Power Electronic Converters

CHAPTER 1 Power Electronic Converters ANDRZEJ M. TRZYNADLOWSKI University of Nevada, Reno, Nevada This introductory chapter provides a background to the subject of the book. Fundamental principles of electric power conditioning are explained using a hypothetical genetic power converter. Ac to dc, ac to ac, dc to dc, and dc to ac power electronic converters are described, including select operating characteristics and equations of their most common representatives. 1.1 PRINCIPLES OF ELECTRIC POWER C O N D I T I O N I N G Electric power is supplied in a "raw," fixed-frequency, fixed-voltage form. For small consumers, such as homes or small stores, usually only the single-phase ac voltage is available, whereas large energy users, typically industrial facilities, draw most of their electrical energy via threephase lines. The demand for conditioned power is growing rapidly, mostly because of the progressing sophistication and automation of industrial processes. Power conditioning involves both power conversion, ac to dc or dc to ac, and control. Power electronic converters performing the conditioning are highly efficient and reliable. Power electronic converters can be thought of as networks of semiconductor power switches. Depending on the type, the switches can be uncontrolled, semicontrolled, or fully controlled. The state of uncontrolled switches, the power diodes, depends on the operating conditions only. A diode turns on (closes) when positively biased and it turns off (opens) when the conducted current changes its polarity to negative. Semicontrolled switches, the SCRs (silicon controlled rectifiers), can be turned on by a gate current signal, but they turn offjust like the diodes. Most of the existing power switches are fully controlled, that is, they can both be turned on and off by appropriate voltage or current signals. Principles of electric power conversion can easily be explained using a hypothetical "genetic power converter" shown in Fig. 1.1. It is a simple network of five switches, SO through $4, of which S1 opens and closes simultaneously with $2, and $3 opens and closes simultaneously with $4. These four switches can all be open (OFF), but they may not be all closed (ON) because they would short the supply source. Switch SO is only closed when all the other switches are open. It is assumed that the switches open and close instantly, so that currents flowing through them can be redirected without interruption. 2 CHAPTER 1 /POWER ELECTRONIC CONVERTERS $1 $3 SOURCE L OAD s 2 ~ FIGURE 1.1 Generic power converter. The genetic converter can assume three states only: (1) State 0, with switches S 1 through $4 open and switch SO closed, (2) State 1, with switches S1 and $2 closed and the other three switches open, and (3) State 2, with switches $3 and $4 closed and the other three switches open. Relations between the output voltage, Vo, and the input voltage, vi, and between the input current, ii, and output current, io, are Vo- 0 vi -vi i i -- 0 io -i o in State in State in State 0 1 2 (1.1) 0 1 2. (1.2) and in State in State in State Thus, depending on the state of genetic converter, its switches connect, cross-connect, or disconnect the output terminals from the input terminals. In the last case (State 0), switch SO provides a path for the output current (load current) when the load includes some inductance, L. In absence of that switch, interrupting the current would cause a dangerous impulse overvoltage, Ldio/dt ~ -cx~. Instead of listing the input-output relations as in Eqs. (1.1) and (1.2), the so-called switching functions (or switching variables) can be assigned to individual sets of switches. Let a = 0 when switch SO is open and a = 1 when it is closed, b -- 0 when switches S 1 and $2 are open and b = 1 when they are closed, and c = 0 when switches $3 and $4 are open and c = 1 when they are closed. Then, v o -- h(b - c)/) i (1.3) and i i -- ct(b - c)i o. (1.4) The ac to dc power conversion in the generic converter is performed by setting it to State 2 whenever the input voltage is negative. Vice-versa, the dc to ac conversion is realized by periodic repetition of the State 1-State 2 - . . . sequence (note that the same state sequence appears for the ac to dc conversion). These two basic types of power conversion are illustrated in Figs. 1.2 and 1.3. Thus, electric power conversion is realized by appropriate operation of switches of the converter. Switching is also used for controlling the output voltage. Two basic types of voltage control are phase control and pulse width modulation. The phase control consists of delaying States 1 1.1 PRINCIPLESOF ELECTRIC POWER CONDITIONING 3 v~ J (a) (b) FIGURE 1.2 Ac to dc conversion in the generic power converter: (a) input voltage, (b) output voltage. and 2 and setting the converter to State 0. Figure 1.4 shows the generic power converter operating as an ac voltage controller (ac to ac converter). For 50% of each half-cycle, State 1 is replaced with State 0, resulting in significant reduction of the rms value of output voltage (in this case, to 1/~/2 ofrms value of the input voltage). The pulse width modulation (PWM) also makes use of State 0, but much more frequently and for much shorter time intervals. As shown in Fig. 1.5 for the same genetic ac voltage controller, instead of removing whole "chunks" of the waveform, numerous "slices" of this waveform are cut out within each switching cycle of the converter. The switchingfrequency, a reciprocal of a single switching period, is at least one order of magnitude higher than the input or output frequency. The difference between phase control and PWM is blurred in dc to dc converters, in which both the input and output frequencies are zero, and the switching cycle is the operating cycle. The dc to dc conversion performed in the genetic power converter working as a chopper (dc to dc converter) is illustrated in Fig. 1.6. Switches S 1 and $2 in this example operate with the duty ratio of 0.5, reducing the average output voltage by 50% in comparison with the input voltage. The duty ratio of a switch is defined as the fraction of the switching cycle during which the switch is ON. To describe the magnitude control properties of power electronic converters, it is convenient to introduce the so-called magnitude control ratio, M, defined as the ratio of the actual useful output voltage to the maximum available value of this voltage. In dc-output converters, the useful output voltage is the dc component of the total output voltage of the converter, whereas in acoutput ones, it is the fundamental component of the output voltage. Generally, the magnitude control ratio can assume values in the - 1 to + 1 range. 4 CHAPTER 1 / POWER ELECTRONIC CONVERTERS (a) Vo STATE 1 STATE 2 STATE 1 . STATE 2 t (b) ' FIGURE 1.3 Dc to ac conversion in the generic power converter: (a) input voltage, (b) output voltage. In practical power electronic converters, the electric power is supplied by voltage sources or current sources. Each of these can be of the uncontrolled or controlled type, but a parallel capacitance is a common feature of the voltage sources while a series inductance is typical for the current sources. The capacitance or inductance is sufficiently large to prevent significant changes of the input voltage or current within an operating cycle of the converter. Similarly, loads can also have the voltage-source or current-source characteristics, resulting from a parallel capacitance or series inductance. To avoid direct connection of two capacitances charged to different voltages or two inductances conducting different currents, a voltage-source load requires a current-source converter and, vice versa, a current-source load must be supplied from a voltage-source converter. These two basic source-converter-load configurations are illustrated in Fig. 1.7. 