In this paper, structural sound and vibration control using passive and semi-active shunt piezoelectric damping circuits is presented. A piezoelectric patch with an electrical shunt circuit is bonded to a base structure. When the structure vibrates, the piezoelectric patch strains and transforms the mechanical energy of the structure into electrical energy, which can be effectively dissipated by the shunt circuit. Hence, the shunt circuit acts as a means of extracting mechanical energy from the base structure. Different types of shunt circuits (such as an RL series circuit, an RL parallel circuit and an RL-C circuit) employed in the passive damping arrangement, are analyzed and compared. Using the impedance method, the general modeling of different shunt piezoelectric damping techniques is presented. The piezoelectric shunt circuit can be seen as an additional frequency-dependence damping of the system. One of the primary concerns in shunt damping is to choose the optimal parameters for shunt circuits. In past efforts most of the proposed tuning methods were based on modal properties of the structure. In this study, a design method based on minimization of the sound power of the structure is proposed. The optimal parameters for shunt circuits are obtained using linear quadratic optimal control theory. Numerical simulations are performed for each of these shunts techniques focusing on minimizing radiated sound power from a clamped plate. It is found that the RL series, RL parallel and pulse-switching circuits have basically the same control performance. The RL-C parallel circuit allows us to reduce the value of the inductance L due to the insertion of an external capacitance C. However, the control performance will be reduced simultaneously. The pulse-switching circuit is more stable than an RL series circuit with regard to structural stiffness variations. Finally, experimental results are presented using an RL series/parallel shunt circuit, RL-C parallel shunt circuit and pulse-switching circuit.