Lectures 10-11 Effect of source inductance on phase controlled AC
ELEC4240/9240 Power Electronics Lectures 10-11 Effect of source inductance on phase controlled AC-DC converters 10.1 Single-phase converters 10.1.1 Overlap in single-phase, CT fully controlled converter T1 Ls iL i1 ip R vi vo L V m ax sin ω t vs T2 Ls i2 Figure 10.1 Single-phase C-T converter with source inductance The presence of source inductance means that commutation of load current from one thyristor to the next, as they are triggered with a firing angle α, can not be instantaneous. This source inductance, Ls, is invariably because of the inductance of the supply lines and the leakage inductance of the input transformer. For this circuit, the overlap of conduction for the duration µ makes the output voltage zero (which is the mean of the overlapping input voltages) during this period . Lectures 10&11 – Overlap in AC-DC Converters 10&11-1 F. Rahman ELEC4240/9240 Power Electronics T1 Ls iL i1 ip R vi vo L V m a x s in ω t vs T2 Ls i2 Figure 10.2 Waveforms in the converter of figure 10.1, commutation overlap and commutation notches, for α = 45°. Lectures 10&11 – Overlap in AC-DC Converters 10&11-2 F. Rahman ELEC4240/9240 Power Electronics T1 Ls iL i1 ip R vi vo L V max sin ω t vs T2 Ls i2 Figure 10.3 Waveforms in the converter of figure 10.1, commutation overlap and commutation notches, for α = 130°. Lectures 10&11 – Overlap in AC-DC Converters 10&11-3 F. Rahman ELEC4240/9240 Power Electronics The part of the input voltage waveform which is missing from the output voltage is given by Vmax sinω t = Ls ∫ α +µ α di dt 10.1 Vmax sin (ω t ) d (ω t ) = ω Ls ∫ Id 0 10.2 di where Vmax is the peak of the input ac voltage and Id is the dc level of the output (load) current. Note that in each half cycle of the above waveform, the input current through the incoming thyristor rises from zero to Id in time µ/ω, starting from the instant of firing. Thus, integrating equation 10.2, −Vmax cos (α + µ ) − cos α = ω Ls I d cos (α + µ ) = cosα − ω Ls Vmax Id 10.3 The commutation or overlap angle µ can be found from expression 10.3. It increases with load current Id at any firing angle. The output dc voltage is then given by Vd = 1 π ∫ π +α α Vmax sin (ω t ) d (ω t ) − 1 π ∫ α +µ α Vmax sin (ω t )d (ω t ) 10.4 = 2Vmax π cos α − ω Ls I π d α = 30° µ° Vd α = 30° Id α = −120° (a) (b) Id Figure 10. 4 Variation of Vd and commutation angle µ with Id and firing angle α. Lectures 10&11 – Overlap in AC-DC Converters 10&11-4 F. Rahman ELEC4240/9240 Power Electronics 10.1.2 Overlap in single-phase, fully controlled bridge converter Without overlap, two thyristors conduct at all times, if turned on. With overlap, all four thyristors conduct during overlap (or commutation of current from the out going pair to the incoming pair of thyristors). vo ip i Ls V maxsin ωt T1 T3 Vd vsi T2 Id Load T4 (a) vo iL vp vp ip ip v si v s -v si (b) Figure 10.5 (a) Single-phase, F-C, bridge converter with source inductance and (b) its waveforms, for α = 45°. Lectures 10&11 – Overlap in AC-DC Converters 10&11-5 F. Rahman ELEC4240/9240 Power Electronics vs vs -vs vo iL vp ip vp ip vsi vs-vsi α =145o Figure 10.5(c) Waveforms in the converter of figure 10.5(a) for α = 145° The part of the input voltage waveform which is missing from the output during overlap is given by Vmax sin ω t = Ls di dt 10.5 As before ∫ α +µ α Vmax sin (ω t ) d (ω t ) = ω Ls cos (α + µ ) = cos α − ∫ Id −Id di = 2ω Ls I d 2ω Ls Id Vmax Lectures 10&11 – Overlap in AC-DC Converters 10.6 10&11-6 F. Rahman ELEC4240/9240 Power Electronics Note that during overlap, the input current i, which flows through Ls, changes by 2Id. The overlap angle µ can be found from the above expression. The angle µ increases when larger load current is commutated. The dc output voltage of the converter with overlap is given by Vd = 1 π = ∫ α +π α 2Vmax π Vmax sin (ω t )d (ω t ) − cos α − 2ω Ls π 1 π ∫ α +µ α Vmax sin (ωt )d (ω t ) Id α = 30° 10.7 µ° Vd α = 30° α = 125° Id Id Figure 10.6(a) Variation of Vd and commutation angle µ with load current and firing angle α for the converter of figure 10.5. Lectures 10&11 – Overlap in AC-DC Converters 10&11-7 F. Rahman ELEC4240/9240 Power Electronics 10.1.3 Overlap in single-phase, half-controlled converter vo ip Ls T1 T3 iL Vd Vmaxsinωt Df D2 LOAD D4 Figure 10.7 The commutation of the load current through the thyristors and the diodes occur in two stages. At first, the load current commutates to the freewheeling diode at the zero crossings of the input voltage. When the incoming thyristor is triggered, the freewheeling load current commutates to this thyristor. The part of the input voltage waveform missing from the output is thus dropped across the source inductance due to the load current rising to Id during the commutation overlap angle µ when a thyristor is triggered. Consequently, the overlap angle µ and the dc output voltage Vd for this converter with source inductance Ls are given by, 1 ω ∫ α +µ α Vmax sin ω td( ω t ) = Ls ∴ cos (α + µ ) = cos α − ω Ls Vmax ∫ Id 0 di Id 10.8 The dc output voltage is given by 1 Vd = π ∴ Vd = ∫ π α Vmax π Vmax sin ω td( ω t ) − [1 + cos α ] − ∫ α +µ α Vmax sin ω td( ω t ) ω Ls Id π Lectures 10&11 – Overlap in AC-DC Converters 10&11-8 10.9 F. Rahman ELEC4240/9240 Power Electronics 500 vs -500 vo iL iDf ip vp vsi vs-vsi For α = 45° Figure 10.8. Waveforms in the H-C converter with source inductance; for α = 45°. Lectures 10&11 – Overlap in AC-DC Converters 10&11-9 F. Rahman ELEC4240/9240 Power Electronics vs vo iL iDf vp ip vsi vs-vsi 0 π 2π 3π For α = 145° Figure 10.9. Waveforms in the H-C converter with source inductance; for α = 145° Lectures 10&11 – Overlap in AC-DC Converters 10&11-10 F. Rahman ELEC4240/9240 Power Electronics 10.2 Three-phase converters 10.2.1 Overlap in a three-phase, CT, fully-controlled converter, Ls ic Ls ia c T3 vcn n van a iL vabi vab vbn T1 Ls b ib R T2 vo L Figure 10.10 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 van and vcn, the output voltage vo is given by vo = van + vcn 2 10.10 The voltage pulse missing from the output voltage waveform is bounded by va and vo. This is given by voL = van − van − vcn van − vcn vacl − l = = 2 2 2 10.11 3 Vmax sin ω t where Vmax is the peak of the line-neutral 2 By expressing vol = voltage. 1 ω ∫ α +µ 3 Vmax 2 α sin ω td (ω t ) = Ls Hence cos (α + µ ) = cos α − 2ω L s 3 V max ∫ Id di 0 2ω L s Id V max l − l I d = cos α − 10.12 from which µ can be found. The dc output voltage is given by 1 Vd = 2π / 3 ∫ 5π +α 6 π 6 +α Vmax sin ω td (ω t ) − Lectures 10&11 – Overlap in AC-DC Converters 3 Vmax 2 10&11-11 ∫ α +µ α sin ω td (ω t ) F. Rahman ELEC4240/9240 = Power Electronics 3 3Vmax 3ω Ls cos α − Id 2π 2π 10.13 Figure 10.11 Waveforms in the circuit of figure 10.10 for α = 45° Lectures 10&11 – Overlap in AC-DC Converters 10&11-12 F. Rahman ELEC4240/9240 Power Electronics Figure 10.12 Waveforms in the circuit of figure 10.10 for α = 130° Lectures 10&11 – Overlap in AC-DC Converters 10&11-13 F. Rahman ELEC4240/9240 Power Electronics 10.2.2 Overlap in 3-φ, fully-controlled bridge converter vo +VD/2 ia van T3 T1 Ls R ib vbn vcn Ls T4 T6 Load Vd ic n iL T5 L T2 −VD/2 Figure 10.13 In this converter, overlap due to source inductance occur 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 Id. 