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DC to AC Conversion (inverter) www.sayedsaad.com www.tkne.net www.sayedsaad.com www.tkne.net Chapter 4 DC to AC Conversion (INVERTER) • General concept • Basic principles/concepts • Single-phase inverter – Square wave – Notching – PWM • Harmonics • Modulation • Three-phase inverter Power Electronics and 1 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net DC to AC Converter (Inverter) • DEFINITION: Converts DC to AC power by switching the DC input voltage (or current) in a pre-determined sequence so as to generate AC voltage (or current) output. • TYPICAL APPLICATIONS: – Un-interruptible power supply (UPS), Industrial (induction motor) drives, Traction, HVDC • General block diagram IDC Iac + + VDC Vac − − Power Electronics and 2 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Types of inverter • Voltage Source Inverter (VSI) • Current Source Inverter (CSI) "DC LINK" Iac + + C VDC Load Voltage − − L ILOAD + IDC Load Current VDC − Power Electronics and 3 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Voltage source inverter (VSI) with variable DC link DC LINK + + + Vs C Vin Vo - - - CHOPPER INVERTER (Variable DC output) (Switch are turned ON/OFF with square-wave patterns) • DC link voltage is varied by a DC-to DC converter or controlled rectifier. • Generate “square wave” output voltage. • Output voltage amplitude is varied as DC link is varied. • Frequency of output voltage is varied by changing the frequency of the square wave pulses. Power Electronics and 4 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Variable DC link inverter (2) • Advantages: – simple waveform generation – Reliable • Disadvantages: – Extra conversion stage – Poor harmonics Vdc2 Higher input voltage Higher frequency Vdc1 Lower input voltage Lower frequency T1 T2 t Power Electronics and 5 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net VSI with fixed DC link INVERTER + + Vin C Vo (fixed) − − Switch turned ON and OFF with PWM pattern • DC voltage is held constant. • Output voltage amplitude and frequency are varied simultaneously using PWM technique. • Good harmonic control, but at the expense of complex waveform generation Power Electronics and 6 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Operation of simple square- wave inverter (1) • To illustrate the concept of AC waveform generation SQUARE-WAVE INVERTERS T1 T3 D1 D3 + VO - VDC IO T4 T2 D2 D4 S1 S3 EQUAVALENT CIRCUIT S4 S2 Power Electronics and 7 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Operation of simple square- wave inverter (2) S1,S2 ON; S3,S4 OFF for t1 < t < t2 vO S1 S3 VDC VDC t + vO − t1 t2 S4 S2 S3,S4 ON ; S1,S2 OFF for t2 < t < t3 vO S1 S3 VDC t2 t3 + vO − t S4 S2 -VDC Power Electronics and 8 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Waveforms and harmonics of square-wave inverter INVERTER OUTPUT Vdc -Vdc FUNDAMENTAL V1 4VDC π 3RD HARMONIC V1 3 5RD HARMONIC V1 5 Power Electronics and 9 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Filtering • Output of the inverter is “chopped AC voltage with zero DC component”.In some applications such as UPS, “high purity” sine wave output is required. • An LC section low-pass filter is normally fitted at the inverter output to reduce the high frequency harmonics. • In some applications such as AC motor drive, filtering is not required. (LOW PASS) FILTER L + + C vO 1 vO 2 LOAD − − vO 1 vO 2 Power Electronics and 10 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Notes on low-pass filters • In square wave inverters, maximum output voltage is achievable. However there in NO control in harmonics and output voltage magnitude. • The harmonics are always at three, five, seven etc times the fundamental frequency. • Hence the cut-off frequency of the low pass filter is somewhat fixed. The filter size is dictated by the VA ratings of the inverter. • To reduce filter