Introduction

High-precision pressure sensors have a wide range of applications in industrial production and medical fields. For example, in the process of oil production, pressure sensors can contribute to better oil flow control and improve production efficiency1,2. Concerning the automotive industry, pressure sensors meet the demands of the condition monitoring of the engine to accomplish its intelligent control3. Besides, pressure sensors can obtain physiological parameters such as the blood pressure and respiratory pressure of patients, which assist doctors in making more accurate diagnosis results4. In aerospace, pressure sensors can precisely capture the minute flow field changes on the aircraft surface, providing vital support for the aerodynamic optimal design5. The pressure scanning valves are known for their high-precision measure for pressure, with accuracies up to 0.03% of the full scale (FS). However, their performance is limited by the scanning speed of electronic switches, which hampers their ability to capture dynamic pressure changes6.

The pressure sensor, fabricated using Micro Electro Mechanical Systems (MEMS) technology, has the advantage of being miniaturized and mass-produced, making it suitable for a wide range of applications. In terms of sensing mechanism, MEMS pressure sensors can be primarily divided into piezoresistive type7, optical fiber type4, and capacitive type8. Optical fiber sensors are immune to the electromagnetic interference but have the disadvantages of damageable fibers and complex signal processing circuits9. Capacitive sensors feature good temperature stability while they are easily affected by parasitic capacitance, resulting in low accuracy10. Compared with optical fiber and capacitive sensors, piezoresistive sensors are susceptible to temperature fluctuation and they demand for external power supply that leads to high power consumption. Nevertheless, they exhibit advantages such as high sensitivity, good linearity, fast dynamic response, and simple signal processing11, which offers the potential to develop miniature high-precision pressure sensors. Piezoresistive pressure sensors, being intrinsically linear, depend on output voltage signals that correlate to the linear calibration curve to ascertain the surface pressures. As a result, linearity plays a crucial role in affecting sensor measurement accuracy, which has always been a focal point of research. Generally, the sensor linearity is enhanced by optimizing the sensing membrane structure. Amphenol Sensor Company introduced a sensor with a double boss structure, whose linearity is 40% higher than that of the square membrane12. Wang et al. developed a double-island beam-diaphragm structure pressure sensor with a nonlinear error of 0.04%FS13. Zou et al. used bulk membrane processing technology to optimize the nonlinear error of pressure sensors to 0.05%FS14. In addition, optimizing the piezoresistor structure is another approach to improving linearity15. Shang et al. suggested that increasing the membrane area could significantly enhance sensor linearity and sensitivity16. Li et al. proposed a method to optimize the number of segments in piezoresistors, developing a sensor with a nonlinear error of 0.0182%FS17. Although the asymmetric Wheatstone bridge can reduce the nonlinear error of the sensor, the piezoresistors with different resistances will lead to a large zero drift18. Therefore, researchers further investigate to enhance the sensor linearity through back-end data processing algorithms19. Jiang et al. employed a Pade approximation-based calibration algorithm to eliminate the cross-sensitivity and unexpected jump points, reducing sensor nonlinear error to 0.01%FS20. Moreover, Huang et al. proposed a hybrid polynomial kernel extremum learning machine algorithm to optimize the nonlinearity, which achieved a measurement accuracy of 0.03%FS20,21. Lu et al. introduced an evolutionary method for designing parameter MEMS structures with maximum degrees of freedom and fabricated a sensor with a linearity of 0.058%FS22. Nevertheless, the back-end data processing can only mitigate nonlinearity to a level of 0.01%, while simultaneously complicating subsequent circuit signal acquisition and processing, thereby compromising the dynamic characteristics of the sensor. As far as we know, means of effectively suppressing the nonlinear error that does not deteriorate the other sensing characteristics, such as the zero drift, dynamic response, and so on, have not been discussed in depth.

