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Abstract

To reveal the engineering relationship among the electrical properties of embroidered conductive lines, the electrical properties and arrangements of conductive yarns, it is necessary to establish their equivalent resistance model. Embroidered conductive lines in textiles are usually fabricated by single-layer (conductive and nonconductive yarn used as upper and lower yarn) or double-layer embroidery technology (conductive yarns used as upper and lower yarn). Several researchers have proposed the simple resistance model for single-layer embroidered conductive line based on geometric structure of single conductive yarn in fabric. However, the double-layer conductive line has the contact resistance periodically interlaced by the upper and lower conductive yarns, and it made its equivalent circuit different from that of single-layer conductive line. In this work, a geometric model was built to describe the trace of conductive yarn in fabric, and in combination with Wheatstone Bridge theory, was applied to establish the equivalent resistance models of double-layer conductive lines with a certain width, consisted of various courses. First, the equivalent resistance model of double-layer conductive lines consisting of single course was proposed to calculate the contact resistance. Then, to obtain the electrical resistance of double-layer conductive lines with a certain width, the equivalent resistance model was extended from single course to multiple courses ( $n \geq 3$ ). Finally, to validate the proposed equivalent resistance model, double-layer conductive lines with different embroidery parameters (stitch length and stitch spacing) on nonwoven fabric were fabricated and evaluated. The experimental results revealed that the proposed model accurately predicted the resistances of double-layer conductive lines.

Keywords

Double-layer conductive lines embroidery parameters geometric model contact resistance

Introduction

Wearable devices are growing rapidly in communications, sports, medical monitoring, and security applications.^1,2 As the essential components of many wearable electronic products, conductive lines play an important role in integrating electronics into fabrics.^3,4 In general, the conductive yarns are used to form conductive line on fabric by three technologies (knitting⁵, weaving,⁶ and embroidering^7,8). Among these technologies, digital embroidery technology has gained tremendous attention in wearable electronics as it has the advantage of scale production and high flexibility and aesthetics and does not need glue or etching.^9,10 Therefore, embroidery technology has been widely used to form textile circuits,¹¹ electrodes,¹² transmission lines,¹³ and sensors.^14,15

In the process of fabricating embroidered conductive lines, the embroidery parameters determine the distribution of conductive yarns and further affect the electrical properties of conductive lines. A number of studies show that the electrical performance of embroidered wearable electronics depends on the conductivity of conductive yarn, stitch type, and stitch spacing.^16–18 Also, the result shows that stitch type has a greater influence than stitch spacing on their electrical properties. By comparing satin stitch (perpendicular to the current direction) and peripheral needle pattern (along the current direction), the conductors embroidered by the peripheral needle pattern have lower resistance.^18–20 However, there is a lack of theoretical analysis on the influence of embroidery parameters on their resistance. On the other hand, Zhang proposes the resistance model of single-layer embroidered conductive line by the peripheral needle pattern according to the geometric structure of conductive yarns in the fabric.²¹ The geometric structure of conductive yarns in fabric is based on the trace of single conductive yarn in fabric; however, it is not able to describe the trace consisted of upper and lower conductive yarns in fabric. Therefore, it is critical to establish a new geometric model to describe the trace consisted of upper and lower conductive yarns and further establish the equivalent resistance model of double-layer embroidered conductive lines.

Additionally, the embroidered electronics can be fabricated by single-layer or double-layer embroidery technology. For the double-layer conductive line, it has the contact resistance periodically interlaced by the upper and lower conductive yarns, which made its equivalent resistance model different from that of single-layer conductive line. This structure of double-layer conductive line indicates that its resistance model depends on the contact resistance and linear resistance of conductive yarns. Several studies have discussed on the contact resistance in wearable textiles. The contact resistance of two overlapping conductive materials for the conductive knitted fabric is inversely proportional to their contact force according to the contact resistance theory.²² Zhang et al.²³ further establish the relationship between the normal force on the yarns and the contact resistance, and show that the contact resistance of two overlapping conductive materials can be obtained empirically. For the contact resistance interlaced by upper and lower conductive yarns, due to its contact state different from that of two overlapping conductive materials, there are currently no related reports. In Linz’s work, the conductive yarn is pierced into the hole of the conductive pad by an embroidered needle to form a loop, and contact the conductive pad to generate the contact resistance. Their contact behavior is simulated by that of conductive yarn loop tied around a cylinder and the metal conductor (acted as the conductive pad) along the height of the cylinder.²⁴ Then, the contact resistance can be obtained according to the measured results of the equivalent structure. Although the contact state of conductive yarn and conductive pad is different from the interlaced contact of upper and lower conductive yarns in embroidery, the method provides us an idea for the equivalent replacement method to obtain the contact resistance. As mentioned above, it is still a problem to obtain the contact resistance interlaced by upper and lower conductive yarns.

