Electronic Instrumentation Errors in Measurements

Transcription

1 Electronic Instrumentation Errors in Measurements * In this presentation definitions and examples from Wikipedia, HowStaffWorks and some other sources were used Lecturer: Dr. Samuel Kosolapov

2 Items to be defined/refreshed/discussed Systematic Error Random Error Example with OA zero offset Normal Distribution Mean and Deviation Accuracy and Precision Absolute and relative error Significant Digits. Scientific notation 2

3 Types of Errors in Measurements Measurement Error (or Observational Error) is the difference (delta) between measured value of quantity and the true value of quantity Two types of Measurement Errors: Systematic Errors and Random Errors 3

4 Systematic Error versus Random Error Systematic Error: Value of the error is the same IF we use the instrument in the same way and in the same case Systematic Error in most cases can be reduced by better design and/or by using special procedure : calibration Random Error: Value of the error vary from measurement to measurement Origin of random error is due to factors that are out of our control. In some cases significance of random error can be reduced by better design and/of by using math operations like averaging and filtration 4

5 Systematic Error : Calibration Suppose we know that Y = A*X Q. How one can evaluate Systematic Error from the experimental graph Suppose we know that Y = A*X Q. What is the Calibration Equation? 5

6 Systematic Error in OA. Voltage offset According to OA equation Vout = A(Vp Vm) Vout must be == 0 Practically, Vout!= 0 Output offset voltage Output-offset voltage represents the internal imbalance of an op-amp (for example in the differential amplifier) The output-offset voltage is defined as the measured output voltage when the input terminals are shorted together The output-offset voltage may be modeled by placing a voltage source A o V IO in series with the output voltage source A o (v + - v - ) 6

7 Systematic Error in OA. Voltage offset Vout = A o (v + - v - ) + A o V IO Systematic Error All OA circuits from TAE must be re-analyzed now by taking into account this systematic error Some high quality OA uses laser trimming of the resistors during production Time of production and price??? 7

8 Internal Offset Null adjust. (Recommended by producer) Problem: this adjustment must be done MANUALLY for every circuit 8

9 External Offset Null adjust. Inverting Amplifier This adjustments can be used when more DC control is needed Problem: this adjustment must be done MANUALLY for every circuit 9

10 External Offset Null adjust. Non-Inverting Amplifier Problem: this adjustment must be done MANUALLY for every circuit + problems with gain 10

11 External Offset Null adjust. Differential Amplifier Problem: this adjustment must be done MANUALLY for every circuit + many additional problems with gain and CMRR 11

12 Example. TLC4502 Self-calibrating precision dual OA Modern solution: combination of Analog and Digital elements TLC4502 achieves precision by automatically nulling its input offset voltage during power up. During the calibration procedure (on power up), the operational amplifier is removed from the signal path and both inputs are tied to GND. Then output voltage is measured by ADC. Measured value is stored at SAR (successive-approximation register) and used to generate compensation voltage (by using DAC). This self-calibrating procedure typically requires 300 ms to complete. After calibration, the calibration circuitry drops out of the signal path, becoming transparent to the user. 12

13 Random Error Random Error: Value of the error vary from measurement to measurement A number of measurements is a must Xmeasured = Xreal + RANDOM_ERROR Averaging: Xmeasured = Xreal + RANDOM_ERROR For Normal Distribution RANDOM_ERROR = 0 Q.Non-trivial question: Does averaging always eliminates random error? A. Not always. Other filters like median filter can be used 13

14 Example of Normal (Gaussian) Distribution 14

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16 Mean and Variance for Normal (Gaussian) Distribution Mean Variance : Mean Square Deviation from Mean s Standard Deviation: Root Mean Square Deviation from Mean (RMS) 16

17 Mean practical calculation for Normal (Gaussian) Distribution Mean Most likely value of mean is the sample mean 17

18 Some statistics results for Normal (Gaussian) Distribution 18

19 C-code for calculating Mean, Variance and Standard Deviation Real-time calculations: (for N measurements already available) Sum = 0 SumSq = 0 for (int i=0; i<n; i++) { Sum += X[i]; SumSq += X[i]*X[i] ; // better then POW!!!! } Mean = Sum / N Variance = s*s = (N*SumSq Sum*Sum) / N 19

20 Accuracy and Precision Accuracy refers to the closeness of a measured value to a standard or known value. For example, if in lab you obtain a weight measurement of 3.2 kg for a given substance, but the actual or known weight is 10 kg, then your measurement is not accurate. In this case, your measurement is not close to the known value. Precision refers to the closeness of two or more measurements to each other. Using the example above, if you weigh a given substance five times, and get 3.2 kg each time, then your measurement is very precise. Precision is independent of accuracy. You can be very precise but inaccurate, as described above. You can also be accurate but imprecise. If on average, your measurements for a given substance are close to the known value, but the measurements are far from each other, then you have accuracy without precision. 20

21 Accuracy and Precision A good analogy for understanding accuracy and precision is to imagine a basketball player shooting baskets. If the player shoots with accuracy, his aim will always take the ball close to or into the basket. If the player shoots with precision, his aim will always take the ball to the same location which may or may not be close to the basket. A good player will be both accurate and precise by shooting the ball the same way each time and each time making it in the basket. 21

22 Accuracy and Precision. Example Accuracy is the difference between the true value and our best estimate of it. While the definition may be clear, the practice is not. How much samples we need to collect? Serious statistics methods must be used Most real-life distributions are not normal 22

23 Absolute and Relative Error For measured x absolute error = Δx Example : absolute error = ±1mm relative error = absolute error / value of thing measured relative error = Δx / x Example: Scientific notation must be used (next slide) 23

24 Accuracy and Precision Significant Digits 0 cm Length of the bar is 3.23 cm. All digits are significant here What is the length of the bar in mkm? mkm. Are all digits here significant? NO!!! Correct record is: 3.23 E4 mkm (Scientific notation) 24

25 Control Questions What have I learned? 25

26 Literature to read 1. TBD 26