In summary, there is a scarcity of direct and effective methods to enhance the linearity of piezoresistive pressure sensors. To address the issue, this work presented a sensitivity matching method for the piezoresistors based on the nonlinear influence mechanism of the sensor. By adjusting the position of the longitudinal piezoresistor on the membrane with uniform stress distribution, the sensitivity of the longitudinal piezoresistor can be the same as that of the transverse piezoresistor. When the sensitivity of the four piezoresistors is the same, the linearity of the sensor can be suppressed to a minimum. Then, a cylindrical absolute pressure sensor of 4 mm diameter with a range of 0–100 kPa is designed and fabricated for validating the nonlinearity suppression method. After the calibration test of the sensor, the application test is carried out in the Mach 4 wind tunnel. The experimental results indicate that the method proposed in this paper can significantly improve the measurement accuracy of pressure sensors for wind tunnel pressure tests, driving innovative advancements in aircraft aerodynamic design.

Theoretical modeling of nonlinear error

The piezoresistive pressure sensor is a kind of linear pressure sensor that can accurately measure the input pressure p through the resistance change of the piezoresistor R. For a linear pressure sensor, the relationship between the piezoresistor R and the pressure p is expressed by the second-order equation.

$$\begin{array}{cc}{R}_{i}={R}_{io}(1+{a}_{i}p+{b}_{i}{p}^{2}) & i=1,2,3,4\end{array}$$
(1)

where Ri is the resistance value of the piezoresistor i under the pressure p. Rio is the initial resistance value of the piezoresistor. ai and bi are constants related to the membrane structure. Utilizing the Wheatstone bridge, the resistance change is converted into the output voltage, which is expressed as20

$${V}_{O}=\frac{{R}_{2}{R}_{3}-{R}_{1}{R}_{4}}{({R}_{1}+{R}_{2})({R}_{3}+{R}_{4})}{V}_{in}$$
(2)

where Vo is the output voltage of the sensor and V in is the input voltage of the sensor. Assuming that the four piezoresistors are equal, by substituting Eq. (1) into Eq. (2), the output voltage Vo of the sensor is expressed by the pressure p

$${V}_{O}={V}_{in}\frac{(1+{a}_{2}p+{b}_{2}{p}^{2})(1+{a}_{3}p+{b}_{3}{p}^{2})-(1+{a}_{1}p+{b}_{1}{p}^{2})(1+{a}_{4}p+{b}_{4}{p}^{2})}{[(2+({a}_{1}+{a}_{2})p+({b}_{1}+{b}_{2}){p}^{2})]\cdot [(2+({a}_{3}+{a}_{4})p+({b}_{3}+{b}_{4}){p}^{2})]}$$
(3)

The quadratic terms of pressure p in the equation can be combined and simplified, so the output voltage is obtained as

$${V}_{O}=\frac{{a}_{2}+{a}_{3}-{a}_{1}-{a}_{4}}{4}\{p+[\frac{{b}_{2}+{b}_{3}-{b}_{1}-{b}_{4}}{{a}_{2}+{a}_{3}-{a}_{1}-{a}_{4}}-\frac{{a}_{2}^{2}+{a}_{3}^{2}-{a}_{1}^{2}-{a}_{4}^{2}}{2({a}_{2}+{a}_{3}-{a}_{1}-{a}_{4})}]{p}^{2}\}{V}_{in}$$
(4)

Equation (4) is a quadratic function of one variable related to pressure p. According to the input and output relationship of the sensor, the sensitivity of the sensor can be deduced as

$$S=\frac{{V}_{out}}{{p}_{in}}=\frac{1}{4}({a}_{2}+{a}_{3}-{a}_{1}-{a}_{4})$$
(5)

For piezoresistive pressure sensors, the sensitivity a2 and a3 of the transverse piezoresistive are negative, and the sensitivity a1 and a4 of the longitudinal piezoresistive are positive. Therefore, the overall sensitivity of the sensor is expressed as

$$S=\frac{1}{4}({a}_{2}+{a}_{3}+|{a}_{1}|+|{a}_{4}|)$$
(6)

