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Fully Integrated Time References

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Mobility-based Time References for Wireless Sensor Networks

Part of the book series: Analog Circuits and Signal Processing ((ACSP))

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Abstract

Measuring the time interval between two events requires the choice of a repetitive and regular phenomenon, such as the oscillation of a pendulum, and then counting how many times this phenomenon takes place between the two events. The science of timekeeping has evolved through the centuries by basically adopting more and more precise and reliable periodic phenomena to keep track of time. From the first attempts using evident astronomical events, such as the motion of the sun and the moon, chronometry evolved by employing periodic phenomena in man-made devices, such as sand motion in hourglasses, oscillations in pendulums, balance wheel rotations in mechanical clocks, electromechanical vibrations in quartz crystal oscillators and absorption or emission of radiations in atomic clocks.

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Notes

  1. 1.

    x(t) is obtained using the Fourier series expansion of m(t), i.e. \(A(t) = S(t)\sqrt{{a}_{1 }^{2 } + {b}_{1 }^{2}}\), \(\phi (t) + \theta (t) = 2\pi {f}_{0}\epsilon (t) +\arcsin \frac{{a}_{1}} {\sqrt{{a}_{1 }^{2 }+{b}_{1 }^{2}}}\), where \({a}_{1} = \frac{2} {T}{\int \nolimits \nolimits }_{0}^{{T}_{0}}m(t)\cos (2\pi {f}_{0}t)\,\textrm{ d}t\), \({b}_{1} = \frac{2} {T}{\int \nolimits \nolimits }_{0}^{{T}_{0}}m(t)\sin (2\pi {f}_{0}t)\,\textrm{ d}t\).

  2. 2.

    Note that phase noise can in general also not be a stationary process, since its statistical property changes with time, such as, for example, its variance that can increase with time. The relation between its standard deviation and its PSD is then not well defined. However, defining jitter as a stationary process with parameter Ï„ justifies the calculations presented in this section, even if not all of them are not formally correct, including the integration of S Ï•(f) to compute the jitter.

  3. 3.

    It can be proven, integrating by parts, that \({\int \nolimits \nolimits }_{0}^{+\infty }\frac{{\sin }^{2}x} {{x}^{2}} \textrm{ d}x ={ \int \nolimits \nolimits }_{0}^{+\infty }\frac{\sin x} {x}\textrm{ d}x\); the latter term can be proven to be equal to π ∕ 2 [7].

  4. 4.

    The linearity of passive components describes the variation of the component properties, i.e. resistance or capacitance, for a change in the voltage applied to the component. A high linearity is sign of robustness to voltage variation and consequently low sensitivity to supply voltage variations.

  5. 5.

    Reference capacitors can be constructed whose capacitance is only dependent on the permittivity and one dimension, such as the Thompson–Lampard capacitor [20, 21]. However, their implementation in an integrated circuit is unpractical.

  6. 6.

    This condition can be fulfilled by simply embedding in the oscillator tank all reactive components of the active circuit, such as the parasitic capacitances.

  7. 7.

    In order to fulfill this condition, the parasitic capacitances introduced by the active circuit must be minimized; those parasitic can be due to junction capacitances and MOS capacitances, which have different temperature coefficient than the tank capacitor and consequently can not be trimmed out by a temperature compensation.

  8. 8.

    A more efficient circuit could be employed by adding a cross-coupled pMOS pair in parallel with the tank and re-using the same bias current of the nMOS pair. This would result in a factor 2 for the lower bound for the bias current but it would not radically change the conclusions.

  9. 9.

    State-of-the-art CMOS temperature sensors can achieve an inaccuracy of the order of 0.1 ∘ C, as shown in [39] and as will be further discussed in Chap. 5.

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Authors and Affiliations

  1. NXP Semiconductors, High Tech Campus 32, Eindhoven, Netherlands

    Fabio Sebastiano & Lucien J. Breems

  2. Delft University of Technology, Mekelweg 4, Delft, Netherlands

    Kofi A. A. Makinwa

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  1. Fabio Sebastiano
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  2. Lucien J. Breems
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  3. Kofi A. A. Makinwa
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Sebastiano, F., Breems, L.J., Makinwa, K.A.A. (2013). Fully Integrated Time References. In: Mobility-based Time References for Wireless Sensor Networks. Analog Circuits and Signal Processing. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3483-2_3

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  • DOI: https://doi.org/10.1007/978-1-4614-3483-2_3

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