Understanding Digital Voltmeter Accuracy

The accuracy of a digital voltmeter is crucial for reliable measurements. It's typically specified as a combination of a percentage of the measured reading and a value representing the least significant digit (LSD) on the selected measurement range. This specification helps users understand the potential deviation between the displayed value and the true value of the voltage.

The accuracy specification provided for this 3½ digit digital voltmeter is $\pm(0.5\% + 1)$ digit. This formula breaks down the total potential error into two parts:

• Percentage of Reading: This component is $0.5\%$ of the actual voltage value being measured. It accounts for errors that scale with the magnitude of the measurement.
• Digit (LSD): This component represents the value of the least significant digit on the meter's current range. It accounts for the inherent quantization error or resolution limit of the digital display.

A "3½ digit" digital voltmeter means its display can show up to 1999 counts (the leading '3' digits can range from 0 to 9, and the '½' digit can only be 0 or 1). For instance, on a 20V range, it might display values from 0.00V up to 19.99V. The "1 digit" in the accuracy specification refers to one count of this least significant digit.

Calculating Measurement Error for 10V DC

To find the error when measuring a 10V DC voltage, we need to calculate both error components and sum them up. The key is determining the value of the "1 digit" component based on the measurement range used.

Determining the Value of '1 Digit' for the Measurement

When measuring 10V DC, the most suitable range on a typical 3½ digit voltmeter would be the 20V range. This allows for good resolution since 10V is within the 0V to 19.99V span.

Let's calculate the value represented by 1 digit on the 20V range:

• A 3½ digit meter typically has a maximum reading of 1999 counts. On the 20V range, this maximum count of 1999 corresponds to the full range voltage of 20V.
• The value of 1 digit (the LSD) is calculated as: $ 1 \text{ digit} = \frac{\text{Full Range Voltage}}{\text{Maximum Display Counts}} $ $ 1 \text{ digit} = \frac{20\text{V}}{1999} $ Calculating this value gives approximately $0.010005\text{V}$. For practical purposes in accuracy calculations, this is usually taken as $0.01\text{V}$ or $10\text{mV}$.

Calculating the Two Components of Error

Now, we apply the voltmeter's accuracy specification $\pm(0.5\% \text{ of reading} + 0.01\text{V})$ to the measured value of 10V:

• Error due to percentage of reading: This is $0.5\%$ of the measured 10V. $ \text{Percentage Error} = 0.5\% \times 10\text{V} $ $ \text{Percentage Error} = 0.005 \times 10\text{V} = 0.05\text{V} $
• Error due to the digit (LSD): This is the value we calculated for 1 digit on the 20V range. $ \text{Digit Error} = 0.01\text{V} $

Total Absolute Error and Final Percentage Error

The total absolute error is the sum of these two error components:

$ \text{Total Absolute Error} = \text{Percentage Error} + \text{Digit Error} $ $ \text{Total Absolute Error} = 0.05\text{V} + 0.01\text{V} = 0.06\text{V} $

The question asks for the error in the measurement, typically expressed as a percentage relative to the measured reading. To find the total percentage error:

$ \text{Total Percentage Error} = \left( \frac{\text{Total Absolute Error}}{\text{Measured Reading}} \right) \times 100\% $ $ \text{Total Percentage Error} = \left( \frac{0.06\text{V}}{10\text{V}} \right) \times 100\% $ $ \text{Total Percentage Error} = 0.006 \times 100\% = 0.6\% $

Therefore, the error in the measurement of 10V DC using this voltmeter is $\pm0.6\%$.