Electronics Guide

The Four Fundamental Equations

Maxwell's equations in their differential form describe the local relationships between fields, charges, and currents:

Gauss's Law for Electric Fields:

div(D) = rho

This equation states that electric flux diverges from electric charge. In EMC terms, it explains why charge accumulation on conductors creates electric fields that can couple to nearby circuits. The displacement field D relates to the electric field E through the permittivity of the medium.

Gauss's Law for Magnetic Fields:

div(B) = 0

This equation confirms that magnetic monopoles do not exist; magnetic field lines always form closed loops. For EMC, this means magnetic fields cannot be blocked by simply placing a barrier in their path. They must be redirected or cancelled, which has profound implications for shielding design.

Faraday's Law of Induction:

curl(E) = -dB/dt

A time-varying magnetic field creates an electric field that circulates around it. This is the basis of magnetic field coupling and inductive interference. Any loop in a circuit acts as a pickup antenna for time-varying magnetic fields, and the induced voltage is proportional to the rate of change of magnetic flux through the loop.

Ampere's Law with Maxwell's Correction:

curl(H) = J + dD/dt

Magnetic fields are created by both conduction currents (J) and displacement currents (dD/dt). Maxwell's addition of the displacement current term was revolutionary: it predicts that time-varying electric fields create magnetic fields, completing the electromagnetic wave mechanism. For EMC, this explains how energy radiates from structures even without direct current flow through space.

Wave Equation Derivation

By combining Maxwell's equations in source-free regions, we derive the wave equations that govern electromagnetic propagation:

del-squared(E) - mu*epsilon*(d-squared(E)/dt-squared) = 0

del-squared(H) - mu*epsilon*(d-squared(H)/dt-squared) = 0

These equations reveal that electromagnetic disturbances propagate as waves at velocity:

v = 1/sqrt(mu*epsilon)

In free space, this gives the speed of light c = 299,792,458 m/s. The wave nature of electromagnetic energy is central to understanding both radiation and propagation phenomena in EMC.

Boundary Conditions

At interfaces between different materials, electromagnetic fields must satisfy boundary conditions derived from Maxwell's equations:

• Tangential E continuity: The tangential component of the electric field is continuous across a boundary (unless surface currents exist)
• Normal D discontinuity: The normal component of D changes by the surface charge density
• Tangential H discontinuity: The tangential component of H changes by the surface current density
• Normal B continuity: The normal component of B is continuous

These boundary conditions are essential for understanding shielding effectiveness, skin effect in conductors, and the behavior of fields at material interfaces. At a perfect conductor surface, the tangential electric field must be zero, forcing all E-field energy to be reflected or absorbed.

EMC Implications

Maxwell's equations have several direct implications for EMC engineering:

• Current loops radiate: Faraday's and Ampere's laws together show that any current loop creates both electric and magnetic fields that propagate outward as electromagnetic waves
• High-frequency currents matter more: The dB/dt and dD/dt terms show that faster-changing fields produce stronger effects, explaining why high-frequency harmonics often dominate EMC problems
• Return currents are essential: Current continuity requires that every signal current have a return path; the geometry of this path determines the loop area and thus the radiation and susceptibility characteristics
• Fields couple to all conductors: Faraday's law shows that any conducting loop in a time-varying field will have voltage induced in it; this is the basis of all EMI pickup

Field Region Definitions

The space around a radiating source is conventionally divided into three regions based on distance r from the source:

Reactive Near-Field Region: Extends from the source to approximately lambda/(2*pi) or about 0.16 wavelengths. In this region, energy is predominantly stored rather than radiated. The electric and magnetic fields are not in phase, and the ratio of E to H varies dramatically with position and source geometry. Field strength typically decreases as 1/r-squared or 1/r-cubed.

Radiating Near-Field (Fresnel) Region: Extends from the reactive near-field boundary to about 2*D-squared/lambda, where D is the largest dimension of the source. In this transitional zone, radiation predominates but the field pattern depends on distance. Phase variations across the source aperture are significant.

