Elastic metamaterials, which have huge potential in wave guiding and attenuation applications, can be built from structures with periodic piezoelectric patch arrays. Passive shunts offer the benefits of simplicity and low cost. In this paper, the effects of the magnitude and phase angle of the shunt impedance on the attenuation constant of a beam with periodic piezoelectric patch arrays were studied in order to determine the optimal shunt that produces the widest and most effective band gaps. The attenuation constants were found to be large when the phase angle is Formula rad and when the magnitude decreases exponentially with the excitation frequency. This corresponds to a negative capacitance circuit, which is the optimal shunt for such systems. The attenuation constant of the system reduces significantly when the impedance deviates from the optimal value suggesting other circuits are less effective. The impedance and band structure of resistive–inductive (R-L), negative capacitance and resistive shunts were investigated. As expected, the negative capacitance circuit produces a large band gap, while the R-L circuit only produces a band gap around its natural frequency. The transmissibilities of a finite system with these circuits demonstrated that vibration transmissions are low within the band gaps. Furthermore, the stability of the negative capacitance circuit built using a dual-output second-generation current conveyor (DO-CCII) was examined by studying the pole diagrams. The system was found to be stable in ideal conditions but unstable when parasitic effects are considered. This suggests that the stability of the system is an important consideration for the implementation of this strategy and the different negative impedance converter designs available in the literature should be studied to find a suitable circuit configuration.