Chapter 17 The Ideal Rectifier


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1 Chapter 17 The Ideal Rectifier 17.1 Properties of the ideal rectifier 17.2 Realization of a nearideal rectifier 17.3 Singlephase converter systems employing ideal rectifiers 17.4 RMS values of rectifier waveforms 17.5 Ideal threephase rectifiers Fundamentals of Power Electronics 1 Chapter 17: The Ideal Rectifier
2 17.1 Properties of the ideal rectifier It is desired that the rectifier present a resistive load to the ac power system. This leads to unity power factor ac line current has same waveshape as voltage i ac = v ac R e i ac R e is called the emulated resistance v ac R e Fundamentals of Power Electronics 2 Chapter 17: The Ideal Rectifier
3 Control of power throughput P av = V 2 ac,rms R e (v control ) i ac Power apparently consumed by R e is actually transferred to rectifier dc output port. To control the amount of output power, it must be possible to adjust the value of R e. v ac R e (v control ) v control Fundamentals of Power Electronics 3 Chapter 17: The Ideal Rectifier
4 Output port model The ideal rectifier is lossless and contains no internal energy storage. Hence, the instantaneous input power equals the instantaneous output power. Since the instantaneous power is independent of the dc load characteristics, the output port obeys a power source characteristic. v ac i ac ac input p= R e (v control ) v control v 2 ac R e (v control ) Ideal rectifier (LFR) p = v ac 2 /R e i v dc output vi=p= v ac 2 R e Fundamentals of Power Electronics 4 Chapter 17: The Ideal Rectifier
5 The dependent power source i i i vi = p p v v p v power source power sink iv characteristic Fundamentals of Power Electronics 5 Chapter 17: The Ideal Rectifier
6 Equations of the ideal rectifier / LFR Defining equations of the ideal rectifier: i ac = vi=p v ac R e (v control ) When connected to a resistive load of value R, the input and output rms voltages and currents are related as follows: V rms V ac,rms = R Re p= v 2 ac R e (v control ) I ac,rms I rms = R Re Fundamentals of Power Electronics 6 Chapter 17: The Ideal Rectifier
7 17.2 Realization of a nearideal rectifier Control the duty cycle of a dcdc converter, such that the input current is proportional to the input voltage: i g i ac dcdc converter 1 : M(d) i v ac v g v C R d i g Controller v g Fundamentals of Power Electronics 7 Chapter 17: The Ideal Rectifier
8 Waveforms v ac V M i g t v V i ac V M /R e t M M min v g V M v ac =V M sin (ωt) v g =V M sin (ωt) M(d) = v v g = M min = V V M V M V sin (ωt) Fundamentals of Power Electronics 8 Chapter 17: The Ideal Rectifier
9 Choice of converter M(d) = v v g = V M V sin (ωt) M M min To avoid distortion near line voltage zero crossings, converter should be capable of producing M(d) approaching infinity Above expression neglects converter dynamics Boost, buckboost, Cuk, SEPIC, and other converters with similar conversion ratios are suitable We will see that the boost converter exhibits lowest transistor stresses. For this reason, it is most often chosen Fundamentals of Power Electronics 9 Chapter 17: The Ideal Rectifier
10 Boost converter with controller to cause input current to follow input voltage i g i ac Boost converter L D 1 i v ac v g Q 1 C v R v control Multiplier X v g R s i g v a PWM v ref = k x v g v control v err G c (s) Compensator Controller Fundamentals of Power Electronics 10 Chapter 17: The Ideal Rectifier
11 Variation of duty cycle in boost rectifier M(d) = v v g = V M V sin (ωt) Since M 1 in the boost converter, it is required that V V M If the converter operates in CCM, then M(d) = 1 1d The duty ratio should therefore follow d=1 v g V in CCM Fundamentals of Power Electronics 11 Chapter 17: The Ideal Rectifier
12 CCM/DCM boundary, boost rectifier Inductor current ripple is i g = v gdt s 2L Lowfrequency (average) component of inductor current waveform is i g Ts = v g R e The converter operates in CCM when i g Ts > i g d< 2L R e T s Substitute CCM expression for d: R e < T s 2L 1 v g V for CCM Fundamentals of Power Electronics 12 Chapter 17: The Ideal Rectifier
13 CCM/DCM boundary R e < T s 2L 1 v g V for CCM Note that v g varies with time, between 0 and V M. Hence, this equation may be satisfied at some points on the ac line cycle, and not at others. The converter always operates in CCM provided that R e < 2L T s The converter always operates in DCM provided that R e > T s 2L 1 V M V For R e between these limits, the converter operates in DCM when v g is near zero, and in CCM when v g approaches V M. Fundamentals of Power Electronics 13 Chapter 17: The Ideal Rectifier
