Space-Vector based Advanced Pulse Width Modulation Techniques for Z-Source Inverter

Space-Vector based Advanced Pulse Width Modulation Techniques for Z-Source Inverter

Mr. Kanan Kumar Das

Lecturer,

Department of Electrical Engineering, Templecity Institute of Technology and Engineering (TITE),

Bhubaneswar, India

Abstract The Performances of an inverter depends on what type of modulation strategies is used to switch. There exists different types of classical and advanced Pulse Width Modulation (PWM) techniques to switch the traditional Voltage Source Inverter (VSI).The advanced PWM techniques are superior to classical PWM techniques in respect of load current ripple, Total Harmonic Distortions, switching losses. The same concept of advanced PWM techniques for VSI is applicable for Z-Source Inverter. This paper proposes certain vector sequences based on advanced PWM techniques for ZSI. The proposed vector sequences are analysed in respect of load/line current ripple based on stator flux ripple concept. The superiority of the advanced PWM techniques over classical is verified by simulation study of a ZSI in Power-Sim (PSim9.1.1) for R-L and different motor load conditions.

Keywords Bus-Clamping, Current Ripple, Pulse Width Modulation, Maximum Boost Control, Z-Source Inverter

1. INTRODUCTION

   Z-source Inverter (ZSI) or Impedance Source Inverter (ISI) is extensively used in different field of application like motor drives [1], electric vehicles [2], photovoltaic system [3], uninterruptible power supply [4] etc. The X-shape lattice type impedance network present in between the DC input source and the inverter power circuit, is responsible for the unique buck-boost capability of ZSI. The Z-source concept can be applied to all DC-to-AC, AC-to-DC, AC-to-AC and DC-to- DC power conversion [5]. There exists little interest in the control methodologies improvement for ZSI. Space vector based PWM techniques for ZSI finds little interest among the researchers. This paper presents an extensive study on space vector based PWM techniques for ZSI.

   In literature, ZSI has been analyzed for different control strategies like Simple Boost Control (SBC) [5], Maximum Boost Control (MBC) [6] and Constant Boost Control (CBC) [7]. The MBC gives the highest boosting factor and lowest voltage stress across the switching devices. The MBC can be implemented in carrier based PWM approach or space vector based PWM approach [6-10]. Extensive study has already been done for carrier based PWM approach for ZSI [6-10]. The space vector based PWM approach with MBC came in picture in the year of 2012 [9]. Further there has a few study of ZSI for Bus-Clamping PWM (BCPWM) technique [10]. BCPWM is one of the advanced PWM techniques introduced for VSI. Further, the advanced Bus clamping PWM (ABCPWM) techniques have been introduced. These ABCPWM techniques are sometimes known as Double Switching PWM techniques as a phase is double switched

   over a sub-cycle. As compared to classical Space vector PWM techniques, the BCPWM and ABCPWM techniques perform better in respect of load current ripple, load current distortions and lesser switching losses. These PWM techniques have been already proved superior performances compare to classical space vector PWM for VSI based motor drive application theoretically as well as experimentally [11-14].

   In motor drive application, ZSI becomes a popular choice for its unique features. But the lesser harmonic distortion in the line current is desirable features for the drive application. For this reason, this paper introduces certain vector sequences with proper shoot-through period based on advanced PWM techniques. The expressions for line current ripple for different sequences have been derived based on the stator flux ripple concept. For the verification of the superiority of advanced PWM techniques, a model of ZSI has been simulated using standard Power-Sim (PSim9.1.1) for R-L and different motor load conditions.

   The organization of this paper is as follows- in section-II, the discussion regarding conventional ZSI. In section-III, the different space vector based advanced PWM techniques for VSI have been presented. In section-IV, the shoot-through state of ZSI is discussed. The space vector based different PWM techniques for ZSI have been proposed in section-V. In section-VI, the Maximum Boost Control for space vector PWM has been presented. In the following section-VII, the theoretical analysis of line current ripple based on stator flux ripple for different proposed vector sequences have been elaborated. The simulation results for R-L load and different motor load conditions for different vector sequences have been presented in section-VIII.

2. CONVENTIONAL Z-SOURCE INVERTER Conventional Z-Source Inverter (ZSI) was introduced in

   the year of 2003 by Prof. P.Z. Peng [5]. The boost capability of ZSI is achieved by the application of shoot-through state which is not possible in VSI or CSI (Current Source Inverter). The power circuit for ZSI is shown in Fig-1. During the shoot- through state, the input diode, Di is in the reverse biased mode and the inductors L1 and L2 get charged by the capacitors C1 and C2. This state is achieved by the application of simultaneously turn on the two switches in a leg or simultaneously turn on the four switches of two legs or simultaneously turn on all the switches of three legs. During the active state, the diode will be in conduction mode and the inductors will discharges to the load side and charge the capacitor. On the zero state, the load side is totally disconnected from the impedance network, the inductors

   discharge to the capacitors. By volt-sec balance across the inductors the different relationships between different quantities can be found out. The different quantities for ZSI are tabulated in Table-1. Where D0 is the shoot-through duty ratio and M is the space vector based modulation index (varies from 0 to 0.866) and Vin is the input DC voltage to the ZSI. The boosting factor, B is defined as the ratio between the Z- network output voltages, Vb to the input DC voltage, Vin. The boosting factor depends on D0 which further depends on the shoot-through duration and sub-cycle period, Ts. Further, D0 depends on the modulation index, M. For different control strategies, the D0 is different. The overall gain of the inverter can be controlled by controlling both modulation index and

   boosting factor.

