Application of the computing environment maple to the calculation of the dynamics of the electromagnets in the complicated systems of forced control
The authors propose a technique for calculating the dynamics of electromagnets operating in complex forced systems. Such forced electromagnets are widely used in electromechanical switching devices, in particular in vacuum contactors, to reduce their size, energy consumption and to increase speed, w...
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Інститут технічних проблем магнетизму НАН України
2019
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| Назва видання: | Електротехніка і електромеханіка |
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| Цитувати: | Application of the computing environment maple to the calculation of the dynamics of the electromagnets in the complicated systems of forced control / E.I. Baida, M. Clemens, B.V. Klymenko, O.G. Korol, P.Ye. Pustovoitov // Електротехніка і електромеханіка. — 2019. — № 3. — С. 18-23. — Бібліогр.: 10 назв. — англ. |
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irk-123456789-1590172019-09-21T01:26:13Z Application of the computing environment maple to the calculation of the dynamics of the electromagnets in the complicated systems of forced control Baida, E.I. Clemens, M. Klymenko, B.V. Korol, O.G. Pustovoitov, P.Ye. Електричні машини та апарати The authors propose a technique for calculating the dynamics of electromagnets operating in complex forced systems. Such forced electromagnets are widely used in electromechanical switching devices, in particular in vacuum contactors, to reduce their size, energy consumption and to increase speed, which indicates the relevance of this topic. A mathematical model of the dynamics of a forced electromagnetic system, which takes into account the peculiarities of behavior in transients of its individual elements – the mechanical system, the magnetic and electrical circuits, taking into account the interaction of the electromagnet with a control device when the apparatus is activated, contains certain signs of scientific novelty and is the purpose of the paper. Розглядається запропонована авторами методика розрахунку динаміки електромагнітів, що працюють у складних форсованих системах. Подібні форсовані електромагніти широко застосовуються в електромеханічних комутаційних апаратах, зокрема у вакуумних контакторах, для зменшення їх розмірів, споживання енергії та для підвищення швидкодії, що свідчить про актуальність даної теми. Математична модель динаміки форсованої електромагнітної системи, що враховує особливості поведінки у нестаціонарних процесах її окремих елементів – механічної системи, магнітного та електричного кіл з урахуванням взаємодії електромагніта з пристроєм керування під час спрацьовування апарата, містить певні ознаки наукової новизни і є метою статті. Рассматривается предложенная авторами методика расчета динамики электромагнитов, работающих в сложных форсированных системах. Подобные форсированные электромагниты широко применяются в электромеханических коммутационных аппаратах, в частности в вакуумных контакторах, для уменьшения их размеров, потребления энергии и для повышения быстродействия, что свидетельствует об актуальности данной темы. Математическая модель динамики форсированной электромагнитной системы, учитывающая особенности поведения в нестационарных процессах ее отдельных элементов – механической системы, магнитной и электрической цепей с учетом взаимодействия электромагнита с устройством управления при срабатывании аппарата, содержит определенные признаки научной новизны и является целью статьи. 2019 Article Application of the computing environment maple to the calculation of the dynamics of the electromagnets in the complicated systems of forced control / E.I. Baida, M. Clemens, B.V. Klymenko, O.G. Korol, P.Ye. Pustovoitov // Електротехніка і електромеханіка. — 2019. — № 3. — С. 18-23. — Бібліогр.: 10 назв. — англ. 2074-272X DOI: https://doi.org/10.20998/2074-272X.2019.3.03 http://dspace.nbuv.gov.ua/handle/123456789/159017 621.316+621.316.53 en Електротехніка і електромеханіка Інститут технічних проблем магнетизму НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Електричні машини та апарати Електричні машини та апарати |
| spellingShingle |
Електричні машини та апарати Електричні машини та апарати Baida, E.I. Clemens, M. Klymenko, B.V. Korol, O.G. Pustovoitov, P.Ye. Application of the computing environment maple to the calculation of the dynamics of the electromagnets in the complicated systems of forced control Електротехніка і електромеханіка |
| description |