Far-Field (Fraunhofer) Region: Beyond the radiating near-field. Here, the fields have settled into their characteristic radiation pattern, which is independent of distance except for the 1/r amplitude variation. The E and H fields are perpendicular, in phase, and their ratio equals the intrinsic impedance of the medium (377 ohms in free space).

Near-Field Characteristics

In the near-field, the field character depends strongly on source type:

Electric (High-Impedance) Sources: Sources with high voltage and low current (like traces with high-impedance loads or electrostatic discharge events) produce fields dominated by the electric component. The wave impedance E/H is much greater than 377 ohms, potentially reaching thousands of ohms close to the source. These fields couple primarily through capacitive mechanisms.

Magnetic (Low-Impedance) Sources: Sources with high current and low voltage (like power supply loops or transformer windings) produce fields dominated by the magnetic component. The wave impedance E/H is much less than 377 ohms, potentially only a few ohms. These fields couple primarily through inductive mechanisms.

The field impedance has critical implications for shielding: electric fields are relatively easy to shield with any conductor, while magnetic fields require high-permeability materials or careful attention to shield construction to be effectively attenuated.

Far-Field Characteristics

In the far-field, all sources produce transverse electromagnetic (TEM) waves with consistent properties:

• E and H fields are perpendicular to each other and to the direction of propagation
• The wave impedance E/H equals 377 ohms (in free space)
• Power density decreases as 1/r-squared (inverse square law)
• Field strength decreases as 1/r
• The radiation pattern is fixed and independent of distance

Far-field conditions simplify many EMC calculations and measurements. Regulatory emissions tests are typically performed in the far-field where results are repeatable and independent of exact source geometry.

Boundary Distance Calculations

For EMC work, determining which region applies to a particular situation is essential. The key boundary is at approximately:

r = lambda/(2*pi) = c/(2*pi*f)

At 30 MHz (the lower limit of most radiated emissions testing), this boundary is about 1.6 meters. At 1 GHz, it shrinks to about 5 cm. This explains why measurements at the standard 3-meter or 10-meter distances are comfortably in the far-field for most regulated frequency ranges, but close-proximity coupling situations require near-field analysis.

For electrically large sources (dimension D comparable to or greater than wavelength), the far-field boundary may be pushed out to:

r = 2*D-squared/lambda

This can be significant for large antenna arrays or extended cable structures at high frequencies.

Transition Region Behavior

The transition from near-field to far-field is gradual, not abrupt. In the transition region:

• Wave impedance gradually approaches 377 ohms from either higher or lower values depending on source type
• The radiation pattern evolves toward its far-field shape
• Both reactive and radiating energy components are significant

For EMC analysis in the transition region, simplified models may introduce significant error. Full-wave electromagnetic simulation or careful measurement is often required for accurate results.

Plane Wave Propagation

The simplest electromagnetic wave is a uniform plane wave, where field vectors are constant across any plane perpendicular to the direction of propagation. While true plane waves only exist at infinite distance from any real source, far-field radiation closely approximates plane wave behavior.

For a plane wave propagating in the positive z-direction in a lossless medium:

E(z,t) = E0 * cos(omega*t - beta*z)

H(z,t) = H0 * cos(omega*t - beta*z)

where beta = omega*sqrt(mu*epsilon) is the phase constant, determining how rapidly the phase changes with distance. The wavelength lambda = 2*pi/beta represents the spatial period of the wave.

Intrinsic Impedance

The intrinsic impedance (or wave impedance) of a medium is the ratio of electric to magnetic field strength for a plane wave:

eta = E/H = sqrt(mu/epsilon)

For free space:

eta0 = sqrt(mu0/epsilon0) = 377 ohms

For other materials, the intrinsic impedance changes. Dielectrics with relative permittivity epsilon-r have:

eta = 377/sqrt(epsilon-r) ohms

Good conductors have very low intrinsic impedance due to their high effective permittivity at EMC frequencies. For copper at 1 MHz, the intrinsic impedance is only about 0.00037 ohms. This enormous mismatch with free space impedance is what makes metals effective shields.

