Power Electroni cs Chapter 2 AC to DC Converters (Rectifiers) Power E l e ct r o n i cs 2 Outline 2.1 Single-phase controlled rectifier 2.2 Three-phase controlled rectifier 2.3 Effect of transformer leakage inductance on rectifier circuits 2.4 Capacitor-filtered uncontrolled rectifier 2.5 Harmonics and power factor of rectifier circuits 2.6 High power controlled rectifier 2.7 Inverter mode operation of rectifier circuit 2.8 Thyristor-DC motor system 2.9 Realization of phase-control in rectifier circuits Power E l e ct r o n i cs 3 2.1 Single-phase controlled (controllable) rectifier 2.1.1 Single-phase half-wave controlled rectifier 2.1.2 Single-phase bridge fully-controlled rectifier 2.1.3 Single-phase full-wave controlled rectifier 2.1.4 Single-phase bridge half-controlled rectifier Power E l e ct r o n i cs 4 2.1.1 Single-phase half-wave controlled rectifier Resistive load T VT R u 1 u 2 u VT u d i d a) 0 ωt 1 π 2π ωt ωt ωt ωt u 2 u g u d u VT α θ 0 b) c) d) e) 0 0 ∫ + =+== π α α α π ωω π 2 cos1 45.0)cos1( 2 2 )(sin2 2 1 2 2 2d U U ttdUU (2-1) Half-wave, single-pulse Triggering delay angle, delay angle, firing angle Power E l e ct r o n i cs 5 2.1.1 Single-phase half-wave controlled rectifier Inductive (resistor-inductor) load u 2 0 ωt 1 π 2π ωt ωt ωt ωt ωt u g 0 u d 0 i d 0 u VT 0 θ α b) c) d) e) f) a) u 1 T VT u 2 u VT u d i d Power E l e ct r o n i cs 6 Basic thought process of time-domain analysis for power electronic circuits The time-domain behavior of a power electronic circuit is actually the combination of consecutive transients of the different linear circuits when the power semiconductor devices are in different states. a) b) VT R L VT R L u 2 u 2 tURi t i L ωsin2 d d 2d d =+ (2-2) < t = αi d = 0 )sin( 2 )sin( 2 2 )( 2 d ?ω?α αω ω ?+??= ?? t Z U e Z U i t L R (2-3) Power E l e ct r o n i cs 7 Single-phase half-wave controlled rectifier with freewheeling diode Maximum forward voltage, maximum reverse voltage Disadvantages: – Only single pulse in one line cycle – DC component in the transformer current Inductive load (L is large enough) VT i a) T u 1 u 2 u VT L R d u d VD i R VD R u 2 u d i d u VT i VT I d I d ωt 1 ωt ωt ωt ωt ωt ωtO O O O O O πα π+α b) c) d) e) f) g) i VD R ddVT 2 II π απ ? = d 2 dVT 2 )( 2 1 ItdII π απ ω π π α ? == ∫ ddVD 2 R II π απ + = d 2 2 dVD 2 )( 2 1 R ItdII π απ ω π απ π + == ∫ + (2-5) (2-6) (2-7) (2-8) Power E l e ct r o n i cs 8 2.1.2 Single-phase bridge fully-controlled rectifier π ωt ωt ωt0 0 0 i 2 u d i d b) c) d) u d (i d ) αα u VT 1,4 Resistive load d R T u 1 u 2 i 2 a b VT 1 VT 3 VT 2 VT 4 u d i a) For thyristor: maximum forward voltage, maximum reverse voltage Advantages: – 2 pulses in one line cycle – No DC component in the transformer current Power E l e ct r o n i cs 9 2.1.2 Single-phase bridge fully- controlled rectifier Resistive load Average output (rectified) voltage (2-9) Average output current (2-10) For thyristor (2-11) (2-12) For transformer (2-13) ∫ + = + == π α αα π ωω π 2 cos1 9.0 2 cos122 )(dsin2 1 2 2 2d U U ttUU 2 cos1 9.0 2 cos122 22d d αα π + = + == R U R U R U I 2 cos1 45.0 2 1 2 ddVT α+ == R U II π απ α π ωω π π α ? +== ∫ 2sin 2 1 2 )(d)sin 2 ( 2 1 2 2 2 VT R U tt R U I π απ α π ωω π π α ? +=== ∫ 2sin 2 1 )()sin 2 ( 1 2 2 2 2 R U tdt R U II Power E l e ct r o n i cs 10 2.1.2 Single-phase bridge fully-controlled rectifier Inductive load (L is large enough) T B b R L a) u 1 u 2 i 2 VT 1 VT 3 VT 2 VT 4 u d i d u 2 O ωt O ωt O ωt u d i d i 2 b) O ωt O ωt u VT 1,4 O ωt O ωt I d I d I d I d I d i VT 2,3 