14 Static input characteristics of the boost converter v g A plot of input current i g vs input voltage v g, for various duty cycles d. In CCM, the boost converter equilibrium equation is v g V =1d The input characteristic in DCM is found by solution of the averaged DCM model (Fig (b)): i g R e (d) p R Beware! This DCM R e (d) from Chapter 10 is not the same as the rectifier emulated resistance R e = v g /i g V Solve for input current: i g = v g R e (d) p V v g with p= v g 2 R e (d) R e (d) = 2L d 2 T s Fundamentals of Power Electronics 14 Chapter 17: The Ideal Rectifier
15 Static input characteristics of the boost converter Now simplify DCM current expression, to obtain 2L i VT g 1 v g s V = d 2 v g V CCM/DCM mode boundary, in terms of v g and i g : 2L i VT g > v g s V 1 v g V Fundamentals of Power Electronics 15 Chapter 17: The Ideal Rectifier
16 Boost input characteristics with superimposed resistive characteristic d = 1 d = 0.8 d = 0.6 d = 0.4 d = 0.2 d = 0 CCM: v g V =1d j g = VT 2L i g s i g =v g /R e DCM CCM DCM: 2L i VT g 1 v g s V CCM when 2L i VT g > v g s V = d 2 v g V 1 v g V m g = v g V Fundamentals of Power Electronics 16 Chapter 17: The Ideal Rectifier
17 R e of the multiplying (average current) controller Solve circuit to find R e : Current sensor gain v ac i g i ac v control Multiplier v g v g R s i g Controller Boost converter L X v a v ref = k x v g v control v err Q 1 D 1 PWM G c (s) Compensator i C v R v a =i g R s when the error signal is small, v a v ref multiplier equation v ref =k x v g v control then R e is R e = v g i g = simplify: v ref k x v control v a R s R e (v control ) = R s k x v control Fundamentals of Power Electronics 17 Chapter 17: The Ideal Rectifier
18 Low frequency system model i g Ts Ideal rectifier (LFR) i Ts i ac p Ts v ac v g Ts R e C v Ts R R e R e = R s k x v control v control R e (v control ) = R s k x v control This model also applies to other converters that are controlled in the same manner, including buckboost, Cuk, and SEPIC. Fundamentals of Power Electronics 18 Chapter 17: The Ideal Rectifier
19 Openloop DCM approach We found in Chapter 10 that the buckboost, SEPIC, and Cuk converters, when operated openloop in DCM, inherently behave as lossfree resistors. This suggests that they could also be used as nearideal rectifiers, without need for a multiplying controller. Advantage: simple control Disadvantages: higher peak currents, larger input current EMI Like other DCM applications, this approach is usually restricted to low power (< 200W). The boost converter can also be operated in DCM as a low harmonic rectifier. Input characteristic is i g Ts = v g v g2 R e vv g R e Input current contains harmonics. If v is sufficiently greater than v g, then harmonics are small. Fundamentals of Power Electronics 19 Chapter 17: The Ideal Rectifier
20 17.3 Singlephase converter systems containing ideal rectifiers It is usually desired that the output voltage v be regulated with high accuracy, using a widebandwidth feedback loop For a given constant load characteristic, the instantaneous load current and power are then also constant: p load =vi=vi The instantaneous input power of a singlephase ideal rectifier is not constant: p ac =v g i g with v g =V M sin (ωt) i g = v g R e so p ac = V 2 M sin R 2 ωt = V 2 M e 2R e 1 cos 2ωt Fundamentals of Power Electronics 20 Chapter 17: The Ideal Rectifier
21 Power flow in singlephase ideal rectifier system Ideal rectifier is lossless, and contains no internal energy storage. Hence instantaneous input and output powers must be equal An energy storage element must be added Capacitor energy storage: instantaneous power flowing into capacitor is equal to difference between input and output powers: p C = de C dt = d 1 2 Cv 2 C dt = p ac p load Energy storage capacitor voltage must be allowed to vary, in accordance with this equation Fundamentals of Power Electronics 21 Chapter 17: The Ideal Rectifier
22 Capacitor energy storage in 1 system p ac P load v c = d 1 2 Cv 2 C dt = p ac p load Fundamentals of Power Electronics 22 Chapter 17: The Ideal Rectifier t
23 Singlephase system with internal energy storage i g Ideal rectifier (LFR) i 2 p load = VI = P load v ac i ac v g R e p ac Ts C v C Dcdc converter v i load Energy storage capacitor Energy storage capacitor voltage v C must be independent of input and output voltage waveforms, so that it can vary according to = d 1 2 Cv 2 C dt = p ac p load This system is capable of Widebandwidth control of output voltage Widebandwidth control of input current waveform Internal independent energy storage Fundamentals of Power Electronics 23 Chapter 17: The Ideal Rectifier
