Chapter THREE-PHASE DIODE BRIDGE RECTIFIER The subject of this book is reduction of total haronic distortion (THD) of input currents in three-phase diode bridge rectifiers. Besides the reduction of the input current THD, the ethods proposed here result in iproeent of the rectifier power factor (PF). To build a foundation to introduce the new ethods, in this chapter a three-phase diode bridge rectifier is analyzed and releant oltage waefors are presented and their spectra deried. Also, logic functions that define states of the diodes in the three-phase diode bridge, tered diode state functions, are defined. Let us consider a three-phase diode bridge rectifier as shown in Fig. -. The rectifier consists of a three-phase diode bridge, coprising diodes D to D6. In the analysis, it is assued that the ipedances of the supply lines are low enough to be neglected, and that the load current I OUT is constant in tie. The results and the notation introduced in this chapter are used throughout the book. First, let us assue that the rectifier is supplied by a balanced undistorted three-phase oltage syste, specified by the phase oltages: A D D D5 i + i I OUT OUT + + + i D D4 D6 B Figure -. Three-phase diode bridge rectifier.
8 Chapter ( t) = V cos ω, (.) and = V cos ωt, (.) 4 = V cos ωt. (.) The aplitude of the phase oltage V equals V =, (.4) V P RMS where V P RMS is the root-ean-square (RMS) alue of the phase oltage. Waefors of the input oltages are presented in Fig. -. Assuing that I OUT is strictly greater than zero during the whole period, in each tie point two diodes of the diode bridge conduct. The first conducting diode is fro the group of odd-indexed diodes { D, D,D5}, and it is connected by its anode to the highest of the phase oltages at the tie point considered. The second conducting diode is fro the group of een-indexed diodes { D, D4,D6}, and it is connected by its cathode to the lowest of the phase oltages. Since one phase oltage cannot be the highest and the lowest at the sae tie for the gien set of phase oltages specified by (.), (.), and (.), two of the phases are connected to the load while one phase is unconnected in each point in tie. This results in an input current equal to zero in the tie interal when the phase oltage is neither axial nor inial. The gaps in the phase currents are the ain reason for introducing the current injection ethods, as they are analyzed in the next chapter. The described operation of the diodes in the diode bridge results in a positie output terinal oltage equal to the axiu of the phase oltages, i.e., (, ) A = ax, (.5), while the oltage of the negatie output terinal equals the iniu of the phase oltages,
. Three-Phase Diode Bridge Rectifier 9.5..5 V. -.5 -. -.5-6 - -4-8 - -6 6 8 4 6.5..5 V. -.5 -. -.5-6 - -4-8 - -6 6 8 4 6.5..5 V. -.5 -. -.5-6 - -4-8 - -6 6 8 4 6 ω t [] Figure -. Waefors of the input oltages. (, ) B = in. (.6), Waefors of the output terinal oltages specified by (.5) and (.6) are presented in Fig. -. These waefors are periodic, with the period equal to one third of the line period; thus their spectral coponents are located at triples of the line frequency. Fourier series expansion of the waefor of the positie output terinal leads to A ( ) = + + V n= 9n n+ cos ( nω t), (.7)
Chapter.5. V A.5. -.5 -. -.5-6 - -4-8 - -6 6 8 4 6.5. V B.5. -.5 -. -.5-6 - -4-8 - -6 6 8 4 6.5. V C.5. -.5 -. -.5-6 - -4-8 - -6 6 8 4 6 ω t [] Figure -. Waefors of the output terinal A and B, and the waefor of C. while the Fourier series expansion of the oltage of the negatie input terinal results in B = + + V cos n = 9n ( nω t). (.8) These Fourier series expansions are used frequently in analyses of arious current injection ethods. Soe useful properties of the Fourier series expansions of the output terinal oltages should be underlined here. First, both Fourier series expansions contain spectral coponents at ultiples of tripled line frequency, i.e., at triples of the line frequency. The corresponding spectral coponents of A and B at odd triples of the line frequency at
. Three-Phase Diode Bridge Rectifier ( ) k ω, where k N, are the sae, haing the sae aplitudes and the sae phases. On the other hand, the corresponding spectral coponents at een triples of the line frequency, at 6k ω, hae the sae aplitudes, but opposite phases. These properties are used in the design of current injection networks described in Chapters 6 and 8. The diode bridge output oltage is gien by OUT =, (.9) A B and its waefor is presented in Fig. -4. The Fourier series expansion of the output oltage is OUT = + V cos k = 6k 6 ( kω t). (.) Since spectra of A and B hae the sae spectral coponents at odd triples of the line frequency, these spectral coponents cancel out in the spectru of the output oltage. Thus, the spectru of the output oltage contains spectral coponents only at sixth ultiples of the line frequency. The DC coponent of the output oltage equals V = V.65V. 4V, (.) OUT P RMS while the Fourier series expansion of the AC coponent of the output oltage is..5 V OUT..5..5. -6 - -4-8 - -6 6 8 4 6 ω t [] Figure -4. Waefor of the output oltage.
