Increasing wheatstone bridge sensitivity

1 MANUSCRIPT ID NUMBER TIM-24-06142 Increasing Wheatstone Bridge sensitivity Josep M. Torrents, Julio Garcia-Calvete  Abstract— Many physical variables of interest (movement, force, torque, power, pressure, flow or temperature) are usually measured with sensors whose output (of an electrical parameter) varies little with respect to those physical variables of interest. Thus, in order to have adequate accuracies and resolutions, it is usually configured in a differential way through an electrical circuit known as a Wheatstone Bridge. This paper presents a new Wheatstone Bridge topology that with square alternating signal excitation increases the traditional sensitivity in voltage difference in the two branches of the Bridge with a sensor element (ε/4) to a sensitivity of (ε/e), where (ε) is the difference in the sensor (as per one) caused by the physical variable of interest. It is analyzed analytically, then simulated with calculation and finally measured in an ad-hoc set-up. Index Terms— AC Bridge circuit, capacitive sensor, direct interface. resistive sensor, sensitivity analysis, square alternating signal, step function, u(t) signal, Wheatstone Bridge. I. INTRODUCTION ANY electrical sensors are based on the fact that their resistance, inductance or electrical capacitance (R, L or C) varies depending on a physical variable of interest such as temperature, force, pressure, etc. In general, this variation of R, L or C is small with respect to its absolute value. To increase the resolution in the measurement of the physical variable of interest, a Wheatstone Bridge [1] is traditionally used, such as the one in Fig. 1. Where V dc is the excitation of the Bridge and R 1 , R 2 , R 3 and R 4 are the resistances of the Bridge (it will also be referred as RR Wheatstone Bridge when the branches have only resistors). At least one of the resistances is a function of the physical variable of interest. Fig. 1. Resistive Wheatstone Bridge (or RR Wheatstone Bridge). When all four resistors of the Bridge have the same value This paper was submitted on June, 24 th , 2024. This work was supported in part by the UPC – Barcelona TECH. Josep M. Torrents is with the Electronic Engineering Department, UPC – Barcelona TECH, C4 Building, Campus Nord, c. Jordi Girona, 1, 08034 Barcelona, SPAIN (e-mail: jos[email protected]). (in equilibrium situation and reference value of the physical variable) occurs: 𝑉𝑉 When the reference physical variable varies, (for example R 3 is a function of the physical variable while R 1 , R 2 and R 4 remain constant) imbalance appears in the Bridge and: 𝑉𝑉 It is easily demonstrated [2] that the maximum sensitivity (maximum change of the difference between V a and V b for a given change of the measurand) to a variation of the physical variable is reached when the following equilibrium situation is given: 𝑅𝑅𝑅𝑅 Fig. 1 represents an example in DC. The result is maintained when the Bridge is excited in AC. In the case of AC excitation, the resistors can be replaced by inductances or capacitances (L or C) [3]. Even in a situation of maximum sensitivity (in equilibrium R 1 =R 2 =R 3 =R 4 ), the difference: |𝑉𝑉| is small compared to the excitation (V dc in Fig. 1). The usual solution is to add a high-gain Instrumentation Amplifier (IA) to amplify this difference. In order to simplify the electronics and save the IA, some authors propose working in square alternating signal excitation (u(t) signal) with direct interface [4] or even amp-less interface [5]. Consider that the variable of interest changes and causes a small change (ε) in one of the R 3 resistances. For example, for (ε=0.01) consider an excitation of V dc =1 V. Given a situation of maximum sensitivity [2] (for a single sensing resistor): 𝑅𝑅𝑅𝑅 and 𝑅𝑅󰇛1𝜀󰇜 Analyzing by Kirchhoff and applying McLaurin series expansion where appropriate, the following solution (of dashed trace in Fig. 3) is reached: |𝑉𝑉| 1 𝑉󰈅 𝑅 𝑅𝑅𝑅󰇛1𝜀󰇜 𝑅𝑅󰇛1𝜀󰇜󰈅󰈅1 2󰇛1𝜀󰇜 󰇛2𝜀󰇜󰈅 󰈏1 2󰇛1𝜀󰇜 2󰇡1𝜀2󰇢󰈏󰈏1 2󰇛1𝜀󰇜󰇡1𝜀2󰇢 2󰈏 𝜀4 󰇛1󰇜 Julio García-Calvete. was with UPC – Barcelona TECH. He is now an independent researcher (e-mail: [email protected]). Color versions of one or more of the Figures in this article are available online at http://ieeexplore.ieee.org M This article has been accepted for publication in IEEE Transactions on Instrumentation and Measurement. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TIM.2025.3542674 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 2 MANUSCRIPT ID NUMBER TIM-24-06142 Then, operating with a numerical example is obtained: |𝑉𝑉| 1 𝑉0.00249 Traditionally, the Wheatstone Bridge was adapted to different measurement needs. For example, by resorting to the three-wire or Siemens method when the sensor is located far from the Bridge circuit [2], or when reactances of different nature (made with L and C components in series and parallels) are combined in the Wheatstone Bridge, taking the names of the engineers who proposed them: Owen, Schering, Maxwell, Wein or Hay [6]. Given the importance of this circuit as a measurement method, very closely related to all types of sensors, various authors have worked and determined aspects of the Wheatstone Bridge to improve its sensitivity and accuracy and thus improve the sensorization of physical variables and the instrumentation that embarks them. In [7] for example, aspects of the working supply and detection of the Wheatstone Bridge are improved. In [8, 9], the sensitivity of the Wheatstone Bridge is increased by considering the power dissipated in the branches of the same. In [10], the accuracy of measurements is improved when vector signals are considered in the excitation of the Wheatstone Bridge. Following these changes of the original Wheatstone Bridge, this paper proposes a new measurement system that increases the sensitivity of the Wheatstone Bridge in a given configuration. The goal of this article is to explore a topology, without changing the sensor (or its sensitivity), that increases the sensitivity of the Bridge. That is, to increase the value of │V a – V b │ without increasing the excitation. II. D EVELOPMENT Nowadays, all sensor systems incorporate some type of digital electronics, such as microprocessors and microcontrollers; although the conditioning of the sensing element often continues to be analog (based on amplifiers). So, it is easy to excite a Wheatstone Bridge with a square or step signal of the u(t) type. In this case, the Wheatstone Bridge AC topology illustrated in Fig. 2 is considered (it will also be referred as RC Wheatstone Bridge when the branches have capacitors and resistors). Fig. 2. RC Wheatstone Bridge shows an increased sensitivity compared to the resistive one with a single sensor element, either R (Fig. 2a) or C (Fig. 2b) interchangeably. Regarding the DC topology shown in Fig. 1, the DC excitation has been changed to a step signal and two resistors, one in each branch, for two capacitors. The increased sensitivity is the same whether it is a resistive or capacitive sensing element if the variation in both of the sensing element remains (ε). The analysis proceeds as in the resistive or RR Wheatstone Bridge from the introduction section, and with the same equilibrium starting situation (ε=0). That is to say: 𝑉𝑉 When a variation appears in the physical variable it causes an imbalance in the Bridge (ε≠0) and consequently: 𝑉𝑉 Analyzing by Kirchhoff, solving the differential equation and applying McLaurin series development where appropriate, the following solution is reached: |𝑉𝑉| 1 𝑉𝑒/𝑒/󰇛󰇛󰇜󰇜𝑒𝑒󰇛󰇜  𝑒𝑒𝑒 𝑒1𝑒  𝑒𝑡𝜀 𝜏 󰇛2󰇜 with τ being the RC time constant and t>0 the time variable at the beginning of the step. The continuous line plots (2) in Fig. 3. In addition, (2) has a maximum for (t = τ) and for that moment: |𝑉𝑉| 1 𝑉󰇻󰇡𝑒󰇢󰇡𝜏𝜀 𝜏󰇢󰇻𝜀/𝑒 󰇛2𝑏󰇜 Higher value than that found with the RR Wheatstone Bridge developed in the introduction section: 𝜀𝑒𝜀4 Then, operating with the numerical example from the introduction section is obtained: |𝑉𝑉| 1 𝑉0.00368 Fig. 3 illustrates this result with the numerical value used in the introduction section (ε=0.01). The dashed line shows the classic result presented in the introduction section (1) and the continuous line shows the result of this topology exciting with a step signal (2). It is observed that for measurements around time (t = τ) the sensitivity of the RC Wheatstone Bridge is higher than the classical topology RR Wheatstone Bridge in DC or also in AC excitation. Fig. 3. Result with the numerical value used in the introduction This article has been accepted for publication in IEEE Transactions on Instrumentation and Measurement. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TIM.2025.3542674 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 3 MANUSCRIPT ID NUMBER TIM-24-06142 section (ε=0.01). In dashed line, the result with a classic resistive or RR Wheatstone Bridge; and in a continuous line, the result of this RC Wheatstone Bridge topology exciting with a step signal. III. E XPERIMENT The good performance of the proposed circuit has been verified by experimenting with two Wheatstone Bridge circuits, Fig. 4a and 4b. For the Bridges, shelf components have been used, but they have been measured with an ohmmeter with an additional capacitance measurement option and they have been chosen so that the differences between them were less than 0.1%. Four equal 10 kΩ resistors and two equal 100 nF capacitors have been chosen. Thus, the time constant in the Bridge with two RC branches, Fig. 4b, is (τ=1.0 ms). For the same imbalance in the two Wheatstone Bridges, another 100 kΩ resistor (which leads to approximately a 10% imbalance) has been connected in parallel to only one of the 10 kΩ resistors, 10 kΩ//100 kΩ= 9.09 kΩ, ε=0.1). In addition, the power supply of the bridge is connected through a P-channel MOSFET that is controlled by the Gate with an optocoupled standard signal. The RDS_ON (Resistance between Drain and Source of the MOSFET) is of the order of tens of milliOhm which does not influence the measurement. Thus, both Wheatstone Bridges have been supplied with a direct voltage V dc =10 V, see Fig. 4a and 4b. Fig. 4a. Classic measurement circuit with a resistive or RR Wheatstone Bridge that measures the imbalance of resistors. In present experiment, an imbalance of one of the resistors (R 1 ). Fig. 4b. Proposed measurement circuit with RC Wheatstone Bridge that improves in a suitable time frame the sensitivity of the classic circuit (Fig. 4a). In present experiment, also an imbalance is caused in one of the resistors (R 2 ). The topology in Fig. 4b is slightly different from that in Fig. 4a. If the topology in Fig. 4b were the same as the topology in Fig. 4a, the capacitors would show charge even if P 0 was OFF. Then, a path must be provided to discharge them. In this case, through the N-channel MOSFET governed by P 1 and the protection resistor of the R p circuit. The R p resistor limits the maximum current that would flow through the transistors if the two MOSFETs were driving at the same time. If R p is small compared to R 1 or R 2 , the discharging time is slightly longer than the charging time. In the experiment R p = 180 Ω. The R&S RTB2004 oscilloscope has been used to generate the optocoupled signal patterns that have excited the MOSFETs (P 0 in Fig. 4a, and P 0 and P 1 in Fig. 4b). The MOSFETs have acted as switches and these signals (P 0 and P 1 ) have properly sequenced (see Fig. 5) the ON/OFF states. In addition, this oscilloscope has measured the imbalance in the Bridges, voltages V a and V b (Fig. 4a and 4b). Both circuits have been excited with a repetitive control pattern in 20 steps with a duration of 1 ms each step. This strategy results in a period that repeats every 20 ms and is equivalent to a square signal frequency of 50 Hz in the Bridge. With balanced Bridges, the difference between V a and V b is lower than the sensitivity of the oscilloscope, except in the transitions of the RC Wheatstone Bridge (Fig. 4b), with glitches below 5 mV. Fig. 5 illustrates the P 0 and P 1 patterns for the RC Wheatstone Bridge. It must be ensured that both switches (P 0 and P 1 ) on the RC Wheatstone Bridge are not closed at the same time (nor glitch transitions). Fig. 5 shows the ON/OFF pattern of the charge (P 0 ) and the discharge (P 1 ), which with the guard time steps (1 and 11) in which the two MOSFETs are OFF, ensures that both MOSFETs are not ON at the same time. As mentioned, it is no possibility that both switches are ON at the same time, not even in the time of a glitch. Because of these safety margins, P 0 has 9 steps in ON and 11 steps in OFF. Therefore, the signal that excites the RR Wheatstone Bridge is not symmetrical (duty cycle slightly below 50%) as can be seen in Fig. 6a. Fig. 5. Power and ground connection and disconnection patterns of the RC Wheatstone Bridge, with an OFF-OFF safety time between the two switches. IV. R ESULTS Fig. 6a shows the measurement of the voltages V a , V b (from Fig. 4a of the RR Wheatstone Bridge) with the oscilloscope. It also shows their difference (V a – V b ). The measurement corresponds to R 2 =R 3 =R 4 =10 kΩ and R 1 =9.09 kΩ, i.e., 10% unbalanced (ε=0.1). The oscilloscope time base has been set to 5 ms/div and vertical This article has been accepted for publication in IEEE Transactions on Instrumentation and Measurement. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TIM.2025.3542674 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 4 MANUSCRIPT ID NUMBER TIM-24-06142 sensitivity of channels 1 and 2 (which have measured V a and V b ) has been set to 1 V/div (the scale on the left in Fig. 6a). Finally, the vertical sensitivity of the mathematical channel (which has drawn the difference between V a and V b ) has been set to 0.1 V/div (the scale on the right in Fig. 6a). The blue and orange traces correspond to the voltages V a and V b with a value of approximately half of the supply voltage (V dc /2). And the gray trace corresponds to the mathematical channel with the calculation difference between V a and V b . Applying (1), a theoretical value of the following is obtained: |𝑉𝑉|𝑉∗𝜀410 𝑉∗0.1 40.25 𝑉 Which corresponds to the amplitude value of the gray trace in Fig. 6a. Fig. 6a. Measured values V a and V b in the resistive or RR Wheatstone Bridge, (blue and orange traces); and measured value of their difference (gray trace) for an imbalance of 10% (ε=0.1). Fig. 6b shows the result of measuring with the oscilloscope the voltages in the RC Wheatstone Bridge of the circuit in Fig. 4b: V a , V b and their difference (V a – V b ) when there is an imbalance equal to the previous case of the unbalanced resistive or RR Wheatstone Bridge by 10% (i.e., ε=0.1). For example, with both capacitors of C = 100 nF, R 1 = 10 kΩ and R 2 = 9.09 kΩ. On the RC Wheatstone Bridge (Fig. 4b circuit), the oscilloscope time base has been set again to 5 ms/div. The vertical sensitivity of channels 1 and 2 (which have measured the V a and V b voltages) has been set to 2 V/div. Finally, the vertical sensitivity of the mathematical channel (which has drawn the difference between V a and V b ) has been set to 0.2 V/div. The blue and orange traces correspond to the voltages V a and V b . In this case, when the switching time is 5 times lower than the time constant (τ=RC=1.0 ms) of the RC Wheatstone Bridge, the voltages V a and V b reach a value close to V dc =10 V. And the gray trace, corresponds to the difference calculation between V a and V b . Applying (2b), for times close to the time constant (τ=RC=1.0 ms) of the RC Wheatstone Bridge, a theoretical maximum value of the following is obtained: |𝑉𝑉|𝑉∗𝜀𝑒10 𝑉∗0.1 2.72 0.37 𝑉 Which corresponds to the maximum amplitude value of the gray trace in Fig. 6b. In summary, at the instant of maximum (t=τ=RC) the RC Wheatstone Bridge topology has a higher sensitivity, 46% higher than the RR Wheatstone Bridge topology (0.37/0.25=1.46). Maximum sensitivity is obtained when measured at the instant t=τ after each switching of the square signal. However, in the branch that causes the imbalance, τ is not known with certainty because it depends on the value of the physical variable that is being measured. For small variations in the value of the sensor element, two strategies can be taken: This article has been accepted for publication in IEEE Transactions on Instrumentation and Measurement. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TIM.2025.3542674 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 5 MANUSCRIPT ID NUMBER TIM-24-06142 1) Measure at the instant t=τ of the reference branch that does not contain any sensor element. Fig. 3 shows gentle slopes of the curve around the maximum. With this, a certain loss of sensitivity can be assumed, although each case must be studied in particular. 2) Measure at several points over a period and choose the value of the maximum (which will be close at time t=τ). This second strategy can also be used for larger variations of the sensing element in the RC Wheatstone Bridge. In addition to measuring voltages, it should be measured in time (t=τ=RC) since the last switching of the RC Wheatstone Bridge power supply to maximize sensitivity as mentioned. If there is only one sensing element (of the four possible in a Wheatstone Bridge) and previous measurements have been stored in an equilibrium situation, a direct interface situation could be considered [4, 5]. In this situation, the IA would no longer be needed (but the Sample&Hold circuit (S&H) and Analog to Digital Converter (ADC) would be needed). This situation will be studied in a future. Fig. 6b. Measured values V a and V b in the RC Wheatstone Bridge, (blue and orange traces); and measured value of their difference (gray trace) for the same imbalance of 10% (ε=0.1). V. C ONCLUSION For many variables of interest that are measured through small changes in the electrical properties of sensors, the Wheatstone Bridge is a widely used circuit that allows them to be measured with sufficient accuracy and resolution. This article has presented a new RC Wheatstone