# MEMS Differential Pressure Sensor with Dynamic Pressure Canceler for Precision Altitude Estimation

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Theory and Requirements

_{0}and T

_{0}are the atmospheric pressure and temperature at h = 0, respectively; c is the gradient of temperature per height (−6.5 × 10

^{−3}K/m); M

_{0}is the average molar mass of air (28.8 g/mol); g is the acceleration due to the Earth’s gravity (9.80 m/s

^{2}); and R is the gas constant (8.31 × 10

^{3}m

^{2}g s

^{−2}K

^{−1}mol

^{−1}). We can use the first-order Taylor expansion around h = 0,

#### 2.2. MEMS Pressure Sensor Element

_{out}, and the internal chamber pressure, P

_{in}. When P

_{out}is atmospheric, the DPS will function as a pressure “change” sensor that can detect an ambient pressure change within a certain time. Hereafter, we define differential pressure as ΔP = P

_{out}− P

_{in}. Unlike the diaphragm-type APS, the cantilever-type DPS allows airflow through the gap around the cantilever until the differential pressure vanishes (P

_{in}reaches P

_{out}). The combination of the DPS and the chamber has been identified as a first-order high-pass filter [19], with the transfer function from P

_{out}to ΔP being τs/(1 + τs), where τ is the time constant of the system (Figure 2b). The time constant is a measure of time it takes for P

_{in}to reach P

_{out}. The differential pressure is transduced to the fractional resistance change, ΔR/R, at the sensor sensitivity rate, k

_{p}, followed by conversion to a voltage and amplification with an amplifier gain, k

_{amp}, before readout. Therefore, the transfer function of the sensing system, G(s), is expressed as

#### 2.3. Sensor Cap

_{0}is the atmospheric (static) pressure, ρ is the air density, and v is the flow velocity. Although experiments have shown slightly different distributions than this theoretical calculation [25], in either case, the pressure around the sphere will reach the atmospheric pressure at a certain angle, θ

_{0}, which is less than 90° (41.9° for laminar flow Equation (4)), regardless of the flow velocity. If we can extract and measure the pressure at this particular point, it will always be equal to the static pressure, even in the wind.

#### 2.4. Discrete Transfer Function Model for Height Estimation

^{−1}(s) = (1 + τs)/k

_{amp}k

_{p}τs. We employed the bilinear z-transform to convert the transfer function of the continuous system into its discrete counterpart,

^{′}is the corrected time constant for the discrete system. Therefore, the transfer function for the discrete system is

_{out}, at data point n (the sampling interval is T) can be calculated from the measured data, S(n), using

## 3. Results

#### 3.1. Fabrication of Sensing Elements

#### 3.2. Sensor System Development

#### 3.3. Sensor System Characterization

_{p}(=(ΔR/R)/ΔP), and the time constant, τ [19]. We evaluated the sensor sensitivity by applying a constant differential pressure across the sensor chip and the time constant by using a step pressure input. The amplification gain, k

_{amp}, in Equation (3) was 250 throughout this study, where the sensor output, as ΔR/R, was measured using a one-gauge Wheatstone bridge circuit (bridge voltage = 1 V) and an instrumentation amplifier with a gain of 1000.

_{p}. The result shows a linear relationship in the measured range, where k

_{p}= −1.80 × 10

^{−4}Pa

^{−1}.

_{out}(t) = P

_{step}u(t) exponentially as k

_{p}P

_{step}exp(−t/τ), where

#### 3.4. Evaluation of the Cap

_{out}(because of the negative k

_{p}). The signal then increased, with noise toward the origin, as P

_{in}followed P

_{out}while the wind was present. After 30 s, when the DC fan was turned off and P

_{out}became the atmospheric pressure, P

_{in}was higher than P

_{out}, resulting in a positive sensor output higher than at the initial state.

^{2}/2, where ρ is the air density. The theoretical pressure distribution (Equation (4)), as well as the reported experimental values [25], are also plotted. As a result, the measured dynamic pressure as a function of θ followed a sinusoidal curve, corresponding to the theoretical prediction. The sensor cap with θ = 50° reduced the dynamic pressure the most successfully; for it, the wind effect was 0.90% of the result of the no-cap condition.

