Download Analogical Modelling and Numerical Simulation of the Single

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ACTA ELECTROTEHNICA
Analogical Modelling and Numerical
Simulation of the Single-Phase Resolver
Nicolae Patachi1, Eudor Flueraş1, Tiberiu Coloşi2
1
Faculty of Electric Engineering, Technical University of Cluj-Napoca, Romania
Faculty of Automation and Computer Science, Technical University of Cluj-Napoca, Romania
2
Abstract: The purpose of this paper is to develop a possible variant of an analogical modelling and a numerical simulation for a
single-phase resolver pursuing a more unified and systematized approach. By knowing the input voltage, the proposed simulated
model calculates the output voltage, the input/output current at any time (sequence), in any position of the single-phase resolver
rotor, in dynamic or static mode of operation. The LabVIEW virtual instrument can display the rotor position any time, too.
Keywords: resolver, analogical modelling, state variables, numerical simulation, Taylor series, LabVIEW
1.
INTRODUCTION
The single-phase resolver can be considered a
transducer, an actuator, a special destination automation
equipment with a complex structure[4]. It has two input
signals x1(t) and x2(t), and an output one y(t), as is
shown schematically in Fig. 1.
Next, for this type of resolver will be elaborate:
 The direct analogical model,
 The analogical model based on the state variable,
 The numerical simulation based on state
variables and Taylor series,
 A software application able to be run in multiple
initial conditions.
The indices “1” and “2’ refer to the rotor primary,
respectively the stator secondary.
The entire material is based on a common variant
which establish that the variables in the left of the
equations are considered as independent variables with
a “cause” task and those from the right to be considered
dependent variables with an “effect” task.
2.
THE DIRECT ANALOGICAL MODEL
The direct analogical model is based on the next
system of three nonlinear differential equations on
voltages [2][7]:
di1 d
+ ( M 12i2 )
dt dt
di
d
− ( M 21i1 ) =
R2i2 + L2 2 + u2
dt
dt
di2
=
u2 RL i2 + LL
dt
u1 = R1i1 + L1 ⋅
Fig.1. The single phase resolver: input and output signals
In this paper the next will be considered:
x1(t) = u1(t) = u1 -the voltage winding rotor,
x2(t)=α(t)=α
-the angular displacement of the rotor
winding
y(t)=u2(t) =u2 -the voltage across the stator winding
i1(t) = i1
-the current flow in the winding rotor,
i2(t) =i2
-the current flow in the stator
winding.
(1)
(2)
(3)
where:
• R1 and L1 are the rotor electrical parameters;
• R2 and L2 are the stator electrical parameters;
• RL and LL are the electrical parameters of the
external load supplied from the external stator
terminals;
© 2015 – Mediamira Science Publisher. All rights reserved
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Volume 56, Number 5, 2015
• Ψ12 = M 12 ⋅ i 2 is magnetic flux from the
secondary winding (2) that passes through the primary
winding (1);
• Ψ21 = M 21 ⋅ i1 is magnetic flux from the
primary winding (1) that passes through the secondary
winding (2).
For an angular displacement α(t) of the rotor that
is mechanically driven from outside, denote:
α = ωα ⋅ t = 2πfα ⋅ t
(4)
where
ωα =
dα
= 2πfα
dt
(5)
and the voltage supply, u1(t), is defined by:
u1 = 2 ⋅ U1ef ⋅ sin(ωu1 ⋅ t )
(6)
where ωu1 = 2πf u1
The movement frequency of the mechanically
driven rotor fα is more less than the voltage frequency
u1(t):
fα « fu1.
For mutual inductances from equations (1) and
(2), associated with Fig.1, it can easily establish the
next relations:
M 21 = M 21max cos a
M 12 = M 12max cos a
(8)
(9)
respectively:
dM 21
da
= − M 21max sin a
dt
dt
dM 12
da
= − M 12max sin a
dt
dt
(10)
(11)
For more rigorousness, the mutual inductances
(M21) and (M12) may be considered slightly different
values.
As a result, with respect to (1), it results the
induced voltage:
dM 12
di
d
( M 12i=
⋅ i2 + M 12 ⋅ 2
2)
dt
dt
dt
(12)
Formally identical, with respect to (2) it results
induced voltage
dM 21
di
d
( M 21=
i1 )
⋅ i1 + M 21 ⋅ 1
dt
dt
dt
(14)
respectively:
d
da
( M 21i1 ) =− M 21max sin a
⋅ i1 +
dt
dt
di
+ M 21max cos a 1
dt
(15)
By introducing the results (13) and (14) in (1)
respectively (2) we obtain:
di1
da
− M 12max sin a
⋅ i2 +
dt
dt
di
+ M 12max cos a 2
dt
di1
da
⋅ i1 − M 21max cos
=
M 21max sin aa
dt
dt
di
= ( R2 + RL )i2 + ( L2 + LL ) 2
dt
di
u 2 = R L i2 + LL 2
dt
u1= R1i1 + L1
(16)
(17)
(18)
The previous system of three differential
equations can be considered as the direct analogical
model of the single-phase resolver expressed by
equations of voltages equilibrium: primary rotor
winding voltage, secondary stator winding voltage and,
respectively, voltage across of the external load
supplied from the secondary stator terminals.
3.
THE ANALOGICAL MODEL BASED ON
THE STATE VARIABLES
By introducing in relation (17) the
 di2 
 dt 


