HIGHPERFORMANCE DRIVES --------------------------------------------------------------------------------------------------------------------------------------- E Levi, 2001 29 2.4.2. Voltage source inverter (VSI) Variable voltage, variable frequency operation of induction machines is realised utilising autonomous inverters, in conjunction with a rectifier and a DC link circuit. The voltage source inverter (VSI) is the most frequently applied power supply source for speed control of induction motors. It can be operated in six-step mode or in PWM mode. Six-step operation will be considered first. Three-phase VSI contains three inverter legs. Input voltage for a three-phase VSI is provided by a three- phase (or single-phase) bridge rectifier with capacitor placed at the output. The capacitor provides smoothing of the DC voltage and, for sufficiently large capacitance, DC voltage at the rectifier output approaches a constant value. It will therefore be assumed that inverter input voltage is constant in all the subsequent analysis. Power circuit of a voltage source inverter is shown in Fig. 2.13. As the inverter itself controls only the frequency of the output voltage when operated with switching frequency equal to output fundamental frequency (six-step mode), a controllable rectifier must be used in order to provide control of the output voltage magnitude (output voltage magnitude is proportional to the input DC voltage). Each switch in the inverter circuit is composed of two back-to-back connected semiconductor devices. One of these two is a controllable switch, while the other one is a diode. The three inverter legs are controlled in such a way that leg voltages constitute three-phase system of square-wave voltages. This means that, assuming that upper transistor in leg A is fired at time instant zero, firing of upper transistor in leg B will take place after 120 degrees, while firing of the upper transistor in leg C will be delayed for another 120 degrees. The conduction of each of the six semiconductor switches is again 180 degrees so that at any time three out of six switches are on and the remaining three switches are off. The resulting output voltage waveforms for line-to-line voltages are quasi-square waves, with two 60 degrees zero intervals and two 120 degrees intervals in which line-to-line voltage equals plus and minus DC voltage, respectively. VSI operated in the 180 degrees conduction mode is therefore usually called six-step inverter. Leg voltages of the inverter are given in Fig. 2.14 with respect to the negative pole of the DC link. Line-to-line voltages applied to the induction machine are obtained directly from leg voltages as p C V DC ABC n IM Rectifier and inverter control Fig. 2.13 - Three-phase voltage source inverter (VSI) fed induction motor drive. HIGHPERFORMANCE DRIVES --------------------------------------------------------------------------------------------------------------------------------------- E Levi, 2001 30 v An V DC v AB V DC v Bn v BC v Cn v CA 0 60 120 180 240 300 360 ωt[°] Leg voltages Line-to-line voltages 2/3 V DC v a 1/3V DC v b Phase to neutral voltages v c Fig. 2.14 - Leg, line-to-line and phase to neutral voltages in VSI fed induction machine. v AB =v An -v Bn v BC =v Bn -v Cn (2.53) v CA =v Cn -v An Line-to-line voltages are shown in Fig. 2.14 as well. Finally, if the machine is star connected, it can be shown that in the system of Fig. 2.13 phase to neutral voltages of the machine (included in the Fig. 2.14) are determined with the following expressions: () () () vv vv vv vv vv vv aAnBnCcn bBnAnCn cCnBnAn =−+ =− + =−+ 23 13 23 13 23 13 // // // (2.54) HIGHPERFORMANCE DRIVES --------------------------------------------------------------------------------------------------------------------------------------- E Levi, 2001 31 Let us consider now situation in a three-phase PWM voltage source inverter (VSI) with regard to space vector theory. From the point of view of the distinct non-zero voltage values that can be obtained, there is no difference between six-step VSI and a PWM VSI. A six-step VSI is therefore analysed. Power circuit of the VSI and associated voltage wave-forms, valid for six-step operation, are those of Figs. 2.13 and 2.14. As can be seen From Fig. 2.14, change in any one of the three leg voltages takes place