Electronics Guide

Cavity Resonant Modes

For a rectangular cavity with dimensions a, b, and c (where a is greater than or equal to b is greater than or equal to c), the resonant frequencies of the TE and TM modes are given by:

f-mnp = (c0/2) * sqrt((m/a)^2 + (n/b)^2 + (p/c)^2)

where c0 is the speed of light, and m, n, p are non-negative integers representing the mode indices. For TE modes, at least two indices must be non-zero, while for TM modes, both m and n must be non-zero but p can be zero.

Each mode has a unique field distribution within the cavity. At resonance, the fields form standing wave patterns with nulls and maxima at specific locations. The spacing between modes in frequency is not uniform; it tends to be larger at lower frequencies and decreases as frequency increases.

Mode Density Calculation

The number of modes below a given frequency f in a cavity of volume V can be approximated by Weyl's formula:

N(f) = (8 * pi * V * f^3) / (3 * c0^3)

Taking the derivative with respect to frequency gives the mode density:

dN/df = (8 * pi * V * f^2) / c0^3

This shows that mode density increases with the square of frequency. At low frequencies, modes are sparse and well-separated. As frequency increases, the number of modes per unit frequency bandwidth grows rapidly, eventually producing a quasi-continuous distribution.

For practical chambers, the mode density determines the lowest usable frequency (LUF). A commonly used criterion requires at least 60 modes per chamber Q bandwidth, or alternatively, three times the first mode resonance frequency. Below the LUF, the statistical assumptions underlying reverberation chamber theory break down, and field uniformity cannot be achieved.

Modal Overlap Factor

Each resonant mode has a finite bandwidth determined by losses in the cavity. The modal bandwidth is related to the Q factor:

delta-f = f / Q

The modal overlap factor (MOF) is the ratio of modal bandwidth to average mode spacing:

MOF = (mode density) * (modal bandwidth) = (8 * pi * V * f^3) / (3 * c0^3 * Q)

When MOF is much greater than 1, modes overlap significantly and the chamber operates in the overmoded regime. This is the desired condition for reverberation chamber operation. When MOF is close to or less than 1, individual modes can be resolved and statistical field uniformity is not achieved.

Lowest Usable Frequency

The lowest usable frequency (LUF) is arguably the most important parameter in reverberation chamber specifications. Several methods exist for determining LUF:

• Mode counting method: Requires a minimum number of modes (typically 60-100) within the chamber bandwidth
• First mode multiple: LUF equals 3 to 6 times the frequency of the first resonant mode
• Statistical uniformity criterion: LUF is the frequency above which field uniformity meets specified requirements
• Mode bandwidth criterion: LUF is where at least 1.5 modes overlap within the 3-dB bandwidth

For a typical chamber of dimensions 3m x 2.5m x 2m, the first resonant mode occurs around 72 MHz. Using the three-times rule, the LUF would be approximately 216 MHz. In practice, LUF is often determined empirically through field uniformity measurements.

Ideal Chamber Assumptions

The statistical theory of reverberation chambers rests on several idealized assumptions:

• Uniform mode excitation: All modes within the measurement bandwidth are excited equally
• Random phase: The phases of individual mode contributions are uniformly distributed and statistically independent
• Sufficient stirring: The stirrer creates enough statistically independent field configurations
• Negligible direct coupling: Energy transfer between transmit and receive antennas occurs only through the reverberant field

When these conditions are satisfied, the field components at any location can be modeled as the sum of many random contributions, leading to specific statistical distributions through the central limit theorem.

Rectangular Component Distribution

Under ideal conditions, each rectangular component of the electric field (Ex, Ey, Ez) at any fixed location follows a zero-mean Gaussian distribution:

p(E-rect) = (1 / sqrt(2*pi*sigma^2)) * exp(-E-rect^2 / (2*sigma^2))

The real and imaginary parts are independent and identically distributed. This applies equally to the magnetic field components. The variance sigma^2 is the same for all six rectangular field components (Ex, Ey, Ez, Hx, Hy, Hz), which is a consequence of isotropy in an ideal chamber.

