Electronics Guide

Instrumentation Uncertainty

Measuring instruments contribute uncertainty through several mechanisms:

• Receiver uncertainty: EMI receivers have inherent accuracy limits. CISPR 16-4-2 treats these as separate input quantities, among them the sine-wave voltage accuracy, the pulse amplitude response, the pulse repetition rate response, and a correction for the effect of the noise floor. Taken together, the receiver typically contributes a standard uncertainty of roughly 1.0 dB to 2.5 dB, the larger figures applying when the signal sits close to the noise floor.
• Transducer calibration: Antennas, current probes, artificial mains networks, and other transducers carry calibration uncertainties from the calibration laboratory. These include both the laboratory's own measurement capability and the error introduced by interpolating between calibration frequencies.
• Cable losses: The uncertainty in cable attenuation enters the result directly, because the measurement equation adds the cable loss to the receiver reading. Calibration uncertainty, temperature coefficients, and connector repeatability all contribute.
• Preamplifier gain: Where a preamplifier is used, its gain uncertainty and noise figure affect the measurement, particularly at low signal levels.
• Mismatch: Imperfect impedance matching between transducer, cable, and receiver produces an error whose sign depends on the phase of the reflection coefficient. That phase is rarely known, so the component is normally treated as a U-shaped distribution.

Instrumentation uncertainties are usually evaluated by Type B methods, drawing on calibration certificates, manufacturer specifications, and reasoned judgment rather than on 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 of arrival. Site validation measurements quantify this contribution but carry uncertainty of their own. Site imperfection commonly contributes between about 1.0 dB and 4.0 dB, depending on chamber quality and frequency, and it is frequently the dominant term in a radiated emissions budget.
• Ambient conditions: Background electromagnetic noise limits low-level measurements. Temperature and humidity affect absorber performance and the behavior of the equipment under test.
• Ground plane effects: For measurements above a ground plane, the surface quality, size, and flatness of the plane affect the result. Ground plane uncertainty matters most for radiated emissions from floor-standing equipment.
• Distance factors: Uncertainty in the measurement distance affects the calculated field strength. Because field strength falls off inversely with distance in the far field, an error of 3 percent in the distance produces 20 log₁₀(1.03) ≈ 0.26 dB of error in the reported field strength.

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 arrangements yield different results for identical products, which is why test standards prescribe cable routing in detail.
• EUT operation: The operating mode of the equipment affects emissions. Variability in reaching an identical operating state contributes uncertainty.
• Antenna positioning: Uncertainty in antenna height, polarization angle, and distance from the equipment under test all propagate into the field measurement.
• Operator variation: Different operators interpret procedures differently, leading to setup variations. Detailed procedures reduce this contribution but do not eliminate it.

Setup uncertainties may be evaluated by repeated measurements with deliberate variations, which is a Type A evaluation, or estimated from experience and analysis, which is a Type B evaluation. The choice of method does not change how the resulting component is combined.

EUT-Related Uncertainty

The equipment under test itself contributes uncertainty:

• Directivity: Most equipment radiates differently in different directions. Unless the maximum emission is found through complete azimuthal and height scanning, directivity introduces uncertainty.
• Operating mode effects: Emissions vary with operating condition even within a single specified test mode.
• Warm-up and stability: Equipment behavior changes during warm-up and may drift over a long measurement.
• Unit-to-unit variation: Production units of the same product emit differently. Where compliance of the population is at issue, this variation belongs in the assessment, though it is properly a question of product statistics rather than of measurement instrumentation uncertainty.

The Law of Propagation of Uncertainty

For a result y that depends on N input quantities x₁, x₂, …, x_N through a functional relationship y = f(x₁, x₂, …, x_N), the combined standard uncertainty u_c(y) follows from a first-order Taylor expansion of f about the input estimates:

u_c²(y) = ∑_i=1^N (∂f/∂x_i)² u²(x_i) + 2 ∑_i=1^N−1 ∑_j=i+1^N (∂f/∂x_i)(∂f/∂x_j) u(x_i, x_j)

The partial derivatives are the sensitivity coefficients, each stating how much the output changes per unit change in the corresponding input. The second term contains the estimated covariances u(x_i, x_j) and accounts for correlation between inputs.

