Various Modeling Methods For The Analysis Of A Three Phase Diode Bridge Rectifier And A Three Phase Inverter

Transcription

1 Various Modeling Methods For The Analysis Of A Three Phase Diode Bridge Rectifier And A Three Phase Inverter Parvathi M. S PG Scholar, Dept of EEE, Mar Baselios College of Engineering and Technology, Trivandrum Dr. Nisha G. K Associate Professor, Dept of EEE, Mar Baselios College of Engineering and Technology, Trivandrum Abstract: This paper presents the modelling of a three phase diode rectifier and a three phase inverter in the Modified Nodal Analysis method and in the Average method. Virtually all electronic devices require DC, so rectifiers are used inside the power supplies of all electronic equipment. Inverters are extensively used in power electronic applications for converting DC to AC. The applications of inverters range from uninterrupted power supplies used in home appliances control to HVDC power transmission. The average model equations for rectifier and inverter are formulated using the Forward Euler method. Keywords: Average Method, Modified Nodal Analysis method, Three phase diode bridge rectifier and Three phase inverter. I. INTRODUCTION As technology grows every day, the study of power systems has shifted its direction to power electronics to produce the most efficient energy conversion. Power electronics is the study of processing and controlling the flow of electric energy by supplying voltages and currents in a form that is suited for user loads [1]. The goals of using power electronics are to obtain the benefit of lower cost, small power loss and high energy efficiency. Because of high energy efficiency, the removal of heat generated due to dissipated energy is lower. A rectifier is a power processor that should give a dc output voltage with a minimum amount of harmonic contents [2]. When a polyphase ac is rectified, the phase-shifted pulses overlap each other which produce a smoother dc output than that made by a single-phase ac rectifier. In [3], the author presents the modeling of single phase, three phase rectifiers and suitable controller design for the modeled rectifier circuit. An average inverter model operating in two complementary modes suitable for micro grid simulation applications is proposed in [4], by taking into account the nonlinear behaviour of the switches, delays in the control loops, and the practical constraints. In [5], the author focuses on a combination of three-phase Voltage Source Inverter (VSI) with a predictive current control to provide an optimized system for three-phase inverter that controls the load current. With the advent of increased use of inverters, various switching models are established. The average method is shown to be an effective method for analysis and controller design in inverters [6] [8]. Conventional switching models and state-space averaging methods which take dead-time effects into consideration have been applied successfully in the inverters [9], [10]. II. MODELING OF THREE PHASE DIODE BRIDGE RECTIFIER The modeling of a three phase diode bridge rectifier can be done in two ways the Modified Nodal Analysis method and the average model method (using transient simulation techniques). On comparing the two methods, the Modified Nodal Analysis method is a bit too long process where, we need to find the nodal equations and the inverse matrix at each switching conditions. On the other hand the latter provides transient equations which enables to obtain the output dc voltage and current, without the calculation of matrices. The main advantage of the average model method of modeling of diode bridge rectifier is that it provides ripple free output. A. MODIFIED NODAL ANALYSIS METHOD The power circuit diagram for a three phase diode bridge rectifier is as shown in the Fig Page 21

2 R on, when the diode conducts and an off-resistance value R off, when the diode is not conducting. For the analysis purpose, the three phase diode bridge rectifier is replaced by its equivalent circuit diagram with its on-state and off-state resistance values. The equivalent circuit diagram is obtained for each conduction period, where two diodes conduct, one diode from the positive group and the other from the negative group. Consider the interval ωt = 0 to 30 and ωt = 330 to 360, where the diodes D5 and D6 conduct. The conducting diodes D5 and D6 are replaced by their on-resistance values R on and the rest of the diodes with their off-resistance values R off. The equivalent circuit diagram will be as in Fig. 3. Figure 1: Circuit Diagram of a three phase diode bridge rectifier Figure 2: Input Output waveforms and the diode conduction period of a three phase diode rectifier The diodes are arranged in three legs. Each leg has two series-connected diodes. The upper diodes D1, D3, D5 constitute the positive group of diodes. The lower diodes D2, D4, D6 form the negative group of diodes. The diodes D1, D3, D5 forming the positive group, would conduct when these experience the highest positive voltage. Likewise, the diodes D2, D4, D6 would conduct when these are subjected to most negative voltage. For a particular interval, a diode from the positive group and a diode from the negative group conduct. No two diodes from the same group conduct together at a time. The waveforms for the three phase diode rectifier and each diode s corresponding conducting period are as shown in the Fig. 2. It is seen from the source voltage waveform that, from ωt = 30 to 150, voltage V a is more positive than the voltages V b and V c. Therefore, the diode D1 connected to line a conducts during this interval [11]. Likewise, from ωt = 150 to 270, voltage V b is more positive therefore, diode D3 conducts. Similarly, diode D5 from the positive group conducts from ωt = 270 to 390 and so on. Conduction of the positive group diodes is shown in Fig. 2 as D5, D1, D3, D5 etc. Similarly, the negative diodes also conduct accordingly. A three phase diode bridge rectifier consists of six diodes arranged in three legs. Each diode has an on-resistance value Figure 3: Equivalent circuit diagram of a three phase diode bridge rectifier for the interval ωt = 0 to 30 and ωt = 330 to 360 (D5 and D6 conduct) Now, applying KCL at the nodes a, b, c and d, we get the following equations. At node a ; At node b ; At node c ; At node d ; Converting the above nodal equations into matrix form which amounts to solve a linear system AX = Z, where, A denotes the admittance matrix or the modified nodal analysis matrix, X denotes the unknown matrix (unknown node voltages and currents) and Z is the input matrix (source voltages). We can obtain the unknown matrix X as X = A -1 Z. Page 22

