A novel technique for direct measurements of contact resistance between interlaced conductive yarns in a plain weave

Abstract

Contact resistance between interlaced conductive yarns will under certain circumstances constitute a problem for sensor applications and electrical routing in interactive textile structures. This type of resistance could alter the effective area of the sensor and introduce hot-spots in the routing.

This paper presents a technique for measuring contact resistances on fabric samples. The samples used are unit cells of plain weave, that is, two conductive (silver-coated) yarns in the warp direction and two in the weft direction. The numerical values for the contact resistance are of the order of R_c ≈ 0.3 Ω. A resistor network made of through-hole film resistors with known values is used for evaluation of the method. The results show that the technique provides values typically within ±1% error compared with the known resistor values. Thus, the method can be used in order to calculate the contact resistances of a woven conductive textile.

Keywords

Testing Fabrication weaving measurement Materials

Introduction

Over the last two decades publications on the use of electrically conductive textile structures, such as sensors, antennas and routing systems, have been published within the engineering and the textile research field.^1–4 Some of these articles bear witness of the successes in producing, for example, a textile electrocardiogram (ECG) electrode⁵ and a patch antenna.² However, one factor that has been overlooked by many is the influence of the contact resistance between the interlaced conductive yarns on the electrical behavior of such conductive textiles. A high contact resistance could induce hot-spots as current flows through the fabric, which might be the case if the fabric is used as a pathway for high currents (e.g. heating units). An uneven distribution of the contact resistance could alter the performance of a sensor or an antenna made up of the fabric.

Some groups have started investigating the contact resistance. Banaszczyk et al.⁶ developed a numerical method for obtaining the current distribution in a fabric consisting of exclusively conductive yarns. They also showed that the existence of this contact resistance disqualified woven and knitted structures as simple isotropic conductors. Furthermore, they also call for a way of experimentally determining the contact resistance.

Experiments have been carried out in order to obtain quantitative data on the contact resistance of crossed yarns of different types.^7–9 Banaszczyk et al.⁸ and Dhawan et al.⁹ use an ordinary four-point voltage and current measurement procedure (henceforth referred to as “direct”), while Banaszczyka et al.⁷ use an “indirect” method in which an infrared thermography of a sample is compared to the numerical modeling of such a sample. The indirect method is indeed a suitable way of determining the contact resistance, and it could be further optimized if real measured values were provided as input. The problem with a measurement of two free-hanging crossed yarns, as in Banaszczyk et al.⁸ and Dhawan et al.,⁹ is that it does not reflect the geometry and therefore not the mechanical situation of a woven sample.

As mentioned above there is a demand for measured values in order to refine the numerical simulations and to aid the use of the indirect method, and as also mentioned the direct measurements of contact resistance made earlier do not reflect the geometry of a woven sample. There is thus a lack of reliable and realistic quantitative values of the contact resistance; furthermore, the distribution of these values is not known. If the distribution of the contact resistance values were known, the numerical simulations would be even more realistic.

In this paper, we present a technique that is direct and uses real pieces of woven fabric as samples. This technique thereby provides a means for obtaining more realistic values of the contact resistance and also the distribution of these values.

In the following section the circuit theoretical relationships used in the experiment are presented. For the experimental part we describe the reference sample, made up of through-hole film resistors, the textile samples, made up of silver-coated multifilament yarns as conductive elements, and staple fiber cotton yarns as separating and upholding the former. We also describe the measurement set-up. Finally, we present the results and a reliability and validity discussion of the method.

Method

Contact resistance

Contact resistance has mainly two causes: a constriction of the current path between two conductors in contact and the presence of insulating films between the conductors. In most practical situations the constriction resistance can be approximated by the Holm resistance given by

R_{c} = \frac{ρ}{2 a}

where ρ is the resistivity and a is the radius of the contacting surface as long as

a << λ

, where λ is the mean free path of the electrons. In the case of contacting spots, whose size is of the same order of magnitude as λ, the Sharvin resistance

R_{c} \frac{4 p_{F}}{N π e 2 (2 a) 2}

p_F, N, e being the Fermi momentum of the electrons, the density of free electrons and the charge of the electron respectively, is the appropriate model for the contact resistance, as shown in by Mikrajuddin et al.¹⁰ This holds for two contacting bodies, whose other dimensions are much larger than the contacting radius, and that are not contaminated by any insulating films. In the case of a multifilament yarn, each macroscopic contact will consist of several filaments in contact. Depending then on the diameter of the filaments and on the surface topology of the filament, the contacting spots could be well above λ, but they could also be in the order of magnitude of λ. Furthermore, they will most probably be contaminated by insulating films. The technique described in this paper cannot be used to determine the microscopic nature of the contact resistances. Rather it serves to provide useful data for engineering applications and for refinement of numerical simulations, as called for by Banaszczyk et al.⁶