Field Magnitude Distributions

The magnitude of each field component follows a Rayleigh distribution:

p(|E-component|) = (|E| / sigma^2) * exp(-|E|^2 / (2*sigma^2))

The total electric field magnitude |E| = sqrt(Ex^2 + Ey^2 + Ez^2) follows a chi distribution with six degrees of freedom. The power density (proportional to |E|^2) follows an exponential distribution for each component and a chi-squared distribution with six degrees of freedom for the total field.

These distributions have practical implications:

• The average power is three times the average power of any single component
• Peak-to-average ratios can be calculated to determine test margins
• Confidence intervals can be established for field level estimates

Statistical Uniformity

Spatial uniformity in a reverberation chamber is described statistically rather than deterministically. For each stirrer position, the field varies significantly with location. However, when averaged over many stirrer positions, the statistical properties of the field become uniform throughout the working volume.

Field uniformity is typically characterized by the standard deviation of maximum field values measured at several locations. IEC 61000-4-21 specifies that this standard deviation should not exceed 3 dB above 400 MHz and allows larger values at lower frequencies approaching the LUF.

The working volume is defined as the region where uniformity requirements are met. This is typically 0.5 to 1 meter away from chamber walls and stirrers to avoid near-field effects and boundary layer phenomena.

Isotropy

Isotropy means the statistical field properties are independent of direction. In an ideal chamber:

• The average power in each polarization is equal
• The field arrives equally from all directions when averaged over stirrer positions
• Equipment orientation does not affect the average received power

This is a significant advantage over anechoic testing, where equipment must be rotated and the antenna repositioned to test all angles of incidence. In a reverberation chamber, isotropy ensures thorough exposure regardless of equipment orientation.

Definition and Physical Meaning

The Q factor is defined as:

Q = 2*pi * (energy stored in chamber) / (energy dissipated per cycle)

Equivalently:

Q = omega * U / P-loss = 2*pi*f * U / P-loss

where U is the total stored electromagnetic energy and P-loss is the power dissipated. A high Q means energy is stored efficiently with low losses, resulting in higher field levels for a given input power.

Loss Mechanisms

Energy losses in a reverberation chamber occur through several mechanisms:

Wall losses: The metallic walls have finite conductivity, causing currents to flow that dissipate energy. The wall Q is:

Q-wall = (3*V) / (2*S*delta)

where V is chamber volume, S is surface area, and delta is the skin depth. Wall Q increases with chamber size and decreases with frequency (due to decreasing skin depth).

Aperture leakage: Energy escapes through imperfect door seals, penetrations, and ventilation openings. This is characterized by a leakage Q:

Q-aperture = (4*pi*V) / (lambda * A-effective)

where A-effective is the effective aperture area. Good shielding is essential to minimize this loss.

Antenna losses: The transmit and receive antennas extract energy from the chamber. For an antenna with efficiency eta:

Q-antenna = (16*pi^2*V) / (eta * lambda^3)

Object loading: Equipment under test and other objects absorb energy, reducing Q. This is often the dominant loss mechanism during testing.

Composite Q Factor

The overall chamber Q combines all loss mechanisms:

1/Q-total = 1/Q-wall + 1/Q-aperture + 1/Q-antenna + 1/Q-load

The lowest individual Q dominates the composite value. For an empty chamber, wall losses often dominate at lower frequencies while aperture and antenna losses become more significant at higher frequencies.

Typical Q values range from 1,000 to 100,000 depending on chamber size, frequency, and loading. Q measurements provide important diagnostic information about chamber performance.

Field Level and Power Relationship

The average power density in a reverberation chamber is related to the input power and Q by:

S-average = P-input * Q * lambda / (2*pi*V)

The average electric field magnitude is:

E-average = sqrt(Q * P-input * eta0 * lambda / (2*pi*V))

where eta0 = 377 ohms is the free-space wave impedance. These relationships show that higher Q and smaller chambers produce higher field levels for the same input power. However, Q typically increases with chamber size (due to lower wall losses), partially offsetting the volume effect.

