Electromagnetic shielding

Last updated September 12, 2024

In electrical engineering, electromagnetic shielding is the practice of reducing or redirecting the electromagnetic field (EMF) in a space with barriers made of conductive or magnetic materials. It is typically applied to enclosures, for isolating electrical devices from their surroundings, and to cables to isolate wires from the environment through which the cable runs (see Shielded cable ). Electromagnetic shielding that blocks radio frequency (RF) electromagnetic radiation is also known as RF shielding.

EMF shielding serves to minimize electromagnetic interference. The shielding can reduce the coupling of radio waves, electromagnetic fields, and electrostatic fields. A conductive enclosure used to block electrostatic fields is also known as a Faraday cage . The amount of reduction depends very much upon the material used, its thickness, the size of the shielded volume and the frequency of the fields of interest and the size, shape and orientation of holes in a shield to an incident electromagnetic field.

Materials used

Typical materials used for electromagnetic shielding include thin layer of metal, sheet metal, metal screen, and metal foam. Common sheet metals for shielding include copper, brass, nickel, silver, steel, and tin. Shielding effectiveness, that is, how well a shield reflects or absorbs/suppresses electromagnetic radiation, is affected by the physical properties of the metal. These may include conductivity, solderability, permeability, thickness, and weight. A metal's properties are an important consideration in material selection. For example, electrically dominant waves are reflected by highly conductive metals like copper, silver, and brass, while magnetically dominant waves are absorbed/suppressed by a less conductive metal such as steel or stainless steel.^[2] Further, any holes in the shield or mesh must be significantly smaller than the wavelength of the radiation that is being kept out, or the enclosure will not effectively approximate an unbroken conducting surface.

Another commonly used shielding method, especially with electronic goods housed in plastic enclosures, is to coat the inside of the enclosure with a metallic ink or similar material. The ink consists of a carrier material loaded with a suitable metal, typically copper or nickel, in the form of very small particulates. It is sprayed on to the enclosure and, once dry, produces a continuous conductive layer of metal, which can be electrically connected to the chassis ground of the equipment, thus providing effective shielding.

Electromagnetic shielding is the process of lowering the electromagnetic field in an area by barricading it with conductive or magnetic material. Copper is used for radio frequency (RF) shielding because it absorbs radio and other electromagnetic waves. Properly designed and constructed RF shielding enclosures satisfy most RF shielding needs, from computer and electrical switching rooms to hospital CAT-scan and MRI facilities.^[3]^[4]

EMI (electromagnetic interference) shielding is of great research interest and several new types of nanocomposites made of ferrites, polymers, and 2D materials are being developed to obtain more efficient RF/microwave-absorbing materials (MAMs).^[5] EMI shielding is often achieved by electroless plating of copper as most popular plastics are non-conductive or by special conductive paint.^[1]

Example of applications

One example is a shielded cable, which has electromagnetic shielding in the form of a wire mesh surrounding an inner core conductor. The shielding impedes the escape of any signal from the core conductor, and also prevents signals from being added to the core conductor. Some cables have two separate coaxial screens, one connected at both ends, the other at one end only, to maximize shielding of both electromagnetic and electrostatic fields.

The door of a microwave oven has a screen built into the window. From the perspective of microwaves (with wavelengths of 12 cm) this screen finishes a Faraday cage formed by the oven's metal housing. Visible light, with wavelengths ranging between 400 nm and 700 nm, passes easily through the screen holes.

RF shielding is also used to prevent access to data stored on RFID chips embedded in various devices, such as biometric passports.^[6]

NATO specifies electromagnetic shielding for computers and keyboards to prevent passive monitoring of keyboard emissions that would allow passwords to be captured; consumer keyboards do not offer this protection primarily because of the prohibitive cost.^[7]

RF shielding is also used to protect medical and laboratory equipment to provide protection against interfering signals, including AM, FM, TV, emergency services, dispatch, pagers, ESMR, cellular, and PCS. It can also be used to protect the equipment at the AM, FM or TV broadcast facilities.

Another example of the practical use of electromagnetic shielding would be defense applications. As technology improves, so does the susceptibility to various types of nefarious electromagnetic interference. The idea of encasing a cable inside a grounded conductive barrier can provide mitigation to these risks.

