Transactions on Electrical Engineering, Vol. 3 (2014), No. 1 23 Field-Circuit Modelling of Self-Excited Induction Generators Miksiewicz Roman Silesian University of Technology, Gliwice, Poland, roman.miksiewicz@polsl.pl Abstract — In the paper computational models of an induction self-excited generator are presented. Circuit models taking into consideration nonlinearity of the magnetic circuit enable calculations of the generator static characteristic at an autonomic operation. The field-circuit model using Maxwell 2D software allows determination of time curves of any electrical variable in different conditions of the generator operation. There are presented basic determined static characteristics using both models and time curves of currents, voltages, torque during self excitation and under symmetrical load and one-phase short circuit, in cooperation with a rectifier system and during connection of the excited generator to network. Keywords — induction generator; squirrel cage motor; cuircuit-field modelling. magnetizing reactance Xm; it depends on magnetizing current Im. This equivalent circuit diagram does not take in consideration iron losses. For the considered induction machine the parameters of the equivalent circuit diagram were determined using the RMxprt Maxwell software. The data of the induction motor are the following: rated power 7,5 kW; rated voltage 380 VAC; rated rotational speed 965 rpm; winding connection delta. On the basis of the circuit calculations characteristics Xm(Im) and values of other parameters of the equivalent circuit were determined; they are : Rs = 1,215 ; X σ s = 2,72 ; Rr' = 2,29 ; X σ' r = 4,42 Io I. INTRODUCTION Both squirrel-cage and slip-ring machines are used as wind generators. Squirrel-cage induction machines are used especially in smaller wind power plants mainly due their greater reliability nevertheless their worse control characteristics. They require to use condensers (Fig. 1) and existence of residual magnetism for their excitation. At autonomic operation the generated voltage frequency depends on rotational speed and load what is the essential disadvantage of the induction squirrel-cage generators. In order to assure constant frequency additional powerelectronics equipment with conversion are used, for example AC-DC-AC. Many publications [1-7; 9] discuss problems related to features of the induction generators at autonomic operation and cooperation with a network. n G 3 Ro C Fig. 1. Circuit diagram of an induction generator at autonomic operation. II. CIRCUIT MODEL The equivalent circuit diagram of the circuit model is presented in Fig. 2. The circuit diagram takes into consideration non-linearity of the magnetic circuit as fr X o Rs Is fr X σs ' Rr fr f r - nr ' fr Xσ r Uo XC fr fr X m Ro Fig. 2. Equivalent circuit diagram of an induction generator for one phase. In the equivalent circuit diagram fr means relative frequency - frequency of the stator voltage in comparison with the rated frequency of the motor. The prepared algorithm enables by use of Mathcad software for determination of static characteristics taking into consideration non-linearity of magnetization characteristics determination of characteristics at a constant rotational speed or constant frequency of voltage at the load [5]. The relative frequency may be determined by comparison of the zero real part of the equivalent impedance seen from terminals of the magnetizing reactance to Re(Z z ) = 0 . The stationary working point is calculated by comparison of the imaginary part of the equivalent impedance to the magnetizing reactance: Im(Z z ) = f r X m . ' Ir Transactions on Electrical Engineering, Vol. 3 (2014), No. 1 A. Self-excitation of the generator In Fig. 3 there are presented results of calculations of line voltage and phase current time curves at constant rotational speed n = 1040 rpm, during self-excitation of the loaded generator. In order to make possible the generator self-excitation during digital calculations nonzero initial conditions (current) in one of phase windings were set. 400 0.05 a) 0.07 0.09 0.11 0.13 0.15 0.11 0.13 0.15 Torque (Nm) − 30 − 60 − 90 − 120 − 150 Time (s) t [s] 30 b) 15 Current (A) i(t) [A] III. FIELD-CIRCUIT CALCULATIONS In the field-circuit calculations the Maxwell software with Transient solver was used. Using this software it is possible to determine various transient states of the generator at autonomic operation at symmetric and nonsymmetric loads and also at cooperation with rectifiers. It is also possible to make calculations for various nonsymmetric states. In 2-dimensional calculations there were taken into consideration the resistance of the stator and leakage inductances of the stator and rotor, determined using RMxprt, the same as in the circuit calculations. The software does not take iron losses into consideration. 