Modeling of samples

A conductive yarn possesses a certain linear resistivity, often expressed in units of Ω/m. Using two yarns in the warp direction and two in the weft direction, a unit cell of a plain weave can be constructed. Such a unit cell can be seen schematically in Figure 1. There will be a finite distance between the yarns in each direction, thus each yarn will possess a finite resistance. In Figure 1 there are four points where the yarns cross each other and at each crossing point one yarn in the warp direction will be in mechanical and electrical contact with one yarn in the weft direction. The electrical contact can be described by a contact resistance at direct currents (DCs). The resistance of each yarn can be divided into three parts along the yarn; one that holds the resistance from the end of the yarn to the first crossing point, one that holds the resistance between the crossing points and one that holds the resistance from the second crossing point to the other end. In this way the unit cell can be modeled as a resistive network comprised of 16 resistors, as depicted in Figure 2. The model in Figure 2 can be redrawn in a planar fashion. This is shown in Figure 3, where the resistors have been given indices that will be explained below in the Indexing section. Note here that it does not matter what microscopic mechanisms are at work in constituting neither the yarn resistance nor the contact resistance, since whatever value is the result it will be contained in one of the model resistors. The technique described here is thus independent of choice of material and also of binding as long as it comes to basic woven structures, as defined by Hatch.¹¹ Nor do other weaving parameters hinder the usage of this technique for determining the contact resistance at the binding points.

Figure 1.

Schematic view of a unit cell.

Figure 2.

The electrical model of a unit cell of a plain weave. The resistors labeled R_y model the yarn and the resistors labeled R_c model the contact resistances.

Figure 3.

The planar view of the equivalent circuit with indices on all resistors and all nodes.

Circuit theoretical calculations

The problem of determining the resistances of a resistive network when the outer node potentials and injected currents are known is an instance of the inverse problem. It is known to be ill-posed. Ohm’s law is applied to all resistors and Kirchhoff’s current law is applied to all nodes. The following boundary condition applies: a current source is present at one of the end nodes and at another node there is a current sink, that is, ground; all other nodes are floating. In the following text U will stand for electrical potential relative to ground, R will stand for electrical resistance and I for electrical current.

Indexing

All node potentials are given indices in the following manner: the first index indicates the direction of the yarn, the second index indicates the column and the third indicates the row, as summarized in Figure 4 and Table 1. Following the above rules for indexing Ohm’s law can be stated for the included resistors as in Table 2 and Kirchhoff’s Current Law can be expressed as in Table 3.

Figure 4.

Graphical explanation of the indices as described in Table 1.

Table 1.

Description of the indices; see also Figure 4 for a graphical explanation

Index	Parameter	Value
i	Direction	1 = along weft (left to right),
		2 = along warp (top to bottom),
		3 = from weft to warp (front to rear)
j	Column	1–4
k	Row	1–4

Table 2.

Expressions for the resistances.

R _1, _j _, _k	(U_1, _j _, _k − U_1, _j _+ 1, _k )/I_1, _j _, _k
R _2, _j _, _k	(U_2, _j _, _k − U_2, _j _, _k _+ 1)/I_2, _j _, _k
R _3, _j _, _k	(U_1, _j _, _k − U_2, _j _, _k )/I_3, _j _, _k

Table 3.

Expressions for the currents in the yarn branches.

i = 1	I_1, _j _, _k = I_1, _j _+ 1, _k + I_3, _j _+ 1, _k
i = 2	I_2, _j _, _k = I_2, _j _, _k _+ 1 – I₃ _,j _, _k _+ 1

Equations

Four measurements are needed to compute for all contact resistances, and along with them the four resistances of the yarn branches between the crossing points (i.e. R₁₂₂, R₂₂₂, R₁₂₃ and R₂₃₂) will also be determined. The four measurements are distinguished from each other by the node of injection and node of rejection and together they form a measurement set. In Figure 5 the four measurements are displayed.

Figure 5.

The four pathways in a measurement set. The grounding point is changed during measurement 4 and the injection node is the same as in measurement 2.

