Shielding

Abstract

In the field of EMC, shields are used to:

• 1. Reduce electromagnetic emissions from a product.

• 2. Increase immunity against electric, magnetic, and/or electromagnetic radiation.

The shielding theory presented in this book is based on the accepted shielding theory for electromagnetic waves, initially proposed by Schelkunoff ((1943) Electromagnetic waves. D. van Nostrand Company Inc, New York, pp 303–312) in 1943. The formulas in this chapter are approximations for shields with high electrical conductivity. Before we jump into the theory of shielding, here are two practical pieces of advice:

• 1. Cables and wires. Every single signal which enters and/or leaves a shielded enclosure must be filtered or shielded. In case the cable is shielded, contact the cable shield 360^∘ with the shielded enclosure.

• 2. Slots and apertures. Slots and apertures reduce the shielding effectiveness SE or even lead to higher emissions than without the shield in case of resonances inside a shielding enclosure Hubing ((2021) EMC Question of the Week: 2017–2020. LearnEMC, LLC, Stoughton). If the linear dimension l [m] of a slot or aperture is larger than λ∕2, the shield is assumed to be useless Ott ((2009) Electromagnetic compatibility engineering. Wiley, New York).

Action expresses priorities.

—Mahatma Gandhi

In the field of EMC, shields are used to:

• 1. Reduce electromagnetic emissions from a product.

• 2. Increase immunity against electric, magnetic, and/or electromagnetic radiation.

The shielding theory presented in this book is based on the accepted shielding theory for electromagnetic waves, initially proposed by Schelkunoff ((1943) Electromagnetic waves. D. van Nostrand Company Inc, New York, pp 303–312) in 1943. The formulas in this chapter are approximations for shields with high electrical conductivity. Before we jump into the theory of shielding, here are two practical pieces of advice:

• 1. Cables and wires. Every single signal which enters and/or leaves a shielded enclosure must be filtered or shielded. In case the cable is shielded, contact the cable shield 360^∘ with the shielded enclosure.

• 2. Slots and apertures. Slots and apertures reduce the shielding effectiveness SE or even lead to higher emissions than without the shield in case of resonances inside a shielding enclosure Hubing ([1] EMC Question of the Week: 2017–2020. LearnEMC, LLC, Stoughton). If the linear dimension l [m] of a slot or aperture is larger than λ∕2, the shield is assumed to be useless Ott ([3] Electromagnetic compatibility engineering. Wiley, New York).

13.1 Shielding Theory

Shielding of electromagnetic waves is usually achieved by:

• Reflection. Let us suppose an electromagnetic wave that im**es on a shield with an intrinsic impedance lower than the wave impedance: $| \underline {\eta }_s| < | \underline {Z}_w|$. In that case, the E-field is partially reflected at the shield’s outer surface, and the H-field is partially reflected at the inner surface (see Fig. 13.2) [5].

• Absorption. When an electromagnetic wave propagates through a lossy media (like a good conductor with α > 0), the amplitudes of the electric E-field and the magnetic H-field decrease exponentially with e ^−αt, where α [1/m] is the attenuation coefficient and t [m] the shield thickness (see Fig. 13.1).

  Fig. 13.1

  

  Attenuation of a sinusoidal electromagnetic wave due to absorption loss. The electromagnetic wave comes from the left and im**es on a thick metal shield in the far-field (symbolic drawing; reflection is not shown here)

  Fig. 13.2

  

  Shield with intrinsic impedance $\underline {\eta }_s$ [Ω] and thickness t [m]. Reflection happens at the left boundary (primarily for the E-field) and the right boundary (primarily for the H-field) [4]

13.2 Shielding Effectiveness

Let us have a look at Fig. 13.2 and let us define a few terms. The shielding effectiveness (SE) describes how well a shield attenuates an incident wave (electrical field strength E _i [V/m], magnetic field strength H _i [A/m]) when propagating through a shield. After passing through a shield, the remaining wave has a field strength of E _t [V/m] and H _t [A/m]. The reflected wave has field strength E _r [V/m] and H _r [A/m]. Shielding effectiveness is now defined as [4]:

$$\displaystyle \begin{aligned} \text{SE}_{E,dB} &= 20 \log_{10}\left|\frac{E_i}{E_t}\right| \end{aligned} $$

(13.1)

$$\displaystyle \begin{aligned} \text{SE}_{H,dB} &= 20 \log_{10}\left|\frac{H_i}{H_t}\right| \end{aligned} $$

(13.2)

where:

• |E _i| = electric field strength of the incident wave to the shield barrier in [V/m]

• |E _t| = electric field strength of the wave when it leaves the shield barrier in [V/m]

• |H _i| = magnetic field strength of the incident wave to the shield barrier in [A/m]

• |H _t| = magnetic field strength of the wave when leaving the shield in [A/m]

• SE_E,dB = shield effectiveness for the E-field in [dB]

• SE_H,dB = shield effectiveness for the H-field in [dB]

The shielding effectiveness of the E-field and the H-field compared:

• SE _{E, dB} = SE _{H, dB}. In the case of a uniform plane wave and identical media on both sides of the shield barrier, the shielding effectiveness for the E-field and the H-field are equivalent [4].

• SE _{E, dB} ≠ SE _{H, dB}. In the case of near-fields and/or different media on both sides of the shield barrier, the shielding effectiveness of the E-field and the H-field are not equivalent [4] because of the different reflection losses R _dB [dB].

