R. W. Erickson Department of Electrical, Computer, and Energy Engineering University of Colorado, Boulder
16.4. Power phasors in sinusoidal systems Apparent power is the product of the rms voltage and rms current It is easily measured simply the product of voltmeter and ammeter readings Unit of apparent power is the volt-ampere, or VA Many elements, such as transformers, are rated according to the VA that they can supply So power factor is the ratio of average power to apparent power With sinusoidal waveforms (no harmonics), we can also define the real power P reactive power Q complex power S If the voltage and current are represented by phasors V and I, then S = VI * = P + jq with I* = complex conjugate of I, j = square root of 1. The magnitude of S is the apparent power (VA). The real part of S is the average power P (watts). The imaginary part of S is the reactive power Q (reactive volt-amperes, or VARs). Fundamentals of Power Electronics 18 Chapter 16: Power and Harmonics in Nonsinusoidal Systems
Example: power phasor diagram The phase angle between the voltage and current, or (ϕ 1 θ 1 ), coincides with the angle of S. The power factor is Imaginary axis Q S = V rms I rms S = VI * power factor = P S = cos ϕ 1 θ 1 ϕ 1 θ 1 ϕ 1 θ 1 P Real axis In this purely sinusoidal case, the distortion factor is unity, and the power factor coincides with the displacement factor. ϕ 1 θ 1 I V Fundamentals of Power Electronics 19 Chapter 16: Power and Harmonics in Nonsinusoidal Systems
Reactive power Q The reactive power Q does not lead to net transmission of energy between the source and load. When Q 0, the rms current and apparent power are greater than the minimum amount necessary to transmit the average power P. Inductor: current lags voltage by 90, hence displacement factor is zero. The alternate storing and releasing of energy in an inductor leads to current flow and nonzero apparent power, but P = 0. Just as resistors consume real (average) power P, inductors can be viewed as consumers of reactive power Q. Capacitor: current leads voltage by 90, hence displacement factor is zero. Capacitors supply reactive power Q. They are often placed in the utility power distribution system near inductive loads. If Q supplied by capacitor is equal to Q consumed by inductor, then the net current (flowing from the source into the capacitor-inductive-load combination) is in phase with the voltage, leading to unity power factor and minimum rms current magnitude. Fundamentals of Power Electronics 20 Chapter 16: Power and Harmonics in Nonsinusoidal Systems
Lagging fundamental current of phasecontrolled rectifiers It will be seen in the next chapter that phase-controlled rectifiers produce a nonsinusoidal current waveform whose fundamental component lags the voltage. This lagging current does not arise from energy storage, but it does nonetheless lead to a reduced displacement factor, and to rms current and apparent power that are greater than the minimum amount necessary to transmit the average power. At the fundamental frequency, phase-controlled rectifiers can be viewed as consumers of reactive power Q, similar to inductive loads. Fundamentals of Power Electronics 21 Chapter 16: Power and Harmonics in Nonsinusoidal Systems
