Menu
s
0 Comments

Control Techniques for Three-Leg Voltage Source Inverter Fed Two-Phase Induction Motors
Electrical Engineering Department
Faculty of Eng., South Valley University,
Qena, Egypt
Abstract—The progress of power electronics together with the regularly cost reduction, two-phase induction motor (TPIM) is considered as a variable speed motor for home applications. Three-leg voltage source inverter (VSI) is used for driving the TPIM. Current control technique is used to regulate the stator currents of TPIM in synchronous rotating reference frame. Switching states for the three-leg VSI is obtained by current control. This paper presents a study of different current control techniques for the three-leg VSI. The motor performance under current control techniques have been demonstrated through computer simulations. The results show the merits and demerits of each controller.

Index Terms—Two-phase induction motors, VSI, Hysteresis current control, Finite control set-model predictive control.

NOMENCLATURE
Vsd / VsqStator voltage in d/q-axis V.

rsd / rsqStator resistance in d/q-axis ?.

isd / isqStator current in d/q-axis A.

?sd / ?sqStator flux linkage in d/q-axis Wb.turns.

rr Rotor resistance ?.

ird / irqRotor current in d-axis A.

?sd / ?sqRotor flux linkage in d/q-axis Wb.turns.

llsd / llsqStator self-inductance in d/q-axis H.

lr Rotor self-inductance H.

lmd / lmqMutual inductance in d/q-axis H.

TeElectromagnetic torque N.m.

TLLoad torque N.m.

JMotor inertia kg.m2.

BViscous friction N.m.sec.

?sSynchronous speed rad/sec.

?rMotor speed rad/sec.

?slSlip speed rad/sec.

PNumber of poles.

NaNumber of turns of auxiliary winding.

NmNumber of turns of main winding.

nTurns ratio
VSI Voltage source inverter.

TPIM Two-phase induction motor.

FCS-MPC Finite control set-model predictive control.

HCC Hysteresis current control.

Introduction
T
he global demand for households and commercial applications such as refrigerators, air conditioners, washing machines, pumps…etc., has been increased significantly in recent years ADDIN EN.CITE <EndNote><Cite><Author>Ibrahim</Author><Year>2015</Year><RecNum>44</RecNum><DisplayText>1, 2</DisplayText><record><rec-number>44</rec-number><foreign-keys><key app=”EN” db-id=”xpwd2swdae20ase59pjvxxxb0axavz5a202s”>44</key></foreign-keys><ref-type name=”Conference Proceedings”>10</ref-type><contributors><authors><author>Ibrahim, Shibal</author><author>Hassan, Muhammad Shoaib</author><author>Zaffar, Nauman Ahmad</author></authors></contributors><titles><title>Networked Control of single-phase induction motor air conditioners for constrained power systems</title><secondary-title>Industrial Electronics (ISIE), 2015 IEEE 24th International Symposium on</secondary-title></titles><pages>524-529</pages><dates><year>2015</year></dates><publisher>IEEE</publisher><isbn>1467375543</isbn><urls></urls></record></Cite><Cite><Author>Reicy</Author><Year>2005</Year><RecNum>45</RecNum><record><rec-number>45</rec-number><foreign-keys><key app=”EN” db-id=”xpwd2swdae20ase59pjvxxxb0axavz5a202s”>45</key></foreign-keys><ref-type name=”Conference Proceedings”>10</ref-type><contributors><authors><author>Reicy, Sh</author><author>Vaez-Zadeh, S</author></authors></contributors><titles><title>Vector control of single-phase induction machine with maximum torque operation</title><secondary-title>Industrial Electronics, 2005. ISIE 2005. Proceedings of the IEEE International Symposium on</secondary-title></titles><pages>923-928</pages><volume>3</volume><dates><year>2005</year></dates><publisher>IEEE</publisher><isbn>0780387384</isbn><urls></urls></record></Cite></EndNote>1, 2.Two-phase induction motors (TPIMs) are commonly used for low power AC motors especially for residential households due to low cost, reliability and robustness ADDIN EN.CITE <EndNote><Cite><Author>Kim</Author><Year>1994</Year><RecNum>46</RecNum><DisplayText>3</DisplayText><record><rec-number>46</rec-number><foreign-keys><key app=”EN” db-id=”xpwd2swdae20ase59pjvxxxb0axavz5a202s”>46</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Kim, Young-Real</author><author>Sul, Seung-Ki</author><author>Park, Min-Ho</author></authors></contributors><titles><title>Speed sensorless vector control of induction motor using extended Kalman filter</title><secondary-title>IEEE Transactions on Industry Applications</secondary-title></titles><periodical><full-title>IEEE Transactions on Industry Applications</full-title></periodical><pages>1225-1233</pages><volume>30</volume><number>5</number><dates><year>1994</year></dates><isbn>0093-9994</isbn><urls></urls></record></Cite></EndNote>3. These motors have a laminated iron core stator such as three-phase motors with only two windings (the main and the auxiliary windings), which are arranged perpendicularly to each other and a squirrel cage rotor ADDIN EN.CITE <EndNote><Cite><Author>Sinthusonthishat</Author><Year>2013</Year><RecNum>48</RecNum><DisplayText>4</DisplayText><record><rec-number>48</rec-number><foreign-keys><key app=”EN” db-id=”xpwd2swdae20ase59pjvxxxb0axavz5a202s”>48</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Sinthusonthishat, Saliltip</author><author>Chuladaycha, Nontawat</author></authors></contributors><titles><title>A simplified modulation strategy for three-leg voltage source inverter fed unsymmetrical two-winding induction motor</title><secondary-title>Journal of Electrical Engineering and Technology</secondary-title></titles><periodical><full-title>Journal of Electrical Engineering and Technology</full-title></periodical><pages>1337-1344</pages><volume>8</volume><number>6</number><dates><year>2013</year></dates><isbn>1975-0102</isbn><urls></urls></record></Cite></EndNote>4 as shown in Fig .1.

