Improved Modulation Strategy Based on Minimum Energy Storage Principle for Electrolytic-Capacitor-Less Six-Switch Converter

An improved modulation strategy based on minimum energy storage for DC-link capacitance reduction in a six-switch AC-AC converter is proposed. The proposed modulation strategy enables the energy on the capacitor to accumulate and release twice each in a complete switching cycle, achieving the effect of “fast charging and discharging”. Meanwhile, the inversion and rectification are modulated synchronously. Hence, there is minimum energy stored in the DC-link capacitor. Then, the time average modeling analysis is presented to take insight analysis. When there is the same voltage ripples constraint on the DC side for the conventional and improved modulation strategies, the six-switch converter under the improved modulation strategy has the much less capacitance value of the storage capacitor and even realizes non-electrolytic capacitance. Therefore, improving the system efficiency, power density, and output waveform quality and extending the system life can be achieved. The operation principle and modulation strategy are discussed in detail. Finally, the simulation model and experimental prototype are built to verify effectiveness of the topology and correctness of the proposed six-switch AC-AC converter modulation strategy.


Introduction
Single-phase AC-DC-AC converters with the same input/output frequencies can be found in multi-electric aircraft [1], such as power interface units, standby power supplies, uninterruptible power supplier, etc. Large DC-link energy storage capacitors are usually used to adsorb pulsating power, which is harmful to the system, but the presence of the large electrolytic capacitor will lead to larger volume, heavier weight, and higher price [2]. So, a great deal of research is devoted to reducing DC-link capacitor size and mitigating second-order power in the single-phase converter.
At present, non-electrolytic capacitive methods can be divided into two categories: active methods and passive methods [3]. The main idea of the passive methods is to use passive components or passive networks instead of large electrolytic capacitors for energy storage. The simplest passive method is to replace the large electrolytic capacitor with multiple small-capacity non-electrolytic capacitors (such as film capacitors) in parallel, but its obvious disadvantages such as low withstand voltage, large volume, and high price limit its practical applications. In [4][5][6][7][8], a resonance unit is used to replace the traditional DC energy storage unit, eliminating large-capacity energy storage components, while also realizing soft-switching, and the dynamic performance of the system is also improved. However, due to the shortcomings of the resonant unit itself, the circuit will generate higher voltage and current stress, and the conduction loss will also increase. In addition, this passive method also has the disadvantages of large volume and heavy weight.
The active methods mainly can be grouped into reducing or transferring the instantaneous power difference. Some control strategies are proposed in [9,10]. The instantaneous

The Topology and Modulation Principle of Six-Switch Converter
The topology of the six-switch converter is shown in Figure 1. Each bridge arm is composed of three switches in series, and there are two sets of input and output. Compared with the traditional single-phase back-to-back AC-DC-AC converter, the proposed converter reduces two switches. In the proposed six-switch converter, the switch in the middle of each bridge arm is shared by the rectifier and inverter, and the voltage v XN is no greater than v AN at any time. As shown in Table 1, each bridge arm of the six-switch converter has three switching states.
In order to meet the switching constraint requirements of the six-switch converter, the modified carrier-based PWM method is applied, and the modulation reference of the upper rectifier and the modulation reference of the lower inverter are compared with a common triangular carrier, respectively, as shown in Figure 2.
The instantaneous value of the upper reference must be no less than the lower reference. The PWM signal of switch ST 1 (ST 3 ) is generated by comparing the upper (lower) reference and the triangular carrier, and the PWM signal of switch ST 2 is generated by logic operation. where v cw is the common triangular carrier, v rA and v rX are the upper and the lower modulation references, respectively.
reference and the triangular carrier, and the PWM signal of switch ST2 is generated by logic operation.

