$R_\text{prec}$ is the amplitude of the resistance oscillation, whose value depends on the biasing condition. The $\textdollar abstime$ function returns the absolute simulation time. Since MTJ STOs have considerable phase noise amplitude, $\varphi(t)$ is used to represent the phase fluctuation (phase noise), reflecting the linewidth of the proposed model. For the timescales of circuit-level simulations (generally on the order of $\mu$s), white frequency noise extending over fluctuation frequencies wider than 1 MHz – 100 MHz [@quinsat2010] will be the noise type dominating the linewidth. Hence, for this application, it is natural and adequate to approximate the frequency noise by white frequency noise alone. The amplitude noise of MTJ STOs is coupled to the phase noise due to the nonlinearity of the governing magneto-dynamic equation (Eq.(1) in [@Tingsu_2014TEDI]). Such a coupling greatly enhances the phase noise, making the impact of the amplitude noise less significant for applications [@quinsat2010]. For this reason, the amplitude noise is not explicitly addressed in this work, but its effect is manifested in the linewidth of the proposed MTJ STO model. Nevertheless, Gaussian amplitude noise could be added to the proposed model by further specifying its standard deviation.

An accurate Verilog-A model of oscillators should memorize its 