A TRIAC is connected in series with the machine as to allow control of the motor speed. Experimental and simulation results of the new model confirm the validity ...
MODELLING OF SINGLE PHASE INDUCTION MACHINES C. Coates 1 , D. W. J. Pulle1 and A. Veltman2 1
School of Electrical Engineering and Computer Science, University of Newcastle, Callaghan, NSW 2308, Australia. 2
Department of Electrical Engineering, Group Control Systems, Eindhoven Technical University, P.O. Box 513, Eindhoven, The Netherlands.
Abstract This paper considers a novel model of a single phase, capacitor start induction motor which is based on the use of an ideal rotating transformer (IRTF). The model is an extension of the IRTF based model already used for modelling three phase and DC machines. The paper outlines the new model and discusses an application of the model. A TRIAC is connected in series with the machine as to allow control of the motor speed. Experimental and simulation results of the new model confirm the validity of the work carried out.
1
INTRODUCTION
Research related to single phase drives has received considerable attention over the years [1], [2]. The reason for this is that this type of machine remains commercially interesting for a range of domestic applications. The modelling of this type of machine has often been carried out with complex models [3] which are not readily transparent to the user. The introduction of the ideal rotating transformer (IRTF) [4] has provided the basis for modelling classical machines (induction, synchronous and DC). The first part of the research was aimed at extending the IRTF based four parameter transient space vector induction machine model to allow its use for a single phase machine. The machine in question has a run and start winding, where the latter is connected to the supply via a capacitor and switch. The switch is there to disconnect the start winding after the startup sequence has been completed. The second part of the work is concerned with a practical application namely the use of the single phase machine with a TRIAC (back to back ‘SCR’s’) in series with the machine as shown in figure (1). This will allow the machine to be speed controlled using a relatively simple device and/or implement a soft-start capability. The IRTF based Simulink model of the motor with TRIAC is discussed in this paper and experimental results are presented which confirm the validity of the work.
Figure 1: Machine with SCR’s
2
IRTF MACHINE MODEL
The original four parameter symbolic IRTF based model, as given in figure (2), shows the rotating transformer module which is characteristic for this type of machine model. The energy
Figure 2: IRTF based induction model balance of this ‘IRTF’ module ∗ is of the form xy ixy R i∗ = dψ + Te dθm , where dψ s
R
s
θm represents the angle between rotor and stator
reference frames. The energy equation shows that the space vectors are tied to their respective stator and rotor (super script ‘xy’) coordinate systems. For example the rotor flux vector may be −jθm R or ψ xy = ψ expressed as ψ R R e , while the ∗ is . A generic torque is given by Te = ψ R
representation of the IRTF module is shown in figure (3). The two pole model is readily expand-
ψrun ψstart
= ψrun
(2c)
= k ψstart
(2d)
where i s = irun + jistart , u R = dψdtR and + jψstart . A winding ratio factor ψ R = ψrun k is introduced in equation (2), to account for the difference in start and run winding number of turns. The resultant symbolic diagram of the two phase machine is given in figure (4). The leakage in-
Figure 4: Extended IRTF model for single phase motor
Figure 3: Generic IRTF model able to include skin effect, rotor saliency, double cage rotors, losses, saturation and homopolar currents [5],[6]. The equation set which corresponds to figure (2) is of the form us s ψ xy ψ R 0
s dψ dt R = Lσis + ψ xy = LM ixy s + iR = Rsis +
xy dψ R = RRixy R + dt
(1a) (1b) (1c) (1d)
The positioning of the inductance LM can be to either side of the IRTF module. The development of the single (in fact two phase) model is readily realized because the space vectors are in fact in a two variable form. For example the current is is in this case of the form is = irun + jistart . An ideal transformer (ITF) needs to be added given that the two winding’s are not identical. This ITF module as defined by equation (2), is in fact a variant of the IRTF, which has a winding factor k = 1 and rotating secondary. irun
istart
= irun
(2a)
= k istart
(2b)
ductance and stator resistance values are different for both windings and therefore defined as: T start T Rsru,st = [Rsrun Rsstart ] , Lru,st = [Lrun ] . σ σ Lσ In this case the magnetizing inductance LM is placed between the ITF and IRTF modules. A squirrel cage rotor is used with an equivalent rotor resistance of RR . No allowance has been made at this stage for skin-effect or magnetic core losses, both can be readily incorporated if needed. The input voltage vector is of the form us = urun + justart , where the voltage across the start winding ustart is (according to figure (1)) of the form ustart = urun − C1 istart dt when the centrifugal switch is closed. The current istart and the corresponding magnetizing current component in the start winding are set to zero when the switch is opened. It is noted that the IRTF R , iR to calculate the now uses the vectors ψ torque Te . The model parameters were obtained from a no load and locked rotor test. Details of the motor and parameters used for the simulation are provided in the appendix.
