Simplify Sensitivity Analysis

Difference vs Derivative Methods

 

Computing Sensitivities can be challenging using Derivatives. This topic walks through an example using both the Derivative Method and the quicker Difference Method using simple algebra.

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SENSITIVITY

We've already learned that Sensitivities (S) show a how specific circuit characteristic will change for a given component change. For a resistor divider, for example, an S = 0.5 indicates a resistance change of 1% would cause a gain change of 1% x 0.5 = 0.5%.

DIFFERENCE METHOD

The Difference method finds the sensitivity of a circuit characteristic y=f(x) over a small interval of component change Δx.

For a small Δx, the method approximates the actual derivative. Also note, no new function is introduced, only the original f(x).

DERIVATIVE METHOD

The Derivative Method finds the instantaneous sensitivity of y=f(x) as the interval Δx approaches 0.

Notice, this method introduces a new function - the partial derivative df(x)/dx - found by applying various rules of differentiation (Chain, Product, Power, etc.).

Advantage - You can develop an intuitive feel for S and possibly minimize the sensitivity.

Disadvantage - deriving partial derivatives can be a a challenging task.

RESISTOR DIVIDER

We’ll showcase a simple, widely used resistor divider with gain K.

DIFFERENCE METHOD

Three steps implement the method

1. Write the original function.
2. Copy the function and increment x.
3. Calculate S

As an example, we’ll find the Sensitivity of K to R2.

For R1=750, R2=250 and an R2 increment of 0.001, we get

Notice how S computes easily. However, no further insight is gained.

 

DERIVATIVE METHOD

Let’s see the Derivative Method in action.

1. Write the original function
2. Derive partial derivative function using rules: Chain, Product, Powers, etc.
3. Calculate S and simplify
   

For R1=750, R2=250, we get

  S = 0.750

The Derivative method confirms the Difference's approximation S = 0.7498. With a smaller increment (say 0.0001 or less), you can get even closer. And notice how S scales with the relative signal across R1!

As you can see, the Derivative method requires some mathematical effort (my dormant calculus skills needed a refresh!). Unfortunately, when analyzing a large number of complex circuits with a multitude of components, the sensitivity calculations could quickly become overwhelming and error prone.

EXCEL FILE

Jump into the hands-on spreadsheet!

• Excel File:  sensitivity-derivative-and-difference-methods-b.xlsx
      Right Click on the filename, select "Save link as...".
• Play in the sandbox, modify values, see the impact on errors.
• Copy to a new file - experiment!

TRY IT!

• What is the Sensitivity for an R Divider with R1=500 and R2=500?
• How about S for R1=900 and R2=100?
• How does S scale with the divider ratio? Select another ratio yourself and try to predict S.

 

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