Mohammed.H. Ali. Figure 1 Scheme of the system with two-level inverter and load [3].

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1 Vol.1 No.1, 2017 مجلد 1 العدد Mathematical Driving Model of Three Phase, Two Level Inverter by (Method of Interconnected Sbsystem) Mohammed.H. Ali Electrical power and machines Engineering Department College of engineering Diyala University Iraq diyala-baqba Mobile : moh80mmed@gmail.com Abstract In this paper describe to mathematical analysis for a three-phase, two level inverter designs. As we know the power electronic devices (inverter) to convert the DC power to AC power (controller on otpt voltage and freqency level). In Indstrial applications, the inverters are sed for adjstable speed (AC Drives). In this paper, the mathematical analyses for inverter design are done by sing Software packages C++ Bilder and visal C++ Langage. For non- linear distortions described by the load power factor in power system networks. The P.F is reverse proportional with the harmonics distortion. Small P.F means mch more of harmonic distortion, and lower power qality for consmers. to improve the P.F, and power qality in this paper the small capacitor installed as part of the rectified the load crrent has power (0 KW with P.F load 0.8), the flctations of the rectified voltage mst not greater than +/- 10%.The power factor proportion of the load power, with Modlation coefficient p. approximately nity. The calclation is achieved with different integrations steps with load power 0KW, 0.8 P.F. all reslts done Based on model and experimental data.. Keywords:- Mathematical analysis, Modeling, three phase - two level inverter, Interconnected Sbsystem. I. INTRODUCTION This Paper describes a model of PWM inverter fed three-phase load. The model needs to be based p by decomposition of a system into sb circits that are copled by means of dependent voltage/crrent sorces. Sch an approach ensres high flexibility in constrction of system models along with acceptable accracy of comptation based on model and experimental data, this model described is bilt p on decomposition of complex system into sb circits interconnected via dependent voltage/crrent sorces [1]-[2].To highlight this method of compter model constrction we shall consider the simplest system with two-level converter and three phase loads (0, 100) KW show on in fig (1)..Compter models of the system with load power of 0 KW and semicondctor converters (SC) are widely sed to facilitate development. Figre 1 Scheme of the system with two-level inverter and load []. This system is decomposed into SC sb circit and three-phase load block. SC is fed by crrent from a DC sorce with resistor and indctance in a circit of rectified voltage there is a capacitor C with crrent. Each leg of SC consists of a transistor along with its anti parallel diode. Transistors and diodes are spposed to be ideal gates. The static energy losses are taken into accont by resistor [4]. States of semicondctor 7

2 Vol.1 No.1, 2017 elements are described by a discrete fnction kin, n=1, 2, : if an open transistor or diode connects n-the phase to a positive pole of the capacitor C then kin=1, and if it connects n-the phase to a negative pole then kin = 0. The transistors in an arm are complementary one to the other: if one transistor in an arm is fired, then the other is trned off. The arms of the bridge attached to the positive pole transfer inpt crrent of the inverter circit, this sorce frther.[5]-[6] II. MATHEMATICAL ANALYSIS FOR THREE PHASE TWO LEVEL INVERTER [7]. The dividing mathematical analysis system into sb circit, interconnected-dependent voltage, and crrent sorces. As a reslt represented sb circit shown in Fig 2. Figre 2 Dividing sb scheme of the system with two-level inverter and load. Eqivalent EMF Phase Inverter: = (1) Frther transformation schemes shown in Fig.. Figre Transformation schemes system with two-level inverter and load. Removal from the EMF phases of the zero seqence components. e0 = ( e1 + e2 + e ) / (2) n = en e0, n = 1,2, Voltage of capacitor C: c 1 c i dt c () Otpt phase voltage of ideal converter withot zero-seqence component: e k k 1 k k n e (4) n i i2 i n in rc, DC voltage sorce crrent id is determined from an eqation: Derivative crrent spply:- d r i in n n n (5) dt l H Crrents of bridge arms: iin kin i i ( k 1) i, n 1,2, (6) n in in n Transistor crrents itn and diodes crrents idn: If i when n 1,2,..., 6 dn Inverter inpt crrent: i i i (7) i1 i2 i DC voltage sorce crrent id is determined from an eqation: d ik k rc d k (8) dt l d r i Where crrent of the protection circit: rc iz k (9) z rz The arms of bridge crrents: iz kini, n i in (1 kin ) i (10) n Crrent of capacitor: i i i i (11) c d z di مجلد 1 العدد

