A Novel Phase Current Reconstruction Method for a Three-Level Neutral Point Clamped Inverter ( NPCI ) with a Neutral Shunt Resistor

Abstract: This paper presents three phase current reconstruction methods for a three-level neutral point clamped inverter (NPCI) by measuring the voltage of a shunt resistor placed in the neutral point of the inverter. In order to accurately acquire the phase currents from the shunt resister, the dwell time of the active voltage vectors need to exceed the minimum time. On the other hand, if the time of active voltage is shorter than the minimum time, the current measurement becomes impossible. In this paper, unmeasurable regions for current are classified into three areas. Area 1 is a region in which both phase currents can be measure. Therefore, it is not necessary to restore the current. In Area 2, only one phase current can be measured. Thus, an estimation or restoration method is needed to measure another phase current. In this paper, the current estimation method using an electrical model of the motor is proposed. Area 3 is the region in which both phase currents can not be measured. In this case, it is necessary to move the voltage vector to the current measurable area by injecting the voltage. In this paper, Area 3 is divided into 36 sectors to inject optimal voltage. The proposed methods have the advantages of high current measurement accuracy and low THD (total harmonic distortion). The effectiveness of the proposed methods are verified through experimental results.


Introduction
Two-level inverters are used in most home appliances, such as washing machines, refrigerators, and air conditioners, due to their simple structure, and high reliability and performance.However, in order to overcome the limitations of the efficiency and harmonics of the two-level inverter, three-level inverters have been recently investigated.The three-level neutral point clamped inverter (NPCI) has a structure characteristic of having a neutral point in the direct current (DC) stage, and thus has excellent electro magnetic interference (EMI) and electro magnetic compatibility (EMC) characteristics, due to a low voltage variation rate when switching [1][2][3][4].Unlike a two-level inverter, each arm of a three-level NPCI consists of four switches and two clamping diodes, as shown in Figure 1.When the DC-link voltage is V dc , the voltage of each capacitor is V dc /2.The neutral point is connected to each phase output node by the clamp diodes and switches.Due to this structural feature, the three-level NPCI can output either V dc /2 or −V dc /2 by turning on the two switches located in the upper or lower side of a phase.When the switches of S x3 and S x4 are turned on, the node voltage of For controlling the AC motor, the controller of the inverter requires the values of the three phase currents, which can be acquired through current sensors or a shunt resistor.A phase current sensing inverter (PCSI), as shown in Figure 2a, is a typical three-phase voltage source inverter with two phase current sensors.It requires at least two current sensors and sensing circuits, which raise the cost of the appliances [5].For this reason, a DC link shunt resistor can be used to measure the phase current, as shown in Figure 2b,c.A multi-shunt inverter (MSI) obtains the phase current from the shunt resistor located in the neutral and bottom of the DC link [6].Measurement of the currents through the shunt resistors is possible when the current flows into the shunt resistor.Since the shunt resistors are located between the neutral and bottom of the DC link, it is possible to measure the phase current only if the active voltage vectors are combined with "O" or "N" switching states.However, although the active vectors are combined with "O" or "N" switching states, the phase current cannot be measured due to the short dwell time of the active vectors.The portion in the space vector diagram where the dwell time of the active vectors is not large enough to measure the current is called the current-unmeasurable area (CUA) [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24].A shunt resistor at the neutral point of the three-level NPCI can be used to measure the phase current, as shown in Figure 2c.This is similar to MSI, but a phase current is only obtainable when applying the state of "O" in the switching combination of the effective voltage vector.NPCI is effective in terms of volume and cost as compared to PCSI and MSI.It also has one current sensing circuit, which reduces the ripples caused by current sensor offset and scaling errors [6,7].However, it has a limited time to sample the phase current in the shunt resistor over one period compared to the PCSI and MSI, so that the unmeasurable areas are widened in the output voltage hexagon.In order to overcome the limitations, some researchers have been interested in NPCI topology for phase current reconstruction.For controlling the AC motor, the controller of the inverter requires the values of the three phase currents, which can be acquired through current sensors or a shunt resistor.A phase current sensing inverter (PCSI), as shown in Figure 2a, is a typical three-phase voltage source inverter with two phase current sensors.It requires at least two current sensors and sensing circuits, which raise the cost of the appliances [5].For this reason, a DC link shunt resistor R shunt can be used to measure the phase current, as shown in Figure 2b,c.A multi-shunt inverter (MSI) obtains the phase current from the shunt resistor located in the neutral and bottom of the DC link [6].Measurement of the currents through the shunt resistors is possible when the current flows into the shunt resistor.Since the shunt resistors are located between the neutral and bottom of the DC link, it is possible to measure the phase current only if the active voltage vectors are combined with "O" or "N" switching states.However, although the active vectors are combined with "O" or "N" switching states, the phase current cannot be measured due to the short dwell time of the active vectors.The portion in the space vector diagram where the dwell time of the active vectors is not large enough to measure the current is called the current-unmeasurable area (CUA) [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24].A shunt resistor at the neutral point of the three-level NPCI can be used to measure the phase current, as shown in Figure 2c.This is similar to MSI, but a phase current is only obtainable when applying the state of "O" in the switching combination of the effective voltage vector.NPCI is effective in terms of volume and cost as compared to PCSI and MSI.It also has one current sensing circuit, which reduces the ripples caused by current sensor offset and scaling errors [6,7].However, it has a limited time to sample the phase current in the shunt resistor over one period compared to the PCSI and MSI, so that the unmeasurable areas are widened in the output voltage hexagon.In order to overcome the limitations, some researchers have been interested in NPCI topology for phase current reconstruction.For controlling the AC motor, the controller of the inverter requires the values of the three phase currents, which can be acquired through current sensors or a shunt resistor.A phase current sensing inverter (PCSI), as shown in Figure 2a, is a typical three-phase voltage source inverter with two phase current sensors.It requires at least two current sensors and sensing circuits, which raise the cost of the appliances [5].For this reason, a DC link shunt resistor can be used to measure the phase current, as shown in Figure 2b,c.A multi-shunt inverter (MSI) obtains the phase current from the shunt resistor located in the neutral and bottom of the DC link [6].Measurement of the currents through the shunt resistors is possible when the current flows into the shunt resistor.Since the shunt resistors are located between the neutral and bottom of the DC link, it is possible to measure the phase current only if the active voltage vectors are combined with "O" or "N" switching states.However, although the active vectors are combined with "O" or "N" switching states, the phase current cannot be measured due to the short dwell time of the active vectors.The portion in the space vector diagram where the dwell time of the active vectors is not large enough to measure the current is called the current-unmeasurable area (CUA) [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24].A shunt resistor at the neutral point of the three-level NPCI can be used to measure the phase current, as shown in Figure 2c.This is similar to MSI, but a phase current is only obtainable when applying the state of "O" in the switching combination of the effective voltage vector.NPCI is effective in terms of volume and cost as compared to PCSI and MSI.It also has one current sensing circuit, which reduces the ripples caused by current sensor offset and scaling errors [6,7].However, it has a limited time to sample the phase current in the shunt resistor over one period compared to the PCSI and MSI, so that the unmeasurable areas are widened in the output voltage hexagon.In order to overcome the limitations, some researchers have been interested in NPCI topology for phase current reconstruction.Previous research has established the effective voltage time by shifting the pulse width modulation (PWM) or injecting the voltage to restore the current [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24].However, these methods cause a high total harmonic distortion (THD) by injecting relatively large voltages.In [6], the minimum voltage injection (MVI) method minimizes voltage distortion and operating noise through THD reduction, but this method does not completely reconstruct the phase current at very high modulation index (MI).In addition, the PWM shifting method of [7] also generates harmonics due to asymmetric voltage modulation.
In this paper, the phase current unmeasurable region is classified into three areas, and the current reconstruction methods are proposed according to each area.First, two phase currents can be measured in Area 1, and the normal operation is executed in this area.On the other hand, only 1 phase current can be acquired in Area 2. In this case, the other phase current can be estimated by combining the q-axis current reference obtained from the speed controller and the electrical model of the motor [10].Lastly, Area 3 is defined as an area where no current can be measured.For measuring the current in this area, the optimal voltage injection method is proposed [12].To realize this, the hexagon of SVPWM is divided into 36 sectors, and the optimum injection voltage according to the sector is calculated.In addition, the current accuracy and THD are compared with the conventional method [6] in various MI conditions.The proposed method is verified through experimental results.

