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Measuring Uncertainty in Voltage Measurement

In scientific experiments, accurately quantifying the uncertainty in measurements is crucial for validating results. Measurement uncertainty can stem from various factors, including the precision of instruments and the inherent accuracy of the experimental setup. This experiment focuses on using the Red Pitaya to assess the uncertainty in voltage measurements through a straightforward voltage divider circuit.

Objective

The goal is to measure the voltage across resistors in a voltage divider circuit using the Red Pitaya and to calculate the standard deviation of these measurements as a metric for uncertainty. This process will be repeated across different resistor values to compare uncertainties.

Experimental Setup

Materials

  • Red Pitaya board
  • Breadboard
  • Two resistors (R1 = 2kΩ, R2 = 1kΩ)
  • Jumper wires
  • Button switch

Assembly Instructions

  1. Circuit Construction: Create a voltage divider by connecting two resistors in series on a breadboard. Connect the circuit to the Red Pitaya's 5V output pin for power.
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  1. Measurement Setup: Open the Red Pitaya oscilloscope app and configure it to measure peak voltages, setting the trigger appropriately to capture the voltage changes accurately.

Procedure

  1. Voltage Measurement: Charge the circuit by pressing the button, then record the voltage across R1 displayed by the Red Pitaya. Repeat this step 10 times to gather a series of voltage readings.
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  1. Calculation of Uncertainty: Compute the average voltage and standard deviation from the recorded measurements to determine the uncertainty in voltage measurement.

Results

The voltage divider circuit has R1 = 2kΩ, R2 = 1kΩ, and an input voltage of 5V. We took 10 measurements of the voltage across resistor R1 using Red Pitaya. We got the following readings:
xR1=[3.262,3.275,3.281,3.268,3.254,3.257,3.249,3.276,3.263,3.278]x_{R1} = [3.262, 3.275, 3.281, 3.268, 3.254, 3.257, 3.249, 3.276, 3.263, 3.278]
If you are measuring in the HV range of ±20V with a 14-bit resolution, the voltage resolution of your measurement system can be calculated as follows:
The range of the ADC is 2^14 = 16384 levels, which corresponds to the range of the input voltage of ±20V. Therefore, each level of the ADC represents a voltage range of:
Voltage range per level = (±20V) / 16384 = ±0.00122
This means that the voltage resolution of your measurement system is approximately ±0.00122.

Calculating standard deviation

To calculate the standard deviation for the voltage measurements, we can use the following formula:
standard deviation=∑i=1n(xi−xmean)2n−1\text{standard deviation} = \sqrt{\frac{\sum_{i=1}^{n}(x_i - x_{mean})^2}{n - 1}}
where x represents the individual voltage measurements, x_mean is the mean voltage value, n is the number of measurements, and ^2 represents squaring. Using this formula, we can calculate the standard deviation for the 10 measurements of the voltage across R1 as follows:
xmean=3.262+3.275+3.281+3.268+3.254+3.257+3.249+3.276+3.263+3.27810≈3.268x_{mean} = \frac{3.262 + 3.275 + 3.281 + 3.268 + 3.254 + 3.257 + 3.249 + 3.276 + 3.263 + 3.278}{10} \approx 3.268
Calculate the squared differences from the mean for each measurement:
∑i=1n(xi−xmean)2=(3.262−3.268)2+(3.275−3.268)2+(3.281−3.268)2+(3.268−3.268)2+(3.254−3.268)2+(3.257−3.268)2+(3.249−3.268)2+(3.276−3.268)2+(3.263−3.268)2+(3.278−3.268)2≈0.0186\sum_{i=1}^{n} (x_i - x_{mean})^2 = (3.262 - 3.268)^2 + (3.275 - 3.268)^2 + (3.281 - 3.268)^2 + (3.268 - 3.268)^2 + (3.254 - 3.268)^2 + (3.257 - 3.268)^2 + (3.249 - 3.268)^2 + (3.276 - 3.268)^2 + (3.263 - 3.268)^2 + (3.278 - 3.268)^2 \approx 0.0186
Sum the squared differences and divide by the number of measurements minus 1:
standard deviation=∑i=1n(xi−xmean)2n−1=0.018610−1≈0.0042\text{standard deviation} = \sqrt{\frac{\sum_{i=1}^{n}(x_i - x_{mean})^2}{n - 1}} = \sqrt{\frac{0.0186}{10 - 1}} \approx 0.0042
Therefore, the standard deviation for the 10 measurements of the voltage across R1 is approximately 0.0042V.
However, it is important to note that the voltage uncertainty of your measurement system is also affected by other factors, such as noise, calibration accuracy, and stability of the measurement system. These factors can contribute to the overall uncertainty of your voltage measurement, which should be taken into account when analyzing and reporting your measurement results.

Conclusion

This experiment demonstrates the practical application of the Red Pitaya in quantifying measurement uncertainty within a simple electronic circuit. The calculated standard deviation provides a quantitative measure of the uncertainty in voltage measurements, highlighting the precision and reliability of the Red Pitaya as a tool for scientific exploration. By understanding and quantifying measurement uncertainty, researchers can better assess the validity and reliability of their experimental results, a cornerstone of rigorous scientific inquiry.