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This study investigates the size effects on the dispersion relation of the transverse waves propagating through micro-beam metamaterial covered by piezoelectric patches that are connected with external shunted circuits. Based on the modified couple stress theory and Hamilton's principle, the transfer matrix model is established for determining the complex band structure which incorporates the length scaling effect and the electromechanical resonance. Numerical results show the size-dependent behavior of the complex band structure in terms of the location, the range, and the decaying level. The differences are indicated between the results obtained by the non-classical model and the classical elasticity-based model and the effects of Poisson's ratio. The connection modes of the piezoelectric layers on the characteristics of transverse wave propagation and attenuation are also examined. Further, the tunable nature is demonstrated via investigating the influences of the unit cell length, the circuit parameters, and the distribution of piezoelectric materials on the attenuation diagram. [ABSTRACT FROM AUTHOR]