3. The application of electromagnetic textile materials

#### 3.1 Antistatic textile materials

The electrostatic phenomenon of cellulose fibers in the processing process is not obvious; but the electrostatic interference of protein fibers is pretty serious. Although the wool fiber has high equilibrium moisture regain, its mass specific resistance is the highest in the natural fiber. The resistivity of synthetic fibers such as polyester, nylon, acrylic, and polypropylene, which are generally high in moisture regain, is as high as 1014 Ω cm, and the accumulation of electrostatic charge is obvious.

#### 3.1.1 Electrostatic mechanism

The material is excited by various energies, causing the electrons to escape from the nucleus. The electrons overcome the binding of the nucleus, and the minimum energy required to escape from the surface of the material is called the work function. Different materials or the same material in different states have different work function. The generation and accumulation of electric charge causes the substance to carry static electricity, and the one that acquires the electron exhibits the negative electric property, and the one that loses the electron exhibits the positive electric property, which generate the electrostatic phenomenon.

The resistivity of conventional textile materials is up to 10<sup>10</sup> Ω cm or more, and the generated charge is not easily dissipated, resulting in very serious electrostatic phenomenon. Therefore, the antistatic properties of textile materials have become an important property having a great influence on the processing of textile materials and the use of textiles.

#### 3.1.2 Antistatic technology for textile materials

The antistatic technology of textile materials includes the preparation of antistatic fibers, the preparation of conductive yarns, and the conductive treatment of textiles.

The shielding effectiveness equals to the ratio of electric field strength E<sup>0</sup> when a

point in space is unshielded to the electric field strength ES of the field after shielding, or the ratio of the magnetic field strength H<sup>0</sup> to the magnetic field strength HS of the field after shielding, or the ratio of the power P<sup>0</sup> to the PS of the field after shielding. The shielding effectiveness (SE) can be expressed as follows:

, or SEH <sup>¼</sup> 20lg j j <sup>H</sup><sup>0</sup>

Arching method is usually used to measure shielding effectiveness. The schematic diagram is shown in Figure 6, which can characterize the shielding effectiveness by measuring the power of the receiving antennas and transmitting antennas.

Woven fabric is composed of warp and weft yarns interlaced vertically with each other according to a certain regularity. It is well known that ordinary fiber yarns are transparent to the EM wave, while yarn containing metal fibers is conductive. As metal fiber yarns are closely spaced, continuous conductive paths can be established easily. For an electromagnetic shielding fabric composed of metal fiber yarns or coated with a metal layer or a functional layer, it has a remarkable and typical mesh structure [7], as shown in Figure 7. Figure 7(a) is a schematic view of the structure of the metal fiber-containing yarn fabric and Figure 7(b) is a structural model diagram of the metal fiber yarn extracted in Figure 7(a). The geometric parameters in the figure correspond to the parameters in the fabric: h is the buckling wave height, similar to the thickness of the fabrics in the data; l is the length of the organization cycle; and a is the arrangement cycle spacing of the metal yarns in the fabric.

The metal fiber yarn fabric is considered as a periodic grid structure model composed of conductive yarns [8], as shown in Figure 8. Assume that one parallel periodic array is composed of the warp yarn (as shown in Figure 8(b)) and the

j j HS

, or SE <sup>¼</sup> 10lg j j <sup>P</sup><sup>0</sup>

j j PS

(5)

SEE <sup>¼</sup> 20lg j j <sup>E</sup><sup>0</sup>

Electromagnetic Function Textiles

DOI: http://dx.doi.org/10.5772/intechopen.85586

3.2.2.1 The metallic yarn grid model

Schematic diagram of arching measuring method for shielding effectiveness.

Figure 6.

187

j j ES

3.2.2 Conductive grid structure of electromagnetic shielding fabrics

### 3.1.2.1 The preparation of antistatic fibers

For textile materials with higher mass specific resistance, surfactants are often added to fibers in fiber factories, which absorb water molecules from the environment and reduce static interference in the yarns. The hydrophobic end of the surfactant molecule is adsorbed on the surface of the fiber; the hydrophilic group is pointed to the outer space [3]. Then, the fiber forms the polar surface and adsorbs water molecules in the air. The surface resistivity of the fiber is reduced, and the charge dissipation is accelerated. The method is simple and easy to make; however, the antistatic effect is poor in durability, and the surfactant is volatile and less resistant to washing.

In order to prepare relatively durable antistatic fiber, the methods are following: (1) Adding the surfactant to a fiber-forming polymer during blend spinning; (2) adding the hydrophilic group by block copolymerization; and (3) adding the hydrophilic group by graft modification in a fiber-forming polymer. These can make the fibers obtain durable hygroscopicity and antistatic properties.

In addition, there are also another methods, including fixing the surfactant to the surface of the fiber with a binder and crosslinking the surfactant on the surface of the fiber to form a film. The effect is similar to applying an antistatic varnish onto the surface of the plastic.

Antistatic fibers are usually blended with ordinary fibers, and a higher content of antistatic fibers is required to achieve a more feasible antistatic effect. The specific ratio between antistatic fibers and ordinary fibers should be based on the resistivity of the ordinary fibers used, the final use environment, and requirements of the products.

#### 3.1.2.2 The preparation of conductive fibers

The electrical resistivity of the conductive fiber is smaller than that of the antistatic fiber, and it has a more significant antistatic effect. And during the blend fabrics with the same antistatic effects, the amount of conductive fiber added is much smaller than that of the antistatic fiber [4]. As long as a few thousandths to a few percent of the conductive yarn is added, the fabric can attain antistatic requirements. So with the widespread use of organic conductive fibers [5, 6], the field of application of antistatic fibers has been gradually reduced.

