Comparative Evaluation of Space Vector Based Pulse Width Modulation Techniques in terms of Harmonic Distortion and Switching Losses V S S Pavan Kumar Hari G Narayanan Department of Electrical Engineering Indian Institute of Science Bangalore, India Department of Electrical Engineering Indian Institute of Science Bangalore, India pavan@ee.iisc.ernet.in gnar@ee.iisc.ernet.in ABSTRACT Voltage source inverter (VSI) fed induction motors are widely used in variable speed applications. The harmonic distortion in the line currents of the inverter depends on the switching frequency and the pulse width modulation (PWM) technique employed to switch the inverter. The inverter switching loss is also strongly influenced by the PWM technique used. This paper compares the harmonic distortion and inverter switching losses due to different PWM techniques at a given average switching frequency. The paper evaluates a few specially designed PWM techniques based on space vector approach. It is shown that these techniques lead to reduced line current distortion as well as switching losses compared to well-known techniques such as sine-triangle PWM and conventional space vector PWM (CSVPWM). Keywords Induction motor drives, voltage source inverter, pulse width modulation, harmonic distortion, space vector, switching losses. 1. INTRODUCTION A voltage source inverter (VSI) is used to supply an AC motor drive with a voltage of variable magnitude and frequency. The schematic of a two level VSI is shown in Fig.1. With a fixed DC input voltage, the semiconductor switches in the inverter are required to be switched in an appropriate fashion to obtain the desired fundamental voltage on the line side. Various PWM techniques have been reported for generation of gating pulses for the devices. The choice of PWM technique strongly influences line current distortion and switching losses in the inverter [1-2]. Sine-triangle PWM (SPWM) is a simple and popular method of PWM. In this method, three sinusoidal modulating waves, shifted by 120◦ from each other, are compared with a high frequency triangular carrier to get the switching pattern for the three phases. A sinusoidal modulating wave is shown in Fig.2(a). Suitable common-mode voltages can be added to the 3-phase sinusoidal modulating waves to achieve better harmonic performance and higher line-side voltage for a given DC bus voltage [1-2]. Examples of such modulating waves are shown in Fig.2(b) to Fig.2(d). Such addition of common-mode components could lead to discontinuous modulating waves as shown in Fig.2(c) and Fig.2(d). With these techniques, a phase remains clamped to a given + S1 C1 VDC S3 D1 S5 D3 D5 O R S4 C2 Y S6 D4 B S2 D6 D2 – Figure 1: A two level voltage source inverter using IGBTs DC bus in certain region of the fundamental cycle. A phase is clamped during the middle 60◦ interval in every half cycle in Fig.2(d), while it is clamped during the middle 30◦ interval in every quarter cycle in Fig.2(c). These techniques are termed as 60◦ clamp PWM and 30◦ clamp PWM respectively [3-5]. The average switching frequency of the inverter with such techniques equals two thirds of the carrier frequency. In addition to the above well known techniques, this paper describes two space vector based PWM techniques [6-8]. The harmonic distortion and switching losses due to various PWM techniques are evaluated and compared. It is shown that the two space vector based techniques result in reduced line current distortion and lower switching losses than other techniques at high modulation indices for a typical induction motor drive application. 2. SPACE VECTOR APPROACH TO PWM Space vector based generation of PWM waveforms is explained in this section. 