Switching Status
reference and the triangular carrier, and the PWM signal of switch ST2 is generated by logic operation.
where vcw is the common triangular carrier, vrA and vrX are the upper and the lower modulation references, respectively. Figure 1. The six-switch converter topology.   0  2  4  6  8  10  12  14  16  18  20   0  2  4  6  8  10  12  14  16  18  20   0  2  4  6  8  10  12  14  16  18  20   0  2  4  6  8  10  12  14  16 18 20 Figure 2. Generation of the single bridge arm PWM signal. Figure 3 shows the modulation references of ST1, ST3, ST4, ST6 in a six-switch converter. It can be observed that due to the relative position changes of the four references in one line frequency cycle, the switching signals of the six switches also change. There-  Figure 3 shows the modulation references of ST 1 , ST 3 , ST 4 , ST 6 in a six-switch converter. It can be observed that due to the relative position changes of the four references in one line frequency cycle, the switching signals of the six switches also change. Therefore, the relative position in a line frequency cycle can be divided into four states (two for the first half cycle and two for the second half cycle) for analysis. Take the first half line frequency cycle of the modulation reference as an example to analyze the working states of the circuit. fore, the relative position in a line frequency cycle can be divided into four states (two for the first half cycle and two for the second half cycle) for analysis. Take the first half line frequency cycle of the modulation reference as an example to analyze the working states of the circuit. There are four working states of the six-switch converter, which are introduced as follows. The rectification state (abbreviated as RS) is shown in Figure 4a, the inversion state (abbreviated as IS) is shown in Figure 4b, the freewheeling state (abbreviated as FS) is shown in Figure 4c, and the both-way state (abbreviated as BS) is shown in Figure 4d. When the relative positions of four modulation references are shown as state A in Figure 3, the six-switch converter has the first three working states in one switching cycle. When the relative positions of four modulation references are shown as state B in Figure 3, the six-switch converter has all four working states in one switching cycle.  There are four working states of the six-switch converter, which are introduced as follows. The rectification state (abbreviated as RS) is shown in Figure 4a, the inversion state (abbreviated as IS) is shown in Figure 4b, the freewheeling state (abbreviated as FS) is shown in Figure 4c, and the both-way state (abbreviated as BS) is shown in Figure 4d. When the relative positions of four modulation references are shown as state A in Figure 3, the six-switch converter has the first three working states in one switching cycle. When the relative positions of four modulation references are shown as state B in Figure 3, the six-switch converter has all four working states in one switching cycle.

Switching Status
fore, the relative position in a line frequency cycle can be divided into four states (two for the first half cycle and two for the second half cycle) for analysis. Take the first half line frequency cycle of the modulation reference as an example to analyze the working states of the circuit. There are four working states of the six-switch converter, which are introduced as follows. The rectification state (abbreviated as RS) is shown in Figure 4a, the inversion state (abbreviated as IS) is shown in Figure 4b, the freewheeling state (abbreviated as FS) is shown in Figure 4c, and the both-way state (abbreviated as BS) is shown in Figure 4d. When the relative positions of four modulation references are shown as state A in Figure 3, the six-switch converter has the first three working states in one switching cycle. When the relative positions of four modulation references are shown as state B in Figure 3, the six-switch converter has all four working states in one switching cycle.  From the aforementioned six-switch converter modulation method, a schematic diagram of six switching states in one switching cycle can be achieved, as shown in Figure 5. The state A in Figure 5a corresponds to the state A in Figure 3, which means the relative positions of the modulation references are consistent. State B in Figure 5b corresponds to state B in Figure 3.
From the aforementioned six-switch converter modulation method, a schematic diagram of six switching states in one switching cycle can be achieved, as shown in Figure 5. The state A in Figure 5a corresponds to the state A in Figure 3, which means the relative positions of the modulation references are consistent. State B in Figure 5b corresponds to state B in Figure 3.  vcw, vrA, and vrX were previously defined on paper, and vrB and vrY are the modulation references of the upper switch and the modulation references of the lower switch of the second bridge arm, respectively. d1, d3, d4, and d6 denote the duty ratios of ST1, ST3, ST4, ST6, and TS is the switching period. From Figure 5a, it can be concluded that when the switching states of ST1~ST6 are {101011}, the six-switch converter is in RS, and the state duration is (d1-d4)TS. When the switching states of ST1~ST6 are {110101}, the six-switch converter is in IS, and the state duration is (d6-d3)TS. The rest of the switching states are in FS, and the state duration is (

Improved Modulation Strategy Analysis of Six-Switch Converter Based on Time Average Modulation
It is assumed the inversion and rectification have the same power factor. Under the traditional six-switch converter control strategy, one switching cycle includes eight working states, followed by FS, RS, FS (BS in state B), IS, FS, IS, FS (BS in state B), RS. Two RS (or IS) are too close, and there is only one FS between them, so that the DC side capacitor is continuously charged (or discharged). A large capacitor is required, which greatly limits practical applications of the six-switch converter. The key point of the proposed improved modulation method is to exchange rectification state and inversion state in the second half of the switching cycle by logic control, so that the order of the eight states in a cycle becomes FS, RS, FS (BS in state B), IS, FS, RS, FS (BS in state B), IS. Therefore, the energy on the capacitor can quickly accumulate and release in a complete switching cycle, achieving the effect of 'fast charging and discharging'. Figure 6 shows a larger version of the DC side capacitor voltage vd under two modulation strategies. Each linear section in the curve corresponds to FS, RS, IS, and BS, respectively.