Propagation in Lossy Media

Real materials have losses that attenuate waves as they propagate. In a medium with conductivity sigma, the propagation constant becomes complex:

gamma = alpha + j*beta

where alpha is the attenuation constant (nepers per meter) and beta is the phase constant (radians per meter). The attenuation causes field amplitude to decay exponentially:

E(z) = E0 * exp(-alpha*z)

For good conductors at EMC frequencies, the skin depth delta characterizes how quickly fields decay:

delta = sqrt(2/(omega*mu*sigma)) = 1/alpha

At one skin depth, the field has decayed to 1/e (about 37%) of its surface value. For copper at 1 MHz, delta is approximately 66 micrometers; at 1 GHz, it is about 2.1 micrometers.

Reflection and Transmission at Boundaries

When an electromagnetic wave encounters a boundary between media with different impedances, part of the wave is reflected and part is transmitted. The reflection coefficient for normal incidence is:

Gamma = (eta2 - eta1)/(eta2 + eta1)

where eta1 is the impedance of the incident medium and eta2 is the impedance of the second medium. The transmission coefficient is:

T = 2*eta2/(eta2 + eta1)

For a wave in air (377 ohms) hitting copper (0.00037 ohms at 1 MHz), nearly all the wave is reflected. The tiny transmitted portion is then rapidly attenuated by absorption. This dual mechanism of reflection and absorption is the basis of electromagnetic shielding.

At oblique incidence, the reflection and transmission coefficients depend on the angle and polarization of the incident wave. Total internal reflection can occur when waves travel from a lower-impedance medium toward a higher-impedance medium at angles exceeding the critical angle.

Standing Waves and Resonance

When waves reflect from boundaries and interfere with incident waves, standing wave patterns develop. In enclosed spaces or along transmission lines, these patterns can create resonances where fields are greatly enhanced at specific frequencies.

Cavity resonance occurs in shielded enclosures when dimensions match half-wavelength multiples. For a rectangular cavity with dimensions a, b, and c:

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

where m, n, and p are mode indices (integers, with at least two non-zero). At resonance, even small excitation can produce large internal fields, potentially causing EMC problems that would not be predicted from simple shielding theory.

Capacitive (Electric Field) Coupling

Capacitive coupling occurs when a time-varying voltage on one conductor induces current in another through the electric field between them. Any two conductors separated by an insulator form a capacitor, and high-frequency voltage changes can couple significant current through even small capacitances.

The coupling current is:

I-coupled = C-mutual * dV/dt

where C-mutual is the capacitance between conductors and dV/dt is the rate of voltage change on the source conductor.

Key factors affecting capacitive coupling:

• Distance: Coupling capacitance decreases roughly as 1/distance for closely spaced conductors
• Parallel length: Longer parallel runs increase mutual capacitance proportionally
• Frequency: Higher frequencies couple more strongly (coupling impedance = 1/(omega*C))
• Dielectric constant: Higher permittivity materials between conductors increase capacitance

Mitigation strategies include increasing separation, reducing parallel run length, using ground planes or guard traces between sensitive signals, and reducing source impedance (which reduces the voltage that can develop).

Inductive (Magnetic Field) Coupling

Inductive coupling occurs when a time-varying current in one circuit creates a magnetic field that induces voltage in another circuit. Any circuit loop acts as both a source of magnetic fields and a pickup antenna for external fields.

The induced voltage is:

V-induced = M * dI/dt

where M is the mutual inductance between circuits and dI/dt is the rate of current change in the source circuit. For small loops, M depends on the loop areas and their relative orientation.

Key factors affecting inductive coupling:

• Loop area: Both source and victim loop areas directly affect coupling; minimizing loop area is the most effective mitigation
• Distance: Magnetic field strength decreases as 1/r-cubed for small loops, making distance effective
• Orientation: Maximum coupling occurs when loop planes are parallel; perpendicular loops have zero magnetic coupling
• Frequency: Higher frequencies couple more strongly (V proportional to dI/dt)

Mitigation strategies include minimizing loop areas (keeping signal and return paths close together), using twisted pairs to cancel magnetic fields, orienting sensitive circuits perpendicular to interference sources, and using magnetic shielding materials.