24 Hold up time Internal energy storage allows the system to function in other situations where the instantaneous input and output powers differ. A common example: continue to supply load power in spite of failure of ac line for short periods of time. Hold up time: the duration which the dc output voltage v remains regulated after v ac has become zero A typical holdup time requirement: supply load for one complete missing ac line cycle, or 20msec in a 50Hz system During the holdup time, the load power is supplied entirely by the energy storage capacitor Fundamentals of Power Electronics 24 Chapter 17: The Ideal Rectifier
25 Energy storage element Instead of a capacitor, and inductor or higherorder LC network could store the necessary energy. But, inductors are not good energystorage elements Example 100V 100µF capacitor 100A 100µH inductor each store 1 Joule of energy But the capacitor is considerably smaller, lighter, and less expensive So a single big capacitor is the best solution Fundamentals of Power Electronics 25 Chapter 17: The Ideal Rectifier
26 Inrush current A problem caused by the large energy storage capacitor: the large inrush current observed during system startup, necessary to charge the capacitor to its equilibrium value. Boost converter is not capable of controlling this inrush current. Even with d = 0, a large current flows through the boost converter diode to the capacitor, as long as v < v g. Additional circuitry is needed to limit the magnitude of this inrush current. Converters having buckboost characteristics are capable of controlling the inrush current. Unfortunately, these converters exhibit higher transistor stresses. Fundamentals of Power Electronics 26 Chapter 17: The Ideal Rectifier
27 Universal input The capability to operate from the ac line voltages and frequencies found everywhere in the world: 50Hz and 60Hz Nominal rms line voltages of 100V to 260V: 100V, 110V, 115V, 120V, 132V, 200V, 220V, 230V, 240V, 260V Regardless of the input voltage and frequency, the nearideal rectifier produces a constant nominal dc output voltage. With a boost converter, this voltage is 380 or 400V. Fundamentals of Power Electronics 27 Chapter 17: The Ideal Rectifier
28 Lowfrequency model of dcdc converter Dcdc converter produces wellregulated dc load voltage V. Load therefore draws constant current I. Load power is therefore the constant value P load = VI. To the extent that dcdc converter losses can be neglected, then dcdc converter input power is P load, regardless of capacitor voltage v c. Dcdc converter input port behaves as a power sink. A low frequency converter model is i 2 p load = VI = P load i C v C P load V v load Energy storage capacitor Dcdc converter Fundamentals of Power Electronics 28 Chapter 17: The Ideal Rectifier
29 Lowfrequency energy storage process, 1 system A complete lowfrequency system model: i g i 2 i ac p ac Ts p load = VI = P load i v ac v g R e C v C P load V v load Ideal rectifier (LFR) Energy storage capacitor Dcdc converter Difference between rectifier output power and dcdc converter input power flows into capacitor In equilibrium, average rectifier and load powers must be equal But the system contains no mechanism to accomplish this An additional feeback loop is necessary, to adjust R e such that the rectifier average power is equal to the load power Fundamentals of Power Electronics 29 Chapter 17: The Ideal Rectifier
30 Obtaining average power balance i g i 2 p load = VI = P load i ac p ac Ts i v ac v g R e C v C P load V v load Ideal rectifier (LFR) Energy storage capacitor Dcdc converter If the load power exceeds the average rectifier power, then there is a net discharge in capacitor energy and voltage over one ac line cycle. There is a net increase in capacitor charge when the reverse is true. This suggests that rectifier and load powers can be balanced by regulating the energy storage capacitor voltage. Fundamentals of Power Electronics 30 Chapter 17: The Ideal Rectifier
31 A complete 1 system containing three feedback loops v ac i g i ac v g Boost converter L D 1 Q 1 i 2 v C C DCDC Converter i Load v v control Multiplier X v g R s i g v a PWM d v ref1 = k x v g v control v err G c (s) Compensator Widebandwidth input current controller v Compensator and modulator v ref3 Widebandwidth output voltage controller v C Compensator v ref2 Lowbandwidth energystorage capacitor voltage controller Fundamentals of Power Electronics 31 Chapter 17: The Ideal Rectifier
