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230 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 231 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) 232 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) 233 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 + Bu1 ) 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 234 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. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. T. Colosi, M. L. Unguresan, E. H. Dulf, R. C. Cordos: „Introduction to Analogical Modeling and Numerical Simulation”. Publishing Galaxia Gutenberg, 2009. N. Galan, G. Constantin, M. Cistelecan: „Maşini Electrice”. Didactic and Pedagogical Publisher, 1981. Rodica Holonec, Radu Munteanu, Jr., „Aplicaţii ale instrumentaţiei virtuale în metrologie electrică”, Mediamira Science Publisher Cluj-Napoca 2003, ISBN 973-9357-47-4 Iudean, M.D., Drăgan I Florin, Munteanu Radu jr., Djerdir Abdesslem, Miraoui Abdellatif - „Evaluation of D.C. Motor Variable Torque Components From Measurement of Stator Angular Vibrations”, The twelfth biennial IEEE conference on electromagnetic field computation (CEFC 2006), Miami, Florida, USA, 2006, pg.302. C. Muresan, D. Iudean, R. Munteanu Jr., V. D. Zaharia – „Discrete Time Signal Phase Shifting Using TFD”, Acta Electrotehnica, vol. 55, number 1-2, 2014, pp. 89-93, ISSN: 2344-5637, ISSN-L: 1841-3323. ***, http://ecoca.eed.usv.ro/teaching/emc/emc2007/cem3.pdf *** http://memm.utcluj.ro/materiale_didactice/msem/3-Fluxuri _si_inductivitati.pdf ***, http://telecom.etc.tuiasi.ro/telecom/staff/vlcehan/discipline %20predate/cem/(3)%20CEM-cuplaj%20inductiv.pdf ***http://www.ubm.ro/sites/cee/images/stories/download/erdeiz /Curs_5_mine.pdf 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]