Common Impedance Coupling

Common impedance coupling occurs when two circuits share a common current path, typically a ground or power connection. Current from one circuit flowing through the shared impedance creates a voltage drop that appears as interference in the other circuit.

The interference voltage is:

V-interference = I-source * Z-common

Even small impedances can cause problems when source currents are large or switching rapidly. A 10 milliohm ground path impedance carrying 1 amp at 100 MHz can produce 10 millivolts of interference with significant harmonic content.

Key factors:

• Shared path length: Longer common paths have more inductance and resistance
• Current magnitude: Higher source currents produce larger interference voltages
• Frequency: Inductance dominates at high frequencies, making fast edges problematic
• Victim sensitivity: Low-level analog circuits are particularly vulnerable

Mitigation includes star grounding (separate return paths for different circuits), ground planes (which provide parallel low-impedance paths), and filtering at the point where circuits share connections.

Radiated Coupling

When source and victim are separated by significant distance (beyond the near-field), coupling occurs through electromagnetic radiation. The source acts as an unintentional antenna, radiating energy that propagates through space and is received by the victim acting as a receiving antenna.

The received power depends on:

• Transmitted power: Related to source current and radiation efficiency
• Source antenna gain: Directional characteristics of the radiating structure
• Path loss: Decreases as (4*pi*r/lambda)-squared (Friis equation)
• Receiver antenna gain: Directional characteristics of the victim structure
• Polarization match: Maximum coupling when source and victim have aligned polarization

Radiated coupling is the mechanism addressed by most EMC emissions and immunity standards. Mitigation includes reducing source radiation (shielding, filtering), increasing path loss (distance, absorbers), and reducing victim antenna efficiency (balanced circuits, shielded cables).

Coupling Path Analysis

Real EMC problems often involve multiple coupling mechanisms operating simultaneously. Systematic analysis requires identifying all potential paths:

1. Identify the source of interference (what circuit or signal creates the disturbance)
2. Identify the victim (what circuit or function is affected)
3. Enumerate possible coupling paths (conducted, capacitive, inductive, radiated)
4. Estimate the coupling efficiency of each path
5. Determine which path dominates at the frequency of concern
6. Apply mitigation to the dominant path

Often, fixing one coupling path reveals another that was previously masked. Effective EMC design addresses all significant paths from the beginning rather than chasing problems sequentially.

Statement of Reciprocity

For electromagnetic systems, reciprocity can be stated as: if a current source I1 at location A produces a voltage V2 at location B, then the same current source I1 at location B will produce the same voltage V2 at location A.

More generally, for any passive, linear, time-invariant system without non-reciprocal elements (like ferrites with bias), the transfer function from any point to any other point is the same in both directions.

Antenna Reciprocity

One of the most important applications is antenna reciprocity: an antenna has the same radiation pattern whether used for transmission or reception. This means:

• The gain in any direction is the same for transmitting and receiving
• The polarization characteristics are identical
• The input impedance is the same

For EMC, this is powerful because unintentional antennas (cables, PCB traces, equipment enclosures) also obey reciprocity. If a structure radiates efficiently at a particular frequency and polarization, it will also be susceptible to interference at that same frequency and polarization.

This allows engineers to:

• Use emissions measurements to predict susceptibility vulnerabilities
• Design immunity tests based on emissions characteristics
• Understand that fixing an emissions problem often improves immunity at the same frequency

Coupling Reciprocity

Reciprocity applies to all linear coupling mechanisms. The mutual capacitance between two conductors is the same regardless of which is considered the source. The mutual inductance between two loops is identical in both directions. The transfer impedance of a cable shield is the same whether interference is coupling in or out.

This allows simplification of coupling analysis: you only need to calculate coupling in one direction, and the result applies equally in the reverse direction.

Measurement Applications

Reciprocity enables several practical measurement techniques:

Shielding effectiveness measurement: Instead of placing a source inside an enclosure and measuring external fields, you can place a source outside and measure the field inside. Both give the same shielding effectiveness value.