size, the PWM switching scheme can be utilised. • In this technique, the harmonics are “pushed” to higher frequencies. Thus the cut-off frequency of the filter is increased. Hence the filter components (I.e. L and C) sizes are reduced. • The trade off for this flexibility is complexity in the switching waveforms. Power Electronics and 11 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net “Notching”of square wave Notched Square Wave Vdc −Vdc Fundamental Component Vdc −Vdc • Notching results in controllable output voltage magnitude (compare Figures above). • Limited degree of harmonics control is possible Power Electronics and 12 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Pulse-width modulation (PWM) • A better square wave notching is shown below - this is known as PWM technique. • Both amplitude and frequency can be controlled independently. Very flexible. 1 pwm waveform desired 1 sinusoid SINUSOIDAL PULSE-WITDH MODULATED APPROXIMATION TO SINE WAVE Power Electronics and 13 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net PWM- output voltage and frequency control Power Electronics and 14 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Output voltage harmonics • Why need to consider harmonics? – Waveform quality must match TNB supply. “Power Quality” issue. – Harmonics may cause degradation of equipment. Equipment need to be “de-rated”. • Total Harmonic Distortion (THD) is a measure to determine the “quality” of a given waveform. • DEFINITION of THD (voltage) ∞ ∞ ∑ (Vn, RMS )2 ∑ (VRMS )2 − (V1, RMS )2 THDv = n = 2 = n=2 V1, RMS V1, RMS where n is the harmonics number. Current THD can be obtained by replacing the harmonic voltage with harmonic current : ∞ ∑ (I n, RMS )2 THDi = n = 2 I1, RMS V In = n Zn Z n is the impedance at harmonic frequency. Power Electronics and 15 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Fourier Series • Study of harmonics requires understanding of wave shapes. Fourier Series is a tool to analyse wave shapes. Fourier Series 1 2π ao = ∫ f (v)dθ π 0 1 2π an = ∫ f (v) cos(nθ )dθ π 0 1 2π bn = ∫ f (v) sin (nθ )dθ π 0 Inverse Fourier ∞ 1 f (v) = ao + ∑ (an cos nθ + bn sin nθ ) 2 n =1 where θ = ωt Power Electronics and 16 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Harmonics of square-wave (1) Vdc θ=ωt π 2π -Vdc 1 π 2π ao = ∫ Vdc dθ + ∫ − Vdc dθ = 0 π 0 π Vdc π 2π an = ∫ cos(nθ )dθ − ∫ cos(nθ )dθ = 0 π 0 π Vdc π 2π bn = ∫ sin (nθ )dθ − ∫ sin (nθ )dθ π 0 π Power Electronics and 17 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Harmonics of square wave (2) Solving, bn = Vdc nπ [ π − cos(nθ ) 0 + cos(nθ ) π 2π ] Vdc = [(cos 0 − cos nπ ) + (cos 2nπ − cos nπ )] nπ Vdc = [(1 − cos nπ ) + (1 − cos nπ )] nπ 2Vdc = [(1 − cos nπ )] nπ when n is even, cos nπ = 1 bn= 0 when n is odd, cos nπ = −1 4Vdc bn= nπ Power Electronics and 18 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Spectra of square wave Normalised Fundamental 1st 3rd (0.33) 5th (0.2) 7th (0.14) 9th (0.11) 11th (0.09) 1 3 5 7 9 11 n • Spectra (harmonics) characteristics: – Harmonic decreases as n increases. It decreases with a factor of (1/n). – Even harmonics are absent – Nearest harmonics is the 3rd. If fundamental is 50Hz, then nearest harmonic is 150Hz. – Due to the small separation between the fundamental an harmonics, output low-pass filter design can be quite difficult. Power Electronics and 19 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Quasi-square wave (QSW) Vdc α α α π 2π -Vdc Note that an = 0. Due to half - wave symmetry, 1 π −α 2Vdc bn = 2 ∫ Vdc sin (nθ )dθ = − cos nθ α [ π −α ] π α nπ 2Vdc = [cos(nα ) − cos n(π − α )] nπ Expanding, cos n(π − α ) = cos(nπ − nα ) = cos nπ cos nα + sin nπ sin nα = cos nπ cos nα Power Electronics and 20 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Harmonics control 2Vdc ⇒ bn = [cos(nα ) − cos nπ cos nα ] nπ 2V = dc cos(nα )[1 − cos nπ ] nπ If n is even, ⇒ bn = 0, 4Vdc If