3 Assuming suitable values for the resistances as R on = 1mΩ, R off = 1MΩ, R L = 10kΩ and substituting in the above matrix equation, the inverse of A matrix (A -1 ) can be obtained using an online inverse matrix calculator. The A -1 matrix (after substituting the resistance values) for the interval taken, ωt = 0 to 30 and ωt = 330 to 360, can be evaluated as below. Figure 4: Simplified circuit model of a three phase diode bridge rectifier Utilizing the general state space averaging method, the switching functions s a, s b and s c for a three phase diode bridge rectifier can be expressed by a Fourier series using the first harmonic terms of the series as shown in the equation (5). During the interval, ωt = 30 to 90, the diodes D1 and D6 will be conducting and the inverse matrix can be obtained as below. The average dc output voltage V pn can be expressed as the equation (6). Also, the dc current equation can be obtained as shown in equation (7). From the equation (6), the left hand side of the equation can be expanded as: Similarly, A -1 matrix for each conducting interval can be obtained. The A -1 matrix and the Z matrix is given as input to the program and the corresponding outputs waveforms are obtained. B. AVERAGE METHOD FOR RECTIFIER MODELING USING TRANSIENT SIMULATION TECHNIQUES In order to make prime decisions in the design procedures, power electronic modeling methods are widely used. The modeled circuit helps to determine the performance of the circuit along with the component values as well. State space average modeling method is a common technique that is utilized to obtain models of power electronic circuits for ac and dc power conversions. The main advantage of this particular method is that the accuracy can be increased and ripple free waveforms can be obtained. A simplified three phase diode bridge rectifier is shown in the Fig. 4. The circuit consists of three ac input sources (V A, V B and V C ), input source reactor (L s ), diodes (D 1 to D 6 ), output inductance (L out ), output filter capacitance (C) and resistance (R) which represents the load on the rectifier [12]. Now, the right hand side of the equation (6) can be solved using the Transient simulation method. In this method, the differential current equation can be converted to its equivalent Forward Euler form and solved to obtain the value for the (n+1) th value of the dc current. From the equation (9), the (n+1) th value of the dc current can be calculated as: where, h is the delta constant value or the time step. Similarly, from the equation (7), the differential voltage equation can be converted to its equivalent Forward Euler form and solved to obtain the value for the (n+1) th value of the dc voltage. Thus, from the equation (11), the (n+1) th value of the dc voltage can be calculated as: The modeling equation for a three phase diode bridge rectifier is obtained using the average state space model and Page 23

