Electronics Guide

Measuring Shielding Effectiveness

Shielding effectiveness is measured by several standardized methods:

• Shielded enclosure method: Transmitting and receiving antennas are placed on opposite sides of the enclosure wall. A reference measurement is taken with the wall out of the path, then the measurement is repeated with the enclosure in place, and the difference is the shielding effectiveness. IEEE Std 299 defines this method for enclosures whose smallest linear dimension is at least 2 m, over the range 9 kHz to 18 GHz. IEEE Std 299.1 extends comparable methods to smaller enclosures.
• Coaxial transmission line method: A planar material sample is clamped in a coaxial holder, and its insertion loss is measured against a reference specimen of the same geometry. ASTM D4935 defines this method for planar materials. It is convenient for material characterization but reveals nothing about seams, apertures, or enclosure geometry.
• Free-space method: The sample is placed between transmitting and receiving antennas and the attenuation is measured. This approach suits thin, flexible materials, although diffraction around the edges of the sample sets a ceiling on the value that can be measured.
• TEM cell method: A transverse electromagnetic cell provides a controlled, well-characterized field environment for measuring the shielding effectiveness of a material sample or a small enclosure.

Each method has its own frequency range, sample size requirement, and accuracy characteristics, and the numbers they return are not interchangeable. The choice depends on the application and on whether a material or a complete enclosure is under evaluation.

Practical Shielding Effectiveness Values

Different applications require different levels of shielding effectiveness:

• 10 to 30 dB: Minimal shielding, adequate for reducing interference in non-critical applications
• 30 to 60 dB: Good shielding, typical of commercial electronic enclosures
• 60 to 90 dB: Excellent shielding, expected of sensitive instrumentation and many military installations
• 90 to 120 dB: Very high shielding, of the order sought in secure screened facilities and in magnetically quiet metrology rooms
• Above 120 dB: Exceptional shielding, achievable only in purpose-built facilities with double-wall construction and rigorous control of every penetration

These bands describe complete installations, not bare material. The intrinsic shielding effectiveness of a solid metal wall is far higher than any of them, often exceeding 200 dB on paper. Real enclosures fall short because seams, apertures, and cable penetrations bypass the metal, and above a few hundred megahertz those leakage paths, rather than the wall, set the result.

Factors Affecting Shielding Effectiveness

Overall shielding effectiveness depends on multiple factors:

• Material properties: Conductivity and permeability determine reflection and absorption
• Shield thickness: Thicker shields provide more absorption
• Frequency: Different mechanisms dominate at different frequencies
• Source type: Electric field, magnetic field, and plane wave sources have different shielding characteristics
• Distance from source: Near-field and far-field shielding behave differently
• Apertures and seams: Openings dominate shielding performance at high frequencies

The total shielding effectiveness results from interactions among these factors, which the following sections examine in turn.

Impedance Mismatch and Reflection

The intrinsic impedance of a medium is the ratio of electric to magnetic field strength for a propagating plane wave. For free space:

η₀ = √(μ₀/ε₀) ≈ 376.73 Ω

EMC practice rounds this to 377 ohms, and this article follows that convention. The value is no longer exact: before the 2019 revision of the SI, μ₀ was defined as 4π × 10⁻⁷ H/m and η₀ followed exactly, whereas μ₀ is now a measured quantity carrying a small experimental uncertainty.

For a good conductor of conductivity σ and permeability μ at angular frequency ω:

η_s = √(jωμ/σ) = (1 + j)√(ωμ/2σ)

The magnitude of this impedance is very small for good conductors. Since |1 + j| = √2, the magnitude reduces to:

|η_s| = √(ωμ/σ)

For copper at 1 MHz, with σ = 5.8 × 10⁷ S/m and μ = μ₀:

|η_Cu| = √((2π × 10⁶)(4π × 10⁻⁷)/(5.8 × 10⁷)) ≈ 3.7 × 10⁻⁴ Ω

The mismatch between free-space impedance, 377 ohms, and conductor impedance, 0.37 milliohms, is a factor of about a million. That mismatch is what causes most of the incident energy to be reflected at the surface.