3 III. MODELING OF CONTROL SYSTEM INVERTER In modeling, circit Fig.1. saw tooth voltage is described by the eqation (12). T on T f. t Where Hz, on on freqency of the reference voltage intermediate variable, The second calclation. (12) Voltage control is determined by the following formlas: t t Iraq J. Electrical and Electronic Engineering Vol.1 No.1, 2017 t y (1) sin( ) (14) y1 y max H t 2 1 max sin( t ) (15) y y H 4 1 max sin( t ) (16) y y H Where t, time in second, anglar freqency voltage reference load by rad/sec, Uy max the amplitde of maximm voltage control. IV. DEFINITION THE CURRENT LOAD CURRENT BY INSTANTANEOUS VALUES IN PHASES In one of the possible constrction of a control, system sed PI controller acting load crrent. The actal operating crrent of three-phase load is determined in the process of calclating the instantaneos variables: i1 i2 i A (17) t y B B ( A B) T (18) i I B (19) A and B intermediate variables, aperiodic filter. PI control at the load crrent: I I If U U If Z I, U U y min yi y YMAX max yi U If U yi I. K I 0 constant time (20) U U, then U Where yi I y max y min. K I 0, then, then U y max U y mix ΔI Deviation of the actal crrent from reference sb phase. Δt Step work system control. U Voltage control amplitde. Uyi Integral component of voltage control. Uin Minimm vale control voltage. Uax Maximm vale control voltage. KIi Coefficient integration for the deviation crrent. KIo Coefficient of crrent deviation. IZ The arms of bridge crrents. Simlation reslt saw tooth voltage described Instantaneos vale of voltage control VSI When sinsoidal PWM represented in eqations (21,22, 2 and 24). * t (21) U sin( ) (22) y1m مجلد 1 العدد y 2 m U sin( ) (2) 75

4 Vol.1 No.1, y m U sin( ) (24) Where t, time in second, reference vale freqency anglar by rad/sec, Uy max, The amplitde maximm voltage P. Sinsoidal PWM with zero seqence represented in eqations (25,26 and 27). U sin( ) 0.1* U sin( ) (25) y1m 2 U sin( ) 0..1* U sin( ) (26) y2m 4 U sin( ) 0.1* sin( ) (27) V. RESULTS OF SIMULATION The Modeling system by interconnected sb circit to calclate transient and steady state models of VSI. For reference load power of 100 KW and power factor 0.5 to 0.8 to hold series calclation. Voltage phase calclation 2 v U m 0.8* v (28) Load crrent calclation PL 0000 I A U * cos 80* 0.5 * (29) U 80 Z I R Z cos 2.166* X Z sin (0) (1) (2) X L mH () مجلد 1 العدد models of electrical drives with semicondctor converter and load by sing C++ bilder programmer. The calclations (for given load power 100KW and power factor load 0.5 to 0.8). Table (1) Definitive inpt data Emf power spply Ei 1000 V. The indctance of power Li H spply Active resistance of the Ri 0.01 Ω power spply Capacity of the capacitor C F Resistance of the capacitor Rc 0.01 Ω battery The resistance of protective Rz 1000 Ω resistor Indctive load Ln H Resistance load Rn 9.24 Ω The amplitde of emf load Enm 0 V The anglar freqency emf load omega rad/s inpt data for control system Freqency of the reference fop 2000 Hz vale Freqency rated of load f1 50 Hz voltage Maximm voltage control Ux 1.8 P Maximm voltage across the Ucmx 1500 Vc capacitor The specified operating load Inz 2.89 А crrent The coefficient of the integral Kii 0.25 P of the crrent load The coefficient of the load crrent Kio P Table (2) represented where P.F. to change (0.5 to 0.8) Calclation reslt for I, R and L by sed formla (28,29,0,1,2 and ). Table (2) reslt for I, R and L( 0.5 to 0.8) VII. ALGORITHM CALCULATIONS PROGRAMS VI. MODULATION SYSTEM CONTROL AND CALCULATION TRANSIENT REGION Inpt data for the program represented a table (1).The development complex of mathematical Figre 4 shows the flowchart programming for calclation and soltion eqation by sing C++ and visal C++ [8]- [9]. 76

