Automatic Control System Based on Field Orientation Control for a Two-Phase Assynchronous Motor Helga Silaghi Department of Electrical Drives and Automation, University of Oradea, Faculty of Electrical Engineering and Information Technology, University Str.1, , 410087 Oradea, Romania, E-mail:
[email protected]
Im q Abstract - This paper presents the indirect field orientation control (FOC) applied to a two-phase assynchronous motor. A modern simulation possibility for FOC of the two-phase assynchronous motor is proposed. Finally, some simulation and experimental results are presented.
ωβ
qθ qβ
*
isq
dβ
usq
irq *
dθ urq
Keywords: Electrical machines, Field orientation control, Two-phase assynchronous motor, PSpice Simulation
* ir β urd
I. INTRODUCTION
ωθ
θ * usd
Re isd d
Fig. 1 Reference senses for voltages and currents.
The two-phase asynchronous motor has a stator with two electrical windings placed in slots and spacely 90° decalated as in figure 1,[6]. One of the windings is an excitation winding permanent fed and the second one is a command winding. The rotor is build for a small inertial torque and also for an increased equivalent rotor resistance in order to linearize the mechanical and adjustable characteristics . Specific problems of a two-phase assynchronous motor in adjustable speed drives, refer at the particularity of the control system and the frequency converter. For identifying the main problems and particulary aspects that can appear in this case, an electrical scheme is proposed. It presents a two-phase assynchronous motor with a vector control frequency converter.
The mathematical model supposes that the rotor quantities are considered to be reported to the stator. Reference senses for voltages and currents correspond to figure 1. The orthogonal axes system dθ-qθ has the rotor as reference and it is rotating in trigonometryc sense with velocity expressed in electrical degrees ωθ=dθ/dt. The orthogonal axes dβ-qβ are rotating in trigonometric sense with velocity expressed in electrical degrees ωβ=dβ/dt. By using Park vectors following expression for stator voltage is obtained:
dψ s us = R ⋅is + s dt
(1)
By projecting on d and q axis the two-phase natural model is obtained:
u II. LINEAR MATHEMATICAL MODEL OF TWO-PHASE ASSYNCHRONOUS MOTOR
= R ⋅i + sd s sd
u
sq
= R ⋅i + s sq
dψ
sd dt
dψ
sq dt
(2)
or written in matrix form:
[u s ]= Rs ⋅ [is ]+ dtd [ψ s ]
For solving the problems concerning the two-phase assynchronous motor, the model based on field orientation control is the most useful ,[6].
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(3)
For the rotor winding system results:
The two-phase assynchronous motor is fed from a current converter CSI. The two stator windings are traversed by stator currents isd and isq , [6]. Based on the two rotoric flux components, the flux analyser AF determines:
dψ
r 0 = R ⋅ ir + r dt
(4)
and by projecting on dθ-qθ axis:
0 = R ⋅i + r rd 0 = R ⋅i + r rq
dψ
rd dt
dψ
rq dt
Ψr = Ψrd2 + Ψrq2 sin β =
(5)
[ ] [ ]
(6)
By raporting these equations to a fixed reference system which is rotating in trigonometryc sense with ωβ, as in figure 3.b, we obtain :
dψ sβ usβ =isβ ⋅R + + j ⋅ p⋅ω ⋅ψ β sβ s dt dψ rβ 0=irβ ⋅R + + j ⋅ p⋅⎛⎜ω −ω ⎞⎟⋅ψ r dt ⎝ β m⎠ sβ
=i
(10)
The unitary value for cos β, which corresponds to zero value for sin β, must be programmed, [5]. For this purpose function unit step u(t-t0), can be used. It can be implemented in PSpice by using Analog Behavioral Modeling option, [7]. The new value for cos β becomes:
(7)
cos β = 1 − u (t ) + u (t ) ⋅ cos β ′
(12)
where cos β’ is:
⎧ 1 t ∈ [0, t 0 ) cos β ′ = ⎨ ⎩cos β t ∈ [t 0 , ∞)
dψ
sdβ ⋅R + − p ⋅ ω ⋅ψ s β sqβ dt dψ sqβ u =i ⋅R + − p ⋅ ω ⋅ψ sqβ sqβ s β sdβ dt (8) dψ rdβ 0 =i ⋅R + + p ⋅ ⎛⎜ω − ω ⎞⎟ ⋅ψ rdβ r m ⎠ rqβ ⎝ β dt dψ rqβ 0 =i ⋅R + + p ⋅ ⎛⎜ω − ω ⎞⎟ ⋅ψ rqβ r m ⎠ rdβ ⎝ β dt sdβ
Ψrd Ψr
i sq = I R sin β + I A cos β
where p is the pole number. By separating the real and the imaginary part for both stator and rotor, results:
u
Ψr
; cos β =
where β is the position angle of the magnetic rotorical flux. The axes transformer TA does the rotation with β angle of the two components: active, IA and reactive, IR. finally, the control signal for the current converter is obtained: i sd = I R cos β − I A sin β (11)
or written in matrix form:
d 0=R ⋅ i + ψ r r dt r
Ψrq
(9)
sdβ
(13)