Acquiring Phase Current from Neutral Shunt Resistor
The operation of each three-level NPC inverter phase leg can be represented by a combination of the three switching states "P", "O", and "N".According to these switching states, the inverter has 27 possible combinations of switching states consisting of 24 effective voltage vectors and 3 zero voltage vectors.Because the NPCI has the shunt resistor at the neutral point, the phase current can be acquired only when the effective voltage vector includes the "O" switching state [6].For example, the A phase current can be measured when the effective vector "O", "N", and "N" is applied to the inverter, as shown in Figure 3.
The 27 switching states of the neutral point clamped (NPC) inverter and the respective measurable phase currents are listed in Table 1.In addition, Figure 4 shows the switching vector in the spatial coordinates of the hexagon.Figure 4 and Table 1 show that the current cannot be measured through the neutral-point shunt resistor during the zero vector (V 0 ) and space vectors represented by V 1 to V 6 .
In this paper, the phase current unmeasurable region is classified into three areas, and the current reconstruction methods are proposed according to each area.First, two phase currents can be measured in Area 1, and the normal operation is executed in this area.On the other hand, only 1 phase current can be acquired in Area 2. In this case, the other phase current can be estimated by combining the q-axis current reference obtained from the speed controller and the electrical model of the motor [10].Lastly, Area 3 is defined as an area where no current can be measured.For measuring the current in this area, the optimal voltage injection method is proposed [12].To realize this, the hexagon of SVPWM is divided into 36 sectors, and the optimum injection voltage according to the sector is calculated.In addition, the current accuracy and THD are compared with the conventional method [6] in various MI conditions.The proposed method is verified through experimental results.