Fig. 1. The structure of two-phase induction motor (TPIM)
Due to cost reduction of the semiconducting devices, voltage source inverters (VSIs) are widely used on variable speed motor drives ADDIN EN.CITE <EndNote><Cite><Author>Lu</Author><Year>2007</Year><RecNum>49</RecNum><DisplayText>5</DisplayText><record><rec-number>49</rec-number><foreign-keys><key app=”EN” db-id=”xpwd2swdae20ase59pjvxxxb0axavz5a202s”>49</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Lu, Haifeng</author><author>Qu, Wenlong</author><author>Cheng, Xiaomeng</author><author>Fan, Yang</author><author>Zhang, Xing</author></authors></contributors><titles><title>A novel PWM technique with two-phase modulation</title><secondary-title>IEEE Transactions on Power Electronics</secondary-title></titles><periodical><full-title>IEEE Transactions on Power Electronics</full-title></periodical><pages>2403-2409</pages><volume>22</volume><number>6</number><dates><year>2007</year></dates><isbn>0885-8993</isbn><urls></urls></record></Cite></EndNote>5. Wide spread of TPIM is due to the advent of power electronics.
For the TPIM, there are four operation modes: 1) split phase mode ADDIN EN.CITE <EndNote><Cite><Author>Ouquelle</Author><Year>2006</Year><RecNum>50</RecNum><DisplayText>6</DisplayText><record><rec-number>50</rec-number><foreign-keys><key app=”EN” db-id=”xpwd2swdae20ase59pjvxxxb0axavz5a202s”>50</key></foreign-keys><ref-type name=”Conference Proceedings”>10</ref-type><contributors><authors><author>Ouquelle, Hassan</author><author>Champagne, Roger</author><author>Sybille, Gilbert</author></authors></contributors><titles><title>Using power electronics to increase performance and extend the applications range of a single-phase induction machine</title><secondary-title>Industrial Electronics, 2006 IEEE International Symposium on</secondary-title></titles><pages>2382-2388</pages><volume>3</volume><dates><year>2006</year></dates><publisher>IEEE</publisher><isbn>1424404975</isbn><urls></urls></record></Cite></EndNote>6, 2) capacitor-start mode ADDIN EN.CITE <EndNote><Cite><Author>Daw</Author><Year>2016</Year><RecNum>51</RecNum><DisplayText>7</DisplayText><record><rec-number>51</rec-number><foreign-keys><key app=”EN” db-id=”xpwd2swdae20ase59pjvxxxb0axavz5a202s”>51</key></foreign-keys><ref-type name=”Conference Proceedings”>10</ref-type><contributors><authors><author>Daw, Nouri Ali</author></authors></contributors><titles><title>Comparison of lab work and simulation results for speed control of single phase induction motor capacitor starting</title><secondary-title>Sciences and Techniques of Automatic Control and Computer Engineering (STA), 2016 17th International Conference on</secondary-title></titles><pages>397-402</pages><dates><year>2016</year></dates><publisher>IEEE</publisher><isbn>1509034072</isbn><urls></urls></record></Cite></EndNote>7 , 3) capacitor-start-capacitor-run mode ADDIN EN.CITE <EndNote><Cite><Author>Thanyaphirak</Author><Year>2013</Year><RecNum>52</RecNum><DisplayText>8</DisplayText><record><rec-number>52</rec-number><foreign-keys><key app=”EN” db-id=”xpwd2swdae20ase59pjvxxxb0axavz5a202s”>52</key></foreign-keys><ref-type name=”Conference Proceedings”>10</ref-type><contributors><authors><author>Thanyaphirak, Veera</author><author>Kinnares, Vijit</author><author>Kunakorn, Anantawat</author></authors></contributors><titles><title>Soft starting control of single-phase induction motor using PWM AC Chopper control technique</title><secondary-title>Electrical Machines and Systems (ICEMS), 2013 International Conference on</secondary-title></titles><pages>1996-1999</pages><dates><year>2013</year></dates><publisher>IEEE</publisher><isbn>1479914479</isbn><urls></urls></record></Cite></EndNote>8 , and 4) main and auxiliary winding mode. For the main and auxiliary winding mode, driven by an inverter, the conventional devices such as starting and running capacitors in addition to the mechanical switch all had been eliminated. This led to increased reliability due to the removal of mechanical devices and also increased system efficiency. Another advantage is the reduction of total cost ADDIN EN.CITE <EndNote><Cite><Author>Young</Author><Year>1996</Year><RecNum>53</RecNum><DisplayText>9</DisplayText><record><rec-number>53</rec-number><foreign-keys><key app=”EN” db-id=”xpwd2swdae20ase59pjvxxxb0axavz5a202s”>53</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Young, C-M</author><author>Liu, C-C</author><author>Liu, C-H</author></authors></contributors><titles><title>New inverter-driven design and control method for two-phase induction motor drives</title><secondary-title>IEE Proceedings-Electric Power Applications</secondary-title></titles><periodical><full-title>IEE Proceedings-Electric Power Applications</full-title></periodical><pages>458-466</pages><volume>143</volume><number>6</number><dates><year>1996</year></dates><isbn>1359-7043</isbn><urls></urls></record></Cite></EndNote>9.

