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

$$\delta f(T) = \frac{\Delta f(T)}{F_0}$$

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

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

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.