Acquiring Phase Current from Neutral Shunt Resistor
The operation of each three-level NPC inverter phase leg can be represented by a combination of the three switching states "P", "O", and "N".According to these switching states, the inverter has 27 possible combinations of switching states consisting of 24 effective voltage vectors and 3 zero voltage vectors.Because the NPCI has the shunt resistor at the neutral point, the phase current can be acquired only when the effective voltage vector includes the "O" switching state [6].For example, the A phase current can be measured when the effective vector "O", "N", and "N" is applied to the inverter, as shown in Figure 3.
The 27 switching states of the neutral point clamped (NPC) inverter and the respective measurable phase currents are listed in Table 1.In addition, Figure 4 shows the switching vector in the spatial coordinates of the hexagon.Figure 4 and Table 1 show that the current cannot be measured through the neutral-point shunt resistor during the zero vector (V0) and space vectors represented by V1 to V6.    −ia ia Effective vector (Small vector) In one period of the three-level SVPWM, the sampling point and measurable phase current are expressed as shown in Figure 5.In this case, the "a" phase current can be measured in the "ONN" switching state and the "c" phase current can be measured in the "OON" switching state.The other phase current is calculated using Equation (1):  Table 1.Switching states and acquiring the phase current from the shunt resistor.

Space Vector Switching State Acquiring Current from the Shunt Resistor
Vector Classification Effective vector (Small vector) In one period of the three-level SVPWM, the sampling point and measurable phase current are expressed as shown in Figure 5.In this case, the "a" phase current can be measured in the "ONN" switching state and the "c" phase current can be measured in the "OON" switching state.The other phase current is calculated using Equation (1): Table 1.Switching states and acquiring the phase current from the shunt resistor.

Space Vector Switching State Acquiring Current from the Shunt Resistor
Vector Classification In one period of the three-level SVPWM, the sampling point and measurable phase current are expressed as shown in Figure 5.In this case, the "a" phase current can be measured in the "ONN" switching state and the "c" phase current can be measured in the "OON" switching state.The other phase current is calculated using Equation (1):

Current Unmeasurable Areas
The sampling time for measuring the accurate current should have minimum delay from the point of switching time.This is to avoid the current ripple component in the current damping process by switching, as shown in Figure 6a.The minimum time T min is determined using Equation (2) [9]: where T dead is the dead time to avoid arm-short of the inverter, T settling is the settling time of the neutral-point current, and T ad is the sample and hold time of the A/D converter.Thus, in order to acquire the phase current properly, the switching time should be greater than T min .Figure 6b shows switching state when the voltage modulation index is changed from Figure 5.In the case of "ONN", since the switching time T a in the "O" state is larger than T min , accurate "a" phase current can be obtained.However, in the case of "OON", since the switching time T c of the "O" state is shorter than T min , it is impossible to obtain current on the "c" phase in this state.The areas where the effective voltage dwell time is less than T min are defined as CUAs (current unmeasurable areas).The CUAs in sector 1 of the three level SVPWM hexagon are shown in Figure 7a.In Figure 7a, Area 1 is a region where all phase currents are measurable, Area 2 is a region where only one phase current is measurable, and Area 3 is defined as a current unmeasurable region.The CUAs in all areas of the SVPWM are shown in Figure 7b.

Current Unmeasurable Areas
The sampling time for measuring the accurate current should have minimum delay from the point of switching time.This is to avoid the current ripple component in the current damping process by switching, as shown in Figure 6a.The minimum time Tmin is determined using Equation (2) [9]: where Tdead is the dead time to avoid arm-short of the inverter, Tsettling is the settling time of the neutralpoint current, and Tad is the sample and hold time of the A/D converter.Thus, in order to acquire the phase current properly, the switching time should be greater than Tmin. Figure 6b shows switching state when the voltage modulation index is changed from Figure 5.In the case of "ONN", since the switching time Ta in the "O" state is larger than Tmin, accurate "a" phase current can be obtained.However, in the case of "OON", since the switching time Tc of the "O" state is shorter than Tmin, it is impossible to obtain current on the "c" phase in this state.The areas where the effective voltage dwell time is less than Tmin are defined as CUAs (current unmeasurable areas).The CUAs in sector 1 of the three level SVPWM hexagon are shown in Figure 7a.In Figure 7a, Area 1 is a region where all phase currents are measurable, Area 2 is a region where only one phase current is measurable, and Area 3 is defined as a current unmeasurable region.The CUAs in all areas of the SVPWM are shown in Figure 7b.In addition, the amplitude of the CUAs ∆V is obtained by Equation (3) [10].
where Vdc is the voltage of the DC link capacitor.Energies 2018, 11, x FOR PEER REVIEW 5 of 18

Current Unmeasurable Areas
The sampling time for measuring the accurate current should have minimum delay from the point of switching time.This is to avoid the current ripple component in the current damping process by switching, as shown in Figure 6a.The minimum time Tmin is determined using Equation (2) [9]: where Tdead is the dead time to avoid arm-short of the inverter, Tsettling is the settling time of the neutralpoint current, and Tad is the sample and hold time of the A/D converter.Thus, in order to acquire the phase current properly, the switching time should be greater than Tmin. Figure 6b shows switching state when the voltage modulation index is changed from Figure 5.In the case of "ONN", since the switching time Ta in the "O" state is larger than Tmin, accurate "a" phase current can be obtained.However, in the case of "OON", since the switching time Tc of the "O" state is shorter than Tmin, it is impossible to obtain current on the "c" phase in this state.The areas where the effective voltage dwell time is less than Tmin are defined as CUAs (current unmeasurable areas).The CUAs in sector 1 of the three level SVPWM hexagon are shown in Figure 7a.In Figure 7a, Area 1 is a region where all phase currents are measurable, Area 2 is a region where only one phase current is measurable, and Area 3 is defined as a current unmeasurable region.The CUAs in all areas of the SVPWM are shown in Figure 7b.In addition, the amplitude of the CUAs ∆V is obtained by Equation (3) [10].
where Vdc is the voltage of the DC link capacitor.In addition, the amplitude of the CUAs ∆V is obtained by Equation (3) [10].
where V dc is the voltage of the DC link capacitor.
Energies 2018, 11, 2616 6 of 18 Sector 1 of the output voltage hexagon is divided into four regions composed of effective voltage vectors for the reference voltage vector V re f duration, as shown in Figure 8.When the reference voltage vector is located as shown in Figure 8, the effective voltage vector and the duration time are expressed as follows: where T a , T b , and T c are the duration time of V 13 , V 7 , and V 14 , respectively.
Energies 2018, 11, x FOR PEER REVIEW 6 of 18 Sector 1 of the output voltage hexagon is divided into four regions composed of effective voltage vectors for the reference voltage vector duration, as shown in Figure 8.When the reference voltage vector is located as shown in Figure 8, the effective voltage vector and the duration time are expressed as follows: where Ta, Tb, and Tc are the duration time of V13, V7, and V14, respectively.