When the inputs are uncorrelated, every covariance is zero, the second term vanishes, and the expression reduces to the root-sum-of-squares form:

u_c(y) = √(∑_i=1^N c_i² u²(x_i))

where c_i = ∂f/∂x_i is the sensitivity coefficient for input x_i. Two conditions are easily confused and deserve separation. The covariance term drops out because the inputs are uncorrelated, not because the model is linear. Linearity governs a different question: whether the first-order expansion is adequate at all. Where f is strongly nonlinear across the range spanned by the input uncertainties, higher-order terms matter and the linearized formula misstates u_c(y).

Sensitivity Coefficients

Sensitivity coefficients quantify how the output uncertainty depends on each input. They may be found analytically by differentiating the measurement equation, numerically by evaluating the change in output for a small change in input, or experimentally by varying an input deliberately and observing the result.

In EMC work the coefficients are often unity, because the measurement equation is a sum of logarithmic quantities. A radiated emissions level, for example, is:

E (dBµV/m) = V_r (dBµV) + AF (dB/m) + a_c (dB)

where V_r is the receiver voltage reading, AF is the antenna factor, and a_c is the attenuation of the connection between the antenna and the receiver. Each coefficient equals 1, so an uncertainty of 1 dB in any term contributes 1 dB to the result. Coefficients depart from unity when the model mixes linear and logarithmic quantities, or when a derived quantity such as power density is calculated from field strength.

Correlation Effects

Correlation between input quantities changes the combined uncertainty, and the direction of the change follows the sign of the product c_ic_jr(x_i, x_j). Where the sensitivity coefficients share a sign, positive correlation raises the combined uncertainty above the root-sum-of-squares value and negative correlation lowers it.

In EMC, correlation arises from:

• Common calibration sources: Where several transducers were calibrated against the same reference, the uncertainty of that reference is common to all of them.
• Shared environmental effects: Temperature acts on several components in the same direction, creating correlation between their errors.
• Shared corrections: Where one cable-loss or site-attenuation figure is applied to several readings, the uncertainty of that single figure is common to every reading it touches.

Where correlation is suspected but not quantified, a laboratory may add the suspect components linearly, which assumes full positive correlation and yields a deliberately conservative figure, or combine them in quadrature where partial cancellation is judged likely. The assumption should be recorded in the budget, because the two treatments can differ by several decibels.

Monte Carlo Propagation

Where the measurement equation is strongly nonlinear, or where an input distribution is far from normal, the linearized law of propagation becomes unreliable. Supplement 1 to the GUM, published as JCGM 101:2008, sets out a Monte Carlo method that propagates the input distributions themselves rather than first-order sensitivity coefficients. The procedure is:

1. Assign a probability distribution to each input quantity.
2. Draw a random sample from each input distribution.
3. Evaluate the measurement equation for each set of samples.
4. Summarize the resulting output distribution as a best estimate, a standard uncertainty, and a coverage interval.

Trial counts on the order of one million are a common starting point, because a coverage interval must be estimated from the tails of the output distribution and therefore converges more slowly than the mean. Monte Carlo propagation is especially useful when the measurement equation contains limits, thresholds, or switching logic that makes the output discontinuous, and it handles asymmetric or bounded output distributions without forcing them into a normal approximation.

Type A Evaluation

Type A evaluation derives the standard uncertainty from repeated observations. For n independent observations of a quantity made under the same conditions, the best estimate is the arithmetic mean, and the standard uncertainty of that mean is:

u = s / √n

where s is the experimental standard deviation of the individual observations. The division by √n is what converts the scatter of single readings into the uncertainty of their average.

Type A evaluation suits quantities that can be measured repeatedly under nominally identical conditions. In EMC, repeatability studies across multiple setups, operators, or measurement runs supply exactly this kind of data, and they are the only practical way to quantify setup and operator contributions.

The evaluation carries n − 1 degrees of freedom. A small sample therefore leaves the uncertainty itself poorly determined, and the penalty appears in the coverage factor: ten observations give nine degrees of freedom and a 95 percent factor of 2.26 rather than 1.960, while three observations give two degrees of freedom and a factor of 4.30. The Welch-Satterthwaite formula carries this effect through to the combined uncertainty.

Type B Evaluation

Type B evaluation derives the standard uncertainty from any source other than a statistical analysis of repeated observations: calibration certificates, manufacturer specifications, physical limits, published data, or reasoned judgment.

Common Type B sources include:

• Calibration certificates: These supply the uncertainty of a calibrated value, normally reported as an expanded uncertainty with its coverage factor.
• Manufacturer specifications: These appear as tolerances, accuracy classes, or maximum errors, and must be converted to standard uncertainties before use.
• Reference data: Published values for physical constants, material properties, and correction factors carry stated uncertainties.
• Experience and judgment: Where quantitative data are unavailable, an experienced practitioner can estimate bounds, which the budget should record as such.