1.2 AC TO DC CONVERTERS Ac to dc converters, the rectifiers, come in many types and can variously be classified as uncontrolled versus controlled, single-phase versus multiphase (usually, three-phase), half-wave versus full-wave, or phase-controlled versus pulse width modulated. Uncontrolled rectifiers are based on power diodes; in phase-controlled rectifiers SCRs are used; and pulse width modulated 1.2 AC TO DC CONVERTERS 5 vi t (a) vo F'x X STATE0 ~STATE 0 STATE1 STATE01 ~STATE 0 t (b) FIGURE 1.4 Phase control of output voltage in the generic power converter operating as an ac voltage controller: (a) input voltage, (b) output voltage. rectifiers require fully controlled switches, such as IGBTs (insulated gate bipolar transistors) or power MOSFETs. The two most common rectifier topologies are the single-phase bridge and three-phase bridge. Both are full-wave rectifiers, with no dc component in the input current. This current is the main reason why half-wave rectifiers, although feasible, are avoided in practice. The single-phase and three-phase diode rectifiers are shown in Fig. 1.8 with an RLE (resistive-inductive-EMF) load. At any time, one and only one pair of diodes conducts the output current. One of these diodes belongs to the common-anode group (upper row), the other to the common-cathode group (lower row), and they are in different legs of the rectifier. The line-to-line voltage of the supply line constitutes the input voltage of the three-phase rectifier, also known as a six-pulse rectifier. The single-phase bridge rectifier is usually referred to as a two-pulse rectifier. In practice, the output current in full-wave diode rectifiers is continuous, that is, it never drops to zero. This mostly dc current contains an ac component (tipple), dependent on the type of rectifier and parameters of the load. Output voltage waveforms of rectifiers in Fig. 1.8 within a single period, T, of input frequency are shown in Fig. 1.9, along with example waveforms of the output current. The average output voltage (dc component), Vo, is given by Vo 2 Vi,p '~ 0.63 Vi,p m (1.5) for the two-pulse diode rectifier and Vo=3 Vi,p ~ 0.95 Vi,p -- 0.95 VLL,p (1.6) 6 CHAPTER 1 /POWER ELECTRONIC CONVERTERS v~ (a) Vo (b) FIGURE 1.5 PWM control of output voltage in the generic power converter operating as an ac voltage controller: (a) input voltage, (b) output voltage. for the six-pulse diode rectifier. Here, Vi, p denotes the peak value of input voltage, which, in the case of the six-pulse rectifier, is the peak line-to-line voltage, VLL,p. In phase-controlled rectifiers shown in Fig. 1.10, diodes are replaced with SCRs. Each SCR must be turned on (fired) by a gate signal (firing pulse) in each cycle of the supply voltage. In the angle domain, ~ot, where 09 denotes the supply frequency in rad/s, the gate signal can be delayed by ~f radians with respect to the instant in which a diode replacing a given SCR would start to conduct. This delay, called afiring angle, can be controlled in a wide range. Firing pulses for all six SCRs are shown in Fig. 1.11. Under the continuous conductance condition, the average output voltage, Vo(con) , of a controlled rectifier is given by Vo(con) -" Vo(unc)cos(0~f) (1.7) where Vo(unc) denotes the average output voltage of an uncontrolled rectifier (diode rectifier) of the same type. It can be seen that cos(~f) constitutes the magnitude control ratio of phasecontrolled rectifiers. Example waveforms of output voltage in two- and six-pulse controlled rectifiers are shown in Fig. 1.12, for the firing angle of 45 ~. As in uncontrolled rectifiers, the output current is basically of the dc quality, with certain tipple. The ripple factor, defined as the ratio of the rms value of the ac component to the dc component, increases with the firing angle. At a sufficiently high value of the firing angle, the continuous current waveform breaks down into separate pulses. The conduction mode depends on the load EMF, load angle, and firing angle. The graph in Fig. 1.13 illustrates that relation for a six-pulse rectifier: for a given firing angle, the continuous conduction area lies below the line representing this angle. For example, for load and firing angles both of 30 ~ the load EMF 1.2 AC TO DC CONVERTERS 7 (a) Vo . (b) FIGURE 1.6 PWM control of output voltage in the generic power converter operating as a chopper: (a) input voltage, (b) output voltage. CONVERTER , , (o) CONVERTER ...... (b) FIGURE 1.7 Two basic source-converter-load configurations: (a) voltage-source converter with a currem-source load, (b) current-source converter with a voltage-source load. 8 CHAPTER1/POWERELECTRONICCONVERTERS iO DA~ ,R DB DA'~ 12c (0) Ic ~o DA2 oc,'I OA'2 )E (b) FIGURE 1.8 Diode rectifiers: (a) single-phase bridge, (b) three-phase bridge. coefficient, defined as the ratio of the load EMF to the peak value of line-to-line input voltage, must not be greater than 0.75. Equation (1.7) indicates that the average output voltage becomes negative when 0~f > 90 ~ Then, as the output current is always positive, the power flow is reversed, that is, the power is transferred from the load to the source, and the rectifier is said to operate in the inverter mode. Clearly, the load must contain a negative EMF as a source of that power. Figure 1.14 shows four possible operating quadrants of a power converter. In Quadrants 1 and 3, the rectifier transfers electric power from the source to the load, while Quadrants 2 and 4 represent the inverter operation. A single controlled rectifier can only operate in Quadrants 1 and 4, that is, with a positive output current. As illustrated in Fig. 1.15a, the current can be reversed using a cross-switch between a rectifier and a load, typically a dc motor. In this way, the rectifier and load terminals can be connected directly or cross-connected. This method of extending operation of the rectifier on Quadrants 2 and 3 is only practical when the switch does not have to be used frequently as, for example, in an electric locomotive. Therefore, a much more common solution consists in connecting two controlled rectifiers in antiparallel, creating the so-called dual converter shown in Fig. 1.15b. 1.2 AC TO DC CONVERTERS 9 t T/2 T (a) o I o r T/2 .... It T (b) FIGURE 1.9 Output voltage and current waveforms in diode rectifiers: (a) single-phase bridge, (b) three-phase bridge. There are two types of dual converters. Figure 1.16 shows the circulating current-free dual converter, that is, a rectifier in which single SCRs have been replaced with antiparallel SCR pairs. This arrangement is simple and compact, but it has two serious weaknesses. First, to prevent an interphase short circuit, only one internal rectifier can be active at a given time. For example, with TB 1 and TCI' conducting, TC2' is forward biased and, if fired, it would short lines B and C. This can easily be prevented by appropriate control of firing signals, but when a change in polarity of the output current is required, the incoming rectifier must wait until the current in the outgoing rectifier dies out and the conducting SCRs turn off. This delay slows down the response to current control commands, which in certain applications is not acceptable. Secondly, as in all single phase-controlled rectifiers, if the firing angle is too large and/or the load inductance is too low, the output current becomes discontinuous, which is undesirable. For instance, such a current would generate a pulsating torque in a dc motor, causing strong acoustic noise and vibration. In the circulating current-conducting dual converter, shown in Fig. 1.17, both constituent rectifiers are active simultaneously. Depending on the operating quadrant, one rectifier works with the firing angle, ~f, 1, less than 90 ~ The other rectifier operates in the inverter mode with the firing angle, czf,2, given by CZf,2 - - f l - (Zf, 1 (1.8) where fl is a controlled variable. It is maintained at a value of about 180 ~ so that both rectifiers produce the same average voltage. However, the instantaneous output voltages of the rectifiers are not identical, and their difference generates a current circulating between the rectifiers. If the rectifiers were directly connected as in Fig. 1.15b, the circulating current, limited by the resistance of wires and conducting SCRs only, would be excessive. Therefore, reactors are 10 CHAPTER 1 /POWER ELECTRONIC CONVERTERS ~0 ,r ;R I'- T~ )E (o) icI T~ T lo ;R TCZ~ --x_ T~ )E o (b) FIGURE 1.10 Phase-controlled rectifiers: (a) single-phase bridge, (b) three-phase bridge. TA n TB N L TC TA' TB' L--..