In this paper, the influence of embroidery parameters on the electrical resistance of double-layer embroidered conductive lines with a certain width, consisted of various courses, was systematically studied and the equivalent electrical resistance models were proposed. The geometric structure model of the upper and lower conductive yarns in fabric was built to describe the structure of double-layer embroidered conductive lines. On the basis of the geometric model, the contact electrical resistance interlaced by of the upper and lower conductive yarns was proposed in terms of the electrical resistance of conductive line consisted of single course, and then was applied to obtain the electrical resistance of double-layer conductive lines with a certain width, consisted of multiple courses ( $n \geq 3$ ). Finally, the proposed resistance model was validated by double-layer conductive lines with different embroidery parameters on nonwoven fabric.

Theoretical model

The establishment of embroidery stitches geometric model

The double-layer embroidered conductive line refers to the conductor formed by the upper and lower conductive yarns through embroidery technology in a certain way, as illustrated in Figure 1. In this work, the conductive yarn is assumed to be uniform and have the same electrical properties. The geometric structure of double-layer conductive lines embroidered by the peripheral needle pattern in fabric can be divided into two parts: the geometric structure of the conductive yarns on the fabric surface shown in Figure 2 and through fabric plane shown in Figure 3.

Figure 1.

Double-layer embroidery technology and embroidered conductive lines.

Figure 2.

Geometric structure of the conductive yarns on the fabric surface.

Figure 3.

Geometric structure of the conductive yarns in the thickness direction of fabric: (a) the schematic diagram; (b) the geometric model.

As shown in Figure 2, the conductive yarn embroiders along one direction to form the conductive lines with a certain length and reciprocates back and forth to form the conductive lines with a certain width, so the conductive line is composed of many stitch lengths and parallel yarns. The number of stitch length in each line can be obtained from²⁵

m = {\begin{matrix} \frac{L}{L_{s}^{'}}, \frac{L}{L_{s}^{'}} i s i n t e g e r \\ ⌊ \frac{L}{L_{s}^{'}} ⌋ + 1, \frac{L}{L_{s}^{'}} i s n^{'} t i n t e g e r \end{matrix}

(1)

where stitch length

L_{s}^{'}

refers to the distance between two adjacent stitches along the stitch direction (X direction) and

L

is the length of the conductive line.

Then, the number of parallel yarns is followed as^[25]

n = (⌊ \frac{W}{d_{1}} ⌋ + 1) \times 2 + 1

(2)

where

W

is the width of the conductive line and stitch spacing

d_{1}

refers to the vertical distance between adjacent stitches in the same embroidery direction.

According to equation (2), the corresponding output stitch spacing $d_{2}$ is

d_{2} = \frac{W}{⌊ \frac{W}{d_{1}} ⌋ + 1}

(3)

From the embroidery process, the embroidered conductive lines with one course consist of many same embroidery stitch units.¹⁶ The schematic diagram and geometric model of an embroidery stitch unit (ABCDEFGA) are shown in Figure 3(a) and (b), respectively. The interlacement (

\hat{A B}

) of the upper and lower conductive yarns forms contact resistance. In addition, the segments (BC, DE, EF, and BG) of the conductive yarn in the fabric can be regarded as straight lines under the embroidery tension. During the embroidery process, the needle goes down through the fabric and out almost at the same position. Therefore, it is considered that the angle

