Electronics Guide

Instrumentation Uncertainty

Measuring instruments contribute uncertainty through several mechanisms:

• Receiver uncertainty: EMI receivers have inherent accuracy limitations including reference level uncertainty, frequency response flatness, detector accuracy, and noise floor effects. Typical receiver uncertainty ranges from 1.0 to 2.5 dB depending on the measurement level relative to noise floor.
• Transducer calibration: Antennas, current probes, LISNs, and other transducers have calibration uncertainties from the calibration laboratory. These uncertainties include both the laboratory's measurement capability and interpolation between calibration frequencies.
• Cable losses: The uncertainty in cable attenuation contributes directly to measurement uncertainty. Cable calibration uncertainty, temperature coefficients, and connector repeatability all contribute.
• Preamplifier gain: If preamplifiers are used, their gain uncertainty and noise figure affect measurement uncertainty, particularly at low signal levels.

Instrumentation uncertainties are typically evaluated as Type B uncertainties, based on calibration certificates, manufacturer specifications, and engineering judgment rather than repeated measurements.

Environmental Uncertainty

The test environment contributes significant uncertainty in EMC measurements:

• Site imperfections: Anechoic chambers have finite absorber performance, creating reflections that vary with frequency and direction. Site attenuation measurements quantify this contribution but have their own uncertainty. Typical site imperfection uncertainty ranges from 1.0 to 4.0 dB depending on chamber quality and frequency.
• Ambient conditions: Background electromagnetic noise affects low-level measurements. Temperature and humidity can affect absorber performance and equipment behavior.
• Ground plane effects: For measurements above a ground plane, surface quality, size, and height affect the measurement. Ground plane uncertainty is particularly significant for radiated emissions from floor-standing equipment.
• Distance factors: Uncertainty in measurement distance affects calculated field strength. A 3% distance error causes approximately 0.26 dB field strength error at 3 meters.

Setup and Procedural Uncertainty

The manner in which measurements are performed introduces additional uncertainty:

• Cable configuration: The arrangement of cables on the equipment under test affects emissions. Different cable configurations can yield different results even for identical products.
• EUT operation: The operating mode of the equipment affects emissions. Variability in achieving identical operating states contributes uncertainty.
• Antenna positioning: Uncertainty in antenna height, polarization angle, and distance from the EUT contribute to field measurement uncertainty.
• Operator variation: Different operators may interpret procedures differently, leading to setup variations. Standardized procedures reduce but do not eliminate this contribution.

Setup uncertainties can be evaluated by repeated measurements with intentional variations (Type A evaluation) or estimated from experience and analysis (Type B evaluation).

EUT-Related Uncertainty

The equipment under test itself contributes uncertainty:

• Directivity: Most equipment radiates differently in different directions. Unless maximum emissions are found through complete spatial scanning, directivity effects introduce uncertainty.
• Operating mode effects: Emissions may vary with operating condition even within the specified test mode.
• Warm-up and stability: Equipment behavior may change during warm-up or exhibit time-dependent variations during measurement.
• Unit-to-unit variation: Different production units of the same product exhibit different emissions. If compliance of the population is at issue, this variation becomes part of the uncertainty assessment.

The Law of Propagation of Uncertainty

For a measurement result y that depends on input quantities x1, x2, ..., xn through a functional relationship y = f(x1, x2, ..., xn), the combined standard uncertainty in y is:

u(y)^2 = sum over i of (partial f/partial xi)^2 * u(xi)^2 + 2 * sum over i<j of (partial f/partial xi)(partial f/partial xj) * u(xi,xj)

The partial derivatives are called sensitivity coefficients, representing how much the output changes per unit change in each input. The second term involves covariances between inputs and accounts for correlations.

For uncorrelated inputs with linear functional relationships, this simplifies to the root-sum-of-squares formula:

u(y) = sqrt[sum of (ci * u(xi))^2]

where ci = partial f/partial xi is the sensitivity coefficient for input xi.

