Lecture 8 - Effect of source inductance on rectifier operation

Transcription

1 Lecture 8 - Effect of source inuctance on rectifier operation Ieal S real rectifier with source inuctance The output DC voltages of the rectifier circuits iscusse so far have been foun by assuming that ioe currents transfer (commutate) from one ioe to another instantaneously. However this can not happen when the AC source has some inuctance. (Change of current through any inuctance must take some time!). This source inuctance is associate with the leakage inuctance of the supply transformer an the inuctance of the AC supply network to the input transformer. The commutation process (or the overlap process) forces more than one ioe or a pair of ioes (in a brige rectifier) to conuct simultaneously, resulting in a rop voltage from the output terminals which is proportional to the loa current. The output c voltage of a rectifier falls with loa current, by an amount which is much larger than aitional voltage rop across the conucting ioes when the current through the ioes increases. The AC source inuctance, which consists of the AC line an the input transformer leakage inuctances, is mostly responsible for the aitional voltage rop. Consier the half-wave ioe rectifier shown below. i s D v s I Df v s = sinωt v i D f Loa Figure 8.1. Half-wave ioe rectifier with source inuctance. Let us assume that the loa current is smooth an ripple-free (i.e., of constant, ue to the highly inuctive loa). Assume also that for ωt > 0, the loa current flows through the rectifier ioe an that for ωt >, it commutates to the free-wheeling ioe D f. This transfer of the loa current between the rectifier an the freewheeling ioes can not however be instantaneous, because of the source inuctance. This transfer takes place over a small commutation or overlap angle, uring which time, the current graually falls to zero in one circuit an it rises to in the other circuit at the same rate. Clearly, the two ioes simultaneously conuct uring the commutation process (). Lecture 8 Effect of overlap on rectifier 8-1 F. Rahman

2 v s i D i Df v i Figure 8.2 Waveforms in the rectifier circuit of figure 8.1 Lecture 8 Effect of overlap on rectifier 8-2 F. Rahman

3 Because of the prolonge conuction of D f, the loa voltage is clampe to zero for 0 < ωt <, resulting in some loss of positive voltage in the waveform. Consequently is reuce, the extent of which epens on, which in turn epens on an. During the process of overlap, all of the ac source voltage rops across, so that for 0 < ωt <, i = = 8.1 v sinωt Ls t Integrating, sinωt( ωt) = ωls i = ωls I (1 cos ) = ωl I 8.3 or, s an ωls cos = 1- I 8.4 The overlap, or commutation angle, can the foun from (4) given an. 1 1 = sin( ωt )( ωt ) sinωt( ωt ) = 1 ωls ωls I = 1 I Figure 8.3 oltage regulation characteristic of the rectifier of figure 8.1 ue to source inuctance Lecture 8 Effect of overlap on rectifier 8-3 F. Rahman

4 Overlap in a brige rectifier ue to source inuctance During the positive half cycle, ioes D1 an D4 carries the loa current. During the negative half cycle, ioes D3 an D2 carry the loa current. During overlap all four ioes carry the loa current. The output voltage uring overlap is zero an all of the supply voltage applies across the source inuctor. i p i s D 1 D 3 sinωt v i Loa N:1 D 2 D 4 Figure 8.4. A ioe brige rectifier with source inuctance v s i s - v i Figure 8.6 Waveforms in the rectifier of figure 8.4 Lecture 8 Effect of overlap on rectifier 8-4 F. Rahman

5 Thus, uring commutation overlap, i ω = 8.6 sin t Ls t sinωt( ωt ) = ωls i = 2ωLs I 0 2ω Ls cos = 1 I 8.7 The c output voltage of the converter is given by = sinωt( ωt ) = sinωt( ωt ) sinωt( ωt ) ωls = 1 I 2 ωls = 1 I Figure 8.5 Regulation characteristic of a 1-phase brige rectifier ue to source inuctance Lecture 8 Effect of overlap on rectifier 8-5 F. Rahman

