Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik Technische Universität München Prof. Dr.-Ing. Ralph Kennel Aricsstr. 21 Email: eat@ei.tum.de Tel.: +49 (0)89 289-28358 D-80333 München Internet: http://eal.ei.tum.de Fax: +49 (0)89 289-28336 Power Electronics Exercise: Block Operation (2013-10-10) 1
1 Theory In modern electric driving technology, inverters are necessary equipment to supply electric power to motors. A power inverter is an electric device that converts direct currents (DC) to alternating currents (AC) with adjustable voltage and frequency. The DC is usually generated from the power grid using a power rectifier. Figure 1 shows this concept. Figure 1. Concept for driving an electric motor The rectifier is not included in the scope of this exercise and will not be described. It is only needed to be notice that it provides a DC voltage. With the standard 400 V three-phase power grid the available DC voltage at the rectifier output is about 600 V. This voltage is used as the input for the rectifier. 1.1 Principle The structure of a simplest inverter is shown in Figure 2. Figure 2. Structure of the inverter It is composed of three bridge arms. Each arm has two switching components, which are IGBTs (insolated-gate biopolar transistor) in the figure. They are numbered from 1 to 6. The reason of this special numbering sequence will be explained later. The switches can be turned on or off by controlling the gate pins of the components. Since the IGBT and most other semiconductor switches conduct only in one direction, the diodes connected in parallel provide the current throughways in the reverse direction. The two capacitors connected in series provide a middle point at the input voltage only for the convenience of analysis. The potential U 0 is used as reference and defined as zero Volt. As it has been known, for driving a three-phase motor, the voltages at output points, u, v and w should have a sinusoidal waveform with 120 phase differences. Since the switches can only be turned on or off, it is not possible to generate sinusoidal voltages using the inverter of Figure 2. Fortunately, a rectangular waveform is an approximation of sinusoidal waves. Its first harmonic is shown in Figure 3 with dashed lines. 2
Figure 3. Voltages and switch control commands for driving an electric machine Such voltages can be used to drive electric motors. Of course, it will not provide a perfect performance. The operation with these rectangular voltages is named Block Operation. Please note that the square wave voltages in the above figure are based on the middle point of the input DC voltage. The actual phase voltages of a three phase load are based on the neutral point of the load and will have difference forms. This will be explained in the exercise part. Let s have a short glance to see how to generate the rectangular voltages using the inverter. Referring to Figure 3, at time 0, the IGBT number 1, 6 and 2 are turned on and others are off. Now phase u has a high voltage and the other two phases have a low voltage. At, the status of the two switches in the middle bridge arm are reversed while all other switches keep unchanged. Thus phase v changes to high voltage and the other two phases keep unchanged. At, another commutation occurs in the left bridge arm, and u voltage changes. When this process continues, three rectangular voltages with 120 phase difference can be generated at the inverter output. During the switching process it must be strictly ensured that the two switches in one bridge arm are not simultaneously turned on at any time. Otherwise short circuit would occur and the power switches could be permanently damaged. 3
1.2 Operation A bridge arm with two switches has four switching status, as shown in Figure 4. In this figure the power switches are represented with simplified symbols. For the analysis, we define the status of a bridge arm with symbol 1 when the upper switch is on while the lower switch is off (status a in the figure), and 0 when reversed (status b ). The other two possibilities are not used during normal operation. Status c is only present when the inverter is out of operation. And status d is forbidden at any time. Figure 4. The four possible status of an inverter bridge arm This definition simplifies the description of the inverter operation. Figure 5 shows the required block operation voltages generated by the three bridge arms. The status of the bridge arms for generating the required voltage levels in a certain time period are given in the middle part of the figure. Here we omit the time period of and start from 30. In the bottom part of the figure, the numbers of the six switches that have on status are listed. From this numbers it is now easily to understand why the special numbering of the switches is used (Figure 2). 4