The shielding effectiveness is the sum of reflection loss R _dB [dB], absorption loss A _dB [dB], and the multiple-reflection loss correction M _dB [dB] (Fig. 13.9).

(13.3)

Assuming a uniform plane wave arriving perpendicular to a solid shield, which does not have any slots or apertures, we can write [4]:

$$\displaystyle \begin{aligned} R_{dB} &= 20 \log_{10}\left|\frac{\left(\underline{Z}_w+\underline{\eta}_s\right)^2}{4\underline{Z}_w\underline{\eta}_s}\right| {} \end{aligned} $$

(13.4)

$$\displaystyle \begin{aligned} A_{dB} &= 20 \log_{10}\left|e^{\alpha t}\right| {} \end{aligned} $$

(13.5)

$$\displaystyle \begin{aligned} M_{dB} &= 20 \log_{10}\left|1-\left(\frac{\underline{Z}_w-\underline{\eta}_s}{\underline{Z}_w+\underline{\eta}_s}\right)^2 e^{-2\alpha t}\right| {}\end{aligned} $$

(13.6)

where:

• R _dB = reflection loss of both boundaries of the shield (attenuation due to reflection of the incident wave at the boundaries of the shield, E-fields are primarily reflected at the first boundary—when entering the shield, whereas H-fields are primarily reflected at the second boundary—when leaving the shield) in [dB]

• A _dB = absorption loss (attenuation due to power converted to heat as the wave propagates through the shield, absorption loss is more critical for H-fields than for E-fields because the H-field reflection loss primarily happens at the second boundary when leaving the shield) in [dB]

• M _dB = multiple-reflection loss correction (applicable for very thin shields t < δ, where absorption loss is low and therefore the energy of the H-field at the second boundary is still significant) in [dB]

• $\underline {Z}_w = \eta _0 \approx {377}\,{{\Omega }}$ wave impedance of the incident wave in [Ω]

• $\underline {\eta }_s =$ intrinsic impedance of the shield material in [Ω]

• t = thickness of the shield in [m]

• α = attenuation coefficient of the shield material in [1/m]

Looking at the equations above, the following can be said:

• Reflection loss R.

  • R dominates at lower frequencies because the absorption loss A is low.

  • R in the near-field differs with changing radiation source impedance.

  • R in the near-field is different from R in the far-field.

• Absorption loss A.

  • A dominates at higher frequencies, where the skin depth is small.

  • A is identical for near-field and far-field radiation.

• Multiple-reflection loss correction M.

  • M can be neglected for good conducting (|η _s|≪|Z _w|) and thick shields (t ≫ δ), where A _dB < 15 dB.

  • M must be considered for thin shields (t ≪ δ), where multiple reflections inside the shield reduce the shielding effectiveness.

13.3 Far-Field Shielding

Figure 13.3 presents the absorption, reflection, and multiple-reflection loss correction of a solid shield in the far-field . It can be seen how the reflection loss dominates for low frequencies and the absorption loss for high frequencies.

Fig. 13.3

Shielding effectiveness of a solid aluminum shield with t = 0.1 mm in the far-field

Fig. 13.4

Reflection loss R _dB [dB] for various wave impedances $\underline {Z}_w$ [Ω]

13.3.1 Reflection Loss R for Far-Field Shielding

The reflection loss R _dB [dB] of a solid shield with good conductivity ($| \underline {\eta }_s| \ll |\eta _0|$) in the far-field can be approximated as:

$$\displaystyle \begin{aligned} R &\approx \left|\frac{\eta_0}{4\underline{\eta}_s}\right| = \sqrt{\frac{\sigma}{4 \pi f \mu_r^{\prime} \epsilon_0}} \end{aligned} $$

(13.7)

$$\displaystyle \begin{aligned} R_{dB} &\approx 20 \log_{10}\left|\frac{\eta_0}{4\underline{\eta}_s}\right| = 20 \log_{10}\left[\sqrt{\frac{\sigma}{4 \pi f \mu_r^{\prime} \epsilon_0}}\right] \end{aligned} $$

(13.8)

where:

• R = reflection loss for a plane wave and a solid shield with good conductivity

• η ₀ = μ ₀∕ε ₀ ≈377 Ω = intrinsic impedance of free-space in [Ω]

• $| \underline {\eta }_s| = |(1+j)\sqrt {\pi f \mu _0\mu _r^{\prime }/\sigma }| = \sqrt {2\pi f \mu _0 \mu _r^{\prime } / \sigma } =$ shield impedance in [Ω]

• σ = specific conductance of the shield material in [S/m]

• f = frequency of the sinusoidal plane wave in [Hz]

• $\mu _r^{\prime } =$ relative magnetic permeability of the shield material

• ε ₀ = 8.854 ⋅ 10⁻¹² F/m = permittivity of vacuum, absolute permittivity

13.3.2 Absorption Loss A for Far-Field Shielding

The absorption loss A _dB [dB] of a solid shield barrier of thickness t [m] and of good conductivity (α = 1∕δ) in the far-field can be approximated as:

$$\displaystyle \begin{aligned} A &\approx e^{\alpha t} = e^{t/\delta} \end{aligned} $$