Improved Modulation Strategy Analysis of Six-Switch Converter Based on Time Average Modulation
It is assumed the inversion and rectification have the same power factor. Under the traditional six-switch converter control strategy, one switching cycle includes eight working states, followed by FS, RS, FS (BS in state B), IS, FS, IS, FS (BS in state B), RS. Two RS (or IS) are too close, and there is only one FS between them, so that the DC side capacitor is continuously charged (or discharged). A large capacitor is required, which greatly limits practical applications of the six-switch converter. The key point of the proposed improved modulation method is to exchange rectification state and inversion state in the second half of the switching cycle by logic control, so that the order of the eight states in a cycle becomes FS, RS, FS (BS in state B), IS, FS, RS, FS (BS in state B), IS. Therefore, the energy on the capacitor can quickly accumulate and release in a complete switching cycle, achieving the effect of 'fast charging and discharging'. According to the input/output power balance theory and supposing VS ≈ VAB ≈ VXY, the Equation (2) can be obtained by vector relations. Where X1 is the input impedance and X2 is the output impedance. θ1 represents the phase angle of rectification modulation reference and θ0 is the power factor angle of inverter load. When the radio of X1/X2 is about 0.2, the unit input power factor can be ensured. According to the input/output power balance theory and supposing V S ≈ V AB ≈ V XY , the Equation (2) can be obtained by vector relations. Where X 1 is the input impedance and X 2 is the output impedance. θ 1 represents the phase angle of rectification modulation reference and θ 0 is the power factor angle of inverter load. When the radio of X 1 /X 2 is about 0.2, the unit input power factor can be ensured.
The voltage fluctuation of the DC side capacitor is caused by the fluctuating current flowing through the capacitor, as shown in Equation (3). When the amplitude of i c becomes smaller, the capacitor which used to keep v d relatively stable also decreases. In order to verify the effectiveness of the improved six-switch converter modulation strategy, the i c under the traditional and improved modulation strategies are calculated and compared respectively.
The equation of Kirchhoff's Current Law at point P of six-switch converter in Figure 1 is: In (4), i 1 (t) and i 4 (t) are currents flowing through switch ST 1 and ST 4 based on the predefined reference direction. The average value of current i 1 (t) can be calculated according to the switching state of the six-switch converter. When the switching states of Therefore, the average current i 1 (t) can be written as: Similarly, when the switching states of ST 4 , ST 5 , ST 6 are {011}, i 4 (t) = 0; when the switching states are {101}, i 4 (t) = i s (t) with the duration of (d 4 + d 6 − 1)T S ; when the switching states are {110}, i 4 (t) = i s (t) − i o (t) with the duration of (1 − d 6 )T S . So, the average current i 4 (t) can be written as: Substitute (5) and (6) into (4), average current i c can be figured out as: Modulation reference for rectifier is defined as: Modulation reference for inverter is defined as: where m r and m i are modulation indices for the rectifier and inverter, respectively. Assuming unity input power factor operation condition, ϕ is phase angle difference between rectifier reference and inverter reference. Duty radios d 1 , d 3 , d 4 , d 6 can be calculated through similar triangle theory: Similarly, Parameters d off1 and d off2 are offset values utilized for moving up/down modulation reference of rectifier/inverter.
