Sensitivity Analysis

How does each component error
impact the circuit error?

 

 

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OVERVIEW

What is Sensitivity (S)?

• S answers the question:
  • How does a change in component parameter (x) produce a change a circuit spec (y)?
• Or we can ask:
  • How Sensitive is y to changes in x?
  • S = ∆y / ∆x

Examples

• Parameters (x):  Voffset, Ibias, Resistor Tol, etc.
• Circuit Specs (y):  Offset, Gain, Filter Bandwidth, etc.

What is S good for?

• Knowing S, you can calculate the circuit error y from actual device error x.
    y = x * S
  • Example:
       vo = voff * S
           = 1mV * 5V/V
           = 5mV
  • voff - amplifier input offset voltage
  • S - amplfier gain (S = vo/voff )
• S depends typically on the circuit only, independent of the actual device or component value.
• What can the magnitude of S tell you?
  • Larger S:  select devices with smaller errors (more precise components, higher grade and cost).
  • Smaller S:  select devices with larger errors (less precice components, lower grade and cost)

 

 

MATHEMATICAL DEFINITIONS

Direct Units	Normalized
Example: Sensitivity of offset voltage to bias current.	Example: Sensitivity of gain to resistor tolerance.
Offset Error Calc	Gain Error Calc
Example: ibias = 100nA S = 1000 (V/A) ∆Voffset = ibias * S            = 100nA * 1000 (V/A)            = 100uV	Example: R1_Tol = 0.1% S = 0.5 ∆K/K = R1_Tol * S       = 0.1% * 0.5       = 0.05%

 

GAIN ERROR EXAMPLE

Suppose you have an amplfier defined by a gain of K = R2/R1+1. How does a change in R2 cause a change in K?

The gain error is defined as a ratio where K is the ideal gain and K' is the actual gain with errors.

The Ideal Gain K and Actual Gain K’ can be written as

 

For a numerical example check out the next topic (Offset and Gain Error Example).

THE MATH: ALGEBRA VS. CALCULUS

• Strictly speaking, S is defined as a mathematical Derivative (dK/K) / (dR/R) which can lead to some messy and difficult Calculus.
• On the other hand, the Difference Method  (ΔK/K) / (ΔR/R) is far easier and efficient to calculate using basic Algebra.
• For small values of ΔR, the Difference Method calculates a similar S as the Derivative Method for all practical purposes.
• You should get a similar S for a ΔR of 0.01, 0.001, or 0.0001.

Check out a real world example Derivative vs. Difference Methods.

 

NEXT UP

Let's apply Sensitivity Analysis to a simple Offset and Gain Error Example.

 

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