Figure 3 presents the equivalent electrical circuit for field orientation control simulation. For simulating a transient process of 1s with a step of 0,2 ms, a simulation time of 40s is obtained. Figure 4 shows the time variation of the components for rotor flux on the two rectangular axes. Figure 5 presents the way in which the appearance of magnetic field establishes the values for cos β and sin β which define the field direction. The studied transitory process simulates the two-phase assynchronous motor starting process for all electromagnetic quantities. Figure 6 presents the time variation of electromagnetic torque. Finally, a simulation of starting the motor, by applying for the first 500ms just the input comand for the magnetic flux, keeping at zero active components IA value. Then, a step of 2A was applied at this input too and the results of this simulation are shown in figure 7 for the currents of the two-phase assynchronous motor. For experimental results the electrical scheme with two Tconected transformers as in figure 8 has been used. For comparisson with simulation results, in figure 9 is presented the electrical equivalent circuit for Pspice simulation.
III. CURRENT CONVERTER CONTROL SIGNAL. SIMULATION AND EXPERIMENTAL RESULTS Specific problems of a two-phase assynchronous motor in adjustable speed drives, refer at the particularity of the control system and the frequency converter. For identifying the main problems and particulary aspects that can appear in this case, the scheme shown in figure 2 is proposed. It presents a two-phase assynchronous motor with a vector control frequency converter.
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Fig. 2 Field orientation control scheme for two-phase assynchronous motor
Fig. 4 Simulation results obtained for rotor flux
Fig. 5 Simulation results for stator currents, sin β and cos β
Fig. 3 Equivalent electrical circuit for PSpice simulation Fig. 6 Electromagnetic torque simulation results
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Fig. 9 – Comparisson between simulated and experimental results for stator current. Fig. 7 – Starting motor simulation results.
Fig. 8 – Electrical scheme for experimental results Fig. 10 – Comparisson between simulated and experimental results for stator voltage.
Fig.8 – Electrical equivalent circuit for Pspice simulation.
Fig. 11 – Stator voltage experimental results
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Figures 11 and 12 show the high content of harmonics in stator voltage and currents.
REFERENCES
Fig. 12 – Stator currents experimental results.
[1]
T. H. Barton, Rectifiers, cycloconverters and AC controllers ( Clarendon Press, Oxford, 1994)
[2]
I. Boldea, S. A. Nasar, Vector control of AC drives (CRC Press, London, 1992)
[3]
R. Crowder, Electric drives and their controls (Oxford Science Publications, 1995)
[4]
T. Maghiar, H. Silaghi, F. Hănţilă, M. Vasiliu, An Efficient Method for Analysis of PM Synchronous Generator Voltage, The 3rd Japan Romania Joint Seminar on Applied Electromagnetics and Mechanical Systems, Felix Spa, 2001
[5]
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[6]
H. Silaghi, Errors in Electromagnetic Torque Determination for Two-Phase Assynchronous Motor, Proc.10th International IGTE Symposium on Numerical Field Calculation in Electrical Engineering , Graz, Sept. 11-13, 2002
[7]
H. Silaghi, Electrical drive systems with induction machine.Data acquisition and Informatical Technics (Treira Publishing, Oradea, 2000)
[8]
J. Thunes, R. Kerkman, D. Schlegel, T. Rowan, Current Regulator Instabilities on Parallel Voltage-Source Inverters, IEEE Transactions on Industry Applications, vol.35, no.1, 1999, 70-78
[9]
A. Ţugulea, Power-flows under non-sinusoidal and nonsymmetric periodic and almost periodic steady-states of electrical power systems,Proc. 6th International Conf. On Harmonics in Power Systems (ICHPS VI), Bologna, Italy, 1999, 388-395.
IV. CONCLUSION The paper presents a modern simulation possibility for the field orientation control of a two-phase assynchronous motor, by using PSpice. The two-phase mathematical model is a natural model for the two-phase assynchronous motor. As a final result, PSpice utilitary is able to simulate the behavior of an automatic control system based on field orientation control for this motor. The main advantage of PSpice simulation is the utilitary’s flexibility and adaptability. The experimental results confirmed the waveforms obtained with simulation.
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