32 Bandwidth of capacitor voltage loop The energystoragecapacitor voltage feedback loop causes the dc component of v c to be equal to some reference value Average rectifier power is controlled by variation of R e. R e must not vary too quickly; otherwise, ac line current harmonics are generated Extreme limit: loop has infinite bandwidth, and v c is perfectly regulated to be equal to a constant reference value Energy storage capacitor voltage then does not change, and this capacitor does not store or release energy Instantaneous load and ac line powers are then equal Input current becomes i ac = p ac v ac = p load v ac = P load V M sin ωt Fundamentals of Power Electronics 32 Chapter 17: The Ideal Rectifier
33 Input current waveform, extreme limit i ac = p ac v ac = p load v ac = P load V M sin ωt THD Power factor 0 v ac i ac t So bandwidth of capacitor voltage loop must be limited, and THD increases rapidly with increasing bandwidth Fundamentals of Power Electronics 33 Chapter 17: The Ideal Rectifier
34 17.4 RMS values of rectifier waveforms Doublymodulated transistor current waveform, boost rectifier: i Q Computation of rms value of this waveform is complex and tedious Approximate here using double integral Generate tables of component rms and average currents for various rectifier converter topologies, and compare t Fundamentals of Power Electronics 34 Chapter 17: The Ideal Rectifier
35 RMS transistor current RMS transistor current is i Q I Qrms = 1 Tac 0 T ac i Q 2 dt Express as sum of integrals over all switching periods contained in one ac line period: t T ac /T s I Qrms = 1 T 1 s i 2 Q dt Tac Ts n =1 nt s (n1)t s Quantity in parentheses is the value of i Q2, averaged over the n th switching period. Fundamentals of Power Electronics 35 Chapter 17: The Ideal Rectifier
36 Approximation of RMS expression T ac /T s n =1 I Qrms = 1 T 1 s i 2 Q dt Tac Ts nt s (n1)t s When T s << T ac, then the summation can be approximated by an integral, which leads to the doubleaverage: I Qrms 1 Tac T ac /T s nt s lim Ts T 1 0 s i 2 n=1 Q (τ)dτ Ts (n1)t s = 1 1 i 2 Q (τ)dτdt Tac Ts 0 T ac t tt s = i Q 2 Ts T ac Fundamentals of Power Electronics 36 Chapter 17: The Ideal Rectifier
37 Boost rectifier example For the boost converter, the transistor current i Q is equal to the input current when the transistor conducts, and is zero when the transistor is off. The average over one switching period of i Q2 is therefore If the input voltage is i Q 2 tt s t = 1 i T s T 2 Q dt s = di 2 ac v ac =V M sin ωt then the input current will be given by i ac = V M Re and the duty cycle will ideally be sin ωt V v ac = 1 1d (this neglects converter dynamics) Fundamentals of Power Electronics 37 Chapter 17: The Ideal Rectifier
38 Boost rectifier example Duty cycle is therefore d=1 V M V Evaluate the first integral: 2 i Q = V 2 M T 2 s R e sin ωt Now plug this into the RMS formula: I Qrms = 1 Tac i Q 2 1 V M V sin ωt sin 2 ωt 0 T ac T s dt = 1 Tac 0 T ac 2 V M 2 R e 1 V M V sin ωt sin 2 ωt dt I Qrms = 2 2 V M Tac 2 sin 2 ωt V M sin 3 R e V 0 T ac /2 ωt dt Fundamentals of Power Electronics 38 Chapter 17: The Ideal Rectifier
39 Integration of powers of sin θ over complete halfcycle n 1 π 0 π sin n (θ)dθ 1 π 0 π sin n (θ)dθ = (n 1) π n (n 1) n if n is odd if n is even 1 2 π π π Fundamentals of Power Electronics 39 Chapter 17: The Ideal Rectifier
40 Boost example: Transistor RMS current I Qrms = V M 2R e 1 8 3π V M V = I ac rms 1 8 3π V M V Transistor RMS current is minimized by choosing V as small as possible: V = V M. This leads to I Qrms = 0.39I ac rms When the dc output voltage is not too much greater than the peak ac input voltage, the boost rectifier exhibits very low transistor current. Efficiency of the boost rectifier is then quite high, and 95% is typical in a 1kW application. Fundamentals of Power Electronics 40 Chapter 17: The Ideal Rectifier
41 Table of rectifier current stresses for various topologies Table 17.2 Summary of rectifier current stresses for several converter topologies rms A verage Peak CCM boost Transistor I ac rms 1 8 3π V M V I ac rms 2 2 π 1 π 8 V M I ac rms 2 V Diode I dc 16 3π V V M I dc 2 I dc V VM Inductor I ac rms I ac rms 2 2 π I ac rms 2 CCM flyback, with n:1 isolation transformer and input filter Transistor, xfmr primary L 1 I ac rms 1 8 3π I ac rms V M nv I ac rms 2 2 π I ac rms 2 1 V n I ac rms 2 2 π I ac rms 2 C 1 I 8 V M ac rms 3π nv 0 I ac rms 2 max 1, V M nv Diode, xfmr secondary I dc π nv V M I dc 2I dc 1 nv V M Fundamentals of Power Electronics 41 Chapter 17: The Ideal Rectifier
