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Frequency and Temperature

Table 1: Balance
Device Granularity Stability Power
Tuning Fork XO Coarse 100ppm $<50 \mu W$
Fox F254 32k Osc Coarse 100ppm $56 \mu W$
AT-Cut Quartz XO Fine 25ppm $200 \mu W$
DS32KHz 32K TCXO Coarse 7.5ppm $750 \mu W$
DS3232 32KHz TCXO Coarse 2ppm $1 mW$
DS4026 10MHz TCXO Fine 1ppm $21 mW$
ACHL 10MHz Osc Fine 30ppm $13.8 mW$
Crystek C3392 10MHz Fine 30ppm $10 mW$
8MHz XCXT Fine 1ppm $1.4mW$
Smart Timer Unit Fine 1ppm $\sim 300 \mu W$

Assume that the frequency of a clock as a function of temperature is denoted by the function $f(T)$. Its nominal frequency is designated as $F_0$, and thus the frequency error is $\Delta f(T) = f(T)-F_0$. The normalized frequency error is calculated as

\begin{displaymath}\delta f(T) = \frac{\Delta f(T)}{F_0}\end{displaymath}

and is usually expressed in the unitless quantity called $ppm$, or Parts Per Million, computed as

\begin{displaymath}\delta f(T) = \frac{f(T) - F_0}{F_0}\times 10^6\end{displaymath}

The multiplication by $10^6$ is a measure of convenience since the frequency error of most commercially available clock sources ranges from tens to hundreds of ppm.

Thermal variation is the single most significant contributor to clock frequency error. Thus, changes in the ambient temperature are reflected in the frequency of a node's clock. The most common way to counterbalance this effect in quartz-crystal based oscillators is to measure the temperature and adjust the crystal oscillator (by changing its load capacitance) using a set of predetermined corrections. Commercial Temperature Compensated Crystal Oscillators (TCXO) embody this approach, but adding thermal measurement and analysis logic not only increases the production cost, but power consumption as well. TCXO's are often unfeasible for the strict low power requirements of WSN's. In Table 1, a representative set of oscillators (including several TCXO's) are compared in terms of stability and power.

Thomas Schmid 2008-11-14