(13.9)

$$\displaystyle \begin{aligned} A_{dB} &\approx 20 \log_{10}e^{t/\delta} = 8.7\cdot \frac{t}{\delta} = 8.7\cdot t \cdot \sqrt{\pi f \mu_0 \mu_r^{\prime} \sigma} {} \end{aligned} $$

(13.10)

where:

• A = absorption loss for a plane wave and a solid shield with good conductivity

• α = 1∕δ = attenuation coefficient of the shield material in [1/m]

• t = thickness of the shield in [m]

• $\delta = 1/\sqrt {\pi f \mu _0 \mu _r^{\prime } \sigma } =$ skin depth for a wave with frequency f in [m]

• f = frequency of the sinusoidal plane wave in [Hz]

• $\mu _r^{\prime } =$ relative magnetic permeability of the shield material

• μ ₀ = 12.57 ⋅ 10⁻⁷ H/m = permeability of vacuum, absolute permeability

• σ = specific conductance of the shield material in [S/m]

13.3.3 Multiple-Reflection Loss Correction M for Far-Field Shielding

The multiple-reflection loss correction M _dB [dB] must be considered for thin shields, where the absorption loss A _dB [dB] is below 15 dB. For a plane wave in the far-field and a solid shield barrier of good conductivity, M _dB [dB] can be approximately calculated as [3]:

$$\displaystyle \begin{aligned} M &\approx 1-e^{-2t/\delta} \end{aligned} $$

(13.11)

$$\displaystyle \begin{aligned} M_{dB} &\approx 20 \log_{10}\left(1-e^{-2t/\delta}\right) {} \end{aligned} $$

(13.12)

where:

• M = multiple-reflection loss correction for a plane wave and a thin solid shield with good conductivity

• t = thickness of the shield in [m]

• $\delta = 1/\sqrt {\pi f \mu _0 \mu _r^{\prime } \sigma } =$ skin depth for a wave with frequency f in [m]

13.4 Near-Field Shielding

Figure 13.4 shows that the reflection loss R _dB [dB] for electric dipole antennas in the near-field is much higher compared to the far-field and compared to magnetic dipoles in the near-field.

Fig. 13.5

Approximate shielding effectiveness of a solid aluminum shield with thickness t = 1 mm. (a) In the far-field. (b) In the near-field of an electric dipole at distance d = 10 mm. (c) In the near-field of a magnetic dipole at distance d = 10 mm

Comparing near-field vs. far-field shielding, we can say:

• Reflection loss. R _dB [dB] is affected by the wave impedance $\underline {Z}_w$ [Ω] (see Eq. 13.4). Therefore, the reflection loss R _dB [dB] for near-fields is different from far-fields and different for low impedance sources (see Sect. 13.4.2) compared to high impedance sources (see Sect. 13.4.1).

• Absorption loss. A _dB [dB] is unaffected by the wave impedance $\underline {Z}_w$ [Ω] (see Eq. 13.5). Therefore, the absorption loss A _dB [dB] is identical for near-field and far-field shielding.

• Multiple-reflection loss correction. M _dB [dB] is affected by the wave impedance $\underline {Z}_w$ [Ω] (see Eq. 13.6). However, the influence of the wave impedance on M _dB [dB] is only minor. Therefore, in most cases, it can be assumed that M _dB [dB] is not affected significantly by the wave impedance.

13.4.1 Near-Field Shielding of Electric Sources

To calculate the approximate absorption loss A _dB [dB] and multiple-reflection loss correction M _dB [dB], Eqs. 13.10 and 13.12 can be used. R _dB,E [dB] is the reflection loss when shielding a near-field wave from an electric dipole (predominant E-field) and it is approximately:

$$\displaystyle \begin{aligned} R_e &\approx \left|\frac{\underline{Z}_{we}}{4\underline{\eta}_s}\right| \end{aligned} $$

(13.13)

$$\displaystyle \begin{aligned} R_{dB,e} &\approx 20 \log_{10}\left|\frac{\underline{Z}_{we}}{4\underline{\eta}_s}\right| = 244 + 10\log_{10}\left(\frac{\sigma}{f^3\epsilon_{rw}^{\prime2}\mu_{rs}^{\prime}d^2}\right) \end{aligned} $$

(13.14)

where:

• $| \underline {Z}_{we}| = 1/(2\pi f \epsilon _{rw}^{\prime } \epsilon _0 d) =$ near-field wave impedance of E-field antenna in [Ω]

• $| \underline {\eta }_s| = |(1+j)\sqrt {\pi f \mu _0\mu _r^{\prime }/\sigma }| = \sqrt {2\pi f \mu _0 \mu _r^{\prime } / \sigma } =$ shield impedance in [Ω]

• σ = specific conductance of the shield material in [S/m]

• f = frequency of the sinusoidal plane wave in [Hz]

• $\mu _{rs}^{\prime } =$ relative magnetic permeability of the shield material

• $\epsilon _{rw}^{\prime } =$ relative electric permittivity of the medium through which the near-field wave is propagating (can be assumed to be 1.0 in most cases)

• d = distance from the source antenna (electric dipole) to the shield surface in [m]

The near-field to far-field boundary and wave impedances are discussed in Sects. 8.3 and 8.5.

