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The paper presents the systematic analysis and synthesis of ideal static convertors with any number of input and output phases. Methods of control of dependent quantities are presented and analysed for DC to DC, DC to AC, AC to DC and AC to AC convertors. The convertor is regarded as a switching matrix which consists of fully controllable ideal switches with no losses that can allow current flow in both directions and they can withstand voltage stresses in both directions such as gate turnoff thyristors or forced commutated thyristors. The analysis is valid for power flow from input to output and vice versa and derived for a complete phasor set for both input and output as well as for noncomplete phasor sets. The analysis of the uncontrolled convertors will show that the phase angle of the switching frequency relative to any fixed reference will dictate the magnitude and direction of power flow in the convertor. Detailed analysis of pulse-width modulation convertors and the nonlinear modulating function convertor are covered in the Appendixes. Itis believed that the analysis presented will be helpful in the design, development and synthesis of any solid-state switching convertor.