#### 3.2 Electromagnetic shielding textile materials

#### 3.2.1 Shielding effectiveness

Electromagnetic shielding is a technical measure to prevent or suppress the transmission of electromagnetic energy by using a shield. The shield used can weaken the electromagnetic field strength generated by the field source in the electromagnetic space protection zone. There are two main purposes for shielding: one is to limit the field source electromagnetic energy leaking out from the area that needs protection and the other is to prevent the external electromagnetic field energy entering into the area protected.

#### Electromagnetic Function Textiles DOI: http://dx.doi.org/10.5772/intechopen.85586

3.1.2 Antistatic technology for textile materials

Electromagnetic Materials and Devices

3.1.2.1 The preparation of antistatic fibers

textiles.

resistant to washing.

the surface of the plastic.

3.2.1 Shielding effectiveness

186

3.1.2.2 The preparation of conductive fibers

3.2 Electromagnetic shielding textile materials

energy entering into the area protected.

of the products.

The antistatic technology of textile materials includes the preparation of antistatic fibers, the preparation of conductive yarns, and the conductive treatment of

For textile materials with higher mass specific resistance, surfactants are often added to fibers in fiber factories, which absorb water molecules from the environment and reduce static interference in the yarns. The hydrophobic end of the surfactant molecule is adsorbed on the surface of the fiber; the hydrophilic group is pointed to the outer space [3]. Then, the fiber forms the polar surface and adsorbs water molecules in the air. The surface resistivity of the fiber is reduced, and the charge dissipation is accelerated. The method is simple and easy to make; however, the antistatic effect is poor in durability, and the surfactant is volatile and less

In order to prepare relatively durable antistatic fiber, the methods are following:

In addition, there are also another methods, including fixing the surfactant to the surface of the fiber with a binder and crosslinking the surfactant on the surface of the fiber to form a film. The effect is similar to applying an antistatic varnish onto

Antistatic fibers are usually blended with ordinary fibers, and a higher content

of antistatic fibers is required to achieve a more feasible antistatic effect. The specific ratio between antistatic fibers and ordinary fibers should be based on the resistivity of the ordinary fibers used, the final use environment, and requirements

The electrical resistivity of the conductive fiber is smaller than that of the antistatic fiber, and it has a more significant antistatic effect. And during the blend fabrics with the same antistatic effects, the amount of conductive fiber added is much smaller than that of the antistatic fiber [4]. As long as a few thousandths to a

Electromagnetic shielding is a technical measure to prevent or suppress the transmission of electromagnetic energy by using a shield. The shield used can weaken the electromagnetic field strength generated by the field source in the electromagnetic space protection zone. There are two main purposes for shielding: one is to limit the field source electromagnetic energy leaking out from the area that needs protection and the other is to prevent the external electromagnetic field

few percent of the conductive yarn is added, the fabric can attain antistatic requirements. So with the widespread use of organic conductive fibers [5, 6], the

field of application of antistatic fibers has been gradually reduced.

(1) Adding the surfactant to a fiber-forming polymer during blend spinning; (2) adding the hydrophilic group by block copolymerization; and (3) adding the hydrophilic group by graft modification in a fiber-forming polymer. These can make the fibers obtain durable hygroscopicity and antistatic properties.

The shielding effectiveness equals to the ratio of electric field strength E<sup>0</sup> when a point in space is unshielded to the electric field strength ES of the field after shielding, or the ratio of the magnetic field strength H<sup>0</sup> to the magnetic field strength HS of the field after shielding, or the ratio of the power P<sup>0</sup> to the PS of the field after shielding. The shielding effectiveness (SE) can be expressed as follows:

$$\text{SE}\_E = 20 \lg \frac{|E\_0|}{|E\_S|}, \text{or } \text{SE}\_H = 20 \lg \frac{|H\_0|}{|H\_S|}, \text{or } \text{SE} = \mathbf{10lg} \frac{|P\_0|}{|P\_S|} \tag{5}$$

Arching method is usually used to measure shielding effectiveness. The schematic diagram is shown in Figure 6, which can characterize the shielding effectiveness by measuring the power of the receiving antennas and transmitting antennas.

### 3.2.2 Conductive grid structure of electromagnetic shielding fabrics

Woven fabric is composed of warp and weft yarns interlaced vertically with each other according to a certain regularity. It is well known that ordinary fiber yarns are transparent to the EM wave, while yarn containing metal fibers is conductive. As metal fiber yarns are closely spaced, continuous conductive paths can be established easily. For an electromagnetic shielding fabric composed of metal fiber yarns or coated with a metal layer or a functional layer, it has a remarkable and typical mesh structure [7], as shown in Figure 7. Figure 7(a) is a schematic view of the structure of the metal fiber-containing yarn fabric and Figure 7(b) is a structural model diagram of the metal fiber yarn extracted in Figure 7(a). The geometric parameters in the figure correspond to the parameters in the fabric: h is the buckling wave height, similar to the thickness of the fabrics in the data; l is the length of the organization cycle; and a is the arrangement cycle spacing of the metal yarns in the fabric.

### 3.2.2.1 The metallic yarn grid model

The metal fiber yarn fabric is considered as a periodic grid structure model composed of conductive yarns [8], as shown in Figure 8. Assume that one parallel periodic array is composed of the warp yarn (as shown in Figure 8(b)) and the

#### Figure 6. Schematic diagram of arching measuring method for shielding effectiveness.

the propagation direction) polarized waves, that is, the vertical polarization wave

ε<sup>0</sup> and μ<sup>0</sup> are the dielectric constant and absolute permeability of free space, respectively. Assume that the metallic yarns are lossy and characterized by the per-

s

where δ is the skin depth (m); μ<sup>0</sup> is permeability of the vacuum (H/m); μ<sup>r</sup> is relative permeability of the conductive yarns; σ is electrical conductivity (S/m); and

As shown in Figure 10, the global coordinate system (x, y, z) and the local one (ξ, ψ, ζ) are introduced. The ζ-axis is parallel to the wires of the array; the x-axis is parallel to the ξ-axis, and they are perpendicular to the plane of the periodic parallel

