General Voltage Feedback Circuit Model in the Two-Dimensional Networked Resistive Sensor Array

To analyze the feature of the two-dimensional networked resistive sensor array, we firstly proposed a general model of voltage feedback circuits (VFCs) such as the voltage feedback non-scanned-electrode circuit, the voltage feedback non-scanned-samplingelectrode circuit, and the voltage feedback non-scanned-sampling-electrode circuit. By analyzing the general model, we then gave a general mathematical expression of the effective equivalent resistor of the element being tested in VFCs. Finally, we evaluated the features of VFCs with simulation and test experiment. The results show that the expression is applicable to analyze the VFCs’ performance of parameters such as the multiplexers’ switch resistors, the nonscanned elements, and array size.


Introduction
The networked resistive sensor array in row-column fashion played a key role in various sensing applications such as tactile sensing [1][2][3][4][5][6][7][8], temperature sensing [8,9], and photoconductive image sensing [10,11], in which every sensitive element had two ends with one end connected to a row line and the other end connected to a column line.Thus, the interconnect line number of the  ×  resistive sensor array was reduced to +.To have individual access of all elements in the twodimensional (2D) networked resistive sensor array, many measurement methods such as the inserting diode method [3], the passive integrators method [5], the same potential method [8,[10][11][12][13], and the voltage feedback method (VFM) [1,2,9,[14][15][16][17][18] were proposed and their performances were evaluated.In these methods, VFM was proven to be an efficient technique.
In the sensor array, parameters such as array size and nonscanned elements affected the measurement error of the element being tested (EBT) for the crosstalk caused by parasitic parallel paths.VFMs with op-amps and many multiplexers were proposed for virtually isolating the EBT from the 2D networked resistive sensor array; thus, the EBT could be tested precisely.Different VFMs [15], such as the voltage feedback non-scanned-electrode (VF-NSE) method, the voltage feedback non-scanned-sampling-electrode (VF-NSSE) method, and the voltage feedback non-scanneddriving-electrode (VF-NSDE) method, had different performances.But the multiplexer's switch resistor induced extra errors on the EBT.Moreover, the operating mechanism of these parameters in different VFMs has not been understood completely.
Performances of three hardware circuits in the 2D networked resistive sensor array were evaluated in [5].In some literatures, performances of the parameters in the 2D networked resistive sensor array were evaluated with the circuit simulation software such as CADENCE-SPCTRE and PSpice [12][13][14][15].Liu et al. [15] compared the VF-NSE method, the VF-NSSE method, and the VF-NSDE method with PSpice.Although circuit simulation was useful for accurate prediction of the EBT's measurement error, it consumed a large amount of manual work.Thus, circuit simulation was not convenient for fast complicated analyses of VFCs in the 2D networked resistive sensor array.With a general mathematical expression of the EBT's effective equivalent resistor ( EQ ), fast evaluation of different parameters in different VFCs could be realized by the mathematical simulation software.Saxena et al. [12,13] analyzed the effect of array size on the EBT's measurement error in the equipotential method and proposed the approximate mathematical expression.The results of the measurement voltage errors in the three VFCs such as VF-NSE circuit, VF-NSSE circuit, and VF-NSDE circuit were given by circuit simulations [15,16].But, in VFCs, the voltage could not directly represent the EBT's resistance value.Less theoretical analysis was implemented in VFCs and the mathematical expression of the  EQ applicable to different VFCs was still lacking.
For this purpose, we analyzed voltage feedback circuits (as shown in Figure 1(a)) in the 2D networked resistive sensor array.In these circuits, the VF methods were utilized and the sample resistors were connected with the power ground.This paper was organized as follows.Section 1 introduced the existing measurement methods.Section 2 provided the general model of voltage feedback circuits and its mathematical expression of the EBT's effective equivalent resistance.Section 3 presented experiment results and discussion.Finally, Section 4 provided conclusions.