n is odd, ⇒ bn = cos(nα ) nπ In particular, amplitude of the fundamental is : 4V b1 = dc cos(α ) π The fundamental , b1, is controlled by varying α Harmonics can also be controlled by adjusting α , For example if α = 30o , then b3 = 0, or the third harmonic is eliminated from the waveform. In general, harmonic n will be eliminated if : 90o α= n Power Electronics and 21 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Example A full - bridge single phase inverter is fed by square wave signals. The DC link voltage is 100V. The load is R = 10R and L = 10mH in series. Calculate : a) the THDv using the " exact" formula. b) the THDv by using the first three non - zero harmonics c) the THDi by using the first three non - zero harmonics Repeat (b) and (c) for quasi - square wave case with α = 30 degrees Power Electronics and 22 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Half-bridge inverter (1) S1 ON Vdc S2 OFF + S1 2 VC1 - − V + Vdc o G 0 t RL + VC2 S2 - Vdc − 2 S1 OFF S2 ON • Also known as the “inverter leg”. • Basic building block for full bridge, three phase and higher order inverters. • G is the “centre point”. • Both capacitors have the same value. Thus the DC link is equally “spilt”into two. Power Electronics and 23 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Half-bridge inverter (2) • The top and bottom switch has to be “complementary”, i.e. If the top switch is closed (on), the bottom must be off, and vice-versa. • In practical, a dead time as shown below is required to avoid “shoot-through” faults. S1 signal + S1 (gate) Ishort Vdc G S2 RL signal − (gate) S2 "Shoot through fault" . td td Ishort is very large "Dead time' = td Power Electronics and 24 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Single-phase, full-bridge (1) • Full bridge (single phase) is built from two half-bridge leg. • The switching in the second leg is “delayed by 180 degrees” from the first leg. VRG Vdc 2 LEG R LEG R' π 2π ωt + Vdc Vdc S1 S3 − 2 VR 'G 2 + - Vdc + Vo - 2 Vdc R R' G π 2π ωt - + Vdc − 2 Vdc S4 S2 Vo 2 Vdc - π 2π ωt Vo = V RG − VR 'G G is " virtual groumd" − Vdc Power Electronics and 25 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Three-phase inverter • Each leg (Red, Yellow, Blue) is delayed by 120 degrees. • A three-phase inverter with star connected load is shown below +Vdc + Vdc/2 S1 S3 S5 − G R Y B iR iY iB + S4 S6 S2 Vdc/2 − ia ib ZR ZY ZB N Power Electronics and 26 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Square-wave inverter waveforms VDC/2 VAD t -VDC/2 VB0 t VC0 t (a) Three phase pole switching waveforms VDC 600 1200 VAB t -VDC (b) Line voltage waveform 2VDC/3 VDC/3 VAPH t -VDC/3 -2VDC/3 (c) Phase voltage waveform (six-step) Interval 1 2 3 4 5 6 Positive device(s) on 3 3,5 5 1,5 1 1,3 Negative devise(s) on 2,4 4 4,6 6 2,6 2 Quasi-square wave operation voltage waveforms Power Electronics and 27 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Three-phase inverter waveform relationship • VRG, VYG, VBG are known as “pole switching waveform” or “inverter phase voltage”. • VRY, VRB, VYB are known as “line to line voltage” or simply “line voltage”. • For a three-phase star-connected load, the load phase voltage with respect to the “N” (star-point) potential is known as VRN ,VYN, VBN. It is also popularly termed as “six- step” waveform Power Electronics and 28 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net MODULATION: Pulse Width Modulation (PWM) Modulating Waveform Carrier waveform +1 M1 0 −1 Vdc 2 0 t0 t1 t 2 t 3 t 4 t5 Vdc − 2 • Triangulation method (Natural sampling) – Amplitudes of the triangular wave (carrier) and sine wave (modulating) are compared to obtain PWM waveform. Simple analogue comparator can be used. – Basically an analogue method. Its digital version, known as REGULAR sampling is widely used in industry. Power Electronics and 29 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net