Offset and Gain Error

Example Calculations

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!

Back to EBA Series

CIRCUIT BLOCK

sch

OFFSET ERROR

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!

Evaluate S with Component Values

For R1 = 10k and R2 = 10k

Calculate Offset Error

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

GAIN ERROR

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

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 more on 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%.

Calculate Gain Error

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

TRY IT!

Jump into the hands-on spreadsheet!

Excel File: Offset_Gain_Sensitivity_and_Errors.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!

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 Difference Method is far easier and efficlent to calculate (ΔK/K) / (ΔR/R).
For small values of ΔR, the Difference Method calculates the same 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.

NEXT UP

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

Back to EBA Series