## Practical Electronics for Inventors, Fourth Edition - Paul Scherz, Simon Monk (2016)

### APPENDIX B. Error Analysis

Reliability estimates of measurements greatly enhance their value. For example, saying the resistance of a resistor is 1000 Ω ± 50 Ω tells you much more than simply stating the resistance is 1000 Ω.

The term *error* is basically interchangeable with *uncertainty*, but does not have the same meaning as *mistake*. Mistakes, such as errors in calculations, should be corrected before estimating the experimental error. In estimating the reliability of a single quantity (such as resistance), you should recognize three different kinds of sources of error:

1. Actual variations of the quantity being measured, for example, resistance changes due to temperature variations. In electronics, for accurate measurements, you should consider the temperature ratings and the corresponding errors specified in the data sheets.

2. Test equipment in error. The errors introduced by test equipment. Make sure that all equipment is calibrated, and be certain to take into account input impedance characteristics, such as the input impedance of a multimeter and oscilloscope.

3. Human error. With digital equipment displays this is less of a problem. A scope with a graphical display will not give you high accuracy (only around 5 percent).

**B.1 Absolute Error, Relative Error, and Percent Error**

If Δ*x* is the *absolute error* (or *uncertainty* with a ± in front) in a measurement whose value is *x*, then Δ*x/x* is called the *relative error* (or *fractional uncertainty*). If we multiply the relative error by 100 percent, we get the *percent error*, or 100% ⋅ (Δ*x/x*). The term *tolerance* is interchangeable with both absolute error and percent error. For example, tolerances in length measurements are typically given in terms of absolute error values, while tolerances in resistances are typically given in percent error.

** Example 1:** What’s the relative error and percent error of 0.125 A ± 0.01 A?

*Answer:*

** Example 2:** What are the relative error and the absolute error or uncertainty of a 3300-Ω resistor with a tolerance of 5 percent? What’s the guaranteed range of resistance?

** Answer:** Here, tolerance represents percent error, so

The absolute error or uncertainty is:

Δ*x* = *x* (relative error) = (3300 Ω)(0.05) = ±165 Ω

The resistor is guaranteed for 3300 Ω ± 165 Ω or guaranteed to be between 3135 Ω and 3465 Ω.

**B.2 Uncertainty Estimates**

When dealing with equations with many independent variables, like the following RC charge response equation

the uncertainty or error in the final result (e.g., current) will depend on the individual uncertainties (e.g., uncertainties in resistance, capacitance, voltage, and time). The propagation of errors can be explained by first analyzing simple arithmetic cases:

1. If the desired result is the *sum* or *difference* of two measurements, the absolute uncertainties add:

Let Δ*x* and Δ*y* be the errors in *x* and *y*, respectively. For the *sum*, we have:

*z* = *x* + Δ*x* + *y* + Δ*y*

and a relative error of

(Δ*x* + Δ*y*)/(*x* + *y*)

Since the signs of Δ*x* and Δ*y* can be opposite, adding the absolute values gives a pessimistic estimate of the uncertainty. If errors have a Gaussian distribution and are independent, they combine in quadrature (square root of the sum of the squares):

For the *difference* of two measurements, we obtain a relative error of:

(Δ*x* + Δ*y*)/(*x* − *y*)

which becomes very large if *x* is nearly equal to *y*. This is an important point to note; you must avoid designing experiments where two large quantities are measured and their difference obtained.

2. If the desired result involves *multiplying* or *dividing* measured quantities, then the relative uncertainty of the result is the sum of the relative errors in each of the measured quantities. The most pessimistic case corresponds to adding the absolute value of each term, since the Δ*x** _{i}* and Δ

*x*

*can be of either sign:*

_{i}Again, if the measurement errors are independent and have a Gaussian distribution, the relative errors will add in quadrature:

3. If the desired result is a *power* of the measured quantity, the relative error in the result is the relative error in the measured quantity multiplied by the power. For example, the uncertainty in

*z = x*^{n}

is:

4. For anything more complex, the following equation gives the general method for finding uncertainty in measurements. For example, if *R* = *f*(*x, y, z*) is the functional relationship between three measurements *x, y, z,* then

gives the uncertainty in *R* when the uncertainties *dx, dy,* and *dz* are known.

Usually, you don’t have to go to this extreme—you can usually get by with the rules for adding, subtracting, multiplying, dividing, and rising to the power. To make life easy on you, a cheat sheet is provided here.

**FORMULAS TO CALCULATE UNCORRELATED ERRORS**

If *A* = ± *a, B* = ± *b*, and *C* = ± *c*, where , , are the measured values of the quantities *A, B, C,* and *a, b, c* are the respective errors, then the calculated values with error (assuming independent variables and Gaussian distribution) give:

** Example 1:** The voltages across two series resistors are measured using two different voltmeters. One digital meter reads 6.24 V ± 0.01 V across the first resistor; a less precise analog meter measures 14.3 V ± 0.2 V. What is the total voltage across the pair, including uncertainty in the result?

** Answer:** We simply add voltages and use Eq. 1 to determine the resultant uncertainty:

** Example 2:** The current through a 180-Ω, 5 percent resistor is 1.256 A ± 0.005 A. Determine the voltage across the resistor with uncertainty included.

** Answer:** First convert the tolerance to absolute error (or uncertainty):

Since the equation for voltage is Ohm’s law, *V* = *I* × *R*, we use Eq. 4: