SaM - MEMORIZE (Resistive Sensors)

List of things to memorize:

SaM - Resistive Sensors

• SaM - Potentiometer
• SaM - Metal Strain Gauge • Passive Strain Sensor
  • SaM - Working Conditions of a Strain Gauge
  • SaM ~ Vibration Measurement Based on Strain Gauge
• SaM - RTD Sensor
• SaM - Thermistor
• SaM - Thermocouples
• SaM - Photoresistance
• SaM - Magneto Resitances
  • SaM - Geometrical Magneto Resistance
  • SaM - Types of Magneto Resistances
  • SaM - Anisotropic Magneto Resistance (AMR) • Easy Axis • Barber Pole • Honeywell
• SaM - Chemical Resistive Sensor

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SaM - Potentiometer:$R(x)=\rho \frac{x}{S}$

• Terminology:
  • $R$ : resistance.
  • $x$ : displacement.
  • $\rho$ : resistivity.
  • $S$ : surface.

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SaM - Metal Strain Gauge • Passive Strain Sensor:

• Principle of operation & Strucutre:
  
  
• Formula:$R=R_0(1+G{\epsilon }_x)$
• G (Gauge) Factor:$G=1+2\nu +c(1-2\nu )$
• Considering the Transverse Sensitivity:$R=R_0\left(1+G{\epsilon }_l+G_T{\epsilon }_T\right)$
• Considering the Temperature Effect:$R=R_0(1+\alpha T+G{\epsilon }_x+G{\epsilon }_{TH}+G_T{\epsilon }_{TH})$Where:${\epsilon }_{TH}=({\lambda }_S-{\lambda }_{TO})\cdot \Delta T$
• Real World Quantities:
  • $G{\epsilon }_x\sim 0.1$ ⇒ variation of resisitance $\left(\frac{\Delta R}{R}\right)$ is really small (bad).
  • $G\in [2÷4]$.
  • $R_0\in [50\text{ohm}÷700\text{ohm}]$.
  • ${\alpha }_{\text{TCR}}\in [10^{-5}°{\text{C}}^{-1}÷10^{-6}°{\text{C}}^{-1}]$ ⇒ low TCR (good).
• Terminology:
  • $G$ : gauge factor.
  • $c$ : coefficient of piezoresisitivity.
  • $\nu$ : Poisson modulus.
  • ${\epsilon }_x$ or ${\epsilon }_l$ : parallel strain.
  • ${\epsilon }_T$ : perpendicular strain.
  • $G_T$ : transverse gauge factor.
  • $\alpha$ or ${\alpha }_{\text{TCR}}$ : the TCR.
  • ${\epsilon }_{TH}$ : thermal exampansion.
  • ${\lambda }_S$ : thermal exampsion coefficients of the substrate.
  • ${\lambda }_{TO}$ : thermal exampsion coefficients of the metal grid.
  • $T$ : temperature.
  • $\Delta T$ : temperature variation.

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See Also

• SaM - MEMORIZE (RTD Sensor)
• SaM - MEMORIZE (Thermistor)
• SaM - MEMORIZE (Thermocouple)

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SaM - Photoresistance

• Workings of a Photoresistance:
  1. To use the “photoelectric effect”, we need a semiconductor with energy gap $(E_g)$ less then the energy of a photon $(E)$.
  2. When a photon hits the material, it is able to free one electron from the valence band $(E_V)$ and promote it into the conduction band $(E_C)$.
  3. So given a certain wavelength $(\lambda )$ we will obatin a response in the form of more free carriers, so more conduction, ==⇒ less resistance==
• Energy of the photon:$E=\hslash \cdot \nu$
• Minimum frequency:${\nu }_{\min }=\frac{E_g}{\hslash }$
• Maximum wavelenght:${\lambda }_{\max }=\frac{c\cdot \hslash }{E_g}$
• Conductance of a semiconductor:$G=(qn{\mu }_n+qp{\mu }_p)\cdot \frac{A}{l}$
• Total resistance of a photoresistance:$R=A_x{E}_{\nu }^{-\alpha }$
• This is more like a detector then a real sensor.
• Real World Measures:
  • Visible light has a wavelength: $\lambda \in [0.4\mu \text{m}÷0.7\mu \text{m}]$
    And a corresponding energy: $E\in [1.78\text{eV}÷2.8\text{eV}]$
  • Cadmium sulfide $(CdS)$ semiconductor has an $E_g=1.8\text{eV}$
  • $PbS$ : lead sulfide, $E_g=0.4\text{eV}$ (used in Infra-Red Cameras).
  • $PbSe$ : lead selenide, $E_g=1.5\text{eV}$ (used in Infra-Red Cameras).
  • $\alpha$ : exponent for the photoresistance formula, is in the range $\alpha \in [0.7,0.9]$
  • For a photoresistance we have usually this ratio: $\frac{R_{\text{ILLUMINATED}}}{R_{\text{DARK}}}=10^{-4}$
  • Photoresistances are slow devices: ${\tau }_{63\%}\approx 10\text{ms}$
• Terminology
  • $E_g$ : energy gap.
  • $\lambda$ : wavelength.
  • $\nu$ : frequency.
  • $c\simeq 3\cdot 10^8\frac{m}{s}$ : speed of light.
  • $\hslash =6.626\times 10^{-34}J\cdot s$ : Planck’s constant.
  • $G=\frac{1}{R}$ : conductance.
  • $q=-1.602\times 10^{-19}\text{C}$ charge of the electron.
  • $A_x$ : ???
  • $E_{\nu }$ : illuminance is measured in lux (measure unit) $\left[\text{lux}=\frac{\text{cd}\cdot \text{sr}}{{\text{m}}^2}\right]$
    • cd : candela (measure unit).
    • sr : steradian (measure unit).
  • $\tau$ : rise time.

