# A Single-Phase Buck-Boost Matrix Converter with Low Switching Stresses

Authors: Naveed Ashraf; Tahir Izhar; Ghulam Abbas
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The suggested single-phase ac-to-ac matrix converter operated with inverting and noninverting characteristics may solve the grid voltage swell and sag problem in power distribution system, respectively. It is also employed as a direct frequency changer for domestic induction heating. The output voltage is regulated through duty cycle control of high frequency direct PWM (DPWM) and indirect PWM (IDPWM) switching devices. The DPWM control switches control the switching states of IDPWM switching devices. The inverting and noninverting characteristics are achieved with low voltage stresses and hence lowdv/dt across the high and low frequency-controlled switches. This reduces their voltage rating and losses. The high voltage overshoot problem in frequency step-up operation is also analyzed. The sliding mode (SM) controller is employed to solve this problem. Pulse selective approach determines the power quality of load voltage. The validity of the mathematically computed values is carried out by modelling the proposed topology in MATLAB/Simulink environment and through hardware results.


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## Body

## 1. Introduction

Single-phase direct ac-to-ac converters with bipolar voltage gain are widely employed as radio frequency induction heating, induction motor driver, and voltage sag and swell compensator. Traditionally, ac voltage and frequency regulation is achieved through indirect dual ac-to-ac converters [1–4]. They have high losses, poor conversation efficiency, and low reliability. Direct ac-to-ac converters [5–7] have simple circuit arrangement, small size, low cost, and high conversion efficiency and are easy to control due to single stage power conversion.The power quality supplied at consumer ends is becoming poor day by day due to rapid, uneven spreading of loads and transient generated by heavy loads [8, 9]. Buck-boost ac voltage controllers regulate the dynamic variation of the load voltage. Voltage sags and swells degrade the load performance and power quality. They are normally analyzed in terms of their phase jump, depth, and duration. The research in [10] shows their existence and impact in power system. The end users have to use power conditioning units for their better power quality, safety, and performance. Dynamic voltage restorers (DVR) as discussed in [11, 12] are used to solve the problem of voltage sags in power distribution system. The DVR based on FACTS devices as in [13–15] stabilizes the disturbances in grid voltage without considering the problems of generated harmonic and power quality.Various direct ac-to-ac voltage regulation techniques through PWM control are discussed in [16]. They are implemented by replacing the unidirectional switching devices with bidirectional devices. They can only operate as buck regulator [17] or boost regulator [18] but not both. The ac-to-ac voltage controller in [19, 20] only solves the voltage sag or swell disturbance but not both due to unipolar voltage gain. The ac power converters with z-source arrangements having buck-boost characteristics are introduced in [21–23]. They only solve the voltage sag problem up to 25% without power storage device. An ac-to-ac power conversion technique based on interphase conversion is developed in [24] to improve the power system disturbances. The solution of voltage sag problem is accomplished by getting ac power from other healthy phases. The analysis in [25] solves the problem of unidirectional voltage gain at the cost of hybrid three-phase transformer. The problems of poor power quality are solved with z-source [26] and quasi-z-source [27] topologies. The current rating of power switches has to be increased due to conduction of high current during short-through period. The current and voltage surges are generated due to short circuiting of source or filtering capacitor through switching action that can damage the switching devices. The coupled inductors as in [28] solve the problems of voltage and current surges, but it increases the size and cost of the switching converters. An ac-to-ac converter with common grounding and bipolar voltage gain is proposed in [29]. It operates in noninverting buck and inverting buck-boost mode to compensate the grid voltage sag and swell problem. It cannot be operated as noninverting boost operation. Its high frequency switches operate at high voltage stresses. It is implemented with eight IGBTs having high switching and conduction losses. It cannot be used as a direct frequency changer (DFC) to control the load frequency.The body diode of MOSFET limits its operating speed as it has poor reverse characteristics. This problem is solved in [30, 31] with series connection of fast recovery diode that prevents its forward biasing. In its inverting mode, two high frequency switches are operated with the voltage stresses of vin + vo. Its low frequency switches operate at vin and vo. A buck-boost matrix converter (MC) with common input and output ground is proposed in [32]. Its noninverting operation is implemented with four high frequency switches with the voltage stresses of vin + vo resulting in highdv/dt. Its low frequency switches operate at vin and vo. High voltage and current surges are generated in the frequency boost operation as they depend upon the voltage stresses. The reverse voltage is developed across the fast recovery diode as MOSFET is on in IDPWM control, and across MOSFET as fast recovery diode is forward biased in DPWM control. The high voltage stresses increase the voltage rating of the diode and MOSFETs. This increases the forward voltage, on-state resistance of diode and MOSFET that result in high conduction losses. The total harmonic distortion (THD) of the output voltage is very high due to nonsinusoidal nature of the output voltage.In this proposed research, a single-phase MC is implemented with both noninverting and inverting characteristics with low voltage stresses across the switching devices, thus having low losses. SM controller is developed to solve the problem of high voltage overshoot and poor power quality in frequency step-up operation. Pulse selected approach is used to find out the harmonics contents of the load voltage. The analysis of the proposed research is validated through the MATLAB/Simulink environment and hardware results.


## 2. Operating Modes of the Proposed Converter

The proposed converter in [33] can also be implemented with noninverting and inverting buck-boost characteristics having low voltage stresses and losses. Each switch is the series combination of a MOSFET “M” and diode “Ds”. Its operation in noninverting and inverting modes with positive and negative input voltage is detailed below.