~ 1 TC' N af FIGURE 1.11 Firing pulses in the phase-controlled six-pulse rectifier. r 2;r 1.2 AC TO DC CONVERTERS 11 Vo (a) Vo \ I to)t 2~" l ~ (b) F I G U R E 1.1 2 Output voltage waveforms in phase-controlled rectifiers: (a) single-phase bridge, (b) three-phase bridge (firing angle of 45~ 1.00 i LL I.,1_ LLI O o I 03 SD 0.75 v-.z LLI I I.,~'~"~1 _'Z. 0.50 ---7 . . . . 0.25 I. . . . . . 1 30 ~ It I. . . . . r I I I I. . . . . .......... I , ~ - , - ,~o . "~ _~_ _" _ _ _ _ o ~ d E) Z l-Z O r 03 s . . . . . 0.00 LL w o ,< O o I I I I -0.25 L -0.50 .... 7- . . . . 03 . . . . I I I 7 --'~0-o -0.75 ...... I I 15 30 I I I. . . . . I I ~t. . . . . o E3 Z I-Z I I -4- o (O -1.00 0 45 60 LOAD ANGLE (deg) FIGURE 1.13 Diagram of conduction modes of a phase-controlled six-pulse rectifier. 75 90 12 CHAPTER 1 /POWER ELECTRONIC CONVERTERS vo FIGURE 1.14 Operating quadrants of a controlled rectifier. placed between the rectifiers and the load, strongly reducing the ac component of the circulating current. The circulating current is controlled in a closed-loop control system which adjusts the angle fl in Eq. (1.8). Typically, the circulating current is kept at the level of some 10% to 15% of the rated current to ensure continuous conduction of both constituent rectifiers. The converter is thus seen to employ a different scheme of operation from the circulating current-free converter. Even when the load consumes little power, a substantial amount of power enters one rectifier and the difference between this power and the load power is transferred back to the supply line by the second rectifier. Reactors L 3 and L 4 can be eliminated if the constituent rectifiers are supplied from isolated sources, such as two secondary windings of a transformer. (o) (u) FIGURE 1.15 Rectifier arrangements for operation in all four quadrants: (a) rectifier with a cross-switch, (b) two rectifiers connected in antiparallel. 1.2 AC TO DC CONVERTERS 13 A B C TATA'_ TB;2 TAI~ TA2~ lo TC TB2" " 1;-'2.~ TB2 TC2"~_L R T C I ~ Tc2~L FIGURE 1.16 Circulating current-free dual converter. Both the uncontrolled and phase-controlled rectifiers draw square-wave currents from the supply line. In addition, the input power factor is poor, especially in controlled rectifiers, where it is proportional to cos(~f). These flaws led to the development of PWM rectifiers, in which waveforms of the supply currents can be made sinusoidal (with certain tipple) and in phase with the supply voltages. Also, even with very low values of the magnitude control ratio, continuous output currents are maintained. Fully controlled semiconductor switches, typically IGBTs, are used in these rectifiers. A voltage-source PWM rectifier based on IGBTs is shown in Fig. 1.18. The diodes connected in series with the IGBTs protect the transistors from reverse breakdown. Although the input current, ia, to the rectifier is pulsed, most of its ac component come from the input capacitors, while the current, iA, drawn from the power line is sinusoidal, with only some tipple. Appropriate control of rectifier switches allows obtaining a unity input power factor. Example waveforms of the output voltage, vo, output current, io, and input currents, ia and iA, are shown in Figs. 1.19 and 1.20, respectively. A. B~ C i TBI/ L1. T L2 o I T 9 TA2 .~TB2 7.y_Tc2 ,[ L3 FIGURE 1.1 7 Circulating current-conducting dual converter. 1-,4 14 CHAPTER 1 /POWER ELECTRONIC CONVERTERS A B C iA i0 i sc" SB" R Vo o FIGURE 1.18 Voltage-source PWM rectifier. The voltage-source P WM rectifier is a buck-type converter, that is, its maximum available output voltage (dependent on the PWM technique employed) is less than the peak input voltage. In contrast, the current-source PWM rectifier shown in Fig. 1.21 is a boosttype ac to dc converter, whose output voltage is higher than the peak input voltage. Figure 1.22 depicts example waveforms of the output voltage and current and the input current of the rectifier. II II II II Vo v . v v -IT r -IT -o3t 2~ FIGURE 1.19 Output voltage and current waveforms in a voltage-source PWM rectifier. 1.3 AC TO AC CONVERTERS 15 Ia 27r FIGURE 1.20 Input current waveforms in a voltage-source PWM rectifier. The amount of ripple in the input and output currents of the PWM rectifiers described depends on the switching frequency and size of the inductive and capacitive components involved. In practice, PWM rectifiers are typically of low and medium power ratings. 1.3 AC TO AC CONVERTERS There are three basic types of ac to dc converters. The simplest ones, the ac voltage controllers, allow controlling the output voltage only, while the output frequency is the same as the input frequency. In cycloconverters, the output frequency can be controlled, but it is at least one order of magnitude lower than the input frequency. In both the ac voltage controllers and cycloconverters, the maximum available output voltage approaches the input voltage. Matrix A B C iA -~ sB E SA'-- FIGURE 1.21 Current-source PWM rectifier. _ 1o 16 CHAPTER 1 /POWER ELECTRONIC CONVERTERS vo _ _ ,,, . _ . - . _ - _ _ _ . _ ./ 9 ~f.ot 2~" FIGURE 1.22 Waveforms of the output voltage and current and the input current in a current-source PWM rectifier. converters are most versatile, with no inherent limits on the output frequency, but the maximum available output voltage is about 15% lower than the input voltage. A pair of semiconductor power switches connected in antiparallel constitutes the basic building block of ac voltage controllers. Phase-controlled converters employ pairs of SCRs, SCR-diode pairs, or triacs. A single-phase ac voltage controller is shown in Fig. 1.23 and example waveforms of the output voltage and current in Fig. 1.24. The rms output voltage, Vo, is given by go -- gi 0~e - 0~f - ~ [sin(2ae) - sin(2af)] (1.9) where ~e denotes the so-called extinction angle, dependent on the firing angle, 0~f, and load angle, ~o. The precise value of ~o is usually unknown and changing. Consequently, as shown in Fig. 1.25, only an envelope of control characteristics, M---f(0~f), where M = V o / V i, can accurately be determined. IA T1 lo T2 FIGURE 1.23 Phase-controlled single-phase ac voltage controller. 1.3 AC TO AC CONVERTERS 17 vo // . V ABI I /A I ., FIGURE 1.24 Output voltage and current waveforms in a phase-controlled single-phase ac voltage controller (firing angle of 60~ The output voltage equals the input voltage, and the current is continuous and sinusoidal, when (zf "-- (/9. This can easily be done by applying a packet of narrowly spaced firing pulses to a given switch at the instant of zero-crossing of the input voltage waveform. The first pulse which manages to fire the switch appears at cot ~ r and the ac voltage controller becomes a static ac switch, which can be turned off by cancelling the firing pulses. Several topologies of phase-controlled three-phase ac voltage controllers are feasible, of which the most common, fully controlled controller, usually based on triacs, is shown in Fig. 1.26. 1.0 ! kk 'l O .,,,. ---L --4- 0.8 "~0 I~ I I I I ..... I.__~11- . . . . . -L . . . . . _J on," 0.8 t.-- m Z o t.u E) I-- 0.4 -. . . . Z (.9 ,< 0.2 I. 4- . . . . . . . . I I I 0.0 ,I 0 30 ..... 7- ! I I I I I -t- - -I . I I I I I I I ..... I , , , , , 60 90 i I_N,~ . I~. i 1 _ -. t - ~ ~ .I I .......... . . . . 120 . , , 150 180 FIRING ANGLE (deg) FIGURE 1.25 Envelope of control characteristics of a phase-controlled single phase ac voltage controller. 