Error Analysis in Electrical and Electronic Measurements – GATE Quick Notes

Introduction to Errors

What is Error?

  • Error = Measured Value - True Value

  • Absolute Error: \(E_a = A_m - A_t\)

  • Relative Error: \(E_r = \dfrac{A_m - A_t}{A_t}\)

  • Percentage Error: \(E_p = \dfrac{A_m - A_t}{A_t} \times 100\%\)

  • Fractional Error: \(E_f = \dfrac{A_m - A_t}{A_m}\)

Types of Errors

Classification of Errors

  1. Systematic Errors (Determinate)

    • Instrumental errors

    • Environmental errors

    • Observational errors

  2. Random Errors (Indeterminate)

    • Unpredictable variations

    • Follow statistical laws

  3. Gross Errors (Blunders)

    • Human mistakes

    • Misreading instruments

Systematic Errors - Details

Instrumental Errors:

  • Inherent shortcomings of instruments

  • Calibration errors

  • Loading effects

  • Aging of components

  • Non-linearity

Environmental Errors:

  • Temperature variations

  • Humidity effects

  • Vibrations

  • External electromagnetic fields

  • Pressure variations

Observational Errors:

  • Parallax error

  • Approximation in reading

Accuracy and Precision

Accuracy vs Precision

Accuracy:

  • Closeness to true value

  • Related to systematic errors

  • \(Accuracy = \dfrac{|\text{True Value - Measured Value}|}{\text{True Value}} \times 100\%\)

Precision:

  • Reproducibility of measurements

  • Related to random errors

  • Measure of scatter in repeated measurements

Key Point: High precision doesn’t guarantee high accuracy!

Resolution and Sensitivity

Resolution and Sensitivity

Resolution:

  • Smallest measurable change in input

  • Determines the fineness of measurement

  • \(Resolution = \dfrac{\text{Full Scale Range}}{\text{Number of Divisions}}\)

Sensitivity:

  • Ratio of change in output to change in input

  • \(Sensitivity = \dfrac{\Delta \text{Output}}{\Delta \text{Input}}\)

  • Higher sensitivity means better measurement capability

Threshold: Minimum input required to produce a detectable output

Statistical Analysis

Statistical Parameters

For n measurements: \(x_1, x_2, x_3, ..., x_n\)

Arithmetic Mean:

\[\bar{x} = \dfrac{1}{n}\sum_{i=1}^{n} x_i\]

Deviation:

\[d_i = x_i - \bar{x}\]

Average Deviation:

\[A.D. = \dfrac{1}{n}\sum_{i=1}^{n} |d_i|\]

Standard Deviation and Variance

Standard Deviation:

\[\sigma = \sqrt{\dfrac{1}{n}\sum_{i=1}^{n} (x_i - \bar{x})^2}\]

Variance:

\[\sigma^2 = \dfrac{1}{n}\sum_{i=1}^{n} (x_i - \bar{x})^2\]

For Sample (n-1):

\[s = \sqrt{\dfrac{1}{n-1}\sum_{i=1}^{n} (x_i - \bar{x})^2}\]

Probable Error

Probable Error

Definition: 50% probability that error lies within probable error limits

Probable Error:

\[P.E. = 0.6745 \times \sigma\]

Probable Error of Mean:

\[P.E._m = \dfrac{0.6745 \times \sigma}{\sqrt{n}}\]

Standard Error of Mean:

\[S.E._m = \dfrac{\sigma}{\sqrt{n}}\]

Limiting Errors

Limiting Errors

Definition: Maximum error that can occur in measurement

For Addition/Subtraction:

\[\Delta R = \pm(\Delta A + \Delta B + \Delta C + ...)\]

For Multiplication/Division:

\[\dfrac{\Delta R}{R} = \pm\left(\dfrac{\Delta A}{A} + \dfrac{\Delta B}{B} + \dfrac{\Delta C}{C} + ...\right)\]

For Power: If \(R = A^n\), then \(\dfrac{\Delta R}{R} = \pm n \times \dfrac{\Delta A}{A}\)

Measurement Error Propagation

Error Propagation Formulas

For function \(R = f(A, B, C, ...)\)

General Formula:

\[\Delta R = \sqrt{\left(\dfrac{\partial R}{\partial A}\Delta A\right)^2 + \left(\dfrac{\partial R}{\partial B}\Delta B\right)^2 + ...}\]

Common Cases:

  • \(R = A + B\): \(\Delta R = \sqrt{(\Delta A)^2 + (\Delta B)^2}\)

  • \(R = A \times B\): \(\dfrac{\Delta R}{R} = \sqrt{\left(\dfrac{\Delta A}{A}\right)^2 + \left(\dfrac{\Delta B}{B}\right)^2}\)

  • \(R = \dfrac{A}{B}\): \(\dfrac{\Delta R}{R} = \sqrt{\left(\dfrac{\Delta A}{A}\right)^2 + \left(\dfrac{\Delta B}{B}\right)^2}\)

Instrument Errors

Instrument Error Classifications

Static Errors:

  • Calibration error

  • Offset error

  • Gain error (Scale factor error)

  • Non-linearity error

  • Hysteresis error

Dynamic Errors:

  • Lag error

  • Loading error

  • Insertion error

Loading Effects

Voltage Measurement:

  • Voltmeter draws current from circuit

  • \(R_{voltmeter} >> R_{circuit}\) (ideally infinite)

  • Loading error \(= \dfrac{R_{circuit}}{R_{circuit} + R_{voltmeter}} \times 100\%\)

Current Measurement:

  • Ammeter adds resistance to circuit

  • \(R_{ammeter} << R_{circuit}\) (ideally zero)

  • Loading error depends on ammeter resistance

Calibration and Standards

Calibration

Definition: Process of comparing instrument readings with known standards

Types:

  • Static calibration (DC conditions)

  • Dynamic calibration (AC/transient conditions)

Calibration Curve:

  • Graph of output vs input

  • Determines linearity

  • Used for error correction

Traceability: Chain of calibrations to national/international standards

Significant Figures

Significant Figures

Rules:

  1. All non-zero digits are significant

  2. Zeros between non-zero digits are significant

  3. Leading zeros are not significant

  4. Trailing zeros after decimal point are significant

  5. Trailing zeros in whole numbers may/may not be significant

Operations:

  • Addition/Subtraction: Result has same decimal places as least precise

  • Multiplication/Division: Result has same significant figures as least precise

Normal Distribution and Confidence Intervals

Normal Distribution

Properties:

  • Bell-shaped curve

  • Mean = Mode = Median

  • 68% data within \(\pm 1\sigma\)

  • 95% data within \(\pm 2\sigma\)

  • 99.7% data within \(\pm 3\sigma\)

Confidence Intervals:

  • 68% confidence: \(\bar{x} \pm \sigma\)

  • 95% confidence: \(\bar{x} \pm 2\sigma\)

  • 99.7% confidence: \(\bar{x} \pm 3\sigma\)

Rejection of Data

Rejection of Outliers

Chauvenet’s Criterion:

  • Reject data if probability of occurrence < \(\dfrac{1}{2n}\)

  • Calculate deviation: \(d = |x_i - \bar{x}|\)

  • If \(\dfrac{d}{\sigma} > \text{threshold}\), reject the data

Q-Test (Dixon’s Test):

\[Q = \dfrac{|\text{suspect value} - \text{nearest value}|}{|\text{highest value} - \text{lowest value}|}\]

  • Compare Q with tabulated values

  • If Q > Q(table), reject the data

Measurement System Performance

Performance Characteristics

Static Characteristics:

  • Accuracy and Precision

  • Resolution and Sensitivity

  • Linearity and Hysteresis

  • Stability and Drift

  • Range and Span

Dynamic Characteristics:

  • Speed of response

  • Lag and Dead time

  • Fidelity

  • Dynamic error

Important Formulas

Key Formulas for GATE

  • Relative Error: \(E_r = \dfrac{A_m - A_t}{A_t}\)

  • Percentage Error: \(E_p = E_r \times 100\%\)

  • Standard Deviation: \(\sigma = \sqrt{\dfrac{\sum(x_i - \bar{x})^2}{n}}\)

  • Probable Error: \(P.E. = 0.6745 \times \sigma\)

  • Standard Error of Mean: \(S.E._m = \dfrac{\sigma}{\sqrt{n}}\)

  • Error Propagation (Product): \(\dfrac{\Delta R}{R} = \sqrt{\left(\dfrac{\Delta A}{A}\right)^2 + \left(\dfrac{\Delta B}{B}\right)^2}\)

  • Resolution: \(\dfrac{\text{Full Scale Range}}{\text{Number of Divisions}}\)

  • Sensitivity: \(\dfrac{\Delta \text{Output}}{\Delta \text{Input}}\)

GATE Tips

GATE Exam Tips

Important Points:

  • Systematic errors can be reduced by calibration

  • Random errors can be reduced by taking multiple readings

  • Accuracy depends on systematic errors

  • Precision depends on random errors

  • Probable error represents \(50\%\) confidence level

  • Standard deviation represents \(68\%\) confidence level

  • Always consider significant figures in final answer

  • Loading effects are crucial in voltage/current measurements

Practice: Error propagation problems are common in GATE

Numerical Example

Sample GATE Problem

Problem: A resistor has a nominal value of \(100~\Omega\) with \(\pm 2\%\) tolerance. A voltage of 10 V with \(\pm 1\%\) accuracy is applied. Find the percentage error in power calculation.

Solution:

  • Power \(P = \dfrac{V^2}{R}\)

  • \(\dfrac{\Delta P}{P} = \sqrt{\left(2\dfrac{\Delta V}{V}\right)^2 + \left(\dfrac{\Delta R}{R}\right)^2}\)

  • \(\dfrac{\Delta P}{P} = \sqrt{(2 \times 1\%)^2 + (2\%)^2}\)

  • \(\dfrac{\Delta P}{P} = \sqrt{4 + 4} = 2.83\%\)