< Sector 1 >
Region 1 Region 2 Region 3 Region 4 From Equation ( 4), the duration time of the effective voltage vectors Ta, Tb, and Tc in each region can be obtained through the equations given in Table 2.When Ta, Tb, and Tc are shorter than Tmin, phase current cannot be obtained from the shunt resistor accurately.Therefore, it is a CUA.

Modified Voltage Modulation Method
The method in [7] with the alternative switching pattern is different from the classical SVPWM.The zero vector is replaced with a pair of effective voltage vectors in order to increase the duration time of the effective voltage vector, which does not ensure Tmin.As a result, only one phase current measurable region (Area 2) can be compensated.If the reference voltage vector is in Area 2, as shown in Figure 9a, phase C current is unmeasurable and the zero vector [OOO] is replaced with a pair of effective voltage vectors V14 [OON] and V17 [NNO], as shown in Figure 10.Therefore, only the current of one phase can be reconstructed during one PWM period due to the variation of the switching time.However, if the reference voltage vector is in Area 3, as shown in Figure 9b, it needs two switching cycles.Therefore, the THD of phase current is high, because a pair of vectors, which are located at opposite positions to each other, are used to make the reference voltage.From Equation (4), the duration time of the effective voltage vectors T a , T b , and T c in each region can be obtained through the equations given in Table 2.When T a , T b , and T c are shorter than T min , phase current cannot be obtained from the shunt resistor accurately.Therefore, it is a CUA.

Region
T a T b T c

Modified Voltage Modulation Method
The method in [7] with the alternative switching pattern is different from the classical SVPWM.The zero vector is replaced with a pair of effective voltage vectors in order to increase the duration time of the effective voltage vector, which does not ensure T min .As a result, only one phase current measurable region (Area 2) can be compensated.If the reference voltage vector V re f is in Area 2, as shown in Figure 9a, phase C current is unmeasurable and the zero vector [OOO] is replaced with a pair of effective voltage vectors V 14 [OON] and V 17 [NNO], as shown in Figure 10.Therefore, only the current of one phase can be reconstructed during one PWM period due to the variation of the switching time.However, if the reference voltage vector V re f is in Area 3, as shown in Figure 9b, it needs two switching cycles.Therefore, the THD of phase current is high, because a pair of vectors, which are located at opposite positions to each other, are used to make the reference voltage.

Minimum Voltage Injection (MVI) Method
In this conventional method [6], the minimum voltage injection method is applied to measure the phase current, as shown in Figure 11. Figure 11a shows the reference voltage vector in Area 3, where two phase currents cannot be obtained.In this case, a constant voltage is added to the reference voltage vector to reconstruct the phase current.The reference voltage moved to measurable Area 1 is defined as .The compensation voltage vector is applied by subtracting the constant voltage which has the same magnitude as the added value to the command voltage vector.This compensation method is shown for one period of the switching pattern in Figure 11b.In this case, both "ONN" and "OON" are less than the minimum time Tmin.So in a half period of modulation, the voltage for the reconstruction is injected, and in the other half period, the compensation voltage is injected to cancel the effect of the injected voltage.However, it is difficult to reconstruct the phase current using the MVI method in the high MI region shown in Figure 12.At this time, the compensation voltage exceeds the linear modulation area.Therefore, it is impossible to reconstruct the phase current near the outermost edge of the hexagon.

Minimum Voltage Injection (MVI) Method
In this conventional method [6], the minimum voltage injection method is applied to measure the phase current, as shown in Figure 11. Figure 11a shows the reference voltage vector in Area 3, where two phase currents cannot be obtained.In this case, a constant voltage is added to the reference voltage vector to reconstruct the phase current.The reference voltage moved to measurable Area 1 is defined as .The compensation voltage vector is applied by subtracting the constant voltage which has the same magnitude as the added value to the command voltage vector.This compensation method is shown for one period of the switching pattern in Figure 11b.In this case, both "ONN" and "OON" are less than the minimum time Tmin.So in a half period of modulation, the voltage for the reconstruction is injected, and in the other half period, the compensation voltage is injected to cancel the effect of the injected voltage.However, it is difficult to reconstruct the phase current using the MVI method in the high MI region shown in Figure 12.At this time, the compensation voltage exceeds the linear modulation area.Therefore, it is impossible to reconstruct the phase current near the outermost edge of the hexagon.