General Approximate Circuit Model of VFCs
In VFCs (Figure 1(a)), the feedback voltage (  ) was equal to   .In the basic circuit, no   was connected with the nonscanned switches and no voltage drove the nonscanned electrodes.In VF-NSSE circuit,   was connected with the non-scanned-sampling switches and it drove the non-scanned-sampling electrodes.In VF-NSDE circuit,   was connected with the non-scanned-driving switches and it drove the non-scanned-driving electrodes.In VF-NSE circuit,   was connected with all nonscanned switches and it drove all nonscanned electrodes.
In VFCs, we could use the  EQ between   and   to represent the EBT's measurement value.The general topological structure of the  EQ was shown in Figure 1(b).Every sampling Node   (the th shared row line) was connected with all  driving nodes (all shared column lines) with every connection of a column adjacent resistor ( 1 , . . .,   ).Every driving Node   (the th shared column line) was connected with all  sampling nodes (all shared row lines) with every connection of a row adjacent resistor ( 1 , . . .,   ).  ( 11 in Figure 1) was the EBT, which connected Node   to Node  The structure represented the basic circuit when no connection existed between   and the nonscanned electrodes; the structure represented VF-NSSE circuit when the connections only existed between   and the non-scannedsampling electrodes; the structure represented VF-NSDE circuit when the connections only existed between   and the non-scanned-driving electrodes; the structure represented VF-NSE circuit when the connections existed between   and the nonscanned electrodes.
If each group of elements such as all adjacent nonscanned-sampling elements ( adj s), all adjacent nonscanned-driving elements ( adj s), all nonadjacent elements ( nadj s), all   s, and all   s had similar resistance values, voltages on NSSE and voltages on NSDE would be similar.So, the  EQ topological structure could be simplified to a general approximate circuit model as shown in Figure 2. When the parameter of each element in VFCs varied a little, the differences of voltages in different NSSE circuit and the differences of voltages in different NSDE circuit were tiny.Thus, the circuit model could also be used to analyze VFCs with one element or one group of elements that varied a little.
In Figure 2   In the three VFCs, the currents such as   ,   , and   were different for the current on parasitic parallel resistors (  ).So, the measurement accuracies of the EBT in the three VFCs were different.In this paper, we hoped to propose a general mathematical expression of the  EQ which could clarify the relation between parameters and the EBT's measurement accuracy in VFCs.
Taking VF-NSE circuit, for example, we solved its  EQ .If the op-amp was an ideal amplifier, then   was equal to   .With star-delta conversion, the part including   ,   , and   in Figure 2 could be converted to the part including  1 ,  1 , and  1 as shown in Figure 3.
Thus,  1 ,  1 , and  1 could be expressed as the following equation, as shown in Figure 3: Thus,  2 and  2 in Figure 3 could be expressed as follows: With star-delta conversion, the part including   ,   , and  1 in Figure 3 could be converted to the part including  1 ,  2 , and  3 as shown in Figure 4(a); the part including  1 ,  2 , and  2 in Figure 3 could be converted to the part including  4 ,  5 , and  6 as shown in Figure 4(b).Thus,  1 ,  2 , and  3 in Figure 4(a) could be expressed as the following equation: Thus,  4 ,  5 , and  6 in Figure 4(b) could be expressed as the following equation: In the case that   was a constant in VFCs, from the port of the sample resistor (  ), the general mathematical expression of the  EQ in VFCs could be expressed as follows: Equation ( 5) was also suitable for the basic circuit, VF-NSSE circuit, and VF-NSDE circuit.Thus, we could use (5) to analyze the performance of parameters in VFCs.

Experiments and Discussion
3.1.Simulation Experiment.For verifying the applicability of the general mathematical expression of the  EQ in VFCs, we simulated (5) in MATLAB and VFCs in NI Multisim 12 for comparison.In NI Multisim, the voltage on   was measured and then the   could be obtained.

3.1.1.
The   Range Effect Simulation.The EBT's resistance value affected its measurement errors.In evaluating the effect of   range, the other parameters were set to constant values.We fixed  and  at 8,  switch at 1 Ω, and the resistance value of   and all nonscanned elements in the 2D resistive sensor array at 10 kΩ.The three VFCs were simulated with   varying in 0.10 kΩ-1000 kΩ. Figure 5 showed the simulation results.
When   varied in 0.10 kΩ-1000 kΩ, the   errors of MATLAB in every VFC were a little bigger than its   errors of the NI simulation with similar changing pattern; the   errors in VF-NSE circuit were the biggest and those in VF-NSDE circuit were the smallest.In Figure 8, when   increased in the three VFCs, the   errors decreased while the difference between the simulation results of MATLAB and those of NI increased.

The Switch Resistor (𝑅 𝑠𝑤𝑖𝑡𝑐ℎ ) Effect Simulation.
In evaluating the effect of  switch , the other parameters were set to constant values.In simulations, we fixed  and  at 8 and fixed the resistance value of   and all elements in the sensor array at 10 kΩ.The three VFCs were simulated with  switch that varied in 0.10 Ω-100 Ω, such as   with   = 1 Ω,   with   = 1 Ω, and   =   .Figure 6 showed the simulation results.
With similar values of the   errors, simulation curves could be divided into four groups: the VF-NSE errors with   =   were similar in MATLAB and in NI, which were the biggest; with   =   , the VF-NSSE errors and the VF-NSDE errors were similar in MATLAB and in NI, which were the second biggest; the VF-NSDE errors with   = 1 Ω and the VF-NSSE errors with   = 1 Ω were similar in MATLAB and in NI, which were the smallest; other errors were similar, which were the third biggest.
From Figure 6, we also found that, with  switch varying in 0.10 Ω-100 Ω, the   errors of MATLAB in every VFC were similar to its   errors of the NI simulation; with the increase of  switch in this simulation, the   errors of the three VFCs had positive coefficients.