Cable coupling measurement: Transfer impedance can be measured by injecting current on the shield and measuring voltage on the inner conductor, or by applying voltage to the inner conductor and measuring shield current. Both methods yield the same transfer impedance.

Antenna calibration: An antenna can be calibrated as either a transmitter or receiver, whichever is more convenient, and the calibration applies to both uses.

Limitations of Reciprocity

Reciprocity does not apply when the system contains non-reciprocal elements:

• Ferrites with DC bias: Circulators and isolators are intentionally non-reciprocal
• Active circuits: Amplifiers, oscillators, and other active devices break reciprocity
• Non-linear elements: Diodes, varistors, and saturating inductors
• Moving or time-varying systems: Rotating machinery, switching power supplies during transients

In these cases, coupling must be analyzed separately for each direction, and emissions performance does not necessarily predict immunity characteristics.

Fundamental Antenna Parameters

Several parameters characterize antenna performance:

Radiation Pattern: The three-dimensional distribution of radiated power as a function of direction. Unintentional antennas typically have complex patterns determined by their geometry and the distribution of currents.

Directivity: The ratio of radiation intensity in a particular direction to the average radiation intensity. A higher directivity means more focused radiation. Simple structures like short wires have low directivity (nearly omnidirectional), while larger, more complex structures can have significant directivity.

Gain: Directivity multiplied by radiation efficiency. Gain accounts for losses in the antenna structure. For unintentional antennas, gain is often low because they are poorly matched and have significant losses.

Radiation Resistance: The equivalent resistance that would dissipate the same power as is radiated. For a short dipole (length l much less than wavelength):

R-rad = 20 * pi-squared * (l/lambda)-squared ohms

This shows radiation resistance is very small for electrically short structures, meaning they are inefficient radiators.

Input Impedance: The impedance seen at the antenna terminals. For unintentional antennas, this is often highly reactive and poorly matched to the driving circuit, further reducing radiation efficiency.

Common Unintentional Antenna Types

Small Loops: Current loops with circumference much smaller than a wavelength behave as magnetic dipoles. Their radiation is maximum perpendicular to the loop plane. Radiation increases with loop area squared and frequency to the fourth power, making high-frequency currents in large loops particularly problematic.

Short Monopoles: A conductor over a ground plane shorter than a quarter wavelength acts as a capacitive monopole. Heatsinks, connector pins, and vertical PCB traces often behave this way. Radiation is maximum perpendicular to the conductor axis and increases with length squared and frequency squared.

Dipoles: Two conductors driven differentially form a dipole. The classic half-wave dipole resonates when its total length equals half the wavelength, at which point it is an efficient radiator. Unintentional dipoles can form from cables with both signal and return conductors when common-mode currents are present.

Slot Antennas: Gaps in a conducting surface, such as ventilation slots or seams in an enclosure, can radiate as slot antennas. By Babinet's principle, a slot in a conductor radiates like a dipole with interchanged E and H fields. Slots resonate when their length is half a wavelength.

Patch and Aperture Antennas: Large conducting surfaces with gaps or feeds can exhibit complex resonant behavior. Enclosure panels, large heatsinks, and other extended conductors may show multiple resonant modes.

Radiation from PCB Structures

PCB traces and their return paths form unintentional antennas. Understanding their radiation characteristics helps in EMC design:

Differential-Mode Radiation: The intended signal current loop radiates as a small loop antenna. Radiation is proportional to current, loop area, and frequency squared. Minimizing loop area by keeping signal and return paths close together dramatically reduces radiation.

Common-Mode Radiation: When equal currents flow on both conductors in the same direction (usually due to ground bounce or imperfect balance), the structure radiates as a monopole or dipole. Common-mode currents are often the dominant radiation source because the effective antenna is much larger than the differential loop.