How it works

Electromagnetic radiation consists of coupled electric and magnetic fields. The electric field produces forces on the charge carriers (i.e., electrons) within the conductor. As soon as an electric field is applied to the surface of an ideal conductor, it induces a current that causes displacement of charge inside the conductor that cancels the applied field inside, at which point the current stops. See Faraday cage for more explanation.

Similarly, varying magnetic fields generate eddy currents that act to cancel the applied magnetic field. (The conductor does not respond to static magnetic fields unless the conductor is moving relative to the magnetic field.) The result is that electromagnetic radiation is reflected from the surface of the conductor: internal fields stay inside, and external fields stay outside.

Several factors serve to limit the shielding capability of real RF shields. One is that, due to the electrical resistance of the conductor, the excited field does not completely cancel the incident field. Also, most conductors exhibit a ferromagnetic response to low-frequency magnetic fields,^{[ citation needed ]} so that such fields are not fully attenuated by the conductor. Any holes in the shield force current to flow around them, so that fields passing through the holes do not excite opposing electromagnetic fields. These effects reduce the field-reflecting capability of the shield.

In the case of high-frequency electromagnetic radiation, the above-mentioned adjustments take a non-negligible amount of time, yet any such radiation energy, as far as it is not reflected, is absorbed by the skin (unless it is extremely thin), so in this case there is no electromagnetic field inside either. This is one aspect of a greater phenomenon called the skin effect. A measure of the depth to which radiation can penetrate the shield is the so-called skin depth.

Magnetic shielding

Equipment sometimes requires isolation from external magnetic fields.^[8] For static or slowly varying magnetic fields (below about 100 kHz) the Faraday shielding described above is ineffective. In these cases shields made of high magnetic permeability metal alloys can be used, such as sheets of permalloy and mu-metal ^[9]^[10] or with nanocrystalline grain structure ferromagnetic metal coatings.^[11] These materials do not block the magnetic field, as with electric shielding, but rather draw the field into themselves, providing a path for the magnetic field lines around the shielded volume. The best shape for magnetic shields is thus a closed container surrounding the shielded volume. The effectiveness of this type of shielding depends on the material's permeability, which generally drops off at both very low magnetic field strengths and high field strengths where the material becomes saturated. Therefore, to achieve low residual fields, magnetic shields often consist of several enclosures, one inside the other, each of which successively reduces the field inside it. Entry holes within shielding surfaces may degrade their performance significantly.

Because of the above limitations of passive shielding, an alternative used with static or low-frequency fields is active shielding, in which a field created by electromagnets cancels the ambient field within a volume.^[12] Solenoids and Helmholtz coils are types of coils that can be used for this purpose, as well as more complex wire patterns designed using methods adapted from those used in coil design for magnetic resonance imaging. Active shields may also be designed accounting for the electromagnetic coupling with passive shields,^[13]^[14]^[15]^[16]^[17] referred to as hybrid shielding,^[18] so that there is broadband shielding from the passive shield and additional cancellation of specific components using the active system.

Additionally, superconducting materials can expel magnetic fields via the Meissner effect.