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, voL = van − van + vcn van − vcn vacl −l = = 2 2 2 from α to α + µ on the vac waveform. Using a similar analysis as in the previous section, it can be shown that for this converter, cos (α + µ ) = cos α − and Vd = 3Vmax l − l π cos α − 2ω Ls Id Vmax l −l 3ω Ls π Id Lectures 10&11 – Overlap in AC-DC Converters 10.14 10.15 10&11-14 F. Rahman ELEC4240/9240 Power Electronics Figure 10.14 Waveforms in the converter of figure 10.13; α = 45° Lectures 10&11 – Overlap in AC-DC Converters 10&11-15 F. Rahman ELEC4240/9240 Power Electronics Figure 10.15 Waveforms in the converter of figure 10.13; α = 130° Lectures 10&11 – Overlap in AC-DC Converters 10&11-16 F. Rahman ELEC4240/9240 Power Electronics 10.2.3 Overlap in 3-φ, half-controlled bridge converter vo +VD/2 ia van Ls T5 ib vbn n T3 T1 Df ic vcn Ls T4 T6 iL iDf R Load Vd L T2 −VD/2 Figure 10.16 It can be shown that Vd = 3Vmax l − l 3ω Ls Id [1 + cosα ] − 2π 2π 10.16 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. Lectures 10&11 – Overlap in AC-DC Converters 10&11-17 F. Rahman ELEC4240/9240 Power Electronics 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 =Vdcmax cosα − XLIL and for fully controlled converters Vd = Vdc max ( 1 + cos α ) − X L I L for half controlled converters. 10.17 10.18 Here Vdcmax is the maximum dc output voltage for the converter circuit and XL is a parameter determined by the input source inductance Ls and the converter circuit. V dm ax α = 0° α = 30° Vd α = 60° α = 75° 0 α = 90° α = 110° α = 130° α = 160° − V dm ax Id Figure 10.17 Voltage regulation characteristic of the fully-controlled converter Lectures 10&11 – Overlap in AC-DC Converters 10&11-18 F. Rahman ELEC4240/9240 Power Electronics 10.3.1 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 nπ Vd max cos n sina × sinn (ωt + µ ) − cos α cos n (ωt + µ ) p vn ( t ) = 2 + n sin (α + µ ) sinnωt − cos (α + µ ) cos nωt n −1 10.19 where α = firing angle µ = commutation angle n = order of the output voltage ripple 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 m, the input current harmonics of a sixpulse converter is given by nπ 2 sin 3 bn = nπ nµ sin × 2 nµ 2 10.20 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), 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. Lectures 10&11 – Overlap in AC-DC Converters 10&11-19 F. Rahman ELEC4240/9240 Power Electronics 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, Vmax. 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 dc output voltage of a fully-controlled converter with discontinuous conduction and input source inductance are thus of the form as in the figure below. Consider the figure below in which a single-phase bridge supplies a load with a back emf of Eb. vo ip T1 T3 Id is R Vmaxsinωt Vd L T2 T4 Eb Figure 10.18 Single-phase bridge rectifier with discontinuous current. Lectures 10&11 – Overlap in AC-DC Converters 10&11-20 F. Rahman ELEC4240/9240 Power Electronics For this circuit, L di + Ri = Vmax sin ω t − Eb dt 10.21 The solution for i is V E − R/L t i = Ae ( ) + max sin( ω t − ϕ ) − b Z R 10.22 2 2 where Z = R + ( ω L ) and 10.23 ωL R ϕ = tan −1 10.24 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 i = max R R Eb sin( ω t − ϕ ) − Vmax Z R Eb − ( ω t −α ) R sin( α − ϕ ) e ω L − + Z Vmax 10.25 Current i falls to zero at angle β which is obtained by equating the above equation to zero. Thus Eb Vmax eRβ / ω L × = e Rα / ω L E cos ϕ sin( α − ϕ ) − b Vmax cos ϕ sin( β − ϕ ) − 10.26 The extinction angle β is found by solving the above transcendental equation. Vd can then be calculated. By solving for β for a firing angle 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 Vd - IL Lectures 10&11 – Overlap in AC-DC Converters 10&11-21 F. Rahman ELEC4240/9240 Power Electronics characteristics of a half controlled and a fully controlled converter are shown in figures below. These also include the effect of commutation overlap. Vd V dmax α = 0° α = 30° α = 60° α = 75° 0 α = 90° α = 110° α = 130° α = 160° − V dmax Id Figure 10.19 Converter Vd vs α and regulation characteristic with source inductance and discontinuous conduction. Lectures 10&11 – Overlap in AC-DC Converters 10&11-22 F. Rahman
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