There are three configurations which use VSI for supplying the main and auxiliary windings of TPIM. These common configurations are 1) two-leg VSI, 2) three-leg VSI and 3) four-leg VSI.
Two-leg two-phase VSI is low cost as it utilizes only four power switches as shown in Fig. 2 (a) . However, the generated two-level pulse width modulation (PWM) pattern has high voltage harmonics. In addition, its performance is very weak as it requires two identical DC voltage sources and if they are not available then it requires two balanced capacitors ADDIN EN.CITE <EndNote><Cite><Author>de Rossiter Corrêa</Author><Year>2000</Year><RecNum>54</RecNum><DisplayText>10</DisplayText><record><rec-number>54</rec-number><foreign-keys><key app=”EN” db-id=”xpwd2swdae20ase59pjvxxxb0axavz5a202s”>54</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>de Rossiter Corrêa, Maurício Beltrão</author><author>Jacobina, Cursino Brandão</author><author>Lima, Antonio Marcus Nogueira</author><author>da Silva, Edison Roberto Cabral</author></authors></contributors><titles><title>Rotor-flux-oriented control of a single-phase induction motor drive</title><secondary-title>IEEE Transactions on Industrial Electronics</secondary-title></titles><periodical><full-title>IEEE Transactions on Industrial Electronics</full-title></periodical><pages>832-841</pages><volume>47</volume><number>4</number><dates><year>2000</year></dates><isbn>0278-0046</isbn><urls></urls></record></Cite></EndNote>10.

The performance of four-leg two-phase VSI is perfect as it doesn’t require capacitor balance or two DC voltage sources as shown in Fig. 2 (b). However it is costly compared to the two legs VSI as it utilizes eight power switches. In this case, more switching devices lead to high power loss and low system reliability ADDIN EN.CITE ;EndNote;;Cite;;Author;Demirkutlu;/Author;;Year;2007;/Year;;RecNum;55;/RecNum;;DisplayText;11;/DisplayText;;record;;rec-number;55;/rec-number;;foreign-keys;;key app=”EN” db-id=”xpwd2swdae20ase59pjvxxxb0axavz5a202s”;55;/key;;/foreign-keys;;ref-type name=”Conference Proceedings”;10;/ref-type;;contributors;;authors;;author;Demirkutlu, Eyyup;/author;;author;Cetinkaya, Suleyman;/author;;author;Hava, Ahmet M;/author;;/authors;;/contributors;;titles;;title;Output voltage control of a four-leg inverter based three-phase UPS by means of stationary frame resonant filter banks;/title;;secondary-title;Electric Machines ;amp; Drives Conference, 2007. IEMDC;apos;07. IEEE International;/secondary-title;;/titles;;pages;880-885;/pages;;volume;1;/volume;;dates;;year;2007;/year;;/dates;;publisher;IEEE;/publisher;;isbn;1424407427;/isbn;;urls;;/urls;;/record;;/Cite;;/EndNote;11.

