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THE CURVE OF SHORT-CIRCUIT TORQUE
Miroslav Urbánek
INTRODUCTION
The Curve of Short-Circuit Torque was investigated for the electric motor train SE 471. The power train consists of two
main subsystems, electrical and mechanical. The electrical subsystem describes a model inverter, a DC circuit and an
asynchronous motor with two stator coils. The mechanical subsystem describes mechanical parts of the motor, gear,
hollow shaft and wheelset. These subsystems work together and interact. Picture 1 shows the basic model and Picture 2
shows the model of the power train.
Picture 1 : basic model
Picture 2 – model of power train
THE ELECTRICAL SUBSYSTEM
The electrical subsystem uses the transformation of 3 phases to 2 main directions x,y. The transformation is called
0,F,B. These coefficients are
k F = k B = k0 =
1
3
. For the asynchronous motor with stator coils a coordinate
system is implemented of rotor and two coordinate systems for the stator.
Coordinate system of stator (xI, yI)
… index IA, rotation velocity is zero
Coordinate system of stator (xI, yI)
… index IB, rotation velocity is zero
Picture 3 – coordinate systems of the electric subsystem
Next, the magnetic flows are defined for the components of rotor and stator coils. The subsystem is simulated
by considering a lot of assumptions, e.g. the parameters of stator coils are the same. The following equations represent
the magnetic flows of individual coils. The magnetic flow is the total of the inductions.
ψ sIA1 = Lsσ isIA1 + Lv isIA2 + Lh iµIA
ψ sIA2 = Lsσ isIA2 + Lv isIA1 + Lh iµIA
ψ rIA = Lrσ irIA + Lh iµIA
(1)
A voltage equation that has default configuration u j = R j ⋅ i j +
dψ j
dt
must be used for the simulation. This
equation for individual coils could be used for all the coil coordinate systems.
Voltage equation for all systems
coordinate system I
coordinate system II
dψ sI1
u sI1 = Rs1 ⋅ i sI1 +
dt
dψ sI2
I
I
u s 2 = Rs 2 ⋅ i s 2 +
dt
I
dψ r
0 = Rr ⋅ irI +
− jω ψ r
dt
dψ sII1
u sII1 = Rs1 ⋅ i sII1 +
+ jωψ sII1
dt
dψ sII2
II
II
u s 2 = Rs 2 ⋅ i s 2 +
+ jωψ sII2
dt
dψ r
0 = Rr ⋅ irII +
dt
coordinate system III
dψ sIII
1
+ jωsψ sIII
1
dt
dψ sIII2
= Rs 2 ⋅ isIII2 +
+ jω sψ sIII2
dt
dψ rIII
= Rr ⋅ irIII +
+ jωrψ rIII (2)
dt
III
usIII
1 = Rs1 ⋅ is1 +
usIII2
urIII
From voltage equations in coordinate system III it was created the complete compensatory scheme of
asynchronous motor with two stator coils.
Picture 4 - complete compensatory scheme of asynchronous motor with 2 stator coils
The torque was formulated from the energetic balance. This balance is described by the equation, which is the
sum of the differentials of the energy. The final torque will be used for the mechanical subsystem. This equation is a
coupler and joins both subsystems together.
M =
Pmech
ωm
{
= − p p k p R e jψ hIII ⋅ irIII
}
(3)
The solution was simulated in coordinate system I, which has the simplest equation. These equations must be
edited for the Matlab software. The variables are derivates and they must express and create a system of 6 equations for
the 6 variables.
dis1x
di
dψ hx
+ Lv s 2 x +
dt
dt
dt
dis1 y
dis 2 y dψ hy
u s1 y = Rs ⋅ is1 y + Lsσ
+ Lv
+
dt
dt
dt
di
di
dψ hx
u s 2 x = Rs ⋅ is 2 x + Lsσ s 2 x + Lv s1x +
dt
dt
dt
dis 2 y
dis1 y dψ hy
u s 2 y = Rs ⋅ is 2 y + Lsσ
+ Lv
+
dt
dt
dt
di
dψ hx
0 = Rr ⋅ irx + Lrσ rx +
+ ω (ψ hy + Lrσ ⋅ iry )
dt
dt
diry dψ hy
0 = Rr ⋅ iry + Lrσ
+
− ω (ψ hx + Lrσ ⋅ irx )
dt
dt
dψ hx
di
di ⎞
⎛ di
= Lh ⋅ ⎜ sx1 + sx 2 + rx ⎟
dt
dt
dt ⎠
⎝ dt
dψ hy
⎛ disy1 disy 2 diry ⎞
⎟⎟
= Lh ⋅ ⎜⎜
+
+
dt
dt
dt
dt
⎝
⎠
u s1x = Rs ⋅ is1x + Lsσ
(4)
THE MECHANICAL SUBSYSTEM
The mechanical subsystem is formed by the torque system of the body. The input of the mechanical subsystem is
short circuit torque from the asynchronous motor with 2 stator coils and the adhesive conditions in contact between
wheel and rail. The output are curves from torsional displacement and rotating velocities. The mechanical subsystem
starts the kinematic analysis.