16.5. Harmonic currents in three phase systems The presence of harmonic currents can also lead to some special problems in three-phase systems: In a four-wire three-phase system, harmonic currents can lead to large currents in the neutral conductors, which may easily exceed the conductor rms current rating Power factor correction capacitors may experience significantly increased rms currents, causing them to fail In this section, these problems are examined, and the properties of harmonic current flow in three-phase systems are derived: Harmonic neutral currents in 3ø four-wire networks Harmonic neutral voltages in 3ø three-wire wye-connected loads Fundamentals of Power Electronics 22 Chapter 16: Power and Harmonics in Nonsinusoidal Systems
16.5.1. Harmonic currents in three-phase four-wire networks a i a ideal 3ø source v an v bn + + n v cn c i c + i n neutral connection nonlinear loads i b Fourier series of line currents and voltages: b Σ Σ Σ i a =I a0 + I ak cos (kωt θ ak ) k =1 i b =I b0 + I bk cos (k(ωt 120 ) θ bk ) k =1 i c =I c0 + I ck cos (k(ωt + 120 ) θ ck ) k =1 v an =V m cos (ωt) v bn =V m cos (ωt 120 ) v cn =V m cos (ωt + 120 ) Fundamentals of Power Electronics 23 Chapter 16: Power and Harmonics in Nonsinusoidal Systems
Neutral current i n =I a0 + I b0 + I c0 + Σ k =1 I ak cos (kωt θ ak )+I bk cos (k(ωt 120 ) θ bk )+I ck cos (k(ωt + 120 ) θ ck ) If the load is unbalanced, then there is nothing more to say. The neutral connection may contain currents having spectrum similar to the line currents. In the balanced case, I ak = I bk = I ck = I k and θ ak = θ bk = θ ck = θ k, for all k; i.e., the harmonics of the three phases all have equal amplitudes and phase shifts. The neutral current is then i n =3I 0 + Σ 3I k cos (kωt θ k ) k = 3,6,9, Fundamentals of Power Electronics 24 Chapter 16: Power and Harmonics in Nonsinusoidal Systems
Neutral currents i n =3I 0 + Σ 3I k cos (kωt θ k ) k = 3,6,9, Fundamental and most harmonics cancel out Triplen (triple-n, or 0, 3, 6, 9,...) harmonics do not cancel out, but add. Dc components also add. Rms neutral current is i n, rms =3 I 0 2 + Σ k = 3,6,9, I k 2 2 Fundamentals of Power Electronics 25 Chapter 16: Power and Harmonics in Nonsinusoidal Systems
Example A balanced nonlinear load produces line currents containing fundamental and 20% third harmonic: i an = I 1 cos(ωt θ 1 ) + 0.2 I 1 cos(3ωt θ 3 ). Find the rms neutral current, and compare its amplitude to the rms line current amplitude. Solution i n, rms =3 i 1, rms = (0.2I 1 ) 2 = 0.6 I 1 2 2 I 2 1 + (0.2I 1 ) 2 = I 1 2 2 1 + 0.04 I 1 2 The neutral current magnitude is 60% of the line current magnitude! The triplen harmonics in the three phases add, such that 20% third harmonic leads to 60% third harmonic neutral current. Yet the presence of the third harmonic has very little effect on the rms value of the line current. Significant unexpected neutral current flows. Fundamentals of Power Electronics 26 Chapter 16: Power and Harmonics in Nonsinusoidal Systems
16.5.2. Harmonic currents in three-phase three-wire networks Wye-connected nonlinear load, no neutral connection: a i a ideal 3ø source v an v bn + + n v cn + i n = 0 c i c + v n'n n' nonlinear loads i b b Fundamentals of Power Electronics 27 Chapter 16: Power and Harmonics in Nonsinusoidal Systems
No neutral connection If the load is balanced, then it is still true that i n =3I 0 + Σ 3I k cos (kωt θ k ) k = 3,6,9, But i n = 0, since there is no neutral connection. So the ac line currents cannot contain dc or triplen harmonics. What happens: A voltage is induced at the load neutral point, that causes the line current dc and triplen harmonics to become zero. The load neutral point voltage contains dc and triplen harmonics. With an unbalanced load, the line currents can still contain dc and triplen harmonics. Fundamentals of Power Electronics 28 Chapter 16: Power and Harmonics in Nonsinusoidal Systems