42 Table of rectifier current stresses continued CCM SEPIC, nonisolated Transistor L 1 C 1 I ac rms 1 8 3π I ac rms V M I 2 2 ac rms V π I ac rms 2 1 V M V I ac rms 2 2 π I ac rms 8 3π V M V 0 I ac rms 2 I ac rms max 1, V M V L 2 Diode I ac rms V M V 3 2 I ac rms 2 I dc π V V M I dc V M V I ac rms V M V 2 2I dc 1 V V M CCM SEPIC, with n:1 isolation transformer transistor L 1 C 1, xfmr primary Diode, xfmr secondary I ac rms 1 8 3π I ac rms I 8 V M ac rms 3π nv V M nv I ac rms 2 2 π I ac rms 2 1 V M nv I ac rms 2 2 π I dc π nv V M I dc I with, in all cases, ac rms = I dc 2 V V M, ac input voltage = V M sin(ω t) dc output voltage = V I ac rms 2 Fundamentals of Power Electronics 42 Chapter 17: The Ideal Rectifier 0 I ac rms 2I dc 2 max 1, n 1 nv V M
43 Comparison of rectifier topologies Boost converter Lowest transistor rms current, highest efficiency Isolated topologies are possible, with higher transistor stress No limiting of inrush current Output voltage must be greater than peak input voltage Buckboost, SEPIC, and Cuk converters Higher transistor rms current, lower efficiency Isolated topologies are possible, without increased transistor stress Inrush current limiting is possible Output voltage can be greater than or less than peak input voltage Fundamentals of Power Electronics 43 Chapter 17: The Ideal Rectifier
44 Comparison of rectifier topologies 1kW, 240Vrms example. Output voltage: 380Vdc. Input current: 4.2Arms Converter Transistor rms current Transistor voltage Diode rms current Transistor rms current, 120V Diode rms current, 120V Boost 2 A 380 V 3.6 A 6.6 A 5.1 A Nonisolated SEPIC Isolated SEPIC 5.5 A 719 V 4.85 A 9.8 A 6.1 A 5.5 A 719 V 36.4 A 11.4 A 42.5 A Isolated SEPIC example has 4:1 turns ratio, with 42V 23.8A dc load Fundamentals of Power Electronics 44 Chapter 17: The Ideal Rectifier
45 17.5 Ideal threephase rectifiers Ideal 3ø rectifier, modeled as three 1ø ideal rectifiers: 3øac input dc output ø a i a R e p a ø b i b R e p b R v ø c i c R e p c Fundamentals of Power Electronics 45 Chapter 17: The Ideal Rectifier
46 Ideal 3 rectifier model Combine parallelconnected power sources into a single source p tot : 3øac input dc output ø a i a R e ø b i b R e p tot = p a p b p c R v ø c i c R e Fundamentals of Power Electronics 46 Chapter 17: The Ideal Rectifier
47 Value of p tot Ac input voltages: v an =V M sin ωt v bn =V M sin ωt 120 v cn =V M sin ωt 240 Instantaneous phase powers: 3øac input ø a ø b ø c i a i b i c R e R e R e p a p b p c dc output R v p a = v 2 an R e R e = V 2 M 2R e 1 cos 2ωt p b = v 2 bn = V 2 M 1 cos 2ωt 240 2R e p c = v cn 2 R e = V 2 M 1 cos 2ωt 120 2R e Total 3ø instantaneous power: p tot =p a p b p c = 3 2 V M 2 R e 2 nd harmonic terms add to zero total 3ø power p tot is constant Fundamentals of Power Electronics 47 Chapter 17: The Ideal Rectifier
48 Instantaneous power in ideal 3 rectifier p tot =p a p b p c = 3 2 V M 2 R e In a balanced system, the ideal 3ø rectifier supplies constant power to its dc output a constant power load can be supplied, without need for lowfrequency internal energy storage 3øac input ø a ø b ø c i a i b i c R e R e R e p tot = p a p b p c dc output R v Fundamentals of Power Electronics 48 Chapter 17: The Ideal Rectifier
49 Threephase rectifiers operating in CCM 3øac input ø a ø b 3øacdc boost rectifier i a i b L L v Q 1 i 1 i 2 i 3 D 1 Q 2 D 2 Q 3 D 3 C dc output Load v ø c i c L 3 v 20 v 10 0 Q 4 Q 5 Q 6 D 4 D 5 D 6 Uses six currentbidirectional switches Operation of each individual phase is similar to the 1ø boost rectifier Fundamentals of Power Electronics 49 Chapter 17: The Ideal Rectifier
50 The 3 acðdc boost rectifier Voltagesource inverter, operated backwards as a rectifier Converter is capable of bidirectional power flow Dc output voltage V must be greater than peak ac lineline voltage V L,pk. Ac input currents are nonpulsating. In CCM, input EMI filtering is relatively easy Very low RMS transistor currents and conduction loss The leading candidate to replace uncontrolled 3ø rectifiers Requires six active devices Cannot regulate output voltage down to zero: no current limiting cannot replace traditional bucktype controlled rectifiers Fundamentals of Power Electronics 50 Chapter 17: The Ideal Rectifier