Equation (4) can be abbreviated as

$$\begin{array}{c}{V}_{o}({P}_{i})=(A{P}_{i}+B{({P}_{i})}^{2}){V}_{in},A=\frac{{a}_{2}+{a}_{3}-{a}_{1}-{a}_{4}}{4},\\ B=\frac{{b}_{2}+{b}_{3}-{b}_{1}-{b}_{4}}{{a}_{2}+{a}_{3}-{a}_{1}-{a}_{4}}-\frac{{a}_{2}^{2}+{a}_{3}^{2}-{a}_{1}^{2}-{a}_{4}^{2}}{2({a}_{2}+{a}_{3}-{a}_{1}-{a}_{4})}\end{array}$$
(7)

where A is the sensitivity of the sensor, and B is the second-order linearity coefficient of the sensor. The nonlinear error of the sensor is defined as23,24

$$N{L}_{i}=\frac{{V}_{o}({p}_{i})-\frac{{V}_{o}({p}_{m})}{{p}_{m}}{p}_{i}}{{V}_{o}({p}_{m})}\times 100 \%$$
(8)

where NLi is the nonlinear error at the pressure measuring point pi. Vo(pi) is the sensor output voltage at the pi, and Vo(pm) is the output voltage at full-scale pressure pm. pi is the pressure measurement point, and pm is the maximum pressure measurement point. By substituting Eq. (7) into Eq. (8), the nonlinear error expression of the sensor under pressure Pi can be obtained as

$$\begin{array}{cc}NL({p}_{i})=\frac{A{p}_{i}+B{{p}_{i}}^{2}-(A+B{p}_{m}){p}_{i}}{A{p}_{m}+B{{p}_{m}}^{2}} & i=1,2,\mathrm{3...}.N\end{array}$$
(9)

It can be seen from Eq. (9) that the nonlinear error NLi of the measuring point pressure pi in the full-scale range is also a quadratic function of the pressure. The maximum nonlinear error in the entire working range occurs at pi = pm /2, and its maximum nonlinear error NL(pi)max is expressed as

$$NL{({p}_{i})}_{\max }=NL(\frac{{p}_{m}}{2})=-\frac{B{p}_{m}}{4A}$$
(10)

When the input pressure pi is equal to -Bpm/4 A, the maximum nonlinear error solution of the sensor can be obtained by substituting the constants A and B of Eq. (7) into Eq. (10)

$$N{L}_{\max }=-\frac{1}{4}\frac{{b}_{2}+{b}_{3}-{b}_{1}-{b}_{4}}{{a}_{2}+{a}_{3}-{a}_{1}-{a}_{4}}{p}_{m}+\frac{{a}_{2}^{2}+{a}_{3}^{2}-{a}_{1}^{2}-{a}_{4}^{2}}{8({a}_{2}+{a}_{3}-{a}_{1}-{a}_{4})}{p}_{m}$$
(11)

Defining \(\bar{{\rm{a}}}=({a}_{2}+{a}_{3}-{a}_{1}-{a}_{4})/4\), the NLmax is expressed as

$$\begin{array}{c}N{L}_{\max }=\frac{1}{4\bar{a}}[{a}_{2}(N{L}_{2})+{a}_{3}(N{L}_{3})-{a}_{1}(N{L}_{1})-{a}_{4}(N{L}_{4})]\\ +\frac{1}{32{\bar{a}}^{2}}({a}_{2}^{2}+{a}_{3}^{2}-{a}_{1}^{2}-{a}_{4}^{2}){p}_{m}\bar{a}\end{array}$$
(12)

The first part of Eq. (12) is the average of the four resistors concerning the linearity of their respective sensitivities. The nonlinear error of the piezoresistive effect ai(NLi) cancels each other out by constituting a Wheatstone bridge25. After ignoring the self-nonlinearity of the piezoresistive effect, the maximum nonlinear error of the sensor is expressed as

$$N{L}_{\max }=\frac{1}{32{\bar{a}}^{2}}({a}_{2}^{2}+{a}_{3}^{2}-{a}_{1}^{2}-{a}_{4}^{2}){p}_{m}\bar{a}$$
(13)

The sensitivity ai of the piezoresistor itself is expressed as an integral form of the membrane stress26.

$${a}_{i}=\frac{{V}_{out}}{{V}_{in}}=0.5{\pi }_{44}(\frac{1}{A}{\iint }_{A}{\sigma }_{x}dxdy-\frac{1}{A}{\iint }_{A}{\sigma }_{y}dxdy)$$
(14)

σx and σy are the stresses in the x and y directions on the membrane. A is the stress integral area of the area where the piezoresistor is located. π44 is the piezoresistive coefficient. It can be seen from Eq. (14) that the sensitivity ai of piezoresistors is related to the stress difference and the integral area of the stress difference. By arranging the position of the piezoresistors on the membrane, the sensitivity of the piezoresistors can be adjusted. When the sensitivity of the four piezoresistors is equal, by substituting a1 = a2 = a3=a4into Eq. (13), the theoretical NLmax is reduced to zero.