Assuming that the diameters of the metallic yarns and periodic spacing are small compared to the wavelength, then the parallel array can be modeled by a homogeneous thin anisotropic sheet with thickness a. Under the condition of the EM wave having normal incidence, the following relations can be written among the propagating field components expressed in the local coordinate system and averaged over

Schematic wire mesh illuminated by a plane wave with normal incidence and transverse magnetic (TM) or

Eζð Þ¼ a Eζð Þ 0 (7) E<sup>ψ</sup> ð Þ¼ a E<sup>ψ</sup> ð Þ 0 (8) Hζð Þ¼ a Hζð Þ 0 (9) H<sup>ψ</sup> ð Þ¼ a H<sup>ψ</sup> ð Þþ 0 J (10)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 π f μ0μrσ

(6)

and the horizontal polarization wave, respectively, as shown in Figure 9.

δ ¼

array, which has yarn orientation ψ, spacing p, and yarn diameter a.

the wire array period, on each side of the sheet.

unit-length impedance Zw (unit: Ω/m). The skin depth is given by

f is frequency (Hz).

Electromagnetic Function Textiles

DOI: http://dx.doi.org/10.5772/intechopen.85586

Figure 9.

189

transverse electric (TE) polarization.

#### Figure 7.

The grid structure of electromagnetic shielding fabric. (a) Typical mesh structure (b) The structure of the metal fiber-containing yarn fabric.

#### Figure 8.

Grid structure model. (a) Grid structure; (b) Period parallel array 1 and (c) Period parallel array 2.

other is composed of the weft yarn (as shown in Figure 8(c)). The two wire arrays are directly cascaded at a certain orientation angle. The contact impedances between the wires of the two arrays are assumed to be negligible, due to the EM coupling between the two grid arrays. In particular, the grid structure model is only composed of metal fiber yarns, that is, the effect of ordinary yarns in fabric is not considered.

The two parallel periodic arrays (as shown in Figure 1) have wire orientation ψ1, ψ2, and (° ); spacing p<sup>1</sup> and p<sup>2</sup> (m); and wire diameter a<sup>1</sup> and a<sup>2</sup> (m), respectively. In most fabric structures, the grid is anisotropic and asymmetric, namely, ψ<sup>1</sup> 6¼ ψ2, p<sup>1</sup> 6¼ p2, and a<sup>1</sup> 6¼ a2.

#### 3.2.2.2 Shielding effectiveness of the grid model

The periodic grid is regarded as a stratified medium made of two periodic parallel arrays at a certain angle. The transmission matrix was established, and the SE for different polarization incident waves was calculated by analyzing the propagation of EM fields passing through the wire mesh. It can be used for the calculation of the SE of isotropic and anisotropic metal wire mesh structures for different polarization.

The grid is excited by a plane wave having normal incidence, and the incident EM wave can be decomposed into TMZ (transverse magnetic: the magnetic field component only in the plane perpendicular to the propagation direction) and TEZ (transverse electric: the electric field component only in the plane perpendicular to

#### Electromagnetic Function Textiles DOI: http://dx.doi.org/10.5772/intechopen.85586

the propagation direction) polarized waves, that is, the vertical polarization wave and the horizontal polarization wave, respectively, as shown in Figure 9.

ε<sup>0</sup> and μ<sup>0</sup> are the dielectric constant and absolute permeability of free space, respectively. Assume that the metallic yarns are lossy and characterized by the perunit-length impedance Zw (unit: Ω/m). The skin depth is given by

$$\delta = \sqrt{\frac{1}{\pi f \mu\_0 \mu\_r \sigma}}\tag{6}$$

where δ is the skin depth (m); μ<sup>0</sup> is permeability of the vacuum (H/m); μ<sup>r</sup> is relative permeability of the conductive yarns; σ is electrical conductivity (S/m); and f is frequency (Hz).

As shown in Figure 10, the global coordinate system (x, y, z) and the local one (ξ, ψ, ζ) are introduced. The ζ-axis is parallel to the wires of the array; the x-axis is parallel to the ξ-axis, and they are perpendicular to the plane of the periodic parallel array, which has yarn orientation ψ, spacing p, and yarn diameter a.

Assuming that the diameters of the metallic yarns and periodic spacing are small compared to the wavelength, then the parallel array can be modeled by a homogeneous thin anisotropic sheet with thickness a. Under the condition of the EM wave having normal incidence, the following relations can be written among the propagating field components expressed in the local coordinate system and averaged over the wire array period, on each side of the sheet.

$$E\_{\zeta}(a) = E\_{\zeta}(\mathbf{0})\tag{7}$$

$$E\_{\boldsymbol{\Psi}}(\boldsymbol{a}) = E\_{\boldsymbol{\Psi}}(\mathbf{0}) \tag{8}$$

$$H\_{\zeta}(\mathfrak{a}) = H\_{\zeta}(\mathfrak{0}) \tag{9}$$

$$H\_{\Psi}(\mathfrak{a}) = H\_{\Psi}(\mathfrak{0}) + \mathfrak{f} \tag{10}$$

#### Figure 9.

other is composed of the weft yarn (as shown in Figure 8(c)). The two wire arrays

The grid structure of electromagnetic shielding fabric. (a) Typical mesh structure (b) The structure of the metal

The two parallel periodic arrays (as shown in Figure 1) have wire orientation ψ1,

The periodic grid is regarded as a stratified medium made of two periodic parallel arrays at a certain angle. The transmission matrix was established, and the SE for different polarization incident waves was calculated by analyzing the propagation of EM fields passing through the wire mesh. It can be used for the calculation of the SE of isotropic and anisotropic metal wire mesh structures for different polarization. The grid is excited by a plane wave having normal incidence, and the incident EM wave can be decomposed into TMZ (transverse magnetic: the magnetic field component only in the plane perpendicular to the propagation direction) and TEZ (transverse electric: the electric field component only in the plane perpendicular to

most fabric structures, the grid is anisotropic and asymmetric, namely, ψ<sup>1</sup> 6¼ ψ2,

); spacing p<sup>1</sup> and p<sup>2</sup> (m); and wire diameter a<sup>1</sup> and a<sup>2</sup> (m), respectively. In

are directly cascaded at a certain orientation angle. The contact impedances between the wires of the two arrays are assumed to be negligible, due to the EM coupling between the two grid arrays. In particular, the grid structure model is only composed of metal fiber yarns, that is, the effect of ordinary yarns in fabric is not

Grid structure model. (a) Grid structure; (b) Period parallel array 1 and (c) Period parallel array 2.

considered.