θ

is nearly equal to

90 °

. In this case, the upper yarn (in blue) and the lower yarn (in yellow) are closely connected along the Z direction (fabric thickness direction) shown in Figure 3(a), which can be regarded as orthogonal interlacing, and in this way, the distance of point A and B is similarly equal to the diameter d of the conductive yarn. Additionally, the theoretical A or B is at the midpoint of the fabric thickness T if the upper and lower conductive yarns have the same tension. The effect of tension fluctuation between the upper and lower conductive yarns is not considered in this work. From the geometric relationship shown in Figure 3(b) as well as the above assumption, the corresponding lengths are expressed as

l_{1} = l_{3} = \frac{T}{2} \times \frac{1}{\sin θ} \approx \frac{T}{2}

(4)

l_{2} = l_{5} = L_{s} - d - 2 \times L_{B c} \times s i n (θ - 90 °) \approx L_{s} - d

(5)

l_{4} = l_{6} = \frac{T}{2} \times \frac{1}{s i n θ} \approx \frac{T}{2}

(6)

Resistances of each part in the embroidery stitch unit model

Based on the above geometric model, single embroidery stitch unit has both length-related resistance $R_{l}$ and contact resistance $R_{c}$ , as shown in Figure 4. $R_{l 1}$ , $R_{l 2}$ , $R_{l 3,} R_{l 4}$ , $R_{l 5}$ , and $R_{l 6}$ represent the length-related resistances. For the length-related resistance of adjacent conductive yarns along the Y direction (perpendicular to the stitch direction), it is much smaller than the equivalent resistance in each line along the X direction and could be neglected. $R_{c 1}$ represents the contact resistance of upper and lower conductive yarns shown in Figure 4(a), which is identical in the entire resistance model. The contact resistance $R_{c 2}$ will be generated between the adjacent conductive yarns along the Y direction when the output stitch spacing $d_{2}$ is less than twice the diameter of conductive yarns ( $d_{2} \leq 2 d$ ). Under the assumption of the uniform distribution of stitches, $R_{c 2}$ is identical in the entire model. The length-related resistance in stitch segment is divided into left ( $R_{k l}$ ) and right parts ( $R_{k r}$ ) by the ratio factor k, shown in Figure 4(b), and the resistances of adjacent yarns are the same. According to the Wheatstone bridge theory, there is no current flowing through the contact resistance $R_{c 2}$ , which is regarded as open. Therefore, the resistance model was further simplified, as shown Figure 4(c).

Figure 4.

Schematic diagram (left) and resistance model (right) of embroidery stitch unit (a) with one course; (b) with multiple courses ( $d_{2} \leq 2 d$ ); (c) with multiple courses ( $d_{2} > 2 d$ ).

Otherwise ( $d_{2} > 2 d$ ), the two adjacent conductive yarns along the Y direction will not contact, as shown in Figure 4(c).

The establishment of resistance model of conductive line

From the resistance model of each embroidery stitch unit, the resistance models of conductive lines with various courses were established. According to Figure 4(a), there will be no contact resistance $R_{c 2}$ if the conductive line is embroidered only with one course ( $n = 1)$ , and then the contact resistance $R_{c 1}$ is obtained. According to equation (2), the conductive line with a certain width is composed of at least three parallel conductive yarns ( $n \geq 3$ ). Herein, the resistance models of conductive lines are discussed according to the conductive yarn’s arrangement along the Y direction ( $n = 1$ and $n \geq 3$ ).

Conductive lines with one course ( $n = 1$ )

The model of conductive line with one course is shown in Figure 5. In the figure, the part enclosed by the red dotted line is the repeating unit along the X direction. The resistance value of the model is represented by

R_{e q} = (m + 1) \times R_{c 1} + m \times \frac{(R_{l 1} + R_{l 2} + R_{l 3}) \times (R_{l 4} + R_{l 5} + R_{l 6})}{R_{l 1} + R_{l 2} + R_{l 3} + R_{l 4} + R_{l 5} + R_{l 6}}

(7)

According to equations (1) and (4–6), equation (7) is represented by

R_{e q} = {\begin{matrix} (\frac{L}{L_{s}^{'}} + 1) \times R_{c 1} + \frac{L}{L_{s}^{'}} \times \frac{T + L_{s}^{'} - d}{2} \times R_{0}, \frac{L}{L_{s}^{'}} i s i n t e g e r \\ (⌊ \frac{L}{L_{s}^{'}} ⌋ + 2) \times R_{c 1} + (⌊ \frac{L}{L_{s}^{'}} ⌋ + 1) \times \frac{T - d + \frac{L}{⌊ \frac{L}{L_{s}^{'}} ⌋ + 1}}{2} \times R_{0}, \frac{L}{L_{s}^{'}} i s n^{'} t i n t e g e r \end{matrix}

(8)

where

R_{0}

represents the resistance per unit length of the conductive yarn.

