The stability of systems with hysteresis, driven by developments in smart material applications, has been an important topic of research over the past two decades. Most results provide sufficient conditions for boundedness of the system states, but do not further investigate the steady state solutions. In this paper, we present an example of a system with hysteresis that possesses self-excited limit cycles. In particular, we consider an integral-controlled system with backlash (also known as play operator). A Newton-Raphson algorithm is formulated to calculate the limit cycles in the system. We then prove that the amplitude and period of these limit cycles have linear relationships to parameters within the system. These results are then confirmed in simulation, where we demonstrate our ability to predict and modify the properties of the limit cycles.