18 CHAPTER1 /POWER ELECTRONIC CONVERTERS A B C T~TB TC FIGURE 1.26 Phase-controlled, fully controlled three-phase ac voltage controller. If switching functions, a, b, and c, are assigned to each triac, output voltages,/)a, Vb, and vc, of the controller are given by [Va][a eli. 1 Vb vc -- --a b -a -b -c vB c (1.10) vc where UA, VB, and vC are line-to-ground voltages of the supply line. Analysis of operation of the fully controlled controller is rather difficult since, depending on the load and firing angle, the controller operates in one of three modes: (1) Mode 1, with two or three triacs conducting; (2) Mode 2, with two triacs conducting; and (3) Mode 3, with none or two triacs conducting. The output voltage waveforms are complicated, as illustrated in Fig. 1.27 for voltage va of a controller with resistive load and a firing angle of 30 ~ The waveform consists of segments of the VA, VAB/2, and VAC/2 voltages. The envelope of control characteristics of the controller is shown in Fig. 1.28. Four other topologies of the phase-controlled three-phase ac voltage controller are shown in Fig. 1.29. If ratings of available triacs are too low, actual SCRs must be used. In that case, an SCR-diode pair is employed in each phase of the controller. Such a half-controlled controller is shown in Fig. 1.29a. If the load is connected in delta, the three-phase ac voltage controller can have the topology shown in Fig. 1.29b. The triacs (or SCR-diode pairs) can also be connected after the load, as in Figs. 1.29c and 1.29d. Similarly to phase-controlled rectifiers, phase-controlled ac voltage controllers draw distorted currents from the supply line, and their input power factor is poor. Again, as in the rectifiers, these characteristics can significantly be improved by employing pulse width modulation. Pulse width modulated ac voltage controllers, commonly called ac choppers, require fully controlled power switches capable of conducting current in both directions. Such switches can be assembled from transistors and diodes; two such arrangements are shown in Fig. 1.30. The P W M ac voltage controller, also known as ac chopper, is shown in Fig. 1.31 in the single-phase version. For simplicity, and to stress the functional analogy to the genetic converter, the bidirectional switches are depicted as mechanical contacts. When the main switch, S1, is 1.3 I I I I ~'i".~l // :/ I_/ I ,4 I Ill I I / \1 "J i I k Ix I I I \ I \1 \ ,' i ! ! ",,! \ ! \ --~--~-4---t--'I., " - \ \ Ilk ,/I ,/ / I ~,, I 'k, i\ iX iN, I \i\\1 i Ih X i \ix !i 'I I 1 30 Ct,f 60 90 19 // 4-----I---- 4 - - - - - ~ - - ~' \,i",1 \/ AC T O A C CONVERTERS I I i~-- I, J t/ , ix/ IX t\I \ i/@ I ./I 4 '\i,. / 1_,,~/ /1"'/ !i'l / I[) I ,~ ,, I 120 150 180 210 240 270 300 330 360 (deg) cot FIGURE 1.27 Waveform of the output voltage in a fully controlled three-phase ac voltage controller (resistive load, firing angle of 30~ chopping, that is, turning on and off many times per cycle, the current drawn from the LC input filter is interrupted. Therefore, another switch, $2, is connected across the load. It plays the role o f the freewheeling switch SO in the generic power converter in Fig. 1.1. Switches S 1 and $2 are operated complementarily: when S 1 is turned on, $2 is turned off and vice versa. Denoting the duty ratio o f switch S1 by D 1, the magnitude control ratio, M, taken as ratio o f the rms output voltage, F o, to rms input voltage, Vi, equals ~/-D1. 1.0 " 'J' ' 1 9 o !-< n,' 0.8 I c..,~. I ~ "--:, \l ) "0 \ . . . . . ~ __.-_ _x'~, . . . . I t I I \ %, I I I\\\ t \..~. i \-~ I_ J i - - - I I I L I .X. I I I I t I ~, \ --I 0 n,' 0.6 l.,-.z O o I..iJ o 0.4 E~ I-z < ___i_ .... .... i- . . . . ..... tI t i i _l . . . . . i i ~---r -.---,.----4 i i i 0.2 I 0.0 ..... 0 \_i___k i\ ..... i i i i i i i i ...... --r i 1 i %%----~ ink i I \\ i I \\ I i I I ..... ,,,, 90 -- i - i. . . . . I I I .... 60 i ~ ~i t--~--~r i \ I I \ t i \k i k!\ t,:,,,t,,,..,,l, 30 I -T. . . . . . . \ t - - - ~ - r - 120 150 180 FIRING ANGLE (deg) FIGURE 1.28 Envelope of control characteristics of the fully controlled three-phase ac voltage controller. 20 CHAPTER 1 /POWER ELECTRONIC CONVERTERS I i] I (a) (b) (c) (d) i FIGURE 1.29 Various topologies of phase-controlled three-phase ac voltage controllers: (a) half-controlled, before-load, (b) delta-connected, before-load, (c) wye-connected after-load, (d) delta-connected, after-load. Example waveforms of the output voltage, vo, and current, io, of the ac chopper are shown in Fig. 1.32. The high-frequency component of the pulsed input current, ia, is mostly supplied by the filter capacitors, so that the current, iA, drawn from the power line is similar to that of the PWM voltage-source rectifier (see Fig. 1.20). Analogously to the single-phase ac chopper in Fig. 1.31, three-phase ac choppers can be obtained from their phase-controlled counterparts by replacing each triac with a fully controlled bidirectional switch. A similar switch must be connected in parallel to each phase load to provide an alternative path for the load current when the load is cut off from the supply source by the main switch. The dual converter in Fig. 1.17 can be operated as a single-phase cycloconverter by varying the firing angle O~f,1 in accordance with the formula ~f, 1(t) -- c o s - 1 [ M sin(cOot)] (1.11) where the magnitude control ratio, M, represents the ratio of the peak value of the fundamental output voltage to the maximum available dc voltage of the constituent rectifiers. The output frequency, oao, must be significantly lower than the supply frequency, 09. Example waveforms of the output voltage of such cycloconverter are shown in Fig. 1.33 for 090/09 = 0.2 and two values of M: 1 and 0.5. Two three-phase six-pulse cycloconverters are shown in Fig. 1.34. The cycloconverter with isolated phase loads in Fig. 1.34a is supplied from a single three-phase source. If the loads are interconnected, as in Fig. 1.34b, individual phases of the cycloconverter must be fed from separate sources, such as isolated secondary windings of the supply transformer. Practical 1.3 AC TO AC CONVERTERS 21 MAIN TERMINAL 1 MAIN TERMINAL 1 CONTROL TERMINALS CONTROL TERMINALS MAIN TERMINAL 2 MAIN TERMINAL 2 (o) (b) FIGURE 1.30 Fully controlled bidirectional power switch assemblies: (a) two transistors and two diodes, (b) one transistor and four diodes. iA II II $1 io i $2 Vo O FIGURE 1.31 Single-phase ac chopper. 22 CHAPTER 1 / P O W E R ELECTRONIC CONVERTERS 2.A ,o 47t" cot FIGURE 1.32 Output voltage and current waveforms in an ac chopper. cycloconverters are invariably high-power converters, typically used in adjustable-speed synchronous motor drives requiring sustained low-speed operation. The matrix converter, shown in Fig. 1.35 in the three-phase to three-phase version, constitutes a network of bidirectional power switches, such as those in Fig. 1.30, connected between each of the input terminals and each of the output terminals. In this respect, the matrix converter Vo I ~ I Ii Ii 11 II II I I 91 II I I II IiIj ~! ~, !! ',! 'iil ',! ki ~I ~I L~" ~"~ g ~IVI~' ~' il U ~; i/!i il ~'MIt , :/~.', I',I',,,/I/~F'\.l~/k/k! ,,,dL~l~li! ~,~v,,/:/,,,/:/',/,/~/~/',/, '*', -),,~ )'~Vx~V~ (a) ~11! I! rilYl~l~l~lYl~ Ill Ii 1 Y ilt tI ?I t'tI Jl't tlt tt~' t't~ tlI tlI' t~~' t~I t~f I , ~ I II I II II I I II II I COot ,,,k~,IliIIl~,I,~," V ~ V ~] ~.I ,I ~I I ~111 I ~11~ ~ 1 1 ~ Y It'd Y x ',~ ~l ~ V I V "~ (b) FIGURE 1.33 Waveforms of output voltage in a six-pulse cycloconverter: (a) M = 1, (b) M = 0.5 (o90/09 = 0.2). 1.3 . . . . AC TO AC CONVERTERS . 23 -b- 7", f (o) - _ ~ (b) FIGURE 1 . 3 4 Three-phase six-pulse cycloconverters: (a) with isolated phase loads, (b) with interconnected phase loads. constitutes an extension of the generic power converter in Fig. 1.1. The voltage of any input terminal can be made to appear at any output terminal (or terminals), while the current in any phase of the load can be drawn from any phase (or phases) of the supply line. An input LC filter is employed to screen the supply system from harmonic currents generated by the converter, which operates in the PWM mode. The load inductance assures continuity of the output currents. Although, with the 9 switches, the matrix converter can theoretically have 512 states, only 27 states are permitted. Specifically, at any time, one and only one switch in each row must be closed. Otherwise, the input terminals would be shorted or the output currents would be interrupted. The voltages, va, vb, and vc, at the output terminals are given by l)b Pc -- / XAb LXAc XBb XBc XCb XCc (1.12) I)B /)C where XAa through Xcc denote switching functions of switches SAa through Scc, and 1)A, I)B, and vc are the voltages at the input terminals. In turn, the line-to-neutral output voltages, Van, Vbn, and Vcn, can be expressed in terms of va, vb, and