   generate the reference vector in continuous fashion, sampling of the reference vector has to be considered inside the hexagon. The reference vector is sampled once in every sub- cycle, Ts. As the positions of the space vectors in hexagon are fixed, the reference vector at any position in a sector can be generated by applying nearby active vectors and zero vectors in such way that the volt-sec balance over a sample period or sub-cycle period can be achieved. The time durations for which the active and zero vectors are applied are known as dwell times. The dwell times remain same for all the sectors. The expressions for dwell times corresponding active vector-1, active vector -2 and null vector- 0/7 in sector-1 are as follows-

   T M Sin(60 – ) T ; T

   M Sin T ;T T T T

   (1)

   1 Sin60 s 2

   Sin60 s 0 s 1 2

   Fig. 1: Power circuit for conventional ZSI

   Table-1: Expression for different quantities for Z-source Inverter

   Index Symbol Expression

   1

   where is the angle between the reference vector and the starting vector of the sector (active vector-1 in sector-1). The range of is 00 to 600. For generating the reference vector, the near-by active vectors and zero vectors need to be applied in a sequence suc that the switching losses in the inverter is minimum and the better quality of output waveforms can be achieved. For that the selection of vector sequence in a sample cycle or sub-cycle period, Ts should be satisfy the following conditions-

   1. At least one zero vector must be present.

      Boosting Factor B

      Output voltage of Z-network V b

      1 – 2 D 0

      Vin

      1 – 2 D0

      1 – D

   2. Two nearby active vectors must be present.

   3. The maximum no of switching must be less than or equal to three.

      Capacitor (C1) Voltage

      Capacitor (C2) Voltage Overall Gain

      Peak Phase Voltage

      VC1

      VC2

      G

      V

      0 Vin

      1 – 2D0

      1 – D 0 V

      1. – 2 D 0 in

      2. M B

      3

      2 M V

   4. In state transition, only one phase should be switched

   5. The dwell times should be divided among vectors in such way that the Volt-Sec balanced can be achieved.

      The reference vector in any sector can be generated by applying different vector sequences which are obeyed the

      ph 3 b

3. ADVANCED PWM TECHNIQUES FOR VSI PWM techniques can be broadly classified into two

   categories namely carrier based PWM techniques and space vector based PWM techniques. Space vector based PWM

   above mentioned conditions (1 to 5). The initial vector in the vector sequences can be any vectors among four vectors in sector-1 (0,1,2 and 7). Fig-3 and Table-2 summarises the different valid and not-valid vector sequences possible in sector-1.

   techniques give better performances compare to carrier based

   PWM techniques like better DC bus utilization, lesser Total Harmonic Distortion (THD) and lesser switching losses. Conventional Space Vector PWM (CSVPWM) is one of the most popular and simplest space vector PWM techniques. In

   (010) V3 0327-7230 V2(110)

   0127-7210

   0347-7430

   Sector 2

   Sector 1

   Sector 3

   space vector approach, each state of the inverter generates a

   V7

   (111)

   V ref

   vector of fixed magnitude and angle with reference in space vector plane. For 2-level VSI, there exists eight vectors which can be generated by eight switching state of VSI. Among eight vectors, six are known as active vectors (corresponding switching state is known as active state) and two zero or null vectors (corresponding switching state is known as zero or

   (011) V4

   V1(100)

   Sector 4

   Sector 6

   (000)

   0167-7610

   0547-7450

   V0

   Sector 5

   null state). Active vectors have fixed magnitude and fixed angle with reference whereas the zero or null vectors have zero magnitude. These eight vectors form a hexagon in space vector plane. The null vectors are located at the origin whereas each corners of the hexagon is occupied by the active vector. The hexagon for 2-level VSI is shown in Fig.2(a). The hexagon has six sectors of 600 spatial duration each. Each sector is formed by two active vectors and two zero vectors. The reference vector corresponding desired output voltages revolves inside the hexagon at a continuous fashion. To

   1. V5

      0567-7650

      (a)

      V6(101)

      721-127

      743-347 034-430

      1. V3

         032-230 (B-)

         723-327 (Y+)

         (R+)

         V2(110)

         (0127-7210) vector sequence is popularly known as conventional space vector PWM. (0121-1210) and (7212- 2127) vector sequence are known as special PWM type-1 or

      2. V4

         Sector 2

         (Y+)

         (R-)

         Sector 3

         V7

         (111)

         Sector 4

         (000)

         (B+)

         V0

         054-450

         (R-)

         Sector 5

         012-210

         (B-)

         Sector 1

         (Y-)

         745-547

         Sector 6

         V1(100)

         Advanced Bus-Clamping PWM (ABCPWM) type-1. Whereas vector sequence (1012-2101) and (2721-1272) are named as special PWM type-2 or Advanced Bus-Clamping PWM (ABCPWM) type-2. In these vector sequences, over a sub- cycle period, one phase is switched double and one phase is switched once and another phase remains clamped. Some times these vector sequences are known as Double Switching PWM sequences.

