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Analysis and design of flux cancellation power-decoupling method for electrolytic-capacitorless three-phase cascaded multilevel inverters

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Abstract

Cascaded multilevel inverter cells produce double-grid frequency ripples that require a large electrolytic capacitor bank for every cell, resulting in reduced system reliability and lifespan. This paper proposes a new generalized power-decoupling methodology that is applicable to any three-phase cascaded multilevel inverter topology. The proposed flux cancellation method is based on forcing a three-phase double-frequency ripple into the core of a three-phase transformer. The flux components from each phase, which are phase-shifted by 120°, cancel each other inside the core. Therefore, no power-decoupling capacitor is required in this method. A three-port bidirectional isolated converter is proposed to cancel the three-phase 120 Hz pulsating power in a single high-frequency (HF) core. High-leakage inductances and imbalances among the ports of a HF transformer are a topographical challenge. The imbalance in leakage inductances can be reduced by improving the winding schemes. However, increased leakage and imbalance among the three ports are unavoidable under high voltages because of the need for higher isolation. A universal solution involves the application of a phase shift-based controller to obtain balanced and reduced voltage ripples among the three DC links. This paper presents the dynamic analysis and controller design procedure. Results of the prototype hardware confirm the suitability of the proposed power-decoupling methods.

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Acknowledgements

This research is financially supported by the National Research Foundation of Korea (NRF) funded through the Korea Government (MEST) under Grant NRF2019R1A2C108460511

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Soongsil University, Seoul, South Korea

    Mohammad Sameer Irfan, Mohamed Atef Tawfik, Ashraf Ahmed & Joung-Hu Park

Authors
  1. Mohammad Sameer Irfan
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  2. Mohamed Atef Tawfik
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  3. Ashraf Ahmed
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  4. Joung-Hu Park
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Corresponding author

Correspondence to Joung-Hu Park.