Due to the high cost of the four-leg VSI and balancing the two capacitors of the two-leg VSI, the three-leg two-phase VSI is preferable. The three-leg can also generate the zero voltage vectors while the two-leg can’t. Three-leg two-phase VSI consists of two power switches per leg. The inverter is connected to the motor windings ADDIN EN.CITE <EndNote><Cite><Author>Meshram</Author><Year>2015</Year><RecNum>56</RecNum><DisplayText>12</DisplayText><record><rec-number>56</rec-number><foreign-keys><key app=”EN” db-id=”xpwd2swdae20ase59pjvxxxb0axavz5a202s”>56</key></foreign-keys><ref-type name=”Conference Proceedings”>10</ref-type><contributors><authors><author>Meshram, Snehil M</author><author>Fadnis, AY</author></authors></contributors><titles><title>Direct Torque Control of three-leg inverter driving two-phase induction motor</title><secondary-title>Power, Communication and Information Technology Conference (PCITC), 2015 IEEE</secondary-title></titles><pages>242-247</pages><dates><year>2015</year></dates><publisher>IEEE</publisher><isbn>1479974552</isbn><urls></urls></record></Cite><Cite><Author>Meshram</Author><Year>2015</Year><RecNum>56</RecNum><record><rec-number>56</rec-number><foreign-keys><key app=”EN” db-id=”xpwd2swdae20ase59pjvxxxb0axavz5a202s”>56</key></foreign-keys><ref-type name=”Conference Proceedings”>10</ref-type><contributors><authors><author>Meshram, Snehil M</author><author>Fadnis, AY</author></authors></contributors><titles><title>Direct Torque Control of three-leg inverter driving two-phase induction motor</title><secondary-title>Power, Communication and Information Technology Conference (PCITC), 2015 IEEE</secondary-title></titles><pages>242-247</pages><dates><year>2015</year></dates><publisher>IEEE</publisher><isbn>1479974552</isbn><urls></urls></record></Cite></EndNote>12 as shown in Fig. 2(c). The proper way to obtain high motor performance with low drive circuit cost will be the three-leg VSI with a simple control technique.

(a)

(b)

(c)
Fig. 2. topologies of inverters for driving TPIM
(a) Two-leg (b) Four-leg (c) Three-leg
In some cases, motor current instead of motor terminal voltages are recommended where current control feedback loop is essential for precise speed or torque control. Several researchers have investigated different control techniques to improve the performance of TPIM. Many current control techniques for power converters have been presented. Hysteresis current controller (HCC) is commonly used in the last decades. It depends on a feedback loop with two level comparators ADDIN EN.CITE <EndNote><Cite><Author>Kazmierkowski</Author><Year>1998</Year><RecNum>38</RecNum><DisplayText>13</DisplayText><record><rec-number>38</rec-number><foreign-keys><key app=”EN” db-id=”xpwd2swdae20ase59pjvxxxb0axavz5a202s”>38</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Kazmierkowski, Marian P</author><author>Malesani, Luigi</author></authors></contributors><titles><title>Current control techniques for three-phase voltage-source PWM converters: A survey</title><secondary-title>IEEE Transactions on Industrial Electronics</secondary-title></titles><periodical><full-title>IEEE Transactions on Industrial Electronics</full-title></periodical><pages>691-703</pages><volume>45</volume><number>5</number><dates><year>1998</year></dates><isbn>0278-0046</isbn><urls></urls></record></Cite></EndNote>13. The switching states are generated when the error exceeds a predefined tolerance band. HCC is simple to implement, fast transient response and independence of machine parameters. Nevertheless, this controller may cause a variable switching frequency and hence produces unwanted harmonics and increased losses ADDIN EN.CITE <EndNote><Cite><Author>Talib</Author><Year>2016</Year><RecNum>39</RecNum><DisplayText>14</DisplayText><record><rec-number>39</rec-number><foreign-keys><key app=”EN” db-id=”xpwd2swdae20ase59pjvxxxb0axavz5a202s”>39</key></foreign-keys><ref-type name=”Conference Proceedings”>10</ref-type><contributors><authors><author>Talib, MHN</author><author>Isa, SN Mat</author><author>Hamidon, HE</author><author>Ibrahim, Z</author><author>Rasin, Z</author></authors></contributors><titles><title>Hysteresis current control of induction motor drives using dSPACE DSP controller</title><secondary-title>Power and Energy (PECon), 2016 IEEE International Conference on</secondary-title></titles><pages>522-527</pages><dates><year>2016</year></dates><publisher>IEEE</publisher><isbn>1509025472</isbn><urls></urls></record></Cite></EndNote>14.
An advanced control technique has been presented such as finite control set-model predictive controller (FCS-MPC). It has been introduced in the last decades as a controller for motor drives. The main concept of FCS-MPC depends mainly on predicting the future behavior of variables. The key parameter of the FCS-MPC is the cost function as it determines the required control action ADDIN EN.CITE <EndNote><Cite><Author>Rodriguez</Author><Year>2013</Year><RecNum>40</RecNum><DisplayText>15</DisplayText><record><rec-number>40</rec-number><foreign-keys><key app=”EN” db-id=”xpwd2swdae20ase59pjvxxxb0axavz5a202s”>40</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>Rodriguez, Jose</author><author>Kazmierkowski, Marian P</author><author>Espinoza, Jose R</author><author>Zanchetta, Pericle</author><author>Abu-Rub, Haitham</author><author>Young, Hector A</author><author>Rojas, Christian A</author></authors></contributors><titles><title>State of the art of finite control set model predictive control in power electronics</title><secondary-title>IEEE Transactions on Industrial Informatics</secondary-title></titles><periodical><full-title>IEEE Transactions on Industrial Informatics</full-title></periodical><pages>1003-1016</pages><volume>9</volume><number>2</number><dates><year>2013</year></dates><isbn>1551-3203</isbn><urls></urls></record></Cite></EndNote>15. It uses the states of VSI and the future behavior is predicted at the next sample. Then the switching state which minimizes the predefined cost function is applied.
The purpose of this paper is to examine and compare the performance of three-leg VSI fed TPIM using different control techniques. This paper is structured as follows; section II presents the dynamic model of TPIM. In section III, the three-leg two-phase VSI construction has been investigated. HCC and FCS-MPC have been discussed in section IV and V, respectively. The simulation results have been analyzed in section VI. Finally, section VII summarizes the main points of the results.