         (R+)

         1. V5

            016-610 761-167

            056-650 (Y-)

            wt=0

            (b)

            V6(101)

            0 1 2

            0 1 0101

            1 0121

            1 2

            7 0127

            2 1012

            0 1 0 1010

            1 0 1210

            7 2 1272

            1. V3

               Sector 2

               V2(110)

               (a) (b)

               Sector 1

               0323-3230 (B-)

               7232-2327 (Y+)

               7212-2127

               0 1 2101

               0 7210

            2. V4

   0343-3430

   7434-4347

   (R-)

   Sector 3

   V7

   (Y+)

   (111)

   2

   1

   2

   7454-4547

   (000)

   0121-1210

   0161-1610

   (B-)

   V1(100)

   1

   2

   7

   2

   7 2 1

   2127

   7

   2721

   (R+)

   7212

   (Y-)

   7616-6167

   (B+)

   7 2727

   (R+)

   Sector 4

   0565-5650 (Y-)

   Sector 6

   0545-5450

   (R-)

   V0

   7 2 7272

   (011) V4

   (001) V5

   4743-3474

   (Y+)

   3034-4303

   (R-)

   Sector 3

   (010) V3

   Sector 4

   5054-4505

   (R-)

   4745-5474

   (B+)

   (001) V5

   Sector 5 wt=0

   (c)

   3032-2303 (B-)

   Sector 2

   V7

   (111)

   (000)

   5056-6505 (Y-)

   V0

   Sector 5 wt=0

   (d)

   7656-6567 (B+)

   V6(101)

   Sector 1

   2723-3272 (Y+)

   2721-1272

   (R+)

   1012-2101

   (B-)

   V2(110)

   1016-6101

   (Y-)

   6761-1676

   (R+)

   6765-5676 (B+)

   Sector 6

   V6(101)

   V1(100)

   (c) (d)

   Fig.3: Different possible vector sequences starting with (a) vector-0, (b) vector-1, (c) vector-2 and (d) vector-7 for 2-level VSI in sector-1

   To reduce the switching loss and current ripple, there exists another two sequences (012-210) and (721-127). These sequences are known as Bus-Clamping PWM (BCPWM). At high modulation index and same average switching frequency, line current ripple is lesser with BCPWM compare to CSVPWM. On the other hand, at high modulation index and same carrier frequency, the switching loss is lesser in case of BCPWM compare to CSVPWM. At high modulation index, ABCPWM is better than BCPWM like reduction in line current ripple. The hexagon for 300 BCPWM is shown in Fig. 2(b), for each quarter of fundamental cycle, one phase gets clamped either to positive DC link or negative DC link. The swapping of vector sequence in subsectors under a sector (like 721-127 is for first subsector in sector-1 and 012-210 is for second subsector in sector-1) gives 600BCPWM. In this PWM, each phase gets clamped middle 600 duration of each fundamental cycle. Fig. 2(c) and 2(d) show the hexagon for 300ABCPWM Type-1 and 300 ABCPWM Type-2

   respectively. The swapping of vector sequence between subsectors in a sector give the 600ABCPWM. From the

   Fig.2: Space Vector Hexagon for 2-level VSI for (a) CSVPWM (0127- 7210), (b)300 BCPWM (012-210), (c) Special PWM-1 (0121-1210) and (d)

   Special PWM-2 (1012-2101)

   sequences it can observe that the number of switching for 012 is 2 where as for the other vector sequences the number of switching is three over a Ts. So for same average switching frequency over the fundamental cycle or line cycle, the sub- cycle period for 012 is 2/3 of that for other sequences. On the other hand, for 012, 0121 and 1012, a hase (here B-Phase) gets clamped, these vector sequences are also known as Clamping Sequences or Discontinuous PWM sequences. It has been already proven that ABCPWM is better than BCPWM and CSVPWM at high modulation indices at same average switching frequency [11-14].

   Table-2: Different vector sequences for 2-level VSI Table-4: Different vector sequences for ZSI

   Name Vector Sequence

   Valid/ not-Valid

   Reason

   Name Vector Sequence

   CSVPWM	0127-7210	Valid	conditions (1-4) are satisfied
   Special PWM 1	0121-1210	Valid	conditions (1-4) are satisfied
   Special PWM 1	7212-2127	Valid	conditions (1-4) are satisfied
   Special PWM 2	1012-2101	Valid	conditions (1-4) are satisfied
   Special PWM 2	2721-1272	Valid	conditions (1-4) are satisfied