Appendix

Appendix

$$H_{{\rm RR1}} = \left( {\omega_{s} \frac{{L_{{\rm YB}1} L_{YR2} }}{{\left( {L_{{\rm YB}1} L_{{\rm YR}1} - L_{{\rm YY}}^{2} } \right)}} + \omega_{s} \frac{{\left( {L_{{\rm YB}1} L_{{\rm YY}} - L_{{\rm YY}}^{2} } \right)}}{{\left( {L_{{\rm YB}1} L_{{\rm YR}1} - L_{{\rm YY}}^{2} } \right)}}} \right),$$
$$H_{{\rm RR2}} = \frac{1}{{\pi L_{R} }}\frac{{L_{{\rm YB}1} L_{YR2} }}{{\left( {L_{{\rm YB}1} L_{{\rm YR}1} - L_{{\rm YY}}^{2} } \right)}}\sin \left( {\emptyset_{R} } \right),$$
$$H_{{\rm RR3}} = \frac{1}{{\pi L_{R} }}\frac{{L_{{\rm YB}1} L_{YR2} }}{{\left( {L_{{\rm YB}1} L_{{\rm YR}1} - L_{{\rm YY}}^{2} } \right)}}V_{dca} \cos \left( {\emptyset_{R} } \right),$$
$$H_{{\rm RR4}} = \left( {\omega_{s} \frac{{\left( { - L_{{\rm YB}1} L_{{\rm eq2}_{\rm Y}} + L_{{\rm YY}} L_{{\rm eq2}_{\rm Y}} } \right)}}{{\left( {L_{{\rm YB}1} L_{{\rm YR}1} - L_{RR}^{2} } \right)}} + \omega_{s} \frac{{\left( {L_{{\rm YB}1} L_{{\rm YY}} - L_{{\rm YY}}^{2} } \right)}}{{\left( {L_{{\rm YB}1} L_{{\rm YR}1} - L_{{\rm YY}}^{2} } \right)}}} \right),$$
$$H_{{\rm RR5}} = \frac{1}{{\pi L_{Y} }}\frac{{\left( { - L_{{\rm YB}1} L_{{\rm eq2}_{\rm Y}} + L_{{\rm YY}} L_{{\rm eq2}_{\rm Y}} } \right)}}{{\left( {L_{{\rm YB}1} L_{{\rm YR}1} - L_{{\rm YY}}^{2} } \right)}}\sin \left( {\emptyset_{Y} } \right),$$
$$H_{{\rm RR6}} = \frac{1}{{\pi L_{b} }}\frac{{\left( { - L_{{\rm YB}1} L_{{\rm eq2}_{\rm Y}} + L_{{\rm YY}} L_{{\rm eq2}_{\rm Y}} } \right)}}{{\left( {L_{{\rm YB}1} L_{{\rm YR}1} - L_{RR}^{2} } \right)}}V_{dcY} \cos \left( {\emptyset_{Y} } \right),$$
$$H_{{\rm RR7}} = \left( {\omega_{s} \frac{{ - L_{{\rm YY}} L_{{\rm YB}2} }}{{\left( {L_{{\rm YB}1} L_{{\rm YR}1} - L_{{\rm YY}}^{2} } \right)}} + \omega_{s} \frac{{\left( {L_{{\rm YB}1} L_{{\rm YY}} - L_{{\rm YY}}^{2} } \right)}}{{\left( {L_{{\rm YB}1} L_{{\rm YR}1} - L_{{\rm YY}}^{2} } \right)}}} \right),$$
$$H_{{\rm RR8}} = \frac{1}{{\pi L_{B} }}\frac{{ - L_{{\rm YY}} L_{{\rm YB}2} }}{{\left( {L_{{\rm YB}1} L_{{\rm YR}1} - L_{{\rm YY}}^{2} } \right)}}\sin \left( {\emptyset_{B} } \right),$$
$$H_{{\rm RR9}} = \frac{1}{{\pi L_{cB} }}\frac{{ - L_{{\rm YY}} L_{{\rm YB}2} }}{{\left( {L_{{\rm YB}1} L_{{\rm YR}1} - L_{{\rm YY}}^{2} } \right)}}V_{dcB} \cos \left( {\emptyset_{B} } \right),$$
$$H_{{\rm RI1}} = \left( { - \omega_{s} \frac{{L_{{\rm YB}1} L_{YR2} }}{{\left( {L_{{\rm YB}1} L_{{\rm YR}1} - L_{{\rm YY}}^{2} } \right)}} - \omega_{s} \frac{{\left( {L_{{\rm YB}1} L_{{\rm YY}} - L_{{\rm YY}}^{2} } \right)}}{{\left( {L_{{\rm YB}1} L_{{\rm YR}1} - L_{{\rm YY}}^{2} } \right)}}} \right),$$
$$H_{{\rm RI2}} = - \frac{1}{{\pi L_{R} }}\frac{{L_{{\rm YB}1} L_{YR2} }}{{\left( {L_{{\rm YB}1} L_{{\rm YR}1} - L_{{\rm YY}}^{2} } \right)}}\cos \left( {\emptyset_{R} } \right),$$
$$H_{{\rm RI3}} = \frac{1}{{\pi L_{a} }}V_{dcR} \frac{{L_{{\rm YB}1} L_{YR2} }}{{\left( {L_{{\rm YB}1} L_{{\rm YR}1} - L_{{\rm YY}}^{2} } \right)}}\sin \left( {\emptyset_{R} } \right) ,$$