after every sixty degrees. Leg voltages have constant values within sixty degrees intervals. Thus it follows that the space vector of leg voltages will have six distinct and discrete values and that, instead of uniformly rotating in the complex plane, it will be jumping from one position to the other. Table 2.1 summarises values of leg voltages in the six sixty degrees intervals, lists switches that are on, and defines a corresponding space vector for each interval. Apart from the six non-zero voltage space vectors, that can be obtained in the six-step mode of operation, two additional vectors (no. 7 and 8) are added at the bottom of the Table. These two vectors can be obtained only in PWM operation of the VSI and they describe the condition when the induction motor terminals are short circuited either through the positive rail of the dc supply (vector 7) or through the negative rail of the dc supply (vector 8). Calculation of the leg voltage space vectors is rather simple. From the definition of the voltage space vector in (2.28) one gets, by substituting individual leg voltages of Table 2.1 for each of the six sixty degrees intervals, the following (note that stationary reference frame is considered at all times; superscript ‘s’ is omitted): () vvavav a j a j s s An Bn Cn =++ = = 2 3 23 43 22 ,exp( ) exp( ) ππ (2.28) () () () () () () vV vVj vVj vVj vVj vVj DC DC DC DC DC DC 12 3 45 6 23 23 3 23 2 3 23 23 4 3 23 5 3 == = == = exp( /) exp( /) exp( ) exp( / ) exp( / ) ππ ππ π (2.55) Table 2.1 - Leg voltages switching state switches on space vector Leg voltage v An Leg voltage v Bn Leg voltage v Cn 1 1,4,6 v 1 V DC 00 2 1,3,6 v 2 V DC V DC 0 3 2,3,6 v 3 0V DC 0 4 2,3,5 v 4 0V DC V DC 5 2,4,5 v 5 00V DC 6 1,4,5 v 6 V DC 0V DC 7 1,3,5 v 7 V DC V DC V DC 8 2,4,6 v 8 000 The two remaining space vectors are identically equal to zero as either all the leg voltages are zero or all the leg voltages have the same value ( 10 2 ++ =aa ). Hence vv 78 0== (2.56) It follows from (2.55) that all the non-zero space vectors have identical amplitudes. However, they are stationary, indicating that only discrete values of the leg voltage space vector are possible in a VSI. In the six-step mode of operation transition from one space vector to the other takes place after each sixty degrees interval (for 50 Hz output, after 3.33 ms). In the PWM mode of operation the non-zero values remain to be given with (2.55). PWM mode adds two more possible vectors, called zero vectors, (2.56). Additionally, transition from one vector to the other takes place at much higher frequency than the HIGHPERFORMANCE DRIVES --------------------------------------------------------------------------------------------------------------------------------------- E Levi, 2001 32 output frequency and each vector is utilised many times in creation of the output voltage of the given frequency. Distinct values of leg voltages (2.55) can be described with a single equation () () vVjk leg DC =− é ë ê ù û ú 23 1 3 exp π k = 1,2,3,4,5,6 (2.57) For k =7andk = 8 leg voltage space vector equals zero. As can be seen from (2.57), time is not present in this equation, confirming that the vector does not travel continuously in time. Frequency is not present either, so that the rate at which certain vector value is applied will be governed by switching frequency in the PWM VSI. Space vector values for the leg voltage are shown in Fig. 2.15. Consider next line-to-line voltages at the output of the inverter, shown in Fig. 2.14. Table 2.2 summarises values of line-to-line voltages in different sixty degree intervals and again lists two more states, obtainable in PWM mode only, when all the line-to-line voltages are zero. There are again six non-zero values of the voltage space vector and two conditions that yield zero value of the voltage space vector. 