Delta-connected load a i a ideal 3ø source v an v bn + + n v cn + i n = 0 c i c deltaconnected nonlinear loads i b b There is again no neutral connection, so the ac line currents contain no dc or triplen harmonics The load currents may contain dc and triplen harmonics: with a balanced nonlinear load, these circulate around the delta. Fundamentals of Power Electronics 29 Chapter 16: Power and Harmonics in Nonsinusoidal Systems
Harmonic currents in power factor correction capacitors PFC capacitors are usually not intended to conduct significant harmonic currents. Heating in capacitors is a function of capacitor equivalent series resistance (esr) and rms current. The maximum allowable rms current then leads to the capacitor rating: esr C rated rms voltage V rms = I rms 2π fc rated reactive power = I 2 rms 2π fc Fundamentals of Power Electronics 30 Chapter 16: Power and Harmonics in Nonsinusoidal Systems
16.6. AC line current harmonic standards US MIL-STD-461B International Electrotechnical Commission Standard 1000 IEEE/ANSI Standard 519 Fundamentals of Power Electronics 31 Chapter 16: Power and Harmonics in Nonsinusoidal Systems
US MIL-STD-461B An early attempt to regulate ac line current harmonics generated by nonlinear loads For loads of 1kW or greater, no current harmonic magnitude may be greater than 3% of the fundamental magnitude. For the nth harmonic with n > 33, the harmonic magnitude may not exceed (1/n) times the fundamental magnitude. Harmonic limits are now employed by all of the US armed forces. The specific limits are often tailored to the specific application. The shipboard application is a good example of the problems faced in a relatively small stand-alone power system having a large fraction of electronic loads. Fundamentals of Power Electronics 32 Chapter 16: Power and Harmonics in Nonsinusoidal Systems
International Electrotechnical Commission Standard 1000 First draft of their IEC-555 standard:1982. It has since undergone a number of revisions. Recent reincarnation: IEC-1000-3-2 Enforcement of IEC-1000 is the prerogative of each individual country, and hence it has been sometimes difficult to predict whether and where this standard will actually be given the force of law. Nonetheless, IEC-1000 is now enforced in Europe, making it a de facto standard for commercial equipment intended to be sold worldwide. IEC-1000 covers a number of different types of low power equipment, with differing harmonic limits. Harmonics for equipment having an input current of up to 16A, connected to 50 or 60 Hz, 220V to 240V single phase circuits (two or three wire), as well as 380V to 415V three phase (three or four wire) circuits, are limited. Fundamentals of Power Electronics 33 Chapter 16: Power and Harmonics in Nonsinusoidal Systems
Low-power harmonic limits In a city environment such as a large building, a large fraction of the total power system load can be nonlinear Example: a major portion of the electrical load in a building is comprised of fluorescent lights, which present a very nonlinear characteristic to the utility system. A modern office may also contain a large number of personal computers, printers, copiers, etc., each of which may employ peak detection rectifiers. Although each individual load is a negligible fraction of the total local load, these loads can collectively become significant. Fundamentals of Power Electronics 34 Chapter 16: Power and Harmonics in Nonsinusoidal Systems