### 2.1. Noninverting Buck-Boost Mode with Positive Input

In this operating mode, the buck-boost of the output voltage is accomplished by varying the duty cycle via the control of switching devicesM2,M6 and M3,M5 as DPWM and IDPWM control, respectively. The switch M4 is switched at the low output frequency. The remaining switches M1, M7, and M8 remain off in this operating mode, thus offering no conduction and switching losses. The current conduction paths for the turn on and turn off period of high frequency switching devices are highlighted in Figures 1(a) and 1(b), respectively. PWM average switching model with duty cycle “d” for positive input voltage based on the analysis in [29] is obtained as follows.(1)diLdt=1Ld.vin-d′vo,hered′=1-ddvodt=1C-d.io+d′.iL-ioAll state variables of (1) in steady state condition are distributed at fundamental frequency (low frequency); their derivative terms can be ignored to find their voltage and current transfer ratios. That is to say,(2)dvin=v01-d⇒v0vin=d1-d(3)iLio=11-d.Figure 1
Current conduction path for various operating modes of the proposed converter for: (a) & (b) positive voltage gain from positive input voltage; (c) & (d) positive voltage gain from negative input voltage; (e) & (f) negative voltage gain from positive input voltage; (g) & (h) negative voltage gain from negative input voltage.
(a) (b) (c) (d) (e) (f) (g) (h)


### 2.2. Noninverting Buck-Boost Mode with Negative Input

The direct high frequency PWM switching ofM3 and M5 with indirect switching of switches M2 and M6 controls the output voltage for negative input voltage. Inductor is charged and its stored energy is released to load during turn on and turn off intervals, respectively. In this operating mode, the low frequency switch M1 remains in conduction state. The three remaining switches M4, M7, and M8 are not required to be turned on. The current paths through which the inductor is charged and power is transferred to load are highlighted in Figures 1(c) and 1(d), respectively. Noninverting averaging PWM model with duty ratio “d” for negative input voltage is realized as follows.(4)didt=1Ld.vin-d′vodvodt=1C-d.io+d′.iL-ioThe voltage and currents gains are computed by solving (4) as their derivative terms are ignored in steady state condition. (5)dvin=v01-d⇒v0vin=d1-d(6)iLio=11-d


### 2.3. Inverting Buck-Boost Mode with Positive Input

The inverting output with buck-boost characteristics is accomplished via the PWM switching of switchesM2,M5 as direct and M3, M6 as indirect control. Such switching stores energy in inductor and then transfers it to load through the current conduction paths as shown in Figures 1(e) and 1(f), respectively. From the remaining four switches, switch M7 is switched at low load frequency and there is no role of switches M1, M4, and M8 in this mode of operation. Inverting averaging PWM model with duty ratio “d” for positive input voltage can be realized as follows.(7)didt=1Ld.vin+d′vodvodt=1C-d.io-d′.iL+ioThe variable of the right-hand sides of (7) can be ignored in steady state due to their low variation to determine the voltage and current transfer ratios or gains. That is to say,(8)dvin=-v01-d⇒v0vin=-d1-d(9)iLi0=-11-d.


### 2.4. Inverting Buck-Boost Mode with Negative Input

The negative input is converted into positive output via the switching action of one low frequency switchM8 and four high frequency switches M3, M6 and M2, M5 as direct and indirect PWM control, respectively. The remaining switches M1, M4, and M7 maintain their off state throughout the operation of this mode. Figures 1(g) and 1(h) show the charging and discharging path of inductor current. Inverting averaging PWM model with duty ratio “d” for negative input voltage is realized as follows.(10)didt=1Ld.vin+d′vodvodt=1C-d.io-d′.iL+ioEquation (10) in steady state is realized to find the voltage and current gains. That is to say,(11)dvin=-v01-d⇒v0vin=-d1-d(12)iLi0=-11-d.The operating modes from “A” to “D” can be used to operate the proposed converter as an ac voltage controller and a direct frequency changer with noninverting and inverting buck-boost characteristics. Next section explores the operation of the proposed converter as an ac voltage controller and direct frequency changer in detail.


## 2.1. Noninverting Buck-Boost Mode with Positive Input

In this operating mode, the buck-boost of the output voltage is accomplished by varying the duty cycle via the control of switching devicesM2,M6 and M3,M5 as DPWM and IDPWM control, respectively. The switch M4 is switched at the low output frequency. The remaining switches M1, M7, and M8 remain off in this operating mode, thus offering no conduction and switching losses. The current conduction paths for the turn on and turn off period of high frequency switching devices are highlighted in Figures 1(a) and 1(b), respectively. PWM average switching model with duty cycle “d” for positive input voltage based on the analysis in [29] is obtained as follows.(1)diLdt=1Ld.vin-d′vo,hered′=1-ddvodt=1C-d.io+d′.iL-ioAll state variables of (1) in steady state condition are distributed at fundamental frequency (low frequency); their derivative terms can be ignored to find their voltage and current transfer ratios. That is to say,(2)dvin=v01-d⇒v0vin=d1-d(3)iLio=11-d.Figure 1
Current conduction path for various operating modes of the proposed converter for: (a) & (b) positive voltage gain from positive input voltage; (c) & (d) positive voltage gain from negative input voltage; (e) & (f) negative voltage gain from positive input voltage; (g) & (h) negative voltage gain from negative input voltage.
(a) (b) (c) (d) (e) (f) (g) (h)


## 2.2. Noninverting Buck-Boost Mode with Negative Input

The direct high frequency PWM switching ofM3 and M5 with indirect switching of switches M2 and M6 controls the output voltage for negative input voltage. Inductor is charged and its stored energy is released to load during turn on and turn off intervals, respectively. In this operating mode, the low frequency switch M1 remains in conduction state. The three remaining switches M4, M7, and M8 are not required to be turned on. The current paths through which the inductor is charged and power is transferred to load are highlighted in Figures 1(c) and 1(d), respectively. Noninverting averaging PWM model with duty ratio “d” for negative input voltage is realized as follows.(4)didt=1Ld.vin-d′vodvodt=1C-d.io+d′.iL-ioThe voltage and currents gains are computed by solving (4) as their derivative terms are ignored in steady state condition. (5)dvin=v01-d⇒v0vin=d1-d(6)iLio=11-d


## 2.3. Inverting Buck-Boost Mode with Positive Input

The inverting output with buck-boost characteristics is accomplished via the PWM switching of switchesM2,M5 as direct and M3, M6 as indirect control. Such switching stores energy in inductor and then transfers it to load through the current conduction paths as shown in Figures 1(e) and 1(f), respectively. From the remaining four switches, switch M7 is switched at low load frequency and there is no role of switches M1, M4, and M8 in this mode of operation. Inverting averaging PWM model with duty ratio “d” for positive input voltage can be realized as follows.(7)didt=1Ld.vin+d′vodvodt=1C-d.io-d′.iL+ioThe variable of the right-hand sides of (7) can be ignored in steady state due to their low variation to determine the voltage and current transfer ratios or gains. That is to say,(8)dvin=-v01-d⇒v0vin=-d1-d(9)iLi0=-11-d.