The authors propose a technique for calculating the dynamics of electromagnets operating in complex forced systems. Such forced electromagnets are widely used in electromechanical switching devices, in particular in vacuum contactors, to reduce their size, energy consumption and to increase speed, which indicates the relevance of this topic. A mathematical model of the dynamics of a forced electromagnetic system, which takes into account the peculiarities of behavior in transients of its individual elements – the mechanical system, the magnetic and electrical circuits, taking into account the interaction of the electromagnet with a control device when the apparatus is activated, contains certain signs of scientific novelty and is the purpose of the paper. |
| format |
Article |
| author |
Baida, E.I. Clemens, M. Klymenko, B.V. Korol, O.G. Pustovoitov, P.Ye. |
| author_facet |
Baida, E.I. Clemens, M. Klymenko, B.V. Korol, O.G. Pustovoitov, P.Ye. |
| author_sort |
Baida, E.I. |
| title |
Application of the computing environment maple to the calculation of the dynamics of the electromagnets in the complicated systems of forced control |
| title_short |
Application of the computing environment maple to the calculation of the dynamics of the electromagnets in the complicated systems of forced control |
| title_full |
Application of the computing environment maple to the calculation of the dynamics of the electromagnets in the complicated systems of forced control |
| title_fullStr |
Application of the computing environment maple to the calculation of the dynamics of the electromagnets in the complicated systems of forced control |
| title_full_unstemmed |
Application of the computing environment maple to the calculation of the dynamics of the electromagnets in the complicated systems of forced control |
| title_sort |
application of the computing environment maple to the calculation of the dynamics of the electromagnets in the complicated systems of forced control |
| publisher |
Інститут технічних проблем магнетизму НАН України |
| publishDate |
2019 |
| topic_facet |
Електричні машини та апарати |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/159017 |
| citation_txt |
Application of the computing environment maple to the calculation of the dynamics of the electromagnets in the complicated systems of forced control / E.I. Baida, M. Clemens, B.V. Klymenko, O.G. Korol, P.Ye. Pustovoitov // Електротехніка і електромеханіка. — 2019. — № 3. — С. 18-23. — Бібліогр.: 10 назв. — англ. |
| series |
Електротехніка і електромеханіка |
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18 ISSN 2074-272X. Electrical Engineering & Electromechanics. 2019. no.3
© Ye.I. Baida, M. Clemens, B.V. Klymenko, O.G. Korol, P.Ye. Pustovoitov
UDC 621.316+621.316.53 doi: 10.20998/2074-272X.2019.3.03
Ye.I. Baida, M. Clemens, B.V. Klymenko, O.G. Korol, P.Ye. Pustovoitov
APPLICATION OF THE COMPUTING ENVIRONMENT MAPLE
TO THE CALCULATION OF THE DYNAMICS OF THE ELECTROMAGNETS
IN THE COMPLICATED SYSTEMS OF FORCED CONTROL
Загальний опис теми дослідження. Розглядається запропонована авторами методика розрахунку динаміки електромагнітів,
що працюють у складних форсованих системах. Подібні форсовані електромагніти широко застосовуються в
електромеханічних комутаційних апаратах, зокрема у вакуумних контакторах, для зменшення їх розмірів, споживання
енергії та для підвищення швидкодії, що свідчить про актуальність даної теми. Математична модель динаміки форсованої
електромагнітної системи, що враховує особливості поведінки у нестаціонарних процесах її окремих елементів – механічної
системи, магнітного та електричного кіл з урахуванням взаємодії електромагніта з пристроєм керування під час
спрацьовування апарата, містить певні ознаки наукової новизни і є метою статті. Методика розрахунку динаміки
форсованих електромагнітів застосовує математичний пакет Maple. В основу розрахунку покладено математичну модель,
яка представляє собою систему нелінійних диференційних рівнянь магнітного і електричного кіл, доповнених рівняннями
руху елементів механічної системи. Застосування пакету Maple, який багато в чому бере на себе складнощі математичного
опису різних процесів, автоматично здійснюючи дуже складні і громіздкі математичні перетворення, дозволяє, уникаючи
складних процесів вибору способу чисельного інтегрування, програмування складних й громіздких рівнянь та процедур їх
чисельного інтегрування, отримувати результати розрахунків у зручній табличній та/або графічній формі, що свідчить про
практичну значущість даної роботи. Наведені у статті результати зіставлення розрахунків з опублікованими раніше
експериментальними даними, свідчать про високу ефективність запропонованих моделей та методик. Бібл. 10, рис. 6.
Ключові слова: електромагніти, динаміка, форсоване керування, комутаційні апарати, вакуумні контактори,
математичний пакет Maple.