After the first two measurements are finished, the values of resistors R₂₂₂ and R₂₃₂ can be determined as

R_{222} = \frac{(U_{234}^{1} - U_{231}^{1}) (U_{224}^{2} - U_{221}^{2}) - (U_{224}^{1} - U_{221}^{1}) (U_{234}^{2} - U_{231}^{2})}{(U_{234}^{1} - U_{231}^{1}) I_{2} - (U_{234}^{2} - U_{231}^{2}) I_{1}}

(1)

and

R_{232} = \frac{(U_{234}^{1} - U_{231}^{1}) (U_{224}^{2} - U_{221}^{2}) - (U_{224}^{1} - U_{221}^{1}) (U_{234}^{2} - U_{231}^{2})}{(U_{224}^{2} - U_{221}^{2}) I_{1} - (U_{224}^{1} - U_{221}^{1}) I_{2}}

(2)

where the upper index indicates the measurement (e.g. while looking at Figure 4,

U_{234}^{1}

means the voltage measured at the bottom end of the rightmost warp yarn during the injection of the current I₁ (see Figure 5) and

U_{234}^{2}

means the same voltage but measured at the second measurement, that is, during the injection of the current I₂). Once the values of these two resistors are known they can be used to determine the currents flowing in each branch of the circuit and hence the values of R₃₂₃, R₃₃₂ and R₃₃₃ can be computed as the voltage drop over them divided by the current through them. The third measurement is needed to obtain the value of R₁₂₃ and the fourth to obtain R₁₂₂ and R₃₂₂.

Experiments

All experiments were conducted in a room where no precautions had been taken in order to eliminate or minimize any unaccounted electric fields, nor was the atmosphere controlled. All electrical equipment was connected to the normal Swedish power grid.

Evaluation circuit

Two evaluation circuits with the same topology as shown in Figure 3 were constructed, using ordinary through-hole film resistors with a labeled uncertainty of either 1% or 5%. The resistor values were measured individually on a NI USB-4065 DMM digital multimeter using a four-wire configuration before they were soldered together. One of the evaluation circuits had resistors in the range of 800–1200 Ω; the large values were used in order to minimize risks of numerical errors. The other evaluation circuit had resistors in the range of 0.15–3.3 Ω. These values were chosen for the reason that preliminary measurements indicated that this was the range of the resistances in the samples.

Data collection and manipulation

A National Instruments data acquisition card (DAQ), NI-USB 6229 BNC, was used for generating currents and reading potentials. A LabView program was designed to carry out these tasks. The DAQ analog outputs of the DAQ-card were used as voltage-controlled current sources. The current produced in this fashion was recorded by the NI USB-4065 DMM connected in series between the ground and the DUT (Device Under Test) and controlled by the LabView program. Each potential reading, as well as each current reading, consists of 70 points sampled at 1 kHz. The mean value of these points is used in the calculations of, for example, Equations (1) and (2). For each measurement set one value for each resistor is obtained. Each datum presented in the results section of this article was obtained by iterating the measurement set 500 times and the listed values are the mean values and standard deviation of these 500 points.

Textile samples

Samples of plain weave with a 1:1 setting were fabricated on a Sulzer PU loom with eight shafts. At the starting point all warp yarns were mercerized cotton Nm 68-2 staple fiber yarns. The conductive yarns were Statex Schieldex dtex 235 2/1 and these replaced the first, 20th and 40th warp stands. The weft was then set to make 19 picks with the same cotton yarn as in the warp and on every 20th pick the Statex Schieldex yarn was used. The weave was then cut into pieces that contained two conducting yarns in the warp and weft directions, respectively. The samples thus formed “blow-ups” of a unit cell of a plain weave with the cotton yarns as a supporting structure and squares of conductive yarns that were well isolated from each other everywhere except at the crossing points. A sample is shown in Figure 6. As mentioned in the Modeling of samples section, the choice of yarn type (monofilament, multifilament or staple fiber) is not crucial for the measurement.

Figure 6.

A textile sample. The darker yarns are the silver-coated ones. The red markings are for fitting of the sample into the measuring device. (Color online only.).

Measurement device

A device was constructed for carrying out the measurements. It consists of a printed circuit board with the copper plating used as electrodes for the free nodes of the samples. In order to compute all eight resistance values the ground reference node needs to be changed during the last measurement. The switching of the reference point is accomplished by driving two relays that at the fourth measurement change the grounding terminal from R₁₁₂ to R₂₃₁. The voltage drops are all measured relative to the high input of the DMM in order not to include any input impedance from the DMM itself. The light-emitting diodes (LEDs) seen in Figure 7 are there to prevent current paths into the output channels that are not currently exciting the network. The dotted line in Figure 7 shows the interface between the sample and the device shows a photograph of the holder and Figure 8 shows the holder with a sample being mounted.

