Calculation of Attenuation of Electromagnetic Waves Penetrating the Shielding Layer

Electromagnetic shielding is a cornerstone of modern EMC (electromagnetic compatibility) engineering. Its purpose is to reduce the intensity of unwanted electromagnetic fields that could interfere with sensitive electronics or violate regulatory limits. The attenuation that a shielding layer provides is governed by three fundamental mechanisms:

• Reflection loss (R) – energy reflected at the interface between the incident medium (usually air) and the shielding material.
• Absorption loss (A) – energy dissipated as heat within the material due to induced eddy currents and magnetic hysteresis.
• Multiple‑reflection loss (B) – additional attenuation caused by waves bouncing back and forth between the two faces of a thin shield before finally being absorbed.

A rigorous understanding of these processes allows designers to predict the shielding effectiveness (SE) of a given material and geometry, and to make informed trade‑offs between cost, weight, and performance.

Shielding Effectiveness: The Core Equation

Shielding effectiveness is defined as the ratio (in decibels) between the field strength measured outside the shield and that measured inside, under identical excitation conditions. When the three loss mechanisms are considered independently, the total SE can be expressed as

SE = R + A + B (dB)

Each term is derived from electromagnetic theory and depends on material properties, geometry, and frequency. The following subsections detail the calculation of each component.

Reflection Loss (R)

Reflection loss arises from the mismatch between the wave impedance of free space (≈ 377 Ω) and the intrinsic impedance of the shielding material. The magnitude of R depends on whether the incident wave is in the far field or near field, and whether the field is predominantly electric or magnetic.

• (f) – frequency (Hz)
• (r) – distance from source to shield (m)
• (\mu_r) – relative permeability of the material
• (\sigma_r) – relative conductivity (with copper as the reference, (\sigma_r = 1))

These empirical expressions capture the fact that highly conductive, high‑permeability materials produce large reflection losses, especially at higher frequencies.

Absorption Loss (A)

Absorption loss is independent of the wave impedance and depends solely on how deeply the wave penetrates the material. The key parameter is the skin depth (\delta):

[
\delta = \frac{1}{\sqrt{\pi f \mu \sigma}}
]

where (\mu = \mu_0 \mu_r) and (\sigma = \sigma_0 \sigma_r). The absorption loss in decibels is then

[
A = 8.686 \frac{t}{\delta} = 131.4, t \sqrt{f,\mu_r,\sigma_r}
]

with (t) expressed in meters. The second form shows that increasing thickness, conductivity, or permeability all raise A, while higher frequencies reduce the skin depth and thus increase absorption.

Multiple‑Reflection Loss (B)

When a shield is thin or the absorption loss is modest (typically (A < 10) dB), a fraction of the transmitted wave can bounce between the two faces before being absorbed. This effect is captured by

[
B = 20 \log_{10}!\left(1 - e^{-2A/8.686}\right)
]

Because the exponential term is always less than one, (B) is negative, indicating that internal reflections actually decrease the overall shielding effectiveness. In most practical cases where (A > 10) dB, the B term can be neglected.

Worked Example

Let us calculate the SE of a 0.2 mm copper foil at 1 MHz in the far‑field.

1. Reflection Loss

[
R = 168 - 10 \log_{10}!\left(\frac{1\times10^6 \times 1}{1}\right)
= 168 - 60
= 108\ \text{dB}
]

2. Absorption Loss

[
A = 131.4 \times 0.2\times10^{-3} \times \sqrt{1\times10^6 \times 1 \times 1}
= 26.28\ \text{dB}
]

3. Multiple‑Reflection Loss

Since (A = 26.28) dB > 10 dB, (B) is negligible.

4. Total Shielding Effectiveness

[
\text{SE} = R + A = 108 + 26.28 = 134.28\ \text{dB}
]

The result demonstrates that at 1 MHz a thin copper foil provides excellent shielding, with reflection loss dominating the attenuation.

Practical Design Considerations

While the equations above give a solid theoretical baseline, real‑world shielding rarely achieves the ideal values. Several non‑idealities must be addressed:

• Gaps and seams – Any opening larger than a small fraction of the wavelength can act as an antenna. A common rule of thumb is to keep gaps less than (\lambda/20). Over‑engineering the seam (e.g., overlapping, using conductive gaskets) mitigates leakage.
• Material imperfections – Surface roughness, oxidation, and solder joints can introduce additional resistance, reducing effective conductivity and increasing skin depth.
• Low‑frequency performance – At low frequencies, the wave impedance of free space is very low, so reflection loss is minimal. Designers must rely on absorption, which requires high‑permeability alloys (e.g., mu‑metal, permalloy) and sufficient thickness.
• Multiple reflections in thin coatings – Conductive paints or sputtered layers often have thicknesses below the skin depth at high frequencies. In such cases, the B term becomes significant and can reduce SE by several decibels.
• Thermal and mechanical constraints – Increasing thickness or using high‑permeability materials may conflict with weight, size, or thermal dissipation requirements. A balanced trade‑off is essential.

Conclusion

Accurate calculation of electromagnetic wave attenuation through a shielding layer is essential for reliable EMC design. By quantifying reflection, absorption, and multiple‑reflection losses, engineers can predict shielding effectiveness, select appropriate materials, and optimize thickness. However, the theoretical models must be complemented with practical measures—tight seams, proper material handling, and frequency‑specific strategies—to ensure that the final product meets real‑world performance targets.