$$H_{{\rm RI4}} = \left( { - \omega_{s} \frac{{\left( { - L_{{\rm YB}1} L_{{\rm eq2}_{\rm Y}} + L_{{\rm YY}} L_{{\rm eq2}_{\rm Y}} } \right)}}{{\left( {L_{{\rm YB}1} L_{{\rm YR}1} - L_{RR}^{2} } \right)}} - \omega_{s} \frac{{\left( {L_{{\rm YB}1} L_{{\rm YY}} - L_{{\rm YY}}^{2} } \right)}}{{\left( {L_{{\rm YB}1} L_{{\rm YR}1} - L_{{\rm YY}}^{2} } \right)}}} \right),$$
$$H_{{\rm RI5}} = - \frac{1}{{\pi L_{R} }}\frac{{\left( { - L_{{\rm YB}1} L_{{\rm eq2}_{\rm Y}} + L_{{\rm YY}} L_{{\rm eq2}_{\rm Y}} } \right)}}{{\left( {L_{{\rm YB}1} L_{{\rm YR}1} - L_{RR}^{2} } \right)}}\cos \left( {\emptyset_{Y} } \right),$$
$$H_{{\rm RI6}} = \frac{1}{{\pi L_{B} }}\frac{{\left( { - L_{{\rm YB}1} L_{{\rm eq2}_{\rm Y}} + L_{{\rm YY}} L_{{\rm eq2}_{\rm Y}} } \right)}}{{\left( {L_{{\rm YB}1} L_{{\rm YR}1} - L_{{\rm YY}}^{2} } \right)}}V_{dcY} \sin \left( {\emptyset_{Y} } \right),$$
$$H_{{\rm RI7}} = \left( {\omega_{s} \frac{{L_{{\rm YY}} L_{{\rm YB}2} }}{{\left( {L_{{\rm YB}1} L_{{\rm YR}1} - L_{{\rm YY}}^{2} } \right)}} - \omega_{s} \frac{{\left( {L_{{\rm YB}1} L_{{\rm YY}} - L_{{\rm YY}}^{2} } \right)}}{{\left( {L_{{\rm YB}1} L_{{\rm YR}1} - L_{{\rm YY}}^{2} } \right)}}} \right),$$
$$H_{{\rm RI8}} = \frac{1}{{\pi L_{B} }}\frac{{L_{{\rm YY}} L_{{\rm YB}2} }}{{\left( {L_{{\rm YB}1} L_{{\rm YR}1} - L_{{\rm YY}}^{2} } \right)}}\cos \left( {\emptyset_{B} } \right) ,$$
$$H_{{\rm RI9}} = - \frac{1}{{\pi L_{B} }}\frac{{L_{{\rm YY}} L_{{\rm YB}2} }}{{\left( {L_{{\rm YB}1} L_{{\rm YR}1} - L_{{\rm YY}}^{2} } \right)}}V_{dcB} \sin \left( {\emptyset_{B} } \right) ,$$
$$H_{{\rm aRv1}} = - \frac{4}{\pi c}\sin \left( {\emptyset_{R} } \right),$$
$$H_{{\rm Rv2}} = \frac{4}{\pi c}\cos \left( {\emptyset_{R} } \right),$$
$$H_{{\rm Rv3}} = - \frac{2}{{R_{R} c}},$$
$$H_{{\rm Rv4}} = - \left( {\frac{4}{\pi c}I_{{pR - {\text{Re}} }} \cos \left( {\emptyset_{R} } \right) + \frac{4}{\pi c}I_{{pR - {\text{Im}} }} \sin \left( {\emptyset_{R} } \right)} \right),$$
$$H_{{\rm Rv5}} = \frac{2}{c},$$
$$H_{{\rm YR}1} = - \left( {z_{Y1} \omega_{s} + z_{Y4} \omega_{s} } \right),$$
$$H_{{\rm YR2}} = - z_{{\rm Yr}1} \frac{1}{{\pi L_{R} }}\sin \left( {\emptyset_{R} } \right),$$
$$H_{{\rm YR3}} = - z_{{\rm Yr}1} \frac{1}{{\pi L_{R} }}V_{{{\text{dcR}}}} \cos \left( {\emptyset_{R} } \right),$$
$$H_{{\rm YR4}} = - \left( {z_{{\rm Yr}2} \omega_{s} + z_{{\rm Yr}4} \omega_{s} } \right),$$
$$H_{{\rm YR5}} = - z_{{\rm Yr}2} \frac{1}{{\pi L_{Y} }}\sin \left( {\emptyset_{Y} } \right),$$
$$H_{{\rm YR6}} = - z_{{\rm Yr}2} \frac{1}{{\pi L_{Y} }}V_{{{\text{dcY}}}} \cos \left( {\emptyset_{Y} } \right),$$
$$H_{{\rm YR7}} = - \left( {z_{{\rm Yr}3} \omega_{s} + z_{{\rm Yr}4} \omega_{s} } \right),$$
$$H_{{\rm YR8}} = - z_{{\rm Yr}3} \frac{1}{{\pi L_{B} }}\sin \left( {\emptyset_{B} } \right),$$