Finally, substitute d 1 , d 3 , d 4 and d 6 into Equation (7): where i s (t) is expressed as I s sinwt, the phase angle of inversion modulation reference is (wt + θ 1 + ϕ), so the output voltage v xy is also (wt + θ 1 + ϕ). Since the output current i o lags behind the inversion output voltage v xy , the output current's phase angle is (wt Parameters doff1 and doff2 are offset values utilized for moving up/down modulation reference of rectifier/inverter. Finally, substitute d1, d3, d4 and d6 into Equation (7): where is(t) is expressed as Is sinwt, the phase angle of inversion modulation reference is (wt + θ1 + φ), so the output voltage vxy is also (wt + θ1 + φ). Since the output current io lags behind the inversion output voltage vxy, the output current's phase angle is (wt + θ1 + φ − θo). θo is the power factor angle of inverter load, thus io(t) can be expressed as Iosin(wt + θ1 + φ − θo). The angle relationship of proposed converter is shown below.
The constant term in Equation (15) is identical to zero based on the assumptions of no switching or capacitor loss.
The constant term in Equation (15) is identical to zero based on the assumptions of no switching or capacitor loss.
Amplitude of current i c can be gotten as below through expansion of (16).
It is observed from (17) that when phase angle ϕ reaches (θ o − θ 1 )/2, minimization of AC power variation can be achieved in DC-link voltage. Therefore, ϕ is equal to (θ o − θ 1 )/2 in this paper and modulation indices for rectifier and inverter are supposed to be approximately the same (m r = m i = m). I cmin under traditional control can be written as: I c under improved control can be attained in the same way. When the switching states of ST 1 , ST 2 , ST 3 are {011}, i 1 (t) = 0; when the switching states are {101}, i 1 (t) = i s (t) with the duration of ((d 1 + d 3 + d 4 + d 6 )/2 − 1)T s ; when the switching states are {110}, i 1 (t) = i s (t) − i o (t) with the duration of ((d 1 − d 3 − d 4 − d 6 )/2 + 1)T s . Therefore, the average current i 1 (t) can be written as: Similarly, when the switching states of ST 4 , ST 5 , ST 6 are {011}, i 4 (t) = 0; when the switching states are {101}, i 4 (t) = i s (t) with the duration of ((d 1 + d 3 + d 4 + d 6 )/2 − 1)T s ; when the switching states are {110}, i 4 (t) = i s (t) − i o (t) with the duration of (1 − d 6 )T s . So, the average current i 4 (t) can be written as: Substituting d 1 , d 3 , d 4 and d 6 into Equation (20), it yields: The constant term in Equation (22) is identical to zero based on the assumptions that no switching or capacitor loss.
Amplitude of current i c can be got as below through expansion of (23).
I c ' can reach its minimum when ϕ = θ o − θ 1 : The above has been obtained I cmin = 0.5m|I o − I s |. Subtracting Equation (18) from (25) to compare the amplitude, it yields: It can be known I cmin > I c min from (26). The calculation result shows that improved modulation strategy reduces the amplitude of I c and the DC side voltage fluctuation. Therefore, a smaller capacitor C d can be used to maintain the relative stability of the DC voltage v d , thereby achieving the purpose of reducing the capacitance.
The derivations of I cmin and I c min have already reflected the difference of modulation strategy. Therefore, whether the improved modulation strategy is effective can be judged by observing the fluctuation of DC side capacitor voltage under the line frequency cycle.

Simulation and Experimental Results
In order to further verify the feasibility and correctness of the proposed modulation strategy, a linear-load typical system with unity input power factor is built on the Matlab/Simulink platform. The simulation results based on traditional and improved modulation strategy, respectively, are compared to prove the validity of the improved strategy. Simulation parameters are shown in Table 2.  Seen from the simulation results, the proposed modulation strategy can reduce DC-link voltage variation under the line frequency. Therefore, DC-link voltage vd can remain relatively stable with a smaller capacitor. At the same time, the THD of improved output voltage drops significantly and the waveform is better.
Through (17) and (24) in the previous section, Ic = Ic' can be attained when phase angle difference is equal to zero. At this point, it should be observed that the DC-link voltage fluctuations are equal under the line frequency cycle, regardless of which modulation strategy is adopted, as shown in Figure 9.  Seen from the simulation results, the proposed modulation strategy can reduce DC-link voltage variation under the line frequency. Therefore, DC-link voltage vd can remain relatively stable with a smaller capacitor. At the same time, the THD of improved output voltage drops significantly and the waveform is better.