Converting Type B data to a standard uncertainty requires an assumption about the underlying distribution:

• Normal distribution: Where the source quotes an expanded uncertainty U with coverage factor k, as a calibration certificate does, the standard uncertainty is u = U/k. A certificate quoting k = 2 therefore yields u = U/2.
• Rectangular distribution: Where only the bounds ±a are known and no value within them is more likely than any other, the variance is a²/3, so u = a/√3 ≈ 0.577a.
• Triangular distribution: Where values near the midpoint are more likely than values near the bounds ±a, the variance is a²/6, so u = a/√6 ≈ 0.408a.
• U-shaped distribution: Where values near the bounds are the most likely, as with mismatch error whose reflection phase is unknown, the variance is a²/2, so u = a/√2 ≈ 0.707a.

The divisor is not a formality. Treating a ±2 dB specification as a 2 dB standard uncertainty overstates that component by a factor of 1.73, and repeating the mistake across a budget can inflate the reported expanded uncertainty by more than a decibel.

Uncertainty Budget

An uncertainty budget tabulates every contributing component, showing:

• The quantity measured or influencing the measurement
• The value or estimate of that quantity
• The standard uncertainty of the quantity
• The probability distribution assumed in evaluating it
• The sensitivity coefficient
• The contribution to the combined uncertainty, equal to the sensitivity coefficient times the standard uncertainty

The budget provides transparency and traceability for the combined uncertainty. It also identifies the dominant contributors, which is what directs effort toward better equipment, tighter procedures, or improved environmental control.

An illustrative budget for a radiated emissions measurement might read as follows. The figures in the third column are standard uncertainties; the distribution column records the assumption already used to derive them, not a further division still to be applied.

Because every sensitivity coefficient is unity and the components are treated as uncorrelated, the combined standard uncertainty is the root sum of squares:

u_c(y) = √(0.8² + 1.2² + 0.3² + 2.0² + 0.5² + 1.0²) = √7.42 = 2.7 dB

The budget makes the priorities plain. Site imperfections account for 4.00 of the 7.42 total variance, or 54 percent, and the antenna factor for a further 19 percent. Cable loss accounts for 1 percent; replacing the cable with a better characterized one would move the combined uncertainty by roughly 0.02 dB. Because components add in quadrature, halving the largest contributor is worth more than eliminating the three smallest ones together.

Coverage Factor Definition

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

U = k u_c(y)

The coverage factor sets the width of the interval and therefore the level of confidence attached to it. For a normal distribution, the GUM tabulates the following pairs:

• k = 1 corresponds to 68.27 percent coverage
• k = 1.645 corresponds to 90 percent coverage
• k = 1.960 corresponds to 95 percent coverage
• k = 2 corresponds to 95.45 percent coverage
• k = 2.576 corresponds to 99 percent coverage
• k = 3 corresponds to 99.73 percent coverage

In EMC testing, k = 2 is the usual choice, and it is conventionally described as giving approximately 95 percent coverage. The qualifier matters. A coverage factor of 2 encloses 95.45 percent of a normal distribution, not 95 percent; the factor that encloses exactly 95 percent is 1.960. The convention is deliberate, since a round factor can be applied uniformly across laboratories, but a report should state the coverage factor used rather than the level of confidence alone. The two are not interchangeable, and they diverge sharply once the effective degrees of freedom are small.

Effective Degrees of Freedom

Where components with different numbers of degrees of freedom are combined, the effective degrees of freedom of the combined uncertainty follow from the Welch-Satterthwaite formula:

ν_eff = u_c⁴(y) / ∑_i=1^N (c_i⁴ u⁴(x_i) / ν_i)

where ν_i is the number of degrees of freedom of component i. The fourth powers throughout are not a typographical flourish. Writing squares in their place is a common slip, and it inflates ν_eff, which in turn understates the coverage factor and the reported uncertainty.

Type B components are commonly assigned infinite degrees of freedom, which expresses the assumption that the stated uncertainty is itself well known. A Type A component drawn from a small sample has few degrees of freedom, and because it enters the denominator divided by ν_i, it can dominate ν_eff even when it is not the largest uncertainty component.