Ek =
(
)
1 13
1
1 6 T
1
2
2
2
2
&
&
(
)
(
)
I
⋅
ϕ
+
m
+
m
⋅
∆
+
m
+
m
⋅
∆
ϕ
+
q& j ⋅ M j ⋅ q& j + ∑ I i ⋅ ϕ& i2
∑
∑
i
i
11
12
8 ,11
9
10
8,9
13
24
2 4444444244444443 2 j = 2
2 = 2, 6
i =7
1
243 1
243
1442443 1i4
2
1
Ep =
3
4
1
1
2
2
k11,12 ⋅ (ϕ12 − ϕ11 ) + k10 ,11 ⋅ (R11⋅ϕ11 − R10ϕ10 − (R11⋅ + R10 )ϕ13 ) +
2 44
2 4444444244444443
1
42444
3 1
1
2
1
1
2
2
+ k9,10 ⋅ (ϕ10 − ϕ9 ) + k8,9 ⋅ (R9⋅ϕ9 − R8ϕ8 + (R9⋅ + R8 )ϕ13 ) +
2 44
2 44444424444443
1
42444
3 1
3
(5)
4
1
1
1
1 6 T
2
2
2
+ k13 ⋅ ϕ13 + k7 ,8 ⋅ (ϕ8 − ϕ7 ) + k5, 7 ⋅ (ϕ7 − ϕ5 ) + ∑ q j ⋅ K j ⋅ q j
2424
2 442443 1
2 442443 2 j = 2
1
3 1
1442443
5
6
7
8
This model is described by 11 dynamic equations in a matrix system and it is considered to be a torque body
which is connected by the torque springs. This equation is
&&(t ) + B ⋅ q& (t ) + K ⋅ q(t ) = f (t )
M ⋅q
where
M, B, K are
(6)
matrices of mass, damping and stiffness. The vector of the generalized coordinates
q = [ϕ 2 , ϕ 3 ....,ϕ13 ] includes torsional displacement of all the bodies and the vector f (t ) includes exciting torque.
T
The vector is torque created by the asynchronous motor and reactions from contact between the wheel and rail. The
contact uses Kalker’s rolling theory. This theory describes dimensionless creep velocities, which are inserted into the
force equations.
Picture 5 shows the mathematical model of the mechanical subsystem.
Picture 5 – the mechanical subsystem
Kalker’s rolling theory
Left wheel (6)
Right wheel (2)
lengthwise direction
6
γ e1 = 6 γ 1 =
v6 e1 − v1e1 v61 − v11
=
v1e1
v11
2
γ e1 = 2γ 1 =
v2 e1 − v1e1 v21 − v11
=
v1e1
v11
cross direction
6
γ e 2 = 6γ 2 =
v6 e 2 − v1e 2 v62 − v12
=
v1e1
v11
2
γ e 2 = 2γ 2 =
v6 e 2 − v1e 2 v22 − v12
=
v1e1
v11
rotation around vertical axis
6
γ e3 = 6γ 3 =
v6 e3 − v1e 3 v63 − v13
=
v1e1
v11
2
γ e3 = 2γ 3 =
v2 e 3 − v1e 3 v23 − v13
=
v1e1
v11
(7)
This picture shows directions of
velocities, when the wheelset moves in
rail.
These force equations use
dimensionless creep velocities and
they are the cross force and spin force.