5 Vol.1 No.1, 2017 مجلد 1 العدد Figre 5 Schemes of characteristic crrent (,, and ) Fig. 4 Algorithm calclations programs The reslts are shown in Fig.(5,6 and 7) and the table () when P.F.=0.5 Table () Harmonic analysis ( Sorce crrent: The rectified voltage: Crrent in the first switch The crrent vale of the crve: The maximm vale of the crve: The minimm vale of the crve: Voltage Inverter 1 phase The crrent vale of the crve: Harmonics coefficient: 0.52 Crrent 1 phase load The crrent vale of the crve: Harmonics coefficient: Voltage control Acting vale of the crve: Harmonics coefficient: Figre 6 Characteristic crrent and voltage of SC (,,, and ) Figre 7 Characteristic of voltage PWM reference at = 2000 Hz (carrier freqency). The reslts shows in Fig.(8, 9and 10) and the table (4) when P.F.=

6 Vol.1 No.1, 2017 مجلد 1 العدد Table (4) Harmonic analysis ( Sorce crrent: The rectified voltage: Crrent in the first switch The crrent vale of the crve: The maximm vale of the crve: The minimm vale of the crve: Voltage Inverter 1 phase The crrent vale of the crve: Harmonics coefficient: Crrent 1 phase load The crrent vale of the crve: Harmonics coefficient: control voltage Acting vale of the crve: Harmonic Act. Of Phase (grad) Harmonics coefficient: Figre 8 Schemes and characteristic crrent (,, and ) Figre10 Characteristic of voltage PWM reference at = 2000 Hz (carrier freqency) The reslts are shown in Fig.(11,12 and 1) and the table (5) when P.F.= 0.7 Table (5) Harmonic analysis ( Sorce crrent: The rectified voltage: Crrent in the first switch The crrent vale of the crve: The maximm vale of the crve: The minimm vale of the crve: Voltage Inverter 1 phase The crrent vale of the crve: Harmonics coefficient: Crrent 1 phase load The crrent vale of the crve: Harmonics coefficient: control voltage Acting vale of the crve: Harmonics coefficient: Figre 9 Characteristic crrent and voltage of SC (,,, and ) 78

7 Vol.1 No.1, 2017 مجلد 1 العدد Table (6) Harmonic analysis ( Figre 11 Schemes and characteristic crrent (,, and ) Sorce crrent: The rectified voltage: Crrent in the first switch The crrent vale of the crve: 2.24 The maximm vale of the crve: The minimm vale of the crve: Voltage Inverter 1 phase The crrent vale of the crve: Harmonics coefficient: Crrent 1 phase load The crrent vale of the crve: Harmonics coefficient: control voltage Acting vale of the crve: Harmonics coefficient: Figre 12 Characteristic crrent and voltage of SC (,,, and ). Figre 14 Schemes and characteristic crrent (,, and Figre 1 Characteristic of voltage PWM. Reference at = 2000 Hz (carrier freqency The reslts shows in Fig.(14, 15 and 16) and the table (6) when P.F.= 0.8 Figre 15 Characteristic crrent and voltage of SC (,,, and ) 79

8 Vol.1 No.1, 2017 مجلد 1 العدد When inpt data for the program represented in Table (8). The development of complex mathematical models of electrical drives with semicondctor converter and load by sing C++ bilder programmer. The series of calclations (for given load power 0KW and power factor load 0.8, exchange capacitor. Figre16 Characteristic of voltage PWM reference at = 2000 Hz (carrier freqency). VIII. MODULATION AND CALCULATION Ø 2 LEVEL INVERTOR. Installation the small vale of capacitance in part of the rectified crrent at a given load power of 0 KW with P.F Table (7) represented parameter of the load: Table (7) given load power of 0 KW with P.F Po, KW,Volt We calclated parameters by the following eqas. 2 V U m 0.8* V Table (8) Definitive inpt data. Emf power spply Ei 1000 V. The indctance of power Li H spply Active resistance of the Ri 0.01 Ω power spply Capacity of the capacitor C F Resistance of the capacitor Rc 0.01 Ω battery The resistance of protective Rz 1000 Ω resistor Indctive load Ln H Resistance load Rn 9.24 Ω The amplitde of emf load Enm 0 V The anglar freqency emf load omega rad/s inpt data for control system Freqency of the reference fop 2000 Hz vale Freqency rated of load f1 50 Hz voltage, Hz Maximm voltage control Ux 1.8 P Maximm voltage across the Ucmx 1500 Vc capacitor The specified operating load Inz 2.89 А crrent The coefficient of the integral Kii 0.25 P of the crrent load The coefficient of the load crrent Kio P The reslts are shown in Fig.(17, 18 and 19) and the table (9) when C=.. 80