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SaM - Geometrical Magneto Resistance

• Current density equation:$\vec{J}=nq\vec{v}=\sigma \vec{E}$Can be rewritten as:$\vec{v}=\mu \vec{E}$Where:$\mu =\frac{\sigma }{nq}$
• Geometrical magneto resistance formula:$\vec{v}=\mu \vec{E}+\mu (\vec{v}\times \vec{B})$
• Terminology:
  • $\vec{J}$ : current density vector, $\left[J\right]=\frac{\text{A}}{{\text{m}}^2}$
  • $n$ : density of carriers.
  • $q$ : charge of an electron $\left(-1.6\cdot 10^{-19}\text{C}\right)$ or of a hole $\left(+1.6\cdot 10^{-19}\text{C}\right)$
  • $\vec{v}$ : velocity of carriers.
  • $\sigma$ : conductivity of the material.
  • $\vec{E}$ : electric field vector.
  • $\mu$ : mobility of carriers.
  • $\vec{B}$ : magnetic field density.

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SaM - Types of Magneto Resistances

Given a certain magnetic field intensity $(H)$ a magneto-resistive material will change its resisitivity.

• An OMR (Ordany Mangeto Resitive) material can change its reisitivity up to a $5\%$.
• There are also other types of mangeto resistances, these are all due to “quantistic effects”:
  • GMR (Giant Magneto Resitance)
  • CMR (Colossal Magneto Resitance)
  • TMR (Tunnel effect Magneto Resitance)

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SaM - Anisotropic Magneto Resistance (AMR) • Easy Axis • Barber Pole • Honeywell

• Materials that present AMR, like: Ni, Fe and Permalloy (Fe +  Ni).
• AMR Formula:$\rho (\theta )={\rho }_{\parallel }-\Delta \rho \cdot {\sin }^2\theta$
• Easy axis $(x)$, an axis taken in the same direction of ${\vec{M}}_r$
• If we have that ${\vec{H}}_{\text{ext}}\ll H_S$ than we can approximate:$\sin \theta \approx \frac{H_y}{H_S}$
• AMR Formula using the approximation:$\rho \simeq {\rho }_{\parallel }-\Delta \rho {\left(\frac{H_y}{H_S}\right)}^2$
• Total resisitance value:$R\simeq R_0\left(1-\frac{\Delta R}{R_0}{\left(\frac{H_y}{H_S}\right)}^2\right)$Where:$\begin{array}{l}\Delta R=\Delta \rho \cdot \frac{l}{S}\\ \Delta \rho ={\rho }_{\parallel }-{\rho }_{\perp }\\ R_0={\rho }_{\parallel }\cdot \frac{l}{S}\end{array}$
• Graph of $\rho (\theta )$ given $\theta$:
  
• At $\theta \approx 45°$ the sensor behaves linearly.
• Rotate the current $45°$ with a barber pole:
  
• Simplified resistance formula for a barber pole:$R(H)\approx R_0\mp \Delta R\frac{H_y}{H_S}$
• Honeywell or resistive bridge:
  