Picture 6 - directions of velocities
s ⋅ψ& ⋅ψ ⎞ 6
⎛ y& r
⎛ λ ψ& ⎞
Ty = 6Ty + 2Ty =− 6f 22 ⋅ ⎜ − 6 ⋅ψ −
⎟− f 23 ⋅ ⎜ 6 − ⎟ −
v ⎠
v⎠
⎝v r
⎝ r
s ⋅ψ& ⋅ψ ⎞ 2
⎛ λ ψ& ⎞
⎛ y& r
− 2f 22 ⋅ ⎜ − 2 ⋅ψ +
⎟− f 23 ⋅ ⎜ 2 + ⎟
v ⎠
v⎠
⎝ r
⎝v r
4
4
(
)
(
(8)
)
M z = − 6Ty + 2Ty ⋅ s ⋅ψ + − 6Tx + 2Tx ⋅ s + 6M z + 2M z =
(
)
(
)
= − 6Ty + 2Ty ⋅ s ⋅ψ + − 6Tx + 2Tx ⋅ s + 6M z + 2M z =
s ⋅ψ& ⋅ψ
v
[
6
⎛ y& r
f 22 ⋅ ⎜ − 6 ⋅ψ −
⎝v r
s ⋅ψ& ⋅ψ
⎛ λ ψ& ⎞ 2
⎛ y& r
⎞ 6
⎟+ f 23 ⋅ ⎜ 6 − ⎟− f 22 ⋅ ⎜ − 2 ⋅ψ +
v⎠
v
⎠
⎝v r
⎝ r
⎛ λ ψ& ⎞⎤
⎞ 2
⎟− f 23 ⋅ ⎜ 2 + ⎟⎥ ⋅
v ⎠⎦
⎝ r
⎠
⎧
⎡⎛ r
⎞ s ⋅ψ& ⎤ 2 ⎡⎛ r2
⎞ s ⋅ψ& ⎤ ⎫
⎛ y& r6
6
− ⋅ψ −
+
−
−
⋅
+
⋅
1
f
s
f
⋅ s ⋅ψ + ⎨− 6f11 ⎢⎜ 6 − 1⎟ +
⎜
⎟
23 ⎜
⎥ 11 ⎢ r
⎥⎬
r
v
v
v
r
⎝
⎝
⎠
⎝
⎠
⎣
⎦⎭
⎣
⎦
⎩
s ⋅ψ& ⋅ψ ⎞ 6
s ⋅ψ& ⋅ψ ⎞ 2
⎛ λ ψ& ⎞ 2
⎛ y& r
⎛ λ ψ& ⎞
−
⎟− f 33 ⋅ ⎜ 2 + ⎟
⎟− f 33 ⋅ ⎜ 6 − ⎟+ f 23 ⋅ ⎜ − 2 ⋅ψ +
v ⎠
v⎠
v ⎠
v⎠
⎝v r
⎝ r
⎝ r
The subsystems join together and include 19 equations
for the 19 variables. The system can simulate defects in
the power drive. e.g. destruction of an inverter and the
resulting short circuit and then short circuit torque. The
next picture shows a possible short circuit between two
phases or phase and electric ground.
Picture 7 – short circuit
(9)
LITERATURE
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[2] Lata M., Konstrukce kolejových vozidel II, Pardubice 2004
[3] Roubíček O., Elektrické motory a pohony, Praha 2004
[4] Novák J., Elektromechanické systémy v dopravě a ve strojírenství, ČVUT Praha 2004
[5] Javůrek J., Regulace moderních elektrických pohonů
[6] Wiedemann E.,Kellenberger W., Konstrukce elektrických strojů, Praha
[7] Danzer J., Elektrická trakce 2, Plzeň 2000
[8] Švígler J.,Modelování kontaktu kola s kolejnicí , Plzeň 2000
[9] Hájek E.,Pružnost a pevnost I ,ČVUT Praha 1981
[10]Zahradník J., Piskač L., Pfeifer V., Formánek J., Elektrická výzbroj obráběcích strojů, ZČU FEL , 2006
[11] Vodrášek. F, Výkonová elektronika-1.svazek ZČU Plzeň, 1994
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[13] Brauer M., Germishuizen J., Jöckel A., Körner O., Dynamic Short Circuits of Traction Drives – Comparison of
Induction Motors with PM Synchronous Motors
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metódy , Žilina, 2005
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1998
[17] Zeman V.,Hlaváč Z., Byrtus M., Švígler J., Modelování torzních kmitů pohonu kolejového vozidla dutým
hřídelem, Výzkumná zpráva VCKV, Plzeň, 2005
[18] Švígler J., podklady k předmětu KME/MDP
[19] Slavík J., Stejskal V., Zeman V., Základy dynamiky strojů, ČVUT 1997
[20] Zeman K., Bartoš V., Matematický model trakčního pohonu Asynchronní motor napájený 2 napěťovými střídači,
Výzkumná zpráva, Plzeň, 1996
[21] Vijay K. G.,Rao V., Dynamics of railway vehikle systéme, Academic publisher, London, 1984
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[23] Kletečka J., Fořt P., Technické kreslení
[24] www.atlaslokomotiv.cz
[25] http://www.vagonka.cz
[26] http://rail.czechian.net/CZ_SK_FR/CZ_RAIL/studenka.htm