Reflection Coefficient Calculation

The reflection coefficient for normal incidence is:

Γ = (η_s − η₀)/(η_s + η₀)

Since η_s ≪ η₀ for good conductors:

Γ ≈ −1

Nearly all of the incident wave is therefore reflected, with a phase reversal. Reflection loss, however, is measured by the wave that is transmitted, and a shield has two boundaries: the front, air-to-metal surface and the rear, metal-to-air surface. Multiplying the transmission coefficients of both boundaries gives the total reflection loss:

R = 20 log₁₀|(η₀ + η_s)²/4η₀η_s| dB

For good conductors, where η_s ≪ η₀, the numerator is dominated by η₀², and this simplifies to:

R ≈ 20 log₁₀(η₀/4|η_s|) dB

Substituting the expression for conductor impedance gives the widely used engineering form, in which conductivity is expressed relative to copper (σ_r), permeability relative to free space (μ_r), and frequency f in hertz:

R = 168 + 10 log₁₀(σ_r/μ_rf) dB

The constant 168 absorbs η₀, the copper reference conductivity, and the unit conversions; it is not the reflection loss at any frequency. For copper (σ_r = 1, μ_r = 1) at 1 MHz the formula returns R = 168 + 10 log₁₀(10⁻⁶) = 108 dB, which agrees with evaluating 20 log₁₀(η₀/4|η_s|) directly. The form shows that reflection loss rises with conductivity but falls with both frequency and permeability. High-conductivity, low-permeability materials such as copper and aluminum therefore give the highest reflection loss, while high-permeability materials such as mu-metal give much less and rely instead on absorption.

Reflection Loss for Common Materials

Evaluating the engineering formula at 1 MHz, using the relative conductivities and permeabilities tabulated in the standard EMC references, gives the following plane-wave reflection losses:

• Copper: About 108 dB (σ_r = 1, μ_r = 1)
• Aluminum: About 106 dB (σ_r ≈ 0.61, μ_r = 1)
• Austenitic stainless steel: About 91 dB (σ_r ≈ 0.02, μ_r ≈ 1, since grades such as 304 are essentially nonmagnetic)
• Cold-rolled steel: About 70 dB, taking σ_r ≈ 0.17 and μ_r ≈ 1,000; the high permeability is what lowers the figure
• Mu-metal: About 50 dB, taking σ_r ≈ 0.03 and the low-frequency value μ_r ≈ 20,000

These are calculated values, not measurements, and they are only as good as the assumed material constants. The permeability of steel and of mu-metal varies widely with grade, frequency, and drive level, and the formula shifts by 10 dB for every decade change in μ_r. The mu-metal figure in particular rests on a low-frequency permeability that the material does not retain at 1 MHz, so treat it as an illustration of the trend rather than as a design number.

Reflection loss falls by 10 dB for each decade increase in frequency. At 1 GHz, copper provides about 78 dB against plane waves. Note that the magnetic materials, despite their excellent performance against low-frequency magnetic fields through absorption and flux shunting, reflect plane waves less effectively than nonmagnetic conductors, because reflection loss scales as σ_r divided by μ_r.

These high values apply to plane waves in the far field. Near-field sources, and magnetic sources in particular, present a much lower wave impedance, which yields far lower reflection loss. This is why magnetic field shielding at low frequencies is so difficult.

Double Reflection at Entry and Exit

When an electromagnetic wave enters a shield, reflection occurs at the front surface. The transmitted portion then propagates through the shield and encounters the back surface, where another reflection occurs. The wave impedance on both sides of the shield is typically the same, that of free space, so the exit reflection coefficient has the same magnitude as the entry coefficient but the opposite sign.

The total reflection loss includes contributions from both surfaces. For a shield with air on both sides:

R_total = R_entry + R_exit

Each boundary contributes approximately the same reflection loss when the shield is thick enough that the wave inside is heavily attenuated before it reaches the second surface. For thin shields the two reflections interact, which produces the multiple reflection effects discussed below.