Minimum Voltage Injection (MVI) Method
In this conventional method [6], the minimum voltage injection method is applied to measure the phase current, as shown in Figure 11. Figure 11a shows the reference voltage vector V re f in Area 3, where two phase currents cannot be obtained.In this case, a constant voltage is added to the reference voltage vector to reconstruct the phase current.The reference voltage moved to measurable Area 1 is defined as V m .The compensation voltage vector V c is applied by subtracting the constant voltage which has the same magnitude as the added value to the command voltage vector.This compensation method is shown for one period of the switching pattern in Figure 11b.In this case, both "ONN" and "OON" are less than the minimum time T min .So in a half period of modulation, the voltage V m for the reconstruction is injected, and in the other half period, the compensation voltage V c is injected to cancel the effect of the injected voltage.

Minimum Voltage Injection (MVI) Method
In this conventional method [6], the minimum voltage injection method is applied to measure the phase current, as shown in Figure 11. Figure 11a shows the reference voltage vector in Area 3, where two phase currents cannot be obtained.In this case, a constant voltage is added to the reference voltage vector to reconstruct the phase current.The reference voltage moved to measurable Area 1 is defined as .The compensation voltage vector is applied by subtracting the constant voltage which has the same magnitude as the added value to the command voltage vector.This compensation method is shown for one period of the switching pattern in Figure 11b.In this case, both "ONN" and "OON" are less than the minimum time Tmin.So in a half period of modulation, the voltage for the reconstruction is injected, and in the other half period, the compensation voltage is injected to cancel the effect of the injected voltage.However, it is difficult to reconstruct the phase current using the MVI method in the high MI region shown in Figure 12.At this time, the compensation voltage exceeds the linear modulation area.Therefore, it is impossible to reconstruct the phase current near the outermost edge of the hexagon.However, it is difficult to reconstruct the phase current using the MVI method in the high MI region shown in Figure 12.At this time, the compensation voltage V c exceeds the linear modulation area.Therefore, it is impossible to reconstruct the phase current near the outermost edge of the hexagon.
Figure 12.Voltage injection method in Area 3 beside a vertex of the hexagon.

Based Method for Current Reconstruction in Area 2
In order to control the constant speed and constant torque of an AC motor, a current controller is essential.In general, a proportional integral (PI) controller is used on the synchronous reference frame of d-q axis [10].Figure 13 shows the block diagram of the synchronous PI controller, and the electrical model of the motor system.Where Ra, La, Kp, and Ki mean the stator resistance, stator inductance, proportional gain, and integral gain, respectively.The proportional and integral gains of the PI controller are calculated using Equation ( 5): where La is the stator inductance, Ra is the stator resistance, and ωcc is the bandwidth of the PI regulator.
Then, the closed-loop transfer function Gc(s) of the block diagram is given by Equation ( 6): According to Equations ( 5) and ( 6), the combination of the PI controller and model of the motor is equivalent to a first low-pass filter whose cutoff frequency is ωcc.In this case, the real d-q axis current can be estimated using the current reference and low pass filter, as shown in Figures 13 and  14.Finally, the estimated three phase currents (îabc) can be obtained from reverse transformation of the estimated d-q axis currents (îdq e ).This estimated current is used to replace the unmeasurable current when the voltage command lies inside Area 2, where only one phase current is acquired.This current estimation method does not need PWM shift or voltage injection for current reconstruction.

Based Method for Current Reconstruction in Area 2
In order to control the constant speed and constant torque of an AC motor, a current controller is essential.In general, a proportional integral (PI) controller is used on the synchronous reference frame of d-q axis [10].Figure 13 shows the block diagram of the synchronous PI controller, and the electrical model of the motor system.Where R a , L a , K p , and K i mean the stator resistance, stator inductance, proportional gain, and integral gain, respectively.The proportional and integral gains of the PI controller are calculated using Equation ( 5): where L a is the stator inductance, R a is the stator resistance, and ω cc is the bandwidth of the PI regulator.Then, the closed-loop transfer function G c (s) of the block diagram is given by Equation ( 6): According to Equations ( 5) and ( 6), the combination of the PI controller and model of the motor is equivalent to a first low-pass filter whose cutoff frequency is ω cc .In this case, the real d-q axis current can be estimated using the current reference and low pass filter, as shown in Figures 13 and 14.Finally, the estimated three phase currents (î abc ) can be obtained from reverse transformation of the estimated d-q axis currents (î dq e ).This estimated current is used to replace the unmeasurable current when the voltage command lies inside Area 2, where only one phase current is acquired.This current estimation method does not need PWM shift or voltage injection for current reconstruction.
Energies 2018, 11, x FOR PEER REVIEW 8 of 18 Figure 12.Voltage injection method in Area 3 beside a vertex of the hexagon.