Mathematical model

Suppose that we have a spherical shell of a (linear and isotropic) diamagnetic material with relative permeability $\mu _{\text{r}}$ , with inner radius $a$ and outer radius $b$ . We then put this object in a constant magnetic field: $\mathbf {H} _{0}=H_{0}{\hat {\mathbf {z} }}=H_{0}\cos(\theta ){\hat {\mathbf {r} }}-H_{0}\sin(\theta ){\hat {\boldsymbol {\theta }}}$ Since there are no currents in this problem except for possible bound currents on the boundaries of the diamagnetic material, then we can define a magnetic scalar potential that satisfies Laplace's equation: ${\begin{aligned}\mathbf {H} &=-\nabla \Phi _{M}\\\nabla ^{2}\Phi _{M}&=0\end{aligned}}$ where $\mathbf {B} =\mu _{\text{r}}\mathbf {H}$ In this particular problem there is azimuthal symmetry so we can write down that the solution to Laplace's equation in spherical coordinates is: $\Phi _{M}=\sum _{\ell =0}^{\infty }\left(A_{\ell }r^{\ell }+{\frac {B_{\ell }}{r^{\ell +1}}}\right)P_{\ell }(\cos \theta )$ After matching the boundary conditions ${\begin{aligned}\left(\mathbf {H} _{2}-\mathbf {H} _{1}\right)\times {\hat {\mathbf {n} }}&=0\\\left(\mathbf {B} _{2}-\mathbf {B} _{1}\right)\cdot {\hat {\mathbf {n} }}&=0\end{aligned}}$ at the boundaries (where ${\hat {n}}$ is a unit vector that is normal to the surface pointing from side 1 to side 2), then we find that the magnetic field inside the cavity in the spherical shell is: $\mathbf {H} _{\text{in}}=\eta \mathbf {H} _{0}$ where $\eta$ is an attenuation coefficient that depends on the thickness of the diamagnetic material and the magnetic permeability of the material: $\eta ={\frac {9\mu _{\text{r}}}{\left(2\mu _{\text{r}}+1\right)\left(\mu _{\text{r}}+2\right)-2\left({\frac {a}{b}}\right)^{3}\left(\mu _{\text{r}}-1\right)^{2}}}$ This coefficient describes the effectiveness of this material in shielding the external magnetic field from the cavity that it surrounds. Notice that this coefficient appropriately goes to 1 (no shielding) in the limit that $\mu _{\text{r}}\to 1$ . In the limit that $\mu _{\text{r}}\to \infty$ this coefficient goes to 0 (perfect shielding). When $\mu _{\text{r}}\gg 1$ , then the attenuation coefficient takes on the simpler form: $\eta ={\frac {9}{2\left(1-{\frac {a^{3}}{b^{3}}}\right)\mu _{\text{r}}}}$ which shows that the magnetic field decreases like $\mu _{\text{r}}^{-1}$ .^[19]

Related Research Articles

An electromagnetic field is a physical field, mathematical functions of position and time, representing the influences on and due to electric charges. The field at any point in space and time can be regarded as a combination of an electric field and a magnetic field. Because of the interrelationship between the fields, a disturbance in the electric field can create a disturbance in the magnetic field which in turn affects the electric field, leading to an oscillation that propagates through space, known as an electromagnetic wave.

Mu-metal is a nickel–iron soft ferromagnetic alloy with very high permeability, which is used for shielding sensitive electronic equipment against static or low-frequency magnetic fields.

Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, electric and magnetic circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar, etc. They describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. The equations are named after the physicist and mathematician James Clerk Maxwell, who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon. The modern form of the equations in their most common formulation is credited to Oliver Heaviside.

<span class="mw-page-title-main">Paramagnetism</span> Weak, attractive magnetism possessed by most elements and some compounds

Paramagnetism is a form of magnetism whereby some materials are weakly attracted by an externally applied magnetic field, and form internal, induced magnetic fields in the direction of the applied magnetic field. In contrast with this behavior, diamagnetic materials are repelled by magnetic fields and form induced magnetic fields in the direction opposite to that of the applied magnetic field. Paramagnetic materials include most chemical elements and some compounds; they have a relative magnetic permeability slightly greater than 1 and hence are attracted to magnetic fields. The magnetic moment induced by the applied field is linear in the field strength and rather weak. It typically requires a sensitive analytical balance to detect the effect and modern measurements on paramagnetic materials are often conducted with a SQUID magnetometer.

A magnetic field is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to the magnetic field. A permanent magnet's magnetic field pulls on ferromagnetic materials such as iron, and attracts or repels other magnets. In addition, a nonuniform magnetic field exerts minuscule forces on "nonmagnetic" materials by three other magnetic effects: paramagnetism, diamagnetism, and antiferromagnetism, although these forces are usually so small they can only be detected by laboratory equipment. Magnetic fields surround magnetized materials, electric currents, and electric fields varying in time. Since both strength and direction of a magnetic field may vary with location, it is described mathematically by a function assigning a vector to each point of space, called a vector field.