IEC-1000: Class A and B Class A: Balanced three-phase equipment, and any equipment which does not fit into the other categories. This class includes low harmonic rectifiers for computer and other office equipment. These limits are given in Table 16.1, and are absolute ampere limits. Class B: Portable tools, and similar devices. The limits are equal to the Table 16.1 limits, multiplied by 1.5. Classes C, D, and E: For other types of equipment, including lighting (Class C) and equipment having a special waveshape (Class D). Fundamentals of Power Electronics 35 Chapter 16: Power and Harmonics in Nonsinusoidal Systems
Class A limits Table 16.1. IEC-1000 Harmonic current limits, Class A Odd harmonics Even harmonics Harmonic number Maximum current Harmonic number Maximum curre 3 2.30A 2 1.08A 5 1.14A 4 0.43A 7 0.77A 6 0.30A 9 0.40A 8 n 40 0.23A (8/n) 11 0.33A 13 0.21A 15 n 39 0.15A (15/n) Fundamentals of Power Electronics 36 Chapter 16: Power and Harmonics in Nonsinusoidal Systems
16.6.2. IEEE/ANSI Standard 519 In 1993, the IEEE published a revised draft standard limiting the amplitudes of current harmonics, IEEE Guide for Harmonic Control and Reactive Compensation of Static Power Converters. Harmonic limits are based on the ratio of the fundamental component of the load current IL to the short circuit current at the point of common (PCC) coupling at the utility I sc. Stricter limits are imposed on large loads than on small loads. The limits are similar in magnitude to IEC-1000, and cover high voltage loads (of much higher power) not addressed by IEC-1000. Enforcement of this standard is presently up to the local utility company. The odd harmonic limits are listed in Table 16.2. The limits for even harmonics are 25% of the odd harmonic limits. Dc current components and half-wave rectifiers are not allowed. Fundamentals of Power Electronics 37 Chapter 16: Power and Harmonics in Nonsinusoidal Systems
IEEE-519 current limits, low voltage systems Table 16.2. IEEE-519 Maximum odd harmonic current limits for general distribution systems, 120V through 69kV Isc/I L n < 11 11 n<17 17 n<23 23 n<35 35 n THD <20 4.0% 2.0% 1.5% 0.6% 0.3% 5.0% 20 50 7.0% 3.5% 2.5% 1.0% 0.5% 8.0% 50 100 10.0% 4.5% 4.0% 1.5% 0.7% 12.0% 100 1000 12.0% 5.5% 5.0% 2.0% 1.0% 15.0% >1000 15.0% 7.0% 6.0% 2.5% 1.4% 20.0% Fundamentals of Power Electronics 38 Chapter 16: Power and Harmonics in Nonsinusoidal Systems
IEEE-519 voltage limits Table 16.3. IEEE-519 voltage distortion limits Bus voltage at PCC Individual harmonics THD 69kV and below 3.0% 5.0% 69.001kV 161kV 1.5% 2.5% above 161kV 1.0% 1.5% It is the responsibility of the utility to meet these limits. Fundamentals of Power Electronics 39 Chapter 16: Power and Harmonics in Nonsinusoidal Systems
Chapter 17 Line-Commutated Rectifiers 17.1 The single-phase full-wave rectifier 17.1.1 Continuous conduction mode 17.1.2 Discontinuous conduction mode 17.1.3 Behavior when C is large 17.1.4 Minimizing THD when C is small 17.2 The three-phase bridge rectifier 17.2.1 Continuous conduction mode 17.2.2 Discontinuous conduction mode 17.3 Phase control 17.3.1 Inverter mode 17.3.2 Harmonics and power factor 17.3.3 Commutation 17.4 Harmonic trap filters 17.5 Transformer connections 17.6 Summary Fundamentals of Power Electronics 1 Chapter 17: Line-commutated rectifiers