   CSVPWM 0

   ST1

   1 ST2 2

   ST3 7

   xxxxxxxxx 0101-1010 not-Valid

   condition 2 is not satisfied

   Special PWM 2

   Special PWM 2

   Special PWM 1

   Special PWM 1 Bus-Clamping PWM

   1. ST1

   2. ST3

   0 ST1

   7 ST3

   0

   0

   7

   1

   2

   ST1

   ST1

   ST3 ST2 ST2 1

   1

   2

   2

   1

   ST2

   ST2 2

   ST2 1

   ST2 1

   ST2 2

   2

   xxxxxxxxx

   7272-2727 not-Valid

   condition 2 is not satisfied

   Bus-Clamping PWM

   7 ST3 2

   ST2 1

   Bus-Clamping PWM 012-210 Valid conditions (1-4) are satisfied

   Bus-Clamping PWM 721-127 Valid conditions (1-4) are satisfied

4. SHOOT THROUGH STATE FOR ZSI

   In ZSI, the shoot-through state is generated in seven possible switching combinations which is listed in Table-3. ST1, ST2 and ST3 are generated by simultaneously turning ON both switches of R-phase leg, Y-phase leg and B-phase leg of inverter respectively. These shoot-through states give minimum number of switching transition that means minimum switching loss. Each of these type of shoot-through has four kind of switching combination i.e. other two phases can switched in four combination. In the Table-3, X denotes either ON (1) or OFF (0).

   Table-3: Shoot-through states for 2-level ZSI

   ST-State	S 1	S4	S3	S6	S5	S2	Sub-State	Switching
   ST1	1	1	X	X	X	X
   ST2	X	X	1	1	X	X	4	Minimum
   ST3	X	X	X	X	1	1
   ST4	1	1	1	1	X	X
   ST5	X	X	1	1	1	1	2	Medium
   ST6	1	1	X	X	1	1
   ST7	1	1	1	1	1	1	0	Maximum

   Similarly, ST4, ST5 and ST6 are generated by simultaneously turning ON of R-Y phases, Y-B phases and B-R phases respectively. These kind of shoot-through states involve more number of switching transitions i.e. switching loss is more as compare to ST1, ST2 and ST3. But the highest switching loss is involved with the shoot-through state ST7 as it is generated by turning ON simultaneously all the switches of

   MBC gives maximum boosting factor and less switching stress as the total zero vector state is replaced by shoot- through state. The space vector hexagon for ZSI is similar to VSI as shown in Fig.2(a). The difference between vector sequences for VSI and ZSI is only the present of shoot through state in ZSI vector sequence. By inserting proper shoot-through, based on the advanced PWM for VSI, new PWM sequences for ZSI can be produced. Table-4 presents the different possible vector sequences for ZSI. It can be observed that for minimize the switching losses, only ST1, ST2 and ST3 shoot-through states have been used in the proposed vector sequences.The division of shoot-through time period should be equal in a Ts. Unlike MBC, in SBC some portion of the traditional zero vector is replaced by shoot- through state. The timing diagrams for different PWM techniques for ZSI under SBC are shown in Fig.4. Fig.4.(a) shows the timing diagram corresponding to CSVPWM. In similar fashion, Fig.4.(b) and 4(c) present the timing diagrams corresponding special PWM-2 and special PWM-1 respectively. The timing diagram corresponding BCPWM is shown in Fig.4 (d). The dark portion in the timing diagrams indicates the shoot-through state duration. As shown from the timing diagrams that the vector sequence is selected in such way that the total number of switching over Ts is less than or equal to 3. The dwell times for different vectors for ZSI are similar to VSI except the zero vector time duration is divided into two equal parts. One part is replaced by total shoot- through state period, TST. The shoot-through time duration is equally divided into different shoot-through states for getting symmetric voltage waveform over Ts. The dwell times for ZSI in sector-1 are presented in (2). Tm is total zero vector plus shoot-through vector time duration. T0 is the time period corresponding zero vector in SBC.

   three phase legs. It involves highest number of switching transitions. So, for minimum switching loss, ST1, ST2 and ST3 are chosen as the desirable shoot-through states.

   T M Sin( 60 -) T ; T

   1 Sin60 s 2

   ;

   M Sin T

   Sin60 s

   (2)

5. SPACE VECTOR PWM FOR ZSI

   In section-3, the different Space Vector based PWM techniques for 2-level VSI have been discussed. All these PWM techniques can be applied to switch ZSI. In [9], the conventional space vector PWM technique for ZSI has been proposed. Few studies have been done for other PWM techniques. A PWM technique for ZSI is implemented considering the following three control strategies- Simple Boost Control (SBC), Maximum Boost Control (MBC) and Constant Boost Control (CBC).