<span class="mw-page-title-main">Coaxial cable</span> Electrical cable type with concentric inner conductor, insulator, and conducting shield

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Electromagnetic or magnetic induction is the production of an electromotive force (emf) across an electrical conductor in a changing magnetic field.

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In atomic physics, hyperfine structure is defined by small shifts in otherwise degenerate electronic energy levels and the resulting splittings in those electronic energy levels of atoms, molecules, and ions, due to electromagnetic multipole interaction between the nucleus and electron clouds.

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Induction heating is the process of heating electrically conductive materials, namely metals or semi-conductors, by electromagnetic induction, through heat transfer passing through an inductor that creates an electromagnetic field within the coil to heat up and possibly melt steel, copper, brass, graphite, gold, silver, aluminum, or carbide.

In electromagnetism, permeability is the measure of magnetization produced in a material in response to an applied magnetic field. Permeability is typically represented by the (italicized) Greek letter μ. It is the ratio of the magnetic induction $to the magnetizing field as a function of the field in a material. The term was coined by William Thomson, 1st Baron Kelvin in 1872, and used alongside permittivity by Oliver Heaviside in 1885. The reciprocal of permeability is magnetic reluctivity.$

In physics and engineering, a constitutive equation or constitutive relation is a relation between two or more physical quantities that is specific to a material or substance or field, and approximates its response to external stimuli, usually as applied fields or forces. They are combined with other equations governing physical laws to solve physical problems; for example in fluid mechanics the flow of a fluid in a pipe, in solid state physics the response of a crystal to an electric field, or in structural analysis, the connection between applied stresses or loads to strains or deformations.

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The vacuum magnetic permeability is the magnetic permeability in a classical vacuum. It is a physical constant, conventionally written as μ₀. It quantifies the strength of the magnetic field induced by an electric current. Expressed in terms of SI base units, it has the unit kg⋅m⋅s⁻²·A⁻². It can be also expressed in terms of SI derived units, N·A⁻².