i VT 1,4 ∫ + === απ α αα π ωω π cos9.0cos 22 )(dsin2 1 222d UUttUU (2-15) Commutation Thyristor voltages and currents Transformer current Power E l e ct r o n i cs 11 Electro-motive-force (EMF) load With resistor a) b) R E i d u d i d O E u d ωt I d O ωt α θ δ Discontinuous current i d Power E l e ct r o n i cs 12 Electro-motive-force (EMF) load With resistor and inductor When L is large enough, the output voltage and current waveforms are the same as ordinary inductive load. When L is at a critical value O u d  E i d ωt ωt π δ α θ =π dmin 2 3 dmin 2 1087.2 22 I U I U L ? ×== πω (2-17) Power E l e ct r o n i cs 13 2.1.3 Single-phase full-wave controlled rectifier a) b) u 1 T R u 2 u 2 i 1 VT 1 VT 2 u d u d i 1 O O α ωt ωt Transformer with center tap Comparison with single-phase bridge fully-controlled rectifier Power E l e ct r o n i cs 14 2.1.4 Single-phase bridge half-controlled rectifier Half-control Comparison with fully-controlled rectifier Additional freewheeling diode O b) u 2 O u d i d I d O O O O O i 2 I d I d I d I d I d α ωt ωt ωt ωt ωt ωt ωt α π?α π?α i VT 1 i VD 4 i VT 2i VD 3 i VD R a b R L u 2 i 2 u d i d VT 1 VT 2 VD 3 VD 4 VD R T Power E l e ct r o n i cs 15 Another single-phase bridge half-controlled rectifier T u 2 VD 3 VD 4 VT 1 VT 2 load Comparison with previous circuit: – No need for additional freewheeling diode – Isolation is necessary between the drive circuits of the two thyristors Power E l e ct r o n i cs 16 Summary of some important points in analysis When analyzing a thyristor circuit, start from a diode circuit with the same topology. The behavior of the diode circuit is exactly the same as the thyristor circuit when firing angle is 0. A power electronic circuit can be considered as different linear circuits when the power semiconductor devices are in different states. The time-domain behavior of the power electronic circuit is actually the combination of consecutive transients of the different linear circuits. Take different principle when dealing with different load – For resistive load: current waveform of a resistor is the same as the voltage waveform – For inductive load with a large inductor: the inductor current can be considered constant Power E l e ct r o n i cs 17 2.2 Three-phase controlled (controllable) rectifier 2.2.1 Three-phase half-wave controlled rectifier (the basic circuit among three-phase rectifiers) 2.2.2 Three-phase bridge fully-controlled rectifier (the most widely used circuit among three-phase rectifiers) Power E l e ct r o n i cs 18 2.2.1 Three-phase half-wave controlled rectifier Resistive load, α = 0o u 2 u a u b u c O ωt 1 ωt 2 ωt 3 u G O u d O O u ab u ac O i VT 1 u VT 1 ωt ωt ωt ωt ωt T R u d i d VT 2 VT 1 VT 3 Common-cathode connection Natural commutation point Power E l e ct r o n i cs 19 Resistive load, α = 30o u 2 u a u b u c O ωt O ωt O ωt O ωt O ωt u G u d u ab u ac ωt 1 i VT 1 u VT 1 u ac T R u d i d VT 2 VT 1 VT 3 Power E l e ct r o n i cs 20 Resistive load, α = 60o T R u d i d VT 2 VT 1 VT 3 ωt ωt ωt ωt u 2 u a u b u c O O O O u G u d i VT 1 Power E l e ct r o n i cs 21 Resistive load, quantitative analysis ? When α≤ 30o, load current i d is continuous. ? When α > 30o, load current i d is discontinuous. αα π ωω π α π α π cos17.1cos 2 63 )(sin2 3 2 1 22 6 5 6 2d UUttdUU === ∫ + + ? ? ? ? ? ? ++= ? ? ? ? ? ? ++== ∫ + ) 6 cos(1675.0) 6 cos(1 2 23 )(sin2 3 2 1 2 6 2d α π α π π ωω π π α π UttdUU (2-18) (2-19) Average load current Thyristor voltages R U I d d = (2-20) 0 30 60 90 120 150 0.4 0.8 1.2 