   Tm Ts T1 T2 ;

   T Tm ; T Tm

   ST 2 0 2

   0 ST1 1

   S1 S4 S3

   S6 S5

   ST2

   2 ST3 7

   1 ST1 0

   S1

   S4 S3

   S6 S5

   T1 T	ST Tm T	ST T1 T	ST T2

   ST1

   1 ST2 2

   comparative study between them can be theoretically analyzed by the measurement of stator flux ripple over the sub-carrier cycle. In this section, the theoretical evaluation of RMS stator flux ripple over a sub-carrier cycle has been done for different vector sequences. As the number of switching for sequence 0- ST-1-ST-2 (ST stands for Shoot-Through) is less compare to

   S2 S2

   other sequences as shown in the earlier section, the sub-carrier

   Tm TST

   TST

   T2 TST Tm

   4 3 3 3 4

   TS

   2 3 2 3 2 3

   TS

   cycle for this sequence is considered as 0.67Ts but others reains same as Ts. But under maximum boost control

   1. (b)

      technique, the zero vectors are replaced by corresponding

      shoot-through vectors. Under MBC, the number of switching

      0 ST1 1

      S1 S4 S3

      S6 S5

      S2

      Tm TST T1

      2 3 2

      ST2 2

      TST T2

      3

      TS

      ST2 1

      TST T1

      3 2

      0 ST1

      S1 S4 S3

      S6 S5

      TST

      0.67Tm

      S2

      2 3

      1

      0.67 T1

      0.67TS

      ST2

      TST

      3

      2

      1. T2

         (ON to OFF or OFF to ON) for ST-1-ST-2-ST and 1-ST-1-

         ST-2 are same and equal to 4. Whereas under MBC, the number of switching (ON to OFF or OFF to ON) for ST-1- ST-2-ST-1 and ST-1-ST-2 are 5 and 3 respectively. For evaluating the theoretical study under same average switching frequency, the sub-carrier cycle for ST-1-ST-2-ST, 1-ST-1- ST-2, ST-1-ST-2-ST-1 and ST-1-ST-2 are 0.8Ts, 0.8Ts, Ts and

         0.6Ts respectively. For evaluating the time integral of error voltage vector, few quantities have been defined along the q- axis and d-axis. The quantities are the product of dwell times and the error voltage vector components along the q and d

         (c) (d)

         Fig.4: Timing diagrams of different PWM techniques for 2-level ZSI under SBC: Timing diagram corresponding to (a) CSVPWM (0127-7210), (b) Special PWM-2 (1012-2101) ,(c) Special PWM-1 (0121-1210) and (d)300 BCPWM (012-210)

6. MAXIMUM BOOST CONTROL

   In Maximum Boost Control (MBC), the total traditional zero vector duration is replaced by corresponding shoot-through state duration. As a result, the boosting factor increases and the voltage stress across switching device decreases. As the space vector hexagon is divided into equal six sector and symmetry to each other, the average shoot-through over fundamental cycle is same as that for one sector. The

   expression for average shoot-through duration (), average shoot-through duty ratio (0) and boosting factor, B under

   axes. The quantities are n1, n2 and nZ corresponding to q-axis component of error voltage vector e1, e2 and e0/7 respectively. e0/7 does not have any component along the d- axis. The quantity along the d-axis for both e1 and e2 is denoted by A. The quantities are tabulated in the Table-6. The

   time integral of error voltage vector is drawn in d-q reference frame for a particular spatial angle, . The stator flux ripple

   vector starts from origin and increasing in a particular direction same as the direction of corresponding error voltage vector. For example, for e1, the stator flux ripple vector will increase in the direction of e1 vector. The tip of stator voltage is forming a geometric shape in the reference frame and comes back to the origin again at the end of the sub-cycle period

   MBC are shown in (3).

   V2 Ve2

   _ 3 3 2M

   2M

   q

   2 3 M

   T ST Ts

   Ts Sin 3

   Ts Sin d 1

   Ts

   Vref

   0 3

   3

   V e0/7

   Ve1

   _ V0/7

   _ T ST

   2 3M ; B

   (3) V 1

   D0

   Ts

   1

   4 3M

   d

7. CURRENT RIPPLE ANALYSIS BASED ON STATOR FLUX RIPPLE

   The error voltage vector is defined as the difference between the applied voltage vector and desired reference

   Fig.5: Error voltage vectors in sector-1

   Table-5:Different error voltage vectors and corresponding q and d

   axis components in sector-1

   vector. The error voltage vector corresponding applied vector

   1, 2 and 0/7 are e1, e2 and e0/7 respectively in sector-1

   Error Vector

   Error Vector

   q-component

   Error Vector

   d-component

   as shown in Fig-5. Table-5 presents the d and q components of

   V e1 V 1 V ref

   V e1q V 1 CosV ref

   V e1d V 1 Sin

   error voltage vectors. The load or line current ripple of the inverter can be measured in term of stator flux ripple. The

   Ve2 V2 Vref

   Ve2q V2 Cos( ) Vref

   3

   V e2d V 2 Sin ( )

   3

   stator flux ripple vector is the time integral of error voltage

   0

   vector. The measurement of stator flux ripple over a sub- carrier cycle is an adequate measurement of line/load current ripple over a sub-carrier cycle [12-15]. As in the previous section, the different vector sequences based on advanced PWM techniques for VSI, have been proposed for ZSI, the

   Vez V0/ 7Vref

   V ezq V ref

   Table-6: Different quantities when reference vector in sector-1 ST

   1 q

   (Error Vector-Sec) q

   (Error Vector-Sec) ST

   d ~

   n1 ( V1

   Cos V

   ref

   ) T1

   A V1

   2

   1

   1

   Sin T

   ST

   n (V Sin ( ) V

   ) T

   A V

   Sin ( ) Td

   2 2

   n z

   3

   V ref

   ref 2

   • T sh

   2 3 2

   0

   ~ 0.33 nZ 0.5n1

   q

   0.67nZ 0.5n1 n3 ~

   0.5 A

   The shape of the geometry is different for different vector sequences. The stator flux ripple vector can be decomposed into d-axis and q-axis components. Further, the components