32 7,8 41Re( α ) 56 Fig. 2.15 - Discrete values of the leg voltage space vector. Table 2.2 - Line-to-line voltages switching state switches on space vector Line-to-line voltage v AB Line-to-line voltage v BC Line-to-line voltage v CA 1 1,4,6 v 1L V DC 0 −V DC 2 1,3,6 v 2L 0V DC −V DC 3 2,3,6 v 3L −V DC V DC 0 4 2,3,5 v 4L −V DC 0V DC 5 2,4,5 v 5L 0 −V DC V DC 6 1,4,5 v 6L V DC −V DC 0 7 1,3,5 v 7L 000 8 2,4,6 v 8L 000 Space vector of line-to-line voltages is again calculated using the definition of the space vector, (2.28). Substitution of individual line-to-line voltages into (2.28) for each of the six sixty degrees intervals produces the following result: HIGHPERFORMANCE DRIVES --------------------------------------------------------------------------------------------------------------------------------------- E Levi, 2001 33 () () () () () () vVj vVj vVj vVj vVj vVj L DC L DC L DC L DC L DC L DC 12 34 5 6 23 3 6) 23 3 2 23 3 5 6) 23 3 7 6) 2 3 3 3 2 2 3 3 11 6) == == == exp( / exp( / ) exp( / exp( / exp( / ) exp( / ππ ππ ππ (2.58) Space vector of line-to-line voltages is again equal to zero in states 7 and 8, vv LL 78 0== (2.59) Line-to-line voltages are therefore characterised with six discrete values, whose amplitude is √3larger than for the leg voltages, and they are shifted in phase by 30 degrees with respect to the corresponding values of the leg voltage space vector. Values of the line-to-line voltage space vector are shown in Fig. 2.16. Space vector of line-to-line voltages, whose discrete values are given in (2.58), can be described with an expression similar to (2.57) () vVjk L DC =− é ë ê ù û ú 2 3 321 6 exp π k = 1,2,3,4,5,6 (2.60) For k =7andk = 8 line-to-line voltage space vector equals zero. As expected, (2.60) is independent of time. Hence the time interval during which the space vector remains in one position is determined with the inverter switching frequency. 2 37,8 1 Re ( α ) 46 5 Fig. 2.16 - Discrete values of the line-to-line voltage space vector. Finally, let us consider phase to neutral voltages of the motor, whose wave-forms are given in Fig. 2.14. Table 2.3 summarises values of phase to neutral voltages for the six sixty degrees intervals. Table 2.3 - Phase to neutral voltages switching state switches on space vector Phase voltage v a Phase voltage v b Phase voltage v c 1 1,4,6 v 1phase (100) (2/3)V DC -(1/3)V DC -(1/3)V DC 2 1,3,6 v 2phase (110) (1/3)V DC (1/3)V DC -(2/3)V DC 3 2,3,6 v 3phase (010) -(1/3)V DC (2/3)V DC -(1/3)V DC 4 2,3,5 v 4phase (011) -(2/3)V DC (1/3)V DC (1/3)V DC 5 2,4,5 v 5phase (001) -(1/3)V DC -(1/3)V DC (2/3)V DC 6 1,4,5 v 6phase (101) (1/3)V DC -(2/3)V DC (1/3)V DC HIGHPERFORMANCE DRIVES --------------------------------------------------------------------------------------------------------------------------------------- E Levi, 2001 34 Applying once more the same procedure, one finds the non-zero values of the space vector of the phase voltages as equal to: () () () () () () vV vVj vVj vVj vVj vVj phase DC phase DC phase DC phase DC phase DC phase DC 12 34 56 23 23 3 23 2 3 23 23 4 3 23 5 3 == == == exp( / ) exp( / ) exp( ) exp( / ) exp( / ) π ππ ππ (2.61) The two remaining space vectors are again identically equal to zero. Hence vv phase phase78 0== (2.62) Values of the space vector of phase voltages are identically equal to the values of the leg voltage space vector. Thus () () vVjk phase DC =− é ë ê ù û ú 23 1 3 exp π k = 1,2,3,4,5,6 (2.63) For k =7andk = 8 phase voltage space vector equals zero. Space vector values for the phase to neutral voltages are shown in Fig. 2.17, which is identical to Fig. 2.15. 32 7,8 41Re( α ) 56 Fig. 6.17 - Discrete values of the phase voltage space vector. As expected for a balanced three-phase system of voltages, line-to-line voltages have √3larger amplitude than phase voltages and are leading corresponding phase voltages by 30 degrees. In all the cases so far the stationary reference frame was under consideration. Suppose now that the space vectors of phase to neutral voltages are to be represented in a synchronously rotating reference frame, ω a = ω, θ ω s t= . Then from (2.63) and (2.30) () () () () vveVjk j vVjkt phase synch phase j DC s phase synch DC s () () exp exp( ) exp == − é ë ê ù û ú − =−− æ è ç ö ø ÷ é ë ê ù û ú − θ π θ π ω 23 1 3 23 1 3 k = 1,2,3,4,5,6 (2.64) This is an interesting result. Recall that for sinusoidal voltage supply space vector is constant in the synchronously rotating frame, while it continuously changes in time in the stationary reference frame. In contrast to this, space vector of VSI is constant for any given k in the stationary reference frame, while it continuously changes in time in synchronously rotating reference frame. VSI is nowadays operated in PWM mode rather than in six-step mode. However, the discrete values of the