1.17 3 2 1 α/(￠) U d  U 2 1- resistor load 2- inductor load 3- resistor-inductor load Power E l e ct r o n i cs 22 Inductive load, L is large enough Load current i d is always continuous. Thyristor voltage and currents, transformer current a b c T R L u 2 u d e L i d VT 1 VT 2 VT 3 u d i a u a u b u c i b i c i d u ac u ab u ac O ωt O ωt O ωt O ωt O ωt O ωt α u VT 1 αα π ωω π α π α π cos17.1cos 2 63 )(sin2 3 2 1 22 6 5 6 2d UUttdUU === ∫ + + (2-18) ddVT2 577.0 1 IIII === (2-23) d VT VT(AV) 368.0 57.1 I I I == (2-24) 3 2RMFM 45.2 UUU == (2-25) Power E l e ct r o n i cs 23 2.2.2 Three-phase bridge fully-controlled rectifier Circuit diagram b a c i d u d VT 1 VT 3 VT 5 VT 4 VT 6 VT 2 d 2 d 1 T n i a load Common-cathode group and common-anode group of thyristors Numbering of the 6 thyristors indicates the trigger sequence. Power E l e ct r o n i cs 24 Resistive load, α = 0o b a T n load i a i d u d VT 1 VT 3 VT 5 VT 4 VT 6 VT 2 d 2 d 1 u 2 u d1 u d2 u 2L u d u ab u ac u ab u ac u bc u ba u ca u cb u ab u ac u ab u ac u bc u ba u ca u cb u ab u ac ú?üyt? u a u c u b ωt 1 O ωt O ωt O ωt O ωt α = 0° i VT 1 u VT 1 Power E l e ct r o n i cs 25 Resistive load, α = 30o b a T n load i a i d u d VT 1 VT 3 VT 5 VT 4 VT 6 VT 2 d 2 d 1 u d1 u d2 α = 30° i a O ωt O ωt O ωt O ωt u d u ab u ac u a u b u c ωt 1 u ab u ac u bc u ba u ca u cb u ab u ac ú?üyt? u ab u ac u bc u ba u ca u cb u ab u ac u VT 1 Power E l e ct r o n i cs 26 Resistive load, α = 60o b a T n load i a i d u d VT 1 VT 3 VT 5 VT 4 VT 6 VT 2 d 2 d 1 α = 60° u d1 u d2 u d u ac u ac u ab u ab u ac u bc u ba u ca u cb u ab u ac u a ú?üyt? u b u c O ωt ωt 1 O ωt O ωt u VT 1 Power E l e ct r o n i cs 27 Resistive load, α = 90o u d1 u d2 u d u a u b u c u a u b ωtO ωtO ωtO ωtO ωt O i a i d u ab u ac u bc u ba u ca u cb u ab u ac u bc u ba i VT 1 b a T n load i a i d u d VT 1 VT 3 VT 5 VT 4 VT 6 VT 2 d 2 d 1 Power E l e ct r o n i cs 28 Inductive load, α = 0o b a T n load i a i d u d VT 1 VT 3 VT 5 VT 4 VT 6 VT 2 d 2 d 1 u d1 u 2 u d2 u 2L u d i d ωtO ωtO ωtO ωtO u a α = 0° u b u c ωt 1 u ab u ac u bc u ba u ca u cb u ab u ac ú?üyt? i VT 1 Power E l e ct r o n i cs 29 Inductive load, α = 30o b a T n load i a i d u d VT 1 VT 3 VT 5 VT 4 VT 6 VT 2 d 2 d 1 u d1 α = 30° u d2 u d u ab u ac u bc u ba u ca u cb u ab u ac ú?üyt? ωtO ωtO ωtO ωtO i d i a ωt 1 u a u b u c Power E l e ct r o n i cs 30 Inductive load, α = 90o b a T n load i a i d u d VT 1 VT 3 VT 5 VT 4 VT 6 VT 2 d 2 d 1 α = 90° u d1 u d2 u ac u bc u ba u ca u cb u ab u ac u ab ú?üyt? u d u ac u ab u ac ωtO ωtO ωtO u b u c u a ωt 1 u VT 1 Power E l e ct r o n i cs Average output voltage For resistive load, When a > 60o, load current i d is discontinuous. Average output current (load current) Transformer current Thyristor voltage and current – Same as three-phase half-wave rectifier EMF load, L is large enough – All the same as inductive load except the calculation of average output current αωω π α π α π cos34.2)(sin6 3 1 2 3 2 3 2d UttdUU == ∫ + + (2-26) ? ? ? ? ? ? ++== ∫ + ) 3 cos(134.2)(sin6 3 2 3 2d α π ωω π π α π UttdUU (2-20) (2-27) dd 2 d 2 d2 816.0 3 2 3 2 )( 3 2 2 1 IIIII == ? ? ? ? ? ? ×?+×= π ππ π (2-28) R EU I ? = d d (2-29) R U I d d = Quantitative analysis 31 Power E l e ct r o n i cs 32 2.3 Effect of transformer leakage inductance on rectifier circuits In practical, the transformer leakage inductance has to be taken into account. Commutation between thyristors thus can not happen instantly, but with a commutation process. R a b c T L u d i c i b i a L B L B L B i k VT 1 VT 2 VT 3 u d i d ωtO ωtO γ i c i a i b i c i a I d u a u b u c α Power E l e ct r o n i cs 33 Commutation process analysis Circulating current i k during commutation Commutation angle Output voltage during commutation u b -u a = 2·L B ·di a /dt i k : 0 I d i a = I d -i k : I d 0 i b = i k : 0 I d 2d d d d bak Bb k Bad uu t i Lu t i Luu + =?