Introduction: Principle of Space Vector PWM Treats the sinusoidal voltage as a constant amplitude vector rotating at constant frequency. Coordinate Transformation ( abc reference frame to the stationary d-q frame) : A three-phase voltage vector is transformed into a vector in the stationary d-q coordinate frame which represents the spatial vector sum of the three-phase voltage. This PWM technique approximates the reference voltage Vref by a combination of the eight switching patterns (V0 to V7). The vectors (V1 to V6) divide the plane into six sectors (each sector: 60 degrees). Vref is generated by two adjacent non-zero vectors and two zero vectors.
PWM – Voltage Source InverterOpen loop voltage control VSI AC vref PWM motorClosed loop current-control AC iref PWM VSI motor if/back
PWM – Voltage Source InverterPWM – single phase Vdc dc vc vPulse width tri vc modulator qq
PWM METHODS Output voltages of three-phase inverter where, upper transistors: S1, S3, S5 lower transistors: S4, S6, S2 switching variable vector: a, b, c
The eight inverter voltage vectors (V0 to V7)
The eight combinations, phase voltages and output line to line voltages
Basic switching vectors and Sectors 6 active vectors (V1,V2, V3, V4, V5, V6) Axes of a hexagonal DC link voltage is supplied to the load Each sector (1 to 6): 60 degrees 2 zero vectors (V0, V7) At origin No voltage is supplied to the load Fig. Basic switching vectors and sectors.
Space Vector Modulation Definition: Space vector representation of a three-phase quantities xa(t), xb(t) and xc(t) with space distribution of 120o apart is given by: x = ( x a ( t ) + ax b ( t ) + a 2 x c ( t ) ) 2 3 a = ej2π/3 = cos(2π/3) + jsin(2π/3) a2 = ej4π/3 = cos(4π/3) + jsin(4π/3)x – can be a voltage, current or flux and does not necessarily has to be sinusoidal
Space Vector Modulation v = ( v a ( t ) + av b ( t ) + a 2 v c ( t ) ) 2 x x ax x 3Let’s consider 3-phase sinusoidal voltage: va(t) = Vmsin(ωt) vb(t) = Vmsin(ωt - 120o) vc(t) = Vmsin(ωt + 120o)
Space Vector Modulation v = ( v a ( t ) + av b ( t ) + a 2 v c ( t ) ) 2 3 Let’s consider 3-phase sinusoidal voltage:At t=t1, ωt = (3/5)π (= 108o)va = 0.9511(Vm)vb = -0.208(Vm)vc = -0.743(Vm) t=t1
Space Vector Modulation v = ( v a ( t ) + av b ( t ) + a 2 v c ( t ) ) 2 3 Let’s consider 3-phase sinusoidal voltage: bAt t=t1, ωt = (3/5)π (= 108o)va = 0.9511(Vm) avb = -0.208(Vm)vc = -0.743(Vm) c
Three phase quantities vary sinusoidally with time (frequency f) ⇒ space vector rotates at 2πf, magnitude Vm
Space Vector Modulation S1 S3 S5 + va - Vdc a + vb - b + vc - n c S4 S6 S2 N We want va, vb and vc to followva* v*a, v*b and v*cvb* S1, S2, ….S6vc*
Space Vector Modulation S1 S3 S5 + va - Vdc a + vb - b + vc - n c S4 S6 S2 van = vaN + vNn N vbn = vbN + vNnFrom the definition of space vector: vcn = vcN + vNn v = ( v a ( t ) + av b ( t ) + a 2 v c ( t ) ) 2 3
Space Vector Modulation =0 2 ( v = v aN + av bN + a 2 v cN + v Nn (1 + a + a 2 ) 3 )vaN = VdcSa, vaN = VdcSb, vaN = VdcSa, Sa, Sb, Sc = 1 or 0 2 ( v = Vdc S a + aS b + a 2 S c 3 ) v = ( v a ( t ) + av b ( t ) + a 2 v c ( t ) ) 2 3
Space Vector Modulation Sector 2  V3  V2 (1/√3)Vdc Sector 3 Sector 1  V1 V4 (2/3)Vdc Sector 4 2 ( v = Vdc S a + aS b + a 2 S c 3 ) Sector 6  V5 Sector 5  V6