3
SIMULATION MODELS
The SCR module shown in figure (1) is modelled with the aid of a switch S and resistance Rof f , as shown in figure(5). The switch S is placed in position 2 when the SCR’s conduct, i.e us = uload , where uload represents the voltage across the machine. The ‘logic’ moves the switch to position 1
u_run 1 u_start Mux
us
1 s
3 −K− 1/LM
Rs_run,Rs_start
In1
psi_R
Out1
1 s
In2
cur switch2 psi_R
In1
i_s
Out1 i_run
In2 i_start
−K− 1/cap
i_R_xy
i_R
i’_s
psi_R
5
1 i_run
Tmek T_e
theta
ITF_s
p 1/J
cur switch1 1 s
4 iR_vec
psi’_R. psi_R_xy
1/Lsig_run,1/ Lsig_start
7.0 R_R
Tel−>Tmek
IRTF_cur
J
2
1/s
6
−> wrmek
wrmek
2
i_start
wmek −>wel 1 s
p
Relay
inv
Tl
−1
Single 4 pole phase IM machine model
Figure 6: Simulink Model machine when the devices moves to its blocking mode. Under these conditions the load voltage is of the form uload = us − Rof f iload , where iload = irun + istart . The value of Rof f = 1 kΩ was chosen to maintain acceptable computation times. The Simulink
is modelled by the ‘cur switch’ modules which sets the variables istart , iM start to zero when the start winding is to be disconnected. The bold signal lines shown are vector lines which represent two T variables, for example [irun istart ] .
4
Figure 5: Generic model of SCR’s model of the machine, as given in figure (6), is a direct interpretation of figure (4). Readily identifiable are the IRTF and ITF modules, which are based on figure (3) and equation (2) respectively. The electrical torque Te from the IRTF module is multiplied by the pole pair number p of the motor, as to arrive at the mechanical torque. The mec mechanical load equations Te p − TL = J dωdt , dθmec ωmec = dt . where ωmec , θmec represent the shaft speed and angle respectively. The electrical rotor angle used by the IRTF is of the form θ = p θmec . A ‘relay’ module is used to control the switch in the start winding. The switch action
RESULTS
A series of measurements were made to verify the new simulation model concept. For this purpose the single phase capacitor start motor (without mechanical load) was initially connected directly on line to the 240 V 50 Hz supply. The simulated and measured results as shown in figures (7) and (8), show good agreement between the currents in the run and start winding bearing in mind that these results were taken at start-up,i.e high slip region, where skin effects are, at present, not accounted for. The simulated results also show the instantaneous shaft torque Tshaf t = p Te , where p = 2 is the number of pole pairs. Note that with the current choice of variables, run winding on the real axis, and start winding on the imaginary axis causes a clockwise rotating (negative mathematical direction) magnetic field, so the accelerating torque is negative in figure (7). The machine with TRIAC was run with a firing angle of 120 degrees and the machine still operating under no-load. Initially the behaviour of this drive at start up was examined by looking at the run and start winding currents during start up. The results as shown in figure (9), (10) again confirm the ability of the model to predict the behaviour of the test motor. A final test was made by considering the machine
Irun [A] simulation
Irun [A] simulation
10 0 −10 0
0.05
0.1
5
0
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0
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0
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0.05 (c) t [s]
0.06
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10
Istart [A] simulation
Istart [A] simulation
(a) t [s] 20 10 0 −10 −20 0
0.05
0.1
5 0 −5 −10
0.15 Torque [Nm] simulation
Torque [Nm] simulation
(b) t [s] 10 0 −10 −20
0
0.05
(c) t [s]
0.1
0 −1 −2 −3
0.15
Figure 7: Simulated results:irun (t) , istart (t) , pTe (t), DOL sequence.