Mathematical Model Of TpimThe two-phase induction motor has two windings where the displacement between these windings is 90 electrical degrees. As the TPIM has asymmetry number of turns in windings, hence they will lead to different resistance and reactance. By neglecting core saturation and iron losses, the complete set of equations that describes the mathematical model of asymmetrical windings in d-q frame as follows ADDIN EN.CITE <EndNote><Cite><Author>de Rossiter Correa</Author><Year>2002</Year><RecNum>57</RecNum><DisplayText>16, 17</DisplayText><record><rec-number>57</rec-number><foreign-keys><key app=”EN” db-id=”xpwd2swdae20ase59pjvxxxb0axavz5a202s”>57</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>de Rossiter Correa, Mauricio Beltrao</author><author>Jacobina, Cursino Brandao</author><author>Lima, Antonio Marcus Nogueira</author><author>da Silva, Edison Roberto Cabral</author></authors></contributors><titles><title>A three-leg voltage source inverter for two-phase AC motor drive systems</title><secondary-title>IEEE Transactions on Power Electronics</secondary-title></titles><periodical><full-title>IEEE Transactions on Power Electronics</full-title></periodical><pages>517-523</pages><volume>17</volume><number>4</number><dates><year>2002</year></dates><isbn>0885-8993</isbn><urls></urls></record></Cite><Cite><Author>de Rossiter Corrêa</Author><Year>2004</Year><RecNum>58</RecNum><record><rec-number>58</rec-number><foreign-keys><key app=”EN” db-id=”xpwd2swdae20ase59pjvxxxb0axavz5a202s”>58</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>de Rossiter Corrêa, Maurício Beltrão</author><author>Jacobina, Cursino Brandão</author><author>Da Silva, Edison Roberto Cabral</author><author>Lima, Antonio Marcus Nogueira</author></authors></contributors><titles><title>Vector control strategies for single-phase induction motor drive systems</title><secondary-title>IEEE Transactions on Industrial Electronics</secondary-title></titles><periodical><full-title>IEEE Transactions on Industrial Electronics</full-title></periodical><pages>1073-1080</pages><volume>51</volume><number>5</number><dates><year>2004</year></dates><isbn>0278-0046</isbn><urls></urls></record></Cite></EndNote>16, 17:
VsdVsq=rsd00rsqisdisq+ddt?sd?sq(1)
00=rr00rrirdirq+ddt?rd?rq+?r01-10?rd?rq(2)
?sd?sq=llsd00llsqisdisq+lmd00lmqirdirq(3)
?rd?rq=lr00lrirdirq+lmd00lmqisdisq(4)
Te=P2(lmq. isq.ird-lmd. isd.irq)(5)
P2(Te-TL)=Jd?rdt+B?r(6)
From equation (5), the torque contains two terms; the average torque and pulsation torque. As reduction in pulsation torque leads to minimization of stator currents unbalance, many researchers with different assumption are focused to reduction in torque pulsation ADDIN EN.CITE <EndNote><Cite><Author>Vaez-Zadeh</Author><Year>2000</Year><RecNum>42</RecNum><DisplayText>18</DisplayText><record><rec-number>42</rec-number><foreign-keys><key app=”EN” db-id=”xpwd2swdae20ase59pjvxxxb0axavz5a202s”>42</key></foreign-keys><ref-type name=”Conference Proceedings”>10</ref-type><contributors><authors><author>Vaez-Zadeh, S</author><author>Langari, H</author></authors></contributors><titles><title>High average-low pulsating torque operation of single phase induction motors</title><secondary-title>Industry Applications Conference, 2000. Conference Record of the 2000 IEEE</secondary-title></titles><pages>1513-1518</pages><volume>3</volume><dates><year>2000</year></dates><publisher>IEEE</publisher><isbn>0780364015</isbn><urls></urls></record></Cite></EndNote>18. By referring all parameters in rotor and stator to d-axis, equations from (1) to (5) in synchronously rotating reference frame which is donated by (r) can be expressed as follows:
VsdrVsqr=rsdr00rsqrisdrisqr+ddt?sdr?sqr+?s0-110?sdr?sqr (7)
00=rrr00rrrirdrirqr+ddt?rdr?rqr+?sl0-110?rdr?rqr(8)
?sdr?sqr=llsdr00llsqrisdrisqr+lmdr00lmqrirdrirqr(9)
?rdr?rqr=lrr00lrrirdrirqr+lmdr00lmqrisdrisqr(10)
Te=P2(lmqr. isqr.irdr-lmdr. isdr.irqr)(11)