Design of quasi-zero nonlinear error sensor

Sensing membrane design

A square chip with a side length of 2 mm is designed and packaged into a cylindrical sensor with a diameter of 4 mm. The chip is fabricated by doping boron ions on a 200-μm-thick N-type silicon wafer. The membrane is an important factor affecting the sensitivity of the sensor. According to the Kirchhoff–Love plate theory, the transverse displacement of the membrane should be limited to less than twenty percent of the membrane thickness to ensure that the membrane stress is linear concerning the applied pressure25.

$$W{}_{\max }=\frac{0.0138p{l}^{4}}{E{h}^{3}} < 0.2t$$
(15)

where l is the side length of the sensing membrane, p is the maximum pressure applied to the membrane, E is Young’s modulus of silicon, and h is the membrane thickness. To ensure the service life and reliability of the membrane, 1/5 of the breaking strength of the silicon is selected as the maximum stress value of the membrane.

$${\sigma }_{\max }=0.308p{l}^{2}/{h}^{2} < 0.2{\sigma }_{m}$$
(16)

where σmax is the breaking strength of silicon. σm is the maximum pressure input in 100 kPa. Therefore, the thickness of the membrane h is 30 μm, and the side length of the membrane l is 1600 μm.

Piezoresistor structure design

To analyze the maximum stress of the membrane, the Finite Element Analysis (FEA) of the pressure sensor is conducted. The physical model of the pressure sensing chip is first established, and then the clamped boundary is applied to the support plane. Afterward, a pressure load of 100 kPa is applied to the sensing surface of the upper chip. FEA of the stress distribution is carried out after mesh division, and the simulation results obtained by FEA are shown in Fig. 1a. In the middle of the membrane edge, there are four regions of the maximum stress difference for arranging piezoresistors. To further determine the piezoresistor position, a stress difference line is extracted on the central line AA of the membrane, and the stress difference curve is shown in Fig. 1b. Meanwhile, the 1-frequency of the membrane is 199.27 kHz obtained by FEA. According to the stress distribution curve, the width of the maximum stress difference region on the membrane is about 20 μm, and the length of the maximum stress difference region is about 150 μm. Considering the redundant design, the dimension parameters of the piezoresistor are determined with a width of 10 μm and a length of 120 μm. However, the transverse piezoresistor a1 and a4 are arranged in the stress concentration area, and part of the longitudinal piezoresistor a2 and a3 extends beyond the stress concentration area. This will undoubtedly reduce the distributed stress on the longitudinal piezoresistor and reduce its sensitivity. To address the issue, the longitudinal piezoresistor is divided into a few sections so that more piezoresistors are distributed in the stress concentration area, as shown in Fig. 1c.

Fig. 1: The FEA of a chip.
figure 1

a Membrane stress distribution. b Stress difference distribution of curve AA. c Current distribution

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The sensitivity of sensors under different segments of the longitudinal piezoresistor is analyzed by FEA. Based on the stress distribution analysis model, the boundary conditions of bound current conservation continue to be set on the piezoresistor and wire. The boron doping concentration was set as 2 × 1018 cm−3 on the piezoresistor and 1 × 1020 cm−3 on the wire. Apply voltage and ground to the input port, and measure the potential at the output port. Figure 2 displays the sensor output voltage under different segments of the longitudinal piezoresistor. Linear fitting is carried out according to the least square method, and the slope of the fitting curve is the sensitivity of the sensor. It can be seen from Fig. 2 that the sensitivity of the sensor increases as the number of segments of the longitudinal piezoresistor increases. This is consistent with the theoretical analysis that the greater the integral value of stress difference, the higher the sensitivity. However, after the piezoresistor is divided into four segments, the improvement effect is only 0.89%, from 1.667 to 1.682 mV/kPa. In practical application, the more piezoresistors are produced, the more process errors will be introduced, so the longitudinal piezoresistor of four segments is selected as the optimal.