=+= (2-30) Power E l e ct r o n i cs 34 Quantitative calculation Reduction of average output voltage due to the commutation process Calculation of commutation angle – I d ,Χ – X B ,Χ – For Α 90 o , Α , Χ dB 0 B 6 5 6 5 B 6 5 6 5 Bbb 6 5 6 5 dbd 2 3 d 2 3 )(d d d 2 3 )(d)] d d ([ 2 3 )(d)( 3/2 1 IXiLt t i L t t i LuutuuU I π ω π ω π ω π ω π π γα π α π γα π α π γα π α === ??=?=? ∫∫ ∫∫ ++ + ++ + ++ + d k k k (2-31) 2 dB 6 2 )cos(cos U IX =+? γαα (2-36) ≤ Power E l e ct r o n i cs 35 Summary of the effect on rectifier circuits d U? d B I X π d B 2 I X π d B 2 3 I X π d B 3 I X π d B 2 I mX π )cos(cos γαα +? 2 Bd 2U XI 2 Bd 2 2 U XI 2 dB 6 2 U IX 2 dB 6 2 U IX m U XI π sin2 2 Bd Single- phase full wave Single- phase bridge Three- phase half- wave Three- phase bridge m-pulse recfifier ? ? Circuits Conclusions – Commutation process actually provides additional working states of the circuit. – di/dt of the thyristor current is reduced. – The average output voltage is reduced. – Positive du/dt – Notching in the AC side voltage Power E l e ct r o n i cs 36 2.4 Capacitor-filtered uncontrolled (uncontrollable) rectifier Emphasis of previous sections – Controlled rectifier, inductive load Uncontrolled rectifier: diodes instead of thyristors Wide applications of capacitor-filtered uncontrolled rectifier – AC-DC-AC frequency converter – Uninterruptible power supply – Switching power supply 2.4.1 Capacitor-filtered single-phase uncontrolled rectifier 2.4.2 Capacitor-filtered three-phase uncontrolled rectifier Power E l e ct r o n i cs 37 2.4.1 Capacitor-filtered single-phase uncontrolled rectifier Single-phase bridge, RC load a) + R C u 1 u 2 i 2 VD 1 VD 3 VD 2 VD 4 i d i C i R u d b) 0 i u d θ δ π 2π ωt i,u d Power E l e ct r o n i cs 38 2.4.1 Capacitor-filtered single-phase uncontrolled rectifier Single-phase bridge, RLC load a) b)  + R C L + u 1 u 2 i 2 u d u L i d i C i R VD 2 VD 4 VD 1 VD 3 u 2 u d i 2 0 δθ π ωt i 2 ,u 2 ,u d Power E l e ct r o n i cs 39 2.4.2 Capacitor-filtered three-phase uncontrolled rectifier Three-phase bridge, RC load a) + a b c T i a R C u d i d i C i R VD 4 VD 6 VD 1 VD 3 VD 5 VD 2 b) O i a u d i d u d u ab u ac 0δθ ωt π π 3 ωt Power E l e ct r o n i cs 40 2.4.2 Capacitor-filtered three-phase uncontrolled rectifier Three-phase bridge, RC load Waveform when ωRC≤1.732 ωt ωt ωt ωt i a i d i a i d O O O O aωRC=``bωRC< 3 3 Power E l e ct r o n i cs 41 2.4.2 Capacitor-filtered three-phase uncontrolled rectifier Three-phase bridge, RLC load a) b) c) + a b c T i a R C u d i d i C i R VD 4 VD 6 VD 1 VD 3 VD 5 VD 2 i a i a O O ωt ωt Power E l e ct r o n i cs 42 2.5 Harmonics and power factor of ` ``rectifier circuits Originating of harmonics and power factor issues in rectifier circuits – Harmonics: working in switching states—nonlinear – Power factor: firing delay angle causes phase delay Harmful effects of harmonics and low power factor Standards to limit harmonics and power factor 2.5.1 Basic concepts of harmonics and reactive power 2.5.2 AC side harmonics and power factor of controlled rectifiers with inductive load 2.5.3 AC side harmonics and power factor of capacitor-filtered uncontrolled rectifiers 2.5.4 Harmonic analysis of output voltage and current Power E l e ct r o n i cs 43 2.5.1 Basic concepts of harmonics and reactive power For pure sinusoidal waveform For periodic non-sinusoidal waveform or where Fundamental component Harmonic components (harmonics) ∑ ∞ = ++= 1 )sincos()( n nno tnbtnaatu ωωω (2-54) ( ) 2 sin( )uut U tω ?