2.1 Inverter states and the space vector plane A three phase quantity can be transformed into a space vector having components along two mutually perpendicular axes. The three phase input voltages at the stator terminals of an induction machine can be transformed to a space vector as shown in Eqn.1, where v s (t) is the stator voltage space vector; vRN , vY N and vBN are the line-to-neutral (3)−+− ++−(2) II III I VREF (7)+++ Vp α (4)−++ +−−(1) (0)−−− 0 IV VI −Vp V (5)−−+ 0 60 120 180 ωt(degrees) 240 300 360 (a) ◦ +−+(6) ωt = 0 Figure 3: Space vectors of a two level voltage source inverter Vp voltages [9]. 0 v s (t) = vRN (t) + vY N (t)ej 2π 3 + vBN (t)ej 4π 3 (1) 3 −Vp 0 60 120 180 ωt(degrees) 240 300 360 (b) Vp 0 A two level VSI has 2 or 8 combinations of switching states, which are known as the states of an inverter. Among these states, the two states 7 and 0 are known as zero states, as they yield zero output voltage. The remaining six states (1 to 6) give a non-zero output voltage, and are known as active states. The voltage vectors corresponding to the eight states are shown in Fig.3. The space vector plane is divided into six sectors by the vectors corresponding to active states. The sectors are marked as I to V I in Fig.3. 2.2 −Vp 0 60 120 180 ωt(degrees) 240 300 360 240 300 360 (c) Vp 0 −Vp 0 60 120 180 ωt(degrees) (d) Figure 2: Modulating waves for (a) sine-triangle PWM, (b) conventional space vector PWM, (c) 30◦ clamp PWM and (d) 60◦ clamp PWM. Peak value of sine wave is Vm . Vp is the peak of triangular = 0.8. wave. Modulation index VVm p Dwell times of inverter states In space vector based PWM, the voltage reference is provided using a revolving voltage reference vector V REF instead of three phase modulating waves. At steady state, the magnitude of this reference vector is proportional to the desired fundamental voltage and its frequency equals the fundamental frequency. The revolving V REF vector is sampled once in every subcycle Ts , which gives the average vector to be applied in the given sub-cycle. This average vector can be best realized by the vectors bounding the sector in which it is present using the principle of volt-second balance. The reference and applied volt-seconds are balanced over a sub-cycle duration, i.e. V REF Ts = V x Tx + V y Ty + V z Tz (2) where Ts is the duration of a sub-cycle; V x and V y are the active vectors bounding the corresponding sector; V z is the zero vector. The vectors V x and V y correspond to the active states x and y respectively. The zero vector V z may correspond to the zero state 0 or the zero state 7. For sector I, x = 1 and y = 2. Tx ,Ty and Tz denote the dwell times of the states/vectors. The expressions for Tx , Ty and Tz are given by Tx = Ty = Tz = VREF sin(60◦ − α) · · Ts VDC sin 60◦ VREF sin α · · Ts VDC sin 60◦ Ts − (Tx + Ty ) (3a) (3b) (3c) 3. Ts =T 0 1 2 7 Tz 2 Tx Ty Tz 2 t (a) Ts = 2 T 3 1 2 2T 3 z 2T 3 x 2T 3 y t (b) T Ts = 2 3 7 2 1 2T 3 z 2T 3 y 2T 3 x t (c) Ts =T 0 1 Tz Tx 2 2 1 Ty Tx 2 t (d) Ts =T 7 2 Tz Ty 2 1 2 Tx Ty 2 t (e) Ts =T 1 0 1 2 Tz Tx 2 Ty t (f) Tx 2 Ts =T 2 7 2 1 Ty 2 Tz Ty 2 Tx t (g) Figure 4: Seven possible sequences in sector I: (a) 0127, (b) 012, (c) 721, (d) 0121, (e) 7212, (f ) 1012 and (g) 2721. where α is the angle measured from the starting of a sector to V REF in anticlockwise direction. 2.3 In the present section, the switching sequences explained in previous section are compared in terms of RMS current ripple and switching energy lost over a sub-cycle. 3.1 0 Possible switching sequences There are numerous possible ways of applying the vectors for corresponding time intervals. The zero vector can be applied using the zero state + + +(7) or the zero state − − −(0). An active state can be applied more than once in a sub-cycle [5-8,10-11]. The various possible sequences in sector I are 0127,012,721,0121,7212,1012 and 2721. These seven sequences are illustrated in Fig.4 [7-8]. In all these sequences, the number of switchings in a sub-cycle is less than or equal to three. Also, only one phase switches during a transition from one state to the next. The sub-cycle