51 Control of switches in CCM 3 acdc boost rectifier Pulsewidth modulation: v 10 v Drive lower transistors (Q 4 Q 6 ) with complements of duty cycles of respective upper transistors (Q 1 Q 3 ). Each phase operates independently, with its own duty cycle. v 20 v 10 Ts = d 1 v Ts 0 0 d 1 T s T s t v v 20 Ts = d 2 v Ts 1 v 12 2 v 20 3 v 10 0 Q 1 Q 4 i 1 i 2 i 3 Q 2 Q D 3 1 D 2 D 3 Q 5 Q 6 D 4 D 5 D 6 0 d 2 T s T s t Fundamentals of Power Electronics 51 Chapter 17: The Ideal Rectifier v 30 v 0 v 30 Ts = d 3 v Ts 0 d 3 T s T s t Conducting Q devices: 1 / D 1 Q 4 / D 4 Q 2 / D 2 Q 5 / D 5 Q 3 / D 3 Q 6 / D 6 0
52 Average switch waveforms Average the switch voltages: v 10 v v 10 Ts = d 1 v Ts v 10 Ts = d 1 v Ts v 20 Ts = d 2 v Ts v 30 Ts = d 3 v Ts Average lineline voltages: v 12 Ts = v 10 Ts v 20 Ts = d 1 d 2 v 23 Ts = v 20 Ts v 30 Ts = d 2 d 3 v 31 Ts = v 30 Ts v 10 Ts = d 3 d 1 v Ts v Ts v Ts v 20 v 30 0 d 1 T s T s t v v 20 Ts = d 2 v Ts 0 d 2 T s T s t v 0 0 Average switch outputside currents: i 1 Ts = d 1 i a Ts i 2 Ts = d 2 i b Ts i 3 Ts = d 3 i c Ts v 30 Ts = d 3 v Ts 0 d 3 T s T s t Conducting Q devices: 1 / D 1 Q 4 / D 4 Q 2 / D 2 Q 5 / D 5 Q 3 / D 3 Q 6 / D 6 0 Fundamentals of Power Electronics 52 Chapter 17: The Ideal Rectifier
53 Averaged circuit model L i a Ts ø a ø b L i b Ts (d 1 d 2 ) v Ts d 1 i a Ts d 2 i b Ts d 3 i c Ts C Load v Ts ø c L (d 2 d 3 ) v Ts (d 3 d 1 ) v Ts i c Ts v 12 Ts = v 10 Ts v 20 Ts = d 1 d 2 v 23 Ts = v 20 Ts v 30 Ts = d 2 d 3 v 31 Ts = v 30 Ts v 10 Ts = d 3 d 1 v Ts v Ts v Ts i 1 Ts = d 1 i a Ts i 2 Ts = d 2 i b Ts i 3 Ts = d 3 i c Ts Q: How to vary d such that the desired ac and dc waveforms are obtained? Solution is not unique. Fundamentals of Power Electronics 53 Chapter 17: The Ideal Rectifier
54 Sinusoidal PWM A simple modulation scheme: Sinusoidal PWM Vary duty cycles sinusoidally, in synchronism with ac line d 1 =D D m sin ωt ϕ d 2 =D D m sin ωt ϕ 120 d 3 =D D m sin ωt ϕ 240 where ω is the ac line frequency D 0 is a dc bias D m is the modulation index For D 0 = 0.5, D m in the above equations must be less than 1. The modulation index is defined as onehalf of the peak amplitude of the fundamental component of the duty cycle modulation. In some other modulation schemes, it is possible that D m > 1. Fundamentals of Power Electronics 54 Chapter 17: The Ideal Rectifier
55 Solution, linear sinusoidal PWM If the switching frequency is high, then the inductors can be small and have negligible effect at the ac line frequency. The averaged switch voltage and ac line voltage are then equal: v 12 Ts = d 1 d 2 v Ts v ab Substitute expressions for duty cycle and ac line voltage variations: 1 2 D m sin ωt ϕ sin ωt ϕ 120 v Ts = V M sin ωt sin ωt 120 For small L, ϕ tends to zero. The expression then becomes 1 2 D mv = V M Solve for the output voltage: V = 2V M D m V = 2 3 V L,pk D m = 1.15 V L,pk D m Fundamentals of Power Electronics 55 Chapter 17: The Ideal Rectifier
56 Boost rectifier with sinusoidal PWM V = 2 3 V L,pk D m = 1.15 V L,pk D m With sinusoidal PWM, the dc output voltage must be greater than 1.15 times the peak lineline input voltage. Hence, the boost rectifier increases the voltage magnitude. Fundamentals of Power Electronics 56 Chapter 17: The Ideal Rectifier
57 Nonlinear modulation Triplen harmonics can be added to the duty ratio modulation, without appearing in the lineline voltages d 1 d 2 d Overmodulation, in v 12 Ts /V which the modulation index 1 D m is increased beyond 1, also leads to undistorted lineline voltages provided that D m The pulse width modulator saturates, but the duty ratio variations contain only triplen harmonics. V = V L,pk is obtained at D m = Further increases in D m cause distorted ac line waveforms Fundamentals of Power Electronics 57 Chapter 17: The Ideal Rectifier ωt
58 Bucktype 3 acðdc rectifier 3øac input i a Q 1 Q 2 Q 3 i L L dc output ø a ø b i b D 1 D 2 D 3 C Load v i c Q 4 Q 5 Q 6 ø c Input filter D 4 D 5 D 6 Can produce controlled dc output voltages in the range 0 V V L,pk Requires twoquadrant voltagebidirectional switches Exhibits greater active semiconductor stress than boost topology Can operate in inverter mode by reversal of output voltage polarity Fundamentals of Power Electronics 58 Chapter 17: The Ideal Rectifier
59 BuckÐboost topology 3øac input ø a i a Q 1 Q 2 Q 3 i L Q 7 D 7 dc output ø b i b D 1 D 2 D 3 L C Load v i c Q 4 Q 5 Q 6 ø c Input filter D 4 D 5 D 6 Fundamentals of Power Electronics 59 Chapter 17: The Ideal Rectifier
60 Cuk topology 3øac input dc output ø a L 1 i a Q 1 Q 2 Q D 3 1 D 2 D 3 C1 L 2 ø b L 1 i b D 7 Q7 C 2 Load v ø c L 1 i c Q 4 Q 5 Q 6 D 4 D 5 D 6 Fundamentals of Power Electronics 60 Chapter 17: The Ideal Rectifier