Figure 5. The switching procedure of the inverter 5
2 Exercises 2.1 Exercise 1 2.1.1 Problem Figure 6 shows an inverter with a three-phase symmetric load. Figure 7 shows the switching status of the three inverter arms based on time. Please calculate the voltage values in different switching status and draw the curves of the following voltages: 1. u u0 (between winding u input point and DC middle point 0) 2. u v0 (between winding v input point and DC middle point 0) 3. u uv (between winding input points u and v, line voltage) 4. u N0 (between neutral point N and DC middle point 0) 5. u un (between winding u input point and neutral point, phase voltage of winding u) 6. u vn (between winding v input point and neutral point, phase voltage of winding v) 7. u wn (between winding w input point and neutral point, phase voltage of winding w) 8. Calculate the space vectors using phase voltages and check the forms of the space vector trajectories. Figure 6. Simplified circuit of an three-phase inverter with load Figure 7. Arm switching status depending on time (phase angle) 6
2.1.2 Solution 1. Depending on the switching status of bridge arm u, u u0 has two values as described in the following equation. 2. u v0 has the same values as u u0. 3. Therefore, u uv depends on the switching of arm u and v: 4. Since, thus, Depends on the switching status of all three arms. The switching circuit can be summarized in only two cases: (1) one arm has status 1 and the other two arms have status 0, and (2) one arm has status 0 and the other two arms have status 1. In steady state, for either cases, we have and 7
Based on the above equations, we have So Please refer to the results of question 1 and 2 for the last step. 5. u un depends on u u0 and u N0 ( ), the combination of which causes four different cases: 6. u vn is similar to u un : 7. u wn is similar to u un : 8
The voltage curves: 8. Space vector instances of the phase voltages (u un, u vn, u wn ): Vector length = 2/3 U dc It is known that the three-phase voltage with sinusoidal form has an ideal circular trajectory, where the space vector circulates around the original point continuously and smoothly. On the contrary, the space vector of the block operation shown in the above figure doesn t have a continuous trajectory. For example during the time period of the space vector stays at the 60 position. And at it jumps to 120 position and stays there for another 60 time period. Such none continuous movement of space vector evokes unexpected effects on the machine. 9
2.2 Exercise 2 2.2.1 Problem Refer to the Figure 6 in Exercise 1, supposing the load is a pure inductive symmetric three-phase load, and the inductance is L, please 1. calculate the phase currents, i u, i v and i w and draw their values depending on time (phase angle), 2. draw the space vector of the current, 3. draw the space vector of an ideal three-phase sinusoidal current, and 4. compare the trajectories of the two current space vectors. 2.2.2 Solution to question 1 The phase current is, where x stands for u, v or w for different windings. I x is a constant during integration. Since the phase voltages are segmented constant values, the currents are segmented lines with constant slopes. And the larger the voltage is, the greater the slope is., where t 0 is the start time of a voltage segment, I x (t 0 ) is the current value at the start time. Using the result of the phase voltages in exercise 1, it is able to calculate the phase currents. They are drawn in the following figure, where I 0 is the maximum current. Figure 8. Currents in a three-phase inductive load using block operation 10
2.2.3 Solution to questions 2 to 4 From the result of the last question, we list the phase current values at some time points in the following table. ωt 30 90 150 210 270 330 i u (x I 0 ) 1/3 2/3 1/3-1/3-2/3-1/3 i v (x I 0 ) -2/3-1/3 1/3 2/3 1/3-1/3 i w (x I 0 ) 1/3-1/3-2/3-1/3 1/3 2/3 The current space vector at the six positions listed in the last table is drawn in the following figure (vector length at the vertex = 2/3 I 0 ). The hexagonal is the trajectory of the current space vector of the block operation, and the circle is the trajectory of the ideal current space vector. Figure 9. Space vector trajectories This seems to be the same as that of the voltage. However, there is essential difference. The trajectory of the current vector is continuous. In the figure, the end of the space vector moves continuously along the hexagonal, not like the voltage, where the space vector only takes six single values. 3 References Valentine, Richard (1998). Motor control electronics handbook. New York: McGraw-Hill (pages from 246) Andrzej M. Trzynadlowski (1994). The field orientation principle in control of induction motors. Assinippi Park: Kluwer academic publishers (chapter 1) 11