Stability of Frequency Locked Loops

Abstract

Passive frequency standards are characterized by the use of a reference resonance to stabilize the frequency of an external probe oscillator. A common configuration is shown in Fig. 1 [1–6]. The probe oscillator generally has phase modulation imposed on the carrier in order to interrogate the resonance with a minimum of offset. The resulting amplitude modulation is demodulated to yield an error curve that is essentially the derivative of the resonance. Although Fig. 1 shows a transmission system, similar schemes are sometimes possible in reflection [5]. The error signal from the demodulator is used to steer the probe signal toward the center of the resonance line [1–9]. For analysis times longer than one period of the modulation cycle, and under the condition that the probe oscillator wanders less than the half width of the error curve in the loop attack time, we can treat this curve as approximately static[1–4]. Near line center the loop error voltage V_D at the synchronous detector, is approximately V_D = k (v ^o_o - v_R) + V_n where k is the slope of the error curve, v_R is the resonance frequency of the reference, and v ^o_o is the open loop frequency of the probe oscillator and V_n is the detector noise. If we now close the loop with gain G(f), it can be shown that the spectral density of fractional frequency fluctuations S_y(f) for the probe source becomes where S_y ^o(f) is the open-loop spectral density of fractional frequency fluctuations of the probe source, Sy^R(f) is that of the reference, and Sy^N(f) is that of the detector and interrogation noise referred to the demodulator output.