# Appendices:

## Period Measurement Resolution

Definition: The granularity of a sensor measurement is the smallest change it can detect in the quantity that it is measuring. Resolution, which is the inverse of granularity, is related to the precision with which the measurement is made.

Units: Hz/Count

Assume that Timer 3 count is recorded on each transition. Then for two frequencies, F1 and F2 the Timer 3 counts are computed by:

1

${F_{TMR3}} = \frac{{PBCLK}}{{T{3_{PS}}}} = \frac{{{{10}^7}}}{{256}}$

2

$T{3_{COUNT1}} = \frac{{{{\rm{F}}_{{\rm{TMR}}3}}}}{{{\rm{F}}1}}$

3

$T{3_{COUNT2}} = \frac{{{{\rm{F}}_{{\rm{TMR}}3}}}}{{{\rm{F}}2}}$

To compute the resolution in Hz per count, assume that the count difference is unity, which results in Eq. 4 or Eq. 5, whichever expresses the desired units.

4

$\left| {T{3_{COUNT1}} - T{3_{COUNT2}}} \right|$ = $Granularity$ = $\left( {{F_{TMR3}}} \right)$ $\cdot$ $\left| {\left( {\frac{1}{{{\rm{F}}2}} - \frac{1}{{{\rm{F}}1}}} \right)} \right|COUNTS/{\rm{Hz}}$

5

$Resolution = \left( {\frac{1}{{{{\rm{F}}_{{\rm{TMR}}3}}}}} \right) \cdot \left( {\frac{1}{{\left| {\left( {\frac{1}{{{\rm{F}}1}} - \frac{1}{{{\rm{F}}2}}} \right)} \right|}}} \right) = \left( {\frac{1}{{{{\rm{F}}_{{\rm{TMR}}3}}}}} \right) \cdot \left( {\frac{{{\rm{F}}1 \cdot {\rm{F}}2}}{{\left| {\left( {{\rm{F}}2 - {\rm{F}}1} \right)} \right|}}} \right)Hz/COUNT$

6

$\left| {F2 - F1} \right| = {\rm{Resolution}} = \left( {\frac{{{\rm{F}}1 \cdot {\rm{F}}2}}{{{{\rm{F}}_{{\rm{TMR}}3}}}}} \right)Hz/COUNT$

If F1 approximately equals F2, at 1 Hz, the resolution is 0.0001024 Hz/ Count. At 200 Hz, the resolution is 1.024 Hz/ count.