Chapter ˆ OUT = V + cos 6 k= 6k ( kω t). (.) Another waefor of interest in the analyses that follow is the waefor of the reaining oltage, C, i.e., the waefor obtained fro segents of the phase oltages during the tie interals when they are neither axial nor inial. A node in the circuit of Fig. - where that oltage could be easured does not exist, in contrast to the waefors of A and B that can be obsered at the diode bridge output terinals. Howeer, the waefor and the spectru of C can be coputed easily using the fact that the su of the instantaneous alues of the phase oltages equals zero, + +. (.) = In each point in tie, one of the phase oltages equals A, another one equals B, while the reaining one equals C. Thus, the output terinal oltages and the reaining oltage add up to zero. This gies the following expression for the reaining oltage : C =, (.4) A B and its spectru is coputed using spectra of A and B, gien by (.7) and (.8), resulting in C = V + k= ( 6k ) cos (( 6k ) ω t). (.5) In the spectru of the reaining oltage the spectral coponents are located at odd triples of the line frequency, since the spectral coponents of A and B at een triples of the line frequency cancel out. Another oltage of interest is the aerage of the output terinal oltages, defined as AV = ( A + B ) = C. (.6) Using the spectru of C, gien by (.5), the spectru of AV is obtained as
. Three-Phase Diode Bridge Rectifier AV = V + k= ( 6k ) cos (( 6k ) ω t). (.7) The spectral coponents of AV are located at odd triples of the line frequency, the sae as in the spectru of C. After the waefors of the rectifier oltages are defined and their spectra deried, waefors of the rectifier currents are analyzed. In the analysis of the rectifier currents, let us start fro the states of the diodes. First, let us define the diode state functions d k for k {,,,4,5,6 } such that dk = if the diode indexed with k conducts, and d k = if the diode is blocked. Values of the diode state functions are suarized in Table -, while the waefors of the diode state functions during two line periods are depicted in Fig. -5. Fro the data of Table - it can be concluded that the rectifier of Fig. - can be analyzed as a periodically switched linear circuit, since the states of the diodes are expressed as functions of the tie ariable. This significantly siplifies the analysis, as seen in Chapter 9, where the discontinuous conduction ode of the diode bridge is analyzed, though with significant atheatical difficulties, since the circuit cannot be treated as a periodically switched linear circuit. After the diode state functions are defined, currents of the diodes can be expressed as i Dk d k ( ω t) I OUT = (.8) for k {,,,4,5,6 } aerage alue:. All of the diode current waefors hae the sae I D = I OUT, (.9) Table -. Diode state functions. d t Segent ( ω ) d ( ω t ) d( ω t ) d4 ( ω t ) d5 ( ω t ) d6 ( ω t ) < t < 6 6 < ω t < < ω t < 8 8 < t < 4 4 < ω t < < t < 6
4 Chapter d ( t ω ) -6 - -4-8 - -6 6 8 4 6 d ( t ω ) -6 - -4-8 - -6 6 8 4 6 d ( t ω ) -6 - -4-8 - -6 6 8 4 6 d ( t 4 ω ) -6 - -4-8 - -6 6 8 4 6 d ( t 5 ω ) -6 - -4-8 - -6 6 8 4 6 d ( t 6 ω ) -6 - -4-8 - -6 6 8 4 6 ω t [] Figure -5. Waefors of the diode state functions. which is of interest for sizing the diodes. The axiu of the reerse oltage that the diodes are exposed to is equal to the axiu of the output oltage and equal to the line oltage aplitude, V = V = V 6. (.) D ax P RMS Using the diode state functions, the rectifier input currents i p, where