13.4.2 Near-Field Shielding of Magnetic Sources

To calculate the approximate absorption loss A _dB and multiple-reflection loss correction M _dB [dB], Eqs. 13.10 and 13.12 can be used. R _dB,M [dB] is the reflection loss when shielding a near-field wave from a magnetic dipole (predominant H-field) and it is approximately:

$$\displaystyle \begin{aligned} R_m &\approx \left|\frac{\underline{Z}_{wm}}{4\underline{\eta}_s}\right| \end{aligned} $$

(13.15)

$$\displaystyle \begin{aligned} R_{dB,m} &\approx 20 \log_{10}\left|\frac{\underline{Z}_{wm}}{4\underline{\eta}_s}\right| = -63 + 10\log_{10}\left(\frac{\sigma f d^2 \mu_{rw}^{\prime2}}{\mu_{rs}^{\prime}}\right) {} \end{aligned} $$

(13.16)

where:

• $| \underline {Z}_{wm}| = 2\pi f \mu _{rw}^{\prime } \mu _0 d =$ near-field wave impedance of H-field antenna in [Ω]

• $| \underline {\eta }_s| = |(1+j)\sqrt {\pi f \mu _0\mu _r^{\prime }/\sigma }| = \sqrt {2\pi f \mu _0 \mu _r^{\prime } / \sigma } =$ shield impedance in [Ω]

• σ = specific conductance of the shield material in [S/m]

• f = frequency of the sinusoidal plane wave in [Hz]

• $\mu _{rs}^{\prime } =$ relative magnetic permeability of the shield material

• $\mu _{rw}^{\prime } =$ relative magnetic permeability of the medium through which the near-field wave is propagating (can be assumed to be 1.0 in most cases)

• d = distance from the source antenna (electric dipole) to the shield surface in [m]

Note: Eq. 13.16 is an approximation and assumes $| \underline {Z}_{wm}| \gg | \underline {\eta }_{s}|$. In case of a negative reflection loss R _dB,m, set R _dB,m = 0 and $M_{dB_m} = 0$. In case R _dB,m is positive and near zero, Eq. 13.16 is in slight error [3].

The near-field to far-field boundary and wave impedances are discussed in Sects. 8.3 and 8.5.

13.4.3 Low-Frequency Magnetic Field Shielding

Low-frequency (f < 100 kHz) magnetic field waves are the most difficult waves to shield because the absorption loss is low due to the low-frequency f [Hz] and the reflection loss is low due to the low wave impedance $\underline {Z}_{w}$ [Ω]. Figure 13.5 illustrates this problem for a solid 1 mm aluminum shield in the far-field and the near-field.

Fig. 13.6

Shielding of low-frequency magnetic field with high-permeability shield material (symbolic)

Therefore, two additional methods for shielding low-frequency magnetic fields are presented here:

• Shields with $\boldsymbol {\mu }_{\mathbf {r}}^{\prime } \gg \mathbf {1.0}$. Use a shielding material with high magnetic permeability $\mu _r^{\prime } \gg 1.0$ (e.g., mu-metal; see Sect. G.1). The drawing in Fig. 13.6 should illustrate how a low-reluctance shield will divert the magnetic flux in an environment with $\mu _r^{\prime }=1$. Note: magnetic permeability decreases with increasing frequency f [Hz] and with increasing magnetic field strength H [A/m] or magnetic flux Φ [Wb], respectively.

  Fig. 13.7

  

  Shielding of low-frequency magnetic field with a shorted turn (conductor loop)

• Shorted turn method. Use a loop conductor which is placed in the magnetic field H [A/m]. The induced current $\underline {I}_{ind}$ [A] in the loop conductor will generate a counter magnetic field that leads to a reduced magnetic field in the vicinity of the loop (see Fig. 13.7).

  Fig. 13.8

  

  Apertures in shields. (a) Single aperture of maximum linear dimension l [m]. (b) Linear array of identical and closely spaced apertures. (c) Multidimensional array of identical and closely spaced apertures

13.5 Slots and Apertures

Slots and apertures can cause considerable leakage if their largest dimension is l > (λ∕10). In addition, they are efficient radiators (yes, radiators!) when their maximum linear dimension l [m] is equal to λ∕2 [2]. Thus, if a slot or aperture has a linear dimension of l ≥ λ∕2, the shielding effectiveness SE_dB [dB] can be assumed to be 0 dB.

At high frequencies (f > 1 MHz), slots and apertures are usually of more concern than the attenuation or reflection loss of a shield, because the sum of attenuation and reflection loss of any reasonably thick (t ≫ δ) and conductive shield ($| \underline {\eta }_s|\ll | \underline {Z}_w|$) is higher than 100 dB (see Fig. 13.5). In other words, in cases where the intrinsic SE of a shielding material is high, the SE is defined by the slots and apertures of the shield.

