Lecture 8 - Effect of source inuctance on rectifier operation 8.1 Rectifier with input source inuctance The output DC voltage an current of 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 L s. (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 of voltage from the output terminals which is proportional to the loa current. The output DC voltage V 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. 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 t > 0, the loa current flows Lecture 8 - Effect of overlap 1 F. Rahman
through the rectifier ioe an that for t >, 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 L s. 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 conuct simultaneously uring the commutation process (). Because of the prolonge conuction of D f, the loa voltage is clampe to zero for 0 < t <, resulting in some loss of positive voltage in the v o waveform. Consequently V is reuce, the extent of which epens on, which in turn epens on L s an. Lecture 8 - Effect of overlap 2 F. Rahman
i s L s D v s = V max sint v s v i i Df D f I Loa Figure 8.1. Half-wave ioe rectifier with source inuctance. v s i D i Df v o v i Figure 8.2 Waveforms in the rectifier circuit of figure 8.1 During the process of overlap, all of the ac source voltage rops across L s, so that for 0 < t <, Lecture 8 - Effect of overlap 3 F. Rahman
v Vmax sint Ls t i 8.1 Integrating, V sint( t ) L i L I 8.2 max s s 0 0 V (1cos ) L I 8.3 or, max s an Ls cos = 1 I V max 8.4 The overlap, or commutation angle, can the foun from (8.4), for given an L s. 1 1 V Vmax sin( t)( t) Vmax sint( t) 2 0 2 0 = Vmax 1 Vmax Ls Ls I 1 I 2 2Vmax 8.5 V max V Figure 8.3 Voltage regulation characteristic of the rectifier of figure 8.1 ue to source inuctance. Lecture 8 - Effect of overlap 4 F. Rahman
8.2 Overlap in a single-phase 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 L s. v o i p L s i s D 1 D 3 v s = V max sint v i V Loa N:1 D 2 D 4 Figure 8.4 Dioe brige rectifier with source inuctance Thus, uring commutation overlap, i 8.6 Vmax sin t Ls t Vmax sint( t) Ls i 2Ls I 0 2 L s cos 1 I V 8.7 max Lecture 8 - Effect of overlap 5 F. Rahman
The DC output voltage of the converter is given by 1 V Vmax sint( t) 1 1 Vmax sint( t ) Vmax sint( t ) 0 0 2Vmax 2Ls 8.8 2V max Ls 1 I Vmax 8.9 Lecture 8 - Effect of overlap 6 F. Rahman
Figure 8.5 Waveforms in the ioe brige rectifier with source inuctance. Lecture 8 - Effect of overlap 7 F. Rahman
Figure 8.5 cont. 2V max V Figure 8.6 Regulation characteristic of a 1-phase brige rectifier ue to source inuctance. Lecture 8 - Effect of overlap 8 F. Rahman
Effect of overlap on three-phase center-tap rectifier In the three-phase, center-tap rectifier of figure 8.7, the loa current starts to commutate to ioe D2 after v b starts to become more positive than v a. This starts from /6 after the zero crossing of v b. During overlap, both ioes D1 an D2 carry the loa current which is assume to remain constant uring the process. During overlap, i v L v t a an s o 8.10 ib vbn Ls vo t 8.11 Assuming that remains constant uring the overlap time, an noting that ia ib I, so that i t a i t b 8.12 Aing the voltage equations an canceling the equal but opposite terms, uring overlap, v o v an 2 v bn 8.13 Lecture 8 - Effect of overlap 9 F. Rahman
v an L s V ai D1 v bn L s D2 v o v cn L s D 3 V Loa n Figure 8.7 Three-phase center-tap rectifier with source inuctance. v an v bn v cn v o v abi i a i b i c Figure 8.8 Waveforms in the rectifier of figure 8.7 Lecture 8 - Effect of overlap 10 F. Rahman
Thus, uring the commutation overlap, the converter output voltage v o is the average of the voltages of the lines unergoing commutation. Once the loa current is fully commutate, v o 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: van Vmax sint vbn Vmax sin t 2 /3 ; v V sin t 4 /3 cn ; max The part of the positive voltage pulse lost ue to overlap starting from angle t = /6 is given by v bn v bn v an v bn v an L i s 2 2 t 8.14 The area (shae) insie the voltage pulse lost ue to overlap is given by 6 vbn v an ( t ) Ls i LsI 2 0 8.15 6 in which v bn has been taken as the reference waveform. Note that (v b - v a ) is the line-line voltage v ba. The integral on the right han sie can be evaluate by shifting the origin by /6 to the right, at the crossing of v bn an v an. Thus Lecture 8 - Effect of overlap 11 F. Rahman
3V max sint( t ) LsI 8.16 0 2 2 Ls 1cos I V, so that 8.17 max ll 2 L cos 1 I s V where V max l-l = 3 V max 8.18 max l l The DC output voltage is V 3 3Vmax 3Ls 2 2 I 3Vmax ll Ls 1 I 2 Vmax ll 8.19 3V maxl l 2 V Ieal V - characteristic, i.e., for L s = 0 V - characteristic, with L s Figure 8.9 Regulation characteristic of the rectifier in figure 8.7 Lecture 8 - Effect of overlap 12 F. Rahman
Figure 8.10 Commutation notch voltage at the input to the rectifier. Lecture 8 - Effect of overlap 13 F. Rahman
Effect of source inuctance on three-phase ioe brige rectifier. v L+ v o = v L+ v L v cn v an v bn L s L s L s i c i a i b v abi D 1 D 3 D 5 V i L R Loa L D 4 D 6 D 2 v L Figure 8.11 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 L s an where all voltages are with respect to the fictitious neutral point. v L+ is the potential of the positive DC-link voltage bus (cathoes of the upper ioes) of the rectifier with respect to the neutral point. Lecture 8 - Effect of overlap 14 F. Rahman
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 V L+ as a result of the overlap is vb va vb va i vb vl vb Ls 2 2 t 8.22 Integrating for the uration of the overlap I 6 vb va ( t) Ls i LsI 2 8.23 0 6 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 3V max sint( t ) Ls I 8.24 0 2 2 Ls 1cos I V max ll, so that 8.25 2 Ls cos 1 I V maxll where V max l-l = 3 V max 8.26 The DC output voltage V is given by Lecture 8 - Effect of overlap 15 F. Rahman
3V max ll 1 Vmax ll V sint t /3 0 2 3V max ll 3Ls 8.27 3Vmax l l Ls V 1 I Vmax l l 8.28 3V max l l V Figure 8.13 Voltage regulation characteristic of the threephase ioe brige rectifier ue to source inuctance. Lecture 8 - Effect of overlap 16 F. Rahman
v a v b v c v o v ABi i a i b i c Commutation notches in v abi Figure 8.12 Waveforms in the rectifier of figure 8.11. Lecture 8 - Effect of overlap 17 F. Rahman