Basic Sensitivity – Dave McGlone

Goal: Determine the effect on a function X(y) due to a change in parameter y. This is known as the sensitivity of X to y.

This is expressed:

$\begin{displaymath}S_y^X \; = \; \frac{y}{\, X \,} \, \frac{\, \partial X \, }{\partial y} \; \; \approx \; \; \frac{ \, \frac{ \, \Delta X \, }{X} \, }{\frac{ \, \Delta y \, }{y}} \; \; \Rightarrow \; \; \frac{y}{\, X \, } \, \frac{\, \Delta X \, }{\Delta y}\end{displaymath}$

Consider a form of usage in Mathematica. Let X be “func”; let y be “m”. The expression equivalency is:

$\begin{displaymath}\frac{y}{\, X \,} \, \frac{\, \partial X \, }{\partial y} \; \; \Rightarrow \; \; \frac{m}{\, func \, } \, \frac{\, \partial (func) \, }{\partial m}\end{displaymath}$

To implement in Mathematica:

$\begin{displaymath}\begin{align}&\text{m = y;} \\&\text{func = X;} \\&\text{result} \; = \; \frac{\text{m}}{\, \text{func} \, } \, \text{D[func,m]}\end{align}\end{displaymath}$

An example … (Note: “ $\text{Log[ ]}$ ” in Mathematica is the natural logarithm, aka “ $\text{ln}$ “)
First define the functions:

$\begin{displaymath}\begin{align}&\text{func1 = x } \mathbb{e}^{\text{-y z } \, (\alpha \, + \, \Delta \alpha) } \\\\&\text{func2 = PowerExpand[Log[func1]] } \rightarrow \, \text{- y z }(\alpha \, + \, \Delta \alpha) \, + \, \text{Log[x]}\end{align}\end{displaymath}$

Then define the expressions:

$\begin{displaymath}\text{s1 = }\frac{\Delta \alpha}{\, \text{func1} \, } \, \text{D[func1,} \, \Delta \alpha\text{] // Simplify} \; \; \rightarrow \; \; \text{-y z }\Delta \alpha\end{displaymath}$

$\begin{displaymath}\text{s2 = }\frac{\Delta \alpha}{\, \text{func2} \, } \, \text{D[func2,} \, \Delta \alpha\text{] // Simplify} \; \; \rightarrow \; \; \frac{\text{y z }\Delta \alpha}{\, \text{y z }(\alpha \, + \Delta \alpha) \, - \, \text{Log[x]}\,}\end{displaymath}$

Consider a basic voltage divider.

The transfer function of this network is:

$\mathcal{T} \; = \; \frac{\, V_{out} \, }{V_{in}} \; = \; \frac{R_2}{\, R_1 \, + \, R_2 \, }$

The sensitivity of this function to changes in R $_1$ is:

$S_{R_1}^{\mathcal{T}} \; = \; -\frac{R_1}{\, R_1 \, + \, R_2 \, }$

and for R $_2$ :

$S_{R_2}^{\mathcal{T}} \; = \; \frac{R_1}{\, R_1 \, + \, R_2 \, }$

The sensitivities are identical except for the sign. The sensitivity of $\mathcal{T}$ to changes in R $_1$ is negative: if R $_1$ increases, the value of $\mathcal{T}$ decreases. However, the value of $\mathcal{T}$ increases if R $_2$ increases. If the two resistors are equal, the sensitivity is 0.5. This implies that a 1% change in either resistor will result in a 0.5% change in $\mathcal{T}$ .

In the limit, if R $_1$ >> R $_2$ , the sensitivity of $\mathcal{T}$ to changes in resistance tends to a value of 1; a 1% change in either resistor causes a 1% change in $\mathcal{T}$ . If R $_2$ >> R $_1$ , the sensitivity of $\mathcal{T}$ to changes in resistance tends to a value of 0; the function is essentially independent of changes in resistance.

The most common variations in resistances are tolerance and temperature. Other possibly significant variations are voltage and load/life effects. Some applications will also require radiation tolerances … but those applications are quite specific – if you need them, you’ll know where and how to get them.

Voltage and aging coefficients are often ignored (and may not even be specified) but may have a significant effect in precision applications.

Some specifications for a precision resistor:

Tolerance: ±100 ppm (0.01%)
Temperature coefficient: ±0.2 ppm/C (-55°C to 125°C)
Power coefficient (self-heating): 5 ppm
voltage coefficient: < 0.1 ppm/V
load/life stability: 50 ppm (after 2000 hrs)
ESD: > 25 kV
thermal stabilization: ±10 ppm
rise time: 1 ns
Inductance: < 0.08 μH

A proper sensitivity analysis for a precision application will take each of these factors into consideration. The analysis of that simple resistor divider is no longer “simple” …

The same principles apply to reactive components and construction techniques as well (for example, flexing of a PCB can change the operational accuracy of a circuit. Many components will have flexibility criteria.)

That’s good for now.

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