Fig. 2
figure 2

The relationship between the segments of the piezoresistor and the sensitivity

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Piezoresistor sensitivity match

The longitudinal piezoresistor is divided into four segments, while the transverse piezoresistor is of one segment. The stress drop gradient in the longitudinal piezoresistor layout area is larger, and the adjustment efficiency is higher. Therefore, the position of the transverse piezoresistor remains unchanged, and the sensitivity match is accomplished by adjusting the position of the longitudinal piezoresistor, as shown in Fig. 3a. The effects of the distance from the longitudinal piezoresistor to the membrane edge on the nonlinear error are investigated utilizing numerical simulation, as shown in Fig. 3b. Among them, the step size is 2 μm, the adjustment range is −2 to 18 μm, and the nonlinear error of the sensor changes from −0.1 to 0.1%. As shown in Fig. 3c, the nonlinear error decreases to 0.001%FS when the distance from the longitudinal piezoresistor to the membrane edge is 5.4 μm. According to Eq. (13), when the sensitivity of the longitudinal piezoresistor is equal to that of the transverse piezoresistor, the linearity of the sensor is reduced to 0.001%FS. The structure and position of the longitudinal piezoresistor are designed to improve the sensitivity of the longitudinal piezoresistor and make it equal to the transverse. This can be achieved when the transverse piezoresistor is the most sensitive. Therefore, the point with the lowest linearity is also the point with the highest sensitivity of the sensor. The specific design results are shown in Fig. 3. Finally, the optimal design of the quasi-zero-nonlinear pressure sensor is completed, whose parameter is shown in Table 1.

Fig. 3: The design of the piezoresistor sensitivity match.
figure 3

a Position adjustment of the longitudinal piezoresistor. b Relationship between input pressure and nonlinear error at different displacement distances. c The relationship between displacement distance and linearity

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Table 1 Sensor parameter design results
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Dynamic performance simulation analysis

To evaluate the dynamic characteristics of the proposed sensor, harmonic analysis based on finite element method (FEM) is conducted. According to the previously reported works, the resonant properties of the diaphragm-type microsensor of FEM simulation are highly consistent with those of experimental results27,28. Therefore, the FEM simulation can provide an estimation of the dynamic characteristics of the proposed sensor.

According to the theory of elastic thin plate, the basic micro equation of transverse vibration of the elastic thin plate under lateral load is expressed as

$$\frac{{\partial }^{4}\omega }{\partial {x}^{4}}+2\frac{{\partial }^{4}\omega }{\partial {x}^{2}\partial {y}^{2}}+\frac{{\partial }^{4}\omega }{\partial {y}^{4}}+\frac{\rho h}{D}\frac{{\partial }^{2}\omega }{\partial {t}^{2}}=\frac{q}{D}(x,y,t)$$
(17)

where D is the bending stiffness of the diaphragm, expressed as Eh3/12(1 − ν2), w is the deflection of the diaphragm along the x and y directions, E is Young’s modulus of silicon material, h is the thickness of the diaphragm, and ν is the Poisson’s ratio of silicon material. In the formula, ρ is the density of silicon, q is the transverse dynamic load borne per unit area and is the fourth-order partial differential equation about the flexion surface function ω (x, y, t). t is the time coordinate. According to FEA, the first-order characteristic frequency of the sensor chip is calculated to be 199.27 kHz29,30, as shown in Fig. 4.

Fig. 4
figure 4

First-order resonant frequency mode

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The output voltage amplitude of the sensor under different frequency pressures is shown in Fig. 5. The output amplitude of the sensor increases by 3 dB at 121.82 kHz, and the output amplitude is maximum at 200.21 kHz, which is consistent with the vibration 1st resonant frequency.