=+ (2-55) ∑ ∞ = ++= 1 )sin()( n nno tncatu ?ωω (2-56) 22 nnn bac += )/arctan( nnn ba=? ?sinnn ca = ?cosnn cb = Power E l e ct r o n i cs 44 Harmonics-related specifications Take current harmonics as examples Content of nth harmonics I n is the effective (RMS) value of nth harmonics. I 1 is the effective (RMS) value of fundamental component. Total harmonic distortion I h is the total effective (RMS) value of all the harmonic components. %100 1 ×= I I HRI n n (2-57) %100 1 ×= I I THD h i (2-58) Power E l e ct r o n i cs 45 Definition of power and power factor For sinusoidal circuits Active power Reactive power Q=U I sin? Apparent power S=UI Power factor λ =cos ? ∫ == π ?ω π 2 0 cos)( 2 1 UItuidP 222 QPS += S P =λ (2-59) (2-60) (2-61) (2-63) (2-62) (2-64) Power E l e ct r o n i cs 46 Definition of power and power factor For non-sinusoidal circuits Active power P=U I 1 cos? 1 Power factor Distortion factor (fundamental-component factor) ν =I 1 / I Displacement factor (power factor of fundamental component) λ 1 = cos? 1 Definition of reactive power is still in dispute. 11 111 coscos cos ?ν? ? λ ==== I I UI UI S P (2-65) (2-66) 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. Power E l e ct r o n i cs 47 Review of the reactive power concept Power E l e ct r o n i cs 48 2.5.2 AC side harmonics and power factor of controlled rectifiers with inductive load Single-phase bridge fully-controlled rectifier u 2 O ωt O ωt O ωt u d i d i 2 b) O ωt O ωt u VT 1,4 O ωt O ωt I d I d I d I d I d i VT 2,3 i VT 1,4 T B b R L a) u 1 u 2 i 2 VT 1 VT 3 VT 2 VT 4 u d i d Power E l e ct r o n i cs 49 AC side current harmonics of single-phase bridge fully-controlled rectifier with inductive load ∑∑ == == +++= "" " ,5,3,1,5,3,1 d d2 sin2sin 14 )5sin 5 1 3sin 3 1 (sin 4 n n n tnItn n I tttIi ωω π ωωω π (2-72) where n=1,3,5,… πn I I n d 22 = (2-73) Conclusions – Only odd order harmonics exist – I n ∝ 1/n – I n / I 1 = 1/n Power E l e ct r o n i cs 50 Power factor of single-phase bridge fully- controlled rectifier with inductive load Distortion factor Displacement factor Power factor ν π == ≈ I I 1 22 09. α?λ coscos 11 == αα π ?νλλ cos9.0cos 22 cos 1 1 1 ≈=== I I (2-75) (2-76) (2-77) Power E l e ct r o n i cs 51 Three-phase bridge fully-controlled rectifier b a T n load i a i d u d VT 1 VT 3 VT 5 VT 4 VT 6 VT 2 d 2 d 1 u d1 α = 30° u d2 u d u ab u ac u bc u ba u ca u cb u ab u ac ú?üyt? ωtO ωtO ωtO ωtO i d i a ωt 1 u a u b u c Power E l e ct r o n i cs 52 AC side current harmonics of three-phase bridge fully-controlled rectifier with inductive load Conclusions – Only 6k±1 order harmonics exist (k is positive integer) – I n ∝ 1/n – I n / I 1 = 1/n ∑∑ = ±= = ±= ?+=?+= ?++??= "" " 3,2,1 16 1 3,2,1 16 dd da sin2)1(sin2sin 1 )1( 32 sin 32 ]13sin 13 1 11sin 11 1 7sin 7 1 5sin 5 1 [sin 32 k kn n k k kn k tnItItn n ItI tttttIi ωωω π ω π ωωωωω π (2-79) 1 6 6 , 6 1, 1, 2, 3, n II IInkk n π π ? = ? ? ? ? ==±= ? ? d d "`` where (2-80) Power E l e ct r o n i cs 53 Power factor of three-phase bridge fully-controlled rectifier with inductive load Distortion factor Displacement factor Power factor 955.0 3 1 ≈== π ν I I α?λ coscos 11 == αα π ?