For a typical PCB trace at 100 MHz:

• A 1 cm trace with a well-defined return path has a loop area of perhaps 10 square mm and radiates weakly
• The same trace with 1 mA of common-mode current acts as a 1 cm monopole and can radiate 20-40 dB more than the differential mode

Cable Radiation

Cables are among the most efficient unintentional antennas in electronic systems. Their length often approaches or exceeds wavelength at EMC frequencies, making them resonant radiators.

Shield Current Radiation: Current flowing on the outside of a cable shield radiates efficiently. This common-mode current may be induced by ground potential differences, coupling through shield discontinuities, or capacitive coupling from internal conductors.

Resonance Effects: A cable becomes a quarter-wave monopole when its length equals lambda/4, and a half-wave dipole when its length equals lambda/2. At these resonant lengths, radiation is greatly enhanced. A 1-meter cable resonates as a quarter-wave monopole at 75 MHz and a half-wave dipole at 150 MHz.

Mitigation: Cable radiation is controlled through proper shield termination, ferrite common-mode chokes, cable length management, and ensuring the shield is at a stable reference potential at both ends.

Enclosure Radiation

Equipment enclosures can radiate through several mechanisms:

• Slot radiation: Ventilation slots, seams, and gaps act as slot antennas
• Aperture leakage: Openings for displays, connectors, or controls allow internal fields to escape
• Surface current radiation: Currents flowing on enclosure surfaces (often from cable shield attachments) radiate as patch antennas
• Resonance: Internal cavity resonances can enhance fields at specific frequencies, increasing radiation through any apertures

Effective enclosure design requires attention to all these mechanisms, treating the enclosure as an antenna system rather than a simple Faraday cage.

Coupled Transmission Line Model

When two transmission lines run parallel, they couple through both electric and magnetic fields. The coupling is characterized by mutual capacitance C-m and mutual inductance L-m per unit length. The governing equations for the coupled system are:

dV1/dz = -L1*dI1/dt - Lm*dI2/dt

dV2/dz = -Lm*dI1/dt - L2*dI2/dt

dI1/dz = -C1*dV1/dt - Cm*dV2/dt

dI2/dz = -Cm*dV1/dt - C2*dV2/dt

These coupled equations show that voltages and currents on one line affect the other through the mutual parameters.

Weak Coupling Approximation

When coupling is weak (Cm much smaller than C1, C2 and Lm much smaller than L1, L2), the coupled line behavior can be approximated by treating the coupling as a perturbation to independent transmission lines. This leads to simple expressions for crosstalk.

For a generator line (carrying the source signal) and a receptor line (the victim):

Near-end crosstalk (NEXT): Coupling at the end of the receptor line nearest the source

V-NEXT = (1/4) * (Cm/C + Lm/L) * (V-source) * length/rise-time

for electrically short lines (length much less than rise-time times velocity).

Far-end crosstalk (FEXT): Coupling at the far end of the receptor line

V-FEXT = (1/2) * (Cm/C - Lm/L) * length * dV-source/dt

The FEXT expression shows that if Cm/C equals Lm/L (the homogeneous medium condition, satisfied in stripline but not microstrip), FEXT vanishes. This is exploited in high-speed digital design to minimize far-end crosstalk.

Even and Odd Mode Analysis

For symmetric coupled lines, the system can be decomposed into even and odd propagation modes:

Even mode: Equal voltages and currents on both lines. The effective capacitance is C-self minus C-mutual, and inductance is L-self plus L-mutual. The even-mode characteristic impedance is:

Z-even = sqrt((L + Lm)/(C - Cm))

Odd mode: Equal and opposite voltages and currents. The effective capacitance is C-self plus C-mutual, and inductance is L-self minus L-mutual. The odd-mode characteristic impedance is:

Z-odd = sqrt((L - Lm)/(C + Cm))

Modal analysis simplifies many coupled line problems because the modes propagate independently. Any excitation can be decomposed into even and odd components, each analyzed separately, then recombined for the total response.

Shield Transfer Impedance

Shielded cables can be modeled using transfer impedance, which quantifies coupling between the shield and internal conductors. The transfer impedance Z-t relates the voltage induced on the internal conductor to the current flowing on the shield:

V-internal = Z-t * I-shield * length

For solid shields, transfer impedance is dominated by the DC resistance at low frequencies and by field penetration through the shield thickness at high frequencies. For braided shields, the porosity of the weave creates additional coupling paths that dominate at higher frequencies.

