Electronics Guide

Normal (Gaussian) Distribution

The normal distribution is the most commonly encountered distribution in EMC work. Many physical quantities that result from the sum of numerous independent factors tend toward normal distributions by the Central Limit Theorem. EMC emissions and immunity levels, when expressed in logarithmic units (dB), often follow approximately normal distributions.

The normal distribution is characterized by two parameters: the mean (mu) representing the center of the distribution, and the standard deviation (sigma) representing its spread. For a normally distributed variable:

• Approximately 68% of values fall within one standard deviation of the mean
• Approximately 95% of values fall within two standard deviations of the mean
• Approximately 99.7% of values fall within three standard deviations of the mean

When analyzing EMC data, it is important to verify that the assumption of normality is reasonable. Graphical methods such as normal probability plots (Q-Q plots) or formal statistical tests like the Shapiro-Wilk test can assess normality. Data that appear non-normal in linear units may become approximately normal when transformed to logarithmic (dB) scale.

Log-Normal Distribution

When the logarithm of a variable follows a normal distribution, the variable itself follows a log-normal distribution. This is common in EMC because many electromagnetic quantities multiply together (coupling factors, attenuation, gain) rather than adding. When multiplied quantities are converted to decibels, they add, often producing normally distributed dB values and hence log-normally distributed linear values.

Log-normal distributions are inherently asymmetric with a long tail toward higher values. This characteristic is physically meaningful in EMC: while emissions levels cluster around typical values, occasional units may exhibit significantly higher emissions due to resonances, manufacturing variations, or other factors that multiplicatively combine.

When working with log-normal data, analysis is often performed in the logarithmic domain where normal distribution methods apply, then results are transformed back to linear units if needed.

Rayleigh and Rice Distributions

The Rayleigh distribution arises in EMC when measuring the magnitude of a signal composed of many random phasors, such as the field strength in a reverberation chamber or multipath propagation environment. When there is no dominant component and all phasors have equal mean amplitude, the magnitude follows a Rayleigh distribution.

The Rice (Rician) distribution generalizes the Rayleigh distribution to include a dominant component. In EMC, Rice distributions apply when measuring fields that include both a direct coupling path and scattered components. The ratio of the dominant component to the scattered component power (the K-factor) determines how closely the distribution resembles Rayleigh (K approaches 0) or approaches a shifted normal distribution (K approaches infinity).

These distributions are particularly important in reverberation chamber testing and in analyzing field uniformity in anechoic chambers with imperfect absorber performance.

Uniform and Triangular Distributions

Uniform distributions apply when a quantity is equally likely to take any value within a defined range. In EMC, uniform distributions often represent manufacturing tolerances when only the tolerance limits are known without additional information about the distribution within those limits.

Triangular distributions provide a simple model when the most likely value and the range are known. They are often used in uncertainty analysis when more detailed distribution information is unavailable but it is reasonable to assume values near the center are more likely than values near the extremes.

Extreme Value Distributions

Extreme value distributions describe the distribution of the maximum (or minimum) of many samples. In EMC, these distributions apply when determining the expected maximum emissions from a large production population or the minimum immunity level across a fleet of products.

The Gumbel distribution (Type I extreme value) is commonly used when the underlying data are normally distributed. If emissions data follow a normal distribution, the maximum emissions from a sample of n units will follow a Gumbel distribution. This has important implications for relating small-sample test results to production population behavior.

Confidence Intervals for the Mean

When estimating the mean of a normally distributed quantity from a sample, the confidence interval depends on the sample mean, sample standard deviation, sample size, and the desired confidence level.

For a sample of size n with mean x-bar and standard deviation s, the 100(1-alpha)% confidence interval for the population mean is:

x-bar +/- t(alpha/2, n-1) * s / sqrt(n)

where t(alpha/2, n-1) is the critical value from the t-distribution with n-1 degrees of freedom. For 95% confidence (alpha = 0.05), this critical value ranges from 12.7 for n=2 to 1.96 as n approaches infinity.

The width of the confidence interval decreases as sample size increases, proportional to 1/sqrt(n). Doubling precision requires quadrupling the sample size. This relationship is crucial when planning EMC test sample sizes: achieving tight confidence intervals with small samples is fundamentally limited.