νλλ cos955.0cos 3 cos 1 1 1 ≈=== I I (2-81) (2-82) (2-83) Power E l e ct r o n i cs 54 2.5.3 AC side harmonics and power factor of capacitor-filtered uncontrolled rectifiers Situation is a little complicated than rectifiers with inductive load. Some conclusions that are easy to remember: – Only odd order harmonics exist in single-phase circuit, and only 6k±1 (k is positive integer) order harmonics exist in three-phase circuit. – Magnitude of harmonics decreases as harmonic order increases. – Harmonics increases and power factor decreases as capacitor increases. – Harmonics decreases and power factor increases as inductor increases. Power E l e ct r o n i cs 55 2.5.4 Harmonic analysis of output voltage and current ? ? ? ? ? ? ? ?= += ∑ ∑ ∞ = ∞ = tn n k U tnbUu mkn mkn n ω π ω cos 1 cos2 1 cos 2 d0 d0d0 u d ωtO π m π m 2π m U 2 2 (2-85) where m m UU π π sin2 2d0 = (2-86) d0 2 1 cos2 U n k b n ? ?= π Output voltage of m-pulse rectifier when α = 0o (2-87) Power E l e ct r o n i cs 56 Ripple factor in the output voltage Output voltage ripple factor where U R is the total RMS value of all the harmonic components in the output voltage and U is the total RMS value of the output voltage Ripple factors for rectifiers with different number of pulses d0 R U U u =γ 2 d0 22 R UUUU mkn n ?== ∑ ∞ = (2-88) (2-89) m 2 3 6 12 ? γ u  % 48.2 18.27 4.18 0.994 0 Power E l e ct r o n i cs 57 Harmonics in the output current )cos( dd n mkn n tndIi ?ω ?+= ∑ ∞ = (2-92) where R EU I ? = d0 d (2-93) 22 )( LnR b z b d n n n n ω+ == (2-94) R Ln n ω ? arctan= (2-95) Power E l e ct r o n i cs 58 Conclusions for α = 0o Only mk (k is positive integer) order harmonics exist in the output voltage and current of m-pulse rectifiers Magnitude of harmonics decreases as harmonic order increases when m is constant. The order number of the lowest harmonics increases as m increases. The corresponding magnitude of the lowest harmonics decreases accordingly. Power E l e ct r o n i cs 59 For α ≠ 0o Quantitative harmonic analysis of output voltage and current is very complicated for α ≠ 0o. As an example, for 3-phase bridge fully-controlled rectifier 0 30 120 150 18060 0.1 0.2 0.3 90 n=6 n=12 n=18 α/(°) U 2L c n 2 ∑ ∞ = ++= kn n n d tncUu 6 )cos( θω d (2-96) Power E l e ct r o n i cs 60 2.6 High power controlled rectifier 2.6.1 Double-star controlled rectifier 2.6.2 Connection of multiple rectifiers Power E l e ct r o n i cs 61 2.6.1 Double-star controlled rectifier Circuit Waveforms When α= 0o T abc L R n i P L P u d i d VT 2 VT 6 VT 4 VT 1 VT 3 VT 5 c ' a ' b ' n 1 n 2 u d1 u a u b u c i a u d2 i a ' u c ' u a ' u b ' u c ' O ωt O ωt O ωt O ωt I d 1 2 I d 1 6 I d 1 2 I d 1 6 Difference from 6-phase half-wave rectifier Power E l e ct r o n i cs 62 Effect of interphase reactor (inductor, transformer) n L R   u d1 L P u b ' u d2 u d n 2 n 1 i P u a VT 1 VT 6 u P 1 2 u p u d1 ,u d2 O O 60 360 ωt 1 ωt ωt b) a) u a u b u c u c ' u a ' u b ' u b ' b b d1d2p uuu ?= )( 2 1 2 1 2 1 d2d1pd1pd2d uuUuuuu +=+=?= (2-98) (2-97) Power E l e ct r o n i cs 63 Quantitative analysis when α = 0o ]9cos 40 1 6cos 35 2 3cos 4 1 1[ 2 63 2 d1 ????+?+= ttt U u ωωω π (2-99) ]9cos 40 1 6cos 35 2 3cos 4 1 1[ 2 63 ])60(9cos 40 1 )60(6cos 35 2 )60(3cos 4 1 1[ 2 63 2 2 d2 ??????= ????°?+°??°?+= ttt U ttt U u ωωω π ωωω π (2-100) ]9cos 20 1 3cos 2 1 [ 2 63 2 p ??????= tt U u ωω π (2-101) ]6cos 35 2 1[ 2 63 2 d ?????