Skin Depth and Field Decay

The skin depth δ is the distance over which the electromagnetic field amplitude decays to 1/e, about 37 percent, of its surface value:

δ = √(2/ωμσ) = √(1/πfμσ)

Here ω is the angular frequency, μ is the permeability, and σ is the conductivity. For copper at 1 MHz:

δ_Cu = √(1/(π × 10⁶ × 4π × 10⁻⁷ × 5.8 × 10⁷)) ≈ 66 µm

The field amplitude at depth z into the shield is:

E(z) = E₀e^−z/δ

This exponential decay is rapid. At one skin depth the field is at 37 percent of its surface value; at two skin depths, 13.5 percent; at three skin depths, 5 percent; at five skin depths, below 1 percent.

Skin Depth for Various Materials

Skin depth varies strongly with material and frequency.

Copper (σ = 5.8 × 10⁷ S/m, μ_r = 1):

• 60 Hz: 8.5 mm
• 1 kHz: 2.1 mm
• 100 kHz: 0.21 mm (210 µm)
• 1 MHz: 66 µm
• 100 MHz: 6.6 µm
• 1 GHz: 2.1 µm

Aluminum (σ ≈ 3.5 × 10⁷ S/m, μ_r = 1):

• 1 MHz: 85 µm
• 100 MHz: 8.5 µm
• 1 GHz: 2.7 µm

Steel (σ ≈ 6 × 10⁶ S/m, μ_r from about 100 to 1,000):

• 60 Hz: 0.84 to 2.7 mm, depending on permeability
• 1 kHz: 0.21 to 0.65 mm
• 1 MHz: 6.5 to 21 µm

High-permeability materials such as steel have much smaller skin depths at low frequencies, which makes them effective against low-frequency magnetic fields. The spread in the steel figures is a reminder that the permeability of a ferromagnetic shield is not a fixed number: it depends on grade, heat treatment, frequency, and the strength of the applied field.

Absorption Loss Formula

The absorption loss A for a shield of thickness t is:

A = 20 log₁₀(e^t/δ) = 8.686(t/δ) dB

Substituting the skin depth gives:

A = 8.686t√(πfμσ) dB

Key observations:

• Absorption loss is proportional to shield thickness
• Absorption loss increases with the square root of frequency
• Higher conductivity and higher permeability both increase absorption
• Each skin depth of material contributes about 8.7 dB of absorption loss

Practical Absorption Loss Values

For a copper shield 1 mm thick:

• At 100 kHz: about 41 dB, since the wall is about 4.8 skin depths
• At 1 MHz: about 131 dB, about 15 skin depths
• At 10 MHz: about 415 dB in theory, a figure no measurement could confirm because other paths dominate long before it

Absorption loss becomes very large at high frequencies, even for thin shields. The practical limit at high frequencies is therefore not material absorption but the apertures and seams that bypass the solid material.

Material Selection for Absorption

When absorption is the primary shielding mechanism, which is the case for thick shields and at high frequencies, material selection follows these guidelines:

• Copper and aluminum: Excellent for general-purpose high-frequency shielding because of their high conductivity
• Steel: Higher absorption per unit thickness at low frequencies because of its permeability, despite its lower conductivity
• Mu-metal and similar nickel-iron alloys: Very effective at low frequencies, where their permeability is highest
• Conductive composites: Modest absorption, useful when weight or moldability outweighs shielding performance

Physics of Multiple Reflections

Consider a wave entering a thin shield:

1. Part of the incident wave reflects at the front surface; the rest enters the shield
2. The wave propagates through the shield and attenuates
3. At the back surface, part of the wave exits; the rest reflects back into the material
4. The internally reflected wave returns to the front surface, where it is again partly reflected and partly transmitted
5. The process continues, each bounce contributing to the total transmitted and reflected fields

For electrically thick shields of many skin depths, the internal reflections are so attenuated that they become negligible. For thin shields they are the largest source of error in an uncorrected estimate.

Multiple Reflection Correction Factor

The correction factor is:

M = 20 log₁₀|1 − Γ²e^−2t/δe^−j2t/δ| dB

Here Γ is the reflection coefficient at the internal metal-to-air boundary and t is the shield thickness. For a good conductor Γ ≈ −1, so Γ² ≈ 1 and the expression depends almost entirely on the ratio t/δ. The real exponential accounts for the attenuation of each internal round trip, and the complex exponential for its phase.