Figure 7.

A schematic view of the measurement device. The dotted line symbolizes the interface between the sample and the device.

Figure 8.

A sample in the process of being mounted.

Results

Results from known resistors

The technique was tested on a resistive network with the same topology as in Figure 3 with resistors ranging from 0.15 to 3.3 Ω. In Table 4, the μ and σ columns to the left display the mean value and standard deviation as measured by four wires using the NI USB-4065 DMM, and the two rightmost columns show the same parameters measured by the new technique. The length of the connecting legs of the resistors was not recorded. The errors are typically within ±0.7%: two exceptions are R₃₂₃ and R₃₃₃, which have errors of 4.7% and 1.4%, respectively. The errors are calculated in such a manner that the values obtained by the new technique are considered approximations to the values measured with the four-wire technique, put in a formal way:

Table 4.

Four-wire measurements and measurement with the new technique of resistors in the low resistance range.

R_ijk	μ four-wire (Ω)	σ four-wire (mΩ)	μ network (Ω)	σ network (mΩ)	ɛ _abs (Ω)	η (−)	δ (%)
122	1.47	2	1.48	3.2	0.01	0.0068	0.68
123	1.47	2	1.48	5.5	0.01	0.0068	0.68
222	3.28	2	3.28	11.8	0	0	0
232	3.28	2	3.28	8.9	0	0	0
322	0.147	0.3	0.148	3.1	0.001	0.00680	0.68
323	0.148	0.3	0.155	3.5	0.007	0.00473	4.73
332	0.147	0.3	0.148	4.2	0.001	0.00680	0.68
333	0.148	0.3	0.150	4.6	0.002	0.0135	1.35

absolute error

ɛ_{abs} = μ_{n . t} - μ_{4 w}

relative error

η = \frac{ɛ_{abs}}{μ_{4 w}}

and percent error

δ = η * 100

where μ _n.t and μ₄ _w stand for the value measured by the new technique and the value measured by the four-wire technique, respectively. When the new technique was tested on large value resistors (R ≈ 1 kΩ) the errors became practically immeasurable.

Results from woven samples

Following the test with the known resistors, the technique was used on the previously described woven samples. Ten samples were prepared and measured. In Table 5 the measured values for one of the samples, sample no 6, are displayed. The mean values of all the samples can be seen in Table 6. Notice that the mean values of the resistances are given in [Ω], while the standard deviations are given in [mΩ]. The values are the ones obtained from the 8 × 500 measured resistance values; to further clarify the meaning of the values, in Figure 9 the two measurements of the same sample are presented. Each of the two times eight histograms consists of 50 equally spaced bins situated between the minimum value and maximum value. The data in Table 5 is part of the data used in the statistical analysis in the Measurement systems capability study section.

Figure 9.

The distributions of the eight resistances of sample no 6 as measured the first and second time. The blue histograms are the datapoints as measured for the first time and the red ones are from the second measurement. (Color online only.)

Table 5.

Comparisons between values of sample no. 6 as measured for the first and second time.

Sample 6	Measurement 1 resistance (Ω)	σ (mΩ)	Sample 6	Measurement 2 resistance (Ω)	σ (mΩ)
R_ijk			R_ijk
122	2.25	5.7		2.24	6.0
123	2.37	10.8		2.37	12.6
222	3.04	10.9		3.05	11.3
232	3.29	17.2		3.30	15.1
322	0.37	6.2		0.28	10.3
323	0.29	8.6		0.71	28.1
332	0.30	12.3		0.35	10.8
333	0.34	10.0		0.39	15.7

Table 6.

The experimental design for the measurement systems analysis and the resulting r_i values