## 2.4. Inverting Buck-Boost Mode with Negative Input

The negative input is converted into positive output via the switching action of one low frequency switchM8 and four high frequency switches M3, M6 and M2, M5 as direct and indirect PWM control, respectively. The remaining switches M1, M4, and M7 maintain their off state throughout the operation of this mode. Figures 1(g) and 1(h) show the charging and discharging path of inductor current. Inverting averaging PWM model with duty ratio “d” for negative input voltage is realized as follows.(10)didt=1Ld.vin+d′vodvodt=1C-d.io-d′.iL+ioEquation (10) in steady state is realized to find the voltage and current gains. That is to say,(11)dvin=-v01-d⇒v0vin=-d1-d(12)iLi0=-11-d.The operating modes from “A” to “D” can be used to operate the proposed converter as an ac voltage controller and a direct frequency changer with noninverting and inverting buck-boost characteristics. Next section explores the operation of the proposed converter as an ac voltage controller and direct frequency changer in detail.


## 3. Operation as a Voltage and Frequency Controller

The operation of the proposed converter can be realized as a voltage controller and direct frequency changer by operating it in noninverting and inverting modes. The detail of operation as ac voltage controller and direct frequency changer is explored below with the help of their corresponding switching sequences.


### 3.1. Matrix Converter as a Noninverting and Inverting AC Voltage Controller

The noninverting operation of proposed converter as an ac voltage controller is obtained by switching the proposed converter in mode “A” and “B” alternatively. Its inverting operation requires the alternative operation of mode “C” and “D”. The situation is depicted in Figures 2(a) and 2(b) indicating the gating sequence of noninverting and inverting buck-boost characteristics with two direct and two indirect PWM control switches.Figure 2
Switching sequences: (a) noninverting buck-boost ac voltage controller; (b) inverting buck-boost ac voltage controller.
(a) (b)


### 3.2. Matrix Converter as a Frequency Changer

The proposed converter can also be implemented for step change of the load frequency. The frequency buck and boost operation are discussed separately with the help of switching waveforms.(1  ) Frequency Buck Operation. In frequency buck operation, the output frequency is considered to be one-half of the input frequency. So, it operates in mode “A” to “D”, respectively, as the required output is noninverted and inverted for first and second cycles of the input voltage as shown in Figure 3.Figure 3
Buck-boost with frequency step-down operation.(2  ) Frequency Boost Operation. In frequency boost operation, the output frequency is considered two times as that of input frequency. The required output is obtained with the operation of the proposed converter in modes “A”, “C”, “D”, and “B”, respectively, as shown in Figure 4.Figure 4
Buck-boost with frequency step-up operation.The analysis from Figures2–4 shows that the low frequency switches have less control effort as compared to high frequency switches. So RMS and average currents through low frequency switches are low than that of high frequency switching devices. Section 4 compares the performance parameters of the proposed converter with the existing converters in detail.


## 3.1. Matrix Converter as a Noninverting and Inverting AC Voltage Controller

The noninverting operation of proposed converter as an ac voltage controller is obtained by switching the proposed converter in mode “A” and “B” alternatively. Its inverting operation requires the alternative operation of mode “C” and “D”. The situation is depicted in Figures 2(a) and 2(b) indicating the gating sequence of noninverting and inverting buck-boost characteristics with two direct and two indirect PWM control switches.Figure 2
Switching sequences: (a) noninverting buck-boost ac voltage controller; (b) inverting buck-boost ac voltage controller.
(a) (b)


## 3.2. Matrix Converter as a Frequency Changer

The proposed converter can also be implemented for step change of the load frequency. The frequency buck and boost operation are discussed separately with the help of switching waveforms.(1  ) Frequency Buck Operation. In frequency buck operation, the output frequency is considered to be one-half of the input frequency. So, it operates in mode “A” to “D”, respectively, as the required output is noninverted and inverted for first and second cycles of the input voltage as shown in Figure 3.Figure 3
Buck-boost with frequency step-down operation.(2  ) Frequency Boost Operation. In frequency boost operation, the output frequency is considered two times as that of input frequency. The required output is obtained with the operation of the proposed converter in modes “A”, “C”, “D”, and “B”, respectively, as shown in Figure 4.Figure 4
Buck-boost with frequency step-up operation.The analysis from Figures2–4 shows that the low frequency switches have less control effort as compared to high frequency switches. So RMS and average currents through low frequency switches are low than that of high frequency switching devices. Section 4 compares the performance parameters of the proposed converter with the existing converters in detail.


## 4. Performance Evaluation of the Proposed Converter

The performance of the proposed converter is evaluated and compared with the existing matrix converters in terms of voltage stresses and power losses.


### 4.1. Voltage Stresses

Voltages are developed across the series connected diode once a high frequency switch is controlled in IDPWM manners as diode is reverse biased. Reverse voltage is developed across the MOSFET in DPWM control as diodes is forward biased. The instantaneous voltage stresses across the low and high frequency-controlled switches are computed in (13), where vin and vo are the input and output voltage, respectively.(13)vM2,M3proposed=vDS2,DS3proposed=vinvM5,M6proposed=vDS5,DS6proposed=vovM1,M4,M7,M8proposed=vin-voThe instantaneous voltage across the switching devices of converters in [30, 31] for inverting and in [32] for noninverting modes is computed in (14) and (15), respectively. (14)vM1,M2=vinvM5,M6=vovM3,M4=vD3,D4=vin+vo(15)vM1,M2=vin+vovD5,D6=vin+vovM4=vovM3=vinSo, the voltage stresses across the MOSFET and series diode in the proposed converter are low and hence they have lowdv/dtproblem.