Общее описание темы исследования. Рассматривается предложенная авторами методика расчета динамики
электромагнитов, работающих в сложных форсированных системах. Подобные форсированные электромагниты
широко применяются в электромеханических коммутационных аппаратах, в частности в вакуумных контакторах,
для уменьшения их размеров, потребления энергии и для повышения быстродействия, что свидетельствует об
актуальности данной темы. Математическая модель динамики форсированной электромагнитной системы,
учитывающая особенности поведения в нестационарных процессах ее отдельных элементов – механической
системы, магнитной и электрической цепей с учетом взаимодействия электромагнита с устройством управления
при срабатывании аппарата, содержит определенные признаки научной новизны и является целью статьи.
Методика расчета динамики форсированных электромагнитов применяет математический пакет Maple. В основу
расчета положена математическая модель, которая представляет собой систему нелинейных дифференциальных
уравнений магнитного и электрического кругов, дополненных уравнениями движения элементов механической
системы. Применение пакета Maple, который во многом берет на себя сложности математического описания
различных процессов, автоматически осуществляя очень сложные и громоздкие математические преобразования,
позволяет, избегая сложных процессов выбора способа численного интегрирования, программирование сложных и
громоздких уравнений и процедур их численного интегрирования, получать результаты расчетов в удобной
табличной и/или графической форме, что свидетельствует о практической значимости данной работы.
Приведенные в статье результаты сопоставления расчетов с опубликованными ранее экспериментальными
данными, свидетельствуют о высокой эффективности предложенных моделей и методик. Библ. 10, рис. 6.
Ключевые слова: электромагниты, динамика, форсированное управление, коммутационные аппараты, вакуумные
контакторы, математический пакет Maple.
Introduction. DC electromagnets are simpler in design
than the AC electromagnets as they have higher reliability
and durability. In terms of initial traction force, size and
mass, however, they lose significantly in comparison to the
the AC electromagnets (which are actually forced
electromagnets), because during the operation of the
electromagnet, the currents in their windings exceed (ten
times or more) the values of currents that are in the
windings after the operation of the electromagnet. Forced
control of DC electromagnets essentially means that during
operation a current flows through a winding, whose value
significantly exceeds the current permissible under long-
term heating, is used to increase the traction force during
operation and to increase the speed of the apparatus. After
operation, the current in the winding and, accordingly, its
magnetomotive force (MMF) are reduced, but the armature
of the electromagnet remains in the final (brought) state,
since, at small gaps, the traction force is usually
superfluous even at small values of MMF.
Forced electromagnetic systems (FEMS) [1] are
widely used in drive systems of low and medium voltage
electromechanical switching devices, in particular in
contactors, which execute switching operations (switching
on and off) the main circuits of powerful electric motors
and some other objects.
The FEMS usually includes the electromagnetic
mechanism (EMM) – the main contact module and the
actuator, which provides the execution of switching operations
by contacts (in contactors, the actuator's role is most often
performed by a forced unpolarized electromagnet with a
rotating spring) and a control device (CD) that performs
changes in the windings‘ control circuit to provide the required
values of currents both during operation and in the final state.
Note, that in the AC electromagnet which is actually forced,
since during the operation the current in the winding is much
greater than the current in the final state, the values of the
current change without any CD due to the difference in the
values of the inductance in the released and final states).
ISSN 2074-272X. Electrical Engineering & Electromechanics. 2019. no.3 19
When designing switching devices with FEMS, it is
necessary to take into account the interaction of the
electromagnet with the control device during the
operation of the apparatus. Therefore, it is necessary to
calculate the dynamics of the electromagnet taking into
account the action of the CD. There is a large number of
publications devoted to the methods of calculating the
dynamics of electromagnets, some of which are listed in
references [2 – 9]. Nonetheless, in our opinion,
insufficient attention is given to the issue of the
interaction of the forced electromagnet with a CD taking
into account the large variety of existing CDs and some
features of operation of forced electromagnets in
switching devices.
The goal of the work is to describe the mathematical
model of FEMS dynamics, which takes into account
peculiarities of behavior in switching devices of individual
elements of FEMS during transients, as well as to build a
technique for calculating the dynamics of forced
electromagnets using the computing environment Maple.