Std order	Run order	Prototype sample	r ₁	r ₂	r ₃	r ₄	r ₅	r ₆	r ₇	r ₈
8	1	P 5	3.461	0.206	2.493	0.138	3.240	0.173	2.712	0.133
3	2	P 3	2.419	0.336	4.138	0.663	2.639	0.398	3.923	0.226
4	3	P 3	2.425	0.328	4.155	0.389	2.623	0.412	3.909	0.181
11	4	P 7	2.364	1.141	2.988	0.537	2.599	0.292	2.782	0.135
2	5	P 2	3.264	0.309	2.682	0.139	3.500	0.255	2.452	0.260
18	6	P 10	2.962	0.225	2.552	0.490	3.559	0.271	2.670	0.802
10	7	P 6	3.040	0.286	2.367	0.336	3.286	0.303	2.248	0.368
1	8	P 2	3.274	0.414	2.696	0.208	3.506	0.295	2.468	0.281
15	9	P 9	2.589	0.277	3.560	0.259	2.731	0.216	3.662	0.236
12	10	P 7	2.374	1.220	3.019	0.480	2.639	0.363	2.814	0.180
5	11	P 4	2.935	0.278	2.294	0.201	3.204	0.272	2.189	0.282
14	12	P 8	2.638	0.224	3.321	0.385	2.747	0.624	3.454	0.397
9	13	P 6	3.054	0.714	2.371	0.393	3.297	0.347	2.237	0.282
13	14	P 8	2.638	0.232	3.319	0.251	2.740	0.537	3.464	0.404
6	15	P 4	2.962	0.393	2.303	0.238	3.221	0.467	2.182	0.153
7	16	P 5	3.519	0.413	2.553	0.323	3.279	0.301	2.751	0.474
16	17	P 9	2.604	0.345	3.585	0.235	2.764	0.197	3.667	0.253
17	18	P 10	2.995	0.199	2.585	0.817	3.583	0.268	2.693	0.599

Linear resistivity

The conductive yarns in sample no 6 were measured between the binding points with an ordinary ruler and a mark was made at each binding point. After that the conductive yarns were pulled out of the fabric. This was done to measure the length of the yarns “in free”, since the path length of sin(t) is approximately 21.6% longer than the straight line, since the arc length, r(t), of a curve f(t) = sin(t) on the interval 0 ≤ t ≤ 2π is

r (t) = \frac{\int_{0}^{2 π} \sqrt{1 + (f (t)) 2} d t}{2 π} = \frac{\int_{0}^{2 π} \sqrt{1 + \cos 2 (t)} d t}{2 π}

and the yarn in a plain weave can (ideally) be parameterized as a sinus curve. If the angular frequency of the yarn, while traversing its path, is not unity, the ratio between path length and ordinate length will of course vary. The same is also valid for variations in amplitude. The measured resistance values were then divided by the length of the yarns between the marks in order to obtain a value of the linear resistivity of the yarn; the results are displayed in Table 7. As can be seen, the free lengths of the warp yarns are much higher than the free length of the weft yarns.

Table 7.

The lengths and resistances of the four yarn resistances of sample 6 together with the resulting linear resistivity of the yarn.

R_ijk	Length IN fabric (cm)	Length free (cm)	Resistance (Ω)	Linear resistivity (Ω/m)
R ₁₂₂	1.8	2	2.25	112.5
R ₁₂₃	1.8	2	2.37	118.5
R ₂₂₂	2.3	2.6	3.34	128.5
R ₂₂₃	2.3	2.7	3.36	124.4

Measurement systems capability study

The purpose of the measurement systems capability study is to quantify the statistical ability of the measurement system to measure and compute the resistance for each the point of interest on any product of the type described in this paper. This includes not only the factors of uncertainty attributed to the measuring device, but also such factors as the nature of the samples, the handling of the samples and environmental parameters.

Looking at such histograms as in Figure 9 one can see that the distribution of the 500 measured points for each of the eight resistances is quite narrow. Also one should keep in mind that if the fabric is to be used as “usual” then it will not be fixed in any particular way and that was the intention when designing the holding jig with a hole in the middle.

As explained in the Experiments section, the measurement station was programmed to first take 70 voltage readings at the six free nodes and repeat this procedure four times with different boundary conditions each time. The 70 voltage readings yield a voltage average for each node. In order to keep the indexing at a reasonable level the notation in Table 8 will be adopted in this section.

Table 8.

Translation of the denoting of the resistances.

R₂₂₂ → r₁

R₃₂₃ → r₂

R₁₂₃ → r₃

R₃₃₃ → r₄

R₂₃₂ → r₅

R₃₃₂ → r₆

R₁₂₂ → r₇

R₃₂₂ → r₈

The order is the same as that in which the LabView program computes the resistances. The contact resistances will thus be even numbered and the yarn resistances will be odd numbered.

The voltage averages are then used to solve for the resistance values, denoted now by r_i, at each point. The station then takes 70 new voltage readings, computes new voltage averages and solves for another set of resistance values. The 70-reading measurement is repeated 500 times for the sample. The 500 resistance values are subsequently averaged, yielding the sample average resistance, denoted by r_i, at each resistance point of the sample.