   0.67 nZ 0.5n1

   0.33nZ

   Tsh T 1 Tsh

   3 2 3

   d

   0.5n1

   Tsh T1

   T2 3 2

   Tsh T1 Tsh

   3 2 3

   0.5A Tsh T1

   T2 3 2

   are drawn with respect to time. The variation of components of stator flux ripple vector with respect to time is different for different vector sequences as shown in Fig-6. The components consist of linear piece of lines and the slope of the line changes at the switching transition of vector sequence.

   TS

   ST

   (c)

   TS

   q

   ST ST q

   ~

   ~

   1

   2 ST

   1 2 d

   ST 0.3nZ 0.6n1

   q

   d ~

   0.6n 0.6n ~

   0.6A

   Z 1

   0.264 n 0.8 n d

   Z 1

   ~ 0.264 nZ

   0.8A

   3Tsh

   0.3nZ

   3T1

   3Tsh

   3T2

   3Tsh

   3T1

   3Tsh

   3T2

   q 0.536 nZ 0.8n1

   10 5 10 5

   10 5 10 5

   ~

   0.6TS

   0.6TS

   4Tsh 0.264 nZ

   15

   0.8T1

   4Tsh

   15

   0.8T2

   4Tsh

   15

   d 4Tsh 15

   0.8T1

   4Tsh 15

   0.8T2

   4Tsh 15

   0 .8T S

   (a)

   1

   ~

   0 .8T S

   q

   (d)

   Fig-6: The variation of stator flux ripple in d-q reference frame over sub cycle and the variation of q and d axis components of stator flux ripple over one sub-cycle with time for (a) ST-1-ST-2-ST, (b) 1-ST-1- ST-2, (c) ST-1-ST-2-ST-1 and (d) ST-1-ST-2

   For evaluating the RMS stator flux ripple over the sub-cycle, the RMS value of individual d and q axis need to be found.

   2 ST

   1

   The square of linear piece of line becomes parabola. The RMS value of stator flux ripple is summation of individual d and q axis RMS value. RMS stator flux ripple- 0127, 1012,

   ST

   d 0121

   and 012

   over a sub-cycle for ST-1-ST-2-ST, 1-ST-1-

   0.8A

   ST-2, ST-1-ST-2-ST-1 and ST-1-ST-2 respectively are expressed in square form in (4a), (4b), (4c) and 4(d) .

   0 .4 n1

   0.8n 0.4n

   2 1 (

   )2

   1 z 0 .4 A

   ~ ~

   q d

   0127 = 3 0.264 1

   3 +

   4T1

   10

   4Tsh

   10

   0.4 n1 0.4nz

   4T1 10

   4Tsh 10

   0.8n2

   4T2

   5

   4T1 10

   4Tsh 10

   4T1

   10

   4Tsh 10

   4T2

   5

   3 [(0.264)2 + (0.264)(0.264 + 0.81 )

   + (0.264 + 0.8 )2] 1 +

   0.8TS

   0.8TS

   1 [(0.264 + 0.8 )2

   1

   (b)

   3

   1

   + (0.264 + 0.81)(0.536

   + 0.8 ) + (0.536 + 0.8 )2] +

   1 1 3

   4(a)

   1 [(0.536 + 0.8 )2 (0.536 + 0.8 )(0.264 )

   3 1

   1

   + (0.264 )2] 1 +

   1 (0.264 )2 + 1 (

   )2 1 + + 2]

   -6

   x 10

   3

   F1012

   2 1

   3

   2 1

   3 0.8 [

   12

   11

   0127

   F

   RMS Stator flux ripple

   1012 = 3 (0.41) 2 + 10

   F012

   9

   1 [(0.4 )2 + (0.4 )(0.4 + 0.4 ) 8

   3 1 1 1

   7

   F0121

   2 6

   + (0.41 + 0.4) ] 2 +

   5

   1 [(0.4 + 0.4 )2 + (0.4 + 0.4 )(0.8 + 0.4 ) 4

   3 1

   1

   1

   1

   4(b) 3

   (degree )

   2

   + (0.81 + 0.4)2] 2 +

   0 10 20 30 40 50 60

   1 [(0.8 + 0.4 )2 (0.8 + 0.4 )(0.4 )

   3 1

   1

   + (0.4)2]

   + 2

   Fig-7: Variation of RMS stator flux ripple with over a sector (Vref=0.8 and Ts=1e-4 sec)

   -6

   x 10

   1 1

   1

   15

   2 9

   3 (0.4)2 + 3 (0.8)2 [ +

   + ]

   8

   8

   RMS Stator flux ripple

   F1012 F

   2 = 1 (0.33 )2 +

   7 0127

   0121 3

   3

   6 F012

   1 [(0.33 )2 + (0.33 )(0.33 + 0.5 ) 5

   3

   1

   4 F0121

   + (0.33 + 0.51)2]