Figure 5.

Resistance model of double-layer embroidered conductive lines with one course (n = 1).

Conductive lines with multiple courses ( $n \geq 3$ )

According to the section “Resistances of Each Part in the Embroidery Stitch Unit Model,” the contact resistance $R_{c 2}$ will be generated between the adjacent yarns along the Y direction when the output stitch spacing $d_{2}$ is less than twice the diameter of conductive yarns, and there is no current flowing through the contact resistance $R_{c 2}$ . Thus, the simplified resistance model of double-layer embroidered conductive line formed by many contacted conductive yarns ( $n \geq 3$ ) is shown in Figure 6, and the area enclosed by the red dotted line along the X direction is a repeating unit. The equivalent resistance is

R_{e q} = \frac{1}{n} \times [(m + 1) \times R_{c 1} + m \times \frac{(R_{l 1} + R_{l 2} + R_{l 3}) \times (R_{l 4} + R_{l 5} + R_{l 6})}{R_{l 1} + R_{l 2} + R_{l 3} + R_{l 4} + R_{l 5} + R_{l 6}}]

(9)

According to above equations (1) and (3–6), the equivalent resistance can be obtained by

R_{e q} = {\begin{matrix} \frac{1}{(⌊ \frac{W}{d_{1}} ⌋ + 1) \times 2 + 1} \times [(\frac{L}{L_{s}^{'}} + 1) \times R_{c 1} + \frac{L}{L_{s}^{'}} \times \frac{T + L_{s}^{'} - d}{2} \times R_{0}], \frac{L}{L_{s}^{'}} i s i n t e g e r; \\ \frac{1}{(⌊ \frac{W}{d_{1}} ⌋ + 1) \times 2 + 1} \times [(⌊ \frac{L}{L_{s}^{'}} ⌋ + 2) \times R_{c 1} + (⌊ \frac{L}{L_{s}^{'}} ⌋ + 1) \times \frac{T - d + \frac{L}{⌊ \frac{L}{L_{s}^{'}} ⌋ + 1}}{2} \times R_{0}], \frac{L}{L_{s}^{'}} i s n^{'} t i n t e g e r \end{matrix}

(10)

Otherwise, the contact resistance

R_{c 2}

will not be generated. At this time, the resistance model of double-layer embroidered conductive line is the same as the simplified resistance model shown in Figure 6 and the equivalent resistance can be obtained by equation (10).

Figure 6.

Simplified resistance model of double-layer embroidered conductive line with multiple courses ( $n \geq 3$ ).

Model validation and evaluation

Preparation of embroidered conductive lines

The silver-plated nylon multifilament yarn for embroidered conductive lines was from Qingdao Hengtong X-silver Specialty Textile Co., Ltd. (Qingdao, China). Its diameter was 0.17 mm and the linear resistance was 389 Ω/m. Cotton,²⁶ nonwoven fabric,²⁷ denim,²⁸ and other fabrics can be used as substrates. In this work, the nonwoven fabric was taken as examples to validate the model. Then, the conductive yarns were embroidered into a 100% polyester nonwoven substrate through a computerized embroidery machine. Polyester nonwoven was from Shandong Jiantong Engineering Technology Co., Ltd. (Dezhou, China). Its fabric thickness was 0.698 mm measured by fabric thickness gauge (YG141N, the second textile equipment Co., Ltd., Changzhou) according to the GB/T3820-1997 standard (Determination of thickness of textiles and textile products).

The double-layer embroidered conductive lines were fabricated by computerized embroidery machine (brother NV 950), as shown in Figure 1. First, the conductive lines ( $d_{2} > 2 d$ ) were fabricated to obtain the contact resistance $R_{c 1}$ . This method can only obtain the contact resistance $R_{c 1}$ of conductive lines with a certain stitch length. Therefore, the conductive lines ( $d_{2} > 2 d$ ) with different stitch lengths (1–6 mm) were fabricated under the same embroidery tension (level 1) to explore the influence of stitch length on contact resistance $R_{c 1}$ .