Figure 9: Simulated results: irun (t) , istart (t) , pTe (t), start up sequence, with α = 120 degrees.
15
6
10
4 Irun [A] measured
Irun [A] measured
1
5 0 −5
0 −2 −4
−10 −15
2
−6 0
0.05
0.1
0.15
0
0.01
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0.05 (a) t [s]
0.06
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0
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0.05 (b) t [s]
0.06
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(a) t [s]
10 Istart [A] measured
Istart [A] measured
20 10 0 −10 −20
5 0 −5 −10
0
0.05
0.1 (b) t [s]
Figure 8: Measured results:irun (t) , istart (t), DOL sequence.
0.15
Figure 10: Measured results: irun (t) , istart (t), start up sequence, with α = 120 degrees.
at no load speed, where the voltage across and the current through the run winding were measured and compared against those obtained with the new model.
3
Irun [A] measured
2 1 0 −1
1.42
1.425
1.43
1.435
1.44
1.445
1.45
1.455
1.46
1.465
1.445
1.45
1.455
1.46
1.465
(a) t [s]
0 300
−2 1.415
Urun [V] simulation
−3 1.415
1.42
1.425
1.43
1.435
1.44 (a) t [s]
1.445
1.45
1.455
1.46
1.465
200
Urun [V] measured
Irun [A] simulation
−2
2
200 100 0 −100 −200 −300
0
1.415
1.42
1.425
1.43
1.435
1.44 (b) t [s]
−200
Torque [Nm] simulation
1.415
1.42
1.425
1.43
1.435
1.44 (b) t [s]
1.445
1.45
1.455
1.46
2
1.465
Figure 12: Measured results: irun (t) , urun (t), no load speed.
1 0
1/3 Hp, 1440 rpm, 240 V, 2.5 A. The following set of simulation parameters were derived −2 for this machine: Leakage inductances: Lrun = 1.415 1.42 1.425 1.43 1.435 1.44 1.445 1.45 1.455 1.46 1.465 σ (c) t [s] 77.3 mH, Lstart = 33.7 mH, magnetizing inducσ tance: LM = 276 mH, stator resistances: Rsrun = start = 11.4 Ω, pole pair number: Figure 11: Simulated results: irun (t) , urun (t) , pTe (t), 10.3 Ω, Rs p = 2, rotor resistance:R R = 7.3 Ω, inertia:J = no load speed. 0.0025 kgm2 , capacitor: C = 200 µF ,ITF winding ratio:k = 0.67. The centrifugal switch on and The results as shown (7), (8) give confidence in off points were taken to be at 25 %, 75 % of the the ability of the new simulation model. synchronous speed respectively.
5
−1
CONCLUSIONS
An extension to an ‘ideal rotating transformer’ (IRTF) based four parameter machine model has been discussed. The new IRTF model described in this paper has been adapted to allow the modelling of single phase induction machines. Details of this new model have been presented. A series of evaluation tests have been carried out which include a DOL start, and the use of a TRIAC connected between motor and supply. Results shown, have confirmed the effectiveness of the new modelling approach. The model presented was applied to a specific single phase induction machine but it is emphasized that the new model can be used for a much wider range of machines.
6
APPENDIX
The machine used for this research was from ‘Crompton Parkinson’, Model P A5144B − P , with the following name plate data:
References [1] A.S. Ba-thunya, R. Khopkar, K. Wei, and H. A. Toliyat. Single phase induction motor drives-a literature survey. Electric Machines and Drives Conference, vol 1, 2000. [2] F. Blaabjerg, F. Lungeanu, K. Skaaug, and A. Aupke. Comparison of variable speed drives for single-phase induction motors. Power Conversion Conference,PCC Osaka 2002, vol 3, 2002. [3] O.Ojo and O. Omozsi. Parameter estimation in single-phase induction machines. IAS Annual Meeting, vol 4, 2001. [4] A. Veltman. The fish method: interaction between AC-machines and switching power converters. Delft University Press, Delft, 1993. [5] A. Veltman and P.P.J. van den Bosch. A universal method for modelling electrical ma-
chines. Proceedings IEE, Electrical Machines and Drives, London, pp. 193-197, 1991. [6] D.W.J. Pulle and A. Veltman. Quantification of homopolar components in machines connected to branch delta type soft-starters. Accepted for publication in EPE2003, -, 2003.
