A simulation model of electrical resistance applied in designing conductive woven fabrics

Abstract

Limited researches have been proposed regarding the theoretical model of conductive woven fabric. In a previous study, one type of simulation model was derived to compute the resistance of conductive woven fabric. This paper proposed another fast estimated method to obtain the electrical resistance of conductive thermal woven fabrics (CTWFs) based on the previous model but design oriented. This new model has a similar predicted effect, for which the maximum deviation is less than 1.2% compared to the previous one. The cover factor was a major factor in this model, which assists designers to comprehend and manage the method rapidly. The results revealed that the proposed fast estimated model was well fitted (P-value < 0.05) and could well simulate the electrical resistance of CTWFs within a certain error variation. According to this model, designers can independently estimate the electrical resistance and design customized products of CTWFs, which will be produced effectively by reducing extra waste of energy and cost.

Keywords

fast estimated model conductive woven fabric woven structures electrical resistance design oriented

The combination of conductive yarns with electronic components and different textile methods has been receiving much attention in wearable electronics researches. Challenges in creating conductive fabric are as follows: (a) as electrical functions are embedded in textiles it is crucial to retain the flexibility and comfort of the fabrics;¹ (b) fibers/yarns and fabrics have to meet special requirements concerning not only conductivity but also processability and wearability;² (c) the electrical property of conductive fabric needs better understanding, such as the issue of electrical resistance in conductive fabric. Before applying electronics to the fabrics by using conductive yarns, we need to model the complex and elastic fabric structures. Previous studies on the electromechanical analysis of conductive yarn in textile-based electronic circuits showed little success in presenting an analytical formulation for equivalent electrical resistance.^4–9 Zhao et al.¹ managed to establish a kind of simulation model of electrical resistance for conductive woven fabric. However, most textile designers or garment designers lack the background knowledge of science and engineering or of other disciplines, and may consider it difficult to design intelligent textile products with a fine aesthetic appearance. Thus, a comprehensive and manageable method that designers can adopt is an urgent requirement.

This paper proposes a fast estimated model of electrical resistance used in designing conductive thermal woven fabrics (CTWFs) with three basic weaving structures – plain weave, twill weave, and satin weave. The weft density of weft silver-coated yarn was changed according to different arrangements. Combined with the result of the previous simulation model, the new model will be an empirical model using only the cover factor as variable. Experiments were designed and conducted to verify the availability of the proposed model. A customized design method of CTWFs can be produced according to this fast estimated model, which can meet the demand of a highly efficient prototype design and conserve costs and energy.

Calculation of the cover factor

The fabric cover factor is used to assess the fabric tightness. The warp cover factor, weft cover factor, and fabric cover factor are the areas covered by warp yarn or weft yarn, or both yarns. It can be expressed as the ratio of the area covered by the warp and weft yarn and the total area of the fabric. Under the condition of the same fabric, the greater the fabric cover factor is, the tighter the fabric is. Figure 1 illustrates the maximum cover of 1/3 twill weave, which means the warp yarns are kept in planes so that their projections are touching each other and the weft yarns interlace in between.³
Figure 1.
Structure diagram of 1/3 twill weave.

Take the following, for example
$L_{wa} = a + b + a = b + t_{we} a$
(1)

$a = \sqrt{(d_{wa} + d_{we}) 2 - h_{wa}^{2}}$
(2)

Because the weft yarn and the warp yarn are in the same surface
$h_{wa} = d_{we}$
(3)

$a = \sqrt{d_{wa}^{2} + 2 d_{wa} d_{we} + d_{wa}^{2} - d_{we}^{2}} = \sqrt{d_{wa}^{2} + 2 d_{wa} d_{we}}$
(4)

$b = (R_{wa} - t_{we}) d_{wa}$
(5)
where R_wa is the cyclic pick of the warp yarn and R_we is the same concept for the weft yarn. t_wa and t_we are the frequencies at which the warp yarn and weft yarn interlace in one cyclic unit.