13.5.1 Single Aperture

If we assume a shield with high intrinsic SE, so that the SE is defined by the slots and apertures of the shielding enclosure, the SE can be calculated based on the maximum linear dimension l [m] (which must be less than λ∕2). In the case of one single aperture, we can write [3]:

$$\displaystyle \begin{aligned} \text{SE}_{dB} \approx 20 \log_{10}\left(\frac{\lambda}{2l}\right) {} \end{aligned} $$

(13.17)

where:

• λ = wavelength of the sinusoidal wave in [m]

• l = maximum linear dimension of one single aperture (l < λ∕2) in [m]

13.5.2 Multiple Apertures

If there is not only one aperture but multiple apertures—as shown in Fig. 13.8b—the shielding effectiveness will be reduced even further. The shielding effectiveness of a linear array of equally and closely spaced apertures n of maximum length l [m]—where the total array length l _array [m] is less than λ∕2—is approximately [3]:

$$\displaystyle \begin{aligned} \text{SE}_{dB} \approx 20 \log_{10}\left(\frac{\lambda}{2l}\right) - 20 \log_{10}\sqrt{n} = 20 \log_{10}\left(\frac{\lambda}{2l\sqrt{n}}\right) {} \end{aligned} $$

(13.18)

where:

• λ = wavelength of the sinusoidal wave in [m]

  Fig. 13.9

  

  Shielding effectiveness in the case of a single slot (l < (λ∕2)) and high intrinsic SE of the shield material

• l = maximum linear dimension of one single aperture with l < λ∕2 in [m]

• n = number of closely spaced, identical apertures in an array with l _array < λ∕2

In the case of a multidimensional array of m rows (and m < n; see Fig. 13.8c), the additional rows (second, third, . . .) will not reduce the SE significantly [3]. Thus, the SE of a multidimensional array of equal-sized apertures is the SE of one single hole, minus the SE reduction of the first row of n apertures (see Eq. 13.18, Fig. 13.10).

Fig. 13.10

Shielding effectiveness reduction of an array of n identical and closely spaced apertures in comparison with one single aperture

Apertures located on different surfaces, which all look in different directions, do not decrease the overall shielding effectiveness because they radiate in different directions [3]. Thus, it is a good idea to distribute apertures around the surface of a shielding enclosure.

13.5.3 Waveguide Below Cutoff

The shielding effectiveness of an aperture can be improved by adding a depth to the aperture. Figure 13.11 shows apertures, which have been extended to waveguides . The concept is as follows: the waveguide has a defined cutoff frequency f _c [Hz], and when a wave with frequency f ≪ f _c enters the waveguide, it will be attenuated. Above the cutoff frequency f _c [Hz], the waveguide is very efficient at passing all frequencies and thus does not improve the SE of the aperture anymore.

Fig. 13.11

Apertures shaped to form a waveguide. (a) Circular aperture with waveguide. Left: cross-sectional view. Right: frontal view. (b) Rectangular aperture with waveguide. Left: cross-sectional view. Right: frontal view

First, we have to determine the cutoff frequency f _c [Hz] of the waveguide aperture, which is given for free-space (air, where v ≈ c) as [3, 4]:

$$\displaystyle \begin{aligned} f_{c,cwg} &\approx \frac{1.8412 \cdot v}{\pi D} = \frac{1.8412 \cdot c}{\pi D} \end{aligned} $$

(13.19)

$$\displaystyle \begin{aligned} f_{c,rwg} &\approx \frac{v}{2w} = \frac{c}{2w} \end{aligned} $$

(13.20)

where:

• f _c,cwg = waveguide cutoff frequency of a circular waveguide in [Hz]

• f _c,rwg = waveguide cutoff frequency of a rectangular waveguide in [Hz]

• v = velocity of the wave in [m/sec]

• $c = 1/(\sqrt {\mu _0\epsilon _0}) = 2.998 \cdot 10^{8}$ m/sec = speed of light

• w = largest inner side length (width) of rectangular waveguide in [m]

• D = inner diameter of circular waveguide in [m]

Now, we can calculate the shielding effectiveness SE_dB of n identical and closely spaced waveguide apertures in case the wave has a frequency f ≪ f _c [4] [3]:

$$\displaystyle \begin{aligned} \text{SE}_{dB,cwg} &\approx 20 \log_{10}\left(\frac{\lambda}{2l\sqrt{n}}\right) + 32 \frac{t}{w} \end{aligned} $$

(13.21)

$$\displaystyle \begin{aligned} \text{SE}_{dB,rwg} &\approx 20 \log_{10}\left(\frac{\lambda}{2l\sqrt{n}}\right) + 27.3 \frac{t}{D} \end{aligned} $$

(13.22)

where:

• SE_dB,cwg = shielding effectiveness of a circular waveguide aperture in [dB]

• SE_dB,rwg = shielding effectiveness of a rectangular waveguide aperture in [dB]

• λ = wavelength of the sinusoidal wave in [m]

• l = maximum linear dimension of one single aperture (l < λ∕2) in [m]

• w = largest inner side length (width) of rectangular waveguide in [m]

• D = inner diameter of circular waveguide in [m]

• t = depth of the waveguide in [m]

• n = number of closely spaced, identical apertures in an array with l _array < λ∕2 (set n = 1 in case of a single aperture)

13.6 Grounding of Shields

First, a disclaimer: this section is about solid shielding enclosures and not about shielded cables. Grounding of cable shields is discussed in Sect. 13.7.4.

A solid shielding enclosure does not need to be grounded to be an effective shield for electromagnetic waves from outside of the enclosure to the inside, and vice versa [3]. However, metallic shields are often grounded for safety reasons and to protect the inside from electrostatic charges.

In case cables are leaving or entering the shield, consider these points:

• Unshielded cables. The shielding enclosure and the circuit ground must be connected with low impedance close to where the cable leaves and/or enters the enclosure. This way, the common-mode voltage which may drive the cable conductors is reduced to a minimum. In addition, every signal which enters and/or leaves a shielded enclosure should be low-pass filtered.