Fig. 5
figure 5

Chip output amplitude ratio under different frequencies

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Sensor fabrication and testing

Sensor prototype fabrication

Using a 200-μm-thick silicon wafer of type (100), a P-type pressure sensor with piezoresistors is prepared by micromachining techniques such as photolithography, ion implantation, sputtering, and ICP etching. An overview of the sensor manufacturing process is shown in Fig. 6a. In Fig. 6i, a 450-nm-thick silicon oxide is grown on both sides of the silicon wafer by dry–wet-dry oxidation. In Fig. 6ii, a heavy ion-doped window is formed by Buffered Oxide Etch (BOE) wet etching. In Fig. 6iii, utilizing ion injection technology, boron ions with a dose of 2 × 1017 cm−2 and energy of 50 keV are first implanted to form a wire. In Fig. 6iv, an oxide layer is generated by dry–wet–dry oxidation again. After re-photoetching, the light ion doping window is formed, in Fig. 6v. In Fig. 6vi, using the ion injection technique, boron with a dose of 1 × 1014 cm−2 and energy of 65 keV is implanted to form a piezoresistor. After annealing in the oxidation furnace, a piezoresistor with a junction depth of 1 μm and a resistivity of 0.04 Ω·cm is formed. Therefore, the resistance value is calculated as 4.8 kΩ. In Fig. 6vii, the photoresist is used as a stripping mask, which is lifted off after sputtering 1 μm aluminum to form a pad. After annealing at 450 °C for 30 min, the interface of silicon and aluminum forms the ohmic contact. In Fig. 6viii, the membrane is etched by Inductively Coupled Plasma (ICP) etching on the back side of the silicon chip. Finally, in Fig. 6ix, through the silicon-glass anode bonding technology, the silicon wafer and the glass (BF33) wafer are closely connected. In Fig. 6b, using the gear scribing technique, the chip is separated from the wafer. The longitudinal piezoresistor divided into four segments is shown in Fig. 6c. After wire bonding and curing, a 4 mm diameter cylindrical sensor prototype is obtained, in Fig. 6d.

Fig. 6: Sensor chip fabrication process.
figure 6

a The diagram of chip fabricated process. b the chip photo. c The piezoresistor photo. d The sensor photo

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Sensor prototype testing

An experimental system is established for calibrating the sensor prototype, as shown in Fig. 7a. A pressure range from 0 to 100 kPa can be generated and controlled by the pressure controller (Fluck, 6270 A). The output voltage of the sensor is collected by a multimeter. By evenly sampling 11 points within the range of 0–100 kPa and repeating back and forth three times, the relationship between average input and output signals collected is shown in Fig. 7b. The fitting slope of the test result is 1.6810 mV/kPa, which is consistent with the theoretical analysis result of 1.6825 mV/kPa. The nonlinear error of the sensor prototype calculated according to Eq. (8) is shown in Fig. 7c. According to the results of the three measurements, the nonlinearity of the sensor is ±0.004%FS, the hysteresis is 0.025%FS, and the repeatability is 0.015%FS. After the sensor is powered on for 14 h, the drift of zero is shown in Fig. 7d. The maximum drift of the sensor is 118.64 μV, accounting for 0.0705%FS of the full-scale output of 168.10 mV. For the surface pressure measurement of the aircraft in the wind tunnel, the maximum duration of incoming air blowing is less than 10 min. After 14 h of zero stability test, the sensor with good zero stability can achieve accurate pressure measurement in the wind tunnel. The characteristics of the sensor and the comparison of the sensor with other sensors are shown in Tables 2 and 3.

Fig. 7: Static performance of sensor.
figure 7

a Photo of a pressure sensor static test. b Comparison of measured and simulated values of sensor output. c Comparison of measured and simulated nonlinear error. d Sensor zero drift test results

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Table 2 Characteristics of the sensor at 25 °C
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Table 3 Comparisons with different fabricated sensors
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Wind tunnel pressure test

Wind tunnel test is an important step in the process of aerodynamic characteristics design of aircraft29. The wind tunnel test comparison of the proposed sensor is depicted in Fig. 8. The input airflow increases to Mach 4 in the contracted section, the flow rate is stable in the test section, and the air inflow is recovered in the collection section. Mach 4 indicates that the object’s velocity is four times the speed of sound. The test sensor is mounted symmetrically with the Kulite sensor and the pressure scan valve on the flat model, which is placed in the test section of the wind tunnel as shown in Fig. 8b. The pressure points arranged symmetrically on the plate model can be approximated as feeling the same pressure. Among them, the pressure scanning valve can be regarded as the real pressure value of the measurement point because the measurement accuracy reaches 0.01%FS. The output pressure of both the Kulite and the test sensor can be compared with the measured value of the pressure scanning valve to verify the accuracy of the measured pressure results. The angle of attack of the flat model is controlled from −2° to 6° under Mach 4 to simulate the flow field pressure of the aircraft at different angles shown in Fig. 8c.

