Active Coupling RC-Oscillator

Abstract

In this chapter, we study the actively cross-coupled RC-oscillator, which consists of two RC-oscillators coupled by transconductance amplifiers. The single and cross-coupled relaxation oscillators were extensively studied and a comprehensive analysis of the cross-coupled RC-oscillator in relaxation regime can be found in the literature. Here, we study the cross-coupled RC-oscillator in nearly sinusoidal regime with low coupling factor. For a strong coupling, amplifiers are approximated by hard limiters, injecting a square wave current into the other oscillator. In oscillators with high-quality factor-resonant tanks (such as the LC-oscillators), the high-order harmonics are filtered out and the injected signal is reduced to the first harmonic of the Fourier series. However, to ensure that low-quality factor oscillators (as the RC-oscillator) are in the sinusoidal regime, a low coupling factor is necessary; hence, the high-order harmonics are not sufficiently attenuated. The assumption of low coupling strength makes the analysis more cumbersome.

We found that for the sinusoidal regime, this quadrature oscillator can be modeled as two van der Pol oscillators (VDPOs) coupled by two transconductances and that the oscillation frequency is insensitive to the mismatches and is given by the quadratic mean between the free-running frequencies of the coupled oscillators. However, contrary to what the theory predicted, simulations reveal that the oscillation frequency depends on the coupling factor. The increase of the coupling factor decreases the oscillation frequency. This is explained by the fact that the input capacitances of the transconductance amplifiers depend on the coupling factor. Due to the Miller effect, the input capacitances of the transconductance amplifiers depend on the bias current. Since the input capacitances are in parallel with the oscillator capacitance, they influence the oscillation frequency.