$$H_{{\rm YR9}} = - z_{{\rm Yr}3} \frac{1}{{\pi L_{B} }}V_{dcc} \cos \left( {\emptyset_{B} } \right),$$
$$H_{{\rm YI1}} = \left( {z_{{\rm Yr}1} \omega_{s} + z_{{\rm Yr}4} \omega_{s} } \right),$$
$$H_{{\rm YI2}} = z_{{\rm Yr}1} \frac{1}{{\pi L_{R} }}\cos \left( {\emptyset_{R} } \right),$$
$$H_{{\rm YI3}} = - z_{{\rm Yr}1} \frac{1}{{\pi L_{R} }}V_{{{\text{dca}}}} \sin \left( {\emptyset_{R} } \right),$$
$$H_{{\rm YI4}} = \left( {z_{{\rm Yr}2} \omega_{s} + z_{{\rm Yr}4} \omega_{s} } \right),$$
$$H_{{\rm YI5}} = z_{{\rm Yr}2} \frac{1}{{\pi L_{Y} }}\cos \left( {\emptyset_{Y} } \right),$$
$$H_{{\rm YI6}} = - z_{{\rm Yr}2} \frac{1}{{\pi L_{b} }}V_{{{\text{dcb}}}} \sin \left( {\emptyset_{Y} } \right),$$
$$H_{{\rm YI7}} = \left( {z_{{\rm Yr}3} \omega_{s} + z_{{\rm Yr}4} \omega_{s} } \right),$$
$$H_{{\rm YI8}} = z_{{\rm Yr}3} \frac{1}{{\pi L_{B} }}\cos \left( {\emptyset_{B} } \right),$$
$$H_{{\rm YI9}} = - z_{{\rm Yr}3} \frac{1}{{\pi L_{B} }}V_{{{\text{dcB}}}} \sin \left( {\emptyset_{B} } \right),$$
$$H_{{\rm Yv1}} = - \frac{4}{\pi c}\sin \left( {\emptyset_{Y} } \right),$$
$$H_{{\rm Yv2}} = \frac{4}{\pi c}\cos \left( {\emptyset_{Y} } \right),$$
$$H_{{\rm Yv3}} = - \frac{2}{{R_{Y} c}},$$
$$H_{{\rm Yv4}} = - \left( {\frac{4}{\pi c}I_{{pY - {\text{Re}} }} \cos \left( {\emptyset_{Y} } \right) + \frac{4}{\pi c}I_{{pY - {\text{Im}} }} \sin \left( {\emptyset_{Y} } \right)} \right),$$
$$H_{{\rm Yv5}} = \frac{2}{c},$$
$$H_{{\rm BR1}} = \left( {\omega_{s} \frac{{L_{{\rm YY}} L_{YR2} }}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}} + \omega_{s} \frac{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YY}} } \right)}}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}}} \right),$$
$$H_{{\rm BR2}} = \frac{1}{{\pi L_{R} }}\frac{{L_{{\rm YY}} L_{YR2} }}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}}\sin \left( {\emptyset_{R} } \right),$$
$$H_{{\rm BR3}} = \frac{1}{{\pi L_{a} }}\frac{{L_{{\rm YY}} L_{YR2} }}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}}V_{dcR} \cos \left( {\emptyset_{R} } \right),$$
$$H_{{\rm BR4}} = \left( {\omega_{s} \frac{{\left( {L_{{\rm YR}1} L_{{\rm eq2}_{\rm b}} - L_{{\rm YY}} L_{{\rm eq2}_{\rm b}} } \right)}}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}} + \omega_{s} \frac{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YY}} } \right)}}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}}} \right),$$
$$H_{{\rm BR5}} = \frac{1}{{\pi L_{Y} }}\frac{{\left( {L_{{\rm YR}1} L_{{\rm eq2}_{\rm Y}} - L_{{\rm YY}} L_{{\rm eq2}_{\rm Y}} } \right)}}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}}\sin \left( {\emptyset_{Y} } \right),$$
$$H_{{\rm BR6}} = \frac{1}{{\pi L_{Y} }}\frac{{\left( {L_{{\rm YR}1} L_{{\rm eq2}_{\rm Y}} - L_{{\rm YY}} L_{{\rm eq2}_{\rm Y}} } \right)}}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}}V_{{{\text{dcY}}}} \cos \left( {\emptyset_{Y} } \right),$$