Nine individual samples were selected at random; each sample was measured two times, producing a total of 144 sample average resistances (18 per resistance point). The 18 measurements were completed in randomized order. The experiment is thus a completely randomized single-factor experiment with n = 2 replicates. The experimental design and resulting sample average resistances are shown in Table 6, where the standard order of the design is in the first column from the left, the actual run order is in the second column, the prototype sample name is in the third, the sample average resistance of the first resistance point (r₁) is in the fourth column, the sample average resistance of the second point (r₂) is in the fifth column, and so on. To clarify, ordering the samples as {P2, P2, P3, P3, P4, P4, … , P10, P10} corresponds to an ordered list {1, 2, 3, 4, … , 18} and the standard order, labeled Std order, in Table 6, is the randomized list {8, 3, 4, 11, … , 17}. This latter randomized list corresponds to the randomized order in which the samples were measured, that is, {P5, P3, P3, P7, … , P10}, and that is labeled Prototype sample in Table 6. The readings from a 10th prototype sample, P1, were discarded due to practical measurement problems, the results of which were discovered in the statistical analysis.

Analysis of variance and residual diagnostics

The statistical analysis was performed by treating the factor prototype sample as random and employing the statistical model

y_{u, v} = μ + τ_{u} + ε_{u, v}

where y_u_, _v denotes the observed

{\bar{r}}_{i}

for prototype u measured for the vth time, μ denotes the overall average of all observed

{\bar{r}}_{i}

in the experiment, τ _u denotes a normally and independently distributed random variable with mean 0 and variance

σ_{τ}^{2}

(i.e. NID(0,

σ_{τ}^{2}

)) representing the variability between different samples and ε_u_, _v denotes an NID(0, σ²) random variable representing the variability between measurements on the same sample. The variances of the two random variables may be estimated by means of a one-way analysis of variance (ANOVA).¹² The ANOVA is shown in Table 9. The quantities in the leftmost column are the usual parameters included in an ANOVA, that is, looking at column 2 (for r₁): SS_ri is the sum of squares of differences between each entry in column 4 of Table 6 and the overall average; SS_error is the sum of squares of differences between each entry of column 4 of Table 6 and the average of r₁ as measured for the first and second time (i.e. the average of r₁ for P2–P10); df_ri is the degrees of freedom for the SS_ri; df_error is the degrees of freedom for the SS_error; MS_ri is the mean squares (SS_ri/df_ri) and likewise for MS_error. The F-value is the ratio between MS_ri and MS_error and the p-value is the probability of obtaining a test statistic at least as extreme as the one observed given that the null hypothesis is true (H₀: μ2 = μ3 = ⋯ = μ10).The R² value is a measure of approximately how much of the variability that is explained by the ANOVA model.

Table 9.

The analysis of variance on the r_i values

	${\bar{r}}_{1}$	${\bar{r}}_{2}$	${\bar{r}}_{3}$	${\bar{r}}_{4}$	${\bar{r}}_{5}$	${\bar{r}}_{6}$	${\bar{r}}_{7}$	${\bar{r}}_{8}$
SS_ri	2.35	1.41	6.23	0.44	2.30	0.20	6.30	0.42
SS_error	0.0029	0.13	0.0035	0.12	0.0028	0.036	0.0019	0.094
df_ri	8	8	8	8	8	8	8	8
df_error	9	9	9	9	9	9	9	9
MS_ri	0.29	0.18	0.78	0.055	0.29	0.025	0.79	0.052
MS_error	0.00032	0.015	0.00039	0.014	0.00031	0.0040	0.00021	0.010
F-value	908.85	12.13	2018.31	4.02	938.56	6.35	3802.20	5.04
p-value	<0.0001	0.0005	<0.0001	0.0266	<0.0001	0.0060	<0.0001	0.0130
σ	0.018	0.12	0.020	0.12	0.018	0.063	0.014	0.10
μ	2.86	0.42	2.94	0.36	3.06	0.33	2.90	0.31
R ²	0.9988	0.9151	0.9994	0.7813	0.9988	0.8496	0.9997	0.8176

The estimate of σ², denoted by

\overset{\land}{σ} 2

; is MS_error, while the estimate of

σ_{τ}^{2}

, denoted by

{\overset{\land}{σ}}_{τ}^{2}

, is

{\overset{\land}{σ}}_{τ}^{2} = \frac{{MS}_{ri} - {MS}_{error}}{n} = \frac{{MS}_{ri} - {MS}_{error}}{2}

n being the number of repeated measurements per sample. Table 10 shows the resulting variance estimates for the eight resistance averages

{\bar{r}}_{i}

, as well as the signal-to-noise ratio:

({\overset{\land}{σ}}_{τ}^{2} / \overset{\land}{σ} 2) .