When ν_eff is finite, the coverage factor for a stated level of confidence comes from the t-distribution rather than the normal distribution. For ν_eff = 10 and 95 percent coverage, k = 2.23 rather than 1.960; for ν_eff = 4 it is 2.78. As ν_eff grows, k approaches the normal value, and by a few dozen degrees of freedom the difference is negligible in practice.

Confidence Levels

The level of confidence expresses the probability that the value of the measurand lies within the expanded uncertainty interval, on the assumption that the uncertainty has been evaluated correctly and that no significant effect has been overlooked. Levels in common EMC use are:

• Approximately 95 percent (k = 2): The standard for most EMC measurements, and the basis on which CISPR tabulates its reference values.
• 99 percent (k = 2.576) or 99.73 percent (k = 3): Used where higher confidence is required, such as for safety-related equipment.
• 90 percent (k = 1.645): Occasionally used for screening or pre-compliance work, where a wider tolerance for error is acceptable.

The choice involves a trade-off. Higher confidence produces a wider interval, which erodes the apparent compliance margin and increases the chance of rejecting conforming product. Lower confidence narrows the interval and increases the chance of passing product that does not conform. Neither risk disappears; the decision rule determines which party carries it.

GUM Requirements

The GUM, published as JCGM 100:2008 and as ISO/IEC Guide 98-3:2008, requires that a complete uncertainty statement include:

• A clear description of the measurand
• The measurement result, corrected for all recognized significant systematic effects
• The expanded uncertainty U
• The coverage factor k used to obtain it
• The level of confidence associated with that coverage factor

Applied to the budget above, whose combined standard uncertainty was 2.7 dB, the expanded uncertainty at k = 2 is 5.4 dB, and the statement reads: "The radiated electric field strength at 100 MHz is 32.5 dBµV/m with an expanded uncertainty of 5.4 dB (k = 2, level of confidence approximately 95 percent)."

CISPR Requirements

CISPR 16-4-2 governs the treatment of measurement instrumentation uncertainty in radio-disturbance measurements. Measurement instrumentation uncertainty, abbreviated MIU, covers the dispersion induced by the measuring system itself, and the standard both publishes reference values for it and prescribes how it enters a compliance decision.

CISPR 16-4-2 distinguishes two quantities:

• U_cispr: A reference expanded uncertainty at k = 2 that CISPR has evaluated and tabulated for each standardized measurement method. In the 2011 edition, U_cispr is 6.3 dB for radiated disturbance field strength from 30 MHz to 1000 MHz measured at an open-area test site or in a semi-anechoic chamber, 5.3 dB for the same measurement in a fully anechoic room, 4.5 dB for disturbance power from 30 MHz to 300 MHz, and 3.4 dB for conducted disturbance voltage at a mains port from 150 kHz to 30 MHz using an artificial mains network. These figures have been revised between editions, so a laboratory should read them from the edition in force rather than from memory.
• U_lab: The expanded MIU at k = 2 that a particular laboratory evaluates for its own measurement system, following the GUM, for the specific method in use.

The decision rule builds the reference uncertainty into the limit and penalizes only laboratories whose uncertainty exceeds it. If U_lab is less than or equal to U_cispr, the measured disturbance is compared directly against the limit with no uncertainty correction. If U_lab exceeds U_cispr, the measured level increased by (U_lab − U_cispr) must not exceed the limit, so the less capable laboratory bears the consequence of its larger uncertainty. The arrangement keeps published limits stable across laboratories while preserving an incentive to maintain measurement capability.

Laboratory Accreditation

Accredited testing laboratories must demonstrate competence in uncertainty evaluation as part of ISO/IEC 17025:2017. Accreditation bodies assess:

• Identification of uncertainty sources
• Appropriate evaluation methods for each source
• Correct combination and reporting
• Metrological traceability of the calibrations the budget relies on
• Competence of personnel in uncertainty evaluation

Interlaboratory comparison programs provide external evidence for uncertainty claims. Proficiency test results that fall within the stated uncertainties support the laboratory's evaluation; results that fall outside them point either to an underestimated component or to an effect the budget has missed altogether.

Compliance Assessment

How measurement uncertainty enters a compliance decision varies by regulatory authority and application. Common approaches include:

• Simple acceptance: The measured value is compared directly with the limit. This is appropriate where the standard has already built an allowance for uncertainty into the limit, as CISPR 16-4-2 does through U_cispr.
• Guard band: A margin, often the expanded uncertainty, is subtracted from the limit before comparison. This gives high confidence in a pass but tightens the effective requirement.
• Shared risk: The acceptance limit is adjusted by a fraction of the uncertainty, dividing the risk between supplier and customer.
• Probabilistic assessment: The result and its uncertainty define a distribution, from which the probability of conformity is calculated directly.