Conversion from 3 phases to 2 phases : For Sector 1,Three-phase line modulating signals (VC)abc = [VCaVCbVCc]Tcan be represented by the represented by the complex vector VC = [VC]αβ = [VCaVCb]Tby means of the following transformation: VC α = 2/3 . [vCa - 0.5(vCb + vCc )] VC β = √3/3 . (vCb - vCc)
Space Vector Modulation Reference voltage is sampled at regular interval, T Within sampling period, vref is synthesized using adjacent vectors and zero vectorsIf T is sampling period, 110 V1 is applied for T1, V2 V2 is applied for T2 Sector 1Zero voltage is applied for therest of the sampling period, T2 V2 T 0 = T − T 1− T 2 T Where, 100 T1 = Ts.|Vc|. Sin (π/3 - θ) T1 V1 T2 = Ts.|Vc|. Sin (θ) V1 T
Space Vector ModulationReference voltage is sampled at regular interval, TWithin sampling period, vref is synthesized using adjacent vectors andzero vectors T0/2 T1 T2 T0/2 V0 V1 V2 V7If T is sampling period, V1 is applied for T1, va V2 is applied for T2 vbZero voltage is applied for therest of the sampling period, vc T0 = T − T1− T2 T T Vref is sampled Vref is sampled
Space Vector Modulation How do we calculate T1, T2, T0 and T7?They are calculated based on volt-second integral of vref1 T 1 To T1 T2 T7 ∫T 0 T 0 ∫ 0 ∫ 0 ∫ v ref dt = v 0 dt + v 1dt + v 2 dt + v 7 dt 0 ∫v ref ⋅ T = v o ⋅ To + v 1 ⋅ T1 + v 2 ⋅ T2 + v 7 ⋅ T7 2 2v ref ⋅ T = To ⋅ 0 + Vd ⋅ T1 + Vd (cos 60o + j sin 60o )T2 + T7 ⋅ 0 3 3 2 2v ref ⋅ T = Vd ⋅ T1 + Vd (cos 60o + j sin 60o )T2 3 3
Space Vector Modulation q T = T1 + T2 + T0,7 110 V2 Sector 1 v ref ⋅ = v ref ( cos α − j sin α ) α 100 2 2 V1 dv ref ⋅ T = Vd ⋅ T1 + Vd (cos 60o + j sin 60 )T2 o 3 3
Space Vector Modulation 2 2v ref ⋅ T = Vd ⋅ T1 + Vd (cos 60o + j sin 60o )T2 3 3 2 1 1T v ref cos α = Vd T1 + Vd T2 T v ref sin α = Vd T2 3 3 3Solving for T1, T2 and T0,7 gives:T1= 3/2 m[ (T/√3) cos α - (1/3)T sin α ]T2= mT sin α where, M= Vref/ (Vd/ √3)
Comparison of Sine PWM and Space Vector PWM Fig. Locus comparison of maximum linear control voltage in Sine PWM and SV PWM.
Comparison of Sine PWM and Space Vector PWM a o b c vao Vdc/2 For m = 1, amplitude of fundamental for vao is Vdc/2∴amplitude of line-line = 3 Vdc 2 -Vdc/2
Comparison of Sine PWM and Space Vector PWM Space Vector PWM generates less harmonic distortion in the output voltage or currents in comparison with sine PWM Space Vector PWM provides more efficient use of supply voltage in comparison with sine PWM Sine PWM : Locus of the reference vector is the inside of a circle with radius of 1/2 V dc Space Vector PWM : Locus of the reference vector is the inside of a circle with radius of 1/√3 Vdc ∴ Voltage Utilization: Space Vector PWM = 2/√3 or (1.1547) times of Sine PWM, i.e. 15.47% more utilization of voltage.
Space Vector ModulationComparison between SVM and SPWMSVM 1We know max possible phase voltage without overmodulation is Vdc 3 ∴amplitude of line-line = Vdc 3 Vdc − Vdc 2 ≈ 15.47% Line-line voltage increased by: x100 3 Vdc 2
1. Power Electronics: Circuits, Devices and Applications by M. H. Rashid, 3rd edition,Pearson2. Power Electronics: Converters, Applications and Devices by Mohan, Undeland andRobbins, Wiley student edition3. Power Electronics Handbook: M.H. Rashid, Web edition4. Modern Power Electronics And Ac Drives: B.K. Bose5. Extended Report on AC drive control, IEEE : Issa Batarseh6. Space vector modulation: Google, Wikipedia ; for figures.