Two consequences follow:

• M is a negative correction for electrically thin shields. Energy that has already crossed the wall once is returned to it and crosses again, so less of the incident energy is kept out than reflection and absorption alone would suggest. Evaluating the expression gives about −12 dB at t = 0.1δ and about −1.3 dB at t = 0.5δ.
• M vanishes for thicker walls. Once t exceeds about one skin depth, where absorption alone has reached roughly 8.7 dB, the round-trip term is small and the correction stays within a decibel of zero. Standard practice is to drop M whenever A exceeds about 10 dB.

The correction matters most for low-impedance magnetic sources, not because its sign reverses but because that is where thin walls and large skin depths coincide. Against a high-impedance electric field or a plane wave, reflection loss is already enormous and the wall is usually many skin depths thick at the frequencies of interest, so the correction is irrelevant.

When Multiple Reflections Matter

Multiple reflection effects are significant when:

• Shield thickness is less than about one skin depth, and the effect grows rapidly as the wall thins further
• Frequencies are low, so the skin depth is large
• Thin foils or plated coatings serve as the shield
• Conductive plastics, whose low conductivity produces large skin depths, serve as the shield

For most practical metal shields above a few hundred kilohertz, absorption loss is large enough that multiple reflections can be ignored. At power-line and audio frequencies, or with thin conductive coatings, the correction must be included for an accurate prediction.

Total Shielding Effectiveness

Schelkunoff's decomposition combines the three terms additively:

SE = R + A + M

where, all in decibels:

• R is reflection loss, always positive
• A is absorption loss, always positive
• M is the multiple reflection correction, negative for thin walls and approaching zero for thick ones

Addition in decibels corresponds to multiplication of the individual attenuation factors in linear terms. The decomposition assumes a plane wave at normal incidence on an infinite, homogeneous, uniform slab, so it describes the wall material rather than the enclosure built from it.

Field Region Boundaries

The boundary between the near-field and far-field regions occurs at approximately:

r = λ/2π = c/2πf

Here r is the distance from the source, λ is the wavelength, and f is the frequency. At 1 MHz this boundary lies about 48 m from the source; at 100 MHz, about 0.48 m; at 1 GHz, about 4.8 cm.

In the near field the wave impedance varies with distance and source type. In the far field it settles at the free-space value of 377 ohms, whatever the source.

Far-Field (Plane Wave) Shielding

In the far field the incident wave is essentially a plane wave with a wave impedance of 377 ohms, and the reflection and absorption formulas given above apply directly. The characteristics are:

• Wave impedance is constant at 377 ohms
• Electric and magnetic fields are in phase and mutually perpendicular
• Both field components are attenuated equally by the shield
• Shielding effectiveness follows from material properties alone

Far-field shielding is conceptually simpler because the source geometry does not enter the wave impedance.

Near-Field Electric Source Shielding

Near an electric field source, which is to say one of high voltage, low current, and high impedance, the wave impedance greatly exceeds 377 ohms. For a small electric dipole at distance r:

Z_wave ≈ (λ/2πr) × 377 Ω = (c/2πfr) × 377 Ω

At distances well inside λ/2π this reaches thousands of ohms. The consequences for shielding are:

• High wave impedance means a larger mismatch with the low-impedance conductive shield
• Reflection loss is higher than for plane waves
• Even thin conductive shields are effective
• Electric fields are comparatively easy to shield

This is why almost any metal enclosure, even one with significant apertures, shields well against a nearby electric field source.

Near-Field Magnetic Source Shielding

Near a magnetic field source, one of high current, low voltage, and low impedance, the wave impedance falls far below 377 ohms. For a small current loop at distance r:

Z_wave ≈ (2πr/λ) × 377 Ω = (2πfr/c) × 377 Ω

At close range this can drop to a fraction of an ohm. The consequences for shielding are:

• Low wave impedance approaches the impedance of the conductive shield itself
• Reflection loss is much lower than for plane waves
• Shielding must rely primarily on absorption and on flux shunting
• Low-frequency magnetic fields are the hardest case in all of shielding

This explains why power-frequency magnetic fields from transformers and motors resist shielding. The skin depth in copper at 60 Hz is about 8.5 mm, so a copper wall of any practical thickness absorbs almost nothing.