Confidence Intervals for Percentiles

EMC specifications often concern percentiles rather than means. For example, a requirement might state that 95% of production units must comply with an emissions limit. Confidence intervals for percentiles are more complex than for means and require larger sample sizes for equivalent precision.

For normally distributed data, a confidence interval for a percentile can be constructed using the tolerance interval formulation. The upper 95/95 tolerance limit (95% confidence that 95% of the population is below the limit) is commonly used in EMC:

Upper limit = x-bar + k * s

where k is a tolerance factor that depends on sample size, confidence level, and the proportion to be covered. For 95/95 tolerance with n=10, k is approximately 3.38; for n=30, k is approximately 2.55; and as n approaches infinity, k approaches 1.645 (the 95th percentile of the standard normal distribution plus a small correction for the estimated standard deviation).

The large k-factors required for small samples underscore the value of increasing sample sizes in EMC testing: a ten-sample test requires margins over 3 standard deviations, while a thirty-sample test requires only about 2.5 standard deviations.

Confidence Intervals for Standard Deviation

Quantifying variability is often as important as quantifying central tendency in EMC. The confidence interval for the standard deviation of a normally distributed quantity follows a chi-square distribution.

For a sample of size n with standard deviation s, the 100(1-alpha)% confidence interval for the population standard deviation sigma is:

sqrt[(n-1)*s^2 / chi^2(alpha/2, n-1)] to sqrt[(n-1)*s^2 / chi^2(1-alpha/2, n-1)]

This confidence interval is asymmetric, being wider on the upper side. This asymmetry is physically meaningful: while one can be fairly confident that the true variability is not much less than observed, it could be substantially more.

Confidence intervals for standard deviation require even larger sample sizes than those for the mean. With n=10, the 95% confidence interval spans roughly a factor of 2 (from about 0.7s to 1.5s). Reducing this uncertainty to a factor of 1.5 requires approximately n=30.

Fundamentals of Hypothesis Testing

A hypothesis test begins with a null hypothesis (H0) representing the default assumption, and an alternative hypothesis (H1) representing what we seek to demonstrate. The test calculates the probability of observing data as extreme as the actual observations if the null hypothesis were true. This probability, called the p-value, is compared against a significance level (alpha) to decide whether to reject the null hypothesis.

Two types of errors can occur:

• Type I error (false positive): Rejecting the null hypothesis when it is true. The probability of Type I error is controlled by the significance level alpha, typically set at 0.05.
• Type II error (false negative): Failing to reject the null hypothesis when it is false. The probability of Type II error is denoted beta, and 1-beta is called the power of the test.

In EMC, these errors have practical interpretations. A Type I error might mean incorrectly concluding a product fails when it actually complies. A Type II error might mean approving a non-compliant product. The relative consequences of these errors should inform the choice of significance level and required sample sizes.

One-Sample Tests

One-sample tests compare sample data against a specified value. The one-sample t-test determines whether a sample mean differs significantly from a hypothesized value, such as an EMC limit.

For compliance testing, the hypothesis structure is typically:

• H0: The true mean equals (or exceeds) the limit
• H1: The true mean is below the limit

This formulation places the burden of proof on demonstrating compliance. The sample must show emissions sufficiently below the limit to provide statistical confidence that the population mean is indeed below the limit.

One-sample tests for percentiles are more complex. Non-parametric approaches using order statistics can test whether a specified proportion of the population meets a criterion without distributional assumptions, but require larger sample sizes.

Two-Sample Tests

Two-sample tests compare two groups to determine whether they differ significantly. Applications in EMC include comparing before/after performance of a design change, comparing two measurement methods, or comparing products from different manufacturing lines.

The two-sample t-test compares means from two groups. When sample sizes are equal and variances can be assumed equal, the test is straightforward. When variances may differ, Welch's t-test provides a more robust approach that does not assume equal variances.

The F-test compares variances between two groups, useful when determining whether a design change affected emissions variability as well as emissions level. However, the F-test is sensitive to non-normality and should be supplemented with graphical assessment of the data.

Multiple Comparisons

When performing multiple hypothesis tests simultaneously, the probability of at least one false positive increases. If 20 independent tests are performed at alpha=0.05, the probability of at least one false positive is approximately 64%, far exceeding the nominal 5% level.