17.1 The single-phase full-wave rectifier i g i L L + D 4 D 1 v g Z i C v R D 3 D 2 Full-wave rectifier with dc-side L-C filter Two common reasons for including the dc-side L-C filter: Obtain good dc output voltage (large C) and acceptable ac line current waveform (large L) Filter conducted EMI generated by dc load (small L and C) Fundamentals of Power Electronics 2 Chapter 17: Line-commutated rectifiers
17.1.1 Continuous conduction mode Large L v g THD = 29% Typical ac line waveforms for CCM : As L, ac line current approaches a square wave i g 10 ms 20 ms 30 ms 40 ms CCM results, for L : distortion factor = I 1, rms I rms = 4 π 2 = 90.0% t THD = 1 distortion factor 2 1 = 48.3% Fundamentals of Power Electronics 3 Chapter 17: Line-commutated rectifiers
17.1.2 Discontinuous conduction mode Small L Typical ac line waveforms for DCM : As L 0, ac line current approaches impulse functions (peak detection) v g i g THD = 145% 10 ms 20 ms 30 ms 40 ms As the inductance is reduced, the THD rapidly increases, and the distortion factor decreases. Typical distortion factor of a full-wave rectifier with no inductor is in the range 55% to 65%, and is governed by ac system inductance. t Fundamentals of Power Electronics 4 Chapter 17: Line-commutated rectifiers
17.1.3 Behavior when C is large Solution of the full-wave rectifier circuit for infinite C: Define cos (ϕ 1 θ 1 ), PF, M 1.0 0.9 0.8 β cos (ϕ 1 θ 1 ) PF THD 200% 150% β 180 135 K L = 2L RT L 0.7 M 100% 90 M = V V m 0.6 0.5 DCM CCM THD 50% 45 0.4 0 0 0.0001 0.001 0.01 0.1 1 10 Fundamentals of Power Electronics 5 Chapter 17: Line-commutated rectifiers K L
17.1.4 Minimizing THD when C is small Sometimes the L-C filter is present only to remove high-frequency conducted EMI generated by the dc load, and is not intended to modify the ac line current waveform. If L and C are both zero, then the load resistor is connected directly to the output of the diode bridge, and the ac line current waveform is purely sinusoidal. An approximate argument: the L-C filter has negligible effect on the ac line current waveform provided that the filter input impedance Z i has zero phase shift at the second harmonic of the ac line frequency, 2 f L. i g i L L D 4 D 1 + v g Z i C v R D 3 D 2 Fundamentals of Power Electronics 6 Chapter 17: Line-commutated rectifiers
Approximate THD f 0 = 1 2π LC R 0 = L C 50 10 THD=30% THD=10% THD=3% Q = R R 0 f p = 1 2πRC = f 0 Q Q 1 THD=1% THD=0.5% 0.1 1 10 100 f 0 / f L Fundamentals of Power Electronics 7 Chapter 17: Line-commutated rectifiers
Example v g THD = 3.6% i g 10 ms 20 ms 30 ms 40 ms t Typical ac line current and voltage waveforms, near the boundary between continuous and discontinuous modes and with small dc filter capacitor. f 0 /f L = 10, Q = 1 Fundamentals of Power Electronics 8 Chapter 17: Line-commutated rectifiers
17.2 The Three-Phase Bridge Rectifier i a i L L ø a + 3ø ac ø b D 1 D 2 D 3 C V dc load R ø c D 4 D 5 D 6 i a (ωt) i L Line current waveform for infinite L 0 90 180 270 360 i L ωt Fundamentals of Power Electronics 9 Chapter 17: Line-commutated rectifiers
17.2.1 Continuous conduction mode Fourier series: Σ i a = 4 nπ I L sin nπ n = 1,5,7,11,... 2 sin nπ 3 Similar to square wave, but missing triplen harmonics sin nωt i a (ωt) 0 i L 90 180 270 360 i L ωt THD = 31% Distortion factor = 3/π = 95.5% In comparison with single phase case: the missing 60 of current improves the distortion factor from 90% to 95%, because the triplen harmonics are removed Fundamentals of Power Electronics 10 Chapter 17: Line-commutated rectifiers