61 Use of three singlephase rectifiers 3øac input ø a ø b Dcdc converter with isolation Dcdc converter with isolation i dc output C v Each rectifier must include isolation between input and output Isolation transformers must be rated to carry the pulsating singlephase ac power p ac Outputs can be connected in series or parallel ø c Dcdc converter with isolation Because of the isolation requirement, semiconductor stresses are greater than in 3ø boost rectifier Fundamentals of Power Electronics 61 Chapter 17: The Ideal Rectifier
62 Some other approaches to threephase rectification Lowharmonic rectification requires active semiconductor devices that are much more expensive than simple peakdetection diode rectifiers. What is the minimum active silicon required to perform the function of 3ø lowharmonic rectification? No active devices are needed: diodes and harmonic traps will do the job, but these require lowfrequency reactive elements When control of the output voltage is needed, then there must be at least one active device To avoid lowfrequency reactive elements, at least one highfrequency switch is needed So let s search for approaches that use just one active switch, and only highfrequency reactive elements Fundamentals of Power Electronics 62 Chapter 17: The Ideal Rectifier
63 The singleswitch DCM boost 3 rectifier 3øac input L 1 i a D 7 dc output ø a D 1 D 2 D 3 ø b L 2 i b Q 1 C v L 3 i c ø c Input filter D 4 D 5 D 6 Inductors L 1 to L 3 operate in discontinuous conduction mode, in conjunction with diodes D 1 to D 6. Average input currents i a Ts, i b Ts, and i c Ts are approximately proportional to the instantaneous input lineneutral voltages. Transistor is operated with constant duty cycle; slow variation of the duty cycle allows control of output power. Fundamentals of Power Electronics 63 Chapter 17: The Ideal Rectifier
64 The singleswitch DCM boost 3 rectifier 3øac input L 1 i a D 7 dc output v an V M ø a D 1 D 2 D 3 ø b L 2 L 3 i b i c Q 1 C v t ø c Input filter D 4 D 5 D 6 i a v an dt s L t Fundamentals of Power Electronics 64 Chapter 17: The Ideal Rectifier
65 The singleswitch 3 DCM flyback rectifier 3øac input dc output ø a T 1 i a D 1 D 2 D 3 T 1 D 7 D 8 D 9 ø b T 2 i b Q 1 T 2 C v T 3 i c T 3 ø c Input filter D 4 D 5 D 6 D 10 D 11 D 12 Fundamentals of Power Electronics 65 Chapter 17: The Ideal Rectifier
66 The singleswitch 3 DCM flyback rectifier This converter is effectively three independent singlephase DCM flyback converters that share a common switch. Since the openloop DCM flyback converter can be modeled as a Loss Free Resistor, threephase lowharmonic rectification is obtained naturally. v an V M t Basic converter has a boost characteristic, but buckboost characteristic is possible (next slide). i a v an dt s L t Inrush current limiting and isolation are obtained easily. High peak currents, needs an input EMI filter Fundamentals of Power Electronics 66 Chapter 17: The Ideal Rectifier
67 3 Flyback rectifier with buckboost conversion ratio T 1 T 2 T 3 3øac input dc output ø a i a D 1 D 2 D 3 D 7 D 8 D 9 ø b i b Q 1 C v i c T 1 T 2 T 3 ø c input filter D 4 D 5 D 6 T 1 T 2 T 3 Fundamentals of Power Electronics 67 Chapter 17: The Ideal Rectifier
68 Singleswitch threephase zerocurrentswitching quasiresonant buck rectifier 3øac input L r i a Q 1 L dc output ø a D 1 D 2 D 3 ø b L r i b D 7 C r C v L r i c ø c Input filter D 4 D 5 D 6 Inductors L r and capacitor C r form resonant tank circuits, having resonant frequency slightly greater than the switching frequency. Turning on Q 1 initiates resonant current pulses, whose amplitudes depend on the instantaneous input lineneutral voltages. When the resonant current pulses return to zero, diodes D 1 to D 6 are reversebiased. Transistor Q 1 can then be turned off at zero current. Fundamentals of Power Electronics 68 Chapter 17: The Ideal Rectifier
69 Singleswitch threephase zerocurrentswitching quasiresonant buck rectifier i a v an R 0 t Input line currents are approximately sinusoidal pulses, whose amplitudes follow the input lineneutral voltages. Lowest total active semiconductor stress of all bucktype 3ø low harmonic rectifiers Fundamentals of Power Electronics 69 Chapter 17: The Ideal Rectifier
70 Multiresonant singleswitch zerocurrent switching 3 buck rectifier 3øac input i a L a Q 1 L d dc output ø a ø b i b L a C r1 v cra D 1 D 2 D 3 L r C v ø c i c L a C r1 C r1 D 4 D 5 D 6 D 7 C r2 Inductors L r and capacitors C r1 and C r2 form resonant tank circuits, having resonant frequency slightly greater than the switching frequency. Turning on Q 1 initiates resonant voltage pulses in v cra, whose amplitudes depend on the instantaneous input lineneutral currents i a to i c. All diodes switch off when their respective tank voltages reach zero. Transistor Q 1 is turned off at zero current. Fundamentals of Power Electronics 70 Chapter 17: The Ideal Rectifier