• Shielded cables. The cable shield should be connected with low impedance to the outside of the shielding enclosure. In this case, the cable shield can be considered as the extension of the shielded enclosure.

Consider the galvanic series to prevent corrosion when connecting dissimilar metals for grounding (e.g., aluminum and steel; see Table H.2). This is especially important in the case of safety grounds.

13.7 Cable Shields

13.7.1 Transfer Impedance Z _t

The transfer impedance $\underline {Z}_t$ [Ω/m] is a property of shielded cables , and it is used to measure the shielding effectiveness of cable shields. The transfer impedance describes the magnetic coupling between the noise current $\underline {I}$ [A] along the shield to the induced voltage $\underline {V}$ [V] to the inner conductor per length l [m]. The smaller the transfer impedance $\underline {Z}_t$ [Ω/m], the more effective the shield.

Figure 13.12 shows the schematic diagram of a transfer impedance measurement setup of a cable shield. $\underline {Z}_t$ [Ω/m] can be calculated as [6]:

$$\displaystyle \begin{aligned} \underline{Z}_t = \frac{\underline{V}}{l\cdot \underline{I}} \end{aligned} $$

(13.23)

where:

• $\underline {V} =$ measured voltage between the center conductor and the inner surface of the shield in [V]

  Fig. 13.12

  

  Transfer impedance $\underline {Z}_t$ [Ω/m] measurement circuit

• $\underline {I} =$ test current in the shield in [A]

• l = length of the shielded cable under test in [m]

The transfer impedance $\underline {Z}_t(f)$ [Ω/m] is a function of frequency f [Hz]. At low frequencies f [Hz], $\underline {Z}_t \cdot l$ is equal the direct current resistance R _DC [Ω] of the shield. At high frequencies (> 1 MHz), $\underline {Z}_t$ [Ω/m] of a solid shield decreases because of the skin effect (current flows at outside layer of the shield), whereas for braided shields, $\underline {Z}_t$ [Ω/m] increases at high frequencies (see Fig. 13.13). Remember: a low transfer impedance $\underline {Z}_t$ [Ω/m] means high shielding effectiveness.

Fig. 13.13

Typical normalized transfer impedance of a solid shield and a braided shield

13.7.2 Cable Shielding Against Capacitive Coupling

Figure 13.14 shows a setup of two closely spaced circuits, where circuit 1 is the noise source and circuit 2 is the victim. The noise is capacitively coupled from circuit 1 to circuit 2, where circuit 2 is shielded and the shield is grounded. Whether the shield is grounded at both ends or just at one end does not matter in this case, because the shielding effectiveness against capacitively coupled noise does not depend on a closed-loop shield current; the shielding effectiveness against capacitively coupled noise does mainly depend on the impedance of the shield to ground (the lower the impedance, the better), the value of the stray coupling capacitance C ₁₂ [F] (the lower, the better), and the victim’s impedance $| \underline {Z}_2|$ (the lower, the better).

Fig. 13.14

Simplified circuit representation of cable shielding against capacitive coupling

In the case of a setup like shown in Fig. 13.14, where the shield is grounded with very low impedance, the noise voltage $\underline {V}_n$ [V] can be calculated as:

$$\displaystyle \begin{aligned} \underline{V}_n = \underline{V}_1 \frac{\frac{1}{\frac{1}{\underline{Z}_2}+j\omega C_{2S}+j\omega C_{2G}}}{\frac{1}{j\omega C_{12}}+\frac{1}{\frac{1}{\underline{Z}_2}+j\omega C_{2S}+j\omega C_{2G}}} = \underline{V}_1 \frac{j\omega C_{12} \underline{Z}_2}{1+j\omega\underline{Z}_2(C_{2G}+C_{2S})} {} \end{aligned} $$

(13.24)

where:

• $\underline {V}_1 =$ voltage of the noise source (circuit 1) in [V]

• $\underline {Z}_2 =$ impedance of the victim circuit to ground in [Ω]

• C _1G = capacitance of circuit 1 (noise source) to ground (neglected here, because C _1G is parallel to the ideal voltage source $\underline {V}_1$) in [F]

• C _2G = capacitance of circuit 2 (victim) to ground in [F]

• C _1S = capacitance of circuit 1 (noise source) to ground (neglected here, because C _1S is parallel to the ideal voltage source $\underline {V}_1$ in case the shield is grounded) in [F]

• C _2S = capacitance of circuit 2 (victim) to ground (parallel to $\underline {Z}_2$ in case the shield is grounded) in [F]

• C ₁₂ = coupling stray capacitance between circuit 1 and circuit 2 in [F]

Equation 13.24 and Fig. 13.15 lead us to the conclusion that effective shielding against capacitive coupling is all about minimizing the coupling stray capacitance C ₁₂ [F]. Thus, the following two points are important to remember when it comes to shielding against electric field coupling:

• Shield grounding. The cable shield must be grounded with low impedance. Otherwise, the total coupling capacitance would increase from C _coupling = C ₁₂ to C _coupling = C ₁₂ + 1∕(1∕C _1S + 1∕C _2S).

  Fig. 13.15

  

  Frequency response of capacitively coupled noise voltage $| \underline {V}_n|$ [V] to a shielded cable (where the shield is a good conductor and grounded)

• Minimize protruding conductors. In order to minimize the coupling capacitance C ₁₂ [F], the length of the shielded conductors that extend beyond the shield must be minimized.