The combination of measurement result, uncertainty, and acceptance limit is formalized as a decision rule, which fixes how the risks of false acceptance and false rejection are shared. ILAC-G8 and JCGM 106 give guidance on selecting and documenting one. Under ISO/IEC 17025:2017, a laboratory that issues a statement of conformity must have the decision rule agreed, apply it consistently, and record it in the test report.

Frequency-Dependent Uncertainty

EMC measurement uncertainty varies with frequency, because antennas, site performance, cables, and mismatch all behave differently across a band. A complete analysis evaluates uncertainty across the frequency range of interest rather than quoting one figure for the whole span.

Typical frequency dependence includes:

• Antenna calibration uncertainty rising at the edges of an antenna's band
• Site imperfection effects varying with frequency and with polarization
• Mismatch effects following the frequency variation of the impedances involved
• Cable loss and its uncertainty rising with frequency

Uncertainty may be reported at specific frequencies, as a single worst-case value across a band, or as a frequency-dependent function. CISPR takes the worst-case approach for its reference values, assigning one figure to each method and band.

Measurement in Reverberation Chambers

Reverberation chambers raise distinct uncertainty questions, because the field is statistical by design. The field strength at any point follows a probability distribution, and a measurement is a statistic computed over a set of stirrer positions rather than a single reading.

Key uncertainty components include:

• Stirrer effectiveness: Insufficient stirring leaves the field statistics short of the assumed ideal, biasing the result.
• Number of independent samples: The uncertainty of the estimated mean falls with the number of independent stirrer positions, and independence, not merely the count of positions, is what matters.
• Loading effects: The equipment under test absorbs energy and alters both the chamber quality factor and its field statistics.
• Lowest usable frequency: Field uniformity degrades toward the bottom of the band. The lowest usable frequency is conventionally placed at three to six times the chamber's lowest resonant frequency, the exact multiple depending on chamber size, loading, and stirring arrangement.

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

Uncertainty in Simulation

Computational electromagnetic results carry uncertainty of their own, arising from:

• Model fidelity: The model simplifies the physical system, and the simplification is rarely quantified.
• Input parameter uncertainty: Material properties, dimensions, and source characteristics are themselves uncertain.
• Numerical uncertainty: Mesh discretization, convergence criteria, and finite numerical precision all affect the result.
• Solver applicability: A solver accurate for one class of problem may be poorly suited to another.

Propagating input uncertainties through a simulation calls for sensitivity analysis or Monte Carlo methods, exactly as for a physical measurement. Model uncertainty is harder to quantify, and validation against measurement remains the practical way to bound it.

Equipment Improvements

Better instrumentation reduces contributions from:

• Receivers with tighter amplitude accuracy and lower noise floors
• Antennas calibrated to lower uncertainty, and calibrated at more frequencies
• Lower-loss cables with well characterized attenuation
• Preamplifiers with more stable gain and lower noise figure
• Better matched components, or attenuator pads, which reduce the mismatch term at the cost of sensitivity

Whether an upgrade repays its cost depends on whether the targeted component is a dominant contributor. Improving a term that supplies 1 percent of the variance yields no measurable benefit.

Environmental Improvements

Test environment quality bears directly on several components:

• Better absorber reduces site imperfection uncertainty, usually the largest single term
• Improved shielding lowers the ambient noise floor and the noise floor correction
• Climate control reduces temperature-driven variation in absorber and equipment behavior
• Ground plane flatness and continuity affect measurements made above the plane

Procedural Improvements

Better procedures reduce setup and operator contributions:

• Detailed written procedures reduce operator-to-operator variation
• Automated positioning reduces uncertainty in antenna height, distance, and turntable angle
• Fixed, documented cable configurations improve repeatability between runs
• Finer spatial and azimuthal scanning finds the true maximum emission more reliably

Statistical Improvements

Additional sampling reduces Type A components:

• More repeated measurements reduce the standard uncertainty of the mean
• More independent stirrer positions improve field estimates in a reverberation chamber
• Repeating the full setup, not only the reading, captures setup-dependent variation

The returns diminish quickly, because the standard uncertainty of a mean falls only as 1/√n: halving it requires four times as many observations, and quartering it sixteen times as many. The useful sample size is the one that brings the Type A term below the dominant Type B terms, beyond which further repetition buys nothing.