Figure 8.

Figure 7.

fiber-containing yarn fabric.

Electromagnetic Materials and Devices

ψ2, and (°

188

p<sup>1</sup> 6¼ p2, and a<sup>1</sup> 6¼ a2.

3.2.2.2 Shielding effectiveness of the grid model

Schematic wire mesh illuminated by a plane wave with normal incidence and transverse magnetic (TM) or transverse electric (TE) polarization.

Figure 10. The periodic array in the local coordinate system and the global coordinate system.

in which J is the current density (A/m<sup>2</sup> ) flowing inside the wire and all quantities are considered to be averaged over the wire array period. The current density J can be related to the tangential average electric field components along the wires. According to the following impedance condition,

$$E\_{\zeta}(\mathbf{0}) = Z\_{\mathbf{S}} \mathbf{J} \tag{11}$$

Φloc ½ �¼

½ � Φ is given by

expression is

191

local coordinate system is as follows:

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DOI: http://dx.doi.org/10.5772/intechopen.85586

½ �¼ R

The grid array transmission matrix ½ � Φ is

½ �¼ YT

1 ZS

1 000 0 100 0 010 1=ZS 001

In the global coordinate system, the transformation matrix of the parallel array

in which the transformation matrix ½ � R from the global coordinate system to the

sin φ cos φ 0 0 � cos φ sin φ 0 0 0 0 sin φ cos φ 0 0 � cos φ sin φ

½ �¼ <sup>Φ</sup> ½ � <sup>U</sup> ½ � <sup>0</sup>

where ½ � U and 0½ � are the bi-dimensional unit and null matrices, respectively,

The total thickness of the wire grid is aM ¼ a<sup>1</sup> þ a2; the orientations of two parallel arrays are ψ<sup>1</sup> and ψ2, respectively, and the effective shunt admittances are ½ � YT<sup>1</sup> and ½ � YT<sup>2</sup> , respectively. The transmission matrix of the grid is the product of

Therefore, the matrix of the boundary condition for the metal grid is given as

The shielding factors of the metal grid of against TMz and TEz polarized waves

3 7 7 7 <sup>5</sup> <sup>¼</sup> ½ � <sup>Φ</sup><sup>M</sup>

and ½ � YT is the effective shunt admittance of the parallel array; its matrix

two parallel array transmission matrixes ½ � Φ<sup>1</sup> and ½ � Φ<sup>2</sup> as shown below:

½ �¼ Φ<sup>M</sup> ½ � Φ<sup>2</sup> ½ �¼ Φ<sup>1</sup>

in which the effective shunt admittance ½ � YM of the grid is

½ � Ezð Þ aM Eyð Þ aM � � ½ � Hzð Þ aM Hyð Þ aM � �

with normal incidence are FTM and FTE, respectively, shown as:

½ � YT ½ � U

� sin <sup>φ</sup> cos <sup>φ</sup> � cos <sup>2</sup><sup>φ</sup> sin <sup>2</sup>φ sin φ cos φ

> ½ � U ½ � 0 ½ � YM ½ � U

> > ½ � Ezð Þ 0 Eyð Þ <sup>0</sup> � � ½ � Hzð Þ 0 Hyð Þ <sup>0</sup> � �

½ �¼ <sup>Φ</sup> ½ � <sup>R</sup> �<sup>1</sup> <sup>Φ</sup>loc ½ �½ � <sup>R</sup> (17)

� � (19)

� � (20)

½ �¼ YM ½ �þ YT<sup>1</sup> ½ � YT<sup>2</sup> (22)

� � (21)

(16)

(18)

(23)

in which the expressions of impedance ZS for the periodic array are given by

$$Z\_S = \left[ Z\_W p \Re + j a \sqrt{\mu/\varepsilon} \right] \tag{12}$$

where μ (H/m) and ε (F/m) are the magnetic permeability and dielectric constant of the substrate, respectively; α is a parameter that depends on the geometry structure of the periodic array and on the wavelength in the substrate λ<sup>W</sup> (m).

$$a = \left(p/\lambda\_{\mathcal{W}}\right) \log\left(p/(a \cdot \pi)\right) \tag{13}$$

Combining Eqs. (10) and (11) yield

$$H\_{\Psi}(a) = H\_{\Psi}(\mathbf{0}) + E\_{\zeta}(\mathbf{0})/Z\_{\mathbb{S}}\tag{14}$$

According to Eqs. (7)–(9) and (14), the boundary condition describing the relation among the EM field components tangential to the conductive grid is obtained, and its matrix expression is

$$\begin{bmatrix} E\_{\zeta}(a) \\ E\_{\psi}(a) \\ H\_{\zeta}(a) \\ H\_{\psi}(a) \end{bmatrix} = [\Phi\_{loc}] \begin{bmatrix} E\_{\zeta}(\mathbf{0}) \\ E\_{\psi}(\mathbf{0}) \\ H\_{\zeta}(\mathbf{0}) \\ H\_{\psi}(\mathbf{0}) \end{bmatrix} \tag{15}$$

in which Φloc ½ � is the transformation matrix of the parallel array in the local coordinate system as follows:

