Offset and Gain Error - Example

How can we apply Sensitivity Analysis to a basic circuit with errors? Let's walk through finding the Offset Error due to voff and the Gain Error due to R2 Tolerance. These are essential concepts for more complex error analysis! It took me a few trips around the rodeo to develop some understanding and skills. Take your time, refresh and repeat!

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CIRCUIT BLOCK

sch

OFFSET ERROR

1. Find Offset Sensitivity (S)

How does a change in voff cause a change in vo?
As you can see from the schematic, it's the same as the non-inverting signal gain!

2. Evaluate S with Component Values

For R1 = 10k and R2 = 10k

3. Calculate Offset Error

For voff = 1mV, the offset error is found as

GAIN ERROR

1. Find Gain Sensitivity (S)

How does a change in R2 cause a change in K?

The gain error is defined as a ratio.

   where
    K - ideal gain
    K' - actual gain with errors

For the non-inverting amp, the Ideal Gain K and Actual Gain K’ can be written as

2. Evaluate S with Component Values

Calculate K and K' for R1 = 10k, R2 = 10k.
Choose an arbitrarily small value of ΔR2, say 1% (0.01) for the S calculation. (See Calculating Sensitivity below.)

Finally, we calculate S as ratio of gain error over resistor error.

What does S tell us?
For a 1% change in R2, the gain will change by 0.5%.
SImilarly, for a 5% change in R2, the gain will change by 2.5%.

3. Calculate Gain Error

For an R2 Tolerance of 0.1%, the gain error is found as

TRY IT!

Play in the Excel file - modify values, see what happens!
Right Click on the filename, select "Save link as...".

Excel File: Offset_Gain_Sensitivity_and_Errors.xlsx

Signal Gain

With R1=R2=10k, what is the circuit gain K?

Offset Error

For voff (1mV), what is the Offset Sensitivity S and Error?
Now change R1 or R2 to 2x or 1/2x the original value. What happens to S and ΔVoffset?
Change voff=1mV to a different value, can you predict the error?

Gain Error

Return values to R1=R2=10k.
For R2 (1%), what is the Gain Sensitivity S and Error?
Now change R1 or R2 to 2x or 1/2x the original value. What happens to S and ΔK/K?
Change R1_Tol = 1% to a different value, can you predict the error?

CALCULATING SENSITIVITY

We chose an arbitrarily small value of ΔR2 to calc S. Why?

Strictly speaking, S is defined as a mathematical Derivative (dR/K) / (dR/R) which can lead to some messy and difficult calculus.
On the other hand, the Delta Method is far easier and efficlent to calculate (ΔK/K) / (ΔR/R).
For small values of ΔR, the Delta Method calculates the same S as the Derivative Method for all practical purposes.
You should get the same S for a ΔR of 0.01, 0.001, or 0.0001.

NEXT UP

How do errors from a circuit block impact the larger Signal Chain?

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