Bridge topology (the branches of the Bridge have been made from capacitors and resistors). For square alternating signal excitation (or u(t) type step excitation) and a combination of resistive or capacitive sensors, this topology improves sensitivity by 46% compared to the classic configuration of the resistive or RR Wheatstone Bridge (with the same excitation amplitude). The acquisition of the measurement must be performed at a time close to the time constant (τ = RC) of the branches of the RC Wheatstone Bridge topology. C ONTRIBUTION First author: Conception of the idea, theoretical developments, design of the experiment and writing. Second author: Design of the experiment and realization of the same. A CKNOWLEDGMENT Authors thank the kindness and willingness to help of the library staff of the UPC – Barcelona TECH BRGF: Montse Aragües, Marta García, Ruth Iñigo, Raquel Lorca, Josep Martínez and Clara Poveda. This article has been accepted for publication in IEEE Transactions on Instrumentation and Measurement. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TIM.2025.3542674 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 6 MANUSCRIPT ID NUMBER TIM-24-06142 R EFERENCES [1] E.O. Doebelin, Measurement Systems: Application and Design, revised ed., McGraw-Hill, Inc., New York, 1975, pp. 605–612. [2] R. Pallás-Areny, J.G. Webster, Sensors and Signal Conditioning, John Wiley and Sons, Inc., New York, 1991, pp. 92–100. [3] E.O. Doebelin, Measurement Systems: Application and Design, revised ed., McGraw-Hill, Inc., New York, 1975, pp. 246–257. [4] J.E. Gaitán-Pitre, M. Gasulla, and R. Pallàs-Areny, “Analysis of a Direct Interface Circuit for Capacitive Sensors,” IEEE Transactions on Instrumentation and Measurement, vol. 58, no. 9, pp. 2931-2937, May 2009, doi: 10.1109/TIM.2009.2016782. [5] T. Miyazaki, S. Nakagawa, and H. Ishikuro, "High-Resolution AutoBalancing Wheatstone-Bridge with Successive Approximation of ΔΣModulated Digitally Controlled Variable Resistor," 26th IEEE International Conference on Electronics, Circuits and Systems (ICECS), pp. 474-477, November 2019, doi: 10.1109/ICECS46596.2019. [6] H. V. Malmstadt, C. G. Enke, and E. C. Toren, Electronics for Scientists. Principles and Experiments for Those Who Use Instruments, 2 nd printing, with corrections, W. A. Benjamin, Inc., New York, 1963, chapter 6. [7] G. Weiss, "Wheatstone Bridge Sensitivity," IEEE Transactions on Instrumentation and Measurement, vol. 18, no. 1, pp. 2-6, March 1969, doi: 10.1109/TIM.1969.4313752. [8] J. E. Maisel, "Optimization of the Wheatstone Bridge Sensitivity," IEEE Transactions on Instrumentation and Measurement, vol. 26, no. 1, pp. 17-21, March 1977, doi: 10.1109/TIM.1977.4314476. [9] E. Takagishi, "Comments on "Optimization of the Wheatstone Bridge Sensitivity", IEEE Transactions on Instrumentation and Measurement, vol. 27, no. 3, pp. 304-306, September 1978, doi: 10.1109/TIM.1978.4314692. [10] E. Takagishi, "On the Balance of an AC Wheatstone Bridge," IEEE Transactions on Instrumentation and Measurement, vol. 29, no. 2, pp. 131136, June 1980, doi: 10.1109/TIM.1980.4314886. Josep M. Torrents received Master Telecommunication Engineering and PhD degrees from the UPC – Barcelona TECH, Barcelona, Spain, in 1989 and 1996, respectively. He is an Associate Professor of Electronic Engineering Department UPC – Barcelona TECH, and he has taught courses in several areas of electronics in EETAC, Castelldefels (Barcelona), ETSEIB, Barcelona, FNB, Barcelona and ETSETB, Barcelona. Currently, he teaches in FNB and ETSETB. From 1998 to 1999, he was a Visiting Scholar at Northwestern University, Evanston, IL. His research includes instrumentation methods and sensors based on electrical impedance and magnetic measurements, mainly applied in ceramic materials. ORCID 0000-0002-8289-6706. Julio Garcia-Calvete received Master Industrial Engineering and PhD degrees from the UPC – Barcelona TECH, Barcelona, Spain, in 2005 and 2011, respectively. He was Assistant Professor of Electrical Engineering Department UPC – Barcelona TECH, and he taught courses in ETSEIT, Terrassa (Barcelona). He also worked for ENDESA from 1982 to 2023. He holds several Patents related with assistant devices for handicap people. He is now an independent researcher. ORCID 0009-0006-4169-9374. This article has been accepted for publication in IEEE Transactions on Instrumentation and Measurement. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TIM.2025.3542674 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/