Mathematical model. The calculation of the FEMS
dynamics, where the electromagnet is only a part of the
control system that contains the power source, the
mechanical system, the forced control device and the
electromagnet itself, is reduced to a solution of coupled
differential equations. These describe the transients in the
circuits of the windings and in the circuits of the forced
control device of these windings, transients in magnetic
circuits of electromagnets taking into account the effect of
eddy currents. They also describe the dynamics of the
moving part of the switching device, namely, the armature
of the electromagnet, the main contacts of the apparatus,
mechanical parts that connect armature with moving
contacts (levers, rods, springs, etc.). In many cases, forced
control systems are designed so that, when the switching
device is operated, the circuit diagrams of the windings
are automatically changed if there are several windings or
if there are certain changes in the circuits of the forced
control device. In the process of performing the on and
off operation of the device, also the mass (the moment of
inertia) of the moving system can significantly change. If
the switching of the main circuits uses vacuum
interrupters, the process of movement is significantly
affected by bellows and the actual vacuum, which
“prevents” the opening of the contacts, pulling the
moving contact to the fixed one. All of the above-
mentioned factors must be taken into account by creating
separate fragments of the mathematical model of the
dynamics of the electromagnet in the composition of the
forced control system, i.e., writing the equation for
transients in the mechanical system, as well as in the
magnetic and electric circuits of the forced
electromagnetic system.
Transients in the mechanical system of the
switching device. Kinematics of switching devices in
many cases are built so that the part of moving elements
carries translational motion, and the other part carries
rotary motion (Fig. 1). Since the angles of rotation of
moving parts in actual apparatus are relatively small and
usually do not exceed 10–15, a rotating motion with a
slight error can be reduced to a translational motion
relative to an element adopted as a basic one. Such
element may be, for example, an actuator whose armature
moves progressively along the axis of the actuator, which
is at a distance ra from the axis of rotation O of the lever,
to which other parts of the switching device are attached
and which operate at different distances (shoulders) from
the axis O. From the initial armature position the path s
passed by the actuator is measured. In calculating the
dynamics of motion reduced to the motion of the base
element (in our case, to the actuator's armature), the
masses of parts moving at different distances from the O
axis must be replaced by reduced masses in accordance
with the condition of maintaining the moment of inertia.
For example, the reduced mass dm of the part with mass
md acting at distance rd from the O axis is given by the
formula:
2
a
2
ddd / rrmm . (1)
If the moment of inertia J of some part, for example of
the lever shown in Fig. 1, is known, then the calculation
of its reduced mass is carried out according to the
formula:
2
a
/
J / rJm . (2)
Thus, the actuator’s mass reduced to the axis of the
actuator's motion for the fragment shown in Fig. 1 will be
equal to:
2
a
2
a
2
ddaa // rJrrmmm . (3)
The forces that counteract the movement of the
actuator’s armature must also be replaced by reduced
forces, which are calculated in accordance with the
condition of maintaining the moment of force. For
example, the force Fs of a spring acting at a distance rs
from the O axis in the calculation of the dynamics of the
actuator's armature motion should be replaced by the
reduced force
asss / rrFF . (4)
rd
rs
ra
Fs
armature (ma)
O
lever (J)
detail (md)
F
actuator
0
s
Fig. 1. A fragment of the mechanical system
of the switching device
A characteristic feature of electromechanical
switching devices is the gradual nature of the forces
opposing the movement of the actuator's armature as well
as the gradual nature of the change in the mass of moving
parts due to the peculiarities of the kinematics of these
devices (Fig. 2).
20 ISSN 2074-272X. Electrical Engineering & Electromechanics. 2019. no.3
el
ec
tr
om
ag
ne
t
ar
m
at
ur
e
li
nk
in
su
la
to
r
re
tu
rn
s
pr
in
g
tr
av
er
se
lo
ck
-n
ut
s
co
nt
ac
t s
pr
in
g
fl
ex
ib
le
c
on
ne
ct
io
n
be
ll
ow
s
m
ov
in
g
co
nt
ac
t
fi
xe
d
co
nt
ac
t
0
sk
s sc
co
nt
ac
t g
ap
fu
ll
a
ct
ua
to
r
ga
p
so (overtravel)
in
it
ia
l p
os
iti
on
(
po
si
tio
n
of
r
es
t)
in
it
ia
l c
on
ta
ct
p
os
it
io
n
en
d
po
si
tio
n
so
Fig. 2. Kinematic circuits of the switching device in three
characteristic positions
During the operation of the switching device, two stages
of the motion of its mechanical system can be observed: the
first stage – from the initial position (position of rest) to the
initial contact position (0 < s < sc) and the second stage –
from the of initial contact position to the end position (sc < s
< sk). At the first stage, the reduced mass of moving parts is
comprised of the reduced mass of the mechanical system mv
and a reduced mass of moving contacts mс. At the second
stage, the moving contacts stop – they are faced with fixed
contacts, therefore the reduced mass of moving parts is
almost abruptly reduced to the value mv
.at
;at
сv
сcv
ssm
ss mm
m (5)
Thus, when the switching device operates, there is a
motion with a variable mass, which is described by such a
system of differential equations:
;
d
d
rFFvm
t
(6)
,
d
d
v
t
s
(7)
where F is the electromagnetic force providing the
actuator's movement, Fr is the reduced force that
counteracts the movement of the actuator’s armature, v is
the speed of the actuator's armature, s is the path passed
by the armature from the beginning of the movement, t is
time.
