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Injection-Locking of Nonharmonic Oscillators

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Injection-Locking in Mixed-Mode Signal Processing

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Abstract

This chapter investigates how Barkhausen criteria can be used to analyze ring oscillators. The modeling of harmonic oscillators and relaxation oscillators is explored with a special attention to the distinct nonlinear characteristics of relaxation oscillators. The representation of a nonharmonic oscillator with a set of harmonic oscillators is presented. The fundamentals of Volterra series are reviewed. The concept of the Volterra elements of a nonlinear element and the Volterra circuits of a nonlinear circuit are introduced and the process of how to obtain them is exemplified. The modeling of voltage comparators is studied. The Volterra circuits of an injection-locked nonharmonic oscillator are derived and the characteristics of the Volterra circuits are investigated. The chapter explores how the Volterra circuit approach can be used to analyze the dual-comparator relaxation oscillator under the injection of a pair of differential currents and how the high-order Volterra circuits of the oscillator contribute to the effective injection signals of the first-order Volterra circuit of the oscillator. Finally, the lock range of the dual-comparator relaxation oscillator is investigated.

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Notes

  1. 1.

    Type-2 PLLs with a PFD are generally preferred over their type-1 counterparts, i.e., charge-pump-less PLLs as they allow the independent tuning of the loop bandwidth and damping factor of the PLLs [65, 103].

  2. 2.

    Practical relaxation oscillators typically employs four transistors operated in an ON/OFF mode, as shown in Fig. 4.7, rather than current sources to charge and discharge the capacitors so as to greatly simplify design as the voltage of the integrating capacitors does not need to vary with time in a linear fashion.

  3. 3.

    More generally, the waveform of the output of a nonharmonic oscillator takes the shape of a trapezoid. The output can also be represented its Fourier series and contains an infinite number of frequency components.

  4. 4.

    The frequency components of \(\hat {V}_o(\omega )\) at negative frequencies are not shown in Fig. 4.10.

  5. 5.

    In general, modified nodal analysis that permits branch currents, along with nodal voltages, is used to formulate the governing equations of an arbitrary circuit [85].

  6. 6.

    The Volterra series expansion of a nonlinear circuit accounts for the impact of the elements with memory such as capacitors and inductors by representing the nodal voltages and branch currents in the form of Volterra series expansion of the nodal voltages and branch currents of the circuit with frequency-dependent coefficients. Nonlinear elements are depicted using their Taylor series expansion.

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Authors and Affiliations

  1. Electrical and Computer Engineering, Ryerson University, Toronto, ON, Canada

    Fei Yuan

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  1. Fei Yuan
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Yuan, F. (2020). Injection-Locking of Nonharmonic Oscillators. In: Injection-Locking in Mixed-Mode Signal Processing. Springer, Cham. https://doi.org/10.1007/978-3-030-17364-7_4

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  • DOI: https://doi.org/10.1007/978-3-030-17364-7_4

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  • Online ISBN: 978-3-030-17364-7

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