vc as [vo] 11211][Va] Vbn Vcn -- -~ - 1 -- 1 2 - 1 /)b -- 1 2 vc 9 (1.13) 24 CHAPTER 1 / POWER ELECTRONIC CONVERTERS AB - - F C ( , vc ( SAo' SAb SAc' l v.n FIGURE 1.35 Three-phase to three-phase matrix converter. [,A] [XAaxAbxAc]Ei] The input currents, iA, iB, and io are related to the output currents, ia, ib, and ic, as iB ic = XBa XBb XBc XCa XCb XCc ib . ic (1.14) Fundamentals of both the output voltages and input currents can successfully be controlled by employing a specific, appropriately timed sequence of the switching functions. As a result of such control, the fundamental output voltages acquire the desired frequency and amplitude, while the low-distortion input currents have the required phase shitt (usually zero) with respect to the corresponding input voltages. Example waveforms of the output voltage and current are shown in Fig. 1.36. For reference, waveforms of the line-to-line input voltages are shown, too. The output frequency, o~o, in Fig. 1.36a is 2.8 times higher than the input frequency, co, while the ~Oo/~Oratio in Fig. 1.36b is 0.7. Respective magnitude control ratios, M, are 0.8 and 0.4. Apart from the conceptual simplicity and elegance, matrix converters have not yet found widespread application in practice. Two major reasons are the low voltage gain, limited to ~ / 2 ~-, 0.866, and unavailability of fully controlled bidirectional semiconductor switches. 1.4 DC TO DC CONVERTERS Dc to dc converters, called choppers, are supplied from a dc voltage source, typically a diode rectifier and a dc link, as shown in Fig. 1.37. The dc link consists of a large capacitor connected 1.4 DC TO DC CONVERTERS 25 Vo \\ /l'x\ ,, /i~<'\ i/",,i~ I~llll~k~ \ //N/N\ i" \ "..4 / '..~!/",'~~lltlltt:ll]ll..-",, . -,, _-- \<iJ. . v ot J411111ll]111. -,o -" (a) / vo '! I111111llri f't IlU I1II Illllll f , ;t :,/,, !1 ,\,, JtA Illt~ 'v v v RttlI~,,' V V V ',~ t ~l'i A ,", it t ~ I\ it I\ 11~ ~ t , -~ II.i \I \I \i ~lgliltillli i.! \, \! \, \, i~llllllllli II>; ,~, X X ~! illIRIIFUIt X ,", ,x, X PllI~IlPtltl \i ~ \k/ V x \Itjx',.,' H II,BllJ I,,t v^ V 'd, \ tV \I L! II Ii t, '~. , , . I , d ,,. ~.J,'IJJ"~.~'~,F4,1^\/\I j \..,,r o , (b) FIGURE 1 . 3 6 Output voltage and current o90/09 = 0.7, M = 0.4. waveforms in a matrix converter: (a) o9o/o9=2.8, M=0.8; (b) across the input terminals of the chopper and, often but not necessarily, a series inductance. The capacitor smooths the dc voltage produced by the rectifier and serves as a source of the highfrequency tipple current drawn by the chopper. The inductor provides an extra screen for the supply power system against the high-frequency currents. All choppers are pulse width modulated, the phase control being infeasible with both the input and output voltages of the dc type. Most choppers are of the step-down (buck) type, that is, the average output voltage, Vo, is always lower than the input voltage, Yi- The first-quadrant chopper, based on a single fully RE C TIFIE R DC LINK T FIGURE 1.37 Dc voltage source for choppers. Et. 9 C) 0 26 CHAPTER 1 / P O W E R ELECTRONIC CONVERTERS R ( )v, 2 FIGURE 1 . 3 8 First-quadrant chopper. controlled switch and a freewheeling diode, is shown in Fig. 1.38. Both the output voltage, vo, and current, io, can only be positive. The average output voltage is given by (1.15) Vo =DV i where D denotes the duty ratio of the switch. The magnitude control ratio, M, is defined here as V o / V i and it equals D. Example waveforms of vo and io are shown in Fig. 1.39, with M changing from 0.5 to 0.75. As in all PWM converters, the output voltage is pulsed, but the output current is continuous thanks to the load inductance. The current ripple is inverse proportional to the switching frequency, fsw. Specifically, the rms value, Io,ar of the ac component of the output current is given by I~162= IMI(1 - IMI) Vi 2 ~/'3tfsw (1.16) where L denotes inductance of the load. The reason for the absolute value, IMI, of the magnitude control ratio appearing in Eq. (1.16) is that this ratio in choppers can assume both the positive and negative values 9In particular, M > 0 indicates operation in the first and third quadrant (see Fig. 1.14), while M < 0 is specific for choppers operating in the second and fourth quadrant. The most versatile dc to dc converter, the four-quadrant chopper shown in Fig. 1.40, can, as its name indicates, operate in all four quadrants. In the first quadrant, switch $4 is turned on all the time, to provide a path for the output current, io, while switch S 1 is chopping with the duty ratio D1. The remaining two switches, $2 and $3, are OFE In the second quadrant, it is switch $2 that is chopping, with the duty ratio D2, and all the other switches are OFE Analogously, in the third quadrant, switch S 1 is ON, switch $3 is chopping with the duty ratio D 3 and, in the fourth quadrant, switch $4 is chopping with the M = 0.50 M = 0.75 Vo Io t FIGURE 1.39 Example waveform of output voltage and current in a first-quadrant chopper. 1.4 .,o - DC TO DC CONVERTERS 27 - $1t $3 D4 (: , [ D3 FIGURE 1.40 Four-quadrant chopper. duty ratio D 4. When a chopping switch is OFF, conduction of the output current is taken over by a respective freewheeling diode, for instance, D1 in the first quadrant of operation. The magnitude control ratio, M, is given by M -- D1 1 - D2 -D 3 D4 - 1 in in in in Quadrant Quadrant Quadrant Quadrant 1 2 3 4. (1.17) If the chopper operates in Quadrants 2 and 4, the power flows from the load to the source, necessitating presence of an EMF, E, in the load. The EMF must be positive in Quadrants 1 and 2, and negative in Quadrants 3 and 4. For sustained operation of the chopper with a continuous output current, the magnitude control ratio must be limited in dependence on the ratio E / V i as illustrated in Fig. 1.41. These limitations, as well as Eq. (1.17), apply to all choppers. Any less-than-four-quadrant chopper can easily be obtained from the four-quadrant topology. Consider, for instance, a two-quadrant chopper, capable of producing an output voltage of both polarities, but with only a positive output current. Clearly, this converter can operate in the first and fourth quadrants. Its circuit diagram, shown in Fig. 1.42, is determined by eliminating switches $2 and $3 and their companion diodes, D2 and D3, from the four-quadrant chopper circuit in Fig. 1.40. i QUADRAN/ QUADRANT2 I -1 .... I 1 '-Vi QUADR " -- 1 FIGURE 1.41 Allowable ranges of the magnitude control ratio in a four-quadrant chopper. 28 CHAPTER 1 /POWER ELECTRONIC CONVERTERS O ~ O FIGURE 1.42 First-and-fourth-quadrant chopper. A step-up (boost) chopper, shown in Fig. 1.43, produces a pulsed output voltage, whose amplitude, /Io,p, is higher than the input voltage. If a sufficiently large capacitor is connected across the output terminals, the output voltage becomes continuous, with Vo ~/Io, p > Vi. When switch S is turned on, the input inductor, Lc, is charged with electromagnetic energy, which is then released into the load by turning the switch off. The magnitude control ratio, M, defined as Vo,p/Vi, in an ideal (lossless) step-up chopper is given by M - 1 (1.18) 1-D where D denotes the duty ratio of the switch. In real choppers, the value of M saturates at a certain level, usually not exceeding 10 and dependent mostly on the resistance of the input inductor. Example waveforms of the output voltage and current in a step-up chopper without the output capacitor are shown in Fig. 1.44. 1.5 DC TO AC CONVERTERS Dc to ac converters are called inverters and, depending on the type of the supply source and the related topology of the power circuit, they are classified as voltage-source inverters (VSIs) and current-source inverters (CSIs). The simplest, single-phase, half-bridge, VSI is shown in Fig. 1.45. The switches may not be ON simultaneously, because they would short the supply source. There is no danger in turning both switches off, but the output voltage, vo, would then depend on the conducting diode, that is, it could not be determined without some current sensing arrangement. Therefore, only two states of the inverter are allowed. Consequently, a single switching function, a, can be assigned to the inverter. Defining it as a= o t~ v~ ~ 0 1 ifSA=ON ifSA=OFF and S A ' = O F F and S A ' = O N , (1.19) ~o R Io Vl t FIGURE 1.44 FIGURE 1.43 Step-up chopper. Output voltage and current waveforms in a stepup chopper (D = 0.75). 