After performing the differentiation operation in (6),
we obtain:
;
d
d
d
d
r
2 FFv
s
m
t
v
m (8)
Function (5) is discontinuous, at its differentiation
there are pulsed functions which makes it practically
impossible to carry out further calculations using (8),
therefore we have applied the approximation of this
function with the use of hyperbolic tangent. As a result,
the following expression is obtained:
2/)))((th1( ccv ssAmmm , (9)
where A is a suitably chosen large constant number.
The function (9) is smooth, allowing for a
differentiation operation, but the mathematical expression
of the derivative is very cumbersome, but the Maple
computing environment does not require the programmer
to perform transformations related to bringing the
equations to a canonical form: this code performs all the
required complex algebraic transformations itself. In the
Maple environment it is only necessary to write
expressions (7), (8), (9) and write the command that
provides the solution of the system of differential
equations.
The force Fr , which counteracts the movement of the
actuator’s armature is formed due to the action of the
rotary and contact springs, the action of forces of the
deformation of the bellows, the action of vacuum and
friction. This force has a step-by-step nature, but since it
does not require differentiation, the corresponding
expression in the code can be written as follows:
.,at )(
;at
cc
ck
34
3
c
c
12
1
r
ssss
ss
FF
F
sss
s
FF
F
F (10)
where F1, F2 are the values of reduced countermeasures
force, respectively, at the beginning and end of the first
stage of the movement of the mechanical system of the
apparatus; F3, F4 are the values of reduced countermeasure
force, respectively, at the beginning and end of the second
stage of the movement of the mechanical system of the
apparatus.
The difference between F3 and F2 must be equal to the
sum of the values of the initial contact forces at all poles of
the apparatus.
Transients in the magnetic circuit. A mathematically
rigorous calculation of transient magnetic field can be carried
out by solving a system of nonlinear partial differential
equations featuring the magnetic vector potential A. Similar
problems are solved relatively simply for 2D plane-parallel
or plane-meridian fields [9]. In such cases, where the total
costs of simulating high-fidelity 3D models is inacceptably
high and even 2D models are not applicable, alternative
techniques may need to be used including those that have
proven themselves well in the past. Such a technique for
magnetic circuits is a method of electromagnetic analogies,
complemented by new features provided by modern
software products, in particular the Maple code.
Figure 3 shows an substitution circuit of the magnetic
circuit of a double-rod electromagnet used in the vast
majority of vacuum and low voltage contactors. In contrast
to the well-known circuit, which is given in many sources, in
particular in [1], in this circuit, on each section of the core,
divided along the axis into n equal parts, the «eddy» MMFs
that arise during transients are introduced:
.,...,2,1,
d
d
cmc nj
t
Φ
GF j
j (11)
ISSN 2074-272X. Electrical Engineering & Electromechanics. 2019. no.3 21
Rma
Λ0
Λ0 Rmc1
Rmc1 Fm
Fmc1
Fmc1
Rmy
Fm
Фa
Фof1 Фcn = Фy
Fma Fmy Ф1 Ф0 = Фa
Rmc2
Rmc2 Fm
Fmc2
Fmc2
Fm
Фof2
Ф2
Rmcj
Rmcj Fm
Fmcj
Fmcj
Fm
Фof3
Фj
Rmcn
Rmcn Fm
Fmcn
Fmcn
Fm
Фofn
Фn
Фofj Фofj+1
Λ
of
Λ
of
Λ
of
Λ
of
Λ
of
Λ
of
Fig. 3. Substitution circuit of the magnetic circuit of a double-rod electromagnet taking into account the effect of eddy currents in cores,
armature and yoke
«Eddy» MMFs also appear in the branches of the
substitution circuit corresponding to the sections of the
armature and yoke:
.