Sensitivity Coefficients

Sensitivity coefficients quantify how output uncertainty depends on each input. They can be determined analytically (by differentiating the measurement equation), numerically (by evaluating the change in output for small changes in input), or experimentally (by intentionally varying inputs and observing output changes).

In EMC measurements, sensitivity coefficients are often unity because the measurement equation involves simple addition (in dB). For example, the measured emissions level is:

E (dBuV/m) = Receiver reading (dBuV) + Antenna factor (dB/m) + Cable loss (dB)

Here, each coefficient is 1, and uncertainties in each term contribute equally to the result. However, when converting between linear and logarithmic scales, or when calculating derived quantities, sensitivity coefficients may differ from unity.

Correlation Effects

Correlations between input quantities affect combined uncertainty. Positive correlations increase combined uncertainty beyond the root-sum-of-squares; negative correlations reduce it.

In EMC, correlations arise from:

• Common calibration sources: If multiple transducers were calibrated by the same reference source, their uncertainties are correlated.
• Shared environmental effects: Temperature affects multiple components similarly, creating correlation.
• Systematic errors: Errors that affect multiple measurements in the same direction create correlation.

When correlation is suspected but not quantified, it is conservative to assume full positive correlation (add uncertainties linearly) for worst-case assessment or to treat as uncorrelated if partial cancellation is likely.

Monte Carlo Propagation

When measurement equations are complex or non-linear, or when input distributions are non-normal, Monte Carlo simulation can propagate uncertainties numerically. The procedure involves:

1. Assign probability distributions to each input quantity
2. Generate random samples from each input distribution
3. Evaluate the measurement equation for each sample set
4. Analyze the resulting output distribution

Monte Carlo propagation is particularly useful when the measurement equation involves limits, thresholds, or switching logic that creates discontinuous behavior. It also handles non-normal input distributions without requiring approximations.

Type A Evaluation

Type A evaluation determines uncertainty from statistical analysis of repeated observations. For n independent observations of a quantity x, the standard uncertainty is the standard deviation of the mean:

u(x) = s(x) / sqrt(n)

where s(x) is the sample standard deviation.

Type A evaluation is appropriate when measurements can be repeated under identical conditions. In EMC, repeatability studies with multiple setups, multiple operators, or multiple measurement runs provide Type A uncertainty data.

The number of degrees of freedom for Type A evaluation equals n-1. Small sample sizes result in large uncertainty in the uncertainty itself, accounted for through the Welch-Satterthwaite formula when combining with other components.

Type B Evaluation

Type B evaluation determines uncertainty from non-statistical sources such as calibration certificates, manufacturer specifications, physical constraints, experience, or engineering judgment.

Common Type B uncertainty sources include:

• Calibration certificates: Provide uncertainty for calibrated values. Typically reported as expanded uncertainty with coverage factor.
• Manufacturer specifications: May be given as tolerances, accuracy classes, or maximum errors. Must be converted to standard uncertainties.
• Reference data: Published values for physical constants, material properties, or correction factors have associated uncertainties.
• Experience and judgment: When quantitative data are unavailable, experienced practitioners can estimate uncertainty bounds.

Converting Type B data to standard uncertainty requires assuming a probability distribution:

• Normal distribution: If uncertainty is given as expanded uncertainty U with coverage factor k, standard uncertainty is u = U/k. Typical k=2 for 95% confidence.
• Rectangular distribution: If only bounds +/-a are known, standard uncertainty is u = a/sqrt(3).
• Triangular distribution: If bounds +/-a are known and center value is most likely, standard uncertainty is u = a/sqrt(6).
• U-shaped distribution: If extreme values are most likely (as with sinusoidal variation), standard uncertainty is u = a/sqrt(2).

Uncertainty Budget

An uncertainty budget tabulates all contributing uncertainty components, showing:

• The quantity measured or influencing the measurement
• The value or estimate of each quantity
• The standard uncertainty for each quantity
• The probability distribution assumed
• The sensitivity coefficient
• The contribution to combined uncertainty (sensitivity coefficient times standard uncertainty)

The uncertainty budget provides transparency and traceability for the combined uncertainty. It identifies dominant contributors, guiding efforts to reduce uncertainty through improved equipment, procedures, or environmental control.