4 the transient simulation techniques. The equations (8), (10) and (12) represent the average model mathematical equations for the rectifier output dc voltage and dc current. III. MODELING OF THREE PHASE INVERTER Similar to the modeling of a three phase diode bridge rectifier, a three phase inverter can also be done in two ways the Modified Nodal Analysis method and the average model method (using transient simulation techniques). The main advantage of the ripple free output in the case of the average model method holds the same for the modeling of a three phase inverter as well. A. MODIFIED NODAL ANALYSIS METHOD A basic three phase inverter is a six-step bridge inverter. It uses a minimum of six thyristors. In inverter terminology, a step is defined as a change in the firing from one thyristor to the next thyristor in the proper sequence. For one cycle of 360, each step would be of 60 interval for a six-step inverter. This means that the thyristors would be gated at regular intervals of 60 in proper sequence so that a three phase ac voltage is synthesized at the output terminals of a six-step inverter. The Fig. 5 shows the power circuit diagram of a three phase bridge inverter using six thyristors and six diodes. Presently, the use of IGBTs in single phase and three phase inverters is on the rise. The basic circuit configuration of inverter, however, remains unaltered with just a small change of replacing the thyristors with IGBTs. A large capacitor connected at the input terminals tends to make the input dc voltage constant. This capacitor also suppresses the harmonics fed back to the dc source. In the Fig. 5, commutation and snubber circuits are omitted for simplicity. The thyristors are numbered in the sequence in which they are triggered to obtain voltages v ab, v bc, v ca at the output terminals a, b, c of the inverter. There are two possible patterns of gating the thyristors. In one pattern, each thyristor conducts for 180 and in the other, each thyristor conducts for 120. But in both these patterns, gating signals are applied and removed at 60 intervals of the output voltage waveform. In three phase inverter, each thyristor conducts for 180 of a cycle. The thyristor pair in each arm, i.e. T 1, T 4 ; T 3, T 6 and T 5, T 2 are turned on with a time interval of 180. The thyristors in the upper group, i.e. T 1, T 3, T 5 conduct at an interval of 120. The conduction time periods for 180 mode 3 phase inverter is as shown in the Fig. 6. It can be understood from the table that in every step of 60 duration, only three thyristors are conducting one from the upper group and two from the lower group or vice versa. Figure 6: Conduction period for a 180 mode 3-phase VSI Similar to the analysis of a three phase diode bridge rectifier, a three phase bridge inverter consists of six thyristors arranged in three legs. Each thyristor has an on-resistance value R on, when the thyristor conducts and an off-resistance value R off, when it is not conducting. For the analysis purpose, the three phase bridge inverter is replaced by its equivalent circuit diagram with its on-state and off-state resistance values. The equivalent circuit diagram is obtained for each conduction period, where three thyristors conduct, one thyristor from the upper group and two from the lower group or vice versa. Consider the interval ωt = 0 to 60, where the thyristors T 1, T 5 and T 6 conduct. The conducting thyristors are replaced by their on-resistance values R on and the rest of the thyristors with their off-resistance values R off. The equivalent circuit diagram will be as in Fig. 7. Figure 7: Equivalent circuit diagram of a three phase bridge inverter for the interval ωt = 0 to 60 (T 1, T 5 and T 6 conduct) Now, applying KCL at the nodes a, b, c and d, we get the following equations. At node a ; At node b ; Figure 5: Circuit Diagram of a three phase bridge inverter using thyristors At node c ; Page 24

5 At node d ; At node o Converting the above nodal equations into matrix form which amounts to solve a linear system AX = Z, where, A denotes the admittance matrix or the modified nodal analysis matrix, X denotes the unknown matrix (unknown node voltages and currents) and Z is the input matrix (source voltages). We can obtain the unknown matrix X as X = A -1 Z. system. Therefore, in case of large scale inverter systems, switching models becomes complicated. Hence, a state space averaging method is formulated which is effective for the analysis and controller design purpose in inverters. A generalized state space averaging model is considered which enhances the fundamental AC voltage and current calculation with desired accuracy and precision. At the same time, this method enables the steady state as well as transient analysis processes [13]. The circuit diagram of a 3 phase 3 wire Voltage source inverter is as shown in the Fig. 8. Assuming suitable values for the resistances as R on = 1mΩ, R off = 1MΩ, R L = 10kΩ and substituting in the above matrix equation, the inverse of A matrix (A -1 ) can be obtained using an online inverse matrix calculator. The A -1 matrix (after substituting the resistance values) for the interval taken, ωt = 0 to 60 can be evaluated as below. Figure 8: Circuit diagram of a three phase three wire voltage source inverter In the Fig. 8, assuming that the loads are 3 phase symmetrical resistive loads in delta connection, whose value are R, we obtain the equations from (7.6) to (7.11). During the interval, ωt = 60 to 120, the diodes T 1, T 2 and T 6 will be conducting and the inverse matrix can be obtained as below. Where, are virtual line currents which can be calculated as mentioned in the equations from (24) to (26). Similarly, A -1 matrix for each conducting interval can be obtained. The A -1 matrix and the Z matrix is given as input to the program and the corresponding outputs waveforms are obtained. B. AVERAGE METHOD FOR 3 PHASE INVERTER MODELING USING TRANSIENT SIMULATION TECHNIQUES Similar to the equation (6) in the modeling of three phase diode bridge rectifier by the average model, it can be written that, Thus, the conventional state equations of three phase inverter can easily be constructed. In the study of inverters and their dynamic performance, various switching models are obtained based on the switching conditions, as in the Modified Nodal Analysis method, which is feasible for a single inverter or a small scale inverter Page 25