Electromagnetic Function Textiles DOI: http://dx.doi.org/10.5772/intechopen.85586

$$\begin{bmatrix} \Phi\_{loc} \end{bmatrix} = \begin{bmatrix} \mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{1} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{1} & \mathbf{0} \\ \mathbf{1}/\mathbf{Z}\_S & \mathbf{0} & \mathbf{0} & \mathbf{1} \end{bmatrix} \tag{16}$$

In the global coordinate system, the transformation matrix of the parallel array ½ � Φ is given by

$$[\Phi] = [\mathbb{R}]^{-1}[\Phi\_{loc}][\mathbb{R}] \tag{17}$$

in which the transformation matrix ½ � R from the global coordinate system to the local coordinate system is as follows:

$$\begin{aligned} [R] = \begin{bmatrix} \sin\varphi & \cos\varphi & 0 & 0\\ -\cos\varphi & \sin\varphi & 0 & 0\\ 0 & 0 & \sin\varphi & \cos\varphi\\ 0 & 0 & -\cos\varphi & \sin\varphi \end{bmatrix} \end{aligned} \tag{18}$$

The grid array transmission matrix ½ � Φ is

$$\begin{bmatrix} \Phi \end{bmatrix} = \begin{bmatrix} \begin{bmatrix} U \end{bmatrix} & \begin{bmatrix} \mathbf{0} \end{bmatrix} \\\begin{bmatrix} Y\_T \end{bmatrix} & \begin{bmatrix} U \end{bmatrix} \end{bmatrix} \tag{19}$$

where ½ � U and 0½ � are the bi-dimensional unit and null matrices, respectively, and ½ � YT is the effective shunt admittance of the parallel array; its matrix expression is

$$\begin{aligned} \, \, [Y\_T] = \frac{\mathbf{1}}{Z\_{\mathbb{S}}} \begin{bmatrix} -\sin\varrho \cos\varrho & -\cos^2\varrho\\ \sin^2\varrho & \sin\varrho \cos\varrho \end{bmatrix} \end{aligned} \tag{20}$$

The total thickness of the wire grid is aM ¼ a<sup>1</sup> þ a2; the orientations of two parallel arrays are ψ<sup>1</sup> and ψ2, respectively, and the effective shunt admittances are ½ � YT<sup>1</sup> and ½ � YT<sup>2</sup> , respectively. The transmission matrix of the grid is the product of two parallel array transmission matrixes ½ � Φ<sup>1</sup> and ½ � Φ<sup>2</sup> as shown below:

$$\begin{aligned} \begin{bmatrix} \Phi\_M \end{bmatrix} = \begin{bmatrix} \Phi\_2 \end{bmatrix} \begin{bmatrix} \Phi\_1 \end{bmatrix} = \begin{bmatrix} \begin{bmatrix} U \end{bmatrix} & \begin{bmatrix} \mathbf{0} \end{bmatrix} \\\begin{bmatrix} Y\_M \end{bmatrix} & \begin{bmatrix} U \end{bmatrix} \end{bmatrix} \end{aligned} \tag{21}$$

in which the effective shunt admittance ½ � YM of the grid is

$$[Y\_M] = [Y\_{T1}] + [Y\_{T2}] \tag{22}$$

Therefore, the matrix of the boundary condition for the metal grid is given as

$$\begin{bmatrix} \begin{bmatrix} E\_x(a\_M) \end{bmatrix} \\ \begin{bmatrix} E\_\mathcal{V}(a\_M) \end{bmatrix} \\ \begin{bmatrix} H\_x(a\_M) \end{bmatrix} \\ \begin{bmatrix} H\_x(a\_M) \end{bmatrix} \end{bmatrix} = \begin{bmatrix} \Phi\_M \end{bmatrix} \begin{bmatrix} \begin{bmatrix} E\_x(\mathbf{O}) \end{bmatrix} \\ \begin{bmatrix} E\_\mathcal{V}(\mathbf{O}) \end{bmatrix} \\ \begin{bmatrix} H\_x(\mathbf{O}) \end{bmatrix} \\ \begin{bmatrix} H\_\mathcal{V}(\mathbf{O}) \end{bmatrix} \end{bmatrix} \tag{23}$$

The shielding factors of the metal grid of against TMz and TEz polarized waves with normal incidence are FTM and FTE, respectively, shown as:

in which J is the current density (A/m<sup>2</sup>

Electromagnetic Materials and Devices

Figure 10.

Combining Eqs. (10) and (11) yield

obtained, and its matrix expression is

coordinate system as follows:

190

According to the following impedance condition,

The periodic array in the local coordinate system and the global coordinate system.

are considered to be averaged over the wire array period. The current density J can be related to the tangential average electric field components along the wires.

in which the expressions of impedance ZS for the periodic array are given by

ZS <sup>¼</sup> ZWp<sup>9</sup> <sup>þ</sup> <sup>j</sup><sup>α</sup> ffiffiffiffiffiffiffi

where μ (H/m) and ε (F/m) are the magnetic permeability and dielectric constant of the substrate, respectively; α is a parameter that depends on the geometry structure of the periodic array and on the wavelength in the substrate λ<sup>W</sup> (m).