Transition Region Effects

Between the near field and the far field, the wave impedance moves gradually between its extreme values, approaching 377 ohms from above for electric sources and from below for magnetic ones. Accurate shielding calculations in this region require knowledge of the actual source geometry and distance.

For practical shielding design:

• If the shield sits well beyond λ/2π, use the plane wave formulas
• If the shield sits well inside λ/2π, identify whether the source is predominantly electric or magnetic
• In the transition region, take the lower of the electric and magnetic field estimates as the conservative answer

Electrostatic Shielding Principles

At DC and very low frequencies, electric field shielding works through the redistribution of charge on the shield surface. When an external electric field impinges on a conductor:

1. Free charges in the conductor redistribute to cancel the external field inside
2. The surface charges create an opposing field that exactly cancels the applied field within the conductor
3. The interior of the conductor, and any cavity it encloses, is field-free

This is the principle of the Faraday cage. A closed conductive enclosure provides complete electrostatic shielding regardless of its thickness or the conductivity of the metal. A mesh or perforated screen is not a closed surface, so it is not perfect, but the field that leaks through the openings decays exponentially with distance into the cage, over a scale set by the mesh pitch. A short way inside a fine mesh, the residual static field is negligible.

Quasi-Static Electric Field Shielding

At frequencies where the shield remains small compared with a wavelength, electric field shielding is still dominated by the Faraday cage effect. The charge redistribution time is short compared with the field oscillation period, so the shield continuously adjusts to cancel the external field.

Key considerations for quasi-static electric field shielding:

• Completeness: Apertures admit field in proportion to their size
• Grounding: The shield must connect to a reference so that charge can flow; an isolated shield can have its potential driven by external fields and then couple that potential into the circuit it surrounds
• Conductivity: Any reasonable conductor is adequate, because the charge redistribution time constant is extremely short
• Thickness: Not critical; very thin foils are effective

High-Frequency Electric Field Shielding

At higher frequencies the wave nature of the field becomes important, and shielding effectiveness follows from reflection and absorption. For an electric field source in the near field, the high wave impedance yields a large reflection loss:

R ≈ 20 log₁₀(Z_wave/4|η_s|) dB

Expressed in the same engineering form used for plane waves, with r the source-to-shield distance in meters and f in hertz:

R_E = 322 + 10 log₁₀(σ_r/μ_rf³r²) dB

For copper 1 m from an electric dipole at 1 MHz, this gives about 142 dB, which matches the direct evaluation of 20 log₁₀(Z_wave/4|η_s|) with Z_wave = 18,000 ohms. Because the near-field electric term falls as the cube of frequency, the figure climbs steeply at lower frequencies and shorter distances, passing 200 dB below roughly 12 kHz at a range of 1 m.

The practical limitations on high-frequency electric field shielding are not the metal but:

• Apertures and seams that allow field leakage
• Cable penetrations that carry the disturbance straight through the barrier
• Cavity resonances that enhance internal fields at particular frequencies

Electric Field Shield Design Guidelines

For effective electric field shielding:

• Ensure continuity: Make the shield as complete as possible and minimize apertures
• Bond seams properly: Provide low-impedance connections at intervals much smaller than a wavelength
• Ground the shield: Connect it to circuit common or earth to provide a charge sink
• Filter cable penetrations: Use feedthrough capacitors or filtered connectors
• Any conductor works: Material selection is rarely critical for electric field shielding alone

Static Magnetic Field Shielding

DC and very low-frequency magnetic fields cannot be shielded by conductivity alone. Unlike electric fields, they penetrate a nonmagnetic conductor as though it were not there. The available approaches are flux shunting and active cancellation.

Flux shunting: High-permeability materials such as mu-metal, permalloy, and soft iron offer a low-reluctance path that draws flux through the shield wall rather than through the protected volume. Effectiveness depends on:

• Permeability of the shielding material, where higher is better
• Cross-sectional area of the shield, since thicker walls carry more flux before saturating
• Completeness of the enclosure, since any gap lets flux leak through
• Avoiding magnetic saturation, which requires the applied field to stay within the material's linear range

Nested shields separated by air gaps outperform a single wall of the same total mass, because each layer sees only the residual field that the previous layer passed. Magnetic shields also need annealing after forming, since the cold work of bending or stamping mu-metal sharply reduces its permeability.