Correction methods address this multiple comparison problem:

• Bonferroni correction: Divide the significance level by the number of comparisons. Simple but conservative.
• Holm-Bonferroni method: A step-down procedure that is less conservative than Bonferroni while still controlling the family-wise error rate.
• False Discovery Rate (FDR): Controls the expected proportion of false positives among rejected hypotheses, appropriate when some false positives are acceptable.

In EMC, multiple comparisons arise when testing at multiple frequencies, multiple configurations, or multiple units. The choice of correction method depends on the consequences of false positives and false negatives in the specific application.

Linear Regression

Simple linear regression fits a straight line to data relating a response variable y to a predictor variable x:

y = a + b*x + error

The coefficients a (intercept) and b (slope) are estimated by least squares, minimizing the sum of squared residuals. Key outputs include:

• Coefficient estimates: The fitted values of a and b, with their standard errors and confidence intervals
• R-squared: The proportion of variance in y explained by x, ranging from 0 (no linear relationship) to 1 (perfect linear relationship)
• Residual analysis: Examination of the differences between observed and fitted values to verify model assumptions

In EMC, linear regression in the log-frequency domain often describes emissions roll-off with frequency. A slope of -20 dB/decade indicates capacitive coupling or single-pole filtering behavior, while -40 dB/decade indicates two-pole behavior.

Multiple Regression

Multiple regression extends linear regression to multiple predictor variables:

y = a + b1*x1 + b2*x2 + ... + bk*xk + error

This enables modeling EMC behavior as a function of multiple factors simultaneously. For example, emissions might depend on clock frequency, supply voltage, load current, and temperature. Multiple regression quantifies the individual contribution of each factor while accounting for the others.

Key considerations in multiple regression include:

• Multicollinearity: When predictor variables are correlated with each other, coefficient estimates become unstable. Variance Inflation Factors (VIF) quantify this problem.
• Variable selection: Including unnecessary variables reduces model precision. Stepwise regression, information criteria (AIC, BIC), or regularization methods (LASSO, ridge regression) guide variable selection.
• Interaction effects: The effect of one variable may depend on another. Including interaction terms (x1*x2) captures these dependencies.

Prediction Intervals

Regression models enable prediction of the response for new values of the predictors. Prediction intervals quantify the uncertainty in these predictions, accounting for both the uncertainty in the fitted model and the inherent variability of individual observations.

Prediction intervals are always wider than confidence intervals for the mean response at the same predictor values. The confidence interval addresses uncertainty in the mean; the prediction interval addresses uncertainty in a single observation.

In EMC, prediction intervals are essential when using regression models for compliance assessment. A prediction interval for emissions at an untested frequency must be narrow enough to provide confidence that the limit will not be exceeded, accounting for both model uncertainty and unit-to-unit variability.

One-Way ANOVA

One-way ANOVA compares means across multiple groups defined by a single factor. For example, comparing emissions from three different production lines or four different component suppliers.

The analysis decomposes total variance into between-group and within-group components. The F-statistic is the ratio of between-group variance to within-group variance. A large F-statistic indicates that the groups differ significantly, meaning the factor affects the response.

Following a significant ANOVA result, post-hoc tests (such as Tukey's HSD or Dunnett's test) identify which specific groups differ from each other, with appropriate adjustment for multiple comparisons.

Two-Way ANOVA

Two-way ANOVA examines two factors simultaneously, along with their potential interaction. For example, analyzing how both operating mode and temperature affect immunity levels.

The analysis partitions variance into:

• Main effect of factor A
• Main effect of factor B
• Interaction between A and B
• Residual (unexplained) variance

An interaction effect means that the effect of one factor depends on the level of the other factor. In EMC, interactions are common: the effect of shielding may depend on frequency, or the effect of grounding scheme may depend on operating current. Identifying and quantifying interactions is essential for developing robust designs.

Random and Mixed Effects

ANOVA factors can be fixed or random. Fixed effects have specific levels of interest (e.g., three specific operating modes). Random effects represent samples from a larger population of levels (e.g., units randomly selected from production).

Mixed models include both fixed and random effects. In EMC, mixed models are valuable when analyzing data with hierarchical structure, such as multiple measurements on each of several units, where unit is a random effect and operating condition is a fixed effect.

Variance components analysis extracts the variance attributable to each random factor, providing direct insight into sources of variability. This supports decisions about where to focus design improvements: reducing between-unit variability requires different strategies than reducing within-unit variability.