1.5 DC TO AC CONVERTERS 29 v~ 0 FIGURE 1.45 Single-phase, half-bridge, voltage-source inverter. the output voltage of the inverter is given by Vo= Vi(a __I) (1.20) where Vi denotes the dc input voltage. Only two values of vo are possible: Vi/2 and -Vi/2. To prevent the so-called shot-through, that is, a short circuit when one switch is turned on and the other has not yet turned off completely, the turn-on is delayed by a few microseconds, called a dead, or blanking, time. The same precaution is taken in all VSIs, with respect to switches in the same leg of the power circuit. The more common, single-phase full-bridge VSI, shown in Fig. 1.46, has two active legs, so that two switching functions, a and b, must be used to describe its operation. Notice that the topology of the inverter is identical to that of the four-quadrant chopper in Fig. 1.40. The output voltage can be expressed in terms of a and b as vo -- Vi(a - b) (1.21) which implies that it can assume three values: Vi, 0, and -Vi. Thus, the maximum voltage gain of this inverter is twice as high as that of the half-bridge inverter. Two modes of operation can be distinguished: the square-wave mode, loosely related to the phase-control mode in rectifiers, and the PWM mode. In the square-wave mode, so named because of the resultant shape of the output voltage waveform, each switch of the inverter is turned on and off only once per cycle of the output voltage. A specific sequence of inverter states is imposed, the state being designated by the decimal equivalent of ab2. For example, if a = 1 and b = 1, the full-bridge inverter is said to be in State 3 because 112 --- 310. The output voltage waveform for the full-bridge inverter in the so-called optimal square-wave mode, which results in the minimum total harmonic distortion of this voltage, is shown in Fig. 1.47. The output current, io, depends on the load, but generally, because of the high content of loworder harmonics (3rd, 5th, 7th, etc.) in the output voltage, it strays substantially from a sinewave. ,C SA DA' SB DB" V, I " S/~ - 0 , FIGURE 1.46 Single-phase, full-bridge, voltage-source inverter. DB 30 CHAPTER 1 / POWER ELECTRONIC CONVERTERS STATE: 0 2 3 1 0 2 3 1 0 Vo l. ~ ' I 4,~.0t FIGURE 1.47 Output voltage waveform in a single-phase, full-bridge, voltage-source inverter in the optimal square-wave mode. Not so in an inverter operating in the PWM mode, which results in a sinusoidal current with high-frequency ripple. Example waveforms of vo and io in a PWM inverter are shown in Fig. 1.48. Three-phase counterparts of the single-phase half-bridge and full-bridge VSIs in Figs. 1.45 and 1.46 are shown in Figs. 1.49a and 1.49b, respectively. The three-phase full-bridge inverter is one of the most common power electronic converters nowadays, predominantly used in ac adjustable speed drives and three-phase ac uninterruptable power supplies (UPSs). The capacitive voltage-divider leg of the incomplete-bridge inverter also serves as a dc link. Two switching functions, a and b, can be assigned to the inverter, because of the two active legs of its power circuit. The line-to-line output voltages are given by IvABII'' ~176 1 VBC -- Vi /)CA 0 1 --1 0 - 89 b (1.22) 1 % f .... 2~ot FIGURE 1.48 Output voltage and current waveforms in a single-phase, full-bridge, voltage-source inverter in the PWM mode. 1.5 iI "~ -- v - DC TO AC CONVERTERS 31 DA" - ~c B tc C IT ~N ~N (o) DC A UAB |B~_B l&c ic C ,,. (b) FIGURF 1.49 Three-phase voltage-source inverters: (a) incomplete-bridge, (b) full-bridge. Ivl I21 llal and the line-to-neutral voltages by VBN VCN --~ --1 2 1 b -- 1 -- 1 1 1 . (1.23) Voltage VAa can assume three values, -Vi, 0, and Vi, but /)BC and /)CA can assume only two values, - V i/2 and Vi/2. The line-to-neutral voltages VAN and VBN can assume four values, - V i / 2 , - V i/6, Vi/6, and Vi/2, and VCN can assume three values, - V i/3, 0, and Vi/3. This voltage asymmetry makes the square-wave operation impractical, and the incomplete-bridge inverter can only operate in the PWM mode. The maximum voltage gain, taken as the ratio of the 32 CHAPTER 1 / POWER ELECTRONIC CONVERTERS maximum available peak value of the fundamental line-to-line output voltage to the input voltage, is only 0.5. Therefore, in spite of the cost savings resulting from the reduced device count, the incomplete-bridge inverter is rarely used in practice. Because of the three active legs, three switching functions, a, b, and c, are associated with the full-bridge three-phase inverter. The line-to-line and line-to-neutral output voltages are given by /)BC - " Vi /)CA and [11 01[a1 0 --1 1 0 - 1 1 b c [vo] i[2 1 ilia ] vBN - - ~ --1 -- 1 VCN 2 -- 1 --1 2 b . C (1.24) (1.25) Each line-to-line voltage can assume three values, - V i, 0, and Vi, and each line-to-neutral voltage five values, - 2 V i / 3 , - V i/3, 0, Vi/3, and 2Vi/3. In the PWM mode, the maximum voltage gain is 1, that is, the maximum attainable peak value of the fundamental line-to-line voltage equals the de supply voltage. As in single-phase inverters, the state of the full-bridge inverter can be defined as the decimal equivalent of abc 2. T h e 5-4-6-2-3-1 ... state sequence, each state lasting one-sixth of the desired period of the output voltage, results in the square-wave mode of operation, illustrated in Fig. 1.50. In this mode, the fundamental line-to-line output voltage has the highest possible peak value, equal to 1.1 Vi, yielding a maximum voltage gain 10% higher than that in the PWM mode. At the same time, in the inverter, there is no possibility of magnitude control of the output voltage, and the voltage waveforms are rich in low-order harmonics, spoiling the quality of output currents. Therefore, in practical energy conversion systems involving inverters, the square-wave mode is used sparingly, only when a high value of the output voltage is necessary. Example waveforms of switching functions and output voltages of the three-phase full-bridge inverter in the PWM mode are shown in Fig. 1.51. The cycle of output voltage is divided here into 12 equal switching intervals, pulses of switching functions located at centers of the intervals. Thus, the switching frequency, fsw, is 12 times higher than the output frequency, f. In practical inverters, the switching frequency is usually maintained constant and independent of the output frequency, at a level representing the best trade-off between the switching losses and quality of the output currents. The so-called voltage space vectors, an idea originally conceived for analysis of three-phase electrical machines, are a useful tool for analysis and control of three-phase power converters as well. Denoting individual phase voltages of an inverter by va, vb, and ve (they can be line-to-line, line-to-ground, or line-to-neutral voltages), the voltage space vector, v, is defined as (1.26) v : v d W jVq where /)d /)q __ 1 --~ 0 ~ -2 /)b t) c " (1.27) Reduction of three phase voltages,/)a,/)b, and ve, to two components, vd and/3q, of the voltage vector is only valid when va + vb + ve = 0. Then, only two of these voltages are independent 1.5 STATE: 2 5 i i I I I I I i I I oat ! I I I I I 1 I I I i I '1, I I I I I L I I I J ! i I I I I , I I ' 1 i I I I I I I I 'i ' ' | I I I I VCN 1 gn" 2 gn" oat oat cot I VBN oat II I v~ 1 I I iI v~ 3 33 i I I VBC DC TO AC CONVERTERS cot , I I i, I i t I I I I I I .~' I I I oat L.._.._~ ~ t 4 x i 5 gn" I I -~X ' ~oat I t 2X FIGURE 1 . 