Recent research reveals the merits of single-stage half-bridge matrix-based (MB) dual-active-bridge (DAB) converter for modular medium-voltage ac (MVAC) to low-voltage dc (LVDC) conversion. The complete set of modulation modes for this ac-dc converter is yet to be analyzed in the literature. Such holistic investigation is necessary before carrying out a design optimization of the ac-dc converter for given voltage and power specifications. The objective of this article is to present a comprehensive analysis of the half-bridge MB-DAB converter's modulation possibilities and optimal design. The exhaustive sets of zero-current-switching (ZCS) and zero-voltage-switching (ZVS) constrained modulation modes for the half-bridge MB-DAB converter are systematically investigated and the analysis reveals the boundaries of operation and feasible modulation schemes. Suitable simulation results illustrating modulation limits of the ac-dc converter reinforce the theoretical analysis. Additionally, an efficiency (η)-power density (ρ) optimal converter design algorithm is presented, which reveals that an all-SiC half-bridge MB-DAB design with ZVS constrained modulation offers the best η - ρ design. The hardware prototype's experimental results validate the converter's operation. © 2021 IEEE.