This paper develops a design method and theoretical analysis for piecewise nonlinear oscillators that have desired circular limit cycles. Especially, the mathematical proof on existence, uniqueness, and stability of the limit cycle is shown for the piecewise nonlinear oscillator. In addition, the relationship between parameters in the oscillator and rotational directions and periods of the limit cycle trajectories is investigated. Then, some numerical simulations show that the piecewise nonlinear oscillator has a unique and stable limit cycle and the properties on rotational directions and periods hold.