duration for sequences 012 and 721 is taken to be 23 of that for the other sequences because there are only two switchings per sub-cycle. This facilitates comparison of all the sequences at the same average switching frequency. ANALYTICAL EVALUATION OF SWITCHING SEQUENCES RMS current ripple over a sub-cycle At any given instant, there is an error between the reference voltage vector (V REF ) and the applied voltage vector. The error voltage vectors corresponding to the vectors V x , V y and V z are illustrated in Fig.5(a). The integral of error voltage vector is the stator flux ripple vector. The trajectory of the tip of the stator flux ripple vector is parallel to the instantaneous error voltage vector [10-11]. For harmonic voltages, an induction motor can simply be modelled by its total leakage inductance. Therefore, the integral of error voltage, or the stator flux ripple, is proportional to the current ripple (both differ by a factor of machine leakage inductance). Hence, flux ripple is a measure of the ripple in line current [10-11]. The trajectory of the tip of stator flux ripple vector for sequence 0127 is shown in Fig.5(a) for VREF = 0.7VDC and α = 20◦ . The stator flux ripple vector can be resloved along the reference axes of a synchronously revolving reference frame, namely q-axis and d-axis. The q-axis is aligned with V REF , while the d-axis lags behind the q-axis by 90◦ . The variation of d-axis and q-axis components of stator flux ripple vector for the sequence 0127 over a sub-cycle is shown in Fig.5(b). The error volt-second quantities Qx , Qy , Qz and D marked in the plot are defined in Eqn.4. Qz Qx Qy D = −VREF · Tz = (cos α − VREF ) · Tx = (cos(60◦ − α) − VREF ) · Ty = sin α · Tx (4a) (4b) (4c) (4d) For sequences 012, 721, 0121, 7212, 1012 and 2721, the locus of the tip of the stator flux ripple vector and the d-axis and q-axis components of the vector for the same reference vector (VREF = 0.7VDC and α = 20◦ ) are shown in Fig.6. The RMS d-axis and q-axis flux ripple over a sub-cycle for a sequence SEQ can be calculated using Eqn.5a and Eqn.5b. The total RMS flux ripple over a sub-cycle for a sequence SEQ is given by Eqn.5c. s Z Ts 1 2 2 (5a) ψedSEQ RM S = ψedSEQ dt Ts 0 s Z Ts 1 2 2 (5b) ψeqSEQ RM S = ψeqSEQ dt Ts 0 q 2 e2 ψeSEQ RM S = ψedSEQ (5c) RM S + ψqSEQ RM S where SEQ = 0127,0121,7212,012,721,1012 or 2721. It is possible to derive closed-form expressions for the RMS flux ripple over a sector for different sequences. Expressions for five sequences (0127, 012, 721, 0121 and 7212) are presented in [11]. The corresponding expressions for 1012 and 2721 are derived here. All these expressions can be written in a generalized form given in Eqn.6. The coefficients in this expression C0 , C1 and C2 for different sequences are given in Table 1. C0 is a function of α for sequences 1012, 2721 and a constant for all other sequences. C1 and C2 are ed ψ Ts D Vy q ey V t O eq ψ Tz 2 Tz 2 Ty Tx e ψ Ve x ez V V REF −Qz 2 Qz +Qx 2 α Vz Vx (a) t O d Qz 2 0127 (b) Figure 5: (a) Error voltage vectors corresponding to applied vectors V x , V y and V z , and the trajectory of the tip of stator flux ripple vector for 0127 (b) Variation of stator flux ripple along d and q axes over a sub-cycle for 0127. ed ψ ed ψ 2T 3 2T 3 t O 2D 3 −2D 3 q q t O e ψ eq ψ 2T 3 z 2T 3 x 2 2T 3 x t O t 2 (Q +Q ) z x 3 d 2 Tz T 3 y eq 3 ψ O d 2Q 3 z 721 2Q 3 z 012 e ψ 2T 3 y 2 (Q +Q ) z y 3 (b) (a) ed ψ ed ψ T T D 2 D 2 q e ψ −D 2 e ψ eq ψ Tz Tx 2 Tx 2 Ty eq ψ −D 2 Ty Tz 2 Ty 2 Tx O t O t O t O q −Qy 2 d d 0121 Qz + Qx 2 −Qx 2 7212 Qz + Qz Qz t Qy 2 (d) (c) ed ψ ed ψ T T t O D D 2 −D 2 q q −D t O e ψ eq ψ Tx 2 Tz Tx 2 e ψ Ty eq ψ Qx 2 d 1012 d t O 2721 O Ty 2 Tz Ty 2 Tx Qy 2 t Qz +Qx (e) Qx +Qz 2 (f) Qy +Qz 2 Qz +Qy Figure 6: Trajectory of the tip of stator flux ripple vector and the variation of flux ripple along d and q axes over a sub-cycle for sequences (a) 012, (b) 721, (c) 