### 4.2. Power Losses

The losses of switching devices come from switching and conduction losses.(1  ) Switching Losses. The controlled switch is a series combination of MOSFET and fast recovery diode. The switching loss of MOSFET depends on switched voltage, current, switching frequency, output capacitance, and rise and fall time. The switching losses of diodes depend on their reverse recovery characteristics. There are no switching losses of diode in a DPWM controlled switch as diode cathode has negative voltage [32]. In the same way, there is no switching loss of MOSFET of IDPWM controlled switch as it is always in conduction state to ensure the continuous inductor current. The switching losses of proposed converter and converters in [31, 32] are the same and computed in (16) and (17), respectively. (16)Pswproposed=fsw4ILvin+voton+toff+Cosvin+vo2+2QRRπvin+vo(17)Psw31=Psw32=fsw4ILvin+voton+toff+Cosvin+vo2+2QRRπvin+vo(2) Conduction Losses. The conduction losses of low switching frequency devices depend on forward bias voltage of fast recovery diode (VF), its on-state resistance (RF), on-state resistance of MOSFET (RM), and magnitude of conducted current (IL) as current conducted by each switch is the same as the inductor. The conduction losses are computed by considering the sinusoidal output current [29] in the form io=io(max)sin(ωt). The conduction losses of the proposed converter and the converter in [31] are computed in (18) and (19), respectively. (18)Pcondproposed=6VFILπ+32IL2RF+RM(19)Pcond31=4VFILπ+2VF∗ILπ+IL2RF+RM+12IL2RF∗+RM∗where VF∗, RF∗, and RM∗ represent the forward voltage, forward resistance of high voltage diode, and forward resistance of MOSFET, respectively.The conduction losses of the proposed converter are low as on-state voltage drops, and resistances of low voltage rating devices are lower than that of high voltage rating devices as evident from (18) and (19).


## 4.1. Voltage Stresses

Voltages are developed across the series connected diode once a high frequency switch is controlled in IDPWM manners as diode is reverse biased. Reverse voltage is developed across the MOSFET in DPWM control as diodes is forward biased. The instantaneous voltage stresses across the low and high frequency-controlled switches are computed in (13), where vin and vo are the input and output voltage, respectively.(13)vM2,M3proposed=vDS2,DS3proposed=vinvM5,M6proposed=vDS5,DS6proposed=vovM1,M4,M7,M8proposed=vin-voThe instantaneous voltage across the switching devices of converters in [30, 31] for inverting and in [32] for noninverting modes is computed in (14) and (15), respectively. (14)vM1,M2=vinvM5,M6=vovM3,M4=vD3,D4=vin+vo(15)vM1,M2=vin+vovD5,D6=vin+vovM4=vovM3=vinSo, the voltage stresses across the MOSFET and series diode in the proposed converter are low and hence they have lowdv/dtproblem.


## 4.2. Power Losses

The losses of switching devices come from switching and conduction losses.(1  ) Switching Losses. The controlled switch is a series combination of MOSFET and fast recovery diode. The switching loss of MOSFET depends on switched voltage, current, switching frequency, output capacitance, and rise and fall time. The switching losses of diodes depend on their reverse recovery characteristics. There are no switching losses of diode in a DPWM controlled switch as diode cathode has negative voltage [32]. In the same way, there is no switching loss of MOSFET of IDPWM controlled switch as it is always in conduction state to ensure the continuous inductor current. The switching losses of proposed converter and converters in [31, 32] are the same and computed in (16) and (17), respectively. (16)Pswproposed=fsw4ILvin+voton+toff+Cosvin+vo2+2QRRπvin+vo(17)Psw31=Psw32=fsw4ILvin+voton+toff+Cosvin+vo2+2QRRπvin+vo(2) Conduction Losses. The conduction losses of low switching frequency devices depend on forward bias voltage of fast recovery diode (VF), its on-state resistance (RF), on-state resistance of MOSFET (RM), and magnitude of conducted current (IL) as current conducted by each switch is the same as the inductor. The conduction losses are computed by considering the sinusoidal output current [29] in the form io=io(max)sin(ωt). The conduction losses of the proposed converter and the converter in [31] are computed in (18) and (19), respectively. (18)Pcondproposed=6VFILπ+32IL2RF+RM(19)Pcond31=4VFILπ+2VF∗ILπ+IL2RF+RM+12IL2RF∗+RM∗where VF∗, RF∗, and RM∗ represent the forward voltage, forward resistance of high voltage diode, and forward resistance of MOSFET, respectively.The conduction losses of the proposed converter are low as on-state voltage drops, and resistances of low voltage rating devices are lower than that of high voltage rating devices as evident from (18) and (19).


## 5. Parameter Design of the Proposed Converter

This section involves the design of inductor, semiconductor switching devices, and input and output capacitors on the basis of the maximum voltage and current stresses as discussed in [29]. The maximum inductor current ignoring ripple during buck-boost operation for constant output power and input voltage can be realized as follows.(20)ILmax=Po1Vin+1Vo=PoVin1+GG=Pod.VinThe peak-to-peak allowable ripples of the state variables of (1) can be expressed by (21) and (22) (21)ΔIL=d.VinLT(22)ΔVo=d.IoCTwhere T is the switching period. The required value of inductor to hold the above peak-to-peak ripple in inductor current is computed as follows.(23)ΔIL≤2IL⇒L≥d2Vin22PofsThe maximum inductor current including ripple component is computed as follows.(24)iLmax=P0dVin+dVin2LfsSimilarly, the required value of output capacitance to hold the peak-to-peak ripple of output voltage is realized as follows.(25)ΔVC≤2Vo⇒C≥Po1-d22fsVin2(26)vomax=Vo+dP02VoCfsThe values of inductor and capacitors are designed by knowing the values of the input voltage, switching frequency, maximum voltage gain, load impedance, peak-to-peak value of inductor current (ki.IL(max)) and output ripple (kv.Vo(max)). (27)Iomax=VinmaxGmaxZL(28)ILmax=Iomax1-d(29)L=Vinmaxd1-dkiILmaxfsThe input capacitance is based on the discontinuous duration of the input current and is computed as follows.(30)Iinmax=VinmaxG2ZL(31)∴Cin=Iinmax1-dkvVinmaxfsOutput capacitance depends on the output ripples and is realized as follows.(32)C=dIomaxkvVomaxfs