A typical CCM waveform i a THD = 31.9% v an v bn v cn 10 ms 20 ms 30 ms 40 ms t Inductor current contains sixth harmonic ripple (360 Hz for a 60 Hz ac system). This ripple is superimposed on the ac line current waveform, and influences the fifth and seventh harmonic content of i a. Fundamentals of Power Electronics 11 Chapter 17: Line-commutated rectifiers
17.2.2 Discontinuous conduction mode i a THD = 99.3% v an v bn v cn 10 ms 20 ms 30 ms 40 ms t Phase a current contains pulses at the positive and negative peaks of the line-to-line voltages v ab and v ac. Distortion factor and THD are increased. Distortion factor of the typical waveform illustrated above is 71%. Fundamentals of Power Electronics 12 Chapter 17: Line-commutated rectifiers
17.3 Phase control Replace diodes with SCRs: Phase control waveforms: ø a i a i L + L + i a α i L v an = V m sin (ωt) 3ø ac ø b Q 1 Q 2 Q 3 v d C V dc load R 0 0 90 180 270 i L ωt ø c Q 4 Q 5 Q 6 v bc v ab v ca v bc v ab v ca v d Average (dc) output voltage: V = 3 π = 3 2 π 90 +α 30 +α 3 V m sin(θ + 30 )dθ V L-L, rms cos α Upper thyristor: Lower thyristor: Q 3 Q 1 Q 1 Q 2 Q 2 Q 3 Q 5 Q 5 Q 6 Q 6 Q 4 Q 4 Fundamentals of Power Electronics 13 Chapter 17: Line-commutated rectifiers
Dc output voltage vs. delay angle α V V L L, rms 1.5 1 0.5 Rectification Inversion V = 3 π = 3 2 π 90 +α 30 +α 3 V m sin(θ + 30 )dθ V L-L, rms cos α 0 0.5 1 1.5 0 30 60 90 120 150 180 α, degrees Fundamentals of Power Electronics 14 Chapter 17: Line-commutated rectifiers
17.3.1 Inverter mode L I L + ø a 3ø ac ø b V + ø c If the load is capable of supplying power, then the direction of power flow can be reversed by reversal of the dc output voltage V. The delay angle α must be greater than 90. The current direction is unchanged. Fundamentals of Power Electronics 15 Chapter 17: Line-commutated rectifiers
17.3.2 Harmonics and power factor Fourier series of ac line current waveform, for large dc-side inductance: Σ i a = 4 nπ I L sin nπ n = 1,5,7,11,... 2 sin nπ 3 sin (nωt nα) Same as uncontrolled rectifier case, except that waveform is delayed by the angle α. This causes the current to lag, and decreases the displacement factor. The power factor becomes: power factor = 0.955 cos (α) When the dc output voltage is small, then the delay angle α is close to 90 and the power factor becomes quite small. The rectifier apparently consumes reactive power, as follows: Q = 3 I a, rms V L-L, rms sin α = I L 3 2 π V L-L, rms sin α Fundamentals of Power Electronics 16 Chapter 17: Line-commutated rectifiers
Real and reactive power in controlled rectifier at fundamental frequency Q S sin α P = I L 3 2 π S = I L 3 2 π V L L rms α V L-L, rms cos α S cos α Q = 3 I a, rms V L-L, rms sin α = I L 3 2 π P V L-L, rms sin α Fundamentals of Power Electronics 17 Chapter 17: Line-commutated rectifiers
17.4 Harmonic trap filters A passive filter, having resonant zeroes tuned to the harmonic frequencies Z s i s Z 1 Z 2 Z 3... i r ac source model Harmonic traps (series resonant networks) Rectifier model Fundamentals of Power Electronics 18 Chapter 17: Line-commutated rectifiers
Harmonic trap Ac source: model with Thevenin-equiv voltage source and impedance Z s (s). Filter often contains series inductor sl s. Lump into effective impedance Z s (s): Z s (s)=z s '(s)+sl s ' i s ac source model Z s Z 1 Z 2 Z 3... Harmonic traps (series resonant networks) i r Rectifier model Fundamentals of Power Electronics 19 Chapter 17: Line-commutated rectifiers
