Errors in the Measurements
The accuracy of the Keithley Model 169 Digital Multimeter for the measurement of DC voltages is given as ±(0.25 % of reading + 1 digit). This means that you should multiply the reading obtained from the meter by 0.25% or 0.0025 and then add one times the value of the least significant digit in the reading to get the maximum expected error value. The actual value of the voltage should be within a range ± that value from the reading.
Example 1. If the meter is on the 20 volt scale and the reading is 2.50 volts the error range can be calculated as follows:
1) First calculate the % error value 0.25% * 2.50 = 0.0025 * 2.50 = 0.00625 volts. This value is due to scaling errors inside the meter.
2) The value of the least significant digit in the reading is 0.01 volts. This error is due to the conversion of the continuous variable to a discrete digital value and is called a quantization error.
3) Adding the quantization error to the scale error gives 1 * 0.01 + 0.00625 = 0.01625 volts. Therefore the maximum error in the given reading will be ±0.01625 volts. This means that the actual voltage lies between 2.50 + 0.01625 = 2.51625 volts and 2.50 - 0.01625 = 2.48375 volts.
or 2.48375 volts < v < 2.5125 volts
Example 2. 20 volt scale reading of 15.00 volts
1) scale error = 0.0025 * 15.00 = 0.0375 volts
2) quantization error = 0.01 volts
3) total error = 0.01+0.0375 = 0.0475
4) range; 14.9525 volts < v < 15.0475 volts
Note: that in the second example the scale error was larger than the quantization error while the quantization error was larger in the first example.
Example 3. Suppose the reading on the 20 volt scale is 1.50 volts.
1) scale error = 0.0025 * 1.50 = 0.00375 volts
2) quantization error = 0.01 volts
3) total error = 0.01+0.00375 = 0.01375
4) range; 1.48625 volts < v < 1.51375 volts
Example 4. with a reading of 1.500 in the 2 volt scale.
1) scale error = 0.0025 * 1.50 = 0.00375 volts
2) quantization error = 0.001 volts
3) total error = 0.001+0.00375 = 0.00475
4) range; 1.49525 volts < v < 1.50475 volts
In example 3 the quantization error is larger then the scale error while in example 4 the quantization error is smaller than the scale error. In these two cases the scale error is the same since the reading is the same. Therefore the total error is lower when the 1.5 volt value is measured on the 2 volt range than when it is measured on the 20 volt range. In general whenever the reading is below one tenth of the full scale value of the meter range you should move down to the next lower range in order to obtain the most accurate measurements.
For measurement of resistance the accuracy is ±(0.2 % of reading + 1 digit) for the middle four ranges, with lower accuracy on the highest and lowest ranges
For DC Current measurements the maximum error is ±(0.75 % of reading + 1 digit). Note that for current measurements there is a column called Maximum Voltage Burden. The voltage burden at currents below full scale decreases in proportion to the current. This gives the value of the circuit voltage that will be lost across the meter when measuring the full scale value for the given range. If this voltage is a significant fraction of the total voltage across the component whose current is being measured, then there will be an additional error in the result due to the fact that the lower voltage across the circuit element will cause the current to be smaller than it was before the meter was inserted in the circuit. Sometimes going to a higher range will give a more accurate reading even though the quantization error will be larger due to the much smaller voltage burden when the reading is below 10% of the full scale reading.
Note that the errors for AC voltage and current measurements are greater than for DC.
Error Propagation in Calculations
When two measurements are combined using an arithmetic operation the result will contain a combination of the errors in the two measurements. The measured values will be represented as the sum of the actual value ± an error. The propagation of the errors will then be observed by performing the arithmetic operation on the measurements in this form.
The worst case error occurs when both of the individual errors have the same sign. Therefore the total error is the sum of the two original errors. The relative error in the sum is approximately the same as the errors in the measurements, if these errors are similar.
With subtraction the errors still add, but since the difference of two measurements is smaller than the sum while the total error will be the same, the relative error will be larger.
If a and b are of about the same magnitude then the difference will be very small and the relative error in the difference can be much greater than the relative error of either of the original values.
For multiplication the relative error is about equal to the sum of the relative errors of the two measurements.
For division the relative error is the same as for multiplication.
Total Error in an Equation
When checking an equation using measured values the effects of all measurement errors must be considered. For example consider Ohm's Law. V = RI
Note: The worst case will be when all three error terms have the same sign. The equation is satisfied to within the expected tolerance of the measuring equipment if the following inequality is satisfied.
Suppose the measurements were:
V = 3.89 V
R = 3.26 kW
I = 1.174 mA
For our meters using the 20 V range for V, The 20 k Ohm range for R, and the 2 mA range for I, the maximum errors would be:
V = 3.89 ± (0.0025*3.89 + 0.01) = 3.89 ± 0.019725 V
R = 3.26 ± (0.002*3.26 + 0.01) = 3.26 ± 0.01652 kW
I = 1.174 ± (0.0075*1.174 + 0.001) = 1.171 ± 0.009805 mA
The calculation then gives:
V = 3.89 ±
0.019725 V = (3.26 ± 0.01652 k Ohms)(1.171 ± 0.009805 mA)
V =
3.89 ± 0.019725 V = (3.26 )(1.174) ± [(3.26)(0.009805) + (1.174)(0.01652)]
V = 3.89 ±
0.019725 V = 3.82724 ± 0.051359 V
Giving the following as the final inequality.
| 3.89-3.82724 | <= |0.019725 + 0.051359|
0.06276 <= 0.071084
0.06 <= 0.07
Therefore Ohm's Law was verified to within the limits of the meter for this case.