Example uncertainty budget for a radiated emissions measurement might include:

Combined standard uncertainty = sqrt(0.8^2 + 1.2^2 + 0.3^2 + 2.0^2 + 0.5^2 + 1.0^2) = 2.7 dB

Coverage Factor Definition

Expanded uncertainty U is obtained by multiplying the combined standard uncertainty by a coverage factor k:

U = k * u(y)

The coverage factor determines the width of the uncertainty interval and the associated confidence level. For a normal distribution:

• k = 1.00 corresponds to approximately 68% coverage
• k = 1.65 corresponds to approximately 90% coverage
• k = 1.96 corresponds to approximately 95% coverage
• k = 2.00 corresponds to approximately 95.4% coverage (commonly used approximation)
• k = 2.58 corresponds to approximately 99% coverage
• k = 3.00 corresponds to approximately 99.7% coverage

In EMC testing, k = 2 is the most common choice, providing approximately 95% confidence that the true value lies within the stated interval.

Effective Degrees of Freedom

When combining uncertainty components with different numbers of degrees of freedom, the effective degrees of freedom of the combined uncertainty is calculated using the Welch-Satterthwaite formula:

v_eff = u(y)^4 / sum over i of [(ci * u(xi))^4 / vi]

where vi is the degrees of freedom for component i.

Type B uncertainties are often assigned infinite degrees of freedom, reflecting the assumption that the stated uncertainty is well characterized. Type A uncertainties from small samples have few degrees of freedom, reducing v_eff and requiring larger coverage factors.

When v_eff is finite, the coverage factor for a given confidence level comes from the t-distribution rather than the normal distribution. For v_eff = 10 and 95% coverage, k = 2.23 rather than 1.96. As v_eff increases, k approaches the normal distribution value.

Confidence Levels

The confidence level expresses the probability that the true value lies within the expanded uncertainty interval, assuming the uncertainty has been properly evaluated. Common confidence levels in EMC include:

• 95% (k approximately 2): Standard for most EMC measurements and required by many standards.
• 99% (k approximately 2.6): Used when higher confidence is required, such as for safety-critical applications.
• 90% (k approximately 1.65): Sometimes used for screening measurements where lower confidence is acceptable.

The choice of confidence level involves trade-offs. Higher confidence requires larger intervals, which may reduce the apparent compliance margin or increase the likelihood of unnecessary failures. Lower confidence increases the risk of incorrect decisions.

GUM Requirements

The Guide to the Expression of Uncertainty in Measurement (GUM, ISO/IEC Guide 98-3) specifies that complete uncertainty statements include:

• A clear description of what was measured
• The measurement result (best estimate of the measurand)
• The expanded uncertainty U
• The coverage factor k used
• The confidence level associated with k

Example statement: "The radiated electric field strength at 100 MHz is 32.5 dBuV/m with an expanded uncertainty of 5.4 dB (k = 2, 95% confidence level)."

CISPR Requirements

CISPR 16-4-2 provides specific guidance for uncertainty in EMC measurements. It defines standard uncertainty values for various measurement conditions (CISPR measurement instrumentation uncertainty, or CMIU) and laboratory uncertainty (LAB uncertainty).

CISPR distinguishes between:

• UCISPR: The uncertainty inherent in the CISPR measurement method itself, determined by CISPR and published in standards.
• Ulab: The additional uncertainty specific to a particular laboratory, determined by each laboratory through its own uncertainty evaluation.

CISPR currently requires laboratories to report measurement results without adjustment for uncertainty, but to state the measurement uncertainty separately. Compliance decisions may consider uncertainty in different ways depending on the regulatory framework.

Laboratory Accreditation

Accredited testing laboratories must demonstrate competence in uncertainty evaluation as part of ISO/IEC 17025 requirements. Accreditation bodies assess:

• Identification of uncertainty sources
• Appropriate evaluation methods
• Proper combination and reporting
• Traceability of calibration uncertainties
• Competence of personnel in uncertainty evaluation

Inter-laboratory comparison programs provide external validation of uncertainty claims. Proficiency test results that fall within stated uncertainties support the laboratory's uncertainty evaluation; results outside stated uncertainties indicate potential problems.