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References

1 2 "Medical Device EMI Shielding". Cybershield. Retrieved 2023-05-02.
↑ "Understanding EMI/RFI Shielding to Manage Interference". Ceptech. Retrieved 2020-04-23.
↑ Seale, Wayne (2007). "The role of copper, brass, and bronze in architecture and design". Metal Architecture. May 2007
↑ "Radio frequency shielding". Copper in Architecture Design Handbook. Copper Development Association Inc. Archived 2020-08-07 at the Wayback Machine
↑ Mohapatra, Prajna P.; Ghosh, Sagnik; Jain, Ashish; Aich, Suman; Dobbidi, Pamu (2023-05-01). "Rare earth substituted lithium ferrite/carbon black ceramic composites for shielding electromagnetic radiation". Journal of Magnetism and Magnetic Materials. 573: 170678. Bibcode:2023JMMM..57370678M. doi:10.1016/j.jmmm.2023.170678. ISSN 0304-8853. S2CID 257867862.
↑ "Metal shields and encryption for US passports". Newscientist.com. Retrieved 18 November 2012.
↑ Vuagnoux, Martin; Pasini, Sylvain (August 2009). Compromising Electromagnetic Emanations of Wired and Wireless Keyboards (PDF). 18th USENIX Security Symposium.
↑ Hobson, P. J.; et al. (2022). "Bespoke magnetic field design for a magnetically shielded cold atom interferometer". Sci. Rep. 12 (1): 10520. arXiv: 2110.04498 . Bibcode:2022NatSR..1210520H. doi:10.1038/s41598-022-13979-4. PMC 9217970 . PMID 35732872. S2CID 238583775.
↑ "MuMETAL" (PDF). Magnetic Shield Corp. 2012. Catalog MU-2. Retrieved 26 June 2016.^{[ permanent dead link ]}
↑ "Trademark Status & Document Retrieval". tsdr.uspto.gov. Retrieved 2017-08-02.
↑ "Interference Technology Magazine Whitepaper on Ferromagnetic Nanocrystalline Metal Magnetic Shield Coatings". Archived from the original on March 15, 2010.
↑ "NMR Magnet Shielding: The seat of the pants guide to understanding the problems of shielding NMR magnets". Acorn NMR. 22 January 2003. Retrieved 27 June 2016.
↑ Packer, M.; Hobson, P.J.; Holmes, N.; Leggett, J.; Glover, P.; Brookes, M.J.; Bowtell, R.; Fromhold, T.M. (2020-11-03). "Optimal Inverse Design of Magnetic Field Profiles in a Magnetically Shielded Cylinder". Physical Review Applied. 14 (5): 054004. arXiv: 2006.02981 . Bibcode:2020PhRvP..14e4004P. doi:10.1103/PhysRevApplied.14.054004. S2CID 221538013.
↑ Packer, M.; Hobson, P.J.; Holmes, N.; Leggett, J.; Glover, P.; Brookes, M.J.; Bowtell, R.; Fromhold, T.M. (2021-06-02). "Planar Coil Optimization in a Magnetically Shielded Cylinder". Physical Review Applied. 15 (6): 064006. arXiv: 2101.01275 . Bibcode:2021PhRvP..15f4006P. doi:10.1103/PhysRevApplied.15.064006. S2CID 230524109.
↑ Liu, C. -Y.; Andalib, T.; Ostapchuk, D. C. M.; Bidinosti, C. P. (2020-01-01). "Analytic models of magnetically enclosed spherical and solenoidal coils". Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment. 949: 162837. arXiv: 1907.03539 . Bibcode:2020NIMPA.94962837L. doi:10.1016/j.nima.2019.162837. ISSN 0168-9002. S2CID 195833040.
↑ Mäkinen, Antti J.; Zetter, Rasmus; Iivanainen, Joonas; Zevenhoven, Koos C. J.; Parkkonen, Lauri; Ilmoniemi, Risto J. (2020-08-14). "Magnetic-field modeling with surface currents. Part I. Physical and computational principles of bfieldtools". Journal of Applied Physics. 128 (6): 063906. arXiv: 2005.10060 . Bibcode:2020JAP...128f3906M. doi:10.1063/5.0016090. ISSN 0021-8979. S2CID 218718690.
↑ Zetter, Rasmus; J. Mäkinen, Antti; Iivanainen, Joonas; Zevenhoven, Koos C. J.; Ilmoniemi, Risto J.; Parkkonen, Lauri (2020-08-14). "Magnetic field modeling with surface currents. Part II. Implementation and usage of bfieldtools". Journal of Applied Physics. 128 (6): 063905. arXiv: 2005.10056 . Bibcode:2020JAP...128f3905Z. doi:10.1063/5.0016087. ISSN 0021-8979. S2CID 218719330.
↑ Royal, Kevin; Crawford, Christopher; Mullins, Andrew; Porter, Greg; Blanton, Hunter; Johnstone, Connor; Kistler, Ben; Olivera, Daniela (2017-09-01). "Hybrid Magnetic Shielding". APS Division of Nuclear Physics Meeting Abstracts. 2017: EA.034. Bibcode:2017APS..DNP.EA034R.
↑ Jackson, John David (10 August 1998). Classical Electrodynamics (third ed.). Section 5.12. ISBN 978-0471309321.

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[:0-1] 1 2 "Medical Device EMI Shielding". Cybershield. Retrieved 2023-05-02.

[2] "Understanding EMI/RFI Shielding to Manage Interference". Ceptech. Retrieved 2020-04-23.

[3] Seale, Wayne (2007). "The role of copper, brass, and bronze in architecture and design". Metal Architecture. May 2007

[4] "Radio frequency shielding". Copper in Architecture Design Handbook. Copper Development Association Inc. Archived 2020-08-07 at the Wayback Machine

[5] Mohapatra, Prajna P.; Ghosh, Sagnik; Jain, Ashish; Aich, Suman; Dobbidi, Pamu (2023-05-01). "Rare earth substituted lithium ferrite/carbon black ceramic composites for shielding electromagnetic radiation". Journal of Magnetism and Magnetic Materials. 573: 170678. Bibcode:2023JMMM..57370678M. doi:10.1016/j.jmmm.2023.170678. ISSN 0304-8853. S2CID 257867862.

[6] "Metal shields and encryption for US passports". Newscientist.com. Retrieved 18 November 2012.

