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Transcription

1 The University of New South Wales School of Electrical Engineering & Telecommunications Lecture 11. Effect of source inductance in three-phase converters 11.1 Overlap in a three-phase, C-T, fully-controlled converter, c L s i c T 3 v cn n v an a L s i a T 1 i L v bn b v ab L s i b v abi T 2 v o R L Figure 11.1 In this circuit, the converter output voltage during overlap is half of the incoming and outgoing voltages. For instance, when T1 is triggered with angle α, after the crossover of v an and v cn, the output voltage v o is given by v o = v an + 2 v cn (11.1) Lecture 11 Overlap in 1 F. Rahman

2 The voltage pulse missing from the output voltage waveform is bounded by v a and v o. This is given by v ol By expressing v v v v v = van = = v ol an cn an cn acl l 3V sinωt max of the line-neutral voltage. 1 ω 3V 2 (11.2) = where V max is the peak α+ μ max Id ω ( ω ) = s α 0 2 sin td t L di 2ω L 2ω L cos cos I cos I 3V V s s Hence ( α + μ) = α d = α max max l l d (11.3) from which μ can be found. The DC output voltage is given by 5 1 π + α 6 3V α + μ max Vd = Vmaxsinωtd ( ωt ) sinωtd ( ωt π ) 2 π /3 + α 2 α 6 3 3Vmax 3ω L = cosα 2π 2π s I d (11.4) Lecture 11 Overlap in 2 F. Rahman

3 (a) Waveforms in the circuit of figure 11.1 for α = 45 Figure 11.2(b). Waveforms in the circuit of figure for α = 45. Lecture 11 Overlap in 3 F. Rahman

4 Figure 11.3 Waveforms in the circuit of figure 11.1 for α = 130 Lecture 11 Overlap in 4 F. Rahman

5 11.2 Overlap in 3-φ, fully-controlled bridge converter +V D /2 v o i a v an L s T 1 T 3 T 5 i L n v bn i b i c V d R Load L v cn L s T 4 T 6 T 2 V D /2 Figure 11.4 In this converter, overlap due to source inductance occurs every 60. When a thyristor connected with the positive voltage is triggered, the positive DC bus voltage becomes the average of the incoming and the outgoing phase voltages. The same happens when one of the thyristors connected with the negative DC bus is triggered. During the commutation angle μ, current in the outgoing line falls gradually to zero, while the current in the incoming line rises to I d. The voltage pulse missing from the output voltage waveform is the difference between the incoming line voltage minus the average of the incoming and the outgoing line voltages. For instance, when thyristor T1 is triggered with angle α, this voltage is given by, Lecture 11 Overlap in 5 F. Rahman

6 v ol v + v v v v = van = = an cn an cn acl l from α to α + μ on the v ac waveform. Using a similar analysis as in the previous section, it can be shown that for this converter, 2ω Ls cos ( α + μ) = cosα I V max l l d (11.5) and 3V 3ω L Vd cos I π π max l l s = α d (11.6) Lecture 11 Overlap in 6 F. Rahman

7 Figure 11.5 Waveforms in the circuit of figure 11.4; α = 45. Lecture 11 Overlap in 7 F. Rahman

8 Figure 11.6 Waveforms in the converter of figure 11.4; α = 130 Lecture 11 Overlap in 8 F. Rahman

9 11.3 Overlap in 3-φ, half-controlled bridge converter +V D /2 v o i a v an L s T 1 T 3 T 5 i Df i L n v bn i b i c D f V d R Load L v cn L s T 4 T 6 T 2 V D /2 Figure 11.7 It can be shown that 3V 3ω L Vd [ 1 cosα ] I 2π 2π max l l s = + d (11.7) The above equation assumes that load current transfers to the free-wheeling diode completely before the next thyristor is triggered, and that the load current transfers from the free-wheeling diode to the incoming thyristor when it is triggered. Lecture 11 Overlap in 9 F. Rahman

10 10.3 Converter voltage regulation due to source inductance. The commutation overlap affects the performance of all converter circuits, including the rectifiers which rely for their operation on natural commutation. The overlap phenomenon affects the operation of the converter in many ways, such as introducing a voltage regulation characteristic. The voltage regulation characteristic of the converter is in general given by Vd = Vdcmaxcosα XLIL (11.8) for fully controlled converters Vd = V dcmax(1+ cos α ) XLIL (11.9) for half controlled converters. Here Vdcmax is the maximum DC output voltage for the converter circuit and XL is a parameter determined by the input source inductance L s and the converter circuit. Lecture 11 Overlap in 10 F. Rahman