d
d a(y)
a(y)ma(y) t
Φ
GF (12)
In (11), (12) and in Fig. 3 the following notation is used:
Gc is the electrical conductance of the equivalent short-
circuited circuit in the eddy current path that arises in the
area with the number j divided into n equal parts of the core
of length lc, Ga(y) is the electrical conductance of the
equivalent short-circuited circuit in the eddy current path
generated in the armature, j is the magnetic flux through
the area of the core with the number j. The variables a, y
are magnetic fluxes through the armature and the yoke,
respectively. The variable Fm denotes the MMF of one coil1,
Fm is the MMF of one part of the coil, divided into n equal
parts (Fm = Fm / n), Rma, Rmy are the magnetic resistance of
the armature and yoke, respectively. The variable Rmсj is the
magnetic resistance of the core section with the number j, 0
is the magnetic conductivity of the working gap, of is the
magnetic conductivity of outflow which falls on one area of
the core. The variables Fmа, Fmy describe the «eddy» MMFs
arising respectively in the armature and yoke; Fmcj is the
«eddy» MMF that occurs in the core section with the number j.
The calculation of the magnetic resistances Rmсj, Rma,
Rmy, the electrical conductances Gc, Ga, Gy, as well as the
magnetic conductance of outflow of is carried out
according to the formulas:
))/(/( cccmc SSΦnlR jj , (13)
))/(/( a(y)a(y)a(y)a(y)ma(y) SSΦlR , (14)
)π8/(сc snlG , (15)
))//(16/( a(y)a(y)a(y)a(y)a(y)a(y) cbbclG s , (16)
nlΛ /сof , (17)
where is the specific magnetic conductivity of outflow;
in the case of two parallel circular cores we have:
1)/(/ln 2
сс
0
dldl
μ
. (18)
The conductances of working gaps can be calculated
by the method of enlarged field tubes:
1 Electromagnet, the circuit of which is shown in Fig. 3, is double-
rod, so it has two identical coils, each of which can have one or
more windings. In the first case, the MMF of the coil is equal to
the MMF of its winding, in the second case, the MMF of the coil
is equal to the sum of MMFs of windings with different number
of turns and different currents in them.
2
2
4
232.163.1
42 p
p
pp
2
p0
0 lδ
d
lδd
δ
dμ
Λ
. (19)
In (13)-(19) we indicate: is the relative magnetic
permeability, i.e., a nonlinear function (for a specific
magnetic material; this function, depending on the magnetic
flux density, is usually given in tabular form.), dc, Sc are the
diameter and cross-sectional area of the core, dp, Sp, lp are the
diameter, cross-sectional area and thickness of the pole tip, l
is the distance between the axes of the cores and is the
working air gap between the pole tip and armature.
The calculation of the magnetic circuit makes it possible
to determine the magnetic fluxes, and thus the traction force
created by the electromagnet. On the other hand, the traction
power determines the movement of the mechanical system,
therefore, processes in the mechanical system and in the
magnetic circuit are interrelated. These processes, however,
are inextricably coupled with processes in electrical circuits,
which we consider below.
Transients in the electric circuit. In this paper,
choosing from a lot of existing systems of forced control,
we consider a system, which is most often used in low
and medium voltage contactors (Fig. 4).
~–
С
VD1
Y2
Y1
H
VD
B
u u0
iB
iB
iH
iH
iC
HB
KM:
iC
LB
LB
LH
LH
(RB) (RH)
(RH)(RB)
Fig. 4. Principle electrical circuit
of a widespread system of forced
control that is used in low and
medium voltage vacuum
contactors and in some SF6
medium voltage contactors [10];
u is the instantaneous value of the
nominative voltage of the control
circuit; u0 is the voltage at the
output of the diode bridge
This forced control system provides power supply from
a DC or AC source:
.ACfor )π2sin(
;DCfor
m tfU
U
u
(20)
where f is the frequency, is the initial phase.
Thus, the voltage at the output of the diode bridge can
be constant or rectified, but in any case, the voltage
reduction due to its drop in the bridge diodes should be
taken into account, which can be especially significant
when powered from the network of ultra-low voltage:
)(2 Bd0 iuuu . (21)
In this paper, the nonlinear characteristic of the diode is
replaced by a piecewise linear dependence: a very large
resistance to Rrd at negative (reverse) currents, at relatively
large positive currents of the voltage drop on the diode is
22 ISSN 2074-272X. Electrical Engineering & Electromechanics. 2019. no.3
considered to be a constant value of Ud0, and at relatively
small positive currents smaller than some value I0, the diode
is considered as a resistor with resistance Ud0 / I0:
.at
;0at /
;0at
00d
000d
rd
d
IiU
IiIiU
iiR
u (22)
Another feature of this system of forced control – it
involves the use of an electromagnet with two coils, each
of which has two windings – booster (B) and holding (H).