According to Eqs. (7)–(9) and (14), the boundary condition describing the relation among the EM field components tangential to the conductive grid is

¼ Φloc ½ �

in which Φloc ½ � is the transformation matrix of the parallel array in the local

Eζð Þ a E<sup>ψ</sup> ð Þ a Hζð Þ a H<sup>ψ</sup> ð Þ a

μ=ε

) flowing inside the wire and all quantities

Eζð Þ¼ 0 ZSJ (11)

h i <sup>p</sup> (12)

α ¼ ð Þ p=λ<sup>W</sup> log ð Þ p=ð Þ a � π (13)

H<sup>ψ</sup> ð Þ¼ a H<sup>ψ</sup> ð Þþ 0 Eζð Þ 0 =ZS (14)

(15)

Eζð Þ 0 E<sup>ψ</sup> ð Þ 0 Hζð Þ 0 H<sup>ψ</sup> ð Þ 0

$$F\_{\rm TM} = \left| E^{\rm TM, i} \right|^2 / \left| E^{\rm TM}(a\_M) \right|^2 \tag{24}$$

P ¼ j j E að Þ <sup>M</sup>

3.2.3 Influencing factors for electromagnetic shielding effectiveness

3.2.3.1 The influence of structural parameters on shielding effectiveness

spacing increases, the shielding effectiveness is significantly reduced.

3.2.3.1.1 The influence of periodic spacing on SE

Figure 11.

193

The shielding effectiveness at different intervals.

SE <sup>¼</sup> 10lg FTM <sup>þ</sup> FTE

For the grid that is isotropic in the ð Þ y; z plane, the SE against the two polarized

In combination with the mesh structure of the electromagnetic shielding fabric,

Samples in which copper filaments are arranged in parallel at different intervals have different SE. As shown in Figure 11, the arrangement periodic intervals of copper filaments are 1, 2, 3, 4, and 5 mm, respectively. The SE of 10–14 GHz is 17– 20, 7–12, 5–10, 2–5, and 2–4 dB, respectively. It can be seen that as the periodic

Metal yarn arrangement interval has an important effect on SE of fabrics. The metal yarn periodic spacing is related to these parameters such as fabric density and

the key factors affecting the electromagnetic shielding effectiveness SE are the structural parameters and material parameters of fabric [9, 10]. Some key 1parameters are as following: the periodic spacing of metal yarns, the conductivity of the yarns (decided by the type of yarns, the type of metal fibers, the content of metal fibers), the diameters of the yarns, the arrangement method, the connection of intersections, the direction of incidence of electromagnetic field, and the frequency.

From Eq. (6), we obtain

Electromagnetic Function Textiles

DOI: http://dx.doi.org/10.5772/intechopen.85586

waves is coincident. It results in

2

2

= 2η<sup>0</sup> ð Þ (36)

(37)

SE <sup>¼</sup> SETM <sup>¼</sup> SETE (38)

$$F\_{\rm TE} = \left| E^{\rm TE, i} \right|^2 / \left| E^{\rm TE}(a\_M) \right|^2 \tag{25}$$

in which ETM,i � � � � and ETE,i � � � � are the amplitude of the incident electric field for the TMz and TEz polarized plane waves, respectively. ETMð Þ aM � � � � and <sup>E</sup>TEð Þ aM � � � � are the amplitude of the corresponding transmitted electrical field. η<sup>0</sup> ½ � is the free space wave impedance and Ei � � is the vector of the incident electric field.

The electric field component on the back faces of the grid expressed in matrix form is given by

$$\left[E(a\_M)\right] = 2\eta\_0^{-1} \left[Z\_{eq}\right] \left[E^\dagger\right] \tag{26}$$

in which

$$\mathbb{E}\left[Z\_{eq}\right] = \left\{ \left[Y\_M\right] + \mathbb{Z}[\eta\_0]^{-1} \right\}^{-1} \tag{27}$$

In the case where the incident wave is the TMz polarized wave, set Ei � � <sup>¼</sup> <sup>0</sup>Ey <sup>i</sup> ½ �<sup>t</sup> . Therefore, the amplitude of the transmitted field is

$$\left|E^{\rm TM}(a\_M)\right| = 2\eta\_0^{-1} \left\{ \left|Z\_{eq}(\mathbf{1}, \mathbf{2})\right|^2 + \left|Z\_{eq}(\mathbf{2}, \mathbf{2})\right|^2 \right\}^{1/2} \left[E\_\mathcal{V}^i\right] \tag{28}$$

Similarly, for a TEz polarized incident wave, set Ei � � <sup>¼</sup> Ey <sup>i</sup> ½ � ; 0 <sup>t</sup> . Therefore, the amplitude of the transmitted field is

$$\left|E^{TE}(a\_M)\right| = 2\eta\_0^{-1}\left\{ \left| Z\_{eq}(\mathbf{1}, \mathbf{1})\right|^2 + \left| Z\_{eq}(\mathbf{2}, \mathbf{1})\right|^2 \right\}^{1/2} \left[ E\_x^i \right] \tag{29}$$

The shielding factors are, respectively

$$F\_{TM} = \frac{\eta\_0^2}{4} \left\{ \left| Z\_{eq}(\mathbf{1}, \mathbf{2}) \right|^2 + \left| Z\_{eq}(\mathbf{2}, \mathbf{2}) \right|^2 \right\}^{-1} \tag{30}$$

$$F\_{\rm TE} = \frac{\eta\_0^2}{4} \left\{ \left| Z\_{eq}(\mathbf{1}, \mathbf{1}) \right|^2 + \left| Z\_{eq}(\mathbf{2}, \mathbf{1}) \right|^2 \right\}^{-1} \tag{31}$$

The SE against TMz and TEz polarized incident waves are, respectively

$$SE^{\rm TM} = \mathbf{10lg} F\_{\rm TM} \tag{32}$$

$$\text{SE}^{\text{TE}} = \mathbf{10lg} F\_{\text{TE}} \tag{33}$$

For isotropic material, the transmitted EM power due to incident TMz or TEz polarized plane waves is equal. Namely

$$\left|E^{\rm TM}(a\_{\mathcal{M}})\right| = \left|E^{\rm TE}(a\_{\mathcal{M}})\right| = \left|E(a\_{\mathcal{M}})\right|\tag{34}$$

For anisotropic materials, the total incident and transmitted power are computed as the average of the two polarizations

$$P^{i} = \left[ \left| E^{\text{TM},i} \right|^2 + \left| E^{\text{TE},i} \right|^2 \right] / (4\eta\_0) \tag{35}$$

