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Ch8 Inverters (converting DC to AC) 8-1 Introduction ․Converting DC to AC ․Applications: adjustable-speed AC motor drives. Uninterruptible power supply (UPS). AC appliances run from an automobile battery. 8-2 The full-bridge converter Fig 8-1 : S1 and S4 should not be closed at the same time, nor should S2 and S3, otherwise, a short circuit would exist across the dc source 8-3 The square-wave Inverter Fig 8-2. (Waveform) An inductive load presents some considerations in designing the switches in full – bridge circuit because the switch current must be bidirectional. io t i f t in t Vdc t T Ae , 0 t R 2 T t Vdc 2 T Be , t T R 2 i f t : forced current LR i n t : natural current io o Vdc In steady – state. Aeo Im in R T Vdc io Beo Im ax 2 R Vdc A Im in R B Im ax Vdc R Vdc Vdc t R I min e , 0 t T 2 R io t T t Vdc Vdc 2 T R Im ax e , t T R 2 T T Vdc Vdc 2 io By symmetry，Imax=－Imin = 2 Im in e R R T Vdc 1 e 2 Im ax Im in R T 2 1 e rms load current : 2 2 Vdc T Vdc io t dt 1 T t Irms R Im in e 2 2 T R dt T 0 0 PL I 2 rms R R load resis tan ce Pdc Vdc Is . If the switches are ideal , then Pdc PL . Fig 8-3 Full – bridge inverter using BJTs 8-4 Fourier Series Analysis With no dc component in the output v0 t Vn sinnwo t n n 1 io t I n sinnwo t n Vn n 1 In Zn In Irms In 1 2 n , rms ( n 1 2 )2 Z n : load impedance at harmonic n. P Pn I n ,rms R 2 n 1 n 1 vo t sin nwot 4Vdc In square wave output n odd n 8-5 Total harmonic distortion (THD) A quality of the AC output voltage or current. Assuming no dc component in the output V 2 n2 n , rms Vrms V12rms 2 , THDV . V1, rms V1, rms THDI ? 8-7 Amplitude and harmonic control By adjusting the interval , the output voltage can be controlled. Fig.8-4 1 2 Vdc2 d wt Vdc 1 Vrms vo t V n sinnwo t n odd ∵Half – wave symmetry Vdc sin nwo t d wo t 2 4Vdc Vn n cos n 4Vdc V1 cos . Harmonic content can also be controlled by adjusting α Harmonic n is eliminated if n 90 90 n Amplitude control and harmonic reduction may not be compatible. To control both amplitude and harmonic, it is necessary to have control over the dc input voltage. Fig 8-5 A graphical representation of the integration in the Fourier series coefficient. 8-8 The half – bridge Inverter Fig 8-7 A square-wave output or bipolar pulse-width-modulated (PWM) output. Vdc Vdc vo or 2 2 The voltage across an open switch is twice the load voltage Vdc 2 Vdc 2 8-9 pulse-width-modulated (PWM) output Advantage：Reduced filter requirements to decrease harmonics and the control of the output voltage amplitude can be realized Disadvantage：more complex control circuits for switches and increased losses due to more frequent switching. Sinusoidal PWM requires: (1). a reference (modulating or control) signal-sinusoidal. (2). a carrier (triangular wave) signal that controls the switching fre. Bipolar switching: Fig 8-8 vo Vdc vo Vdc, v sin e vtri vo Vdc, v sin e vtri S1 and S2 are on when Vsine>Vtri S3 and S4 are on when Vsine < Vtri Unipolar Switching： One (First) Fig 8-9 vo Vdc , 0, Vdc S1 is on when Vsine>Vtri S2 is on when -Vsine<Vtri S3 is on when -Vsine>Vtri S4 is on when Vsine<Vtri Another (second) Fig 8-10 S1 is on when Vsine>Vtri (high fre) S4 is on when Vsine<Vtri (high fre) S2 is on when Vsine> 0 (low fre) S3 is on when Vsine< 0 (low fre) 8-10 PWM definitions and considerations. f