Factorial Designs

Full factorial designs test all combinations of factor levels. A 2^k factorial tests k factors at two levels each, requiring 2^k experimental runs. These designs estimate all main effects and all interactions without confounding.

For example, a 2^3 factorial with factors A (clock frequency: low/high), B (supply voltage: low/high), and C (load: light/heavy) requires 8 runs and estimates:

• Three main effects (A, B, C)
• Three two-factor interactions (AB, AC, BC)
• One three-factor interaction (ABC)

Fractional factorial designs reduce the number of runs by confounding higher-order interactions with main effects or lower-order interactions. A 2^(k-p) fractional factorial requires only 2^(k-p) runs. The resolution of the design indicates the level of confounding: Resolution III designs confound main effects with two-factor interactions; Resolution IV designs confound two-factor interactions with each other but not with main effects.

Screening Designs

When many factors may potentially affect EMC performance, screening designs efficiently identify the most important factors. Plackett-Burman designs can screen up to k-1 factors in k runs (where k is a multiple of 4), estimating main effects under the assumption that interactions are negligible.

Definitive screening designs offer a more robust alternative, accommodating some two-factor interactions and quadratic effects while remaining economical. These designs are particularly useful in early-stage EMC investigations when the factor-response relationships are unknown.

Response Surface Methodology

Response Surface Methodology (RSM) extends DoE to characterize the relationship between factors and response over a continuous region, not just at discrete levels. RSM designs support fitting polynomial models that describe how the response varies across the factor space.

Central Composite Designs (CCD) and Box-Behnken designs are common RSM approaches. They enable fitting quadratic response surfaces:

y = b0 + sum(bi*xi) + sum(bii*xi^2) + sum(bij*xi*xj) + error

RSM is valuable in EMC for optimization problems: finding the combination of design parameters that minimizes emissions or maximizes immunity margin. The fitted response surface identifies both the optimal settings and the sensitivity of performance to deviations from optimal.

Basic Monte Carlo Method

The basic Monte Carlo approach involves:

1. Define the model relating inputs to outputs
2. Specify probability distributions for each input variable
3. Generate random samples from each input distribution
4. Evaluate the model for each sample set
5. Analyze the resulting output distribution

For EMC problems, the model might be a circuit simulation, an analytical coupling model, or a system-level interference calculation. Inputs include component values, geometry, material properties, and environmental factors. Outputs include emissions levels, immunity margins, or interference probabilities.

The number of Monte Carlo samples determines the precision of the results. For estimating the mean, precision improves as 1/sqrt(N) where N is the number of samples. Estimating tail probabilities (such as the probability of exceeding an emissions limit) requires substantially more samples because these events occur rarely.

Variance Reduction Techniques

Advanced Monte Carlo techniques improve efficiency by reducing the number of samples required for a given precision:

• Latin Hypercube Sampling: Ensures samples are spread evenly across the input space, improving coverage compared to purely random sampling.
• Importance Sampling: Concentrates samples in regions that contribute most to the quantity of interest, particularly useful for rare-event probability estimation.
• Control Variates: Uses correlated auxiliary quantities with known expectations to reduce variance in the quantity of interest.
• Antithetic Variates: Pairs negatively correlated samples to reduce variance.

In EMC, importance sampling is particularly valuable when estimating the probability of exceeding emissions limits, which may occur in only a small fraction of the parameter space.

Sensitivity Analysis

Monte Carlo results support sensitivity analysis identifying which input variables most affect output variability. Several approaches exist:

• Correlation analysis: Examines correlations between inputs and outputs across the Monte Carlo samples.
• Regression analysis: Fits a regression model to the Monte Carlo data, with standardized coefficients indicating relative sensitivity.
• Variance-based sensitivity (Sobol indices): Decomposes output variance into contributions from individual inputs and their interactions.

Sensitivity analysis guides design decisions by identifying which parameters most need to be controlled or optimized. In EMC, this might reveal that emissions are sensitive to PCB trace length but insensitive to component tolerance, directing attention to geometry control rather than component selection.

Bayes' Theorem

Bayes' theorem relates the posterior probability (belief after seeing data) to the prior probability (belief before data) and the likelihood (probability of the data given the parameter):

P(theta|data) proportional to P(data|theta) * P(theta)

In words: Posterior is proportional to Likelihood times Prior.