PWM types • Natural (sinusoidal) sampling (as shown on previous slide) – Problems with analogue circuitry, e.g. Drift, sensitivity etc. • Regular sampling – simplified version of natural sampling that results in simple digital implementation • Optimised PWM – PWM waveform are constructed based on certain performance criteria, e.g. THD. • Harmonic elimination/minimisation PWM – PWM waveforms are constructed to eliminate some undesirable harmonics from the output waveform spectra. – Highly mathematical in nature • Space-vector modulation (SVM) – A simple technique based on volt-second that is normally used with three-phase inverter motor- drive Power Electronics and 30 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Natural/Regular sampling MODULATION INDEX = M I : Amplitude of the modulating waveform MI = Amplitude of the carrier waveform M I is related to the fundamental (sine wave) output voltage magnitude. If M Iis high, then the sine wave output is high and vice versa. If 0 < M I < 1, the linear relationship holds : V1 = M I Vin where V1, Vin are fundamental of the output voltage and input (DC) voltage, respectively. −−−−−−−−−−−−−−−−−−−−−−−−−−−− MODULATION RATIO = M R (= p ) Frequency of the carrier waveform MR = p = Frequency of the modulating waveform M R is related to the " harmonic frequency". The harmonics are normally located at : f = kM R ( f m ) where f m is the frequency of the modulating signal and k is an integer (1,2,3...) Power Electronics and 31 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Asymmetric and symmetric regular sampling T +1 M1 sin ω mt sample point t T 3T 5T π 4 4 4 4 −1 Vdc 2 asymmetric sampling t t0 t1 t2 t3 symmetric sampling V − dc 2 Generating of PWM waveform regular sampling Power Electronics and 32 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Bipolar and unipolar PWM switching scheme • In many books, the term “bipolar” and “unipolar” PWM switching are often mentioned. • The difference is in the way the sinusoidal (modulating) waveform is compared with the triangular. • In general, unipolar switching scheme produces better harmonics. But it is more difficult to implement. • In this class only bipolar PWM is considered. Power Electronics and 33 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Bipolar PWM switching ∆ modulating ∆ δ= carrier 4 waveform waveform π 2π kth pulse π 2π δ 1k δ 2k αk Power Electronics and 34 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Pulse width relationships ∆ modulating ∆ δ= carrier 4 waveform waveform π 2π kth pulse π 2π δ 1k δ 2k αk Power Electronics and 35 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Characterisation of PWM pulses for bipolar switching ∆ δ0 δ0 δ0 δ0 + VS 2 δ 2k δ1k V − S 2 αk The kth PWM pulse Power Electronics and 36 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Determination of switching angles for kth PWM pulse (1) AS2 v Vmsin( θ ) AS1 + Vdc 2 Ap1 Ap2 V − dc 2 Equating the volt - second, As1 = Ap1 As 2 = Ap 2 Power Electronics and 37 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net PWM Switching angles (2) The average voltage during each half cycle of the PWM pulse is given as : Vdc δ1k − (2δ o − δ1k ) V1k = 2 2δ o V δ − δ o V = dc 1k = β1k s 2 δ o 2 δ1k − δ o where β1k = δ o Similarly, Vdc δ 2k − δ o V2 k = β 2 k ; where β 2 k = δ 2 o The volt - second supplied by the sinusoid, αk As1 = ∫ Vm sin θdθ = Vm [cos(α k − 2δ o ) − cos α k ] α k − 2δ o = 2Vm sin δ o sin(α k − δ o ) Power Electronics and 38 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Switching angles (3) Since, sin δ o → δ o for small δ o , As1 = 2δ oVm sin(α k − δ o ) Similarly, As 2 = 2δ oVm sin(α k + δ o ) The volt - seconds of the PWM waveforms, Vdc Vdc Ap1 = β1k 2δ o ; Ap 2 = β 21k 2δ o 2 2 To derive the modulation strategy, Ap1 = As1; Ap 2 = As 2 Hence, for the leading edge Vdc β1k 2δ o = 2δ oVm sin(α k − δ o ) 2 Vm ⇒ β1k = sin(α k − δ o ) (Vdc 