0121, (d) 7212, (e) 1012 and (f ) 2721. ψeSEQ SEQ C0(SEQ) 0127 1 12 0121 1 3 7212 1 3 012 1 3 721 1 3 1 3 1012 2721 RM S = q 2 3 4 C0(SEQ) VREF Ts2 + C1(SEQ) VREF Ts2 + C2(SEQ) VREF Ts2 Table 1: Coefficients C0 , C1 and C2 for all the sequences a = sin (60◦ + α) and b = sin (60◦ − α) C1(SEQ) C2(SEQ) » » ` ´ ` ´ 1 32 1 4 √ ab4 + a3 b2 − a2 b3 a3 b − 2ab3 − a2 b2 9 9 3 3 3 – – ` ´ + 32 −2b3 − 3ab2 − a2 b + b − a +a2 + 4b2 − 2ab » ` 2 2 ´ ˆ 3 ˜ 2 2 1 1 2 √ a b − ab3 + 2a3 b −b − ab − 2a b − 12a + 3b 3 9 9 3 – ´ ` + 13 4a2 − 2ab + b2 C1(0121) (60◦ − α) 3 1 √ 3 ˆ8 3 C2(0121) (60◦ − α) a3 + 4ab2 − 8a2 b − 4b3 − 6a + 6b 16a5 − 2b5 + 20ab4 − 56a4 b + 74a3 b2 – −60a2 b3 + 24a2 b − 3ab2 − 9b3 + 18b − 27a 2√ 27 3 (1 − ab) C0(1012) (60◦ − α) = = ER + EY SEQ + EB SEQ |iY 1 | |iB1 | |iR1 | + nY · + nB · (7) nR · Im Im Im SEQ where iR1 iY 1 iB1 ω φ 8 9 ` −15a2 b2 + 11ab3 + 14a3 b − 4a4 − 4b4 – +4 (a − b)2 1 9 » 2 3 ` = Im sin(ωt + φ) = Im sin(ωt − 120◦ + φ) = Im sin(ωt − 240◦ + φ) = 2π · fundamental frequency = power factor angle The values of nR , nY and nB for various sequences in sector I are given in Table 2 along with the corresponding duration of sub-cycle. In any sector, ωt can be written in terms of α (for sector I, ωt = 90◦ + α). Therefore, the normalized ´ −16a4 − 6b4 + 38a3 b + 25ab3 − 41a2 b2 – +12a2 − 18ab + 7b2 C1(1012) (60◦ − α) The switching energy loss in a sub-cycle in an inverter leg is directly proportional to the current in that phase and the number of switchings of that phase in the given sub-cycle. This is based on the assumption that VDC and the device switching times remain constant. The total switching loss in the inverter in a sub-cycle is the sum of switching losses in the three phases. If nR , nY and nB are the number of switchings of phases R, Y and B, respectively, in a given subcycle for a sequence SEQ, the normalized switching energy loss in the inverter in that sub-cycle is given by SEQ 1 3 » C2(012) (60◦ − α) ´ C2(1012) (60◦ − α) Table 2: Number of switchings of the phases in a sub-cycle for different sequences in sector I SEQ 0127 0121 7212 012 721 1012 2721 Switching energy lost in a sub-cycle ESU B ˜ C1(012) (60◦ − α) » functions of α for all the sequences. As seen from the equations, the RMS flux ripple over a sub-cycle for a given sequence is a function of VREF , α and Ts . These expressions can be used to compare the RMS current ripple over a sub-cycle for different sequences. 3.2 (6) nR 1 1 0 1 0 2 0 nY 1 2 2 1 1 1 1 nB 1 0 1 0 1 0 2 Ts T T T 2 T 3 2 T 3 T T switching energy loss over a sub-cycle is a function of α and power factor angle φ. 4. HYBRID PWM TECHNIQUES The modulating waves corresponding to CSVPWM, 30◦ clamp PWM and 60◦ clamp PWM were shown in Fig.2(b), 2(c) and 2(d). These techniques are illustrated in the space vector domain in Fig.7. In CSVPWM, sequence 0127-7210 is applied in sector I. For 60◦ clamp PWM, sequence 721-127 is applied in the first half of the sector and sequence 012-210 is applied in the second half of the sector. The sequences for 30◦ clamp PWM are applied in a way opposite to that of 60◦ clamp PWM. This section describes two hybrid PWM techniques which employ sequences 0127, 0121 and 7212. 4.1 2 0127 0,7 1 (a) 2 2 012 721 721 012 30◦ 30◦ 0,7 1 (b) 0,7 1 (c) Figure 7: PWM techniques considered for evaluation - 1 : (a) CSVPWM (b) 60◦ clamp PWM (c) 30◦ clamp PWM. RM S F lux Ripple over a subcycle 16·10−6 14·10−6 12·10−6 10·10−6 8·10−6 6·10−6 4·10−6 0127 2·10−6 0121 7212 0 0 5 10 15 20 25 30 35 40 45 50 55 60 α(degrees) Figure 8: RMS flux ripple over a sub-cycle Vs α for sequences 0127, 0121 and 7212 (VREF = 0.866 p.u.). Comparison of sequences in terms of RMS current ripple The RMS current ripple over a sub-cycle corresponding to sequences 0127, 0121 and 7212 is compared here. Based on the expressions for RMS flux ripple over a sub-cycle derived in section 3.1, the sequences can be compared with each other and their individual zones of superior performance can be identified. The RMS flux ripple over a sub-cycle is plotted in Fig.8 for the whole range of α. Here, VREF = 0.866p.u.