Based Method for Current Reconstruction in Area 2
In order to control the constant speed and constant torque of an AC motor, a current controller is essential.In general, a proportional integral (PI) controller is used on the synchronous reference frame of d-q axis [10].Figure 13 shows the block diagram of the synchronous PI controller, and the electrical model of the motor system.Where Ra, La, Kp, and Ki mean the stator resistance, stator inductance, proportional gain, and integral gain, respectively.The proportional and integral gains of the PI controller are calculated using Equation ( 5): where La is the stator inductance, Ra is the stator resistance, and ωcc is the bandwidth of the PI regulator.
Then, the closed-loop transfer function Gc(s) of the block diagram is given by Equation ( 6): According to Equations ( 5) and ( 6), the combination of the PI controller and model of the motor is equivalent to a first low-pass filter whose cutoff frequency is ωcc.In this case, the real d-q axis current can be estimated using the current reference and low pass filter, as shown in Figures 13 and  14.Finally, the estimated three phase currents (îabc) can be obtained from reverse transformation of the estimated d-q axis currents (îdq e ).This estimated current is used to replace the unmeasurable current when the voltage command lies inside Area 2, where only one phase current is acquired.This current estimation method does not need PWM shift or voltage injection for current reconstruction.

Proposed Method for Current Reconstruction in Area 3
In order to calculate the optimum injection voltage according to the switching sector of SVPWM, the conventional sector is classified into 36 switching sectors in this paper.These sectors can be represented by 18 straight lines, as shown in Figure 15.These straight lines can be obtained by using the two points of the hexagon.As a result, the 36 switching sectors can be defined by three straight lines.If the reference voltage vector V re located as shown in Figure 15, straight Lines 5, 9, and 13 must be satisfied to discriminate the switching sector.