## 6. Harmonic Analysis

The output voltage waveform in variable frequency operation is nonsinusoidal due to generated harmonics. A pulse selective approach as discussed in [34, 35] is employed to compute the harmonic contents analytically by decomposing a complex nonsinusoidal to its parent sinusoidal waveforms. As the frequency of the output voltage is changed in discrete steps, a step change of 25 Hz is taken for harmonic analysis. The resulted computed harmonics coefficients are summarized in Tables 1 and 2. Through the analysis of Table 1, a generalized close form is developed in (33). (33)α=2f0-fsfs,δ=δ,ifδ=2f0fs=evenδ+1otherwisebn=∑κ=1δ/2-14Vmπ-1κ+1sinnπκ/δ-2πκ/δn-2-sinnπκ/δ+2πκ/δn+2+4Vm2πsinnπα/2n-2-sinnπα/2n+2an=0Table 1
Computed harmonic coefficients in variable frequency operation.
Output Frequency (Hz) Harmonic Coefficients Output Frequency (Hz) Harmonic Coefficients 75 x = 4 V m π ( n - 2 ) s i n n π 3 - 2 π 3 + 1 2 s i n n π 2 100 x = 4 V m π ( n - 2 ) s i n n π 4 - 2 π 4 + 1 2 s i n 2 n π 2  - 4 V m π ( n + 2 ) s i n n π 3 + 2 π 3 + 1 2 s i n n π 2 - 4 V m π ( n + 2 ) s i n n π 4 + 2 π 4 + 1 2 s i n 2 n π 2 a n = 0 a n = 0 b n = K x , for  n = o d d 0 , o t h e r w i s e b n = K x , for  n = 4,8 , 12,16 , … 0 , o t h e r w i s e  125 x = 4 V m π ( n - 2 ) s i n n π 5 - 2 π 5 - s i n 2 n π 5 - 4 π 5 + 1 2 s i n 3 n π 2 150 x = 4 V m π ( n - 2 ) s i n n π 6 - 2 π 6 - s i n 2 n π 6 - 4 π 6 + 1 2 s i n 4 n π 2  - 4 V m π ( n + 2 ) s i n n π 5 + 2 π 5 - s i n 2 n π 5 + 4 π 5 + 1 2 s i n 3 n π 2 - 4 V m π ( n + 2 ) s i n n π 6 + 2 π 6 - s i n 2 n π 6 + 4 π 6 + 1 2 s i n 4 n π 2 a n = 0 a n = 0 bn=Kx,forn=1,3,5,7,…0,otherwise b n = K x , for  n = 2,6 , 10,14 , … 0 , o t h e r w i s e  175 x = 4 V m π ( n - 2 ) s i n n π 7 - 2 π 7 - s i n 2 n π 7 - 4 π 7 200 x = 4 V m π ( n - 2 ) s i n n π 8 - 2 π 8 - s i n 2 n π 8 - 4 π 8  + s i n 3 n π 7 - 6 π 7 + 1 2 s i n 5 n π 2 + s i n 3 n π 8 - 6 π 8 + 1 2 s i n 6 n π 2 - 4 V m π ( n + 2 ) s i n n π 7 + 2 π 7 - s i n 2 n π 7 + 4 π 7 - 4 V m π ( n + 2 ) s i n n π 8 + 2 π 8 - s i n 2 n π 8 + 4 π 8 + s i n 3 n π 7 + 6 π 7 + 1 2 s i n 5 n π 2 + s i n 3 n π 8 + 6 π 8 + 1 2 s i n 6 n π 2 a n = 0 a n = 0 bn=Kx,forn=1,3,5,7,…0,otherwise b n = K x , for  n = 4,8 , 12,16 , … 0 , o t h e r w i s e  225 x = 4 V m π ( n - 2 ) s i n n π 9 - 2 π 9 - s i n 2 n π 9 - 4 π 9 250 x = 4 V m π ( n - 2 ) s i n n π 10 - 2 π 10 - s i n 2 n π 10 - 4 π 10  + s i n 3 n π 9 - 6 π 9 - s i n 4 n π 9 - 8 π 9 + 1 2 s i n 7 n π 2 + s i n 3 n π 10 - 6 π 10 - s i n 4 n π 10 - 8 π 10 + 1 2 s i n 8 n π 2  - 4 V m π ( n + 2 ) s i n n π 9 + 2 π 9 - s i n 2 n π 9 + 4 π 9 - 4 V m π ( n + 2 ) s i n n π 10 + 2 π 10 - s i n 2 n π 10 + 4 π 10  + s i n 3 n π 9 + 6 π 9 - s i n 4 n π 9 + 8 π 9 + 1 2 s i n 7 n π 2 + s i n 3 n π 10 + 6 π 10 - s i n 4 n π 10 + 8 π 10 + 1 2 s i n 8 n π 2 a n = 0 a n = 0 bn=Kx,forn=1,3,5,7,…0,otherwise b n = K x , for  n = 2,6 , 10,14 , … 0 , o t h e r w i s eTable 2
Harmonic analysis for variable frequency operation with duty ratio of 0.2.
Harm.Order. f o = 25 Hz f o = 75 Hz f o = 100 Hz f o = 150 Hz Mathematical Values Simulated Values Mathematical Values Simulated Values Mathematical Values Simulated Values Mathematical Values Simulated Values Mag. PhaseAngle Mag. PhaseAngle Mag. Phase Angle Mag. PhaseAngle Mag. Phase Angle Mag. PhaseAngle Mag. Phase Angle Mag. PhaseAngle 1 54.97 0 54.87 0 15.39 0 15.51 0 0 0 0 0 0 0 0 0  2 0 0 0 0 0 0 0 0 0 0 0 0 14.18 180 14.18 180  3 33.10 0 33.10 0 52.90 0 52.90 0 0 0 0 0 0 0 0 0  4 0 0 0 0 0 0 0 0 54.17 0 54.17 0 0 0 0 0  5 7.88 180 7.88 180 2.35 180 2.36 180 0 0 0 0 0 0 0 0  6 0 0 0 0 0 0 0 0 0 0 0 0 53.75 0 53.75 0  7 3.67 0 3.67 0 18.01 180 18.01 180 0 0 0 0 0 0 0 0  8 0 0 0 0 0 0 0 0 22.06 180 22.07 180 0 0 0 0  9 2.14 180 2.15 180 18.90 0 18.90 0 0 0 0 0 0 0 0 0  10 0 0 0 0 0 0 0 0 0 0 0 0 17.91 180 17.91 180  11 1.41 0 1.41 0 6.92 180 6.93 180 0 0 0 0 0 0 0 0  12 0 0 0 0 0 0 0 0 14.18 0 14.19 0 0 0 0 0  13 1.0 180 1.0 180 5.51 180 5.50 180 0 0 0 0 0 0 0 0  14 0 0 0 0 0 0 0 0 0 0 0 0 8.95 180 8.96 180  15 0.74 0 0.75 0 8.98 0 8.98 0 0 0 0 0 0 0 0 0  16 0 0 0 0 0 0 0 0 10.50 180 10.51 180 0 0 0 0  17 0.58 180 0.58 180 3.19 180 3.19 180 0 0 0 0 0 0 0 0  18 0 0 0 0 0 0 0 0 0 0 0 0 16.12 0 16.13 0  19 0.46 0 0.46 0 4.67 180 4.67 180 0 0 0 0 0 0 0 0  20 0 0 0 0 0 0 0 0 8.35 0 8.36 0 0 0 0 0  21 0.37 180 0.38 180 7.26 0 7.26 0 0 0 0 0 0 0 0 0  22 0 0 0 0 0 0 0 0 0 0 0 0 7.16 180 7.15 180  23 0.31 0 0.31 0 3.18 180 3.18 180 0 0 0 0 0 0 0 0  24 0 0 0 0 0 0 0 0 6.95 180 6.94 180 0 0 0 0  25 0.26 180 0.26 180 2.84 180 2.84 180 0 0 0 0 0 0 0 0  26 0 0 0 0 0 0 0 0 0 0 0 0 5.11 180 5.12 180  27 0.22 0 0.22 0 5.11 0 5.11 0 0 0 0 0 0 0 0 0  28 0 0 0 0 0 0 0 0 5.95 0 5.94 0 0 0 0 0  29 0.18 180 0.18 0 2.11 180 2.11 180 0 0 0 0 0 0 0 0  30 0 0 0 0 0 0 0 0 0 0 0 0 9.59 0 9.59 0