71 Multiresonant singleswitch zerocurrent switching 3 buck rectifier v cra ~ i a R 0 t Inputside resonant voltages are approximately sinusoidal pulses, whose amplitudes follow the input currents. Input filter inductors operate in CCM. Higher total active semiconductor stress than previous approach, but less EMI filtering is needed. Low THD: < 4% THD can be obtained. Fundamentals of Power Electronics 71 Chapter 17: The Ideal Rectifier
72 Harmonic correction Nonlinear load ø a 3ø ac ø b ø c Harmonic corrector Fundamentals of Power Electronics 72 Chapter 17: The Ideal Rectifier
73 Harmonic correction An active filter that is controlled to cancel the harmonic currents created by a nonlinear load. Does not need to conduct the average load power. Total active semiconductor stress is high when the nonlinear load generates large harmonic currents having high THD. In the majority of applications, this approach exhibits greater total active semiconductor stress than the simple 3ø CCM boost rectifier. Fundamentals of Power Electronics 73 Chapter 17: The Ideal Rectifier
74 17.6 Summary of key points 1. The ideal rectifier presents an effective resistive load, the emulated resistance R e, to the ac power system. The power apparently consumed by R e is transferred to the dc output port. In a threephase ideal rectifier, input resistor emulation is obtained in each phase. In both the singlephase and threephase cases, the output port follows a power source characteristic, dependent on the instantaneous ac input power. Ideal rectifiers can perform the function of lowharmonic rectification, without need for lowfrequency reactive elements. 2. The dcdc boost converter, as well as other converters capable of increasing the voltage, can be adapted to the ideal rectifier application. A control system causes the input current to be proportional to the input voltage. The converter may operate in CCM, DCM, or in both modes. The mode boundary is expressed as a function of R e, 2L/T s, and the instantaneous voltage ratio v g /V. A welldesigned average current controller leads to resistor emulation regardless of the operating mode; however, other schemes discussed in the next chapter may lead to distorted current waveforms when the mode boundary is crossed. Fundamentals of Power Electronics 74 Chapter 17: The Ideal Rectifier
75 Summary of key points 3. In a singlephase system, the instantaneous ac input power is pulsating, while the dc load power is constant. Whenever the instantaneous input and output powers are not equal, the ideal rectifier system must contain energy storage. A large capacitor is commonly employed; the voltage of this capacitor must be allowed to vary independently, as necessary to store and release energy. A slow feedback loop regulates the dc component of the capacitor voltage, to ensure that the average ac input power and dc load power are balanced. 4. RMS values of rectifiers waveforms can be computed by double integration. In the case of the boost converter, the rms transistor current can be as low as 39% of the rms ac input current, when V is close in value to V M. Other converter topologies such as the buckboost, SEPIC, and Cuk converters exhibit significantly higher rms transistor currents but are capable of limiting the converter inrush current. Fundamentals of Power Electronics 75 Chapter 17: The Ideal Rectifier
76 Summary of key points 5. In the threephase case, a boosttype rectifier based on the PWM voltagesource inverter also exhibits low rms transistor currents. This approach requires six active switching elements, and its dc output voltage must be greater than the peak input linetoline voltage. Average current control can be used to obtain input resistor emulation. An equivalent circuit can be derived by averaging the switch waveforms. The converter operation can be understood by assuming that the switch duty cycles vary sinusoidally; expressions for the average converter waveforms can then be derived. 6. Other threephase rectifier topologies are known, including sixswitch rectifiers having buck and buckboost characteristics. In addition, threephase lowharmonic rectifiers having a reduced number of active switches, as few as one, are discussed here. Fundamentals of Power Electronics 76 Chapter 17: The Ideal Rectifier
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