Electronic – Question about the accuracy of integrated resistors


I'm studying the implementation of resistors in integrated technology. In particular the books deals with the accuracy of the obtained resistors. It says:

Integrated resistor formula

Then it says:

"We observe that the value of the resistor depends on four parameters that are
achieved with independent technological steps. Therefore, to estimate their
accuracy we assume them to be statistically independent."
and it comes out with the following equation:

Integrated resistor total error

The question is: what is the meaning of this equation? How is it mathematically derived?

Best Answer

\$R\$ depends on \$L\$, \$W\$, \$\overline{\rho}\$ and \$x_j\$.

If the errors of those 4 parameters are statistically independent, then the variance of \$R\$ will be the sum of the variances of those 4 parameters, hence the summation.

Another way to look at it is by taking partial derivatives of \$R\$ with respect to each parameter, then dividing by \$R\$ and squaring. Then you can add it up all together because all variations are statistically independent. Otherwise you couldn't just add them up, you should take into account any cross-correlation between parameters.

And just in case you're wondering how it's possible to add errors that are in the denominator of the equation (parameters \$W\$ and \$x_j\$): we can do it because there is a first order approximation involved.

$$ \frac{1}{W+\Delta W} = \frac{\frac{1}{W}}{1+\frac{\Delta W}{W}} \approx \frac{1}{W} \left( {1-\frac{\Delta W}{W}} \right) \text{ if } \vert{\frac{\Delta W}{W}}\rvert \ll 1 \text{ thus}\\ R+\Delta R \big\vert_{\text{due to }\Delta W} = \frac{L \overline{\rho}}{(W+\Delta W)x_j}= \frac{L \overline{\rho}}{Wx_j}\left( {1-\frac{\Delta W}{W}} \right) = R - R\frac{\Delta W}{W} \\ \Delta R \big\vert_{\text{due to }\Delta W} = - R\frac{\Delta W}{W} \\ {\frac{\Delta R}{R}} \Bigg\vert_{\text{due to }\Delta W} = - \frac{\Delta W}{W} $$

After squaring, the minus sign disappears. The same applies to the error contributed by \$x_j\$.