13.7.3 Cable Shielding Against Inductive Coupling

Figure 13.16 shows a setup of two closely spaced circuits, where circuit 1 is the noise source and circuit 2 is the victim. The noise is inductively coupled from circuit 1 to circuit 2, where circuit 2 is shielded, and the shield is grounded at both ends. If the shield is made of a nonmagnetic material and the shield would only be connected to ground at one end, there would be no shielding effect, because the shielding effectiveness against inductively coupled noise depends on the induced shield current $\underline {I}_S$ [A] (in the case of a nonmagnetic shield material).

Fig. 13.16

Simplified circuit representation of cable shielding against inductive coupling

The shielding effect due to the induced shield current is explained like this:

• 1. Noise voltage induction. Current $\underline {I}_1$ [A] of the noise source induces via mutual inductance M ₁₂ [H] a noise voltage $\underline {V}_{12}$ [V] in the conductor 2.

• 2. Shield current induction. Current $\underline {I}_1$ [A] of the noise source induces via mutual inductance M _1S [H] a voltage $\underline {V}_{1S}$ [V] in the shield. This voltage $\underline {V}_{1S}$ [V] leads to the shield current $\underline {I}_S [A]$, which flows along the cable shield and through the ground.

• 3. Noise voltage compensation. The induced shield current $\underline {I}_S$ [A] induces a so-called compensation voltage $\underline {V}_{S2}$ [V] in the conductor 2. Ideally, the compensation voltage $\underline {V}_{S2}$ [V] has the opposite polarity to the induced noise voltage $\underline {V}_{12}$ [V]. Thus, the induced shield current $\underline {I}_S$ [A] leads to a shielding effect.

The induced noise voltage $\underline {V}_{12}$ [V] from circuit 1 (noise source) to circuit 2 (victim) is:

$$\displaystyle \begin{aligned} \underline{V}_{12} = j \omega M_{12} \underline{I}_1 \end{aligned} $$

(13.25)

where:

• M ₁₂ = Φ ₁₂∕i ₁(t) = mutual inductance between circuit 1 and circuit 2 in [H]

• $\underline {I}_1 =$ current of the noise source (circuit 1) in [A]

• ω = 2πf = angular frequency of the sinusoidal noise signal in [rad/sec]

The induced shield voltage $\underline {V}_{1S}$ [V] from circuit 1 (noise source) to the shield—which must be grounded at both ends—is:

$$\displaystyle \begin{aligned} \underline{V}_{1S} = j \omega M_{1S} \underline{I}_1 \end{aligned} $$

(13.26)

where:

• M _1S = Φ _1S∕i ₁(t) = mutual inductance between circuit 1 and the shield in [H]

• $\underline {I}_1 =$ current of the noise source (circuit 1) in [A]

• ω = 2πf = angular frequency of the sinusoidal noise signal in [rad/sec]

If we model the shield as simple RL-series-circuit, we can write the shield current $\underline {I}_S$ [A], induced by the noise current $\underline {I}_1$ [A] from circuit 1, as:

$$\displaystyle \begin{aligned} \underline{I}_{S} = \frac{\underline{V}_{1S}}{R_S + j\omega L_S} \end{aligned} $$

(13.27)

where:

• $\underline {V}_{1S} =$ induced shield voltage to shield due to $\underline {I}_1$ in circuit 1 (noise source) in [V]

• R _S = resistance of the shield current loop in [Ω]

• L _S = inductance of the shield current loop in [H]

• ω = 2πf = angular frequency of the sinusoidal noise signal in [rad/sec]

The mutual inductance M _S2 [H] between the shield and any inner conductor is equal to the shield inductance L _S [H] [3]. Therefore, the induced compensation voltage $\underline {V}_{S2}$ [V] to the shielded conductor 2 due to the shield current $\underline {I}_1$ [A] can be written as:

$$\displaystyle \begin{aligned} \underline{V}_{S2} = j\omega M_{S2} \underline{I}_S = j\omega L_S \underline{I}_S \end{aligned} $$

(13.28)

where:

• M _1S = mutual inductance between the shield and any inner conductor in [H]

• L _S = inductance of the shield current loop in [H]

• $\underline {I}_S =$ induced shield current due to current $\underline {I}_1$ in circuit 1 in [A]

• ω = 2πf = angular frequency of the sinusoidal noise signal in [rad/sec]

If the mutual inductance M ₁₂ [H] of circuit 1 to circuit 2 is equal to the mutual inductance M _1S [H] of circuit 1 to the shield around circuit 2 (because the loop area and orientation of the shield and the inner conductors are nearly identical), we can calculate the total induced noise voltage $\underline {V}_n$ [V] in circuit 2 (victim) as:

$$\displaystyle \begin{aligned} \underline{V}_{n} &= \underline{V}_{12} - \underline{V}_{S2} = j \omega M_{12} \underline{I}_1 - j\omega M_{S2} \underline{I}_S = j \omega \left( M_{12} \underline{I}_1 - L_S \underline{I}_S \right) \end{aligned} $$

(13.29)

$$\displaystyle \begin{aligned} &= j \omega \left(M_{12} \underline{I}_1 - L_S \frac{\underline{V}_{1S}}{R_S + j\omega L_S}\right) = j \omega \left(M_{12} \underline{I}_1 - \frac{j\omega M_{1S} \underline{I}_1}{R_S/L_S + j\omega}\right) \end{aligned} $$