   1 +

   2 3

   1 [(0.33 + 0.5 )2 + (0.33 + 0.5 )(0.67 + 0.5 ) 2

   3 1

   1

   1

   1

   + (0.67 + 0.5 )2] + 0

   Vref

   1

   3

   4(c)

   0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

   1 [(0.67 + 0.5 )2 + (0.67 + 0.5 )(0.67 + 0.5

   Fig-8: Variation of RMS stator flux ripple with reference vector V

   3 1

   1

   1

   0 -4

   ref

   ( =30 and Ts=1e sec)

   + ) + (0.67 + 0.5

   + )2] 2 +

   It can be observed from 4(a), 4(b), 4(c) and 4(d) that the

   2

   1

   1 2

   RMS stator flux ripple depends on the spatial angle ,

   3 [(0.67 + 0.51 + 2)2 (0.67 + 0.51 + 2)(0.51 )

   + (0.5 )2] +

   1 3

   reference voltage magnitude, Vref and sub-cycle period Ts. Fig- 7 shows the variation of RMS stator flux ripple for different vector sequences for a spatial angle variation from 00 to 600 for Vref=0.8 and Ts=1e-4 sec. It can be observed that RMS

   1 (0.5 )2 1

   1 2 1

   2

   stator flux ripple is lowest for ST-1-ST-2-ST-1 (F0121) in

   3 1 2 + 3 [ + 2 + 4 ]

   4

   2 = 1 (0.3 )2 +

   middle range of . After that vector sequence ST-1-ST-2 gives the lesser RMS stator flux ripple compare to other

   012 3

   1 2

   2

   2 1

   sequences. Further the Fig-8 shows the variation of RMS

   stator flux ripple for a range of Vref from 0 to maximum value

   3 [(0.3)

   + (0.3)(0.3 + 0.61 ) + (0.3 + 0.61) ]

   +

   0.866 for spatial angle, = 300 and sub-cycle period, Ts = 1e- 4 sec. It can be observed that at higher value of Vref (at higher modulation index), vector sequence ST-1-ST-2-ST-1 has

   1 [(0.3 + 0.6 )2 + (0.3 + 0.6 )(0.6

   + 0.6 )

   lowest ripple compare to other vector sequences.

   3 1

   1 1

   + (0.6 + 0.6 )2] +

   4(d)

8. SIMULATION RESULTS

   1

   2

   In this section, the simulation study of proposed PWM

   1 [(0.8 + 0.4 )2 (0.8 + 0.4 )(0.4 )

   techniques have been done and verified with the theoretical

   3 1

   1

   analysis. For the simulation study, standard power electronics

   + (0.4 )2] +

   circuit simulator, Power-Sim software (PSim9.1.1) have been

   1 (0.6

   + 0.6 )2 + 1 (

   2

   )2 1

   1.5

   2

   selected. All the PWM techniques have been implemented in this software with appropriate logical approach. On the other

   3

   1

   3 0.6

   [ + + ]

   hand, for power circuit, the inductors and capacitors for Z- network have been selected according the specified current and voltage ripples. The simulation has been done considering R-L load and different motor load conditions.

   1. Simulation study with 3-phase R-L load

      With the following specifications, the simulation study for R- L load has been done-

      1. Input DC voltage, Vin=162V

      2. Load resistance R=100 .

      3. Load inductance, L=20mH

      4. Switching frequency, fsw=10kHz

      5. Fundamental frequency, f1=50Hz

         The simulation study has been done considering variable modulation index and variable boosting factor condition.

         8

         7.5

         1.5

         Load Current (A)

         1

         0.5

         0

         -0.5

         -1

         -1.5

         1.5

         Load Current (A)

         1

         0.5

         0

         THD=6.26%

         t(sec)

         1.1 1.11 1.12 1.13

         (c)

         7

         Line current THD(%)

         6.5

         6

         5.5

         5

         4.5

         1-ST-1-ST-2

         ST-1-ST-2

         ST-1-ST-2-ST

         -0.5

         -1

         -1.5

         THD=4.18%

         t(sec)

         1.1 1.11 1.12 1.13

         (d)

         4

         3.5

         3

         8

         ST-1-ST-2-ST-1

         1 1.5 2 2.5 3 3.5 4 4.5

         Boosting Factor , B

         (a)

         Fig.-10: Load current and its THD for R-L load for (a) ST-1-ST-2-ST, (b) ST-1-ST-2-ST-1, (c) 1-ST-1-ST-2 and (d)ST-1-ST-2 (modulation index=0.8)

         THD=12.87%

         t(sec)

         No Load Current (A)

         3

         2

         1

         0

         -1

         -2

         7.5

         7

         Line current THD(%)

         6.5

         6

         5.5

         5

         4.5

         ST-1-ST-2-ST

         1-ST-1-ST-2

         ST-1-ST-2

         -3 14.04 14.05 14.06 14.07

         (a)

         THD=4.014%

         t(sec)

         No Load Current (A)

         3

         2

         1

         0

         4

         3.5

         3

         ST-1-ST-2-ST-1 -1

         -2

         0.55 0.6 0.65 0.7 0.75 0.8 0.85

         Modulation index, M

         (b)