Bucking coils: Active cancellation drives currents through coils to create an opposing field. This provides high attenuation over a localized region but requires power, sensing, and control circuitry.

Low-Frequency AC Magnetic Field Shielding

At power-line and low audio frequencies, magnetic field shielding relies on both flux shunting and eddy current effects.

Eddy current shielding: A time-varying magnetic field induces eddy currents in a conductive material, and those currents create an opposing field that partly cancels the applied one. Effectiveness depends on:

• Frequency, since a faster rate of change drives larger currents
• Conductivity, since a better conductor sustains larger currents
• Shield geometry, since eddy currents need closed paths; a slit that interrupts the circulating current destroys the effect

At 60 Hz the skin depth in copper is about 8.5 mm, so a 1 mm copper wall contributes only about 1 dB of absorption. For effective shielding at that frequency:

• Use high-permeability material for flux shunting
• Consider multiple nested shields
• Increase the separation between source and victim, since the near-field magnetic strength of a small loop falls as the cube of distance
• Accept that very high attenuation is impractical

High-Frequency Magnetic Field Shielding

As frequency rises, eddy current shielding becomes increasingly effective. Once the skin depth is small compared with the shield thickness, even ordinary conductors attenuate magnetic fields substantially:

• At 100 kHz the skin depth in copper is about 0.21 mm, so a 1 mm wall gives about 41 dB of absorption
• At 1 MHz the skin depth is about 66 µm, so a 0.5 mm wall gives about 66 dB
• At 10 MHz the skin depth is about 21 µm, so even a 25 µm foil contributes roughly 10 dB on its own

High-permeability materials can add to high-frequency magnetic shielding through their reduced skin depth, but their permeability falls with frequency, and by a few megahertz the nickel-iron alloys retain little of their low-frequency advantage.

Magnetic Shield Design Guidelines

For effective magnetic field shielding:

• Identify the frequency range: The approach varies dramatically with frequency
• For DC and low frequencies: Use high-permeability material in complete enclosures, and consider nested layers
• For the kilohertz range: Combine high permeability for flux shunting with enough thickness for eddy currents
• For high frequencies: Any good conductor several skin depths thick works well
• Avoid saturation: Calculate whether the applied field will saturate a high-permeability wall
• Anneal after forming: Restore the permeability that fabrication removes
• Minimize gaps: Any opening in a magnetic shield lets flux leak through
• Consider active cancellation: For extreme low-frequency requirements, bucking coils may be the only practical answer

Defining Wave Impedance

Wave impedance Z is the ratio of electric field strength to magnetic field strength:

Z = E/H

For a plane wave in free space, this equals the intrinsic impedance:

Z = η₀ = √(μ₀/ε₀) ≈ 377 Ω

For waves from localized sources, the wave impedance varies with distance and source type.

Wave Impedance of Elementary Sources

Electric dipole (rod antenna, high-voltage trace):

At distance r from a small electric dipole:

Z = 377 [1 + 1/(jkr) + 1/(jkr)²]/[1 + 1/(jkr)] Ω

where k = 2π/λ = ω/c is the wavenumber. In the near field, where kr ≪ 1, the squared term dominates and the expression collapses to:

|Z| ≈ 377/kr = (λ/2πr) × 377 Ω

This greatly exceeds 377 ohms at close range.

Magnetic dipole (current loop, transformer):

At distance r from a small current loop the bracketed terms exchange places:

Z = 377 [1 + 1/(jkr)]/[1 + 1/(jkr) + 1/(jkr)²] Ω

which in the near field collapses to:

|Z| ≈ 377kr = (2πr/λ) × 377 Ω

This falls far below 377 ohms at close range. Both expressions converge on 377 ohms once kr grows past unity, which is the far-field limit.