Substituting into the above equation
$L_{wa} = (R_{wa} - t_{we}) d_{wa} + t_{we} \sqrt{d_{wa}^{2} + 2 d_{wa} d_{we}}$
(6)

In the same way
$L_{we} = (R_{we} - t_{wa}) d_{we} + t_{wa} \sqrt{d_{we}^{2} + 2 d_{wa} d_{we}}$
(7)

According to the definition of the cover factor
$K_{wa}' = \frac{R_{wa} d_{wa}}{L_{wa}}$
(8)

$K_{we}' = \frac{R_{we} d_{we}}{L_{we}}$
(9)

If the warp density is d_X, the weft density is d_Y, then $K_{wa}'$ and $K_{we}'$ can be calculated as follows:
1/1 plain weave
$R_{wa} = R_{we} = 2, t_{wa} = t_{we} = 2,$
(10)

$K_{wa}' = \frac{2 d_{X}}{2 \sqrt{d_{X}^{2} + 2 d_{X} d_{Y}}} = \frac{d_{X}}{\sqrt{d_{X}^{2} + 2 d_{X} d_{Y}}}$
(11)

$K_{we}' = \frac{2 d_{Y}}{2 \sqrt{d_{Y}^{2} + 2 d_{X} d_{Y}}} = \frac{d_{Y}}{\sqrt{d_{Y}^{2} + 2 d_{X} d_{Y}}}$
(12)

n/m twill weave
$R_{wa} = R_{we} = n + m, t_{wa} = t_{we} = 2,$
(13)

$K_{wa}' = \frac{(n + m) d_{X}}{(n + m - 2) d_{X} + 2 \sqrt{d_{X}^{2} + 2 d_{X} d_{Y}})}$
(14)

$K_{we}' = \frac{(n + m) d_{Y}}{(n + m - 2) d_{Y} + 2 \sqrt{d_{Y}^{2} + 2 d_{X} d_{Y}})}$
(15)

q ends satin weave
$R_{wa} = R_{we} = q, t_{wa} = t_{we} = 2,$
(16)

$K_{wa}' = \frac{{qd}_{X}}{(q - 2) d_{X} + 2 \sqrt{d_{X}^{2} + 2 d_{X} d_{Y}})}$
(17)

$K_{we}' = \frac{{qd}_{Y}}{(q - 2) d_{Y} + 2 \sqrt{d_{Y}^{2} + 2 d_{X} d_{Y}})}$
(18)

Fast estimated model of electrical resistance

Three basic structures of woven fabrics, plain weave, twill weave, and satin weave, are designed at certain inches in width and certain inches in length. Regular yarns C were used both in the weft and in the warp, as base material. As illustrated in Figure 2, at the left- and right-hand edges of the fabric, several picks of conductive yarn B replaced the warp yarn C to serve as the power supply in the conductive path. Yarn A was woven with yarn C as heating panels at picks according to different arrangements: every pick, every other picks, every fifth picks, etc.
Figure 2.
Three basic structures of woven fabric.

Almost all the designed fabric above cannot reach the maximum cover and, due to the density change, there exists spacing in every adjacent yarns, as shown in Figures 3(a) and (b). However, for calculation convenience, the structure model can be slightly transformed into Figures 3(c) and (d), with which the internal spacing length L_i will be introduced to the model. Since N_waC represents the picks of weft yarn C, the cyclic unit can be calculated
$C_{wa} = \frac{N_{waC}}{R_{wa}}$
(19)

Figure 3.
Modified structure of woven fabric for calculation purposes.

The internal spacing length L_i will be
$L_{i wa} = \frac{W - N_{waC} * d_{wa}}{N_{waC}}$
(20)

where W represents the fabric width.