output voltage are still those obtainable in six-step mode, with the addition of the possibility of connecting all the three phases simultaneously to either positive or negative rail of the DC link. Operation of a VSI in PWM mode yields two substantial benefits, when compared to operation in 180 degrees conduction mode. A diode rectifier can be used instead of a controllable rectifier, since the HIGHPERFORMANCE DRIVES --------------------------------------------------------------------------------------------------------------------------------------- E Levi, 2001 35 inverter is now capable of controlling both the frequency and the rms value of the fundamental component of the output voltage. Additionally, higher harmonics of the voltage are now of substantially higher frequencies, meaning that current is much closer to a true sine waveform. When a VSI is operated in the PWM mode, switching frequency is substantially higher than the output fundamental frequency. One special type of PWM, that is nowadays extremely frequently applied, is the so-called ‘voltage space vector modulation’. This PWM method is frequently used in vector controlled drives of so-called ‘voltage-fed’ type, as will be discussed later. Once when the space vectors have been introduced, it becomes now possible to explain the principle of the voltage space vector modulation. The control system will always generate a reference voltage space vector, that corresponds to the ideal sinusoidal three-phase supply of certain frequency and amplitude. Hence the reference voltage space vector is in the stationary reference frame (superscript ‘s’ is omitted once more): vVe Ve s jt j * ==22 ω α (2.65) On the other hand, the PWM inverter can generate only six discrete non-zero voltage vectors and two zero voltage vectors. It is therefore not possible to directly impose the required reference voltage vector (2.64). However, the reference value of the voltage space vector can be obtained on average, during one switching period, by imposing the two neighbouring available space vectors and a zero space vector for appropriate time intervals during the switching period. Consider the situation shown in Fig. 2.18. The reference voltage space vector is shown in a particular instant of time as being positioned in the first sextant of the plane. During the switching period this desired reference value can be achieved on average by imposing the available space vectors 1 and 2 and the zero voltage space vector for appropriate time intervals. 32 7,8 v s * 41Re( α ) 56 2 bv 2 v s * 7,8 α 1Re( α ) av 1 Fig. 2.18 - The principle of voltage space vector modulation. AccordingtoFig.2.18, vavbv s * =+ 12 (2.66) Let the switching period be T s . Similarly, let the two zero voltage vectors be denoted as v o .Inorderto achieve during one switching period on average required reference voltage (2.65) by means of (2.66), it is necessary to impose non-zero voltage vectors 1 and 2 for the times aT s and bT s , respectively. Zero voltage vector will be imposed for the remainder of the switching period, i.e. for time interval cT s . Therefore a + b + c = 1 (2.67) Proportions of the switching period during which appropriate switching vectors are imposed are governed by the amplitude and phase of the reference voltage space vector. It can be shown that HIGHPERFORMANCE DRIVES --------------------------------------------------------------------------------------------------------------------------------------- E Levi, 2001 36 () a V V b V V cab DC DC = − = =−− 3 2 23 23 3 2 2 23 1 sin sin sin sin πα π α π (2.68) where √2Vandα are the amplitude and the phase of the reference space vector in particular instant. In essence, using voltage space vector modulation, one applies the following for the considered case: vavbvcv Ve a V b V e c s j DC DC j * =++ = æ è ç ö ø ÷ + æ è ç ö ø ÷ +⋅ 120 3 2 2 3 2 3 0 απ (2.69) Substitution of (2.68) into (2.69) yields () () () () () () () 2 2 3 2 2 3 3 2 21 23 33 2 3 33 2 3 33 3 3 2 3 33 2 3 3 2 3 Ve V a be Ve V V V j ej ej ej je j DC j j DC DC j j j j απ α α α α α π πα α π πα α π παπααπ απ πα απ α α =+ =−+ é ë ê ê ù û ú ú =−+ =−++ =+=+= sin sin sin exp sin sin exp sin cos cos sin sin cos sin sin sin cos sin sin (cos sin ) (2.70) Desired reference voltage space vector is therefore achieved on average, during one switching