A good quality coaxial cable might have Z-t of 10 milliohms/meter at low frequencies, rising to 100 milliohms/meter or more at 1 GHz due to braid coupling. This increase with frequency often makes cables more susceptible to high-frequency interference than low-frequency signals.

Ground Plane Effects

Transmission lines over a ground plane (as in most PCB designs) have their coupling characteristics modified by the ground:

• Image currents in the ground plane create an effective return path that reduces inductance
• The ground plane provides partial shielding, reducing mutual inductance between lines
• Gaps or slots in the ground plane can dramatically increase coupling by forcing return currents to detour around the discontinuity
• Multi-layer boards allow ground planes between signal layers, providing excellent isolation

Proper ground plane design is one of the most effective ways to control both emissions and crosstalk in PCB systems.

Waveguide Modes

In hollow metallic waveguides, electromagnetic energy propagates in distinct modes, each with a characteristic field pattern and cutoff frequency:

Transverse Electric (TE) modes: The electric field has no component in the direction of propagation. The magnetic field has both transverse and longitudinal components.

Transverse Magnetic (TM) modes: The magnetic field has no component in the direction of propagation. The electric field has both transverse and longitudinal components.

Each mode has a cutoff frequency below which it cannot propagate. Below the lowest cutoff frequency, electromagnetic energy is attenuated exponentially rather than propagating. This is the principle behind waveguide-below-cutoff filters used to provide EMC protection for ventilation apertures.

For a rectangular waveguide with dimensions a and b (a greater than b), the cutoff frequencies are:

f-c-mn = (c/2)*sqrt((m/a)-squared + (n/b)-squared)

The dominant mode (lowest cutoff) is TE10 with cutoff at c/(2a). Signals below this frequency are attenuated at approximately 27 dB per waveguide width of length.

Cavity Modes

Enclosed metallic cavities support standing wave resonances at discrete frequencies. These cavity modes are important in EMC because they can amplify internal fields and create hot spots of field intensity.

For a rectangular cavity of dimensions a, b, and c, resonant frequencies are:

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

where m, n, p are mode indices (integers with at least two being non-zero).

Cavity resonances can cause:

• Large internal field variations with position and frequency
• Reduced shielding effectiveness at resonant frequencies
• Coupling between circuits that are well-isolated at non-resonant frequencies
• Complex frequency response in shielded enclosures

Mitigation strategies include adding absorbing materials to reduce Q-factor, breaking up large cavities into smaller compartments, and avoiding placing sensitive circuits at field maximum locations.

Transmission Line Modes

Multi-conductor transmission line systems support multiple propagation modes. Understanding these modes is essential for controlling EMC in cable systems and PCB interconnects.

TEM mode: The fundamental mode in two-conductor transmission lines where both E and H are transverse to the direction of propagation. This is the desired signal mode in most applications.

Common-mode versus differential-mode: In a two-conductor line over a reference plane, the differential mode (equal and opposite currents in the two conductors) is the intended signal mode. The common mode (equal currents in the same direction) is typically undesired interference that radiates efficiently.

Higher-order modes: At sufficiently high frequencies, waveguide-type modes can propagate on PCB traces, coaxial cables, and other structures. These modes have different velocities and field patterns than the TEM mode, causing signal distortion and potentially increased radiation.

Modal Conversion

Mode conversion occurs when energy transfers between modes, typically at discontinuities or asymmetries in the propagation structure:

• Differential to common-mode conversion: Occurs at imbalances such as unequal trace lengths, asymmetric vias, or connector transitions. Even small imbalances can convert a fraction of differential signal energy to common mode, which then radiates
• TEM to higher-order mode conversion: Occurs at high frequencies where structures can support multiple modes. Connectors, bends, and vias can excite higher-order modes that cause resonances or radiation
• Coupling between cable modes: External interference can couple to the common mode of a shielded cable, then convert to differential mode through shield imperfections, appearing as interference on the signal conductors

Mode conversion is often the hidden culprit in EMC problems. A circuit that appears well-designed for its intended mode may still have problems due to converted energy in unwanted modes.