Therefore, the modified formula for cover factor is
$K_{wa}' = \frac{R_{wa} d_{wa}}{L_{wa} + L_{i wa}}$
(21)

$K_{we}' = \frac{R_{we} d_{we}}{L_{we} + L_{i we}}$
(22)

$K_{wa}' = \frac{(N + M) d_{X}}{(N + M) 2) d_{X} + 2 \sqrt{d_{X}^{2} + 2 d_{X} d_{Y}}) + \frac{W - N_{waC} * d_{wa}}{N_{waC}}}$
(23)

$K_{we}' = \frac{(N + M) d_{Y}}{(n + m - 2) d_{Y} + 2 \sqrt{d_{Y}^{2} + 2 d_{X} d_{Y}}) + \frac{W - N_{weC} * d_{we}}{N_{weC}}}$
(24)
when N = 1, M = 1, plain weave; N = n, M = m, twill weave; and N = 1, M = q – 1, satin weave.

This formula may appear complex, but most of the components are constant numbers, which can be easily calculated.

According to the special design, yarn A was woven with yarn C at specific picks. The core of the fast estimated model is to gather the cover factor of yarn A. By means of the cover factor of yarn C, the target value can be calculated. As demonstrated in Figure 3(e), the relation of diameter between yarn A and yarn C can be described as
$d_{we} = n * d_{A}$
(25)

Thus, the cover factor of yarn A is
$K_{weA}' = \frac{(N + M) d_{Y}}{(n + m - 2) d_{Y} + 2 \sqrt{d_{Y}^{2} + 2 d_{X} d_{Y}}) + \frac{W - N_{weC} * d_{we} * n^{- 1}}{N_{weC}}}$
(26)

According to the different arrangement of yarn A and different density of the fabric, the picks of yarn A used can be calculated as shown in Table 1. Six samples are selected to use in the fast estimated model set-up. The weft density of yarn A is listed in Table 2. All the samples are fabricated according to the parameters shown in Table 1.
Table 1.
Weaving samples for different weft densities and conductive yarn arrangements in the experiment

Sample
S1
S3 S5 S9 S11

Weft density Interval picks +0 +2 +4 +20 +50

25 Picks of total yarn A 147 49 29 7 2

30 177 59 35 8 3

35 206 68 41 9 4

The bold values in Table 1 are the selected samples that listed in the Table 2.

Table 2.
Weft density of yarn A of selected samples

Sample 35S1 25S1 30S3 25S5 30S9 25S11

Weft density of yarn A (picks/inch) 35 25 10 5 1.43 0.49

New sample code K35 K25 K10 K5 K1.5 K0.5

Table 3.
Cover factor and electrical resistance of conductive thermal woven fabrics

Sample K_P R_P (Ω) K_T R_T (Ω) K_S R_S (Ω)

K35 0.5131 3.870 0.5297 3.700 0.5384 3.610

K25 0.3731 5.420 0.3825 5.180 0.3870 5.060

K10 0.1486 13.510 0.1527 12.910 0.1549 12.610

K5 0.0736 27.490 0.0755 26.260 0.0763 25.650

K1.5 0.0202 99.670 0.0207 95.210 0.0210 92.980

K0.5 0.0051 398.660 0.0052 380.830 0.0053 371.920

Table 4.
Analysis of variance table of curve fitting

Sum of squares df Mean square F Sig.

Plain weave

Regression 15.681 1 15.681 330,619.465 .000

Residual .000 4 .000

Total 15.681 5

Twill weave

Regression 15.741 1 15.741 755,414.803 .000

Residual .000 4 .000

Total 15.741 5

Satin weave

Regression 15.715 1 15.715 1,268,518.710 .000

Residual .000 4 .000

Total 15.715 5

Table 5.
Sample design information

Structure Weft density and arrangement

30S1 (K30) 30S2 (K15) 35S5 (K7)

Plain 3 3 3

Twill 3 3 3

Satin 3 3 3

The fabric tightness is in direct proportion to the cover factor, so the equation can be described as
$R = {mK}^{n}$
(27)

Since the fast estimated model is an experience model based on previous data, the data we use to establish the model is listed in Table 3. The K values are calculated by the method in the previous part. The R values are calculated by the method in previous research.¹

Taking all the data into equation (27), the fast estimated model to simulate the electrical resistance of CTWFs is
$R_{KP} = 1.994 K^{- 0.997}$
(28)

$R_{KT} = 1.968 K^{- 0.999}$
(29)

$R_{KS} = 1.944 K^{- 0.998}$
(30)

The figures of the curve fit are demonstrated in Figure 4. The R value of each fit is all 1.000. The analysis of variance (ANOVA) table (Table 4) indicates that the P-values are less than 0.001, which means the results are considered statistically extremely significant and the curves are well fitted.
Figure 4.
Curve Fitting for plain weave, twill weave, and satin weave.