Time Constant and Settling Time

The chamber time constant is related to Q:

tau = Q / (pi*f)

This determines how quickly the field builds up after excitation begins and how long it takes to decay after excitation stops. For swept-frequency or stepped-frequency measurements, the dwell time at each frequency must be several time constants to allow the field to reach steady state.

At 1 GHz with Q = 10,000, the time constant is about 3 microseconds. For pulsed immunity testing, pulse widths must be long compared to the time constant to achieve full field levels.

Purpose of Stirring

Stirring serves several critical functions:

• Mode mixing: Changes the relative amplitudes and phases of the excited modes
• Statistical sampling: Creates independent field configurations for averaging
• Uniformity improvement: Reduces spatial variations when averaged over stirrer positions
• Polarization randomization: Ensures all polarizations are represented in the averaged field

Without stirring, the field would exhibit fixed standing wave patterns with nulls and maxima at specific locations, making the chamber unsuitable for standardized testing.

Mechanical Stirring

Mechanical stirring uses rotating metal paddles or tuners to change the boundary conditions. As the stirrer rotates, it scatters the electromagnetic field in different directions, creating a time-varying field configuration.

Effective mechanical stirrers have several characteristics:

• Size: Should be electrically large (at least one wavelength) at the lowest frequency of operation
• Asymmetry: Non-symmetric designs provide better mode mixing than symmetric paddles
• Multiple axes: Stirrers rotating about different axes improve uniformity
• Continuous variation: The boundary conditions should change continuously throughout rotation

Stirrer designs include flat paddles, Z-fold configurations, asymmetric irregular shapes, and corrugated surfaces. Multiple stirrers are often used in large chambers.

Frequency Stirring

Frequency stirring (or electronic stirring) varies the excitation frequency over a band to create effectively different mode configurations. Because each mode has a slightly different resonant frequency, changing frequency changes which modes are excited and their relative phases.

Frequency stirring bandwidth must be large enough to span several mode spacings but small enough that the device under test response is approximately constant. Typical bandwidths range from 1% to 10% of the center frequency.

Frequency stirring can be combined with mechanical stirring to increase the number of independent samples. This is particularly useful at lower frequencies where mode density is limited.

Source Stirring

Source stirring uses multiple transmit antennas at different locations, exciting the chamber sequentially or simultaneously with controlled phase relationships. Each source position creates a different mode distribution, providing additional independent samples.

Source position stirring is effective because the coupling coefficients from the source to each mode depend strongly on source location. Moving the source can therefore create significantly different mode configurations.

Number of Independent Samples

The number of independent samples available from stirring determines the measurement uncertainty. Independence is assessed through the correlation between samples:

rho(delta-theta) = mean(E(theta) * E(theta + delta-theta)) / mean(E(theta)^2)

Samples are considered independent when their correlation is below a threshold (typically 0.37 or 1/e). The number of independent samples N-independent affects the standard deviation of power estimates:

sigma-estimate = sigma-true / sqrt(N-independent)

For a tuner with 360 positions, the number of independent samples might range from 10 to 200 depending on frequency and stirrer design. Higher frequencies and better stirrers yield more independent samples.

Absorption Cross Section

The absorption cross section quantifies how much power an object absorbs from the reverberant field:

P-absorbed = sigma-a * S-incident

where sigma-a is the absorption cross section (in square meters) and S-incident is the power density. The absorption cross section depends on object size, shape, material properties, and frequency.

Absorption cross section can be measured by comparing chamber Q with and without the object:

sigma-a = (2*pi*V / lambda) * (1/Q-loaded - 1/Q-empty)

This parameter is useful for characterizing test objects and predicting their effect on chamber performance.

Loading Factor

The loading factor describes the relative power absorbed by objects compared to other losses:

LF = Q-empty / Q-loaded

A loading factor of 1 indicates negligible loading (object absorbs little power). Loading factors of 2 or more indicate significant absorption that must be accounted for in field calculations.

Standards typically specify maximum allowable loading. IEC 61000-4-21 requires loading not to exceed 1.5 dB change in chamber insertion loss, corresponding to a loading factor of about 1.4.