## 7. Sliding Mode Controller

The sliding mode (SM) control is a more suitable feedback approach in direct ac-to-ac converters having inherent variable structure. The buck-boost system is a nonminimum phase system with respect to output voltage regulation [36]. The output voltage is indirectly controlled through the control of inductor current. The output voltage regulation through inductor current control based on sliding control theory, ensures the fast and dynamic response. The basic key is to design a SM control law that ensures the arrival of the system’s states at equilibrium point. The instantaneous inductor current in the proposed converter is unidirectional for positive and negative input voltage. The error signal is generated by comparing it with the reference current; i.e.,(34)e=iref-iL(35)∴dedt=direfdt-diLdt.The error and its derivative are selected as state variablesx1 and x2, which are rewritten as follows.(36)x1=iref-iL(37)x2=dx1dt=direfdt-diLdt


### 7.1. Sliding Surface

The following switching function is selected to govern the switching states of the proposed converter:(38)μ=121+signS(39)μ=1S→0+0S→0-where “S” is the trajectory of the state variable. The error signal is selected as sliding surface to generate the desire control (gating) signals for high frequency switching devices of the proposed converter.(40)S=iref-iL(41)∴dSdt=direfdt-diLdtThe existence of the SM control requires the test of transversality, reachability, and equivalent control condition.(1  ) Transversality Condition. It describes the controllability of the system ensuring that system dynamics are affected from the SM control [37]. It means that control variable should be in the derivative form of the sliding surface. That is to say,(42)ddμdSdt≠0.(2  ) Reachability Condition. It describes the ability of the system to reach the sliding surface [37]. (43)limS→0+S>0limS→0-S<0,limS→0+S•<0limS→0-S•>0(3  ) Equivalent Control Condition. It determines the local stability of the system and ensures that system remains in the sliding surface once it enters the sliding surface [37]; i.e.,(44)dSdtμ=μeq=0,0<μeq<1The required conditions to test the existence of the SM control are tested for all operating modes of the proposed converter and are tabulated in Table3, ensuring the transversality, reachability, and equivalent control condition.Table 3
Stability analysis for inverting and noninverting operation.
Non-Inverting Operation Inverting Operation v i n > 0  &  v o > 0 v i n < 0  &  v o < 0 v i n > 0  &  v o < 0 v i n < 0  &  v o > 0   d i L d t =  μ v i n L - ( 1 - μ ) v 0 L d i L d t =  - μ v i n L + ( 1 - μ ) v 0 L d i L d t =  μ v i n L + ( 1 - μ ) v 0 L d i L d t = -  μ v i n L - ( 1 - μ ) v 0 L   d ( s ) d t = - μ v i n L  + ( 1 - μ ) v 0 L d ( s ) d t = + μ v i n L  - ( 1 - μ ) v 0 L d ( s ) d t = - μ v i n L  - ( 1 - μ ) v 0 L d ( s ) d t = μ v i n L  + ( 1 - μ ) v 0 L   d ( s ) d t = - v i n L S → 0 +   ( μ = 1 ) + v o L S → 0 -   ( μ = 0 ) d ( s ) d t = v i n L S → 0 +   ( μ = 1 ) - v o L S → 0 -   ( μ = 0 ) d ( s ) d t = - v i n L S → 0 +   ( μ = 1 ) - v o L S → 0 -   ( μ = 0 ) d ( s ) d t = v i n L S → 0 +   ( μ = 1 ) v o L S → 0 -   ( μ = 0 )   d d μ d ( s ) d t = - v i n L  - v 0 L d d μ d ( s ) d t = v i n L  + v 0 L d d μ d ( s ) d t = - v i n L  + v 0 L d d μ d ( s ) d t = v i n L  - v 0 L   μ e q = v o v o + v i n μ e q = v o v o + v i n μ e q = v o v o - v i n μ e q = v o v o - v i n