Figure 5.
CCI sampling loom and the weaving experiment.

Experimental details

In the experiment, yarn C is 100% cotton yarn with yarn count of 20/2 Ne S ply. Yarn A is 22/1 dtex single filament silver-coated conductive yarn and yarn B is another silver-coated conductive yarn of 235/34 dtex 2-ply. The electrical resistances of each conductive yarn are 72.6 and 1.1 Ω per cm, while the diameters are 0.005 and 0.290 mm, respectively. The inner fiber of yarn A and yarn B is nylon 6 and nylon 66, respectively. Each kind of sample has three pieces manufactured by a CCI tech automatic dobby sampling loom (Figure 5) in three weaves: plain weave, twill weave, and satin weave. The head type is a gripper head with a speed of around 25 r/min.

As displayed in Table 5, the experiment selects three kinds of fabric with specific weft density and arrangement, which are 30S1, 30S2, and 35S5 with cover factors of 30, 15, and 7, respectively. 30S1 means the fabric has yarn A in every pick with a weft density of 30 picks/inch. 30S2 means the fabric has yarn A in every other pick with a weft density of 30 picks/inch. 35S5 means the fabric has yarn A in five picks with a weft density of 35 picks/inch. The warp density is maintained at 40 ends/inch.

The design and fabrication of the samples are displayed in Figure 6. All samples are tested in a control room under the KSON control system with an air pressure of 1 atm, relative humidity of 65 ± 2%, and temperature of 23 ± 1℃. For measurement purposes, all samples are placed inside the control room for 24 h before testing and none of them are treated with washing or ironing before testing. The samples are aligned on an insulated hard board, the electrical resistance of which was measured by the four-probe method with a Keithley 2010 multimeter.
Figure 6.
Sample design and fabrication.

Result and discussion

Comparison between simulated and measured results

Table 6 lists the simulated result calculated by the fast estimated model. Figure 4 illustrates the comparison between the simulated values and the measured values. The error bar represents the standard deviation. The variations of the 30S1(K30) sample are 12.36% for plain, 1.93% for twill, and 5.43% for satin. The variations of the 30S2(K15) sample are 13.99% for plain, 10.05% for twill, and 3.27% for satin. The variations of the 35S5(K7) sample are 20.72.11% for plain, 12.78% for twill, and 13.95% for satin. The error percentage decreased in the K31 and K15 samples, but increased in the K7 sample when the structure changed. In addition, as the cover factor decreased, the error percentage apparently increased.
Table 6.
Comparison of simulated value and measured value

Sample K_P RK_P (Ω) RM_P (Ω) K_T RK_T (Ω) RM_T (Ω) K_S RK_S (Ω) RM_S (Ω)

30S1 (K30) 0.4459 4.461 5.090 0.4582 4.292 4.990 0.4646 4.178 5.270

30S2 (K15) 0.2217 8.954 9.130 0.2278 8.626 9.590 0.2310 8.391 9.620

35S5 (K7) 0.1021 19.397 20.510 0.1054 18.630 19.260 0.1072 18.054 20.980

The bold values emphasized the R value simulated in fast estimated model.

Suppose that R_M = A + B * R_S (20), where intercept A represents the deviation of the simulated value while coefficient B represents the degree of linear fit. In Figure 7, the linear regression analysis indicates that all the coefficients B are close to 1, which means the models are quite fit to the measurement.
Figure 7.
Linear regression analyses of the simulated and measured values.

The ANOVA table (Table 7) indicates that all the P-values are less than 0.05, which means the results are considered statistically significant and the models are well fitted. In addition, the values of R² are 1.000, 0.996, and 1.000. The closer to 1 the figure is, the better the fit is.
Table 7.
Analysis of variance table of linear regression

Sum of squares df Mean square F Sig.