Electromagnetic Function Textiles DOI: http://dx.doi.org/10.5772/intechopen.85586

$$P = |E(\mathfrak{a}\_M)|^2 / (2\eta\_0) \tag{36}$$

From Eq. (6), we obtain

FTM <sup>¼</sup> <sup>E</sup>TM,i � � � � 2

FTE <sup>¼</sup> ETE,i � � � � 2

wave impedance and Ei � � is the vector of the incident electric field.

amplitude of the corresponding transmitted electrical field. η<sup>0</sup> ½ � is the free space

½ �¼ E að Þ <sup>M</sup> <sup>2</sup>η�<sup>1</sup>

<sup>0</sup> Zeqð Þ <sup>1</sup>; <sup>2</sup> � � � � 2

<sup>0</sup> Zeqð Þ <sup>1</sup>; <sup>1</sup> � � � � 2

> Zeqð Þ <sup>1</sup>; <sup>2</sup> � � � �

Zeqð Þ <sup>1</sup>; <sup>1</sup> � � � �

The SE against TMz and TEz polarized incident waves are, respectively

For isotropic material, the transmitted EM power due to incident TMz or TEz

� <sup>¼</sup> ETEð Þ aM � � �

For anisotropic materials, the total incident and transmitted power are com-

<sup>2</sup> h i

<sup>þ</sup> <sup>E</sup>TE,i � � � �

The electric field component on the back faces of the grid expressed in matrix

� � <sup>¼</sup> ½ �þ YM <sup>2</sup> <sup>η</sup><sup>0</sup> ½ ��<sup>1</sup> n o�<sup>1</sup>

In the case where the incident wave is the TMz polarized wave, set Ei � � <sup>¼</sup> <sup>0</sup>Ey

<sup>0</sup> Zeq

in which ETM,i �

form is given by

in which

� �

Electromagnetic Materials and Devices

� and ETE,i � � �

TMz and TEz polarized plane waves, respectively. ETMð Þ aM

Zeq

� <sup>¼</sup> <sup>2</sup>η�<sup>1</sup>

� <sup>¼</sup> <sup>2</sup>η�<sup>1</sup>

FTM <sup>¼</sup> <sup>η</sup><sup>2</sup> 0 4

FTE <sup>¼</sup> <sup>η</sup><sup>2</sup> 0 4

<sup>E</sup>TMð Þ aM

<sup>P</sup><sup>i</sup> <sup>¼</sup> ETM,i � � � � 2

� �

�

puted as the average of the two polarizations

Similarly, for a TEz polarized incident wave, set Ei � � <sup>¼</sup> Ey

Therefore, the amplitude of the transmitted field is

ETMð Þ aM

ETEð Þ aM � � �

The shielding factors are, respectively

polarized plane waves is equal. Namely

192

� �

�

amplitude of the transmitted field is

<sup>=</sup> ETMð Þ aM �

<sup>=</sup> ETEð Þ aM � � � �

� �

�

<sup>þ</sup> Zeqð Þ <sup>2</sup>; <sup>2</sup> � � � �

<sup>þ</sup> Zeqð Þ <sup>2</sup>; <sup>1</sup> � � � �

<sup>2</sup> n o<sup>1</sup>=<sup>2</sup>

<sup>2</sup> n o<sup>1</sup>=<sup>2</sup>

<sup>2</sup> <sup>þ</sup> Zeqð Þ <sup>2</sup>; <sup>2</sup> � � � �

<sup>2</sup> <sup>þ</sup> Zeqð Þ <sup>2</sup>; <sup>1</sup> � � � �

SETM <sup>¼</sup> 10lgFTM (32)

SETE <sup>¼</sup> 10lgFTE (33)

� ¼ j j E að Þ <sup>M</sup> (34)

= 4η<sup>0</sup> ð Þ (35)

<sup>2</sup> n o�<sup>1</sup>

<sup>2</sup> n o�<sup>1</sup>

�

� are the amplitude of the incident electric field for the

� �

<sup>2</sup> (24)

<sup>2</sup> (25)

� and <sup>E</sup>TEð Þ aM � � �

� � Ei � � (26)

Ei y h i

> Ei z

<sup>i</sup> ½ � ; 0 <sup>t</sup>

� are the

(27)

<sup>i</sup> ½ �<sup>t</sup> .

(28)

(30)

(31)

. Therefore, the

� � (29)

$$SE = 10 \lg \left( \frac{F\_{T\mathcal{M}} + F\_{TE}}{2} \right) \tag{37}$$

For the grid that is isotropic in the ð Þ y; z plane, the SE against the two polarized waves is coincident. It results in

$$\text{SE} = \text{SE}^{\text{TM}} = \text{SE}^{\text{TE}} \tag{38}$$

### 3.2.3 Influencing factors for electromagnetic shielding effectiveness

In combination with the mesh structure of the electromagnetic shielding fabric, the key factors affecting the electromagnetic shielding effectiveness SE are the structural parameters and material parameters of fabric [9, 10]. Some key 1parameters are as following: the periodic spacing of metal yarns, the conductivity of the yarns (decided by the type of yarns, the type of metal fibers, the content of metal fibers), the diameters of the yarns, the arrangement method, the connection of intersections, the direction of incidence of electromagnetic field, and the frequency.

#### 3.2.3.1 The influence of structural parameters on shielding effectiveness

#### 3.2.3.1.1 The influence of periodic spacing on SE

Samples in which copper filaments are arranged in parallel at different intervals have different SE. As shown in Figure 11, the arrangement periodic intervals of copper filaments are 1, 2, 3, 4, and 5 mm, respectively. The SE of 10–14 GHz is 17– 20, 7–12, 5–10, 2–5, and 2–4 dB, respectively. It can be seen that as the periodic spacing increases, the shielding effectiveness is significantly reduced.