Fig. 4. Modeling of TPIM
Three-leg two-phase VSI
The structure of three-leg two-phase VSI is shown in Fig. 3. The inverter consists of six unidirectional power switches (Q1, Q2, Q3, Q4, Q5, and Q6) where two switches in each leg. There are eight possible combinations for the three upper switches which provide output voltage in the main and auxiliary winding of TPIM as illustrated in Table. I. The ON and OFF states between upper power switches (Q1, Q3, and Q5) and lower power switches (Q2, Q4, and Q6) are complementary.

TABLE I. Switching States Of The Three-Leg VSI
Vectors Switching States VabVcbQ1 Q3 Q5 V0 0 0 0 0 0 0
V1 Vdc1 0 0 Vdc0
V2 ?2Vdc?45° 1 0 1 VdcVdcV3 Vdc?90° 0 0 1 0 VdcV4 ?2Vdc?255° 0 1 0 -Vdc-VdcV5 Vdc?180° 0 1 1 – Vdc0
V6 Vdc?270° 1 1 0 0 – VdcV7 0 1110 0

Fig. 3. Three-leg two-phase VSI
Hysteresis Current Control
The TPIM is more complex than three phase machine due to the fact of asymmetry of its windings. A modified TPIM which is presented by Correa ADDIN EN.CITE <EndNote><Cite><Author>de Rossiter Corrêa</Author><Year>2000</Year><RecNum>60</RecNum><DisplayText>10</DisplayText><record><rec-number>60</rec-number><foreign-keys><key app=”EN” db-id=”xpwd2swdae20ase59pjvxxxb0axavz5a202s”>60</key></foreign-keys><ref-type name=”Journal Article”>17</ref-type><contributors><authors><author>de Rossiter Corrêa, Maurício Beltrão</author><author>Jacobina, Cursino Brandão</author><author>Lima, Antonio Marcus Nogueira</author><author>da Silva, Edison Roberto Cabral</author></authors></contributors><titles><title>Rotor-flux-oriented control of a single-phase induction motor drive</title><secondary-title>IEEE Transactions on Industrial Electronics</secondary-title></titles><periodical><full-title>IEEE Transactions on Industrial Electronics</full-title></periodical><pages>832-841</pages><volume>47</volume><number>4</number><dates><year>2000</year></dates><isbn>0278-0046</isbn><urls></urls></record></Cite></EndNote>10 in order to eliminate some of asymmetrical component by referring rotor variables to the main winding axis. From equation (10), the electromagnetic torque will be;
Te=P.lmqr2.lrr( isqr.?rdrn- isdr.?rqr)(12)
The current supplied to the machine needs to be oriented along the rotor flux vector in other word, the d-axis is oriented along the rotor flux vector then,
?rdr=?rr, ?rqr=0(13)
Therefore (12) is reduced to:
Te=P.lmqr2.n.lrr isqr.?rr(14)
From equations (8), (10) and (13) we find that,
isdr=?rrlmdr(15)
?sl=isqr.lmqr.rrr?rr.lrr(16)
The block diagram which represents the control strategy for hysteresis current control (HCC) is shown in Fig. 4. The main idea of HCC technique is based on the common leg of VSI where,
icom=-(imain +iaux )(17)
By using a hysteresis band for each current, the switching states for the power switches can be generated by comparing isq*, isd* and ic* with the three current imain, iaux. and icom , respectively.