= t U u ω π (2-102) Power E l e ct r o n i cs 64 Waveforms when α > 0o b = α b = α b = α u d u d u d ωt O ωtO ωtO u a u b u c u c ' u a ' u b ' u b u c u c ' u a ' u b ' u b u c u c ' u a ' u b ' αcos17.1 2UUd = Power E l e ct r o n i cs 65 Comparison with 3-phase half-wave rectifier and 3-phase bridge rectifier Voltage output capability – Same as 3-phase half-wave rectifier – Half of 3-phase bridge rectifier Current output capability – Twice of 3-phase half-wave rectifier – Twice of 3-phase bridge rectifier Applications Low voltage and high current situations E l e ct r o n i cs Power To increase the output capacity To improve the AC side current waveform and DC side voltage waveform Larger output voltage: series connection Larger output voltage: parallel connection 2.6.2 Connection of multiple rectifiers Connection of multiple rectifiers Power E l e ct r o n i cs 67 Phase-shift connection of multiple rectifiers Parallel connection M L T VT 1 2 c 1 b 1 a 1 c 2 b 2 a 2 L P 12-pulse rectifier realized by paralleled 3-phase bridge rectifiers Power E l e ct r o n i cs 68 Phase-shift connection of multiple rectifiers Series connection C · L RB A  * · · * * 0 30￠MBHHJOH 3 i A c 1 b 1 a 1  c 2 b 2 a 2 i ab2 u a2b2 u a1b1 i a1 i d u d I II I II 0 a) b) c) d) i a1 I d 180° 360° i a2 i ab2 ' i A I d i ab2 ωt ωt ωt ωt 0 0 0 I d 2 3 3 3 I d 3 3 I d I d3 2 3   I d3 2 3  I d3 3 I d 1 3 12-pulse rectifier realized by series 3-phase bridge rectifiers Power E l e ct r o n i cs 69 Quantitative analysis of 12-pulse rectifier Volgate – Average output voltage Parallel connection: Series connection: – Output voltage harmonics Only 12m harmonics exist Input (AC side) current harmonics – Only 12k±1 harmonics exist Connection of more 3-phase bridge rectifiers – Three: 18-pulse rectifier (20o phase difference) – Four: 24-pulse rectifier (15o phase difference) αcos34.2 2 UU = d 4.6782cosU α= d Power E l e ct r o n i cs 70 Sequential control of multiple series-connected rectifiers L i a) ú ? ü u 2 u 2 u 2 I d VT 11 VT 13 VT 14 VT 12 VT 21 VT 23 VT 24 VT 22 VT 31 VT 33 VT 34 VT 32 u d b) c) i I d 2I d u d O α π+α load Circuit and waveforms of series-connected three single-phase bridge rectifiers Power E l e ct r o n i cs 71 2.7 Inverter mode operation of rectifiers Review of DC generator-motor system c) b)a) MG MG MG E G E M I d R ∑ E G E M I d R ∑ E G E M I d R ∑ ∑ ? = R EE I MG d ∑ ? = R EE I GM d should be avoided Power E l e ct r o n i cs 72 Inverter mode operation of rectifiers Rectifier and inverter mode operation of single-phase full-wave converter ∑ ? = R EU I Gd d ∑ ? = R UE I dM d a) b) R +  Energy M 1 0 2 u 10 u 20 u d i d L VT 1 VT 2 u 10 u d u 20 u 10 α O O ωt ωt I d i d U d >E M E M M R +  1 0 2 u d i d L VT 1 VT 2 u 10 u d u 20 u 10 O O ωt ωt I d i d U d <E M α E M i VT 1 i VT 2 i VT 1 i VT 2 i VT 1 i VT 2 i VT 2 i d =i VT +i VT 1 2 i d =i VT +i VT 1 2 i VT 1 i VT 2 i VT 1 Energy Power E l e ct r o n i cs 73 Necessary conditions for the inverter mode operation of controlled rectifiers There must be DC EMF in the load and the direction of the DC EMF must be enabling current flow in thyristors. (In other word E M must be negative if taking the ordinary output voltage direction as positive.) α > 90o so that the output voltage U d is also negative. dM UE > Power E l e ct r o n i cs 74 Inverter mode operation of 3-phase bridge rectifier u ab u ac u bc u ba u ca u cb u ab u ac u bc u ba u ca u cb u ab u ac u bc u ba u ca u cb u ab u ac u bc u a u b u c u a u b u c u a u b u c u a u b u 2 u d ωtO ωtO β = π 4 β = π 3 β = π 6 β = π 4 β = π 3 β = π 6 ωt 1 ωt 3 ωt 2 Inversion angle (extinction angle) β α + β =180o Power E l e ct r o n i cs 75 