expression from (16), after calculus it results:
di1
= a11i1 + a12 i2 + b1u1
dt
(19)
where:
respectively:
d
da
=
( M 12i2 ) M 12max sin a
⋅ i2 +
dt
dt
di
+ M 12max cos a 2
dt
−
a11 =
(13)
M 12max M 21max
da
sin aa
cos
− R1
L2 + LL
dt
M
M
L1 − 12max 21max cos 2 a
L2 + LL
(20)
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ACTA ELECTROTEHNICA


da R2 + RL
cos 
+
M 12max  sin aa
dt L2 + LL


a12 =
M 12max M 21max
cos 2 a
L1 −
L2 + LL
1
b1 =
M
M
L1 − 12max 21max cos 2 a
L2 + LL
For the load circuit, supplied with u2(t), from the
secondary stator terminals, it can be written:
(21)
(22)
 di 
If the result  1  expressed in (19) is introduced
 dt 
in the system formed with (17) and (18), after calculus,
can be express the next:
di2
= a 21i1 + a 22 i2 + b2 u1
dt
(23)
where
a21 = −
⋅ a11 − M 21max sin
M 21max cos aa
L2 + LL
da
dt (24)
1
a22 =
−
[ M 21max cos a ⋅ a12 + ( R2 + RL )]
L2 + LL
1
−
b2 =
M 21max cos a ⋅ b1
L2 + LL
(25)
i21 = −
(26)
4.
(27)
THE NUMERICAL SIMULATION BASED
ON THE STATE VARIABLES AND
TAYLOR SERIES
a12 i10
b
⋅
+ 1 ⋅ u10
a22 i20
b2
fα « fu1
(29)
or
x = A ⋅ x + B ⋅ u1
The numerical simulation based on the state
variables and Taylor series is ground on the important
observation that
(28)
then from (19) and (23) it results:
i11
a
= 11
i21
a21
(31)
that represent the only one linear differential
equation, expressed from state variable (i20). The other
two equations of the system, (19) and (23), respectively
the matrix equation (30) are nonlinear.
Therefore, due to the fact that equations (29) and
(30) are nonlinear, the classical study – by using state
variables - of stability, controllability and observability
cannot be employed.
However, the analogical model based on the state
variables expressed by (29) or (30) and (31), will be the
support for the next chapter.
dα
If α is constant and
= 0 , than the coefficients
dt
a11, a12, a21, a22, b1 and b2 from (20)...(26) will be
constants and the single-phase resolver will be linear,
equivalent with a single-phase transformer. In this case,
for α=0, the mutual electromagnetic coupling becomes
maximum and for α=π/2 this mutual electromagnetic
coupling is null.
Because in literature, is generally considered that
M12max=M21max, (they have values very close) to
simplify the calculation M12max and M21max have the
same value in the program[6][7][8][9].
Denoting:
di1 
= i1 = i11
dt
di2 
= i2 = i21
dt
RL
1
⋅ i20 +
⋅ u 20 ,
LL
LL
(30)
that represent the analogical model based on the
state variables of the single-phase resolver. The two
state variables are the currents i1(t) = i10 and i2(t) = i20
and the input signal is u1(t) = u10. For this signals the
second index corresponds with the order of the
derivative in respect with time. The second input signal,
representing the angular displacement α(t), is contained
in the expressions of the all coefficients (a11, a12, a21,
a22, b1 and b2), that highlights the nonlinear structure of
the single-phase resolver.
(32)
where, for example, fα = 1Hz, and fu1 =
400÷10.000 Hz.
As a result, by successive derivation of (30) with
respect to time, up to order six, inclusive, it results:
x(1) = A ⋅ x + B ⋅ u1
(33)
x(2) = A ⋅ x(1) + B ⋅ u1(1)
(34)
...........................................................
x(6) = A ⋅ x(5) + B ⋅ u1(5)
(35)
where:
dx (1) du1 (2) d 2 x (2) d 2u1
=
x
=
, u1
=
,x
=
, u1
dt
dt
dt 2
dt 2
(1)
and so on.
Considering the current time sequence (k) for
which corresponds the moment tk=k·Δt to the time and
(k-1) the regressive sequence time (previous sequence)
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Volume 56, Number 5, 2015
for which corresponds the moment tk-1=(k-1)·Δt, it
results the Taylor series approximation (limited, for
example up to six order derivation with respect to time)
xk =
x k −1 +
∆t
∆t 2
( Ax + Bu1 ) k −1 +
( Ax + Bu1 ) k −1 +
1!
2!
∆t 3
∆t 6