(13.30)

$$\displaystyle \begin{aligned} &= j \omega \underline{I}_1 \left(M_{12} - \frac{j\omega M_{12}}{R_S/L_S + j\omega}\right) = j \omega M_{12} \underline{I}_1 \left(\frac{R_S/L_S}{R_S/L_S + j\omega}\right) {} \end{aligned} $$

(13.31)

where:

• M ₁₂ = Φ ₁₂∕i ₁(t) = mutual inductance between circuit 1 and circuit 2 in [H]

• $\underline {I}_1 =$ current of the noise source (circuit 1) in [A]

• R _S = resistance of the shield current loop in [Ω]

• L _S = inductance of the shield current loop in [H]

• ω = 2πf = angular frequency of the sinusoidal noise signal in [rad/sec]

Equation 13.31 and Fig. 13.17 lead us to the conclusion that shielding against magnetic field coupling is all about minimizing the mutual inductance M ₁₂ [H] and minimizing the shield current loop resistance R _S [Ω] (given the shield is connected at both ends of the cable and is of nonmagnetic material):

• Minimizing M ₁₂. The mutual inductance M ₁₂ [H], which leads to the magnetic field coupling, can be minimized most effectively by reducing the current loop area of the victim’s circuit (see Sect. 12.1.3). This is why differential signals along a twisted pair (minimal current loop area) are robust against magnetic field coupling.

  Fig. 13.17

  

  Frequency response of inductively coupled noise voltage $| \underline {V}_n|$ [V] to a shielded cable vs. an unshielded cable (where the shield is a good conductor, nonmagnetic, and grounded at both ends)

• Minimizing R _S. Minimizing R _S [Ω] means minimizing the shield resistance, the shield termination resistance (connection to ground), and any resistance in the ground loop. Therefore, for maximum shielding effectiveness against magnetic field coupling, all resistances in the shield current loop must be minimized.

Equation 13.31 shows that in the case of nonmagnetic shielding material ($\mu _r^{\prime }=1$), there is no shielding effect for low-frequency magnetic fields and that the shielding effectiveness increases with increasing frequency (in case the shielding is grounded at both ends).

13.7.4 Grounding of Cable Shields

To be most effective, cable shields should be connected to the outside of the shielded enclosure if available or to circuit ground otherwise. The connection(s) of the shield to chassis/ground should be of low impedance. The most effective termination is a 360^∘ connection of the shield. This means that the shield is contacted around the entire circumference to ground (e.g., with a conductive cable clamp). Avoid pigtail ground connections because they are of high impedance at high frequencies and noise can be picked up by pigtails.

Question: connecting a cable shield to ground at both ends of the cable or only at one end? Answer: it depends:

• Shield grounded at one end. Grounding the shield at only one end prevents low-frequency (f < 100 kHz) noise current to flow along the shield, which could couple noise to the inner conductors. However, at higher frequencies, a current flow cannot be prevented effectively due to the stray capacitance of the shield to other conductors and ground.

  In addition, grounding at only one end leads to the fact that the shield could act as an antenna for high frequencies where the shield is an effective antenna (e.g., at l = λ∕4). Therefore, shield grounding at only one end is only recommended for low-frequency applications with short cables, where the primary goal is the protection against electric E-fields.

• Shield grounded at both ends. Whenever in doubt, connect the cable shield at both ends to chassis/ground. Shield grounding at both ends is recommended for high-frequency (f > 100 kHz), digital circuit applications and when the cable length l [m] is longer than l > λ∕20, where λ [m] is the wavelength of the highest significant frequency of the functional signal inside the cable. In addition, grounding at both ends acts as an additional measure against magnetic H-field coupling, because the magnetic field induces a current to the shield, which induces a voltage in the shielded conductor, which ideally cancels out the induced noise voltage (see Sect. 13.7.3).

  In the case of a solid shield, noise currents with frequency f > 1 MHz flow primarily at the shield’s outer surface. This effect leads to the circumstance that a coaxial cable acts like a triaxial cable (a cable with two shields), where the signal return current flows at the inner surface of the shield and the noise current at the outer surface. This means that there is nearly any common impedance coupling in coaxial cables for frequencies above f > 1 MHz.

• Hybrid shield grounding. Figure 13.18 shows the principle of the hybrid cable shield grounding . This method of grounding enables a selective current flow through the shield. For high-frequency signals, both sides are connected to ground, which enables the high-frequency current to flow through the shield, and due to the skin effect, the high-frequency current flows at the outer surface of the shield. Thus, the inner conductor is protected against high-frequency magnetic fields. On the other hand, low-frequency signals (DC or mains power frequency signals of 50 Hz or 60 Hz) are blocked by the capacitor C _G [F] in Fig. 13.18. Hybrid cable shield grounding is a good option to minimize DC and low-frequency shield currents in case of high voltage differences at both ends of the cable while maintaining a good SE against high-frequency noise.

  Fig. 13.18

  

  Hybrid grounding of a cable shield

13.8 Summary

Figure 13.19 presents a proposal for the design workflow of a shielded enclosure. Table 13.1 compares cable shield terminations at one end vs. both ends.

Fig. 13.19

Proposed workflow for designing a shielded enclosure

Table 13.1 Termination of cable shields to ground: one end vs. both ends