Fig. 8: Sensor wind tunnel test under Mach 4.
figure 8

a Wind tunnel pressure test diagram at Mach 4. b Photo of the flat model in the test section. c Diagram of the Angle of Attack

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After airflow blowing, the pressure scanning valve outputs the surface true pressure value of the flat model. The measured pressure of the Kulite sensor and the proposed sensor needs to be calculated by the fitting curve based on their output voltage. The real-time voltage curve of the output of the Kulite sensor and the proposed sensor is shown in Fig. 9a. At the moment of the arrival of the Mach 4 inflow, the output of the sensor drops instantaneously, indicating that the upper surface of the plate model is subjected to an upward force of less than atmospheric pressure, providing lift for the aircraft. For the first 35 s, the flat model stabilized in the Mach 4 flow field. In 35–73 s, the angle of attack of the plate model goes from 0° to −2°, then up to 6°, and finally down to 0°. Within 75–83 s, the Mach 4 flow field disappears, and the sensor output returns to its initial value. Figure 9a shows that the pressure on the surface of the flat model gradually increases with the increase of the angle of attack, which indicates that the lift force of the surface of the flat model decreases with the increase of the angle of attack30,31. As can be seen from Fig. 9a, the output voltage of sensor 2 is significantly smaller than that of sensors 1 and 3. This is because the zero point of sensor 2 is −10.85 mV, which is significantly less than that of sensor 1 of −2.850 mV and sensor 3 of −1.675 mV. This could be attributed to fabrication process variations. However, the variation of the output voltage of sensor 2 shows the same tendency as the other sensors when subjected to different angles of attack. Figure 9b illustrates the difference in the surface pressure values measured by the proposed sensor, the Kulite sensor, and the pressure scanning valve. As shown in Fig. 9c, the measurement error of the proposed sensor at the 4° attack angle is significantly higher than at other angles. The mounting height of the sensor on the plate, the tilt angle of the sensor mounting surface, and the pollution of the incoming flow may cause the sensor to produce a small vortex structure or turbulence phenomenon in the incoming airflow at a certain angle of attack. These problems could cause changes in the pressure introduced into the pressure sensor used for measurement, which may lead to a large deviation between the measurement results of the sensor and the true value. Figure 9c shows the comparison of the measurement error between the proposed sensor and the Kulite sensor during the wind tunnel experimental testing. According to Fig. 9c, the measurement error of the proposed sensor is ±0.98%, which is superior to that of the Kulite sensor with a value of ±1.24%.

Fig. 9: Test results of the proposed sensor in the Mach 4 wind tunnel.
figure 9

a The output voltage comparison results of the Kulite sensor and test sensor. b Test sensor, Kulite Sensor, and pressure scanning values output comparison. c Measurement errors of the test sensor and Kulite sensor under different angles of attack

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After the sensor is placed at room temperature for three months, the zero drift and sensitivity of the sensor are calibrated again. The test results show that the zero drift voltage of the sensor within 14 h is 110 μV, the full-scale output is 168.09 mV, and the sensitivity is 1.6824 mV/kPa. The zero drift is reduced from 118.64 to 110 μV, which may be caused by the gradual release of packaging stress over time. The full-scale output of the sensor has only changed by 0.0059%, and the sensitivity has changed by 0.0832% compared with that before the wind tunnel test. It can be considered that the sensor has good stability.

Conclusion

To meet the demand for high-precision pressure measurement of aircraft model surface in aerospace, industrial, and medicine. The proposed sensor prototype, based on theoretical research, incorporates a piezoresistor sensitivity matching method to minimize the nonlinear error of the sensor. The static performance test of the 4 mm diameter sensor prototype demonstrates outstanding results, with a nonlinear error of 0.004%FS, repeatability of 0.015%, hysteresis of 0.025%FS, zero drift of 0.0701%FS, sensitivity of 1.682 mV/kPa, and a 3 dB frequency from 0 to 121.82 kHz. Validation in a Mach 4 wind tunnel reveals a deviation of ±0.98% between the proposed sensor and true pressure, indicating high measurement accuracy and significant application potential for pressure measurement in aerospace, industrial, and medical fields.