Poynting Vector and Power Flow

The Poynting vector S describes the direction and magnitude of electromagnetic power flow:

S = E cross H

The magnitude of S (in watts per square meter) represents the power density flowing through a surface perpendicular to S. The integral of S over a closed surface gives the total power flowing out of (or into) the enclosed volume.

For EMC analysis, the Poynting vector helps:

• Identify power flow paths in complex structures
• Understand how energy enters or leaves shielded enclosures
• Calculate the power coupled through apertures
• Visualize how energy flows from sources to victims

Radiation Efficiency and Power Balance

For any radiating structure, energy conservation requires:

P-input = P-radiated + P-dissipated + P-stored-rate-of-change

In steady state (sinusoidal excitation), the stored energy oscillates but does not change on average, so:

P-input = P-radiated + P-dissipated

The radiation efficiency is:

eta = P-radiated/P-input = R-rad/(R-rad + R-loss)

For electrically small antennas (including most unintentional radiators), radiation resistance is very small, so efficiency is low. Most power is dissipated as heat rather than radiated. This limits the maximum possible radiation from a given structure but does not eliminate the EMC concern; even inefficient radiation may exceed limits.

Coupling Energy Limits

Energy conservation places fundamental limits on coupling. The coupled power cannot exceed the source power:

P-coupled less than or equal to P-source

More practically, coupled power is limited by the coupling efficiency, which depends on the geometry, frequency, and impedance matching:

P-coupled = P-source * coupling-coefficient

The coupling coefficient ranges from 0 (no coupling) to 1 (perfect coupling, rarely achieved in practice). Understanding that coupling cannot exceed source power helps identify unrealistic calculations or measurements.

For radiated coupling in the far field, the maximum coupling occurs when source and receiver are both matched to their antennas and point their maximum gain directions at each other. The Friis equation gives:

P-received/P-transmitted = G-t * G-r * (lambda/(4*pi*r))-squared

This shows how received power decreases with distance squared and increases with antenna gains and wavelength squared (lower frequency).

Shielding Energy Balance

When electromagnetic energy encounters a shield, conservation requires:

P-incident = P-reflected + P-transmitted + P-absorbed

Shielding effectiveness SE is defined in terms of the transmitted power (or field):

SE (dB) = 10*log(P-incident/P-transmitted)

A high-SE shield has very little transmitted power because most is either reflected or absorbed. Understanding the relative contributions of reflection and absorption helps in shield design:

• Reflection dominates for good conductors because of impedance mismatch with free space
• Absorption dominates in thick shields or high-permeability materials
• Multiple reflections within the shield can either enhance or reduce effectiveness depending on phase

Quality Factor and Stored Energy

The quality factor Q of a resonant structure relates stored energy to dissipated energy per cycle:

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

High-Q resonances (low loss) store large amounts of energy, making them potentially problematic for EMC:

• High-Q cavity resonances in shielded enclosures amplify internal fields
• High-Q resonances in filters can ring when excited by transients
• High-Q antenna resonances are efficient radiators at their resonant frequency

Adding loss (damping) reduces Q and the stored energy, mitigating these effects at the cost of some power dissipation.

Energy Methods in EMC Design

Energy-based thinking helps guide EMC design decisions:

• Minimize source energy: Reduce switching speeds and current amplitudes where possible to reduce the energy available for interference
• Control energy flow paths: Use ground planes, shields, and filters to direct energy flow along intended paths
• Dissipate unwanted energy: Use absorbers, ferrites, and resistive elements to convert unwanted energy to heat
• Avoid energy storage: Minimize resonances that can store and amplify energy
• Match impedances appropriately: Use intentional mismatches to reflect energy where desired, and matches to absorb energy

By tracking energy through the system and ensuring all energy is accounted for, engineers can verify their EMC analysis and identify potential problems.