Effects on Field Statistics

Moderate loading generally does not significantly affect field statistics or uniformity, though it does reduce field levels. However, heavy loading can:

• Reduce Q below critical values needed for overmoded operation
• Create local field perturbations near the absorbing object
• Affect the number of independent stirrer samples
• Introduce direct coupling paths between antennas

Large equipment under test must be positioned carefully to maintain working volume uniformity and avoid blocking the stirrer or transmit antenna.

Loading Compensation

When Q changes due to loading, the input power must be adjusted to maintain the desired field level:

P-loaded = P-empty * (Q-empty / Q-loaded)

Some test standards require measuring chamber loss at each test frequency and adjusting power accordingly. This ensures consistent field exposure regardless of loading variations.

Calibration Objectives

Chamber calibration serves to:

• Determine the power required to achieve specified field levels
• Verify statistical uniformity throughout the working volume
• Characterize the lowest usable frequency
• Assess stirrer effectiveness and number of independent samples
• Establish measurement uncertainty budgets

Field Uniformity Verification

Field uniformity is typically verified by measuring received power at multiple locations using a reference antenna. The IEC 61000-4-21 procedure specifies:

• Measurements at 8 locations on a volume enclosing the working area
• Three orthogonal antenna orientations at each location
• Full stirrer rotation at each position
• Recording maximum received power for each configuration

The standard deviation of the 24 maximum values (8 positions times 3 orientations) characterizes uniformity. Above 400 MHz, this should not exceed 3 dB.

Chamber Calibration Factor

The chamber calibration factor (CCF) relates input power to average electric field:

CCF = E^2 / P-input (V^2 per meter^2 per watt)

CCF is determined by measuring received power with a reference antenna of known efficiency and applying the appropriate statistical factors. The CCF allows test engineers to set the power level needed to achieve a specified field strength.

Reference Antenna Requirements

Calibration antennas should have:

• Known, stable efficiency over the frequency range
• Omnidirectional or well-characterized directional pattern
• Good impedance match to minimize mismatch uncertainty
• Appropriate size for the working volume

Typical reference antennas include log-periodic antennas, horn antennas, and specially designed reverberation chamber probes. Antenna factors must be known and traceable to national standards.

Spatial Correlation

The spatial correlation function describes how field values at two locations separated by distance d are related:

rho(d) = mean(E(r) * E(r+d)) / mean(E(r)^2)

In an ideal reverberation chamber, the spatial correlation follows a sinc function:

rho(d) = sin(k*d) / (k*d)

where k = 2*pi/lambda. Correlation first reaches zero at d = lambda/2. This means measurement points should be separated by at least half a wavelength to be approximately independent.

Frequency Correlation

The frequency correlation describes the relationship between field values at different frequencies:

rho(delta-f) = 1 / sqrt(1 + (delta-f / delta-f-c)^2)

where delta-f-c is the correlation bandwidth, approximately equal to the modal bandwidth f/Q. For frequencies separated by more than a few times the correlation bandwidth, the field values are essentially independent.

This relationship is exploited in frequency stirring, where sweeping frequency provides independent samples that can be averaged to improve measurement accuracy.

Stirrer Position Correlation

The correlation between field values at different stirrer positions depends on stirrer design and frequency. Well-designed stirrers create low correlation between adjacent positions, maximizing the number of independent samples per rotation.

The number of independent samples can be estimated from the correlation function:

N-independent = (number of positions) / integral(rho^2(theta) d-theta)

Alternatively, the correlation can be estimated from measured data by computing the autocorrelation of power values versus stirrer position.

Implications for Measurement Uncertainty

The standard uncertainty in field or power estimates decreases with the number of independent samples:

u(E) / E = 1 / sqrt(2*N)

u(P) / P = 1 / sqrt(N)

For 100 independent samples, the relative uncertainty in mean power is 10%. To reduce uncertainty to 1% would require 10,000 samples. Practical measurements typically use 50-200 independent samples, accepting uncertainties of 7-14%.

Combining mechanical stirring, frequency stirring, and source position stirring can significantly increase the available independent samples and reduce measurement time while maintaining acceptable uncertainty.