2) Power Electronics and 39 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net PWM switching angles (4) The voltage ratio, Vm MI = is known as modulation (Vdc 2 ) index or depth. It varies from 0 to 1. Thus, β1k = M I sin(α k − δ o ) Using similar method, the trailing edge can be derived : β 2 k = M I sin(α k − δ o ) Substituting to solve for the pulse - width, δ1k − δ o β1k = δo ⇒ δ1k = δ o [1 + M I sin(α k − δ o )] and δ 2k = δ o [1 + M I sin(α k + δ o )] Power Electronics and 40 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net PWM Pulse width Thus the switching angles of the kth pulse is : Leading edge : α k − δ1k Trailing edge : α k + δ1k The above equation is valid for Asymmetric Modulation, i.eδ1k and δ 2k are different. For Symmetric Modulation, δ1k = δ 2k = δ k ⇒ δ k = δ o [1 + M I sin α k ] Power Electronics and 41 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Example • For the PWM shown below, calculate the switching angles for all the pulses. carrier 2V waveform 1.5V π 2π modulating waveform 1 2 3 4 5 6 7 8 9 π t13 t15 t17 t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t14 t16 t18 2π α1 Power Electronics and 42 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Harmonics of bipolar PWM Assuming the PWM waveform is half - wave symmetry,harmonic content of each (kth) PWM pulse can be computed as : 1T bnk = 2 ∫ f (v) sin nθdθ π 0 2 k 1k Vdc α −δ = ∫ − sin nθdθ π α −2δ 2 k o 2α k +δ 2 k V + ∫ dc sin nθdθ π α −δ 2 k 1k 2α k + 2δ o V + ∫ − dc sin nθdθ π α +δ 2 k 2k Which can be reduced to : Vdc bnk = − {cos n(α k − 2δ o ) − cos n(α k − δ1k ) nπ + cos n(α k + δ 2 k ) − cos n(α k − δ1k ) + cos n(α k + δ 2 k ) − cos n(α k + 2δ o )} Power Electronics and 43 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Harmonics of PWM Yeilding, 2V bnk = dc [cos n(α k − δ1k ) − cos n(α k − 21k ) nπ + 2 cos nα k cos n 2δ o ] This equation cannot be simplified productively.The Fourier coefficent for the PWM waveform isthe sum of bnk for the p pulses over one period, i.e. : p bn = ∑ bnk k =1 The slide on the next page shows the computation of this equation. Power Electronics and 44 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net PWM Spectra M = 0.2 Amplitude M = 0.4 1.0 0.8 M = 0. 6 0.6 Depth of Modulation 0.4 M = 0.8 0.2 0 M = 1.0 p 2p 3p 4p Fundamental NORMALISED HARMONIC AMPLITUDES FOR SINUSOIDAL PULSE-WITDH MODULATION Power Electronics and 45 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net PWM spectra observations • The amplitude of the fundamental decreases or increases linearly in proportion to the depth of modulation (modulation index). The relation ship is given as: V1= MIVin • The harmonics appear in “clusters” with main components at frequencies of : f = kp (fm); k=1,2,3.... where fm is the frequency of the modulation (sine) waveform. This also equal to the multiple of the carrier frequencies. There also exist “side-bands” around the main harmonic frequencies. • The amplitude of the harmonic changes with MI. Its incidence (location on spectra) is not. • When p>10, or so, the harmonics can be normalised as shown in the Figure. For lower values of p, the side-bands clusters overlap, and the normalised results no longer apply. Power Electronics and 46 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Bipolar PWM Harmonics 0.2 0.4 0.6 0.8 1.0 h MI 1 0.2 0.4 0.6 0.8 1.0 MR 1.242 1.15 1.006 0.818 0.601 MR +2 0.016 0.061 0.131 0.220 0.318 MR +4 0.018 2MR +1 0.190 0.326 0.370 0.314 0.181 2MR +3 0.024 0.071 0.139 0.212 2MR +5 0.013 0.033 3MR 0.335 0.123 0.083 0.171 0.113 3MR +2 0.044 0.139 0.203 0.716 0.062 3MR +4 0.012 0.047 0.104 0.157 3MR +6 0.016 0.044 4MR +1 0.163 0.157 0.008 0.105 0.068 4MR +3 0.012 0.070 0.132 0.115 0.009 4MR+5 0.034 0.084 0.119 4MR +7 0.017 0.050 Power Electronics and 47 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Bipolar PWM harmonics calculation example Note : for full bridge