period. As time goes by, phase of the reference vector will change, so that coefficients a, b and c will have different values for different switching instants. Similarly, neighbouring vectors that are applied will change as well, from 1 and 2 to 2 and 3, then to 3 and 4, further to 4 and 5, and finally to 5 and 6 during one period of the output frequency. An important consideration is that for sextants other than the first, angle α needs to be substituted with α - 60 degrees for the second sextant, α - 120 degrees for the third sextant, α - 180 degrees for the fourth sextant, α - 240 degrees for the fifth sextant and α - 300 degrees for the sixth sextant in equations (2.68). Example: A three phase induction machine is fed from PWM VSI controlled using voltage space vector modulation. The inverter switching frequency is 10 kHz and the DC bus voltage at the inverter input is 530 V. The inverter output frequency is 40 Hz. Determine which of the space vectors will be applied and for how long when the reference voltage rms value is 190 V, for the following time instants: 1 ms, 5 ms, 7 ms and 12 ms. Solution: The inverter switching frequency is 10 kHz. Hence one switching period is 0.1 ms. The stator voltage reference space vector has the amplitude of √2 x 190 = 268.7 V. Its phase is α=ω t=2π 40 t = 251.33 t and for the given time instants has the following values: t=1ms α = 0.25133 rad or 14.4 degrees t=5ms α = 1.25665 rad or 72 degrees t=7ms α = 1.75931 rad or 100.8 degrees t=12ms α = 3.016 rad or 172.8 degrees As one sixth of the output voltage period (which is 25 ms for 40 Hz) is 4.1666 ms, then t = 1 ms means operation in the first sextant, t = 5 ms, corresponds to operation in the second sextant, t = 7 ms HIGHPERFORMANCE DRIVES --------------------------------------------------------------------------------------------------------------------------------------- E Levi, 2001 37 is still in the second sextant, while t =12 ms is in the third sextant. The following voltage vectors will than be applied: t = 1 ms zero voltage vector plus vectors 1 and 2 t = 5 ms zero voltage vector plus vectors 2 and 3 t = 7 ms zero voltage vector plus vectors 2 and 3 t = 12 ms zero voltage vector plus vectors 3 and 4 Coefficient a will always apply to the first of the two neighbouring vectors in anticlockwise direction (direction of rotation of the reference space vector). Using expressions for calculation of the coefficients a, b and c () () () a V V b V V cab DC DC = − = − =− === =−− 3 2 23 23 15 268 7 530 60 0866 0878 60 3 2 2 23 15 268 7 530 0866 0878 1 sin sin . .sin . .sin sin sin . .sin . .sin πα π α α α π α α one gets: t=1ms α= 14.4 degrees a = 0.627 b = 0.218 c = 0.154 t=5ms α - 60 = 12 degrees a = 0.652 b = 0.182 c = 0.165 t=7ms α - 60 = 40.8 degrees a = 0.288 b = 0.574 c = 0.138 t=12ms α - 120 = 52.8 degrees a = 0.110 b = 0.699 c = 0.190 It is possible to determine the maximum value of the output fundamental rms voltage using the expressions for calculation of the coefficients a, b and c. Consider this same example, in which DC link voltage remains to be 530 V. For the maximum output voltage, duration of the application of the zero voltage vector will be zero (i.e. c = 0). Observe the instant for which α = 30 degrees. In this instant reference voltage is exactly in the middle of the first sextant. Hence space vectors 1 and 2 need to be applied for the same time duration for maximum output voltage in this instant (i.e. a = b =0.5).Thenit follows that () () () () () 223 22305025053223 1 2 3 2 3 2 2 1 3 3 2 1 2 2 1 3 1 6 0 408 0 408 216 2 30 60 30 30 30 30 Ve V e Ve V j V j Ve V j Ve Ve V V V V j DC j j DC DC j DC j DC j DC DC DC max max max max max max ./ ( ) . = =++= + =+ æ è ç ö ø ÷ = Þ == == 0.5+ 0.5 V Locus of the maximum achievable voltage in VSI with space vector PWM is shown in Fig. 2.19. v smax * 4 Fig. 2.19 - Non-zero voltage vectors of the inverter and the achievable maximum output voltage locus. . transition from one vector to the other takes place at much higher frequency than the HIGH PERFORMANCE DRIVES ---------------------------------------------------------------------------------------------------------------------------------------. 2.13 - Three-phase voltage source inverter (VSI) fed induction motor drive. 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