9 Vol.1 No.1, 2017 مجلد 1 العدد Table (9) Harmonic analysis. crrent spply 0.27 The voltage of the capacitor battery The average vale of the crve: Maximm vale of the crve: Maximm vale of the crve: crrent capacitor bank Un1 load voltage In1 load crrent The crrent vale of the crve: Power Spply: = 0.27* = 1268,49W Load Power: = *78.56*2.804*0.8 = W Load Power Ref.: Рref = 0000 W = ( ) = 6,7542 Cos ( ) = Figre 19 The rectified voltage when С= Frther calclation: we sed another vales for capacitor changing from 2 to The capacity of the capacitor bank C, Table 10 represented reslts of calclation Voltage of the capacitor bank (maximm vale) V Ripple of the rectified voltage % Crrent of capacitor bank (rms vale of the crve), A , , , , , , Figre 17 Capacitor crrent and load voltage Fig. 20. Represented the relation between ripple of the rectified voltage and capacity of the capacitor bank IX. CONCLUSION Figre 18 The rectified voltage when С=2 In this paper, the Mathematical Driving Model of Three Phase, Two Level Inverter designs is done. The degree of linear load is described. Power factor is the proportion of power at the first harmonic of the crrent of the total power 81

10 Vol.1 No.1, 2017 consmed by the load. For each nonlinear distortion have P.F and are introdced. In this paper The flctations of the rectified voltage does not greater than +/- 10% with installation of capacitor bank C > 0.00 Micro F. It is shown that an approach of taking into accont inflence of crrent distribtion in the rotor on starting characteristics sed in bilding p the model proved applicable for evalation of HF energy losses cased by PWM SC. This Mathematical model of electric drives done with synchronos machines has been developed. It can se in the real time mode with the help of personal compters. The system is intended for debgging transistor drive microprocessor-based control nits and based on se of mathematical models. REFERENCES [1] M. Pronin, "Simlation and analysis of the system with mltiphase indction generator and mlti-stage active rectifier", Electrotechnika, Rssia, p , no. 5, [2] M.Pronin. A. Vorontsov,"Dependence of crrent plsations of mlti-phase electrical machine on redction of winding pitch and scheme of semicondctor converter". EPE- PEMC, Portoroz, Slovenia, [] C. Cester, A.Kedos -Leboc, and B. Cornt, "Iron loss nder practical working conditions of a PWM powered indction motor". Magnetics, IEEE Transactions, Vol., Isse 5, Part 2, pp , [4] A. J.Moses, F. Anayi,"Effect of PWM Voltage Excitation in Iron Loss of Inverter Fed Motors", IEEE Transactions on Magnetics,vol. 40 (2), no 2, pp , [5] A.Rderman, R. Welch," Electrical Machine PWM Loss Evalation Basics", EEMODS 2005, 5 8 Heidelberg / Germany, September [6] N.Serov, P.Kalachikov, M. Pronin, and A.Vorontsov," Electro-transmission of dmp trcks BelA Z with capacity p to 16 t // An. Mining eqipment and electro mechanics", Rssia, no5, p , [7] М. В.Пронин, "Электроприводы и системы с электрическими машинами и полупроводниковыми преобразователями مجلد 1 العدد (моделирование, расчет, применение)." ЕА Крутякова СПб, Силовыемашины Электросила, Rssia, p 252,2004. [8] Г. Корн, Т. Корн," Справочник по математике для научных работников и инженеров". М., «Наука», p 82 с, [9] Пронин, Михаил Васильевич, and А. Г. Воронцов." Силовые полностью управляемые полупроводниковые преобразователи (моделирование и расчет). СПб, ОАО Электросила", Rssia,