Proposed Method for Current Reconstruction in Area 3
In order to calculate the optimum injection voltage according to the switching sector of SVPWM, the conventional sector is classified into 36 switching sectors in this paper.These sectors can be represented by 18 straight lines, as shown in Figure 15.These straight lines can be obtained by using the two points of the hexagon.As a result, the 36 switching sectors can be defined by three straight lines.If the reference voltage vector is located as shown in Figure 15, straight Lines 5, 9, and 13 must be satisfied to discriminate the switching sector.
), Line 13: ≥ 0 For simple injection voltage calculation, the reference voltage vector ( , ) that rotates the output voltage hexagon with electrical angle , is transformed as shown in Figure 16. is transformed to the shifted reference voltage vector ( , ) in sector 0 by Equation ( 8).
The range of the shifted sector 0 is from −π/6 to π/6.As a result, the injected voltage is only calculated in the shifted sector 0.
As shown in Figure 17a, the shifted sector is symmetrical with respect to the d-axis.Therefore, when calculating the injection voltage, only the positive part needs to be calculated.In the negative region, the negative sign can be added to the magnitude of the q component., are the vector components for moving to the measurable region.After calculating the injection voltage, is added to , and reverse-transformation is executed.This voltage vector is defined as the measurement voltage vector .
is induced during the first half period of the modulation, and the compensation voltage is induced during the other half period.The relation between , , and is given by Equation ( 9).For simple injection voltage calculation, the reference voltage vector V re f (V s ds ,V s qs ) that rotates the output voltage hexagon with electrical angle θ e , is transformed as shown in Figure 16.V re f is transformed to the shifted reference voltage vector V t dqs (V t ds ,V t qs ) in sector 0 by Equation ( 8). ( where The range of the shifted sector 0 is from −π/6 to π/6.As a result, the injected voltage is only calculated in the shifted sector 0. As shown in Figure 17a, the shifted sector is symmetrical with respect to the d-axis.Therefore, when calculating the injection voltage, only the positive part needs to be calculated.In the negative region, the negative sign can be added to the magnitude of the q component.V i dqs V i ds , V i qs are the vector components for moving V t dqs to the measurable region.After calculating the injection voltage, V i dqs is added to V t dqs , and reverse-transformation is executed.This voltage vector is defined as the measurement voltage vector V s m .V s m is induced during the first half period of the modulation, and the compensation voltage V s c is induced during the other half period.The relation between V re f , V s m , and V s c is given by Equation ( 9).
Energies 2018, 11, 2616 10 of 18 voltage vector is defined as the measurement voltage vector .is induced during the first half period of the modulation, and the compensation voltage is induced during the other half period.The relation between , , and is given by Equation ( 9).In order to recover the phase current by injecting the optimal voltage, the shifted sector 0 for the positive q-axis is divided into different parts, as shown in Figure 18a When the reference voltage vector is located in Part.1_1, as shown in Figure 19, the optimal voltage is injected, and the measurement voltage vector is pushed to the measurement point.At this time, the measurement point is on the boarder of Area 2. The magnitude of the injected voltage ( , ) can be calculated by using both the measurement point and , as shown in Equation (10).
Part. 1_1 When the reference voltage vector is located in Part.1_2, as shown in Figure 20, the measurement voltage vector is moved to the measurement line by the optimal voltage injection.The measurement line is defined by Equation ( 11), with two points (2∆V/√3, 0) and (√3∆V, ∆V), as the boarder of Area 2. The magnitude of the injected voltage is obtained by Equation ( 12), which gives the minimum distance from the point to the measurement line through the Pythagorean theorem, as shown in Equation (13).The optimal voltage injection method is used to classify each part of Area 3 in more detail, and then bring to the closest measurement line or measurement point for reconstructing In order to recover the phase current by injecting the optimal voltage, the shifted sector 0 for the positive q-axis is divided into different parts, as shown in Figure 18a, according to the MI.As shown in Figure 18b, each part is more finely divided to restore the phase current by injecting the optimal voltage.In order to recover the phase current by injecting the optimal voltage, the shifted sector 0 for the positive q-axis is divided into different parts, as shown in Figure 18a When the reference voltage vector is located in Part.1_1, as shown in Figure 19, the optimal voltage is injected, and the measurement voltage vector is pushed to the measurement point.At this time, the measurement point is on the boarder of Area 2. The magnitude of the injected voltage ( , ) can be calculated by using both the measurement point and , as shown in Equation (10).
Part. 1_1 When the reference voltage vector is located in Part.1_2, as shown in Figure 20, the measurement voltage vector is moved to the measurement line by the optimal voltage injection.The measurement line is defined by Equation (11), with two points (2∆V/√3, 0) and (√3∆V, ∆V), as the boarder of Area 2. The magnitude of the injected voltage is obtained by Equation (12), which gives the minimum distance from the point to the measurement line through the Pythagorean theorem, as shown in Equation ( 13).The optimal voltage injection method is used to classify each part of Area 3 in more detail, and then bring to the closest measurement line or measurement point for reconstructing the phase currents.At this time, the measurement line or measurement point is at the boarder of Area When the reference voltage vector V re f is located in Part.1_1, as shown in Figure 19, the optimal voltage is injected, and the measurement voltage vector V s m is pushed to the measurement point.At this time, the measurement point is on the boarder of Area 2. The magnitude of the injected voltage V i dqs (V i ds , V i qs ) can be calculated by using both the measurement point and V t dqs , as shown in Equation (10). Part.1_1 When the reference voltage vector is located in Part.1_2, as shown in Figure 20, the measurement voltage vector is moved to the measurement line by the optimal voltage injection.The measurement line is defined by Equation (11), with two points (2∆V/ √ 3, 0) and ( √ 3∆V, ∆V), as the boarder of Area 2. The magnitude of the injected voltage is obtained by Equation (12), which gives the minimum Part.2_4 Part.2_4A resistor of 10 Ω and inductor of 5 mH were used as the load.The resistance of the shunt resistor was 0.2 Ω and its power capacity was 3 W.In addition, the minimum time T min needed to acquire the precise phase current from the shunt resistor was set to be 4.5 µs.The reconstructed current from the shunt resistor using the proposed algorithm was compared with an actual current measured using two current sensors in order to verify its precision.In the experiment, the inverter system was operated by the V/F open-loop control algorithm.The system was controlled with a fixed operating frequency of 12 Hz and variable output voltage to change the MI.The experimental results were classified in 6 cases to compare with the conventional MVI method.Figure 24a,b shows the experimental results with the condition of MI = 0.1 ( = 3.56 V).In this case, the inverter was operated in Area 3 where the two currents cannot be measured from the shunt resistor.As presented in Figure 24b, the injected d-q voltage for the current reconstruction is smaller than the voltage in Figure 24a.The experimental results of Figure 24c,d was measured when the inverter was operated in Area 1 and 2 with MI of 0.4 ( = 13.86V).In Area 1, it is possible to obtain the two phase currents from the shunt resistor, and only one current can be acquired in Area 2. As shown in Figure 24c, the conventional method injects the q-axis voltage to reconstruct the phase current.In contrast, the proposed method can reconstruct the phase current without the voltage injection, as presented in Figure 24d, because the phase current is reconstructed using the estimated current.In Figure 24e,f, is passed through Area 1 and 3 when the MI is 0.6 ( = 20.78V).As shown in Figure 24f, a smaller voltage is injected to reconstruct the phase current in the proposed method in comparison with the conventional method.The waveforms presented in Figure 25e,f was measured when the MI was 0.97 ( = 33.6V).In this case, the trajectory of the voltage reference is near the inscribed circle of the space vector hexagon.In the case of the conventional method, the phase current cannot be reconstructed even if the voltage injection method is used, because the compensated voltage reference exceeds the hexagon.On the other hand, the proposed method can reconstruct the phase current, as presented in Figure 25f, because the voltage reference can be compensated on the boundary of the hexagon.
The accuracy of the phase currents reconstructed using the conventional and proposed methods were calculated to show the superiority of the proposed algorithm.The equation for the accuracy of the phase current is expressed in Equation ( 24).In the experiment, the inverter system was operated by the V/F open-loop control algorithm.The system was controlled with a fixed operating frequency of 12 Hz and variable output voltage to change the MI.The experimental results were classified in 6 cases to compare with the conventional MVI method.Figure 24a,b shows the experimental results with the condition of MI = 0.1 (V re f = 3.56 V).In this case, the inverter was operated in Area 3 where the two currents cannot be measured from the shunt resistor.As presented in Figure 24b, the injected d-q voltage for the current reconstruction is smaller than the voltage in Figure 24a.The experimental results of Figure 24c,d was measured when the inverter was operated in Area 1 and 2 with MI of 0.4 (V re f = 13.86V).In Area 1, it is possible to obtain the two phase currents from the shunt resistor, and only one current can be acquired in Area 2. As shown in Figure 24c, the conventional method injects the q-axis voltage to reconstruct the phase current.In contrast, the proposed method can reconstruct the phase current without the voltage injection, as presented in Figure 24d, because the phase current is reconstructed using the estimated current.In Figure 24e,f, V re f is passed through Area 1 and 3 when the MI is 0.6 (V re f = 20.78V).As shown in Figure 24f, a smaller voltage is injected to reconstruct the phase current in the proposed method in comparison with the conventional method.The waveforms presented in Figure 25e,f was measured when the MI was 0.97 (V re f = 33.6V).In this case, the trajectory of the voltage reference is near the inscribed circle of the space vector hexagon.In the case of the conventional method, the phase current cannot be reconstructed even if the voltage injection method is used, because the compensated voltage reference exceeds the hexagon.On the other hand, the proposed method can reconstruct the phase current, as presented in Figure 25f, because the voltage reference can be compensated on the boundary of the hexagon.
The accuracy of the phase currents reconstructed using the conventional and proposed methods were calculated to show the superiority of the proposed algorithm.The equation for the accuracy of the phase current is expressed in Equation (24).
compared according to the MI, and shown in Figure 26a.The accuracy of the proposed method is higher than the conventional method in the whole range of the MI.The comparison results for the THD of the phase current between the conventional method and the proposed method are presented in Figure 26b.
Energies 2018, 11, x FOR PEER REVIEW 16 of 18 reconstruction is low.The difference in accuracy between the conventional method and the proposed method is compared according to the MI, and shown in Figure 26a.The accuracy of the proposed method is higher than the conventional method in the whole range of the MI.The comparison results for the THD of the phase current between the conventional method and the proposed method are presented in Figure 26b.The THD result for the proposed method is lower than the result for the conventional method in the whole range of the MI, because the THD of the phase current is proportional to the injected voltage.On average the THD of the proposed method is improved by 44.5 % compared with that of the conventional method.