(1.0p.u. = VDC ) and Ts = 100µs. It is evident from Fig.8 that for VREF = 0.866p.u., sequences 0121 and 7212 yield lesser ripple than 0127 in the range of 5◦ ≤ α ≤ 55◦ (approximately). Similarly, for different values of VREF , the range of α over which these sequences result in less ripple will be different. These sets of VREF and α values for which a sequence gives the least ripple indicate the zone of superior performance of that particular sequence. It can be observed from Fig.8 that the flux ripple given by 0121 and 7212 are close to each other, and 0121 results in less ripple in the first half of the sector while 7212 yields less ripple in the second half. Based on this discussion, a hybrid PWM technique can be designed to give reduced current ripple, which is shown in Fig.11(a). This technique is referred to as the three-zone hybrid PWM technique [6,8]. If the tip of reference voltage vector in sector I falls in the region marked 0127 in Fig.11(a), sequence 0127-7210 is applied. Similarly sequences 0121-1210 and 7212-2721 are applied in regions marked 0121 and 7212. Sequences in other sectors are applied accordingly. 4.2 Comparison of sequences in terms of switching losses In a similar manner, various switching sequences can be compared in terms of switching losses based on the switching energy lost in each sub-cycle. The normalized switching energy loss in a sub-cycle ESU B SEQ is a function of α and power factor angle φ. The normalized switching energy loss in a sub-cycle is plotted for sequences 0127, 0121 and 7212 over the whole range of α in Fig.9 (VREF = 0.866 p.u., Im = 1.0 and φ = 30◦ lagging) and Fig.10 (VREF = 0.866 p.u., Im = 1.0 and φ = 0◦ ). From Fig.9 and Fig.10, it can be observed that 7212 produces lower switching losses than 0127 for all values of α, for power factors 0.866 lagging and unity. Whereas 0121 gives better performance in terms of switching losses only for some values of α at different power factors. For lagging power factor operation, which is typical in induction motor drives, using 7212 instead of 0121 will be a better option in terms of switching loss performance. This will not significantly affect the current ripple due to the PWM technique as 0121 and 7212 have closer characteristics in terms of flux ripple (Fig.8). Therefore, the threezone hybrid PWM technique in Fig.11(a) can be modified into a simpler technique shown in Fig.11(b) to give better performance in terms of switching losses. This technique is referred to as the two-zone hybrid PWM technique [7]. T otal Harmonic Distortion ΨT HD 0.0175 N ormalized Switching Energy loss ESU B SEQ 2.75 2.5 2.25 2 1.75 1.5 1.25 1 2 3 0.0125 1 0.01 4,5 CSVPWM(1) 0.0075 60◦ Clamp PWM(2) 30◦ Clamp PWM(3) 0.005 Three zone PWM(4) Two zone PWM(5) 0.0025 0127 0.75 0.015 5 10 15 0121 20 25 30 35 40 45 50 F undamental F requency (Hz) 7212 0.5 0 5 10 15 20 25 30 35 40 45 50 55 60 Figure 12: THD in stator flux ΨT HD for different PWM Techniques Vs fundamental frequency. α(degrees) Figure 9: Normalized switching energy loss over a sub-cycle Vs α for sequences 0127, 0121 and 7212 (VREF = 0.866 p.u., Im = 1.0 and φ = 30◦ lagging). 1.4 2 1.2 N ormalized ΨT HD N ormalized Switching Energy loss ESU B SEQ 3 2.25 2 1.75 1 0.8 CSVPWM(1) 0.6 4,5 60◦ Clamp PWM(2) 30◦ Clamp PWM(3) 0.4 1.5 Three zone PWM(4) Two zone PWM(5) 1.25 0.2 1 5 10 15 20 25 30 35 40 45 50 F undamental F requency (Hz) 0127 0.75 0121 7212 0.5 0 5 10 15 20 25 30 35 40 45 50 55 Figure 13: Normalized ΨT HD for different PWM Techniques Vs fundamental frequency (Normalization is w.r.t. CSVPWM). 60 α(degrees) Figure 10: Normalized switching energy loss over a sub-cycle Vs α for sequences 0127, 0121 and 7212 (VREF = 0.866 p.u., Im = 1.0 and φ = 0◦ ). 2 5. 