#### 3.5. Height Estimation Method and Demonstration

_{0}= 101 kPa and T

_{0}= 14.8 °C, which reduced Equation (2) to

## 4. Discussion

_{0}.

_{0}, and experiences a positive dynamic pressure, the effective angle of the hole on the opposite side will be greater than θ

_{0}and, therefore, will experience a countering negative dynamic pressure because the pressure distribution near θ = θ

_{0}will nearly be linear. After the pressures are combined within the sensor cap, the dynamic pressure should still be canceled. However, when the angle grows too much, the pressure distribution will no longer be linear. In the case of imperfectly turbulent flow, for example, the flow separates from the surface at approximately θ = 80° [25], giving a sudden change in pressure. This situation corresponds to a tilt angle of 30° from the streamline for the 50° cap, which might be the maximum acceptable angle.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Concept of using a sensor cap to cancel dynamic pressure. At the bottom of the cap is a MEMS cantilever-type DPS attached to a printed circuit board and connected to an air chamber.

**Figure 2.**Concept of MEMS cantilever pressure sensor: (

**a**) schematic and (

**b**) transfer function model.

**Figure 3.**MEMS cantilever differential pressure sensor: (

**a**) fabrication process, (

**b**) overall appearance, and (

**c**) SEM photograph.

**Figure 4.**(

**a**) Cap design and (

**b**) fabricated sensor cap joined with PCB using O-rings and showing the tube and connector to a syringe to allow variable-volume tests.

**Figure 5.**System calibration curves. (

**a**) Sensitivity of sensor chip and (

**b**-

**i**,

**b**-

**ii**) dynamic characteristics of the system (time-series response (

**b**-

**i**) and obtained time constant vs. chamber volume (

**b**-

**ii**)). Error bars in (

**b**-

**ii**) correspond to the mean value ± the standard deviation for ten measurements.

**Figure 6.**Evaluation of sensor cap: (

**a**) experimental setup, (

**b**) sensor output as the fan was turned on and off, (

**c**) effect of dynamic pressure, and (

**d**) noise vs. angle of incidence. In (

**c**), the gray dashed line corresponds to the theoretical distribution on a sphere (Equation (4)), and the blue and green dashed lines correspond to experimental values obtained for laminar and turbulent flows, respectively [25].

**Figure 7.**Height estimation under 5 m/s wind conditions: (

**a**) experimental setup, (

**b**) measured and low-pass filtered (L.P.F.) sensor output, and (

**c**) height estimations vs actual height profile.

Method | Range | Differential Accuracy |
---|---|---|

Ultrasound ToF [30] | 30 m | 2 cm |

RF ToF [27] | 40 m | 2.1 cm |

Radar [31,32] | ≈200 m | <5 cm |

Absolute Pressure Sensor [33] | −730 m–9500 m ^{1} | 23 cm ^{2} |

This Work | 200 m ^{3} | 2.8 cm |

^{1}Values of 30–110 kPa, converted with Equation (1).

^{2}A value of 3.9 Pa, converted with Equation (10).

^{3}A 1% error range when using Equation (2); structurally unlimited.

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## Share and Cite

**MDPI and ACS Style**

Yasunaga, S.; Takahashi, H.; Takahata, T.; Shimoyama, I.
MEMS Differential Pressure Sensor with Dynamic Pressure Canceler for Precision Altitude Estimation. *Micromachines* **2023**, *14*, 1941.
https://doi.org/10.3390/mi14101941

**AMA Style**

Yasunaga S, Takahashi H, Takahata T, Shimoyama I.
MEMS Differential Pressure Sensor with Dynamic Pressure Canceler for Precision Altitude Estimation. *Micromachines*. 2023; 14(10):1941.
https://doi.org/10.3390/mi14101941

**Chicago/Turabian Style**

Yasunaga, Shun, Hidetoshi Takahashi, Tomoyuki Takahata, and Isao Shimoyama.
2023. "MEMS Differential Pressure Sensor with Dynamic Pressure Canceler for Precision Altitude Estimation" *Micromachines* 14, no. 10: 1941.
https://doi.org/10.3390/mi14101941