. Three-Phase Diode Bridge Rectifier 5 p {,,}, can be expressed as i ( d ( ω t) d ( t) ) = I. (.) p OUT p p ω Waefors of the input currents are presented in Fig. -6. The input currents hae the sae RMS alue, equal to 6 I RMS = I OUT. (.) The output power of the rectifier is.5..5 i I OUT. -.5 -. -.5-6 - -4-8 - -6 6 8 4 6.5..5 i I OUT. -.5 -. -.5-6 - -4-8 - -6 6 8 4 6.5..5 i I OUT. -.5 -. -.5-6 - -4-8 - -6 6 8 4 6 ω t [] Figure -6. Waefors of the input currents.
6 Chapter P OUT = VOUT IOUT = VIOUT = P IN (.) and it is the sae as the input power P IN, since losses in the rectifier diodes are neglected in the analysis and there are no other eleents in the circuit of Fig. -. The apparent power obsered at the rectifier input is S = V I = V I. (.4) IN P RMS RMS OUT Fro the rectifier input power gien by (.) and the rectifier apparent power gien by (.4), the power factor at the rectifier input is obtained as P = IN PF = =.9549. (.5) S IN This alue for the power factor is reasonably good, and satisfies alost all of the power factor standards. It is significantly better than the power factor alue of the rectifier with the capacitie filter connected at the output, which forces the rectifier to operate in the discontinuous conduction ode. The result is also good in coparison to single-phase rectifiers. Thus, the power factor alue of (.5) is not soething to worry about. The paraeter of the rectifier of Fig. - on which attention is focused is total haronic distortion (THD) of the input currents. To copute the THD alues of the input currents, the RMS alue of the input current fundaental haronic is deterined as I 6 RMS = I OUT. (.6) The fundaental haronics of the input currents are displaced to the corresponding phase oltages for ϕ =, (.7) which results in the displaceent power factor (DPF): DPF = cosϕ =. (.8) The THD of the input currents is deterined applying
. Three-Phase Diode Bridge Rectifier 7 RMS RMS RMS I I THD =, (.9) I resulting in THD = 9 =.8%. (.) This THD alue is considered relatiely high, and its reduction is of interest in soe applications. Efficient ethods to reduce the THD alue of the input currents in three-phase diode bridge rectifiers are the topic of this book. Soe standards liit aplitudes of particular haronic coponents of the input currents. Thus, the spectru of the input currents is a topic of interest. The input currents can be expressed by Fourier series expansions of the for + ( C, n S, n ) () t = I + I cos( nω t) + I sin( nω t) i = I DC DC + n= + n n= I cos( nω t ϕ ), n (.) where I DC = i() t d( ω t), (.), = i() t cos( nωt) d( ω t), (.) IC n, = i() t sin( nω t) d( ω t), (.4) I S n n C, n I S, n I = I +, (.5)
8 Chapter and I S, n tan ϕ n =. (.6) I C, n In the case of the input current of the first phase, specified by (.) for p =, the haronic coponents are I, (.7), DC = I, C, n n n = sin + sin I n OUT, (.8) thus I ; (.9), S, n = I n n = sin + sin n, n I OUT (.4) and n n ϕ n = sgn sin + sin. (.4) Waefors of the input currents of the reaining two phases of the rectifier are displaced in phase for in coparison to one to another, according to i ( ωt) = i ωt = i ωt +. (.4) Thus, all of the input currents share the sae aplitude spectru but hae different phase spectra, as can be deried by applying the tie-displaceent property for the Fourier series expansions in coplex for.