## 7.1. Sliding Surface

The following switching function is selected to govern the switching states of the proposed converter:(38)μ=121+signS(39)μ=1S→0+0S→0-where “S” is the trajectory of the state variable. The error signal is selected as sliding surface to generate the desire control (gating) signals for high frequency switching devices of the proposed converter.(40)S=iref-iL(41)∴dSdt=direfdt-diLdtThe existence of the SM control requires the test of transversality, reachability, and equivalent control condition.(1  ) Transversality Condition. It describes the controllability of the system ensuring that system dynamics are affected from the SM control [37]. It means that control variable should be in the derivative form of the sliding surface. That is to say,(42)ddμdSdt≠0.(2  ) Reachability Condition. It describes the ability of the system to reach the sliding surface [37]. (43)limS→0+S>0limS→0-S<0,limS→0+S•<0limS→0-S•>0(3  ) Equivalent Control Condition. It determines the local stability of the system and ensures that system remains in the sliding surface once it enters the sliding surface [37]; i.e.,(44)dSdtμ=μeq=0,0<μeq<1The required conditions to test the existence of the SM control are tested for all operating modes of the proposed converter and are tabulated in Table3, ensuring the transversality, reachability, and equivalent control condition.Table 3
Stability analysis for inverting and noninverting operation.
Non-Inverting Operation Inverting Operation v i n > 0  &  v o > 0 v i n < 0  &  v o < 0 v i n > 0  &  v o < 0 v i n < 0  &  v o > 0   d i L d t =  μ v i n L - ( 1 - μ ) v 0 L d i L d t =  - μ v i n L + ( 1 - μ ) v 0 L d i L d t =  μ v i n L + ( 1 - μ ) v 0 L d i L d t = -  μ v i n L - ( 1 - μ ) v 0 L   d ( s ) d t = - μ v i n L  + ( 1 - μ ) v 0 L d ( s ) d t = + μ v i n L  - ( 1 - μ ) v 0 L d ( s ) d t = - μ v i n L  - ( 1 - μ ) v 0 L d ( s ) d t = μ v i n L  + ( 1 - μ ) v 0 L   d ( s ) d t = - v i n L S → 0 +   ( μ = 1 ) + v o L S → 0 -   ( μ = 0 ) d ( s ) d t = v i n L S → 0 +   ( μ = 1 ) - v o L S → 0 -   ( μ = 0 ) d ( s ) d t = - v i n L S → 0 +   ( μ = 1 ) - v o L S → 0 -   ( μ = 0 ) d ( s ) d t = v i n L S → 0 +   ( μ = 1 ) v o L S → 0 -   ( μ = 0 )   d d μ d ( s ) d t = - v i n L  - v 0 L d d μ d ( s ) d t = v i n L  + v 0 L d d μ d ( s ) d t = - v i n L  + v 0 L d d μ d ( s ) d t = v i n L  - v 0 L   μ e q = v o v o + v i n μ e q = v o v o + v i n μ e q = v o v o - v i n μ e q = v o v o - v i n


## 8. Simulation Results

The proposed converter is simulated in MATLAB/Simulink environment to find the voltage stresses across the high and low frequency-controlled switches. The noninverting and inverting modes are simulated as a frequency controller with the duty ratio of 0.5. Typical circuit parameter values used in simulation circuit as tabulated in Table4 are computed by using (29), (31), and (32) and by considering their reactive power rating.Table 4
Circuit parameters.
Typical Parameter Value Chopping Frequency (fs) 25 kHz  Filtering Inductance (L) 1 mH  Input Capacitance (Cin) 1μF  Output Capacitance (C) 4.7μF  Output Resistance (R) 40 Ω  MOSFETRDSon⁡ 0.08 Ω  Forward Voltage Drop (VF) 2 V  Source Voltage 230VRMS


### 8.1. Frequency Buck Operation

The frequency step-down operation with output frequency of 25 Hz is accomplished by operating the proposed converter in noninverting and inverting mode in consecutive cycles of the input voltage waveform. Figure5 shows the voltage stresses across the high and low frequency switching devices in noninverting and inverting mode.Figure 5
Voltage stresses across (a)M2(DS2); (b) M3(DS3); (c) M5(DS5); (d) M6(DS6); (e) M1; (f) M4; (g) M7; (h) M8.
(a) (b) (c) (d) (e) (f) (g) (h)It is evident from Figure5 that voltage stresses across the high frequency switches M2(DS2), M3(DS3) and M5(DS5), M6(DS6) are restricted to vin and vo, respectively. The voltage stresses across the low frequency switches M1, M4, M7, and M8 are vin-vo. But the voltage stresses in inverting mode of [29–31] are raised to a high value of vin+vo.


### 8.2. Frequency Boost Operation

In frequency step-up operation, the output is forced to be changed from inverting to noninverting and vice versa to get the required output. This action generates the high voltage surges in the output voltage that can cause the high voltage stresses and highdv/dt across the switching devices. Figures 6 and 7 explore these problems for proposed and converter in [31], respectively.Figure 6
Voltage stresses across (a)M2(DS2); (b) M3(DS3); (c) M5(DS5); (d) M6(DS6); (e) M1; (f) M4; (g) M7; (h) M8.
(a) (b) (c) (d) (e) (f) (g) (h)Figure 7
Voltage stresses across (a)M1(DS1); (b) M2(DS2); (c) M5(DS5); (d) M6(DS6); (e) M3(DS3); (f) M4(DS4).
(a) (b) (c) (d) (e) (f)It is clear from the Figures6 and 7 that maximum voltage stresses of high frequency switching devices in proposed converter are vin and vo. But in [31], they are increased to vin+vo and hence they have the problem of highdv/dt.The nonsinusoidal nature of the output voltage degrades power quality of the output voltage. SM feedback controller is designed to improve the power quality and to reduce the voltage overshoot problem across the switching devices as shown in Figures8 and 9, respectively.Figure 8
Output voltage waveform with SM control.Figure 9
Voltage stresses in noninverting and inverting mode with SM control: (a)M2(DS2); (b) M3(DS3); (c) M5(DS5); (d) M6(DS6); (e) M1; (f) M4; (g) M7; (h) M8.
(a) (b) (c) (d) (e) (f) (g) (h)