In addition, to verify the resistance model of the double-layer embroidered conductive lines with multiple courses, two sets of conductive lines ( $L = 52 m m$ , $W = 3.2 m m$ ) with different stitch spacings and stitch lengths were fabricated on nonwoven fabric. The first group of conductive lines were embroidered by conductive yarns with a fixed stitch length and different stitch spacings. The stitch length was fixed at 4 mm and the stitch spacings ranged from 0.2 mm to 2 mm. The second group of samples were embroidered by conductive yarns with a fixed stitch spacing and different stitch lengths. The stitch spacing was fixed at 0.4 mm and the stitch lengths ranged from 2 mm to 6 mm.

Characterization and measurement

The resistances of embroidered conductive lines were measured by a digital multimeter according to the GB/T 351-2019 standard (metallic materials-Resistance measurement method). The resistances of conductive lines in this work were measured on the rigid, nonconductive board along the direction of the stitch length as shown in Figure 7, all resistance measurements of embroidered conductive lines were repeated five times.

Figure 7.

Test diagram of conductive line.

Validation and discussion

Contact resistance caused by the interlacement of the upper and lower conductive yarns

In the section The establishment of resistance model of conductive line equations (8) and (10) show the relationship between the equivalent resistance of conductive lines with various courses and contact resistance. In order to simplify the calculation of contact resistance $R_{c 1}$ caused by the interlacement of the upper and lower conductive yarns, the contact resistance $R_{c 2}$ of the adjacent yarns along the Y direction should be eliminated. It was a good way to realize it when the resistances of the conductive lines with one course were measured. The contact resistance can be obtained by transferring equation (8)

R_{c 1} = {\begin{matrix} \frac{R_{e q} - \frac{L}{L_{s}^{'}} \times \frac{T + L_{s}^{'} - d}{2} \times R_{0}}{\frac{L}{L_{s}^{'}} + 1}, \frac{L}{L_{s}^{'}} i s i n t e g e r \\ \frac{R_{e q} - (⌊ \frac{L}{L_{s}^{'}} ⌋ + 1) \times \frac{1}{2} \times (T - d + \frac{L}{⌊ \frac{L}{L_{s}^{'}} ⌋ + 1}) \times R_{0}}{⌊ \frac{L}{L_{s}^{'}} ⌋ + 2}, \frac{L}{L_{s}^{'}} i s n^{'} t i n t e g e r \end{matrix}

(11)

Figure 8 shows the contact resistance of the upper and lower conductive yarns with different stitch lengths. The results showed that the contact resistance did not fluctuate greatly with the stitch length, but fluctuated around a certain value (0.26 Ω). According to contact theory, the contact resistance depends on the number of contact points, the contact pressure, material hardness, and the electrical resistivity. For the conductive lines fabricated by the same conductive yarns under the same embroidery tension, the interlacement of the upper and lower conductive yarns have the same material hardness, the same electrical resistivity, the same pressure as well as the same contact points for the same contact diameter.^22,29 It demonstrated that the contact resistance was a fixed value and the average value of contact resistances was taken as the contact resistance in this work, which was 0.264 Ω.

Figure 8.

Contact resistance of the upper and lower conductive yarns with different stitch lengths.

Resistance of conductive lines with different stitch spacings

By inserting the resistance per unit length and related geometric parameters into equation (10), the theoretical resistances of double-layer embroidered conductive lines with different stitch spacings are obtained. Their theoretical and experimental values are shown in Figure 9. It can be seen that the simulated results were very close to the expertimental results and the trends of two data sets were generally consistent while there was a difference between the predicted results and experimental results. The difference is due to the neglect of fabric structure in the model. In fact, the fabric will be compressed under the interlacement of the upper and lower conductive yarns to reduce its thickness, which makes the simulated total length of the conductive yarns in each line larger than the actual yarn. It indicates that the simulated equivalent resistance in each line was larger than the measured values, which further made the simulated equivalent resistances larger than the experimental values according to equation (10). Additionally, it is observed that the resistances of the conductive lines increased in accordance with the increase of the stitch spacings when the stitch length was fixed. The circuit of the conductive lines is composed of many resistances (equivalent resistance in each line) in parallel, and equivalent resistance in each line is composed of many small resistances in series. In the parallel circuit, the reciprocal of the total resistance is equal to the sum of the reciprocal of each sub-resistance, and then the total resistance will be smaller if the more sub-resistances connected in parallel (Ohm’s law). The equivalent resistance of conductive lines in each line kept unchanged when the stitch length was fixed. As the stitch spacing increased, the number of parallel yarns decreased, and the number of the sub-resistances in parallel decreased and made an increase of their equivalent resistance.