The Array Size Effect Simulation.
Parameters of array size such as the number of sampling lines () and the number of driving lines () were proved to have similar influences on the performance of the 2D networked resistive sensor arrays [15].In simulations, we fixed some parameters including the resistance value of the sample resistor and all elements in resistive sensor array at 10 kΩ,  or  at 8, and  switch at 1 Ω;  or  was one of 8, 14, 26, 50, 98, 194, and 386. Figure 7 showed the simulation results.With the change of array size, the   errors of MATLAB in every VFC were similar to its   errors of the NI simulation.With the increase of  or  in VF-NSE circuit, the   errors had significant positive coefficients.In VF-NSSE circuit, the   errors also had significant positive coefficients with the increase of , whereas the   errors had small negative coefficients with the increase of .In VF-NSDE circuit, the   errors had tiny negative coefficients with the increase of , whereas the   errors had significant positive coefficients with the increase of .With the increase of  or , the   errors in VF-NSE circuit were bigger than those of VF-NSSE circuit and VF-NSDE circuit.

The Nonscanned Element Effect Simulation.
All nonscanned elements in resistive sensor arrays affected the measurement error of the EBT, in which the adjacent nonscanned elements played the significant role [13,[16][17][18].In simulations, we fixed some parameters including the resistance value of all other elements and   at 10 kΩ,  and  at 8, and  switch at 1 Ω.The three VFCs were simulated with the resistance value of one nonscanned element that varied in 3 kΩ-20 kΩ, such as  adj ,  adj , and  nadj .Figure 8 showed the simulation results.
When one nonscanned element varied in 3 kΩ-20 kΩ, the   errors of MATLAB in every VFC were a little bigger than its   errors of the NI simulation with similar changing pattern.From Figure 8, we also found that the   errors in VF-NSE circuit were about twice those in VF-NSSE circuit  and VF-NSDE circuit.With one nonadjacent element varying from 3 kΩ to 20 kΩ, the changes of the   errors in the three VFCs were insignificant.With one  adj or one  adj varying from 3 kΩ to 20 kΩ, the changes of the   errors in VF-NSE circuit were significant, while those in VF-NSSE circuit and VF-NSDE circuit were insignificant; the   errors in VF-NSE circuit were the biggest and the   errors in VF-NSDE circuit were the smallest.

The Test
Experiment.An experimental device with an 8 × 16 element array was designed for verifying the features of three VFCs.In the device, OPA2340 (from the datasheet, the offset voltage, the bias current, the gain-bandwidth, and the gain are equal to 150 V, 0.2 pA, 5.5 MHz, and 100 dB, resp.) was used as the op-amp, and ADG884 (from the datasheet, the on-resistance, the on-resistance match between channels, and the on-resistance flatness are equal to 0.41 Ω, 0.05 Ω, and 0.15 Ω, resp.) was used as the multiplex switch.
In experiments, the EBT was replaced by a precision resistance box with its smallest step resistance value at 0.1 Ω, the adjacent elements were also replaced by precision resistance boxes, and the nonadjacent elements were precision resistors with each resistance value at 4.7 kΩ.

3.2.1.
The   Range Effect Experiment.In the   range effect experiment, all adjacent elements were set to the same resistance value of 4.7 kΩ, 10 kΩ, or 20 kΩ.The resistance value of the EBT was varied from 2 kΩ to 20 kΩ; the results were shown in Figure 9.
With the increase of the EBT's resistance value, we found from Figure 9 that all curves of the measurement error had similar negative coefficients.With the same resistance value of the adjacent elements at each   , the EBT's measurement error value of VF-NSE circuit was the biggest, the measurement error value of VF-NSDE circuit was the second biggest, and the measurement error value of VF-NSSE circuit was the smallest.With the increase of the adjacent elements' resistance values, the EBT's measurement error values increased in three VFCs.

3.2.2.
The   Effect Experiment.The adjacent elements in resistive sensor arrays played a significant role in affecting the measurement error of the element being tested [16][17][18].In this experiment, the resistance value of the EBT was fixed at 4.7 kΩ.All adjacent elements were set to the same resistance value of 2 kΩ, 5 kΩ, 10 kΩ, 15 kΩ, or 20 kΩ; the measurement results of the EBT were shown in Figure 10.
With the increase of the adjacent elements' resistance values, we found from Figure 10 that the EBT's measurement values increased in three VFCs.With the same resistance value of the adjacent elements at each   , the EBT's measurement value of VF-NSE circuit was the biggest, the measurement value of VF-NSDE circuit was the second biggest, and the measurement value of VF-NSSE circuit was the smallest.