6 Effect of overlap on three-phase center-tap rectifier In the three-phase, center-tap rectifier of figure below, the loa current starts to commutate to ioe D2 from ωt = 0 + when v b starts to become more positive than v a. During overlap, both ioes D1 an D2 carry the loa current which is assume to remain constant uring the process. v an D 1 v bn D2 v cn D3 Loa n Figure 8.7 Three-phase center-tap rectifier with source inuctance v an v bn v cn v abi i a i b i c Figure 8.7 Figure 8.8 Waveforms in the rectifier of figure 8.7 Lecture 8 Effect of overlap on rectifier 8-6 F. Rahman

7 During overlap, ia van = Ls + vo 8.10 t ib vbn = Ls + vo 8.11 t Assuming that remains constant uring the overlap time, an noting that ia + ib = I, so that ia t ib = t Aing the voltage equations an canceling the equal but opposite terms, v o van + vbn =, uring the overlap process Thus, uring the commutation overlap, the converter output voltage is the average of the voltages of the lines unergoing commutation. Once the loa current is fully commutate, jumps up to the potential v b. Form the ieal output voltage waveform, the area boune by v b an (v a +v b )/2 is lost ue to overlap of two conucting ioes. In the following analysis, the line-neutral voltages are: v sinωt an = ; v = sin( ωt 2 /3) ; v = sin( ωt 4 /3) bn The part of the positive voltage pulse lost ue to overlap starting from angle ωt = /6 is given by v bn v bn + v an v bn v an = = L i s t The area (shae) insie the voltage pulse lost ue to overlap is given by + 6 vbn van ( ωt ) = ωls i= ωlsi Note that (v b - v a ) is the line-line voltage v ba. The integral on the right han sie by shifting the origin by /6 to the left. Thus cn Lecture 8 Effect of overlap on rectifier 8-7 F. Rahman

8 3 sinωt( ωt ) = ωls I ω Ls 1 cos = I, so that 8.17 l l 2ω Ls cos = 1 I where l-l = l l The c output voltage is 3 3 3ωL I 3 1 ωl I s l l s = = l l l l 2 Figure 8.9 Regulation characteristic of the rectifier in figure 8.7 Lecture 8 Effect of overlap on rectifier 8-8 F. Rahman

9 Effect of source inuctance on three-phase ioe brige rectifier v L+ = v L+ v L v an i a D 1 D 3 D 5 i L v cn v bn i c i b v abi D 4 D 6 D 2 R Loa L v L Figure 8.10 Three-phase ioe brige rectifier with source inuctance As for the three-phase CT rectifier, the voltage equations are ia va = Ls + vl+ t 8.20 ib vb = Ls + vl+ t 8.21 when D1 an D3 are in overlap ue to the source inuctance an where all voltages are with respect to the fictitious neutral point. v L+ is the potential of the positive voltage bus (cathoes of the upper ioes) of the rectifier with respect to the neutral point. As before, uring each overlap, the positive an negative c buses have voltages which are average values of the commutating line-line potentials. During the commutation overlap of ioes D1 an D3, the positive rail voltage is (v b + v a )/2, an the positive voltage lost from L+ as a result of the overlap is vb + va vb va i vb vl+ = vb = = Ls t Integrating for the uration of the overlap + 6 vb va ( ωt) ωls i ωlsi = = Lecture 8 Effect of overlap on rectifier 8-9 F. Rahman

10 v a v b v c v ABi i a i b i c Commutation notches in v abi Figure 8.11 Waveforms in the rectifier of figure 8.10 Lecture 8 Effect of overlap on rectifier 8-10 F. Rahman

11 Note again that (v b - v a ) is the line-line voltage. The integral in the right han sie by shifting the origin by /6 to the left. Thus 3 sinωt( ωt ) = ωls I ω Ls 1 cos = I, so that 8.24 l l 2ω Ls cos = 1 I where l-l = l l The c output voltage is given by L = sin t t = I ω ( ω ) 8.26 l l l l l l ω s / l l ωls = 1 I l l l l Figure oltage regulation characteristic of the three-phase ioe brige rectifier ue to source inuctance Lecture 8 Effect of overlap on rectifier 8-11 F. Rahman