Figure 9.

Simulated and experimental resistance of conductive lines with different stitch spacings.

From Figure 9, it is worth noting that the resistance of conductive line is approximately 0.35 Ω when the stitch spacing is 0.2 mm, and the resistance is approximately 2.62 Ω when the stitch spacing is 2 mm. The resistances of conductive lines increased up to around six times in accordance with the increase of the stitch spacing from 0.2 mm to 2 mm, i.e. the change of the stitch spacing makes a multiple increase of the resistance of conductive lines. Moreover, it also can be seen that the resistances of conductive lines are almost the same when their stitch spacings are 1.2 mm, 1.4 mm and 1.6 mm or 1.8 mm, and 2 mm. According to equation (2), the number of parallel yarns along the Y direction depends on the $⌊ \frac{W}{d_{1}} ⌋$ . For the conductive lines with the same width of 3.2 mm, their $⌊ \frac{W}{d_{1}} ⌋$ are the same when the stitch spacings are in the range of 1.2––1.6 mm, which makes their corresponding number of parallel yarns the same and further makes the total resistance close for the conductive lines with fixed stitch lengths.

In addition, the measured resistance of the conductive lines with different stitch spacings and its fitting curve based on the equation (10) are shown in Figure 10. The fitting equation is

y = \frac{1}{0.152 \times ⌊ \frac{3.2}{x} ⌋ + 0.211}

(12)

where

y

and

x

are the resistances of conductive lines and the stitch spacing, respectively. From the data fitting, the fitting gave a good fit to the measured data with 0.982 of regression coefficient. The ANOVA was used to test the significance of the regression equation. The significance level was set as 0.05. The F-value (F = 1246.96) was larger than the F-critical value (

F_{0.05} (2,8) = 19.4

), and the p-value (p < .000) was lower than 0.05 (α). Therefore, the equivalent resistance model of conductive lines based on the geometric model can be verified and used to predict its resistance with stitch spacings.

Figure 10.

Measured resistance of embroidered conductive lines with different stitch spacings and its fitting curve.

Resistance of conductive lines with different stitch lengths

Similarly, by taking the resistance per unit length and related geometric parameters into equation (10), the theoretical resistance of double-layer embroidered conductive lines can be obtained. Figure 11 shows the simulated and experimental resistances of conductive lines with different stitch lengths. It can be seen that there was a difference between the simulated results and the experimental results. In the resistance model, the compression of fabric thickness probably makes the simulated total length of the conductive yarns in each line larger than the actual yarn, so that the simulated equivalent resistance in each line is larger than the measured values. In addition, the resistance of the conductive lines decreased in accordance with the increase of the stitch lengths when the stitch spacing was fixed. In the series circuit, the total resistance is equal to the sum of each sub-resistance, so the total resistance will be larger if the more sub-resistances connected in series (Ohm’s law). The number of resistances (equivalent resistance in each line) in parallel kept unchanged when the stitch spacing was fixed. As the stitch length increased, the number of stitch length in each line decreased accordingly to reduce the number of contact resistances in each line, which made the equivalent resistance in each line decrease and further resulted in the decrease of their equivalent resistance according to equation (10).

Figure 11.

Simulated and experimental resistances of conductive lines with different stitch lengths.