Discussion.
In VFCs, the test experiment and the simulation experiment also proved that the EBT's measurement error value decreased with the decrease of the adjacent elements' resistance values.Thus, the adjacent elements with smaller resistance values had greater influence on the measurement precision of the EBT.The EBT with bigger resistance value was more susceptible to be affected by the adjacent elements.In three VFCs, the EBT's measurement  error value of VF-NSE circuit was bigger than that of VF-NSSE circuit and that of VF-NSDE circuit.Equation ( 5) was conducted under the assumption that each group of elements such as all  adj s, all  adj s, all  nadj s, all   s, and all   s had similar resistances.With the change of   ,  switch , array size, and one nonscanned element, the simulation results of (5) in every VFC were a little bigger than its results of circuit simulation with similar changing pattern.Thus, (5) could be used as a general approximate expression to evaluate VF-NSE circuit, VF-NSSE circuit, and VF-NSDE circuit under the conditions that each group of elements such as all adjacent non-scanned-sampling elements, all adjacent non-scanned-driving elements, all nonadjacent elements, all non-scanned-sampling switch resistors, and all non-scanned-driving switch resistors had similar resistances.When some elements in each element group have bigger variation in resistances, (5) may also be useful to estimate their effect on the EBT's measurement errors.
From (5) in the three VFCs, the increase of the multiplexers' switch resistance value increases the EBT's measurement value, whereas the parameters in the sensor array such as array size, element being tested, and nonscanned element have complicated effects on the EBT's measurement value, which is often known as the crosstalk caused by parasitic parallel paths.
From the simulation results, the effect of parasitic parallel paths in the three VFCs could be significantly weakened with bigger resistance value of nonscanned adjacent element and smaller resistance value of element being tested.Larger number in parameters, such as the sampling line number and the driving line number in VF-NSE circuit, the sampling line number in VF-NSSE circuit, and the driving line number in VF-NSDE circuit, might also be useful to weaken the effect of parasitic parallel paths.But it should be verified in future work.
If other parameters were fixed, as parasitic parallel paths and the switch resistor had opposite influence on the values of the EBT's measurement errors, we could find that the following: when the EBT was in the range of smaller resistance value, the three VFCs had similar performance in accessing the EBT; when the EBT was in the range of middle resistance value, VF-NSSE circuit and VF-NSDE circuit had better performances than VF-NSE circuit in accessing the EBT; when the EBT was in the range of bigger resistance value, VF-NSE circuit had a better performance than VF-NSSE circuit and VF-NSDE circuit in accessing the EBT.
As the other elements in resistive sensor array caused parasitic parallel paths, with smaller resistance values, they would have more significant effects on the EBT's errors in VFCs.VFCs would have good performances in the resistive sensor arrays, such as Thru Mode Force Sensing Resistor (FSR) array and Shunt Mode FSR array, in which the element without pressure on it had a big resistance value and the element with pressure on it had a small resistance value [4].At the same time, a precision op-amp in VFCs always had a limited ability on driving current, for example, tens of mA for OP07.Thus, VFCs would have good performances in the resistive sensor arrays, in which every element had a bigger minimum resistance such as one thousand ohms.

Conclusion
In this paper, a general model and its general mathematical expression of the effective equivalent resistor of the element being tested were proposed for fast evaluating the effects of different parameters in different VFCs such as VF-NSE circuit, VF-NSSE circuit, and VF-NSDE circuit.Experiment results show that the model and the mathematical expression can well illustrate part effects of different parameters of various VFCs in the 2D networked resistive sensor array.VF-NSE circuit has a better performance for the element being tested in the range of big resistance value while VF-NSSE circuit and VF-NSDE circuit have better performances for the element being tested in the range of middle resistance value.Three VFCs have similar performance for the element being tested in the range of small resistance value.

Figure 2 :
Figure 2: General approximate circuit model of VFCs in the 2D networked resistive sensor array.

Figure 3 :
Figure 3: Star-delta conversion result of the part including   ,   , and   .

Figure 5 :
Figure 5: Simulation results of the   range effect in VFCs.

RFigure 6 :
Figure 6: Simulation results of the  switch effect in VFCs.

MFigure 7 :
Figure 7: Simulation results of the array size effect in VFCs.

Figure 8 :
Figure 8: Simulation results of the one nonscanned element effect in VFCs.

RFigure 9 :
Figure 9: Results of the   range effect in VFCs.

Table 1 :
Parameters in the general circuit model of VFCs.