Filter transfer function H(s)= i s(s) i R (s) = Z 1 Z 2 Z s + Z 1 Z 2 or H(s)= i s(s) i R (s) = Z s Z 1 Z 2 Z s Z s i s Z 1 Z 2 Z 3... i r ac source model Harmonic traps (series resonant networks) Rectifier model Fundamentals of Power Electronics 20 Chapter 17: Line-commutated rectifiers
Simple example L s Q p R 0p R 1 i s R 1 Z 1 Z s R 0p L s C 1 ωl 1 L 1 i r 1 ωc 1 f p 1 2π L s C 1 f 1 = 1 2π L 1 C 1 C 1 ωl s Z s Z 1 R 01 = L 1 C 1 Fifth-harmonic trap Z 1 Q 1 = R 01 R 1 R 1 Fundamentals of Power Electronics 21 Chapter 17: Line-commutated rectifiers
Simple example: transfer function Series resonance: fifth harmonic trap 1 Q p f p 40 db/decade Parallel resonance: C 1 and L s Parallel resonance tends to increase amplitude of third harmonic Q 1 f 1 L 1 L 1 + L s Q of parallel resonance is larger than Q of series resonance Fundamentals of Power Electronics 22 Chapter 17: Line-commutated rectifiers
Example 2 L s i s R 1 R 2 R 3 L 1 L 2 L 3 i r C 1 C 2 C 3 5 th harmonic trap Z 1 7 th harmonic trap Z 2 11 th harmonic trap Z 3 Fundamentals of Power Electronics 23 Chapter 17: Line-commutated rectifiers
Approximate impedance asymptotes Z s Z 1 Z 2 Z 3 ωl s 1 ωc 1 f 1 ωl 1 1 ωc 2 f 2 ωl 2 1 ωc 3 f 3 ωl 3 Q 1 R 1 Q 2 R 2 Q3 R3 Fundamentals of Power Electronics 24 Chapter 17: Line-commutated rectifiers
Transfer function asymptotes 1 f 1 Q 1 f 2 Q 2 f 3 Q 3 Fundamentals of Power Electronics 25 Chapter 17: Line-commutated rectifiers
Bypass resistor Z 1 Z s ωl s f bp R bp R n R n R bp f p 1 ωc 1 ωl 1 f 1 R bp Z s Z 1 L n L n C b C n C n 1 f p 40 db/decade f 1 f bp 20 db/decade Fundamentals of Power Electronics 26 Chapter 17: Line-commutated rectifiers
Harmonic trap filter with high-frequency roll-off L s R 5 R 7 R bp L 5 L 7 C b C 5 C 7 Fifth-harmonic trap Seventh-harmonic trap with highfrequency rolloff Fundamentals of Power Electronics 27 Chapter 17: Line-commutated rectifiers
17.5 Transformer connections Three-phase transformer connections can be used to shift the phase of the voltages and currents This shifted phase can be used to cancel out the low-order harmonics Three-phase delta-wye transformer connection shifts phase by 30 : ωt a 3 : n a' a a' T 1 30 T 1 b T 1 T 2 T 3 n' b' T 1 T 2 c' T 3 n' c T 3 T 2 c' c T 3 b T 2 b' Primary voltages Secondary voltages Fundamentals of Power Electronics 28 Chapter 17: Line-commutated rectifiers
Twelve-pulse rectifier a i a i a1 1:n L I L T 1 + 3øac source b c T 1 T 2 T 3 T 3 T 2 n' i a2 3 : n v d dc load T 4 T 4 T 5 T 6 n' T 6 T 5 Fundamentals of Power Electronics 29 Chapter 17: Line-commutated rectifiers
Waveforms of 12 pulse rectifier i a1 i a2 ni L ωt 90 180 270 360 ni L Ac line current contains 1st, 11th, 13th, 23rd, 25th, etc. These harmonic amplitudes vary as 1/n 5th, 7th, 17th, 19th, etc. harmonics are eliminated i a ni L 1+ 2 3 3 ni L 3 ni L 1+ 3 3 Fundamentals of Power Electronics 30 Chapter 17: Line-commutated rectifiers
Rectifiers with high pulse number Eighteen-pulse rectifier: Use three six-pulse rectifiers Transformer connections shift phase by 0, +20, and 20 No 5th, 7th, 11th, 13th harmonics Twenty-four-pulse rectifier Use four six-pulse rectifiers Transformer connections shift phase by 0, 15, 15, and 30 No 5th, 7th, 11th, 13th, 17th, or 19th harmonics If p is pulse number, then rectifier produces line current harmonics of number n = pk ± 1, with k = 0, 1, 2,... Fundamentals of Power Electronics 31 Chapter 17: Line-commutated rectifiers