2 7212 0127 (a) 0121 5.1 0127 1 0,7 (b) Figure 11: PWM techniques considered for evaluation - 2: (a) Three-zone hybrid PWM (b) Two-zone hybrid PWM EVALUATION OF PWM TECHNIQUES IN TERMS OF HARMONIC DISTORTION AND SWITCHING LOSSES In section III, evaluation of sequences at a sub-cycle level is presented. This forms the basis for design and evaluation of various PWM techniques. Comparative evaluation of PWM techniques is presented here. The techniques considered for evaluation are conventional space vector PWM (CSVPWM), 60◦ clamp PWM, 30◦ clamp PWM, three-zone hybrid PWM and two-zone hybrid PWM. These techniques are summarized in Fig.7 and Fig.11. 7212 0,7 1.0 1 Evaluation of harmonic distortion The mean square flux ripple over a sub-cycle can be averaged over a sector to give the mean square flux ripple over a e RM S , for sector. Thus, the RMS flux ripple over a sector Ψ a given VREF and Ts is given by s Z π 3 3 2 e RM S = ψeRM (8) Ψ S SEQ (α) dα π 0 where SEQ is the sequence applied in that sub-cycle, which 5 2 1.2 4 1 1.0 3 0.8 0.6 Two zone PWM(5) CSVPWM(1) 60◦ Clamp PWM(2) Three zone PWM(4) 0.4 30◦ Clamp PWM(3) Lagging p.f. 0.2 −90 −75 −60 −45 −30 −15 Leading p.f. 0 15 30 45 60 75 90 P ower f actor angle (degrees) N ormalized Switching P ower Loss N ormalized Switching P ower Loss 1.4 1.4 1.2 3 1.0 4 depends on the PWM technique employed. The total harmonic distortion (THD) in current IT HD is defined as the ratio of RMS value of the ripple current to RMS value of the fundamental component of the total current. s 2 2 IRM IeRM S S − I1RM S = (9) IT HD = 2 I1RM S I1RM S As current ripple is proportional to stator flux ripple, the harmonic distortion in stator flux can be taken as the analytical measure of harmonic distortion in line current. The THD in stator flux ΨT HD is given by ΨT HD = e RM S Ψ Ψ1RM S 5 CSVPWM(1) 0.6 60◦ Clamp PWM(2) 30◦ Clamp PWM(3) 0.4 Three zone PWM(4) Two zone PWM(5) 5 10 15 20 25 30 35 40 45 50 F undamental F requency (Hz) Figure 15: Normalized switching power loss for different PWM techniques Vs fundamental frequency at p.f.= 0.866 lagging (Normalization is w.r.t. CSVPWM). depends on the PWM technique employed. For calculating the normalized switching power loss, the normalized switching energy loss needs to be multiplied by the sampling frequency. The normalized switching power loss is calculated for the four PWM techniques considered and it is once again normalized w.r.t. CSVPWM. The variation of this normalized switching power loss over the full range of power factor angle is shown in Fig.14 for all the PWM techniques. The variation of normalized switching power loss for different PWM techniques with fundamental frequency at a power factor 0.866 lagging is shown in Fig.15. (10) where Ψ1RM S is the RMS value of the fundamental component of stator flux, which is directly related to VREF . Hence, for a given switching frequency, ΨT HD can be calculated for various PWM techniques over the range of VREF . Fig.12 shows ΨT HD for different PWM techniques as a function of fundamental frequency. As mentioned in section 2, the sampling interval Ts = 23 T for sequences 012 and 721, while Ts = T for other sequences. The duration T is chosen as 200µs, yielding an average switching frequency 1 fsw = 2T = 2.5kHz. Fig.13 shows normalized total harmonic distortion for different PWM techniques. The normalization here is with respect to the THD corresponding to CSVPWM. 5.2 2 0.8 0.2 Figure 14: Normalized switching power loss for different PWM Techniques Vs Power factor angle for VREF = 0.866p.u. (1.0 p.u. = VDC ). Normalization is w.r.t. CSVPWM. 