Metal yarn arrangement interval has an important effect on SE of fabrics. The metal yarn periodic spacing is related to these parameters such as fabric density and

Figure 11. The shielding effectiveness at different intervals.

Figure 12. The shielding effectiveness of metal fiber yarns in parallel and grid arrangement.

tightness. And as the fabric tightness and density increase, the spacing of the metal fibers reduces, the electromagnetic wave transmission decreases, and the SE increases.

yarns are the commonly used yarn. The type of yarns affects the electromagnetic

Stainless steel filaments, core-spun yarns, blended yarns, and twisted yarns composed of stainless steel/cotton with a stainless steel content of 30% are arranged in a grid sample with a periodic spacing of 2 mm. The SE is shown in Figure 14. At the same spacing, the blended yarns have the best shielding effectiveness, while the shielding effectiveness of stainless steel filaments, core spun yarns, and twisted yarns are equivalent. At the frequency of 8–16 GHz, the SE of the former is about 7 dB higher than that of the latter, and both decrease with increasing frequency.

The metal fibers used in the fabric are different, and the different electrical conductivity of the metal may affect the electrical resistivity of the yarns and the

parameters of the yarns and fabrics, resulting in the difference in SE.

The shielding effectiveness of metal fiber yarns in parallel and grid arrangement.

3.2.3.2.2 The effect of the material of the metal fibers on SE

Figure 13.

Electromagnetic Function Textiles

DOI: http://dx.doi.org/10.5772/intechopen.85586

Figure 14.

195

The shielding effectiveness of different types of yarns.

#### 3.2.3.1.2 The influence of the way of arrangement on SE

The woven fabric is interwoven from the yarns of two systems that are perpendicular to each other. Therefore, it is possible to introduce functional fibers into the parallel structure in only one system, or functional yarns are introduced to both systems to form a grid structure.

The stainless steel core spun yarn and the blended yarn were arranged in parallel as a sample with a spacing of 2 mm, and the distance of the grid structure sample is 2 mm in both vertical and horizontal directions. The shielding effectiveness is shown in Figure 12. It can be seen that the SE of the yarn model of the parallel arrangement structure is the same to the grid arrangement structure, and as the frequency increases, the SE gradually decreases.

#### 3.2.3.1.3 The influence of intersection conduction on SE

Separating the yarns of the horizontal and vertical systems in the sample with a thin insulating plate is seen as a nonconducting state. The bare copper wire is arranged in two states with conduction and nonconduction at the intersection, and the periodic intervals of the grid samples are 1, 2, 3, 4, and 5 mm, respectively, the shielding effectiveness is shown in Figure 13. It can be seen that under the same periodic spacing, the shielding effectiveness curves of the copper wire mesh model samples almost coincide in the two states of conduction and nonconduction.

#### 3.2.3.2 Effect of material parameters on shielding effectiveness

The material parameters mainly include the way of forming the metal yarns, the material of the metal fibers, and the content of the metal fibers.

#### 3.2.3.2.1 The effect of the method of forming yarns on SE

Metal monofilaments can be used to make fabrics after they have been formed into yarns by a certain yarn forming method. For metal filaments, core yarns and twisted yarns are the common yarns. However, for metal staple fibers, blended

#### Figure 13.

tightness. And as the fabric tightness and density increase, the spacing of the metal fibers reduces, the electromagnetic wave transmission decreases, and the SE increases.

The woven fabric is interwoven from the yarns of two systems that are perpendicular to each other. Therefore, it is possible to introduce functional fibers into the parallel structure in only one system, or functional yarns are introduced to both

The stainless steel core spun yarn and the blended yarn were arranged in parallel as a sample with a spacing of 2 mm, and the distance of the grid structure sample is 2 mm in both vertical and horizontal directions. The shielding effectiveness is shown in Figure 12. It can be seen that the SE of the yarn model of the parallel arrangement structure is the same to the grid arrangement structure, and as the

Separating the yarns of the horizontal and vertical systems in the sample with a

The material parameters mainly include the way of forming the metal yarns, the

Metal monofilaments can be used to make fabrics after they have been formed into yarns by a certain yarn forming method. For metal filaments, core yarns and twisted yarns are the common yarns. However, for metal staple fibers, blended

thin insulating plate is seen as a nonconducting state. The bare copper wire is arranged in two states with conduction and nonconduction at the intersection, and the periodic intervals of the grid samples are 1, 2, 3, 4, and 5 mm, respectively, the shielding effectiveness is shown in Figure 13. It can be seen that under the same periodic spacing, the shielding effectiveness curves of the copper wire mesh model samples almost coincide in the two states of conduction and nonconduction.

3.2.3.1.2 The influence of the way of arrangement on SE

The shielding effectiveness of metal fiber yarns in parallel and grid arrangement.

frequency increases, the SE gradually decreases.

3.2.3.1.3 The influence of intersection conduction on SE

3.2.3.2 Effect of material parameters on shielding effectiveness

3.2.3.2.1 The effect of the method of forming yarns on SE

material of the metal fibers, and the content of the metal fibers.

systems to form a grid structure.

Electromagnetic Materials and Devices

Figure 12.

194

The shielding effectiveness of metal fiber yarns in parallel and grid arrangement.

yarns are the commonly used yarn. The type of yarns affects the electromagnetic parameters of the yarns and fabrics, resulting in the difference in SE.

Stainless steel filaments, core-spun yarns, blended yarns, and twisted yarns composed of stainless steel/cotton with a stainless steel content of 30% are arranged in a grid sample with a periodic spacing of 2 mm. The SE is shown in Figure 14. At the same spacing, the blended yarns have the best shielding effectiveness, while the shielding effectiveness of stainless steel filaments, core spun yarns, and twisted yarns are equivalent. At the frequency of 8–16 GHz, the SE of the former is about 7 dB higher than that of the latter, and both decrease with increasing frequency.