5 0 Waveforms of switching functions and output voltages in a three-phase, full-bridge inverter in the squarewave operation mode. variables, so that the amount of information carried by Va, Vb, and ve is the same as that carded by v a and Vq. If Vb vc - - Vp COS(COt+~p ~2 7n 0) c o s ( c o t + (p - 7 , (1.28) then [v,] 3 Vq - - 2 Vp sin(cot + qg) ' (1.29) that is, u 3 VpeJ(Cot+q,) (1.30) Thus, as the time, t, progresses, the voltage space vector, v, revolves with the angular velocity co in a plane defined by a set of orthogonal coordinates d and q. 34 CHAPTER 1 /POWER ELECTRONIC 1~ i CONVERTERS i i i i i i , I I i I i i I I , I , I I 11711-11rllnlllln , , , , , Ir-ill-i' , , II I i I I I I I i n n',n; ',o', I i I ci" i i n n n i I i i i I ' I i I i llqln ,b I I i I i I' Ilmt I I I v.. flfi i-ll-IlI-IOIN ' ' . . Fill. ~v,l . . ' dUU' I I I u,,u u!u, ,,u-u u u,,,~I! I I I vB c i i I , , ~U',lGn',l-il-ili-! NIFI IG',R HI I v~, IHHI I [ f I I ' ' , , iu u[u uiuu iuuiuu,7 V~, l In ' nlr'l r-iIn r'ltn n l _r'H-1 1 1 ~ I--lll ~ r H! "1 r I i l ' - l l - l l Iu i I VB N I [ ui i I [ i . Ll~ll4.u -i--Ii i I I I v~. ] l _ t r 4 L n n / . _ . lu 1 I 0 FIGURE I UlU I I I I ! n. I I ! ul[.,i I~ t i t I I r'tn ul. I i i I r'lrll iu I I I I n fl 1 i I I I i l l I I IH H,I.IU,~, 9 1 I I I I I I I I r ~ I I !n. i I ,. I I I i 1. I I ~iu ~ ~ . ~ l l j i h l l r 4 I i l I o~t mt nI i i I I I I Ili-tr-lil~~lhrll!nn ul cot I I I I t u I,_Jzu ~1 I I I t ~ - t n ~ I...qu I I ;rt" I I ulu ul I I i I I I I 1 I1 I1 I u ui I I I O3 . I i t cot -I I I I I cot 2:rt" 1.51 Waveforms of switching functions and output voltages in a three-phase, full-bridge inverter in the PWM mode. In application to VSIs, the revolving voltage vector describes the fundamental output voltages. Each state of the inverter produces a specific stationary voltage space vector, and the revolving vector, v, which is to follow a reference vector, v*, must be synthesized from the stationary vectors in a time-averaging process. The maximum possible value of v determines the maximum voltage gain of the inverter. The four stationary voltage vectors, V 0 through V 3, of the three-phase incomplete-bridge inverter, corresponding to its four allowable states, are shown in Fig. 1.52 in the per-unit format. The input voltage, Fi, is taken as the base voltage. In the process of pulse width modulation, the vector, v, of fundamental output voltage is synthesized as 3 v -- ~ DsV s (1.31) S=O where D s denotes the duty ratio of State S (S = 0 . . . 3). Each switching interval, a small fraction of the period of output voltage, is divided into several nonequal subintervals, constituting durations of individual states of the inverter. Not all four states must be used; three are enough. The vectors employed depend on the angular position of the synthesized vector v. Since the sum of all the duty ratios involved equals 1, the maximum available voltage vector to be generated is limited. It can be shown that the circle shown in Fig. 1.52 represents the locus of that vector. In other words, the maximum magnitude of v is ~ / 4 Vi, which corresponds to the peak value of the line-to-line fundamental output voltage being equal to one-half of the dc supply voltage of the inverter. 1.5 DC TO AC CONVERTERS 35 Jq v. ",,'7 1 ! 1 d , -~ -7 \ . \\\ v0 v2 FIGURE 1.52 Space vectors of line-to-neutral output voltage in a three-phase incomplete-bridge voltage-source inverter. Space vectors of line-to-line output voltage of the full-bridge inverter are shown in Fig. 1.53. There are six nonzero vectors, V1 through V6, whose magnitude equals the dc input voltage, Fi, and two zero vectors, V 0 and V 7. In general, 7 (1.32) v - - ~ DsV s, S=0 but in practice, only a zero vector and two nonzero vectors framing the output voltage vector are used. For instance, the vector v in Fig. 1.53 is synthesized from vectors V 2 and V 3 and a zero vector, V0 or V 7. The radius of circular locus of the maximum output voltage vector indicates a voltage gain twice as high as that of the incomplete-bridge inverter. Specifically, the maximum jq .... i ___ I/6 I E, ' FIGURE 1.53 Space vectors of line-to-neutral output voltage in a three-phase full-bridge voltage-source inverter. 36 CHAPTER1 / POWER ELECTRONIC CONVERTERS available peak value of the fundamental line-to-line output voltage equals Vi. Notice that going clockwise around the vector diagram yields the state sequence 4-6-2-3-1-5..., characteristic for square-wave operation with positive phase sequence, A-B-C. The counterclockwise state sequence 4-5-1-3-2-6..., would result in a negative phase sequence, A-C-B. The voltage-source inverters described can be termed "two-level," because each output terminal, temporarily connected to either of the two dc buses, can only assume two voltage levels. Recently, multilevel inverters have been receiving increased attention. Using the same power switches, they have higher voltage ratings than their two-level equivalents. Also, their output voltage waveforms are distinctly superior to these in two-level inverters, especially in the square-wave mode. The most common, three-level neutral-clamped inverter is shown in Fig. 1.54. Each leg of the inverter is comprised of four semiconductor power switches, S 1 through $4, with freewheeling diodes, D 1 through D4, and two clamping diodes, D 5 and D6, that prevent the dc-link capacitors from shorting. The total of 12 semiconductor power switches implies a high number of possible inverter states. In practice, 27 states are employed only, as each leg of the inverter is allowed to assume only the three following states: (1) S 1 and $2 are ON, $3 and $4 are OFF, (2) $2 and $3 are ON, S 1 and $4 are OFF, and (3) S 1 and $2 are OFF, $3 and $4 are ON. It can be seen that the dc input voltage, Vi, is always applied to a pair of series-connected switches, which explains the m m ~ b I (, _ T T vcT FIGURE 1.54 Three-level neutral-clamped inverter. 1.5 DC TO AC CONVERTERS 37 already-mentioned advantage of multilevel inverters with respect to the voltage rating. It can be up to twice as high as the rated voltage of the switches. Stationary space vectors of output voltage of the three-level inverter are shown in Fig. 1.55. The maximum voltage gain of the inverter is the same as that of the two-level full-bridge inverter, but the availability of 27 stationary voltage vectors allows for higher quality of the output voltage and current. For instance, the square-wave mode of operation results in five-level line-to-line output voltages and seven-level line-to-neutral voltages. Waveforms of output voltages in the three-level inverter operating in the square-wave mode are shown in Fig. 1.56. PWM techniques produce output currents with very low tipple even when medium switching frequencies are employed. All the inverters described so far are hard-switching converters, switches of which turn on and off under nonzero voltage and nonzero current conditions. This results in switching losses, which at high switching frequencies can be excessive. High rates of change of voltages (dv/dt) and currents (di/dt) cause a host of undesirable side effects, such as accelerated deterioration of stator insulation and rotor beatings, conducted and radiated electromagnetic interference (EMI), and overvoltages in cables connecting the inverter with the motor. Therefore, for more than a decade, significant research effort has been directed toward development of practical softswitching power inverters. In soft-switching inverters, switches are turned on and off under zero-voltage or zero-current conditions, taking advantage of the phenomenon of electric resonance. As is well known, transient currents and voltages having ac-type waveforms can easily be generated in lowresistance inductive-capacitive (LC) dc-supplied circuits. Thus, a resonant LC circuit (or circuits) constitutes an indispensable part of the inverter. Two classes of soft-switching inverters have emerged. In the first class, no switches are added to initiate the resonance. A representative of this class, the classic resonant dc link (RDCL) inverter with voltage pulse clipping, is shown in Fig. 1.57. The resonant circuit consists of an inductor, Lr, and capacitor, Cr, placed between the dc link (not shown) and the inverter. Low values of the inductance and capacitance make the resonance Jq / / ,2" \ ~ \ v0= u13= v2s= O \ I .o~/ "% ',m -~',\ -~-~/1\. kx \ ','t, J ~~".Z7 ~"0# .~,., - - ',",,. / i 02 _j~ Oil r,' d /,,1 f/ ',.4,'020 FIGURE 1.55 Space vectors of line-to-neutral output voltage in a three-level neutral-clamped inverter. 38 CHAPTER 1 / P O W E R ELECTRONIC CONVERTERS STATE:18 21 24 15 6 7 8 5 2 11 20 19 18 cot C cot I I !, I I cot I L I'-'I t. I I cot I I---1 N I I 9 I i 1 , cot , i r'- I---1 ~N , I, 9 , , i 2C mt 9 FIGURE 1 . 