## 8.1. Frequency Buck Operation

The frequency step-down operation with output frequency of 25 Hz is accomplished by operating the proposed converter in noninverting and inverting mode in consecutive cycles of the input voltage waveform. Figure5 shows the voltage stresses across the high and low frequency switching devices in noninverting and inverting mode.Figure 5
Voltage stresses across (a)M2(DS2); (b) M3(DS3); (c) M5(DS5); (d) M6(DS6); (e) M1; (f) M4; (g) M7; (h) M8.
(a) (b) (c) (d) (e) (f) (g) (h)It is evident from Figure5 that voltage stresses across the high frequency switches M2(DS2), M3(DS3) and M5(DS5), M6(DS6) are restricted to vin and vo, respectively. The voltage stresses across the low frequency switches M1, M4, M7, and M8 are vin-vo. But the voltage stresses in inverting mode of [29–31] are raised to a high value of vin+vo.


## 8.2. Frequency Boost Operation

In frequency step-up operation, the output is forced to be changed from inverting to noninverting and vice versa to get the required output. This action generates the high voltage surges in the output voltage that can cause the high voltage stresses and highdv/dt across the switching devices. Figures 6 and 7 explore these problems for proposed and converter in [31], respectively.Figure 6
Voltage stresses across (a)M2(DS2); (b) M3(DS3); (c) M5(DS5); (d) M6(DS6); (e) M1; (f) M4; (g) M7; (h) M8.
(a) (b) (c) (d) (e) (f) (g) (h)Figure 7
Voltage stresses across (a)M1(DS1); (b) M2(DS2); (c) M5(DS5); (d) M6(DS6); (e) M3(DS3); (f) M4(DS4).
(a) (b) (c) (d) (e) (f)It is clear from the Figures6 and 7 that maximum voltage stresses of high frequency switching devices in proposed converter are vin and vo. But in [31], they are increased to vin+vo and hence they have the problem of highdv/dt.The nonsinusoidal nature of the output voltage degrades power quality of the output voltage. SM feedback controller is designed to improve the power quality and to reduce the voltage overshoot problem across the switching devices as shown in Figures8 and 9, respectively.Figure 8
Output voltage waveform with SM control.Figure 9
Voltage stresses in noninverting and inverting mode with SM control: (a)M2(DS2); (b) M3(DS3); (c) M5(DS5); (d) M6(DS6); (e) M1; (f) M4; (g) M7; (h) M8.
(a) (b) (c) (d) (e) (f) (g) (h)


## 9. Experimental Setup

An experimental setup is constructed to validate the simulation results (see Figure10). The hardware consists of eight MOSFETs (IRF840) as power electronic switches along with RHG3040 fast recovery diodes. The filter capacitor at the input side and the output side is 1 μF and 4.7 μF, respectively, while the coil has an inductance of 1 mH. Low frequency switching signals are generated by using STM microcontroller. The synchronization of the low frequency control signals with input voltage is accomplished with zero-crossing detection of the input voltage. Two complementary high frequency PWM switching signals are generated with SM controller. The gate driving circuits are implemented with hybrid chips (EXB840) with isolated dc supplies. All the experimental results are recorded using Rigol (DS1052E) oscilloscope in which red color represents inputs and blue color represents outputs.Figure 10
Hardware setup.As already remarked, the proposed converter may be employed as an ac voltage controller. Figure11 shows the output of the proposed converter in noninverting and inverting operating modes with a voltage gain of 0.5.Figure 11
Input-output voltage in (a) noninverting mode and (b) inverting mode.
(a) (b)Similarly, the proposed converter may be employed as a direct frequency changer (DFC). Figure12 shows the conversion of 50 Hz input frequency to 25 Hz and 100 Hz output frequency, respectively. High surges in output voltage result in high surges in input current (see Figure 13).Figure 12
Input and output voltage: (a) frequency step-down operation; (b) frequency step-up operation.
(a) (b)Figure 13
Input current waveform in frequency step-up operation.As can be observed from Figures12 and 13, in open-loop, once the output is changed from noninverting to inverting and vice versa, it generates the surges in output voltage and input current. This problem may result in failure of the switching devices. The surges in output voltage and input current can be eliminated by introducing SM controller into the loop. Figure 14 shows the output voltage and input current of the closed loop system in frequency boost operation of 100 Hz. The experimental results indicate that the generated surges have been successfully eliminated by the controller with regulated output voltages. The total harmonic distortion in the input current is found to be less than 3%.Figure 14
Frequency step-up operation with feedback controller: (a) input and output voltage; (b) input current.
(a) (b)


## 10. Conclusion

This research is focused on a direct ac-to-ac buck-boost matrix converter having bipolar voltage gain to cope with the power system disturbances such as voltage sag and swell by ensuring low voltage stresses and hence lowdv/dtacross low and high frequency-controlled switches This results in low conduction losses due to their low forward voltage and low on-state resistance. The proposed converter can also change the load frequency in variable frequency drive system and radio frequency induction heating. The power quality of the output voltage in variable frequency mode is analyzed by finding the harmonic contents through pulse selective approach. The problem of generated voltage and current surges in frequency boost operation is analyzed. A robust sliding mode controller is designed to suppress these high surges of output voltage and input current. This also improves the power quality of the output voltage with reduced THD and improved power factor. The dynamic stability including transversality, reachability, and equivalent control condition of the proposed converter is verified for its all operating modes. The detailed simulation and experimental results validate the performance of the proposed controlled converter.



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