Wave Impedance and Reflection Loss

Reflection loss depends on the mismatch between the source wave impedance and the shield impedance:

R = 20 log₁₀|(Z_wave + η_s)²/4Z_waveη_s| dB ≈ 20 log₁₀(Z_wave/4|η_s|) dB

For high wave impedance, characteristic of electric field sources:

• Large mismatch with the low-impedance conductor
• High reflection loss
• Easy to shield

For low wave impedance, characteristic of magnetic field sources:

• Small mismatch with the low-impedance conductor
• Low reflection loss
• Difficult to shield through reflection alone

Practical Wave Impedance Values

Wave impedance varies over orders of magnitude with frequency and distance. At 1 MHz, where λ = 300 m, a distance of 1 m gives kr = 2π(1)/300 = 0.021, and therefore:

Electric dipole at 1 MHz, 1 meter distance:

|Z| ≈ 377/0.021 ≈ 18,000 Ω

Magnetic dipole at 1 MHz, 1 meter distance:

|Z| ≈ 377 × 0.021 ≈ 7.9 Ω

Either source in the far field, at any distance well beyond λ/2π:

Z ≈ 377 Ω

A spread of more than three orders of magnitude between the two near-field cases, at the same frequency and the same distance, explains why electric and magnetic field shielding demand different approaches.

Implications for Shield Design

Understanding wave impedance guides shield design decisions:

• Mixed-source environments: Design for the worst case, which is normally the magnetic source
• Frequency-dependent behavior: A shield adequate against far-field sources may fail against a near-field magnetic source at the same frequency
• Distance matters: Moving the shield away from a magnetic source raises the wave impedance and improves reflection loss
• Material selection: For magnetic sources, permeability is worth more than conductivity

Fundamental Skin Depth Formula

The skin depth δ is given by:

δ = √(2/ωμσ) = √(ρ/πfμ)

where:

• ω = 2πf is the angular frequency in radians per second
• μ = μ₀μ_r is the absolute permeability in H/m
• σ is the conductivity in S/m
• ρ = 1/σ is the resistivity in ohm-meters
• f is the frequency in hertz

The formula assumes the good-conductor approximation, σ ≫ ωε, which holds for every metal throughout the EMC spectrum.

Skin Depth Calculation Examples

Copper at room temperature, with σ = 5.8 × 10⁷ S/m and μ_r = 1:

δ_Cu ≈ 0.066/√f m, with f in hertz

δ_Cu ≈ 66/√f µm, with f in megahertz

Example values:

• At 1 kHz: δ = 2.1 mm
• At 1 MHz: δ = 66 µm
• At 1 GHz: δ = 2.1 µm

Aluminum at room temperature, with σ ≈ 3.5 × 10⁷ S/m and μ_r = 1:

δ_Al ≈ 0.085/√f m, with f in hertz

Steel, taking σ ≈ 6 × 10⁶ S/m and μ_r ≈ 1,000, a permeability that varies widely in practice:

δ_steel ≈ 0.0065/√f m, with f in hertz

At that permeability, steel has about one-tenth the skin depth of copper despite its much lower conductivity, because skin depth scales inversely with the square root of the permeability-conductivity product. At the lower end of the range, near μ_r = 100, the advantage shrinks to roughly one-third.

Temperature Effects on Skin Depth

Conductivity falls as temperature rises, which increases skin depth:

σ(T) = σ₂₀/[1 + α(T − 20)]

Here σ₂₀ is the conductivity at 20 degrees Celsius, T is the temperature in degrees Celsius, and α is the temperature coefficient of resistance, about 0.004 per degree Celsius for both copper and aluminum. At 100 degrees Celsius:

• Copper conductivity drops to about 76 percent of its room-temperature value
• Skin depth grows by about 15 percent, since it varies as the inverse square root of conductivity

For shields that run hot, whether from ambient conditions or from the dissipation of the equipment inside, this reduction belongs in the calculation.

Permeability Effects on Skin Depth

Magnetic materials have much smaller skin depths because of their permeability. That permeability is not a constant, however: it depends on both frequency and field amplitude.