11 V dmax V d 0 V dmax α = 0 α = 30 α = 60 α = 75 α = 90 α = 110 α = 130 α = 160 Figure 11.8 Voltage regulation characteristic of the fullycontrolled converter. I d Lecture 11 Overlap in 11 F. Rahman

12 Other effects A. Output voltage ripple The output voltage waveform of the converter is modified by the loss of voltage pulses due to overlap. Analysis of the output voltage waveforms shows that the output ripple frequencies remain the same as for the converter without overlap. Only the output ripple voltage amplitudes are affected. The overlap actually reduces the amplitudes of the ripples. For a p-pulse converter the output ripple voltages, assuming continuous conduction, are given by v () t = Vdmax cos nsina sinn t cos cosn t n 2 nπ p ( ω + μ) α ( ω + μ) n 1 + nsin( α + μ) sinnωt cos( α+ μ) cosnωt where (11.10) α = firing angle μ = commutation angle n = order of the output voltage ripple Lecture 11 Overlap in 12 F. Rahman

13 B Input current harmonics The input current waveforms of the converter have gentler rise and fall of current, rather than the abrupt changes in the converter without overlap. It is obvious that the harmonic amplitudes of the input current are reduced as a result of overlap. Assuming linear transitions, of duration μ, the input current harmonics of a six-pulse converter is given by b n nπ nμ 2sin sin 3 2 = I nπ nμ 2 d (11.21) C. Commutation notches During commutation overlap, simultaneous conduction of two thyristors makes the line to line voltage of the two input lines under commutation zero. Other line to line voltages are found by obtaining the difference of their potentials taking into account the commutation. If the converter input voltage terminals are shared with other loads, (these voltages are invariably used as signals which control the triggering of the thyristors), and then adequate filter circuits must be used to reduce the commutation notches to acceptable levels. If these notches are not adequately filtered out, firing angles vary from thyristor to thyristor, leading to uneven output voltage ripples and thyristor currents. Lecture 11 Overlap in 13 F. Rahman

14 10.4 Converter characteristics with discontinuous load current In analyzing ac-dc converter circuits we have so far assumed that the load current was continuous, i.e., positive at all times. This allowed the output voltage to be described in terms of the firing angle α and the input AC voltage, V max. When load current becomes discontinuous, the output voltage becomes either zero or equal to the voltage of the active DC source in the load. In general, it results in an increased output DC voltage (because of the removal of part of the negative voltage of the AC source in the load). The waveforms of DC output voltage and current of a single-phase fully-controlled converter without input source inductance and with discontinuous conduction is shown in the figure below. It is assumed that the load has a constant a back emf of Eb. Lecture 11 Overlap in 14 F. Rahman

15 v o i p i s T 1 T 3 I d R V max sinωt V d L T 2 T 4 E b Figure 11.9 Single-phase bridge rectifier with discontinuous current. Lecture 11 Overlap in 15 F. Rahman

16 For this circuit, di L + Ri = Vmax sinωt E dt b (11.22) The solution for i is ( ) V i = Ae + sin( ωt ϕ ) Z E R/L t max b R (11.23) where 2 2 Z = R + ( ωl) (11.24) 1 ωl ϕ = tan and R (11.25) The instantaneous load current i is obtained by noting that i = 0 at the effective firing angle α which is either the actual firing angle (when Eb < Vmaxsinα) or the angle sin 1(Eb/Vmax) Thus V max R E i = sin( ωt ϕ ) R Z V b max + V R E ( t ) b R ω α sin( α ϕ ) e ωl max Z (11.26) Current i falls to zero at angle β which is obtained by equating the above equation to zero. Thus Lecture 11 Overlap in 16 F. Rahman

17 e E cosϕ sin( β ϕ ) V R β / ωl max R α / ωl = e Eb cosϕ sin( α ϕ ) (11.27) Vmax The extinction angle β is found by solving the above transcendental equation. Vd can then be calculated. By solving for β for firing angles α, the Vd - IL characteristic of the converter for different load parameter values (R, L and Eb) can be obtained. The boundary between continuous and discontinuous conduction is a semicircle. The V d - IL characteristics of a fully controlled converter are shown in figure These also include the effect of commutation overlap. b Lecture 11 Overlap in 17 F. Rahman

18 V dmax V d 0 V dmax α = 0 α = 30 α = 60 α = 75 α = 90 α = 110 α = 130 α = 160 Figure Converter V d vs α and regulation characteristic with source inductance and discontinuous conduction. I d Lecture 11 Overlap in 18 F. Rahman