carrier f (1) Frequency modulation ratio： m f tri f reference f sin e The Fourier series of the PWM output voltage has a fundamental fre. which is the same as that of the reference signal. Harmonic frequencies exist at and around multiple of the switching fre. A simple low-pass filter can be effective in removing those (harmonics). Higher losses in switches Vm , reference Vm , sin e (2) Amplitude modulation ratio： ma Vm , carrior Vm , tri If ma 1 , then V1 maVdc , (linearly). If ma 1 , the amplitude of the output increases with m a , but not linearly. (3). Swithes：carrying current in either direction. Feedback.diode allowing for switching times in the control. (4). Reference voltage：sinusoidal, non-sinusoidal 8-11 PWM harmonics Bipolar switching：(Fig 8-8) If mf =odd integer, the PWM output then exhibits odd symmetry v o t Vn sinnwo t n 1 For the k-th pulse of the PWM output. (Fig 8-11) 2 k k k 1 Vdc sin nwot d wo t Vdcsin nwot d wot k Vnk k k 2Vdc [cosn k cos n k 1 2 cos n( k k )] n 4 T Vn 2 vo t sin nwot d wo t , , T 2 T 0 p Vnk p :p pulse over T/2. k 1 Normalized frequency spectrum： Fig 8-12 Normalized Fourier coefficients ： n V V Table 8-3 dc Unipolar switching：(Fig 8-9) If mf =even integer, some harmonics that were in the spectrum for the bipolar scheme are absent. (seeing Fig 8-13 and Table 8-5) 8-13 Three-phase inverters Six-step Inverter：Fig 8-17 v AB , v BC , vCA 為 Vdc, 0, or Vdc v AB v A0 v B 0 , v BC v B 0 vC 0 , vCA vC 0 v A0 Because of the six steps in the output waveform for the line-to-neutral voltage resulting from the six switching transitions per period. v A0 v AN vN 0 (1) vB 0 vBN vN 0 (2) v A0 , v B 0 , vC 0 為+Vdc or 0 vC 0 vCN v N 0 (3) v Ao v Bo vCo 1 (1)+(2)+(3) v No 3 ( v AN v BN vCN 0 ) v Ao vBo vCo 2 1 v AN v Ao v No 3 3 vBo vCo v Ao 2 1 vBN vBo v No 3 3 vCo v Ao vBo 2 1 vCN vCo v No 3 3 4Vdc Vn ,L L cos n n 6 2Vdc 2 Vn ,L N 2 cos n cos n 3n 3 3 Where n=1, 6k 1 , k=1、2… (fundamental) (harmonics) THD V 31 % for line-to-line and line-to-neutral voltages. THD I is load dependent and is smaller for a R-L load. Output fre. can be controlled by changing the switching fre.. Output voltage can be controlled by adjusting the DC input voltage. PWM three-phase Inverters Fig 8-18 S1 is on when v A ,ref vtri S2 is on when vC ,ref vtri S3 is on when v B ,ref vtri S4 is on when v A ,ref vtri S5 is on when vC ,ref vtri S6 is on when v B ,ref vtri Harmonics will be minimized if the carrier fre. is chosed to be an odd triple multiple of the reference fre. (that is 3,9,15,…times the reference). For line-to-line voltage, Vn 3 An 3 Bn 3 2 2 n n An 3 Vn sin sin ….. 三相 2 3 n n Bn 3 Vn sin sin 2 3 P Vn Vnk (參考P.313) …… 單相 k 1 Table 8-8 Significant amplitude coefficients 8-15 Induction motor speed control 2 Synchronous speed： S P S r S1ip： S S ω：electric fre. P：number of poles ωr：rotor speed If the applied electrical fre. is changed, the motor speed will change proportionally. To avoid the magnetic flux in the air gap saturated, V constant should be held . Fig 8-19 f The six-step inverter can be used for this application if the dc input is adjustable 8-20 Fig If the DC source is not controllable, a DC-DC converter may be inserted between the DC source and the inverter. The PWM inverter is also another selection for this application.
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