The prior distribution encodes existing knowledge before testing. In EMC, prior knowledge might come from simulation results, similar products, physics-based bounds, or expert judgment. As test data accumulate, the posterior distribution narrows, reflecting increased knowledge about the parameter.

Prior Selection

Choosing appropriate priors requires balancing informativeness with objectivity:

• Uninformative priors: Express minimal prior knowledge, letting data dominate the analysis. Uniform distributions or Jeffreys priors are common choices.
• Weakly informative priors: Provide soft constraints based on physical reasonableness without strongly influencing conclusions.
• Informative priors: Incorporate specific prior knowledge, such as distributions derived from historical data or physics-based models.

In EMC, physics provides natural constraints: emissions cannot be negative, shielding effectiveness cannot exceed theoretical limits, and coupling cannot occur without a physical path. Priors that encode these constraints improve inference efficiency without introducing bias.

Bayesian Inference in EMC

Bayesian methods offer several advantages for EMC applications:

• Sequential updating: As each test unit is measured, the posterior distribution updates, providing a continuously refined estimate. Testing can stop when the posterior provides sufficient certainty.
• Incorporating prior information: Simulation results, supplier data, or historical measurements can formally contribute to the analysis.
• Direct probability statements: Bayesian results directly express the probability that a parameter lies in a given range, rather than the more convoluted interpretation of frequentist confidence intervals.
• Decision integration: Bayesian decision theory provides a coherent framework for making optimal decisions under uncertainty.

For compliance assessment, Bayesian methods can calculate the probability that a product meets specifications, incorporating all available evidence. This is more directly useful than p-values, which only indicate whether the data are consistent with non-compliance.

Computational Methods

Modern Bayesian analysis relies on computational methods to handle complex models:

• Markov Chain Monte Carlo (MCMC): Generates samples from the posterior distribution through a random walk that converges to the target distribution. Allows Bayesian inference for arbitrarily complex models.
• Hamiltonian Monte Carlo: An advanced MCMC method that uses gradient information for more efficient sampling in high dimensions.
• Variational inference: Approximates the posterior with a simpler distribution, trading accuracy for speed in large-scale problems.

Software packages such as Stan, PyMC, and JAGS make Bayesian analysis accessible for practical EMC applications. The choice of method depends on model complexity, data volume, and computational resources.

Sample Size Determination

Determining appropriate sample sizes is crucial for both testing efficiency and result reliability. Sample size depends on:

• Desired confidence level and precision
• Expected variability in the measured quantity
• Effect size to be detected (for hypothesis tests)
• Cost and time constraints

Power analysis calculates the sample size needed to detect a specified effect with specified power. For comparing two groups with expected difference d and common standard deviation s, detecting the difference with 80% power at alpha=0.05 requires approximately n = 16*(s/d)^2 per group.

For percentile estimation (common in EMC compliance), larger samples are required. Estimating the 95th percentile with 95% confidence that at least 95% of the population is below the limit requires a minimum of 59 samples if all pass, with the formula n >= ln(1-confidence)/ln(percentile).

Software Tools

Statistical analysis for EMC applications can be performed using various software platforms:

• General-purpose statistics packages: R, Python (SciPy, StatsModels), MATLAB Statistics Toolbox, Minitab, JMP
• Monte Carlo simulation: Crystal Ball, @Risk, Python (NumPy, SciPy), MATLAB
• Bayesian analysis: Stan, PyMC, JAGS, WinBUGS
• Design of experiments: JMP, Minitab, Design-Expert

Many EMC test equipment manufacturers also provide built-in statistical analysis capabilities. However, understanding the underlying statistical methods remains essential for proper application and interpretation.

Documentation and Reporting

Statistical EMC analyses should be thoroughly documented, including:

• Clear statement of the question addressed and hypotheses tested
• Description of the data, including sample size, measurement conditions, and any data transformations
• Justification for statistical methods chosen
• Verification that method assumptions are satisfied
• Complete results including point estimates, confidence intervals, and p-values where applicable
• Practical interpretation in EMC terms, not just statistical significance

ISO/IEC Guide 98-3 (GUM) and CISPR 16-4-2 provide guidance on reporting uncertainty in EMC measurements. Adhering to these standards ensures results are properly understood and can be compared across laboratories.