This paper describes a set of piecewise constant waveforms which, when used as the method of dispatch of power to a first-order system, offers distinct advantages over other piecewise constant signal methods. The waveforms are similar to those which are produced by binary rate multipliers and the technique is called ‘Binary Rate Modulation’. The stability of the resulting discrete closed loop system is investigated using a circle criterion adaptation, applied to a multiple nonlinearity, with a multiple pulse output in each nonlinearity and by a proposed ultimate linear technique. The gains for asymptotic stability and limit cycling are found from the techniques. Results in accordance with the predicted values are obtained from observed system responses. The two stability techniques can be easily applied to systems using pulse width or pulse frequency modulation output drives to first-order systems.