Question 29.2: Determine the effective number of bits for a resistor-string......
Determine the effective number of bits for a resistor-string DAC, which is assumed to be limited by the INL. The resistors are passive poly resistors with a known relative matching of 1%, and V_{\mathit{R E F}}=5~\mathrm{V}.
Using Eq. (29.10), the maximum INL will be
\left|IN L\right|_{m a x}=\frac{V_{R E F}}{2^{N}}.\sum_{k=1}^{2^{N-1}}\ \frac{\Delta R_{k}}{R}={\frac{V_{R E F}}{2^{N}}}\cdot{\frac{2^{N-1}\cdot\Delta R_{k}}{R}}={\frac{1}{2}}\;L S B\cdot2^{N}\cdot(\%\ \mathrm{matching})=0.01\,V_{R E F} (29.10)
\left|I N L\right|_{m a x}=0.005\cdot V_{R E F}=0.025~\mathrm{V}Since we know that this maximum INL should be equal to ½ LSB in the worst case,
{\frac{1}{2}}L S B={\frac{5}{2^{N+1}}}=0.025\ \mathrm{V}and solving for N yields
N=\mathrm{log_{2}}\left(\frac{5}{0.025}\right)-1=6.64\mathrm{~bits}This means that the resolution for a DAC containing a resistor string matched to within 1% will be, at most 6 bits.