An obvious note is that the signal-to-noise ratios for the contact resistance averages (even numbered) are significantly lower than those of the yarn resistance (odd numbered). This is to be expected, since the nature of the contact resistance is such that it is very sensitive to changes in the environment. As mentioned in the Contact resistance section, the contact resistance is dependent on the contacting pressure between the contact members (i.e. filaments) and since the contacting pressure is a result of the tension of the yarns it is bound to change as soon as it is disturbed. One should also bear in mind the results displayed in Table 4, which indicate that the system is capable of measuring even resistances of the order of R ≈ 0.2 Ω in a satisfactory way.

Table 10.

The variance estimates for r_i.

Variance type	${\bar{r}}_{1}$	${\bar{r}}_{2}$	${\bar{r}}_{3}$	${\bar{r}}_{4}$	${\bar{r}}_{5}$	${\bar{r}}_{6}$	${\bar{r}}_{7}$	${\bar{r}}_{8}$
$\overset{\land}{σ} 2$	0.00032	0.015	0.00039	0.014	0.00031	0.0040	0.00021	0.010
${\overset{\land}{σ}}_{τ i}^{2}$	0.14	0.083	0.39	0.021	0.14	0.011	0.39	0.021
Signal-to-noise ratio	450	5.5	1000	1.5	470	2.7	1900	2.1

A diagnostic check of the underlying assumptions of the ANOVA revealed that there was a slight increase in resistance measured as the experiment progressed. That is, the further into the experiment any sample was measured, the larger the average resistance tended to be. This was true across all resistance points, effectively ruling out the possibility of this phenomenon being attributed to random chance. However, the magnitude of this effect was relatively small for the yarn resistances, much due to the already high signal-to-noise ratios of the measurement system variances compared to the sample variances. The effect of time (or experimental progression) is illustrated in Figure 10, in which the residuals of ${\bar{r}}_{i}$ are plotted versus the run order of the experimental measurements. The corresponding plots for all the other resistance points display similar appearance.

Figure 10.

Residuals versus run order for ${\bar{r}}_{i}$ .

The capability of the proposed measurement system is found by comparing the absolute values of the resistances measured to the measurement system errors. For the yarn resistances, the largest measurement system error variance is $σ_{3}^{2}$ = 0.00039, which yields the conservative estimate of the measurement error σ = √0.00039 = 0.020 Ω. For the contact resistances the corresponding measurement error is σ = √0.015 = 0.12 Ω. The resulting 95% confidence intervals for samples of size n for the respective yarn and contact resistance points are

{\bar{r}}_{yarn, i} = \overset{\land}{\bar{}} r_{i} \pm 1.96 \frac{0.02}{\sqrt{n}} = \overset{\land}{\bar{}} r_{i} \pm \frac{0.0392}{\sqrt{n}}

and

{\bar{r}}_{contact, i} = \overset{\land}{\bar{}} r_{i} \pm 1.96 \frac{0.12}{\sqrt{n}} = \overset{\land}{\bar{}} r_{i} \pm \frac{0.2352}{\sqrt{n}}

where r_i is the estimated sample average resistance and t_α/2, _n has been substituted for 1.96, assuming large sample sizes. As an example, consider repeated measurement of a particular prototype 20 times, each time unmounting and remounting the prototype onto the jig. In that case n = 20 and the confidence interval for the yarn resistance points is approximately

\overset{\land}{\bar{}} r_{i}

± 0.0392/√20 =

\overset{\land}{\bar{}} r_{i}

± 0.0088 Ω and

\overset{\land}{\bar{}} r_{i}

± 0.053 Ω for the contact resistance points.

The repeatability ( $\overset{\land}{σ}$ ) of the contact resistance is approximately six times larger than that of the yarn resistance, resulting in a six times wider confidence interval on any estimate of the mean resistance. This difference in precision is probably due to contact resistance variability arising when unmounting and remounting the prototypes, not measurement technique variability. This claim is supported by the fact that the yarn resistance measurement and the contact resistance measurement were obtained after the same unmounting and remounting action. The capability of the technique is therefore at least as good as the confidence interval of the mean of the yarn resistance, and variability above that should be considered due to between-measurement product variability.