Booster windings are wound by a wire of a relatively
large diameter and have a large MMF. These windings
operate short-term – during the operation of the
electromagnet, when the control auxiliary contact KM is
closed, and as a result they are connected in series,
connected to the power supply to the control circuit, and
generate the MMF that is sufficient to allow the contactor
to operate. After operation of the contactor, the contact
KM opens and the retaining windings are connected in
series with the booster windings. These are wound with a
relatively thin wire. As a result, they have considerably
greater resistance and considerably less MMF, which at a
small gap is sufficient to hold the armature of the
electromagnet in the pulled state. Capacitor C provides
efficient arc extinguishing at interruption of high current
that passed through the booster windings, and the diode
VD2 prevents countercurrent in the holding windings
when the contactor is braked, and in some cases even
blocks the operation of the apparatus.
This electric circuit is often considered to be too
complicated for programming, but its description is much
simpler if the parallel connection of the capacitor C with the
auxiliary control contact KM, which is closed when the
coordinate s of the armature stroke does not exceed the value
of the coordinate sa at which the opening of this contact
occurs, to replace with one capacitor with very high
capacitance CM when s sa, and at s > sa the capacitance of
this capacitor becomes equal to its nominal value C0:
.at
;at
a0
aM
ssC
ssC
C (23)
In this case, we obtain the following differential
equations for electric circuits:
;
d
d
2
d
d
22 C
0
B
B
BBB0 u
t
Φ
N
t
i
LiRu (24)
);(
d
d
2
d
d
22 Hd
0
H
H
HHHC iu
t
Φ
N
t
i
LiRu (25)
,
d
d
HB
C ii
t
u
C (26)
In these three equations, the unknown values are
currents iB, iH, voltage uC and magnetic flux 0, but the
latter is determined when solving the corresponding
magnetic circuit equation.
Note that the Maple code does not require
representing the task of integrating systems of differential
equations in the form of a system solved with respect to
the first derivatives. It independently chooses the method
and step of integration, as well as independently performs
complex algebraic transformations, which makes it an
valuable tool for solving complex problems of forced
systems dynamics calculation.
Model validation on a full specimen.
Data of experimental studies of industrial samples of
vacuum contactors are presented in one of the previous
papers [10], where forced control systems were used as in
Fig. 4. In particular, the oscillograms of the dynamic
characteristics of the currents in the booster and holding
windings were obtained at the power supply of the control
system from DC and AC sources. The experimental data
were compared with the results of calculations performed
using a simplified model of the magnetic circuit, which was
considered as a circuit with lumped parameters (Fig. 5).
Rma
Λδ Λδ
Λofe
Rmc
Rmc
Fm
Fmc
Fmc
Rmy
Fm
Фa
Фof
Фy = Фc
Fma
Fmy
Ф1 = Фc
Ф0 = Фa
yoke (y)
armature (a)
ΛδΛδ
Фy
Фc
Фa Fma
Fmy
Fmc
Фc
Fmc
Λofe
Фof
ly
lclc
la
co
re
(
c)
co
re
(
c)
Fig. 5. A sketch of a double-rod magnetic core of the
electromagnet and its simplified substitution circuit
The equations compiled for two circuits with magnetic
fluxes 0 and 1 have the following form:
;/)(
2
)/(
d
d1
ofe01
δ
2
aa0
0
a
ФФ
Ф
lSΦH
t
Ф
R
(27)
t
Ф
GGwiwi
d
d
)2(22 1
ycHHBB
ofe01cc1yy1 /)()/(2)/( ФФlSФHlSФH (28)
The calculations are carried out in accordance with the
following input data, the designations of which
correspond to those given in Fig. 5 and in explications of
the formulas (5), (10) and (11)–(26): la = 65 mm; ly = 65 mm;
lc =59 mm; l = 64 mm; dc = 26 mm; dp = 30 mm; δ = 5 mm;
ay = 6 mm; aa = 5 mm; by = 40 mm; ba = 32 mm; lp = 3 mm;
sk = δ; sc = 3,5 mm; sa = 4,5 mm; mv = 2,2 kg; mc = 0,6 kg;
F1 = 161 N; F2 = 172 N; F3 = 221 N; F4 = 253 N; LB = 5 mH;
LH = 18 mH; RB = 56 ; RH = 900 ; s = 20108
m;
Rd = 1108
; U0 = 1 V; I0 = 1 A, CM= 1 F; C0 = 1 F.