Through (17) and (24) in the previous section, Ic = Ic' can be attained when phase angle difference is equal to zero. At this point, it should be observed that the DC-link voltage fluctuations are equal under the line frequency cycle, regardless of which modulation strategy is adopted, as shown in Figure 9. Through (17) and (24) in the previous section, I c = I c can be attained when phase angle difference is equal to zero. At this point, it should be observed that the DC-link voltage fluctuations are equal under the line frequency cycle, regardless of which modulation strategy is adopted, as shown in Figure 9. Seen from the simulation results, the proposed modulation strategy can reduce DC-link voltage variation under the line frequency. Therefore, DC-link voltage vd can remain relatively stable with a smaller capacitor. At the same time, the THD of improved output voltage drops significantly and the waveform is better.
Through (17) and (24) in the previous section, Ic = Ic' can be attained when phase angle difference is equal to zero. At this point, it should be observed that the DC-link voltage fluctuations are equal under the line frequency cycle, regardless of which modulation strategy is adopted, as shown in Figure 9.  When only the DC-link capacitance C d is changed, the DC-link ripple coefficient, amplitude and THD of output voltage under traditional modulation strategy and improved modulation strategy are shown in Tables 3 and 4.  Figure 10 intuitively depicts the trends of ripple coefficient changing with C d under two modulation strategies, respectively. As C d increases, the DC-link voltage fluctuation should reduce. However, when the capacitor is small, DC-link voltage waveform under the line frequency cycle is mainly caused by the oscillation under the switching cycle. In fact, the magnitude of voltage fluctuation under the switching cycle still shows a decline trend. 7 600 0.53 34.09 0.39% Figure 10 intuitively depicts the trends of ripple coefficient changing with Cd under two modulation strategies, respectively. As Cd increases, the DC-link voltage fluctuation should reduce. However, when the capacitor is small, DC-link voltage waveform under the line frequency cycle is mainly caused by the oscillation under the switching cycle. In fact, the magnitude of voltage fluctuation under the switching cycle still shows a decline trend. In order to further prove the theoretical analysis of the proposed converter, an experimental prototype is made in laboratory. Specifications of the prototype are the same as Table 2. The prototype block diagram, experimental setup, and the top-view of the proposed converter are shown in Figure 11a-c, respectively. Notice that the converter is In order to further prove the theoretical analysis of the proposed converter, an experimental prototype is made in laboratory. Specifications of the prototype are the same as Table 2. The prototype block diagram, experimental setup, and the top-view of the proposed converter are shown in Figure 11a-c, respectively. Notice that the converter is operated in an open loop and the driving signal is generated by RCP and FPGA EP4CE10F17C8. The switching control signals of the upper, middle, and lower switches on the same bridge arm generated by RCP and FPGA are shown in Figure 12.  Figure 12 shows that the switch sequence is consistent with the theoretical analysis and simulation results. The final output waveforms are shown in Figure 13. The switching control signals of the upper, middle, and lower switches on the same bridge arm generated by RCP and FPGA are shown in Figure 12.  Figure 12 shows that the switch sequence is consistent with the theoretical analysis and simulation results. The final output waveforms are shown in Figure 13.   Figure 12 shows that the switch sequence is consistent with the theoretical analysis and simulation results. The final output waveforms are shown in Figure 13.  The THD of the output voltage under the traditional modulation strategy is 1.38%, and correspondingly, the value under the improved modulation strategy is 0.92%, which proves that the proposed modulation strategy can reduce the DC side voltage ripple coefficient under the line frequency, thereby improving the output voltage waveform.
When the phase angle φ equals to zero, according to theoretical derivation, the ripple coefficients under the traditional and improved modulation strategies should be equal. The experimental results are shown in Figure 15.
In Figure 15a,b, the position and spacing of cursors A and B are equal. It can be observed that the voltage fluctuations under the traditional and improved modulation strategies are almost the same (the delay of the controller itself cannot be ignored).
The stable operation of the system can be strictly ensured when the ripple coefficient is less than 2%. When the ripple factors under traditional strategy and improved strategy are both about 2%, the required DC-link capacitors are shown in Figure 16.