         -3 12.82 12.83 12.84 12.85

         (b)

         THD=13.65%

         t(sec)

         3

         No Load Current (A)

         Fig-9: Variation of Simulated line current THD for different (a) boosting 2

         factor, (b) modulation index after applying different PWM vector sequence for

         R-L load (R=100 , L=20mH) 1

         0

         1.5

         Load Current (A)

         1

         0.5

         0

         -0.5

         -1

         -1.5

         1.5

         Load Current (A)

         1

         0.5

         0

         -0.5

         -1

         -1.5

         THD=5.26%

         t(sec)

         1.1 1.11 1.12 1.13

         (a)

         THD=3.60%

         t(sec)

         -1

         -2

         -3 13.32 13.33 13.34 13.35

         (c)

         No Load Current (A)

         3

         2

         1

         0

         -1

         -2

         -3 13.16 13.17 13.18 13.19

         (d)

         THD=6.999%

         t(sec)

         1.1 1.11 1.12 1.13

         Fig.11: No Load Motor current and its THD for (a) ST-1-ST-2-ST, (b) ST-1- ST-2-ST-1, (c) 1-ST-1-ST-2 and (d)ST-1-ST-2 (modulation index=0.8)

         The selection of the X-network inductors and capacitors by considering the following specifications-

         1. Network is symmetric i.e. L1=L2 and C1=C2

         2. Current ripple of the inductors is taken 5% of the input current.

         3. Voltage ripple across the capacitors is 0.1% of capacitor voltage.

         The calculated values of inductors and capacitors for Z- network are 18.466mH and 429µF.

         Fig-9 presents the variations of line current THD for different values of modulation index and boosting factor for different PWM techniques with R=100 , L=20mH. It can be observed that ST-1-ST-2-ST-1 and ST-1-ST-2 vectors give the minimum THD compare to others two vector sequences. Further, at high modulation index, ST-1-ST-2-ST-1 is better than ST-1-ST-2 as shown in Fig.-9(b). Fig.-10 shows the line current waveforms for the specified R-L load conditions for different vector sequences. It is clearly observed that the quality of waveform is better for ST-1-ST-2-ST-1 compare to other vetor sequences. Further, the measured THD of line current is lesser compare to that for other PWM techniques. After this, vector sequence ST-1-ST-2 gives lesser THD. Only vector sequence 1-ST-1-ST-2 gives higher THD compared to conventional space vector PWM sequence ST-1- ST-2-ST. This simulation study validates the theoretical study which has been presented in the previous section.

   2. Simulation Study With 3-Phase Motor Load

      A three phase Squirrel-Cage Induction motor with following specifications has been selected and simulated with ZSI under no-load, half full load and full load conditions under different PWM techniques. The specifications are-

      1. Rated power=2HP

      2. Rated RMS line to line voltage=208V

      3. Stator resistance=4.2

      4. Rotor resistance=3

      5. Stator inductance=0.001H

      6. Rotor inductance=0.001H

      7. Magnetizing inductance=0.041H

      8. Moment of inertia=0.7kg-m2

      9. No of pole=4

      10. Operating frequency=50Hz

   The simulation study has been done under modulation index

   0.8. The calculated value of input voltage to the ZSI is 243.46V. It can be observed from Fig-11 that the vector sequence ST-1-ST-2-ST-1 gives the better quality of no load current waveform compare to other vector sequences. Under different load conditions, the THD values of load current for different sequences are tabulated in Table-7. It can be concluded that for any load conditions vector sequence ST-1- ST-2-ST-1 gives the lowest load current THD compare to others. After vector sequence ST-1-ST-2-ST-1, ST-1-ST-2 give the lowest load current THD. So, the theoretical analysis what has been done in the earlier sections is validated with the simulation study of motor load under different load conditions.

   Table7: Load current THD under different load conditions under different PWM techniques

   Load 1-ST-1-ST-2 Condition	ST-1-ST-2-ST	ST-1-ST-2	ST-1-ST-2-ST-1
   No 13.65% Load	12.87%	6.999%	4.014%
   Half 11.8247% Load	11.484%	7.086%	3.8063%
   Full 6.7897% Load	6.3142%	5.213%	3.9934%

9. CONCLUSION

This paper proposes certain vector sequences based on advanced Space Vector based PWM techniques. By Psim simulation study of a ZSI, the effectiveness of the proposed techniques have been verified for R-L and different motor load conditions. The theoretical analysis and comparison of different sequences have been done by the concept of stator flux ripple which represents the line current ripple. The simulation study is validated with theoretical study. It has been observed that the propose sequences give lesser line current distortions and switching losses (as clamping is present) compare to classical SVPWM except 1-ST-1-ST-2 sequence. Further, same PWM techniques can be applied to other type of ZSI topologies like quasi-ZSI. Experimental validation of the proposed techniques can be future work. Further, hybrid PWM techniques for more reduction of output distortion for ZSI can be implement and verified in future. This study gives a new direction to the researcher in this area for implementing different real-time Space Vector based PWM techniques for different ZSIs.

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