FCS-MPC & The Discrete Model Of TPIM
This section explains the FCS-MPC and the discrete time model of the system. The main concept of FCS-MPC depends mainly on the topology of three-leg two-phase VSI in other word; it is used to determine directly the switching states of three-leg VSI. There are six active states and two non-active states for the three-leg VSI as presented in Table I. The TPIM model contains some parameters which need to be eliminated. In order to eliminate the pulsation torque in equation (5), the effect of rotor current (irq) can be neglected. By Take the assumption as rotor current is constant (ird); toque equation will have the average term which depends on stator current. According to the above assumption, the dynamic model expressed in equations (1) and (3) can now be expressed as follows:
VsdVsq=rsd00rsqisdisq+ddtllsd00llsqisdisq(18)
The required control function here is the currents of the main and auxiliary windings at a future sample time. By using the forward Euler method with a sample time Ts, the derivative term in (18) can be approximated as follows:
di(t)dt?ik+1-i(k)Ts(19)
The discrete model of TPIM can be expressed as follows:
isd(k+1)isq(k+1)=Tsllsd00TsllsqVsdVsq+1-Ts.rsdllsd001-Ts.rsqllsqisd(k)isq(k) (20)
For the required control of TPIM, the cost function is;
g=isd*(k+1)-isd(k+1)2+isq*(k+1)-isq(k+1)2(21)
The eight switching states, which are given in Table I, can be applied at the next instant (k+1), hence the drive performance can be optimized using the cost function. The complete model for TPIM controlled by FCS-MPC is shown in Fig. 4.

Simulation results
The MATLAB/SIMULINK is used to demonstrate the main characteristics and performance of TPIM fed by a three-leg VSI. A 0.25 hp, 110 V, 50 Hz, 4-pole TPIM machine and the motor parameters are presented in Table II. Fig. 8 and Fig. 9 show the simulation results of the TPIM with change in reference rotor speed and load torque under HCC and MPC, respectively.
TABLE. II. Parameters Of Tpim In Matlab/Simulink
Main winding parameters:
Stator resistance (rsq)2.02 ?
Self-inductance (llsq)7.40 mHMutual-inductance (lmd=lmq)0.1172 H
Auxiliary winding parameters:
Stator resistance (rsd)7.14 ?
Self-inductance (llsd)8.50 mHRotor winding parameters:
Rotor resistance (rr)4.12 ?
Self-inductance (lr)5.60 mHMachine characteristic:
Inertia (J) 0.0146 Kg.

No.of pole pairs (P) 2
Turns ratio (n=Na/Nm) 1.18
PI controller gains Proportional gain (Kp) 5 Integral gain (Ki) 1 Rate limiter 5 N.m For HCC, Fig. 8(a) shows the rotational speed of the motor changes from 1500 rpm to 750 rpm at 1.5 sec. From this figure it is seen that actual motor speed tracks the reference speed. Fig. 8(b) shows the electromagnetic and load torque. As expected the electromagnetic torque changes according to change in load torque. The current of the main and auxiliary winding is shown in Fig. 8(c). The asymmetry of the motor can be verified according to the difference in amplitudes of stator currents.

Fig. 9(a) shows the speed response in FCS-MPC. Fig. 9(b) shows the electromagnetic torque. The electromagnetic torque also changes according to load torque. Fig. 9(c) presents the currents of main and auxiliary windings.

Torque ripple factor (Trf) is used to measure increase or decrease in output torque as the motor shaft rotates over one complete revolution which calculated as follow:
Trf=Tmax-TminTavg*100(22)
Comment on torque ripples
Apply higher load torque (4Nm) , Compare switching frequency
Compare power losses and efficiency, FFT for currents

(a) Rotational speed motor

(b) Electromagnetic & load torque

(c) FFT analysis for the electromagnetic torque

(d) Main & auxiliary currents
Fig. 8 performance of TPIM under HCC

(a) Rotational speed motor

(b) Electromagnetic & load torque

(c) FFT analysis for the electromagnetic torque

(d) Main & auxiliary currents
Fig. 9 performance of TPIM under FCS-MPC
Conclusions
In this paper, two current control techniques of the stator currents of TPIM were presented. TPIM was driven by the three-leg VSI. Several simulation results have been investigated to verify the strength of each controller and compared under the same conditions. From these results; it is possible to verify that the HCC has faster response in motor speed compared with FCS-MPC ?????. However, FCS-MPC provides better dynamic performance ?????. in torque with fewer ripples.