1 Evaluation of switching losses In order to evaluate the switching loss performance of different PWM techniques, the total switching loss in the inverter has to be calculated. The average switching loss per phase over a fundamental cycle is equal to the total switching loss (three phases together) averaged over a sector. Presently, the normalized switching loss is used for comparison. The normalized switching energy loss averaged over a sector for a given power factor angle φ is given by Z π 3 3 ESU B SEQ (α) dα (11) EAV = π 0 where SEQ is the sequence applied in that sub-cycle, which 5.3 Discussion It is evident from Fig.12 and Fig.13 that all the techniques discussed give a better performance in terms of THD compared to CSVPWM when the drive is operating at higher modulation indices. The frequency range of superior performance is typically 35Hz to 50Hz. The 30◦ clamp PWM gives the best performance in terms of THD over the range of 35Hz to 45Hz. The two-zone and three-zone PWM techniques are almost identical in THD performance. These lead to a reduction of 42% in THD when compared to CSVPWM at 50Hz. Referring to the normalized switching loss curve in Fig.14, the two-zone PWM technique is the best in terms of switching loss for lagging power factor angles ranging from 0◦ to 60◦ . It gives 30% reduction in switching loss over CSVPWM at a power factor angle close to 10◦ lagging. For leading power factors ranging from unity to 0.707, 60◦ clamp PWM is the best technique, giving a maximum reduction of 22% at unity p.f. 6. EXPERIMENTAL RESULTS CSVPWM, two-zone PWM and three zone PWM techniques are implemented using ALTERA CycloneII based FPGA platform [12]. A 2.2kW, 415V, 50Hz, squirrelcage induction motor is operated at no load from a 10kVA IGBT based two level VSI with a DC bus voltage of 600V. The average switching frequency used for the PWM techniques is 2.5kHz. The experimental motor current waveforms are T otal Harmonic Distortion(%) 11 10 1 9 3 2 8 7 CSVPWM(1) Three Zone PWM(2) 6 5 30 Two Zone PWM(3) 32 34 36 38 40 42 44 46 48 50 F undamental F requency (Hz) Figure 17: Measured THD in Line Current for different PWM Techniques Vs fundamental frequency at an average switching frequency of 2.5kHz. (a) CSVPWM at 50Hz (b) Three Zone PWM (Using 0127,0121 and 7212) at 50Hz shown in Fig.16. The THD measurements of these waveforms show that at 50Hz, the THD in line current for CSVPWM is 10.07%. The three-zone technique gives a THD of 6.00%, which is 40.4% less compared to CSVPWM. This is the best reduction in THD among the techniques discussed. The two-zone technique gives a THD of 6.43%, which is 36.15% less compared to CSVPWM. These techniques yield superior performance at high modulation indices. The typical range of superior performance is 35-50 Hz. The measured THD values over this range of frequencies is shown in Fig.17. It is observed that the reduction in harmonic distortion starts at 38Hz, and increases towards the rated operating point. 7. CONCLUSION A method of evaluating different PWM techniques in terms of harmonic distortion as well as switching losses is presented. The analysis for any technique can be pursued along similar lines. Four different hybrid PWM techniques are compared considering CSVPWM as the benchmark. The two-zone and three-zone techniques are proved to be better than CSVPWM, both in terms of THD as well as switching losses. While the distortion due to two-zone and three-zone PWM are comparable, the former results in a significantly reduced switching loss at high lagging power factors among the techniques considered. Hence, two-zone PWM technique is well-suited for motor drive applications. (c) Two Zone PWM (Using 0127 and 7212) at 50Hz Figure 16: Line current waveforms of the motor at no load. The measured THD values are: (a) 10.07%, (b) 6.00% and (c) 6.43%. 8. REFERENCES [1] J. Holtz, “Pulse width modulation for electronic power conversion”, Proc. IEEE, vol. 82, no. 8, pp. 1194-1214, Aug. 1994. [2] D. G. Holmes and T. A. Lipo, Pulse Width Modulation for Power Converters: Principles and Practice, IEEE Press and John Wiley Interscience, New York, 2003. [3] J. Kolar, F. C. Zach and H. 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