In the samples used in this paper, the yarn resistances are of the order of 3 Ω so the precision is approximately <±0.3%. The contact resistances are of the order of 0.3 Ω, so the precision for them is <±18%.

Discussion and conclusion

A new method for determining the contact resistance between interlaced electro-conductive yarns has been described and demonstrated. The results suggest that the method can be used successfully.

Even though no statistical experiment was conducted on the reference sample the results presented in Table 4 are typical and did not change. The data in Table 5 also shows in a clear way that the nature of the yarn resistance is more stable and less influenced by outer mechanical impact than the contact resistance. It is also worth noting that the technique used here gives standard deviations that are two orders of magnitude smaller than the mean value in the case of the contact resistances. This means that at every measurement the values are indeed to be trusted. Furthermore, the resistance of the yarns as measured with the new technique also corresponds very well to the linear resistivity stated by the manufacturer, as shown in Table 7; this can also be taken as an indication of the correctness of the technique. The difference in free yarn length between the yarns can be explained by observing that the two yarns corresponding to resistances R₂₂₂ and R₂₂₃ are both warp yarns and the yarns corresponding to resistances R₁₂₂ and R₁₂₃ are both weft yarns.

Since the computed values of the contact resistances are dependent on the measurement of two of the yarn resistances, this could pose a problem if these latter resistances were to be very unstable, but as shown in the Results and Measurement systems capability study sections it can be done with the type of yarn used in this study.

The results from the measurement systems capability study show that the contact resistance values vary a lot, in absolute figures, as compared to the yarn resistances. This is to be expected, because the yarn resistance comes about mainly from the conductive properties of the coating on the filaments in the yarns. There is no reason to believe that the coating changes in any significant way during the mounting–dismounting–remounting procedure in the experiment. Nor will the part of the yarn resistance that can be attributed to the contacts between the filaments be of any significance here. On the contrary then, the electrical contacts between the warp and weft yarns are very sensitive to mechanical disturbance. These contact resistances are the result of a huge number of microscopic real contacting spots that come about only because of the load between the crossing yarns. The load will change, even with the slightest breeze on the sample and even more so while being handled during the mount–dismount–remount procedure, and thus also the contact resistance will change. This goes to show that in a more or less real-life situation the contact resistance of interlaced multifilament silver-coated yarns will vary. This variation will depend on yarn, binding and environmental factors, and a measurement will be necessary each time one wishes to make a new fabric with conductive yarns. This variation implies that when modeling a fabric consisting of conductive yarns it is not enough to assign a single value to the contact resistance but one should rather allow for a variation of it. As also shown in the Measurement systems capability study section, if the behavior of the contacts is to be determined a large number of measurements are needed. The technique presented here will then in a rather quick way give an estimate of the distribution of the contact resistance.

As an example, if using the same type of conductive yarns as in this paper and a loom is set to make 20 picks per cm and the warping is equally spaced, then the yarn resistance will be in the range of

R_{y} = 110 \times \frac{1}{20} cm = 0.05 Ω

and the contact resistance will be, as shown above, in the range of

0.15 Ω \leq R_{c} \leq 0.75 Ω

Then the ratio between contact resistance and yarn resistance will be in the range of

2.7 \leq \frac{R_{c}}{R_{y}} \leq 14

This would, according to Banaszczyk et al.,⁶ result in an anisotropic current distribution in the fabric. On the other hand, the model of the fabric would hardly be appropriate in that situation since the yarns would not make contact only at the binding points. Two adjacent yarns would be in (electrical) contact with each other also between the binding points. It is hard to make any estimation as to the magnitude of such resistances without making any measurements, but intuitively one would expect a lower contacting pressure at such sites and thus an even higher contact resistance.

The results shown in this paper further strengthen Banaszczyk et al.’s assumption that a woven textile cannot automatically be modeled as a homogeneous and isotropic sheet regarding electrical properties, and that when designing applications with these textiles the current distribution is a factor to bear in mind.

Footnotes

Funding

This work was supported by the KK foundation (grant number 2009/0254).

Acknowledgments

The invaluable help of tekn. lic. H Odelius at the Department of Applied Physics at Chalmers University of Technology with the construction of the measuring device, of L Urholm and L Norberg for practical tips and tricks also at Applied Physics at Chalmers and of JÅ Wiman at Gothenburg University, are gratefully acknowledged. Furthermore thanks are due to F Wennersten and R Högberg at the Swedish School of Textiles for help with designing and producing the woven samples.

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