6 The differential voltage and current equations can be converted to its equivalent Forward Euler form and substituting the equations (27) to (29) in the Forward Euler equations we obtain the value for the (n+1) th value of the corresponding ac voltage and ac current. In the case of the average model of the three phase diode bridge rectifier, the parameter specifications used for the simulation are as in the Table 1. SYMBOL PARAMETERS VALUE UNIT V abc Input Voltage 415 V f Fundamental frequency 50 Hz L Filtering Inductor 0.05 H C Filter Capacitor 0.05 F R Load Resistance 2 Ω h Time step 100 µs Table 1: Parameters for rectifier simulation in average model The dc voltage and dc current waveforms at the output of the rectifier for the specified parameters are as shown in the Fig. 10. The switching function is given by: where k = a, b, c ; m is the modulation ratio taken as 0.5 ; where is the initial phase angle. Thus, the modelling equations are formed for a three phase inverter in average model using the Transient simulation techniques. IV. RESULTS AND DISCUSSIONS The output waveforms obtained by simulating the MNA model of rectifier with the parameter specification as R on =1 mω, R off =1 MΩ and R L =10 kω is as shown in the Fig. 9. It depicts rectifier output voltage with amplitude 100 V. Figure 10: Output dc voltage and dc current value of a 3 phase rectifier,v dc = 687 V and I dc = 344 A (X axis: 1 unit = 0.5s and Yaxis: 1 unit = 100 units of voltage and current) The output waveform obtained in the MNA method of modelling of inverter with parameter specifications as R on = 1mΩ, R off = 1MΩ and R L = 10kΩ is shown in the Fig. 11. The figure depicts the phase voltage in the MNA method with input amplitude of 70 V. Figure 9: Rectifier output voltage waveform in MNA method (X axis: 1 unit = 0.002s & Yaxis: 1 unit = 10V) Figure 11: Inverter output phase voltage waveform in MNA method (X axis: 1 unit = 0.005s & Yaxis: 1 unit = 20V) Page 26

7 In the case of the average model of a three phase inverter, the parameter specifications used for simulation are as in the Table 2. SYMBOL PARAMETERS VALUE UNIT V dc Input Voltage 100 V m Modulation Ratio f Fundamental frequency 50 Hz φ 0 Initial phase angle 0 rad/s L Filtering Inductor 1.0 mh r L ESR of filtering inductor Ω C Filter Capacitor 1.0 mf R Load Resistance 10 Ω h Time step 100 µs Table 2: Parameters for inverter simulation in average method The simulation waveform for the input dc voltage to the inverter is as shown in the Fig. 12. The simulation waveform for phase voltages at the output of the inverter is as shown in the Fig. 14. The figure shows the ac phase voltages with a magnitude of (2V dc /3 = V) for the given input dc of 100 V. Figure 14: Output phase voltage waveform for the dc input of 100 V (X axis: 1 unit = 0.1s and Y axis: 1 unit = 20 V V. CONCLUSION In this paper, the modeling of three-phase diode bridge rectifier and a three-wire inverter is done based on the Modified Nodal Analysis method and in the state space average method. The simulation waveforms obtained from the two methods are discussed. It is observed from the paper that the average model of the rectifier and inverter is more effective and efficient as it gives ripple free waveforms, thus the use of filters for smoothening the ripples can be eliminated. REFERENCES Figure 12: Input dc voltage of magnitude 100 V to the inverter (X axis: 1 unit = 0.1s and Y axis: 1 unit = 50V) The simulation waveform for phase currents at the output of the inverter is as shown in the Fig. 13. The figure shows the ac phase currents with a magnitude of 20 A for the given input dc of 100 V Figure 13: Output phase current waveform for the dc input of 100 V (X axis: 1 unit = 0.1S and Y axis: 1 unit = 10 A) [1] Mohan N., Undeland T and Robbins W.(2002). Power Electronics: Converters, Application and Design, John Wiley and Sons Inc. [2] Rashid Muhammad H.(2004) Power Electronics Circuits, Devices, and Applications, New Jersey: Pearson Prentice Hall. [3] Nagisetty Sridhar and R. Kanagaraj. (2015). Modeling and Simulation of Controller for Single Phase and Three Phase PWM Rectifiers. Indian Journal of Science and Technology, 8(32), DOI: /ijst/2015/v8i32/ [4] Zeljko Jankovic, Bora Novakovic, Vijay Bhavaraju and Adel Nasiri. (2014). Average modeling of a three-phase inverter for integration in a microgrid. IEEE Energy Conversion Congress and Exposition (ECCE), [5] Ali M. Almaktoof, A. K. Raji, and M. T. E. Kahn. (2014). Modeling and Simulation of Three-Phase Voltage Source Inverter Using a Model Predictive Current Control. International Journal of Innovation, Management and Technology, 5(1). [6] Runxin Wang, Jinjun Liu. (2009). Redefining a New- Formed Average Model for Three-Phase Boost Rectifiers/Voltage Source Inverters. The 24th Applied Page 27

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