24 0.05 0.07 0.09 − 15 a) 200 Voltage (V) − 30 0 0.05 0.1 0.15 0.2 Time (s) 400 c) − 200 Voltage (V) 200 − 400 Time (s) 30 0.05 0.07 0.09 0.11 0.13 b) − 200 Current (A) 20 10 − 400 0 0.05 0.1 0.15 Time (s) 0.2 t [s] Fig. 4. Torque, phase current and line voltage time curves at rotational speed change (n1 = 1040 rpm) (n2= 1140 rpm). − 10 − 20 . − 30 Time (s) t [s] Fig. 3. Line voltage and phase current time curves during self-excitation of the generator. B. Transient state at rotational speed change. In Fig. 4 curves for transient state (a generator with resistance load and separate operation) are presented at discrete change of the rotational speed from n = 1040 rpm to n = 1140 rpm. There were presented the torque, phase current and line voltage time curves. In the stationary state preceding voltage frequency change was of 49.74 Hz and after the speed change it increased to f = 54.42 Hz. C. Transient state at single-phase short circuit. For the presented field-circuit model it may be performed calculations for an unsymmetrical load. In Fig. 5 there are presented results of calculations (electromagnetic torque, phase currents, terminal voltages) for the following conditions of the generator operation: constant rotational speed n = 1040 rpm; symmetrical load by receiver resistance Ro=30 Ω and next single-phase short circuit of the receiver. As a result of the unsymmetrical short circuit large pulsating component arises in the torque time curve. 0.15 Transactions on Electrical Engineering, Vol. 3 (2014), No. 1 25 600 0.06 0.08 0.1 0.12 0.14 0.16 a) − 40 Voltage (V) Torque (Nm) a) − 80 − 120 400 200 − 160 0.06 − 200 0.07 Current (A) Current (A) 15 0.1 0.12 0.08 0.09 t [s] b) 0.08 0.09 40 b) 30 0.06 0.08 Time (s) Time (s) 0.14 20 0.06 0.07 0.16 − 20 − 15 − 40 Time (s) − 30 c) Time (s) Current (A) t [s] 600 c) 300 Voltage (V) 20 10 0.06 0.07 0.08 0.09 − 10 0.06 0.08 0.1 0.12 0.14 0.16 − 20 Time (s) − 300 0.06 0.07 0.08 − 600 Time (s) Fig. 5. Torque, phase current and line voltage time curves before and after the single phase short circuit. Torque (Nm) d) − 50 − 100 − 150 Time (s) Fig. 7. Steady state of: a) line voltage, b) phase currents, c) line currents, d) electromagnetic torque time curves. Fig. 6. Circuit diagram of a generator with rectifiers of the Schematic Capture module. D. Cooperation of the generator with rectifiers The circuit diagram of the Schematic Capture module of the Maxwell software is presented in Fig. 6. Calculations were performed for the resistance load Ro=30 Ω, capacitors C=90 µF, at the constant rotational speed n = 1040 rpm. Exemplary voltage at the receiver, phase current of the winding and line current time curves 0.09 Transactions on Electrical Engineering, Vol. 3 (2014), No. 1 E. Switching on of excited generator to a network The Maxwell software enables calculations of various transient states. It was considered the case of the nonexcited generator connection to a network and setting a chosen torque. The circuit diagram of the circuit system is presented in Fig. 8. 200 a) 0.05 Torque (Nm) before rectifiers and electromagnetic torque are presented in Fig. 7. Knowledge of these curves allows accurate determining of their RMS values and permissible load. 26 0.1 0.15 0.2 0.25 − 200 − 400 − 600 − 800 − 1 × 10 3 Time (s) Fig. 8. The circuit diagram of the generator in the Schematic Capture module at connection to a network. In Fig. 9 there are presented the torque, rotational speed and current time curves of the excited generator (initial rotational speed of 1040 rpm) during its switching on to a network and loading by torque T = 100 Nm. As can be seen from presented time curves no problems occur with synchronization while large switching currents can be expected. Speed (rpm) b) 1.1 × 10 3 1.05 × 10 3 1 × 10 3 950 900 0.05 0.1 0.15 0.2 0.25 Timet [s] (s) 300 c) 200 0.05 0.1 0.15 0.2 0.25 0.2 0.25 − 100 − 200 − 300 Time (s) 400 d) Current (A) IV. STATIC CHARACTERISTICS OF THE GENERATOR Static characteristics of the generator may be determined on the basis of above calculated parameters of the equivalent circuit diagram and on the basis of the field-circuit calculations for stationary operational conditions. In Fig. 10 there are presented external characteristics U = f(I) determined using both above mentioned methods for two values of capacitors at the constant rotational speed n = 1040 rpm and resistance load. On the basis of the circuit calculations it is easy to determine static characteristics at a constant frequency what is more troublesome for the field-circuit calculations and it requires repeated time-consuming recalculations. Comparing the determined static characteristics using both methods it may be noticed slight differences between them. It influences accuracy of the equivalent circuit diagram parameter determination using the circuit method and stability of these parameters, however from practical point of view it should be used for initial determination of capacitors and generator windings circuit calculations. The circuit calculations give greater variability of voltage at generator terminals. Current (A) 100 200 0.05 0.1 0.15 − 200 − 400 Time (s) Fig. 9. Time curves : a) electromagnetic torque, b) speed, c) phase currents, d) line currents at switching on the excited generator to a network. Transactions on Electrical Engineering, Vol. 3 (2014), No. 1 V. SUMMARY The circuit and circuit-field calculations show good conformity of calculation results. Therefore it is possible to use the circuit model in the initial stage of designing. The prepared field-circuit model allows for calculations of various transient states types and analysis of results both from point of view of the generator and for various systems. 500 a) U [V] Voltage (V) 400 300 200 100 REFERENCES C=90 uF C=70 uF 0 5 10 Current (A) 15 500 b) 400 Voltage (V) 300 200 100 C=90 uF C=70 uF 0 2 × 10 3 3 4 × 10 6 × 10 3 8 × 10 3 1 × 10 4 Power (W) P [W] 500 c) U [V] Voltage (V) 400 300 200 100 C=90 uF C=70 uF 0 5 10 15 Current (A) I [A] 500 d) 400 Voltage (V) 27 300 200 100 C=90 uF C=70 uF 0 2 ×10 3 4 × 10 3 6 ×10 3 8 ×10 3 1 ×10 PPower [W] (W) Fig. 10. 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