Inversion failure and minimum inversion angle Possible reasons of inversion failures – Malfunction of triggering circuit – Failure in thyristors – Sudden dropout of AC source voltage – Insufficient margin for commutation of thyristors Minimum inversion angle (extinction angle) β min =δ +γ+θ￡   2-109 L a b c +  M u d i d E M L B L B L B VT 1 VT 2 VT 3 o u d O O i d ωt ωt u a u b u c u a u b p β γ β <γ α γ β β >γ i VT 1 i VT 2 i VT 3 i VT 1 i VT 2 i VT 3 i VT 1 i VT 3 Power E l e ct r o n i cs 76 2.8 Thyristor-DC motor system 2.8.1 Rectifier mode of operation 2.8.2 Inverter mode of operation 2.8.3 Reversible DC motor drive system (four-quadrant operation) Power E l e ct r o n i cs 77 2.8.1 Rectifier mode of operation Waveforms and equations u d O i d ωt u a u b u c α u d O i a i b i c i c ωt E U d i d R UIREU dMd ?++= ∑ (2-112) π2 3 B MB X RRR ++=∑ where (for 3-phase half-wave) Waveforms of 3-phase half-wave rectifier with DC motor load Power E l e ct r o n i cs 78 Speed-torque (mechanic) characteristic when load current is continuous nCE eM = (2-113) For 3-phase half-wave αcos17.1 2UUd = UIRUE dM ???= ∑αcos17.1 2 e d e C UIR C U n ?+ ?= ∑αcos17.1 2 (2-115) O n a 1 <a 2 <a 3 a 3 a 2 a 1 I d (R B +R M + ) I d C e 3X B 2π (2-114) For 3-phase bridge For 3-phase half-wave d ee I C R C U n ∑ ?= αcos24.2 2 (2-116) Power E l e ct r o n i cs 79 Speed-torque (mechanic) characteristic when load current is discontinuous EMF at no load (taking 3-phase half-wave as example) For α≤ 60o For α> 60o discontinuouts mode continuous mode E E 0 E 0 ' O I dmin I d (0.585U 2 ) ( U 2 ) 2 )60cos(2 20 D ?= αUE 20 2UE = For 3-phase half-wave Power E l e ct r o n i cs 80 Speed-torque (mechanic) characteristic when load current is discontinuous For different α The point of EMF at no load is raised up. The droop rate becomes steer. (softer than the continuous mode) O a 3 a 2 a 1 I d boundary discontinuous mode DPOUJOVPVTNPEF a 5 a 4 E 0 E For 3-phase half-wave (α 1 < α 2 < α 3 ≤ 60o, α 5 > α 4 > 60o) Power E l e ct r o n i cs 81 2.8.2 Inverter mode of operation Equations – are just the same as in the rectifier mode of operation except that U d , E M and n become negative. E.g., in 3-phase half-wave – Or in another form UIRUE dM ???= ∑αcos17.1 2 (2-114) e d e C UIR C U n ?+ ?= ∑αcos17.1 2 (2-115) (2-122) (2-123) eC n ?= 1 ∑ + RIU dd β cos 0 )0 cos( ∑ +?= RIUE d dM β rectifier mode n α 3 α 2 α 1 I d α 4 β 2 β 3 β 4 β 1 α =β = π 2 α in creasin g β in creasin g inverter mode Speed-torque characteristic of a DC motor fed by a thyristor rectifier circuit Power E l e ct r o n i cs 82 2.8.3 Reversible DC motor drive system (4-quadrant operation) L converter 1 converter 2 E M M a b c Back-to-back connection of two 3- phase bridge circuits +  +  +  +  +  +  +  +  AC source converter2 converter2converter2 converter 1 converter2 +TT reverse motoring converter 2 inverting forward braking(regenerating) +n I d I d U dα M E M I d M E M ME M I d M E M n U dβ U dα U dβ O AC source AC source AC source Energy Energy Energy Energy converter 1 converter1 converter1 converter 1 rectifying forward motoring converter 2 rectifying converter 1 inverting reverse braking(regenerating) Power E l e ct r o n i cs 83 4-quadrant speed-torque characteristic of Reversible DC motor drive system converter 1converter 2 n α 3 α 2 α 1 I d α 4 β 2 β 3 β 4 β 1 α =β = π 2 α '=β '= π 2 β' 3 β' 2 β' 1 β' 4 α' 2 α' 3 α' 4 α' 1 α 1 =β ' 1 ; α ' 1 =β 1 α 2 =β ' 2 ; α ' 2 =β 2 α in cr e a s i n g ' β in creasing ' α increas i ng β i n cr easi n g