SaM - MEMORIZE (Primary Sensors)

List of things to memorize:

SaM - Primary Sensors

• SaM - Definition of Primary Sensor
• SaM - Types of Primary Sensors
• SaM - Different Definitions for Preassure
• SaM - Liquid Columns
• SaM - Deformable Structures
• SaM - Burdón Tubes
• SaM - MEMS Membranes
• SaM - Orifices
• SaM - Pitot Tube
• SaM - Anemometry or Hot Wire
• SaM - Introduction to Ultrasounds
  • SaM - Doppler Effect for Primary Sensors
  • SaM - Time of Flight Principle for Primary Sensors
• SaM - Fan or Turbine
• SaM - MEMS Cantilever
• SaM - Load Cells or Pillar
• SaM - Gyroscope
• SaM - Accelerometer
  • SaM - Non-Idealities of an Accelerometer
  • SaM - Transfer Funtion of a Mounted Accelerometer
• SaM - Primary Sensor for Temperature (Skip)

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SaM - Definition of Primary Sensor

Sensor = Primary Sensor + Real Sensor The typical structure of a device is: Sensor ⇒ Front-End Electronics ⇒ A/D Converter

• ==A primary sensor transform a non-electrical quantity into another (more “convinent”) non-electrical quantity==.
• ==A sensor (or in this case called “real sensor”) instead, transforms a non-electrical quantity into an electrical quantity==.

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SaM - Different Definitions for Preassure

• differential pressure (so the difference between two pressures)

• absolute pressure, so it’s like taking $p_1$ as $0$ pascal

• gauge pressure, which is a differential pressure where $p_1$ is $1$ atmosphere

• There are many different applications and solutions for primary sensors for pressure, and they depend essentially on the range (preassure ranges) can be very different

• The unit for pressure that should be used is Pascal.

• Actually also atmosphere is used: one atmosphere is a little more than 100 kilopascal.

• It is also used bar, a bar is exactly 100 kilopascal.

• PSI which is pound per square inch, which corresponds to $6.89\cdot 10^{3}Pa$.

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SaM - Liquid Columns

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SaM - Deformable Structures

• Formula:$p=\frac{k\cdot \Delta x}{A}$

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SaM - Burdón Tubes

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SaM - MEMS Membranes

• Formula:${\overline{D}}_z=\alpha \cdot \Delta p$Where: (NOT IMPORTANT, too complicated)$\alpha =\frac{(3-{\nu }^2)R^4}{16E{t}^2},{\overline{D}}_z=\frac{1}{3}{D}_z(0),\Delta p=p_{\text{above}}-p_{\text{under}}$

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SaM - Orifices

• “Orifices” in italian: “Orifizi”.
• Bernoulli equation:$\frac{1}{2}\rho v^2+p+\rho gh=\text{const.}$
• Orifices equation:$v_2=\sqrt{\frac{p_2-p_1}{\rho \left(\frac{A_2^2}{A_1^2-1}\right)}}$

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SaM - Pitot Tube

• Structure:
  (One tube has static preassure, used as reference, the other is open and its preassure varies due to the velocity of the fluid)
  
• Using the bernoulli equation, if we measure the difference in height:$\frac{1}{2}\rho v_1^2+p_{\text{atm}}+\rho g{h}_1=p_{\text{atm}}+\rho g{h}_2$Or if we measure the difference in preassures:$\frac{1}{2}\rho v_1^2+p_1+\rho g{h}_{\text{static}}=p_2+\rho g{h}_{\text{static}}$Where:
  • $p_{\text{atm}}=p_1$.
  • $h_1=h_2=h_{\text{static}}$. So:$v=\sqrt{\frac{2(p_2-p_1)}{\rho }}$

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SaM - Anemometry or Hot Wire

• We heat something and then we sense the temperature of other devices close to this heater, to understand which is the convection factor, the convection behavior of the fluids surrounding this heat source

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SaM - Doppler Effect for Primary Sensors

$f_0^{\prime }=f_0\left(1-f_D\right)$Where:$f_D=\frac{2v\cos \theta }{c}$

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SaM - Time of Flight Principle for Primary Sensors

• Formulas:$c_w=\frac{L}{2}\frac{T_{UP}+T_{DOWN}}{T_{UP}\cdot T_{DOWN}}$And:$v=\frac{L}{2}\frac{T_{UP}-T_{DOWN}}{T_{UP}\cdot T_{DOWN}}$
• Structure:
  (There is an error in the following formulas)
  

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SaM - MEMS Cantilever

• Formula:$\text{new\_quantity}=\alpha \cdot F$More specifically:
  • $\epsilon ={\alpha }_{\text{stress}}\cdot F$
  • $D_z(x)=\Delta z={\alpha }_{\text{deflection}}\cdot F$
• It is a differential primary sensor:
  

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SaM - Load Cells or Pillar

• Formula:${\epsilon }_l=\frac{F}{E\cdot A}$
• It is a differential sensor:
  

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SaM - Gyroscope

• Coriolis aceleration formula:$\vec{a}=-2\vec{\Omega }\times \vec{v}$
• Simplest case (angular velocity perpendicular of the velocity):$a_y=-2{\Omega }_z{v}_x$

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SaM - Accelerometer

• Calculations:
  1. Find the lumped parameter model:
     
  2. Find the Thevenim Equivalent:
     
     Simplifying the Thevenim resistance, since $m_B\gg m_S$, so we exclude it from the load, since a parallel between $m_S\parallel m_B\simeq m_S$.
  3. Calculate:$H(s)=\frac{m_S}{m_{S}s+\lambda +\frac{k}{s}}$
  4. Transform into Bode form:$\frac{R(s)}{A(s)}=\frac{m_S}{k}\frac{1}{\frac{m_S}{k}{s}^2+\frac{\lambda }{k}s+1}$
• Mounted and Unmounted accelerometer:
  
• Mounted Behaviour if $m_B\gg m_S$.

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SaM - Non-Idealities of an Accelerometer

• Gravity:
  • If the sensor does NOT rotate ⇒ offset (can be corrected).
  • If the sensor rotates ⇒ variable offset (cannot be corrected).
• Not mounted in a “solid” way:
  
  

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SaM - Transfer Funtion of a Mounted Accelerometer

$\frac{R(s)}{a(s)}=H(s)\frac{\frac{m_s}{K}}{\frac{m_s}{K}{s}^2+\frac{\lambda }{K}s+1}$

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SaM - Primary Sensor for Temperature

(Skipped)