. Three-Phase Diode Bridge Rectifier 9 To illustrate the operation of the diode bridge rectifier and to copare its real operation with the deried odel, waefors of an experiental rectifier are recorded and presented. The experiental rectifier operates with a phase oltage RMS alue of VP RMS = V, corresponding to the input oltage aplitude of V = 4 V. The output current range is < I OUT < A, resulting in an output power of up to.5 kw. Experientally recorded waefors of the phase oltages, accopanied by the input currents, are presented in Fig. -7. Fro the waefors, it can be concluded that the oltage syste is balanced, but the oltages are slightly distorted in the for of two typical deiations: flattened sinusoid tops and notches. The flattened tops of the waefors are caused priarily by single-phase rectifiers with capacitie filtering, typical for electronic equipent, and this type of distortion is not caused by the analyzed rectifier. Howeer, the notches are caused by the nonzero line ipedance and coutations in the diode bridge. This coutation effect can also be obsered in a finite slope of the input current waefors during the rising and falling edges, coinciding with the notches in the corresponding phase oltages. Waefors of the output oltage and the output current are presented in the botto row of Fig. -7. The oltage waefor is different fro the waefor presented in Fig. -4 around the inius of the oltage, due to the notches in the phase oltages. Again, this effect is caused by nonzero ipedance of the supply lines. In the output current waefor, the output current ripple at the sixth ultiple of the line frequency can be obsered. This ripple slightly affects the input current waefors. To illustrate dependence of the input oltage and the input current waefors on the output current, and to deterine liits of the accepted rectifier odel, waefors of the phase oltage and the input current are presented in Fig. -8 for IOUT = 4 A, I OUT = 7 A, and I OUT = A. The first effect to be obsered is an increase in the duration of the notches at the phase oltage waefor by increases of the output current, caused by finite ipedance of the supply lines. The second effect is increased output current ripple at I OUT = A, which is caused by saturation of the filter inductor core. Other than these two effects that are not captured by the accepted rectifier odel, the rectifier behaior is within expected liits. In Table -, dependence of the THD of the input oltage, THD of the input current, the input power, the apparent power at the rectifier input, and the rectifier power factor are presented. The oltage waefor is oderately distorted, which slightly increases the output current, due to the increased duration of the notches. The input current THD is slightly lower than predicted by (.), which is caused by the nonzero ipedance of the supply lines. This a
Chapter i i i OUT i OUT Figure -7. Experientally recorded waefors of the input oltages, the input currents, the output oltage, and the output current. Voltage scale = 5 V/di. Current scale = 5 A/di. Tie scale =.5 s/di. ipedance slightly soothens the input current waefor during the diode state transitions, resulting in a lower THD. The power factor at the rectifier input is close to the expected alue, gien by (.5). Fro the experiental data it can be concluded that the rectifier odel adequately describes the rectifier operation. Howeer, the supply line
. Three-Phase Diode Bridge Rectifier, I OUT = 4 A i, I OUT = 4 A, I OUT = 7 A i, I OUT = 7 A, I OUT = A i, I OUT = A Figure -8. Experientally recorded waefors of the phase oltage and the input current for I OUT = 4 A, I OUT = 7 A, and I OUT = A. Voltage scale = 5 V/di. Current scale = 5 A/di. Tie scale =.5 s/di. Table -. Dependence of the rectifier paraeters on I OUT. I OUT THD ( ) THD ( i ) P IN S IN PF 4 A.7 % 9.47 % 99.8 W 946.8 VA.96 7 A.4 % 8.65 % 544. W 67.8 VA.965 A.4 % 7.94 %. W. VA.9588 inductance and the output current ripple ight slightly affect the rectifier operation, and these phenoena are not included in the rectifier odel. Application of the current injection ethods will reoe the notches fro the phase oltages and ake the inductance of the supply lines irreleant. Thus, the output current ripple will reain the only parasitic effect to be concerned about.
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