• The initial relative permeability of mu-metal and similar nickel-iron alloys runs into the tens of thousands at low field levels and low frequencies
• Permeability falls as frequency rises, beginning in the kilohertz range for the highest-permeability alloys; ferrites hold theirs to much higher frequencies but start from far lower values
• Permeability collapses as the field approaches saturation, and mechanical stress from forming reduces it as well

For accurate calculations with magnetic materials:

1. Obtain permeability against frequency curves from the manufacturer rather than using a nominal figure
2. Verify that field levels stay well below saturation
3. Use the permeability appropriate to the operating frequency and field level

Skin Depth in Composite and Coated Materials

Many practical shields use composite materials or thin conductive coatings.

Conductive coatings include zinc or nickel plating on steel, copper plating on plastic, and conductive paints:

• If the coating is much thicker than its skin depth, the coating alone determines the shielding
• If the coating thickness is comparable to its skin depth, both coating and substrate contribute
• Calculate the skin depth in both materials and compare each with its thickness

Conductive composites include carbon-filled plastics and metal-flake loaded polymers:

• Conductivity ranges from a few siemens per meter for carbon-filled compounds to several thousand for heavily metal-loaded ones, still orders of magnitude below bulk metals
• Skin depths can reach centimeters at low frequencies, so absorption is negligible there
• These materials are useful mainly at high frequencies, and mainly where molding a conductive housing is cheaper than fabricating metal

Using Skin Depth in Shield Design

Practical guidelines follow directly from the 8.7 dB contributed by each skin depth of thickness:

• For more than 40 dB of absorption: Make the wall thicker than about 5 skin depths
• For more than 20 dB of absorption: Make the wall thicker than about 2.5 skin depths
• Minimum practical thickness: At least 1 skin depth, below which the multiple reflection correction degrades the result
• Consider the frequency range: Evaluate skin depth at the lowest frequency of concern, where it is largest
• Account for variation: Material property tolerances and plating thickness tolerances both move the answer

Apertures and Seams

Any opening in a shield allows electromagnetic leakage. For a slot much shorter than a wavelength, the shielding effectiveness of the aperture is approximately:

SE_aperture ≈ 20 log₁₀(λ/2ℓ) dB

Here ℓ is the longest dimension of the aperture, not its area. This is the point that catches designers out: a long hairline seam leaks as badly as a square opening of the same length, so narrowing a slot buys almost nothing while shortening it buys a great deal. A 3 cm slot gives only about 14 dB at 1 GHz, where λ = 30 cm, and the expression reaches 0 dB when the slot is half a wavelength long and the aperture resonates.

For high shielding effectiveness:

• Keep every aperture dimension small compared with a wavelength at the highest frequency of concern
• Replace one large opening with many small ones, since many small apertures leak far less than a single aperture of the same total area
• Bond seams at intervals well below λ/10, using gaskets or fasteners to break one long slot into many short ones
• Use waveguide-below-cutoff construction for ventilation

A ventilation opening formed as a tube or a honeycomb cell behaves as a waveguide operated below its cutoff frequency. Well below cutoff the attenuation constant approaches π/a nepers per meter, where a is the width of the guide, which corresponds to about 27 dB for every length of tube equal to that width. A honeycomb panel whose cells are a few millimeters across and several times that in depth therefore supplies very high attenuation across the EMC range while passing air freely.

Cable Penetrations

Cables passing through a shield can defeat it entirely by carrying interference across the barrier as a common-mode current. Proper treatment requires:

• Filtering at the boundary, using feedthrough capacitors, filtered connectors, or common-mode chokes
• Bonding cable shields to the enclosure wall through a full 360-degree connection rather than a pigtail, since a pigtail's inductance destroys the bond above a few megahertz
• Making the filter or the bond where the cable crosses the wall, not somewhere inside
• Minimizing cable length inside the shielded volume

Material and Construction Quality

Practical shielding effectiveness depends on:

• Material conductivity and uniformity, including the integrity of plated or painted coatings
• Surface treatment, since corrosion and anodizing both raise contact resistance across a joint
• Galvanic compatibility between mating metals, since dissimilar pairs corrode and the joint degrades over the life of the product
• Joint design, including overlap, gasket compression, and fastener spacing
• Assembly workmanship, which determines whether the design intent survives production