Plain weave

Regression 127.641 1 127.641 563.150 .027

Residual .227 1 .227

Total 127.867 2

Twill weave

Regression 106.049 1 106.049 2047.686 .014

Residual .052 1 .052

Total 106.101 2

Satin weave

Regression 131.480 1 131.480 1174.586 .019

Residual .112 1 .112

Total 131.592 2

A comparison of the previous model and the fast estimated model is given in Figure 8. RK represents the result using the fast estimated model; RS represents the simulated value using the previous model; RM represents the measured value. The accuracy of the fast estimated model is literally the same as that of the previous model. However, both models have deviation compared to their measured values. There may be two reasons for this. The first one is that the predicted models have their limitations in precision due to the calculation method, no matter the length or cover factor, only considering ideal circumstances. The second reason may be the deviation that is brought from the measurement of the electrical resistance.
Figure 8.
Comparison of the previous model and the fast estimated model.

Equivalent fabric with similar electrical resistance

According to the testing results both in the previous and current experiment, equivalent samples have been noticed due to their similar electrical resistance. The same structures with different weft density or different parameters in both are the target data. Take P25, T25, and S25, for example, shown below, where most of the samples find very similar substitutes while others may be affected by the arrangement limitation of yarn A for unmatched data. In Table 8, the selected samples are listed to support the concept of equivalent fabric, which means that through parameter adjustment, the designer can substitute one design sample with another. The design option, fabric quality, and production cost may meet the needs of different requirements.
Table 8.
Selected equivalent samples with weft density of 25 picks/inch (color online only)

Equivalent samples make the concept of customized design realizable, as shown in Figure 9. At the same size with the same electrical resistance, it is possible to choose woven fabric with a different structure and weft density. If the structure maintains constant, design A can be substituted by arranging less yarn A in the fabric and increasing the weft density. For example, with the same structure in the satin weave, S25-S5 has the same electrical resistance as S30-S6. S6 has less yarn A, which can reduce the usage of silver-coated yarn, thus reducing the cost only by revising a larger weft density. If the structure needs to be replaced by another one, the arrangement of yarn A will be raised while the weft density decreases, such as a loose (25/picks/inch) twill with yarn A embedded in every three picks or a tight (30/picks/inch) satin with yarn A in every four picks. If the density is fixed, a substitute design can also be obtained by arranging more yarn A to reduce the deviation caused by the structure alternating. As shown in Table 8, the red colored values are very similar, which can be adopted as substitute samples. The blue colored ones have the exact same values, which can be perfectly switched as equivalent samples.
Figure 9.
Concept of customized design of conductive thermal woven fabric.

After calculating all the cover factors of yarn A, some interesting phenomena occurred. As highlighted in blue, Table 9 shows that the cover factor of 25 picks/inch is almost the same as that with 30 picks/inch. Referred to Table 8, all samples S5 with a weft density of 25 picks/inch have similar electrical resistance of samples S6 of 30 picks/inch, despite their different structures. In addition, as colored in blue in Table 9, the values of the cover factor are almost the same, which means that the electrical resistance of those samples can be switched to equivalent samples with some adjustment in structure. In summary, samples with lower density in certain numbers of yarn A can be replaced by samples with higher density in lower numbers of yarn A combining with structure adjustment, which makes it controllable to the electrical resistance of the thermal woven fabric within the same size, thus reducing the usage of silver-coated yarn and thereby reducing the cost.
Table 9.
Cover factor of yarn A of all samples (color online only)