As shown in Figure 11, it is worth noting that the resistance of conductive line is approximately 0.95 Ω when the stitch length is 2 mm, and the resistance is approximately 0.68 Ω when the stitch length is 6 mm. The resistance of conductive line decreased by 28% with the increase of the stitch length from 2 mm to 6 mm; therefore, the resistance could be changed in a small range by adjusting stitch length. Moreover, it also can be seen that the resistances of conductive lines are almost the same when their stitch stitchs are 5.2 mm and 5.6 mm. According to equation (1), the number of stitch length in each line along the X diretion is determined by $⌊ \frac{L}{L_{s}^{'}} ⌋$ . For the conductive lines with the same length of 52 mm, their $⌊ \frac{L}{L_{s}^{'}} ⌋$ are the same when the stitch lengths are 5.2 mm and 5.6 mm, so the total resistance is close to that for the condutive lines with fixed stitch spacings.

In combination with the influence of stitch spacing and stitch length on the resistance of conductive line, the impact of stitch spacing on the resistance of conductive line was greater than that of the stitch length. In conclusion, when a certain resistance of conductive line was required, the stitch spacing should be determined first according to the electrical properties of conductive yarns and then adjusted the stitch length.

In addition, the measured resistances of the conductive lines with the change of stitch lengths and its fitting curve based on the equation (10) are shown in Figure 12, and the fitting equation can be divided into two parts.

Figure 12.

Measured resistances of embroidered conductive lines with different stitch lengths and its fitting curve.

For the first part, when $L / L_{s}^{'}$ is not integer, the fitting equation is

y = 0.012 \times ⌊ \frac{52}{x} ⌋ + 0.595

(13)

where

y

and

x

are the resistances of conductive lines and the stitch length, respectively. This fitting gave a good fit to the measured data, where the regression coefficient is 0.923. The F-value (9723.79) was larger than the F-critical value (

F_{0.05} (2,6) = 5.14

), and the p-value (p < .000) was lower than 0.05 (α). Therefore, the equivalent resistance model of conductive lines based on the geometric model was verified and used to predict its resistance with stitch lengths when the

L / L_{s}^{'}

was not integer.

When $L / L_{s}^{'}$ is integer, the fitting equation is

y = \frac{0.886}{x} + 0.509

(14)

where

y

and

x

are the resistances of conductive lines and the stitch length, respectively. This fitting gave a good fit to the measured data, where the regression coefficient is 0.944. The F-value (817.38) was larger than the F-critical value (

F_{0.05} (2.1) = 200

), and the p-value (p = .022) was lower than 0.05 (α). Thus, the equivalent resistance model of conductive lines based on the geometric model was verified and used to predict its resistance with design stitch lengths when the

L / L_{s}^{'}

was integer.

Conclusion

In this paper, the geometric structure of the upper and lower conductive yarn in fabric was built to simulate the structure of double-layer embroidered conductive lines. Based on the geometric structure, the resistance models of double-layer embroidered conductive lines with different number of courses ( $n = 1$ , $n \geq 3$ ) were proposed. The results showed that the proposed resistance model could be used to calculate contact resistance and predict the equivalent resistance of embroidered conductive line with multiple courses ( $n \geq 3$ ) very well. Additionally, it is observed that the stitch length has little influence on the contact resistance of the upper and lower conductive yarns, and the influence of stitch spacing on the resistances of conductive lines is greater than that of stitch length. Generally, the proposed equivalent resistance model provides a theoretical basis for the engineering design of embroidered electronics.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the Fundamental Research Funds for the Central Universities (2232021G-01) and the Shanghai Natural Science Foundation (20ZR1400500) and the Fundamental Research Funds for the Central Universities (CUSF-DH-D-2018033).

ORCID iDs

Yaya Zhang

Huating Tu

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The equivalent resistance model of double-layer embroidered conductive lines on nonwoven fabric

Abstract

Keywords

Introduction

Theoretical model

The establishment of embroidery stitches geometric model

Resistances of each part in the embroidery stitch unit model

The establishment of resistance model of conductive line

Conductive lines with one course ( n = 1 )

Conductive lines with multiple courses ( n ≥ 3 )

Model validation and evaluation

Preparation of embroidered conductive lines

Characterization and measurement

Validation and discussion

Contact resistance caused by the interlacement of the upper and lower conductive yarns

Resistance of conductive lines with different stitch spacings

Resistance of conductive lines with different stitch lengths

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References

Conductive lines with one course ( $n = 1$ )

Conductive lines with multiple courses ( $n \geq 3$ )