energy spread actually in one of the electric engine – in
the transformer.
The widespread perception is that the electromagnetic field energy spread from the transformer primary
winding through the ferromagnetic core to the secondary winding while energy of the electric field in primary winding transform to the energy of magnetic field in
the core and then transform to the energy of the electric
field in secondary winding. Actually the energy supplied to the primary winding coming out through the
lateral primary winding surfaces perpendicular to these
surfaces. The size of the energy which get to the space
between the primary and secondary transformer winding is r/(r + 1)-part of the energy supplied to the
primary winding and size of the energy which get to
the space of the transformer core is only 1/(r + 1)-part
of the energy supplied to the primary winding per time
unit. That part of the electromagnetic field energy
which get into the transformer core rise the magnetic
flux only.
Acknowledgement. The paper has been prepared by the support of Slovak grant projects
KEGA No. 005TUKE-4/2012, KEGA No. 014TUKE4/2013.
REFERENCES
1. Haňka, L.: Electromagnetic Field Theory.
SNTL/ALFA, Praha, 1982
2. Myslík, J.: Electromagnetic Field. BEN technická literatura, Praha
3. Kvasnica, J.: Electromagnetic Field Theory.
ACADEMIA, Praha, 1985
PAPER PREPARATION GUIDELINES
4. Tirpák, A.: Electromagnetics. POLYGRAFIA
SAV, Bratislava, 1999
5. Mayer, D.: Electromagnetic Field Theory.
Edičné stredisko VŠSB, Plzeň, 1981
6. Neveselý, M.: Electrical Engineering III/2.
VŠDS, Žilina, 1995
7. Špaldonová,D., Neveselý, M., Šuriansky, J.:
Electromagnetic Field Transmission and Transfor-
mation in the Electric Engines. VA SNP, Liptovský
Mikuláš, 1999
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ИССЛЕДОВАНИЕ ПОТОКА ЭЛЕКТРОМАГНИТНОЙ ЭНЕРГИИ
В ТРАНСФОРМАТОРАХ
Спальданова Д., д.ест.н., PhD, доц.
Технический университет Кошице
ул. Парк Коменскего, 3, м. Кошице, 04200, Словакия. Е-mail: [email protected]
В статье рассмотрен вопрос исследования потока энергии в электромагнитном поле на основе анализа
элементарных электрических составляющих и электромагнитных преобразователей на основе теории
Максвелла. Для иллюстрации возможностей применения данной теории для задач расчёта потока
электромагнитной энергии был выбран броневой трансформатор, как один из наиболее часто используемых
электромагнитных преобразователей. Во-первых, были получены выражения для описания напряжённости
электрического поля E(t) и напряжённости магнитного поля H(t) сердечника трансформатора и его обмоток.
Во-вторых, была рассчитана плотность переданной электромагнитным полем энергии за единицу времени, то
есть плотность мощности, известную как вектор Пойнтинга, для обмоток и сердечника трансформатора. На
следующем шаге была повторно рассчитана переданная электромагнитным полем энергия за единицу времени,
т.е. переданная мощность, для сердечника и обмоток трансформатора.
Ключевые слова: теория электротехники, теория электромагнитного поля, энергия электромагнитного
поля, поток энергии электромагнитного поля.