$$H_{{\rm BR7}} = \left( {\omega_{s} \frac{{ - L_{{\rm YR}1} L_{{\rm YB}2} }}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}} + \omega_{s} \frac{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YY}} } \right)}}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}}} \right),$$
$$H_{{\rm BR8}} = \frac{1}{{\pi L_{B} }}\frac{{ - L_{{\rm YR}1} L_{{\rm YB}2} }}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}}\sin \left( {\emptyset_{B} } \right),$$
$$H_{{\rm BR9}} = \frac{1}{{\pi L_{B} }}\frac{{ - L_{{\rm YR}1} L_{{\rm YB}2} }}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}}V_{{{\text{dcB}}}} \cos \left( {\emptyset_{B} } \right)$$
$$H_{{\rm BI1}} = \left( { - \omega_{s} \frac{{L_{{\rm YY}} L_{YR2} }}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}} - \frac{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YY}} } \right)}}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}}\omega_{s} } \right)$$
$$H_{{\rm cBI2}} = - \frac{1}{{\pi L_{R} }}\frac{{L_{{\rm YY}} L_{YR2} }}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}}\cos \left( {\emptyset_{R} } \right)$$
$$H_{{\rm BI3}} = \frac{1}{{\pi L_{R} }}\frac{{L_{{\rm YY}} L_{YR2} }}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}}V_{{{\text{dcR}}}} \sin \left( {\emptyset_{R} } \right)$$
$$H_{{\rm BI4}} = \left( { - \omega_{s} \frac{{\left( {L_{{\rm YR}1} L_{{\rm eq2}_{\rm Y}} - L_{{\rm YY}} L_{{\rm eq2}_{\rm Y}} } \right)}}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}} - \frac{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YY}} } \right)}}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}}\omega_{s} } \right)$$
$$H_{{\rm YI5}} = - \frac{1}{{\pi L_{Y} }}\frac{{\left( {L_{{\rm YR}1} L_{{\rm eq2}_{\rm Y}} - L_{{\rm YY}} L_{{\rm eq2}_{\rm Y}} } \right)}}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}}\cos \left( {\emptyset_{Y} } \right)$$
$$H_{{\rm BI6}} = \frac{1}{{\pi L_{Y} }}\frac{{\left( {L_{{\rm YR}1} L_{{\rm eq2}_{\rm Y}} - L_{{\rm YY}} L_{{\rm eq2}_{\rm Y}} } \right)}}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}}V_{{{\text{dcY}}}} \sin \left( {\emptyset_{Y} } \right)$$
$$H_{{\rm BI7}} = \left( {\omega_{s} \frac{{L_{{\rm YR}1} L_{{\rm YB}2} }}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}} - \frac{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YY}} } \right)}}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}}\omega_{s} } \right)$$
$$H_{{\rm BI8}} = \frac{1}{{\pi L_{B} }}\frac{{L_{{\rm YR}1} L_{{\rm YB}2} }}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}}\cos \left( {\emptyset_{B} } \right)$$
$$H_{{\rm BI9}} = - \frac{1}{{\pi L_{c} }}\frac{{L_{{\rm YR}1} L_{{\rm YB}2} }}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}}V_{{{\text{dcB}}}} \sin \left( {\emptyset_{B} } \right)$$
$$H_{{\rm Bv1}} = - \frac{4}{\pi c}\sin \left( {\emptyset_{B} } \right)$$
$$H_{{\rm Bv2}} = \frac{4}{\pi c}\cos \left( {\emptyset_{B} } \right)$$