• Formula of the Honeywell:$V_{\text{TH}}={\alpha }_r\cdot V$We can also calculate the relative sensitivity $({\alpha }_r)$ as: $\frac{V_{\text{TH}}}{V}=\frac{\Delta R}{R_0}\frac{H_y}{H_S}$Remember that:$\Delta R=\left({\rho }_{\parallel }-{\rho }_{\perp }\right)\cdot \frac{l}{S}$
• Magnetic angle sensors formula:$\tan \left(2\theta \right)=\frac{V_{\text{THA}}}{V_{\text{THB}}}$Where:
  • $A$ or $V_{\text{THA}}$, refers to a honeywell bridge sensing along the $x$ axis.
  • $B$ or $V_{\text{THB}}$ refers to the honeywell bridge sensing along the $y$ axis.
• Terminology
  • $\Delta \rho ={\rho }_{\parallel }-{\rho }_{\perp }$
    • ${\rho }_{\parallel }$ is the resisitivity when $\theta =0°$.
    • ${\rho }_{\perp }$ is the resisitivity when $\theta =90°$.
  • $\theta$ is the angle formed by $\vec{J}$ and $\vec{M}$, where:
    • $\vec{J}$ : current density vector, $\left[J\right]=\frac{\text{A}}{{\text{m}}^2}$
    • $\vec{M}$ is the total magnetization vector, accounting for the external magnetic filed ${\vec{H}}_{\text{ext}}$, and for the magnetization of the material ${\vec{M}}_r$, $\left({\vec{M}}_r={\mu }_0\mathcal{X}{\vec{H}}_{\text{ext}}\right)$
    • ${\vec{H}}_{ext}=<H_x,H_y>$
  • $H_S$ is the saturation field, it is a property of the material.
  • $\rho$ : resisitvity.
  • $V_{\text{TH}}=V_{+}-V_{-}$ is the output.
  • ${\alpha }_r$ is the relative sensitivity, and for this structures it is usually $10^{-3}$.
  • $V$ is the input voltage.
  • $\vec{B}={\mu }_0{\vec{H}}_{\text{ext}}$ is the external magnetic field density.

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SaM - Chemical Resistive Sensor

• Structure:
  1. Thick film ceramic resistances (an inert material).
  2. Two metal electrodes.
  3. An active layer (a pourus material, ~ex.: like a mox film).
  4. Thick film metal resisitance on the backside (a heater).
• Topside view:
  
• Backside view:
  
• Principle of Operation:
  1. This sensor is based on absorption.
     Meaning the active layer is a granular material that trap or absorb, different compounds.
  2. After being absorbed on the active layer, there is a transfer of charge from it (from the metal oxide - mox film) to the gas, and vice versa.
     ~Ex.: oxygen tends to take an electron from the metal oxide, and it traps this electron, this negative charge, on the surface.
  3. We have a plus charge region under the surface, like in a diode.
  4. When many of these “diode-granes” are put togheter they resist the flow of current, so we have increased the reisitance of the sensor.
• Sensitivity:$\alpha =\frac{dR}{d[C]}$
• Sensitivity to a different compound:${\alpha }_x=\frac{dR}{d[x]}\ne 0$⇒ ==Chemical Resistive Sensor are non-selective devices==.
• Power used to opertate this device:$P_W=R_H\cdot I^2=G\left(T_S-T_A\right)$
• Conductivity of the device:$G=G_0{e}^{-\frac{q^2{N}_S^2}{2\epsilon K{T}_S{N}_D}}$
• Other things to memorize:
  • Chemical Resistive Sensor are non-selective devices.
  • This is more like a detector than a real sensor.
  • Why we need a heater.
  • Why the temperature of the sensor should be kept constant.
  • These sensors are not so reproducible.
  • The humidity (water vapor), always present in the environment, can influence the resitance value.
• Real World Measures:
  • Thick film : $t\sim 100\mu \text{m}$ ($t$ : thickness).
  • Thin film : $t<10\mu \text{m}$ ($t$ : thickness).
  • $T_S\in [150°\text{C},450°\text{C}]$.
• Terminology:
  • inert : meaning it doesn’t interact with chemical compounds.
  • $[C]$ : specific compund the sensor is designed for.
  • $[x]$ : non-spefic compund the sensor is NOT designed for $([x]\ne [C])$.
  • $P_W$ : power of the overall device.
  • $G=\frac{1}{R}$ : conductance.
  • $q{V}_S$ : height of the potential barrier between each grain of the substrate.
  • $N_S$ : density of the absorbed and charged chemical compound.
  • $T_S$ : temperature of the sensor.
  • $N_D$ : density of donors.
  • $k=1.38\cdot 10^{-23}\frac{\text{J}}{\text{°K}}$ : Boltzmann’s constant.
  • $\epsilon$ : electric permittivity $(\epsilon ={\epsilon }_0{\epsilon }_r)$
    • ${\epsilon }_0$ : electric permittivity of the void $\left({\epsilon }_0=8.85\cdot 10^{-12}\frac{{\text{C}}^2{\text{m}}^2}{\text{N}}\right)$
    • ${\epsilon }_r$ : relative electric permittivity