5 6 Output voltage waveforms in a three-level neutral-clamped inverter in the square-wave mode. FIGURE 1 . 5 7 Resonant dc link inverter. 1.5 DC TO AC CONVERTERS 39 frequency several orders of magnitude higher than the output frequency of the inverter. The resonance is initiated by turning on, for a short period of time, both switches in a leg of the inverter bridge. This loads L r with electromagnetic energy, which is then released into Cr when one of the switches in question is turned off. Because of the low resistance of the resonant circuit, the voltage across the capacitor acquires a sinusoidal waveform. Should the clamping circuit, based on switch S and capacitor Co, be inactive, the peak value of the output voltage would approach 2Vi. When the voltage drops to zero, the freewheeling diodes of the inverter become forward biased and they short the dc buses of the power circuit. This creates zero-voltage conditions for inverter switches, allowing lossless switching of these switches. Output voltage waveforms in the RDCL inverter consist of packets of narrowly spaced resonant pulses. The clamping circuit clips these pulses in order to increase voltage density. A packet of clipped voltage pulses is shown in Fig. 1.58. It represents the equivalent of a single rectangular pulse of output voltage in a hard-switching inverter. It can be seen that the voltage pulses have low dv/dt, which alleviates certain undesirable side effects of inverter switching. In the other class of soft-switching inverters, auxiliary switches trigger the resonance, so that the main switches are switched under zero-voltage conditions. One phase (phase A ) o f such a converter, the auxiliary resonant commutated pole (ARCP) inverter, is shown in Fig. 1.59. To minimize high dynamic stresses on main switches SA and SA', a resonant snubber, based on the inductor L A and capacitors CA1 and CA2, is employed. The resonance is initiated by turning on the bidirectional switch, composed of auxiliary switches SA1 and SA2 and their antiparaUel diodes. The auxiliary switches are turned on and off under zero-current conditions. ARCP inverters, typically designed for high-power applications, provide highly efficient power conversion. In contrast to the RDCL inverter, whose output voltage waveforms consist of packets of resonant pulses, the ARCP inverter is capable of true pulse width modulation, but with the voltage pulses characterized by low dv/dt. Typically, IGBTs or GTOs are used as main switches, while MCTs or IGBTs serve as auxiliary switches. The voltage source supplying a VSI provides a fixed dc input voltage. A battery pack or, more often, a rectifier (usually uncontrolled) with a dc link, such as the dc source for choppers shown in Fig. 1.37, is used. Consequently, polarity of the input current depends on the direction of power transfer between the source and the load of the inverter, which explains the necessity of the freewheeling diodes. Voltage sources are more "natural" than current sources. However, a fair representation of a current source can be obtained combining a current-controlled rectifier % r FIGURE 1.58 Packet of output voltage pulses in a resonant dc link inverter. 40 CHAPTER 1 /POWER ELECTRONIC CONVERTERS FIGURE 1.59 One phase of the auxiliary resonant commutated pole inverter. and a large series inductor, as illustrated in Fig. 1.60. Such a source can be used for supplying dc power to a current-source inverter. In that case, it is the input current, I i, that is maintained constant, and reversal of the power transfer is accompanied by a change in polarity of the input voltage. The three-phase current-source inverter, shown in Fig. 1.61, differs from its full-bridge voltage-source counterpart by the absence of freewheeling diodes, which, because of the unidirectional input current, would be superfluous. Analogously, the single-phase CSI can be obtained from the single-phase full-bridge VSI in Fig. 1.46 by removing the freewheeling diodes. In contrast to VSIs, both switches in the same leg of a CSI can be ON simultaneously. Because of the current-source characteristics of the supply system, no overcurrent will result. It is interruption of the current that is dangerous, because of the large dc-link inductance. Therefore, when changing the state of the inverter, both switches in one phase are kept closed for a short period of time, an analogy to the dead time in VSIs. I, DC LINK rv-v'v'x__.._e > L ,1 c3 t.d (.) t'r" Z 8= (.p (..) () CURRENT FEEDBACK FIGURE 1.60 Supply arrangement for the current-source inverter. 1.5 ~,a I AI ib DC TO AC CONVERTERS 41 T ic B C "Use FIGURE 1.61 Three-phase current-source inverter. Because both switches in the same phase can be ON or OFF simultaneously and one switch can be ON while the other is OFF, switching functions for individual phases would have to be of the quaternary type (having values of 0, 1, 2, or 3). This would be inconvenient; therefore, binary switching functions, a, a', b, b', c, and c', are assigned to individual switches, SA through SC', instead. This makes for 64 possible states of the inverter, of which, however, only 9 are employed, to avoid the uncertainty resulting from supplying a multipath network from a current source. Only one path of the output current is permitted, for example, from terminal A to terminal C, that is, with iA = --ic. The output line currents in the three-phase CSI can be expressed as E'iil,i([!l [:il) (1.33) Defining the state of a CSI as aa'bb'cc'2, only States 3, 6, 9, 12, 18, 24, 33, 36, and 48 are used, States 3, 12, and 48 producing zero output currents. Space vectors of the CSI are shown in Fig. 1.62. To facilitate interpretation of individual vectors, terminals of the inverter passing the output current are also marked. For example, current vector i33, produced in State 33, represents the situation when the load current flows between terminals A and C, that is, iA = Ii and ic = -/i. Analogously to the VSI, when the state sequence corresponding to sequential current vectors i36, i33, i9, i24, i18, i 6. . . . , is imposed, with each state lasting one-sixth of the desired period of the output voltage, the CSI operates in the square-wave mode illustrated in Fig. 1.63. In practice, the rate of change of the current at the leading and trailing edge of each pulse is limited. Still, the load inductance generates spikes of the output voltage at these edges which, in addition to the nonsinusoidal shape of current waveforms, constitutes a disadvantage of CSIs. Sinusoidal currents (with a tipple) are produced in the PWM CSI, which is obtained by adding capacitors 42 CHAPTER 1 /POWER ELECTRONIC CONVERTERS 13- !n = 148 - 0 BC 124 /-/'' . ~" x -.. 13~ 33 d 136 -j ~ 16 FIGURE 1.62 Space vectors of line current in a three-phase current-source inverter. between the output terminals. These capacitors shunt a part of the harmonic content of squarewave currents, so that the load currents resemble those of the VSI. 1.6 CONCLUSION The described variety of power electronic converters allows efficient conversion and control of electrical power. Pulse width modulated converters offer better operating characteristics than the phase-controlled ones, but the very process of high-frequency switching creates undesirable side effects of its own. It seems that phase-controlled rectifiers and ac voltage controllers will 33 STATE: 36 9 24 18 6 lli cot I I I I cot I I 1 2 5 ~r 2~r FIGURE 1.63 Output current waveforms in a current-source inverter in the square-wave mode. 1.6 CONCLUSION 43 maintain their presence in power electronics for years to come. The same observation applies to hard-switching converters, although the soft-switching ones will certainly increase their market share. Switching functions, which stress the discrete character of power electronic converters, and space vectors of voltage and current, are convenient tools for analysis and control of these converters. It is also worth noting that the progress in speed and efficacy of information and power processing in modem PWM converters makes them nearly ideal power amplifiers.