Sample S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11

Interval Picks
+0 +1 +2 +3 +4 +5 +10 +15 +20 +25 +50

D_w
ρ 1 2 3 4 5 6 11 16 21 26 51

25 N_WE 147 73 49 36 29 24 13 9 7 5 2

30 177 88 59 44 35 29 16 11 8 6 3

35 206 103 68 51 41 34 18 12 9 7 4

25 picks/inch
30 picks/inch 35 picks/inch

KweA
RKp
KweA
RKt
KweA
RKs
KweA
RKp
KweA
RKt
KweA
RKs
KweA
RKp
KweA
RKt
KweA
RKs

S1 0.373 5.328 0.383 5.140 0.387 5.014 0.446 4.461 0.458 4.292 0.465 4.178 0.513 3.879 0.530 3.713 0.538 3.606

S2 0.185 10.707 0.190 10.342 0.192 10.083 0.222 8.955 0.228 8.627 0.231 8.392 0.257 7.741 0.265 7.421 0.269 7.202

S3 0.124 15.932 0.128 15.402 0.129 15.009 0.149 13.340 0.153 12.862 0.155 12.506 0.169 11.711 0.175 11.236 0.178 10.900

S4 0.091 21.665 0.094 20.957 0.095 20.416 0.111 17.873 0.114 17.241 0.115 16.760 0.127 15.602 0.131 14.977 0.133 14.525

S5 0.074 26.877 0.075 26.010 0.076 25.333 0.088 22.453 0.091 21.670 0.092 21.060 0.102 19.394 0.105 18.625 0.107 18.060

S6 0.061 32.457 0.062 31.423 0.063 30.600 0.073 27.083 0.075 26.149 0.076 25.408 0.085 23.374 0.087 22.456 0.089 21.770

S7 0.033 59.811 0.034 57.977 0.034 56.422 0.040 49.000 0.041 47.366 0.042 45.997 0.045 44.067 0.046 42.389 0.047 41.068

S8 0.023 86.299 0.023 83.713 0.024 81.439 0.028 71.193 0.028 68.870 0.029 66.854 0.030 66.019 0.031 63.558 0.031 61.552

S9 0.018 110.872 0.018 107.604 0.018 104.655 0.020 97.797 0.021 94.667 0.021 91.866 0.022 87.950 0.023 84.720 0.024 82.023

S10 0.013 155.064 0.013 150.595 0.013 146.418 0.015 130.284 0.016 126.186 0.016 122.418 0.017 112.993 0.018 108.898 0.018 105.405

S11 0.005 386.595 0.005 376.143 0.005 365.374 0.008 260.027 0.008 252.197 0.008 244.497 0.010 197.407 0.010 190.465 0.010 184.252

The bold values emphasized the K value simulated in fast estimated model.

Significance of the fast estimated model

Both the simulated model and the fast estimated model can satisfy the predicted the target electrical resistance. The comparison of the data is listed in Table 10. It is notable that the accuracy of the predication is similar.
Table 10.
Comparison of simulated value and measured value

Sample RK_P (Ω) RS_P (Ω) Error percentage (%) RK_T (Ω) RS_T (Ω) Error percentage (%) RK_S (Ω) Rs_S (Ω) Error percentage (%)

30S1 (K30) 4.461 4.500 0.9 4.292 4.300 0.2 4.178 4.200 0.5

30S2 (K15) 8.954 9.060 1.2 8.626 8.660 0.4 8.391 8.450 0.7

35S5 (K7) 19.397 19.450 0.3 18.630 18.580 −0.3 18.054 18.140 0.5

Table 11.
Difference between the fast estimated model and the previous simulated model

Fast estimated model – design oriented

Designer ⇒ only cover factor matters ⇒ comprehensible, manageable ⇒ easy math ⇒ customized design ⇒ substitute design ⇒ reduce sampling ⇒ energy conservation ⇒ cost reduction

Simulation model – production oriented

Technician ⇒ length of conductive yarn ⇒ usage of conductive materials ⇒ cost of conductive materials ⇒ guide design and production ⇒ cost reduction ⇒ energy conservation

The main difference between these two models is their different purposes as listed in Table 11. The proposed fast estimated model is design oriented and is suitable for the designer to adopt, since a number of fashion and textile designers have trouble understanding the mechanism and complex math during the design and development of new textile products, not to mention the thermal textile, which requires a multi-disciplinary background. It is impractical and a waste of energy and time to make them master the technology and knowledge in a short training time. Thus, a more comprehensible and manageable method is keenly required. With the fast estimated model and concepts similar to that, the designer mainly needs to focus on the textile and product rather than the complex calculation of electrical resistance or others, which can reduce the sampling, thus saving energy and money. The experience and cover factor can meet most of their needs in designing thermal products.