References
ADDIN EN.REFLIST 1S. Ibrahim, M. S. Hassan, and N. A. Zaffar, “Networked Control of single-phase induction motor air conditioners for constrained power systems,” in Industrial Electronics (ISIE), 2015 IEEE 24th International Symposium on, 2015, pp. 524-529.2S. Reicy and S. Vaez-Zadeh, “Vector control of single-phase induction machine with maximum torque operation,” in Industrial Electronics, 2005. ISIE 2005. Proceedings of the IEEE International Symposium on, 2005, pp. 923-928.3Y.-R. Kim, S.-K. Sul, and M.-H. Park, “Speed sensorless vector control of induction motor using extended Kalman filter,” IEEE Transactions on Industry Applications, vol. 30, pp. 1225-1233, 1994.4S. Sinthusonthishat and N. Chuladaycha, “A simplified modulation strategy for three-leg voltage source inverter fed unsymmetrical two-winding induction motor,” Journal of Electrical Engineering and Technology, vol. 8, pp. 1337-1344, 2013.5H. Lu, W. Qu, X. Cheng, Y. Fan, and X. Zhang, “A novel PWM technique with two-phase modulation,” IEEE Transactions on Power Electronics, vol. 22, pp. 2403-2409, 2007.6H. Ouquelle, R. Champagne, and G. Sybille, “Using power electronics to increase performance and extend the applications range of a single-phase induction machine,” in Industrial Electronics, 2006 IEEE International Symposium on, 2006, pp. 2382-2388.7N. A. Daw, “Comparison of lab work and simulation results for speed control of single phase induction motor capacitor starting,” in Sciences and Techniques of Automatic Control and Computer Engineering (STA), 2016 17th International Conference on, 2016, pp. 397-402.8V. Thanyaphirak, V. Kinnares, and A. Kunakorn, “Soft starting control of single-phase induction motor using PWM AC Chopper control technique,” in Electrical Machines and Systems (ICEMS), 2013 International Conference on, 2013, pp. 1996-1999.9C.-M. Young, C.-C. Liu, and C.-H. Liu, “New inverter-driven design and control method for two-phase induction motor drives,” IEE Proceedings-Electric Power Applications, vol. 143, pp. 458-466, 1996.10M. B. de Rossiter Corrêa, C. B. Jacobina, A. M. N. Lima, and E. R. C. da Silva, “Rotor-flux-oriented control of a single-phase induction motor drive,” IEEE Transactions on Industrial Electronics, vol. 47, pp. 832-841, 2000.11E. Demirkutlu, S. Cetinkaya, and A. M. Hava, “Output voltage control of a four-leg inverter based three-phase UPS by means of stationary frame resonant filter banks,” in Electric Machines & Drives Conference, 2007. IEMDC’07. IEEE International, 2007, pp. 880-885.12S. M. Meshram and A. Fadnis, “Direct Torque Control of three-leg inverter driving two-phase induction motor,” in Power, Communication and Information Technology Conference (PCITC), 2015 IEEE, 2015, pp. 242-247.13M. P. Kazmierkowski and L. Malesani, “Current control techniques for three-phase voltage-source PWM converters: A survey,” IEEE Transactions on Industrial Electronics, vol. 45, pp. 691-703, 1998.14M. Talib, S. M. Isa, H. Hamidon, Z. Ibrahim, and Z. Rasin, “Hysteresis current control of induction motor drives using dSPACE DSP controller,” in Power and Energy (PECon), 2016 IEEE International Conference on, 2016, pp. 522-527.15J. Rodriguez, M. P. Kazmierkowski, J. R. Espinoza, P. Zanchetta, H. Abu-Rub, H. A. Young, and C. A. Rojas, “State of the art of finite control set model predictive control in power electronics,” IEEE Transactions on Industrial Informatics, vol. 9, pp. 1003-1016, 2013.16M. B. de Rossiter Correa, C. B. Jacobina, A. M. N. Lima, and E. R. C. da Silva, “A three-leg voltage source inverter for two-phase AC motor drive systems,” IEEE Transactions on Power Electronics, vol. 17, pp. 517-523, 2002.17M. B. de Rossiter Corrêa, C. B. Jacobina, E. R. C. Da Silva, and A. M. N. Lima, “Vector control strategies for single-phase induction motor drive systems,” IEEE Transactions on Industrial Electronics, vol. 51, pp. 1073-1080, 2004.18S. Vaez-Zadeh and H. Langari, “High average-low pulsating torque operation of single phase induction motors,” in Industry Applications Conference, 2000. Conference Record of the 2000 IEEE, 2000, pp. 1513-1518.