[7] Vuagnoux, Martin; Pasini, Sylvain (August 2009). Compromising Electromagnetic Emanations of Wired and Wireless Keyboards (PDF). 18th USENIX Security Symposium.

[8] Hobson, P. J.; et al. (2022). "Bespoke magnetic field design for a magnetically shielded cold atom interferometer". Sci. Rep. 12 (1): 10520. arXiv: 2110.04498 . Bibcode:2022NatSR..1210520H. doi:10.1038/s41598-022-13979-4. PMC 9217970 . PMID 35732872. S2CID 238583775.

[9] "MuMETAL" (PDF). Magnetic Shield Corp. 2012. Catalog MU-2. Retrieved 26 June 2016.^{[ permanent dead link ]}

[10] "Trademark Status & Document Retrieval". tsdr.uspto.gov. Retrieved 2017-08-02.

[11] "Interference Technology Magazine Whitepaper on Ferromagnetic Nanocrystalline Metal Magnetic Shield Coatings". Archived from the original on March 15, 2010.

[12] "NMR Magnet Shielding: The seat of the pants guide to understanding the problems of shielding NMR magnets". Acorn NMR. 22 January 2003. Retrieved 27 June 2016.

[13] Packer, M.; Hobson, P.J.; Holmes, N.; Leggett, J.; Glover, P.; Brookes, M.J.; Bowtell, R.; Fromhold, T.M. (2020-11-03). "Optimal Inverse Design of Magnetic Field Profiles in a Magnetically Shielded Cylinder". Physical Review Applied. 14 (5): 054004. arXiv: 2006.02981 . Bibcode:2020PhRvP..14e4004P. doi:10.1103/PhysRevApplied.14.054004. S2CID 221538013.

[14] Packer, M.; Hobson, P.J.; Holmes, N.; Leggett, J.; Glover, P.; Brookes, M.J.; Bowtell, R.; Fromhold, T.M. (2021-06-02). "Planar Coil Optimization in a Magnetically Shielded Cylinder". Physical Review Applied. 15 (6): 064006. arXiv: 2101.01275 . Bibcode:2021PhRvP..15f4006P. doi:10.1103/PhysRevApplied.15.064006. S2CID 230524109.

[15] Liu, C. -Y.; Andalib, T.; Ostapchuk, D. C. M.; Bidinosti, C. P. (2020-01-01). "Analytic models of magnetically enclosed spherical and solenoidal coils". Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment. 949: 162837. arXiv: 1907.03539 . Bibcode:2020NIMPA.94962837L. doi:10.1016/j.nima.2019.162837. ISSN 0168-9002. S2CID 195833040.

[16] Mäkinen, Antti J.; Zetter, Rasmus; Iivanainen, Joonas; Zevenhoven, Koos C. J.; Parkkonen, Lauri; Ilmoniemi, Risto J. (2020-08-14). "Magnetic-field modeling with surface currents. Part I. Physical and computational principles of bfieldtools". Journal of Applied Physics. 128 (6): 063906. arXiv: 2005.10060 . Bibcode:2020JAP...128f3906M. doi:10.1063/5.0016090. ISSN 0021-8979. S2CID 218718690.

[17] Zetter, Rasmus; J. Mäkinen, Antti; Iivanainen, Joonas; Zevenhoven, Koos C. J.; Ilmoniemi, Risto J.; Parkkonen, Lauri (2020-08-14). "Magnetic field modeling with surface currents. Part II. Implementation and usage of bfieldtools". Journal of Applied Physics. 128 (6): 063905. arXiv: 2005.10056 . Bibcode:2020JAP...128f3905Z. doi:10.1063/5.0016087. ISSN 0021-8979. S2CID 218719330.

[18] Royal, Kevin; Crawford, Christopher; Mullins, Andrew; Porter, Greg; Blanton, Hunter; Johnstone, Connor; Kistler, Ben; Olivera, Daniela (2017-09-01). "Hybrid Magnetic Shielding". APS Division of Nuclear Physics Meeting Abstracts. 2017: EA.034. Bibcode:2017APS..DNP.EA034R.

[19] Jackson, John David (10 August 1998). Classical Electrodynamics (third ed.). Section 5.12. ISBN 978-0471309321.

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