+
( Ax + Bu1 ) k −1 + ... +
( Ax(5) + Bu1(5) ) k −1
3!
6!
(37)
according to the relations (37) respectively (18) at time
tk-1 and it displays them on the Waveform Graph. The
angle α=α(t) is calculated and displayed too. All of
these are calculated depending on the primary and
secondary circuit parameters (of the rotor and stator),
on the initials conditions (the relative position of the
rotor to the stator) and on the u1 and α input signals (the
parameters which define this signals).
The integration by Taylor series was used in the
program [1] and the flowchart is shown in Fig. 2.
The Δt is denoted as the advance iterative step and
is considered small enough, i.e.:
Δt ≤ 0,01·Tu1 = 0,01·(1/fu1)
For
example,
∆t ≤ 2,5 ⋅10
if
(38)
fu1=400
Hz,
it
results
−5
seconds, a completely negligible value
compared to T α=1/f α=1 sec.
As a result, with the vector-matrix from
relationship (37) the numerical simulation based on
state variables and Taylor series of the single-phase
resolver is solved.
5.
SOFTWARE APLICATION FOR THE
NUMERICAL SIMULATION OF THE
STATE VECTOR
For the numerical simulation based on state
variables and Taylor series of the single-phase resolver
the graphical programming software LabVIEW from
National Instruments was used[3][5]. The program
calculates the state vector (i1 and i2) and the voltage u2
Fig.3. The Block Diagram.
Fig.2. The flowchart of the program
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ACTA ELECTROTEHNICA
The Front Panel contains the structural and signal
parameters as data inputs and the graphs for displaying
the calculated signals.
The Block Diagram (figure 3) contains SubVI’s
like those of “calculation of coefficients” or
“calculation of derivatives”, etc.
In the figure 4 are shown the evolutions in respect
with time of different signals: the output voltage u2
across the stator winding (a); the voltage winding rotor
u1 and the output voltage u2 on the same graph (b); the
i1 and i2 currents (c) and the angle α (d). All of these
signals are shown during a complete revolution of the
rotor, with the exception of (b) which is a waveform
chart and it’s displaying the signals for a few numbers
of points.
From figure 4a and figure 4d it can be observed
that the output voltage, u2, across the stator winding
has a maximum amplitude when α = 180° or α=360°
Fig.4. The Front Panel with input parameters and output signals.
Volume 56, Number 5, 2015
and when the angle α is 90° or 270° it has a minimum
one, which depends on fα. Figure 4b shows that the u2
voltage amplitude decreases when the angle α is
approaching to 450° (a complete revolution). In this
graph the amplitude of the u2 voltage was increased by
100 times, so it can be observed the waveform near to
zero.
The i2 current (figure 4c) has the same evolution
in time, but out of phase with the output voltage u2. The
phase difference depends on load inductance.
6.
CONCLUSION
This work shows a method of analogical
modelling and numerical simulation based on state
variables and Taylor series of the single-phase resolver
in LabVIEW. Main advantage of this method is that it
can increase the accuracy of the output voltage (u2) by
increasing the order of derivatives. Certainly the step Δt
will be increased properly too and as well is no need of
large volume of calculations. Also it is not necessary to
display all the calculated values of u2, but may be taken
and displayed only k, ranging from 10 to 10 or from
100 to 100. In this work we used 6th order derivatives,
inclusive, and the step of integration was:
1
.
∆t =
100 fu1
The program provides a good flexibility because it
can be declared and modified many state parameters
and structural parameters in a wide range such as:
electrical parameters for the rotor and stator, the angle
of rotation of the rotor- α or signal parameters. The
chosen load resistance at 100 Ω, can be modified too.
Therefore, it was operated with different
frequencies (i.e. 1000 Hz, 2000 Hz) with conclusions
235
based, logical, justifying the validity of the work
mentioned above.
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***, http://ecoca.eed.usv.ro/teaching/emc/emc2007/cem3.pdf
*** http://memm.utcluj.ro/materiale_didactice/msem/3-Fluxuri
_si_inductivitati.pdf
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Prof.dr.ing. Nicolae PATACHI,
Drd.ing. Eudor FLUERAS
Department of Ekectrical Engineering and Measurements
Faculty of Ekectrical Engineering
Technical University of Cluj-Napoca, Romania
[email protected]
Prof.dr.ing. Tiberiu COLOSI
Automation Department
Faculty of Automation and Computer Science
Technical University of Cluj-Napoca, Romania
[email protected]