Conclusions
This paper proposed a phase current reconstruction method using a neutral shunt resistor, which lowers both the cost and volume of a three-level inverter system.
First, the paper proposes the method for dividing the switching sectors, the rotated sectors, and the region.Each region is separated by the equation of a straight line.
In addition, in Area 2 where there is only one measurable phase current, the remaining phase current is reconstructed by the estimation method.The electrical model of the motor and PI current controller are used to estimate the current.
Finally, the optimal voltage injection method is used in Area 3, where no current is acquired through the shunt resistor (CUAs).Area 3 is divided and defined as parts in more detail.When the reference is located on Area 3 of the hexagon, the phase currents are reconstructed by moving the reference vector to the border of Area 1 or Area 2.
The proposed method utilized a small magnitude injection voltage compared to the conventional method, and hence results in better performance in terms of THD and accuracy of the estimated currents.The validity of the proposed algorithm is proven by the experiments.

Figure 3 .
Figure 3.Current path of neutral shunt resistor when effective vector "O", "N", and "N" is applied to the inverter.

Figure 3 .
Figure 3.Current path of neutral shunt resistor when effective vector "O", "N", and "N" is applied to the inverter.

Figure 5 .
Figure 5.One period of the SVPWM method and sampling point.

Figure 5 .
Figure 5.One period of the SVPWM method and sampling point.Figure 5.One period of the SVPWM method and sampling point.

Figure 5 .
Figure 5.One period of the SVPWM method and sampling point.Figure 5.One period of the SVPWM method and sampling point.

Figure 6 .
Figure 6.T min and T s of the SVPWM method.(a) Minimum time (T min ); (b) one PWM period (T s ).

Figure 13 .
Figure 13.Block diagram of the PI controller and electrical model of the motor system.

Figure 14 .
Figure 14.Block diagram of acquiring estimated current.

Figure 12 .
Figure 12.Voltage injection method in Area 3 beside a vertex of the hexagon.

Figure 13 .
Figure 13.Block diagram of the PI controller and electrical model of the motor system.

Figure 14 .
Figure 14.Block diagram of acquiring estimated current.

Figure 13 .
Figure 13.Block diagram of the PI controller and electrical model of the motor system.

Figure 13 .
Figure 13.Block diagram of the PI controller and electrical model of the motor system.

Figure 14 .
Figure 14.Block diagram of acquiring estimated current.Figure 14.Block diagram of acquiring estimated current.

Figure 14 .
Figure 14.Block diagram of acquiring estimated current.Figure 14.Block diagram of acquiring estimated current.

Figure 18 .
Figure 18.Detailed view of the parts of the positive shifted sector 0. (a) Part1-Part4; (b) segmentation of Parts.

Figure 18 .
Figure 18.Detailed view of the parts of the positive shifted sector 0. (a) Part1-Part4; (b) segmentation of Parts.

Figure 18 .
Figure 18.Detailed view of the parts of the positive shifted sector 0. (a) Part1-Part4; (b) segmentation of Parts.

Figure 19 .Figure 20 .
Figure 19.Optimal voltage injection method in Part.1_1.(a) Reference voltage vector on Part.1_1;(b) detailed view of the left one.

Figure 20 .
Figure 20.Optimal voltage injection method in Part.1_2.(a) Reference voltage vector on Part.1_2;(b) detailed view of the left one.

Figure 23 .
Figure 23.Configuration of hardware system and control board.

Figure 23 .
Figure 23.Configuration of hardware system and control board.

Figure 26 .
Figure 26.Comparison of the conventional and proposed methods according to the variation in the MI.(a) Phase current Accuracy; (b) total harmonic distortion (THD) of phase currents.

Table 1 .
Switching states and acquiring the phase current from the shunt resistor.

Table 2 .
Dwell times of the voltage vector according to the regions.

Table 2 .
Dwell times of the voltage vector according to the regions.