On the other hand, the previous simulated model is production oriented and is suitable for technicians in industry. It can predict the length of conductive yarn that will be used in production, which can help technicians during weaving preparation. It can also estimate the cost of conductive yarn, which is fairly expensive, in order to reduce waste in the weaving phase. It can adjust the design and production in advance according to the calculation as well, which can save energy and reduce emissions.

Conclusion

This paper proposed a design-oriented fast estimated method to obtain the electrical resistance of CTWFs based on the previous production-oriented model. The cover factor was a major factor in this model. The results revealed that the proposed fast estimated model was well fitted and could well simulate the electrical resistance of CTWFs within a certain error variation. Compared to the previous model, this new model has similar accuracy. Based on this model, conductive woven fabric and equivalent fabric can be used as a substitute for different applications at optimum conditions. Designers can estimate the electrical resistance; thus, the customized design of CTWFs can be produced effectively with minimum waste.

Sample	S1	S3	S5	S9	S11
25	Picks of total yarn A	147	49	29	7	2
30	177	59	35	8	3
35	206	68	41	9	4

Sample	35S1	25S1	30S3	25S5	30S9	25S11
Weft density of yarn A (picks/inch)	35	25	10	5	1.43	0.49
New sample code	K35	K25	K10	K5	K1.5	K0.5

Sample	K_P	R_P (Ω)	K_T	R_T (Ω)	K_S	R_S (Ω)
K35	0.5131	3.870	0.5297	3.700	0.5384	3.610
K25	0.3731	5.420	0.3825	5.180	0.3870	5.060
K10	0.1486	13.510	0.1527	12.910	0.1549	12.610
K5	0.0736	27.490	0.0755	26.260	0.0763	25.650
K1.5	0.0202	99.670	0.0207	95.210	0.0210	92.980
K0.5	0.0051	398.660	0.0052	380.830	0.0053	371.920

	Sum of squares	df	Mean square	F	Sig.
Plain weave
Regression	15.681	1	15.681	330,619.465	.000
Residual	.000	4	.000
Total	15.681	5
Twill weave
Regression	15.741	1	15.741	755,414.803	.000
Residual	.000	4	.000
Total	15.741	5
Satin weave
Regression	15.715	1	15.715	1,268,518.710	.000
Residual	.000	4	.000
Total	15.715	5

Structure	Weft density and arrangement
Plain	3	3	3
Twill	3	3	3
Satin	3	3	3

Sample	K_P	RK_P (Ω)	RM_P (Ω)	K_T	RK_T (Ω)	RM_T (Ω)	K_S	RK_S (Ω)	RM_S (Ω)
30S1 (K30)	0.4459	4.461	5.090	0.4582	4.292	4.990	0.4646	4.178	5.270
30S2 (K15)	0.2217	8.954	9.130	0.2278	8.626	9.590	0.2310	8.391	9.620
35S5 (K7)	0.1021	19.397	20.510	0.1054	18.630	19.260	0.1072	18.054	20.980

	Sum of squares	df	Mean square	F	Sig.
Plain weave
Regression	127.641	1	127.641	563.150	.027
Residual	.227	1	.227
Total	127.867	2
Twill weave
Regression	106.049	1	106.049	2047.686	.014
Residual	.052	1	.052
Total	106.101	2
Satin weave
Regression	131.480	1	131.480	1174.586	.019
Residual	.112	1	.112
Total	131.592	2

	Sample	S1	S2	S3	S4	S5	S6	S7	S8	S9	S10	S11
	Interval Picks	+0	+1	+2	+3	+4	+5	+10	+15	+20	+25	+50
D_w	ρ	1	2	3	4	5	6	11	16	21	26	51
25	N_WE	147	73	49	36	29	24	13	9	7	5	2
30	177	88	59	44	35	29	16	11	8	6	3
35	206	103	68	51	41	34	18	12	9	7	4