Compliance Assessment

How measurement uncertainty is used in compliance decisions varies by regulatory authority and application. Common approaches include:

• Simple acceptance: The measured value is compared directly to the limit without considering uncertainty. Used when standards incorporate uncertainty in the limit itself.
• Guard band approach: A margin equal to the expanded uncertainty is added to the measured value (or subtracted from the limit) before comparison. Provides high confidence of compliance but increases the effective stringency.
• Shared risk approach: The limit is adjusted by a fraction of the uncertainty, sharing the risk between supplier and customer. Various risk-sharing ratios are used.
• Probabilistic approach: The measurement result and uncertainty define a probability distribution; compliance probability is assessed directly.

ILAC-G8 provides guidance on the role of measurement uncertainty in compliance assessment. The approach should be agreed upon before testing and documented in the test report.

Frequency-Dependent Uncertainty

EMC measurement uncertainty varies with frequency due to frequency-dependent contributions from antennas, site performance, and other factors. A complete uncertainty analysis should evaluate uncertainty across the frequency range of interest.

Typical frequency dependence includes:

• Antenna calibration uncertainty often increases at frequency band edges
• Site imperfection effects vary with frequency
• Mismatch effects depend on frequency-varying impedances
• Cable losses and their uncertainties increase with frequency

Uncertainty may be reported at specific frequencies, averaged across bands, or as a frequency-dependent function.

Measurement in Reverberation Chambers

Reverberation chambers present unique uncertainty considerations due to their statistical nature. The field strength at any point follows a probability distribution, and measurements rely on statistical processing (typically averaging across stirrer positions).

Key uncertainty components include:

• Stirrer effectiveness: Insufficient stirring leads to non-ideal field statistics
• Number of samples: Uncertainty in the mean field decreases with more stirrer positions
• Loading effects: The EUT affects chamber statistics
• Lowest usable frequency: Below approximately 3 times the fundamental chamber resonance, field uniformity degrades

IEC 61000-4-21 provides methods for validating reverberation chamber performance and assessing associated uncertainties.

Uncertainty in Simulation

Computational electromagnetic simulation results also have uncertainty, arising from:

• Model fidelity: The simulation model inevitably simplifies the real system
• Input parameter uncertainty: Material properties, dimensions, and other model inputs have uncertainties
• Numerical uncertainty: Discretization, convergence criteria, and numerical precision affect results
• Solver validation: The solver's accuracy for the problem type

Propagating input uncertainties through simulation requires methods such as sensitivity analysis or Monte Carlo simulation. Model validation against measurements helps quantify model uncertainty.

Equipment Improvements

Better instrumentation can reduce uncertainty contributions from:

• More accurate receivers with lower noise floors
• Antennas with tighter calibration uncertainty
• Lower-loss cables with better characterized attenuation
• More stable preamplifiers with lower noise figure

The cost-effectiveness of equipment upgrades depends on whether the targeted component is a dominant uncertainty contributor. Improving a small contributor yields little benefit to combined uncertainty.

Environmental Improvements

Test environment quality directly affects several uncertainty components:

• Better absorbers reduce site imperfection uncertainty
• Improved shielding reduces ambient noise effects
• Climate control reduces temperature-related variations
• Ground plane quality affects measurements above the plane

Procedural Improvements

Better procedures can reduce setup and operator-related uncertainty:

• More detailed test procedures reduce operator variation
• Automated positioning systems reduce placement uncertainty
• Standardized cable configurations improve repeatability
• More comprehensive spatial scanning finds maximum emissions more reliably

Statistical Improvements

Increased sampling reduces Type A uncertainty components:

• More repeated measurements reduce standard error of the mean
• More stirrer positions in reverberation chambers improve field estimates
• Multiple test setups average out setup-dependent variations

However, increased sampling has diminishing returns (uncertainty decreases as 1/sqrt(n)) and increased cost. The optimal sample size balances uncertainty reduction against resources.