Cross-coupled LC oscillator

A parallel LC circuit can naturally generate a sinusoidal signal at its resonance frequency: : \omega_0 = \frac{1}{\sqrt{LC}} In an ideal lossless case, the oscillation would persist indefinitely. In practice, however, parasitic resistances in the reactive elements dissipate energy cycle after cycle, causing the amplitude to decay and the oscillation to eventually cease. Such losses can be represented by an equivalent resistance R_0 placed in parallel with the LC network.For sustained oscillation to occur, two fundamental conditions must be satisfied—commonly referred to as the Barkhausen criteria, leading to the conditions: : \mathcal{L}(\omega_\text{m}) =\frac{1}{2} \cdot \frac{kT}{R_0} \cdot \frac{(1+F)}{C^2 \omega_\text{m}^2 A_0^2 / 2} where k is the Boltzmann constant, T is the absolute temperature in Kelvin, A0 is the differential oscillation amplitude across the tank, R0 models the tank losses, C is the tank capacitance, and ωm is the frequency offset. The factor F, called excess noise factor, accounts for the phase noise introduced by the transconductance stage and depends on the specific topology and technology used. Leeson's model provides useful results but does not capture the mechanisms by which transistor electronic noise is converted into phase noise. To account for this, it is necessary to adopt approaches that consider the time-varying nature of the oscillator, such as the Impulse Sensitivity Function (ISF) method. Figure of merit and efficiency Phase noise is not the only figure of interest in an LC oscillator. The power consumption required to achieve a given phase noise level is also a critical parameter. A commonly used figure of merit (FoM) captures this trade-off and is expressed as: : \mathrm{FoM} = -10 \log_{10} \left[ \mathcal{L}(\omega_\text{m})P_{\mathrm{DC}} \left( \frac{\omega_\text{m}}{\omega_\text{osc}} \right)^2 \right] = -10 \log_{10} \left( \frac{kT}{2} \right) + 10 \log_{10} Q^2 - 10 \log_{10} \left( \frac{1 + F}{\eta} \right) where \mathcal{L}(\omega_\text{m})denotes the phase noise at an offset frequency ωm, PDC is the power consumption, and ω0 is the oscillation frequency. By rearranging the expression, the figure of merit can also be written in terms of key oscillator design parameters, such as the tank quality factor Q, the excess noise factor F, and the efficiency η. == CMOS cross-coupled oscillators ==

In CMOS technology, the most common cross-coupled LC oscillator topologies use either a single or a dual transconductor to provide the negative resistance required to sustain oscillation. In the single-transconductor configuration, one pair of MOSFETs injects current into the tank during only half of the oscillation cycle. This kind of operation is usually referred as class B. In the dual-transconductor configuration—often referred to as complementary class B—nMOS and pMOS transistors are arranged symmetrically to alternately source and sink current, generating a differential square-wave current. In the single-transconductor configuration, the current efficiency is ηI = 2/π ≈ 0.64. Thanks to the complementary transconductor it increases to in the complementary configuration. As for the voltage efficiency ηV, it is limited by the voltage headroom required by the bias transistor. In a single nMOS configuration ηV typically reaches up to 0.66, while in the complementary case it is reduced to around 0.4 due to the voltage headroom required by the complementary structure. The overall efficiency η is approximately 0.5 in the complementary configuration, while it is slightly lower in the single-transconductor case, where it reaches about 0.42. Advanced topologies Various techniques can be employed to reduce the excess noise factor and/or increase the efficiency of the oscillator, with the goal of improving its phase noise performance and overall figure of merit. In complementary topologies, this approach can increase the overall efficiency up to approximately 0.8, compared to about 0.5 in configurations without tail filtering. Class C CMOS oscillator In a class C oscillator, the transistors operate in saturation (e.g. drain-to-source voltage is higher than gate-to-source voltage minus threshold voltage for nMOS), and a capacitor is placed in parallel with the current source to enable more effective current injection into the tank. It allows the current efficiency ηI to approach 1 in a single-transconductor topology. However, operation in saturation limits the maximum achievable oscillation amplitude, i.e. the voltage efficiency. Despite this limitation, the overall efficiency can reach approximately 0.77. Class D CMOS oscillator In a class D oscillator, the transistors operate as switches, leveraging the advantages of CMOS technology scaling. This mode of operation enables a single-ended oscillation amplitude that can exceed the supply voltage by a factor of approximately three, resulting in a voltage efficiency ηV of about 3. Consequently, the overall efficiency can reach approximately 0.82. == See also ==