$$H_{{\rm Bv3}} = - \frac{2}{{R_{B} c}}$$
$$H_{{\rm Bv4}} = - \left( {\frac{4}{\pi c}I_{{pB - {\text{Re}} }} \cos \left( {\emptyset_{B} } \right) + \frac{4}{\pi c}I_{{pB - {\text{Im}} }} \sin \left( {\emptyset_{B} } \right)} \right)$$
$$H_{{\rm Bv5}} = \frac{2}{c}$$
$$L_{{\rm eq1}_{\rm R}} = L_{R} \left( {\frac{1}{{L_{m} }} + \frac{1}{{L_{R} }}} \right)$$
$$L_{{\rm eq1}_{\rm Y}} = L_{Y} \left( {\frac{1}{{L_{m} }} + \frac{1}{{L_{Y} }}} \right)$$
$$L_{{\rm eq1}_{\rm B}} = L_{B} \left( {\frac{1}{{L_{m} }} + \frac{1}{{L_{B} }}} \right)$$
$$L_{{\rm eq2}_{\rm R}} = \frac{{L_{R} }}{{L_{m} }}$$
$$L_{{\rm eq2}_{\rm Y}} = \frac{{L_{Y} }}{{L_{m} }}$$
$$L_{{\rm eq2}_{\rm B}} = \frac{{L_{B} }}{{L_{m} }}$$
$$L_{{\rm YR}1} = \left( {L_{{\rm eq1}_{\rm Y}} L_{{\rm eq1}_{\rm R}} - 1} \right)$$
$$L_{{\rm YB}1} = \left( {L_{{\rm eq1}_{\rm Y}} L_{{\rm eq1_{\rm B}}} - 1} \right)$$
$$L_{{\rm RY2}} = L_{{\rm eq1}_{\rm Y}} L_{{\rm eq2_{\rm R}}}$$
$$L_{{\rm YB}2} = L_{{\rm eq1}_{\rm Y}} L_{{\rm eq2}_{\rm B}}$$
$$L_{{\rm YY}} = \left( {L_{{\rm eq1}_{\rm Y}} - 1} \right)$$
$$z_{{\rm Yr}1} = \frac{1}{{L_{{\rm eq1}_{\rm Y}} }}\left( {\frac{{L_{{\rm YB}1} L_{YR2} }}{{\left( {L_{{\rm YB}1} L_{{\rm YR}1} - L_{{\rm YY}}^{2} } \right)}} + \frac{{L_{{\rm YY}} L_{YR2} }}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}}} \right)$$
$$z_{{\rm Yr}2} = \frac{1}{{L_{{\rm eq1}_{\rm Y}} }}\left( {\frac{{\left( { - L_{{\rm YB}1} L_{{\rm eq2}_{\rm Y}} + L_{{\rm YY}} L_{{\rm eq2}_{\rm Y}} } \right)}}{{\left( {L_{{\rm YB}1} L_{{\rm YR}1} - L_{{\rm YY}}^{2} } \right)}} + \frac{{\left( {L_{{\rm YR}1} L_{{\rm eq2}_{\rm Y}} - L_{{\rm YY}} L_{{\rm eq2}_{\rm Y}} } \right)}}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}} - L_{{\rm eq2}_{\rm Y}} } \right)$$
$$z_{{\rm Yr}3} = \frac{1}{{L_{{\rm eq1}_{\rm Y}} }}\left( {\frac{{ - L_{{\rm YY}} L_{{\rm YB}2} }}{{\left( {L_{{\rm YB}1} L_{{\rm YR}1} - L_{{\rm YY}}^{2} } \right)}} + \frac{{ - L_{{\rm YR}1} L_{{\rm YB}2} }}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}}} \right)$$
$$z_{{\rm Yr}4} = \frac{1}{{L_{{\rm eq1}_{\rm Y}} }}\left( {\frac{{\left( {L_{{\rm bcYB1}} L_{{\rm bbYY}} - L_{{\rm bbYY}}^{2} } \right)}}{{\left( {L_{{\rm bcYB1}} L_{{\rm baYR1}} - L_{{\rm bbYY}}^{2} } \right)}} + \frac{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YY}} } \right)}}{{\left( {L_{{\rm YY}}^{2} - L_{{\rm YR}1} L_{{\rm YB}1} } \right)}} - 1} \right).$$

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Irfan, M.S., Tawfik, M.A., Ahmed, A. et al. Analysis and design of flux cancellation power-decoupling method for electrolytic-capacitorless three-phase cascaded multilevel inverters. J. Power Electron. 21, 321–341 (2021). https://doi.org/10.1007/s43236-020-00196-3

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  • DOI: https://doi.org/10.1007/s43236-020-00196-3

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