The results of some calculations are shown in Fig. 6.
These results are plotted directly on the oscillograms, which
makes it possible to assess the adequacy of the proposed
technique and the high accuracy of the calculations
performed.
Conclusions.
1. Utilization of the computing environment Maple for
mathematical modelling of the dynamics of forced
electromagnetic systems allows to significantly accelerate
the process of simulation, save time and significant material
resources, while providing an acceptable accuracy of the
results.
2. The adequacy of the developed model is confirmed by
experimental studies, which showed a good coincidence of
the results of the mathematical and the full-scale
experiments, and, characteristically, this coincidence is
observed in the conditions of complex shapes of curves of
currents in windings.
ISSN 2074-272X. Electrical Engineering & Electromechanics. 2019. no.3 23
u
i
DC, booster winding
calculation results
u
i
DC, holding winding
calculation results
u
i
AC, booster winding
calculation results
u
i
AC, holding winding
calculation results
Fig. 6. Comparison of experimental data obtained on industrial samples of vacuum contactors with calculation results using
a simplified model of the magnetic circuit, which is considered as a circuit with lumped parameters
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Cai Zhaowen. Research on the dynamic calculation model for a DC
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8. Tatevosian A.S. Dinamika elektromagnitov [Dynamics of
electromagnets]. Omsk, Omsk State Technical University Publ.,
2016. 148 p. (Rus).
9. Bajda Ye.I., Klymenko B.V., Pantelyat M.G., Yelanskyi Yu.A.,
Trichet D., Wasselynck G. Peculiarities of calculating the dynamics
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Series: Problems of Improvement of Electrical Machines and
Apparatus. Theory and Practice, 2018, no. 32 (1308), pp. 34-40.
doi: 10.20998/2079-3944.2018.32.06. (Ukr).
Received 07.02.2019
Ye.I. Baida1, M. Clemens2, B.V. Klymenko1, O.G. Korol1,
P.Ye. Pustovoitov1
1 National Technical University «Kharkiv Polytechnic Institute»,
2, Kyrpychova Str., Kharkiv, 61002, Ukraine,
phone +380 57 7076281, e-mail: b.v.klymenko@gmail.com
2 University of Wuppertal,
Rainer-Gruenter-Straße 21, 42119 Wuppertal, Germany,
phone +49 202 439-1924, e-mail: clemens@uni-wuppertal.de
Application of the computing environment Maple to the
calculation of the dynamics of the electromagnets in the
complicated systems of forced control.
General description of the research topic. The authors propose a
technique for calculating the dynamics of electromagnets operating
in complex forced systems. Such forced electromagnets are widely
used in electromechanical switching devices, in particular in
vacuum contactors, to reduce their size, energy consumption and to
increase speed, which indicates the relevance of this topic. A
mathematical model of the dynamics of a forced electromagnetic
system, which takes into account the peculiarities of behavior in
transients of its individual elements – the mechanical system, the
magnetic and electrical circuits, taking into account the interaction
of the electromagnet with a control device when the apparatus is
activated, contains certain signs of scientific novelty and is the
purpose of the paper. The technique of calculating the dynamics of
forced electromagnets uses the computing environment Maple. The
calculation is based on a mathematical model, which is a system of
nonlinear differential equations of the magnetic and electric
circuits, supplemented by the equations of motion of the elements of
a mechanical system. The use of the computing environment
Maple, applied here to automatically perform the mathematical
transformations, allows avoiding the complicated processes of
choosing the numerical integration method, programming of
complex and cumbersome equations and numerical integration
procedures, to obtain results of calculations in convenient tabular
and/or graphic form. This specifically indicates the practical
significance of this work. The results of the comparison of
calculations with previously published experimental data presented
in the paper indicate the high efficiency of the proposed models and
techniques. References 10, figures 6.
Key words: electromagnets, dynamics, forced control, switching
devices, vacuum contactors, computing environment Maple.
How to cite this article:
Baida Ye.I., Clemens M., Klymenko B.V., Korol O.G., Pustovoitov P.Ye. Application of the computing environment
Maple to the calculation of the dynamics of the electromagnets in the complicated systems of forced control. Electrical
engineering & electromechanics, 2019, no.3, pp. 18-23. doi: 10.20998/2074-272X.2019.3.03.
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