It can be observed from Figure 16 that when the voltage ripple is about 2%, the six-switch converter adopting the traditional modulation strategy requires capacitance value of 1800 uF, while the improved one requires capacitance value of 500 uF. The capacitance value is greatly reduced, and long-life film capacitors can be used to replace the large electrolytic capacitors to realize the electrolytic capacitor-less solution of the six-switch converter. The THD of the output voltage under the traditional modulation strategy is 1.38%, and correspondingly, the value under the improved modulation strategy is 0.92%, which proves that the proposed modulation strategy can reduce the DC side voltage ripple coefficient under the line frequency, thereby improving the output voltage waveform.
When the phase angle ϕ equals to zero, according to theoretical derivation, the ripple coefficients under the traditional and improved modulation strategies should be equal. The experimental results are shown in Figure 15.
In Figure 15a,b, the position and spacing of cursors A and B are equal. It can be observed that the voltage fluctuations under the traditional and improved modulation strategies are almost the same (the delay of the controller itself cannot be ignored).
The stable operation of the system can be strictly ensured when the ripple coefficient is less than 2%. When the ripple factors under traditional strategy and improved strategy are both about 2%, the required DC-link capacitors are shown in Figure 16.
strategy are both about 2%, the required DC-link capacitors are shown in Figure 16.
It can be observed from Figure 16 that when the voltage ripple is about 2%, the six-switch converter adopting the traditional modulation strategy requires capacitance value of 1800 uF, while the improved one requires capacitance value of 500 uF. The capacitance value is greatly reduced, and long-life film capacitors can be used to replace the large electrolytic capacitors to realize the electrolytic capacitor-less solution of the six-switch converter. Figure 15. Capacitor voltage waveforms in line frequency cycle when φ = 0°: (a) traditional modulation strategy; (b) improved modulation strategy.

Conclusions
This paper presents an improved modulation strategy for a compact single-phase six-switch AC-AC converter which effectively reduces the number of switches. The proposed modulation strategy allows the six-switch converter to achieve 'fast charging and discharging' in a complete switching cycle. The energy on the DC-link capacitor obtained through the rectifier can be quickly output to the load through the inverter to achieve minimum energy storage. Therefore, under the same DC-link voltage ripple constraint, the capacitance value can be reduced and the system power density can be improved. Based on the time average modeling analysis, the effectiveness of the proposed strategy is fully proved in theory. Finally, the experimental results show that ripple factor drops to 2.20% from 3.49% while THD reduces from 1.38% to 0.92% under improved modulation strategy. The improved converter requires only one-third of the capacitance required by the traditional modulation strategy when the ripple factor is limited to 2%.
Under RL load, there may be different factors for inversion and rectification, the input power and output power will become synchronous and bring about twice frequency power in the DC-link capacitor. Next work will be focused on.   It can be observed from Figure 16 that when the voltage ripple is about 2%, the sixswitch converter adopting the traditional modulation strategy requires capacitance value of 1800 µF, while the improved one requires capacitance value of 500 µF. The capacitance value is greatly reduced, and long-life film capacitors can be used to replace the large electrolytic capacitors to realize the electrolytic capacitor-less solution of the six-switch converter.

Conclusions
This paper presents an improved modulation strategy for a compact single-phase six-switch AC-AC converter which effectively reduces the number of switches. The proposed modulation strategy allows the six-switch converter to achieve 'fast charging and discharging' in a complete switching cycle. The energy on the DC-link capacitor obtained through the rectifier can be quickly output to the load through the inverter to achieve minimum energy storage. Therefore, under the same DC-link voltage ripple constraint, the capacitance value can be reduced and the system power density can be improved. Based on the time average modeling analysis, the effectiveness of the proposed strategy is fully proved in theory. Finally, the experimental results show that ripple factor drops to 2.20% from 3.49% while THD reduces from 1.38% to 0.92% under improved modulation strategy. The improved converter requires only one-third of the capacitance required by the traditional modulation strategy when the ripple factor is limited to 2%.
Under RL load, there may be different factors for inversion and rectification, the input power and output power will become synchronous and bring about twice frequency power in the DC-link capacitor. Next work will be focused on.