Sample	RK_P (Ω)	RS_P (Ω)	Error percentage (%)	RK_T (Ω)	RS_T (Ω)	Error percentage (%)	RK_S (Ω)	Rs_S (Ω)	Error percentage (%)
30S1 (K30)	4.461	4.500	0.9	4.292	4.300	0.2	4.178	4.200	0.5
30S2 (K15)	8.954	9.060	1.2	8.626	8.660	0.4	8.391	8.450	0.7
35S5 (K7)	19.397	19.450	0.3	18.630	18.580	−0.3	18.054	18.140	0.5

Footnotes

Declaration of conflicting interests

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

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is financially supported by the Research Grants Council (RGC) of Hong Kong, China (Project Number: PolyU 154031/14H).

References

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Sample		S1	S3	S5	S9	S11
Weft density	Interval picks	+0	+2	+4	+20	+50
25	Picks of total yarn A	147	49	29	7	2
30		177	59	35	8	3
35		206	68	41	9	4

	25 picks/inch						30 picks/inch						35 picks/inch
	KweA	RKp	KweA	RKt	KweA	RKs	KweA	RKp	KweA	RKt	KweA	RKs	KweA	RKp	KweA	RKt	KweA	RKs
S1	0.373	5.328	0.383	5.140	0.387	5.014	0.446	4.461	0.458	4.292	0.465	4.178	0.513	3.879	0.530	3.713	0.538	3.606
S2	0.185	10.707	0.190	10.342	0.192	10.083	0.222	8.955	0.228	8.627	0.231	8.392	0.257	7.741	0.265	7.421	0.269	7.202
S3	0.124	15.932	0.128	15.402	0.129	15.009	0.149	13.340	0.153	12.862	0.155	12.506	0.169	11.711	0.175	11.236	0.178	10.900
S4	0.091	21.665	0.094	20.957	0.095	20.416	0.111	17.873	0.114	17.241	0.115	16.760	0.127	15.602	0.131	14.977	0.133	14.525
S5	0.074	26.877	0.075	26.010	0.076	25.333	0.088	22.453	0.091	21.670	0.092	21.060	0.102	19.394	0.105	18.625	0.107	18.060
S6	0.061	32.457	0.062	31.423	0.063	30.600	0.073	27.083	0.075	26.149	0.076	25.408	0.085	23.374	0.087	22.456	0.089	21.770
S7	0.033	59.811	0.034	57.977	0.034	56.422	0.040	49.000	0.041	47.366	0.042	45.997	0.045	44.067	0.046	42.389	0.047	41.068
S8	0.023	86.299	0.023	83.713	0.024	81.439	0.028	71.193	0.028	68.870	0.029	66.854	0.030	66.019	0.031	63.558	0.031	61.552
S9	0.018	110.872	0.018	107.604	0.018	104.655	0.020	97.797	0.021	94.667	0.021	91.866	0.022	87.950	0.023	84.720	0.024	82.023
S10	0.013	155.064	0.013	150.595	0.013	146.418	0.015	130.284	0.016	126.186	0.016	122.418	0.017	112.993	0.018	108.898	0.018	105.405
S11	0.005	386.595	0.005	376.143	0.005	365.374	0.008	260.027	0.008	252.197	0.008	244.497	0.010	197.407	0.010	190.465	0.010	184.252

Fast estimated model – design oriented
Designer	⇒ only cover factor matters ⇒ comprehensible, manageable ⇒ easy math ⇒ customized design ⇒ substitute design ⇒ reduce sampling ⇒ energy conservation ⇒ cost reduction
Simulation model – production oriented
Technician	⇒ length of conductive yarn ⇒ usage of conductive materials ⇒ cost of conductive materials ⇒ guide design and production ⇒ cost reduction ⇒ energy conservation

A simulation model of electrical resistance applied in designing conductive woven fabrics – Part II: fast estimated model

Abstract

Keywords

Calculation of the cover factor

Fast estimated model of electrical resistance

Experimental details

Result and discussion

Comparison between simulated and measured results

Equivalent fabric with similar electrical resistance

Significance of the fast estimated model

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References