single - phase bipolar PWM, vo = vRR, = vRG − vR 'G = 2vRG The harmonics are computed from : (VˆRG )n VDC 2 as a function of M I Example : In the full - bridge single phase PWM inverter, VDC = 100V, M I = 0.8, M R = 39. The fundamentalfrequency is 47Hz. Calculate the values of the fundamental - frequency voltage and some of the dominant harmonics. Power Electronics and 48 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Three-phase harmonics: “Effect of odd triplens” • For three-phase inverters, there is significant advantage if p is chosen to be: – odd and multiple of three (triplens) (e.g. 3,9,15,21, 27..) – the waveform and harmonics and shown on the next two slides. Notice the difference? • By observing the waveform, it can be seen that with odd p, the line voltage shape looks more “sinusoidal”. • The even harmonics are all absent in the phase voltage (pole switching waveform). This is due to the p chosen to be odd. Power Electronics and 49 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Spectra observations • Note the absence of harmonics no. 21, 63 in the inverter line voltage. This is due to p which is multiple of three. • In overall, the spectra of the line voltage is more “clean”. This implies that the THD is less and the line voltage is more sinusoidal. • It is important to recall that it is the line voltage that is of the most interest. • Also can be noted from the spectra that the phase voltage amplitude is 0.8 (normalised). This is because the modulation index is 0.8. The line voltage amplitude is square root three of phase voltage due to the three-phase relationship. Power Electronics and 50 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Waveform: effect of “triplens” Vdc π 2π 2 V RG Vdc − 2 Vdc 2 VYG Vdc − 2 Vdc V RY − Vdc p = 8, M = 0.6 Vdc 2 V RG Vdc − 2 Vdc 2 VYG Vdc − 2 Vdc VRY − Vdc p = 9, M = 0.6 ILLUSTRATION OF BENEFITS OF USING A FREQUENCY RATIO THAT IS A MULTIPLE OF THREE IN A THREE PHASE INVERTER Power Electronics and 51 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Harmonics: effect of “triplens” Amplitude 0.8 3 (Line to line voltage) 1.8 1.6 1.4 1.2 1.0 0.8 0. 6 B 0. 4 41 43 61 65 83 85 19 23 37 47 59 67 79 89 0. 2 0 A 21 63 83 85 19 23 41 43 61 65 81 87 39 45 59 67 79 89 37 47 57 69 77 91 Harmonic Order Fundamental COMPARISON OF INVERTER PHASE VOLTAGE (A) & INVERTER LINE VOLTAGE (B) HARMONIC (P=21, M=0.8) Power Electronics and 52 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Comments on PWM scheme • It is desirable to push p to as large as possible. • The main impetus for that when p is high, then the harmonics will be at higher frequencies because frequencies of harmonics are related to: f = kp(fm), where fm is the frequency of the modulating signal. • Although the voltage THD improvement is not significant, but the current THD will improve greatly because the load normally has some current filtering effect. • In any case, if a low pass filter is to be fitted at the inverter output to improve the voltage THD, higher harmonic frequencies is desirable because it makes smaller filter component. Power Electronics and 53 Drives (Version 2): Dr. Zainal Salam, 2002 www.sayedsaad.com www.tkne.net Example The amplitudes of the pole switching waveform harmonics of the red phase of a three-phase inverter is shown in Table below. The inverter uses a symmetric regular sampling PWM scheme. The carrier frequency is 1050Hz and the modulating frequency is 50Hz. The modulation index is 0.8. Calculate the harmonic amplitudes of the line-to-voltage (i.e. red to blue phase) and complete the table. Harmonic Amplitude (pole switching Amplitude (line-to number waveform) line voltage) 1 1 19 0.3 21 0.8 23 0.3 37 0.1 39 0.2 41 0.25 43 0.25 45 0.2 47 0.1 57 0.05 59 0.1 61 0.15 63 0.2 65 0.15 67 0.1 69 0.05 Power Electronics and 54 Drives (Version 2): Dr. Zainal Salam, 2002

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