SaM - MEMORIZE (Capacitive Sensors)

List of things to memorize:

SaM - Capacitive Sensors

• SaM - Types of Capacitive Sensors
• SaM - Capacitive Displacement Sensors
• SaM - Capacitive Proximity Sensors
• SaM - MEMS Capacitive Pressure Sensor
• SaM - Capacitive Accelerometer
• SaM - Capacitive Level Sensor
• SaM - Capacitive Humidity Sensor
• SaM - Capacitive (Ferroelectric) Temperature Sensor
• SaM - Readout Electronics for Capacitive Sensors (FM & AM Oscillators)
  • SaM - Square Wave Oscillator
  • SaM - FM (Frequency Modulation) Based on Oscillators
  • SaM - AM (Amplitude Modulation) Based on Oscillators
  • SaM - Complete AM (Amplitude Modulation) Circuit
  • SaM - AM Modulation with 2 Sensors • Carrier Amplifier
    • SaM - Carrier Amplifier
    • SaM - Low Pass Filter
  • SaM - Switched Capacitors
  • SaM - Sigma-Delta Converter (AD Converter)
• SaM - High Frequency (AC) Signals
• SaM ~ Examples of Capacitive Sensors

Link to original

---

SaM - Types of Capacitive Sensors

• Capacitive sensor:
  • displacement or proximity sensors.
  • level sensors.
  • temperature sensors.
• Capacitive primary sensors:
  • Membranes (MEMS)
  • Deformable strucutre (MEMS).
  • Cantilever (MEMS).
  • Capacitive accelerometer.
  • Preassure sensors.
• Advantages ofcapacitive sensors:
  • They are non-contact.
  • Good resolution for small ranges, in the order $\sim [10^{-5}÷10^{-6}]$.
  • Very simple structure, the sensor is just a metallic plate, it doesn’t matter which kind of metal, usually, and it doesn’t matter the shape, two metallic plate are enough to build a capacitance.
• Disadvantages of capacitive sensors:
  • Small capacitance value.
  • They can work only for small ranges (small distances), this is one of the reasons because they are called proximity sensors.
    We can write this simple relationship in order understand:
    ==The ratio: range (maximum distance that can be measured), diameter of the probe (thinking about a circular probe, a circular plate), is roughly==:$\frac{\text{range}}{\text{probe’s diameter}}=\frac{1}{8}$
  • Capacitive proximity sensors are sensitive to: dirt, humidity, and to extreme temperature and extreme preassure (found in industrial applications).
• Parassitic resistance at low frequency and parasisitic capacitances at high frequencies:
  
• Example with parasisitic resisitances $(R_1,R_2)$:
  
• Output $(V_o)$ at regime:$V_{\infty }=\frac{R_2}{R_1+R_2}{V}_{DC}$(Depends ONLY on the parasistic reisistances, really bad).
• Output $(V_o)$ at the start:$V_0=\frac{C}{C+C_R}{V}_{DC}$(Depends ONLY on the sensor and reference capacitance, good).
• Real World Measures:
  • Capacitice sensor measure a value of: $C=1\text{pF}$. $\left(\text{Remember}:1\text{p}=10^{-12}\right)$
    We measure with a frequency of $100\text{kHz}$.
    ⇒ The resulting impedance $|Z|=\frac{1}{\omega C}$ is equal to $10\text{M}\text{ohm}$, which is realy large.
    ⇒ This high impedance will be a problem: if the front end electronic it’s not well designed, it will load the source.
    Moreover, with high impedance sources, you will have electrical noise.
  • Capacitive sensor are small $\sim 1{\text{cm}}^2$.
  • They are used really close to the target (talking about proximity sensors) $\sim 1\text{mm}$.

---

SaM - Capacitive Displacement Sensors

• Simplest capacitive displacement sensors:
  
  
  

---

SaM - Capacitive Proximity Sensors

• Structure:
  
• Basic formula:$C=\epsilon \frac{A}{d}$
• Graph:
  
• With a large, conductive target, which is also not grounded:
  
• Measured capacitance formula:$C_{\text{MEAS}}=\frac{C_S{C}_{GT}}{C_S+C_{GT}}\approx C_S$If the sensor is small (if its area $A$ is small) relative to the large target ⇒ $C_S\ll C_{GT}$ and $C_{MEAS}\simeq C_S$.
  ⇒ ==So it’s like having a grounded conductive target (good)==.
• Proximity sensor with non-conductive target:
  (Can be done if the electric permittivity of the target is different from the one of the vacuum)
  
• Real World Measures:
  • A circular capacitive proximity sensor with radius $r=10\text{ mm}$, measuring an object $1\text{ mm}$ away will assume a capacitance of about $2.7\text{ pF}$.
  • Impedance of a capacitive sensor:$|Z|=\frac{1}{\omega C_0}$==Small capacitance means high impedance==.
    • This can be mitigated by operating this sensor at high frequency.
      HOWEVER: we know that the higher the frequency the most critical is the design of the electronics
    • We have to cope between these two requirements: keep this impedance small enough to be measured and to operate at a not too high frequencies.

---

SaM - MEMS Capacitive Pressure Sensor

• Structure:
  
• Both the membrane and the fixed plate are conductive and act as the plates of the sensor.
• Formula:$C_0(1+x)$Where: $x$ is a linear function of the differential preassure:$x=\alpha (P_1-P_2)$

---

SaM - Capacitive Accelerometer

• Structure:
  
• Formula:$C=\frac{C_0}{1+\alpha \cdot a_0}$
• Terminology:
  • $C_0$ : capacitance value at rest.
  • $a_0$ : constant accerelation.
  • $\alpha$ : constant sensitivity $\left(\alpha =\frac{M_S}{k\cdot d_0}\right)$
    • $M_S$ : sismic mass.
    • $k$ : elastic constant.
    • $d_0$ : distance between the plate, at rest.

---

SaM - Capacitive Level Sensor

• Structure:
  
• Formula for non-ionic liquid:$\begin{array}{lcc}C& =& C_1+C_2\\ & =& \frac{2\pi {\epsilon }_{1}h}{\ln \left(\frac{b}{a}\right)}+\frac{2\pi {\epsilon }_0(l-h)}{\ln \left(\frac{b}{a}\right)}\end{array}$
• Formual for ionic liquids:$C=\frac{{\epsilon }_{0}A}{d}$Where:
  
• Non-ionic liquids act like an insulator, like hetanol and water.

---

SaM - Capacitive Humidity Sensor

• Structure:
  
• Formula:$C=\frac{A}{d}\epsilon (\text{RH})$
• Real World Measures:
  • Variation of capacitance due to the humidity percentage:$\frac{\Delta C}{C}=1.7\frac{\text{pF}}{\text{RH\%}}$
  • Typical capacitance value:$375\text{pF}+1.7\frac{\text{pF}}{\text{RH\%}}\cdot \text{RH\%}$(_This is a measurable capacitance, it is big enough, good)

---

SaM - Capacitive (Ferroelectric) Temperature Sensor

• Structure:
  
• Ferroelectric material’s electric permittivity:$\epsilon =\frac{K}{T-T_C}$
• Real World Masures:
  • These temperature primary sensors are really rare, since there are many other types of sensors, among which RTD sensors are very effective.
• Terminology:
  • $K$ : constant depending on the material, we have not seen it in details.
  • $T$ : temperature (as always in $\left[°K\right]$)
  • $T_c$ : Curie temperature.

---

SaM - Readout Electronics for Capacitive Sensors (FM & AM Oscillators)

• We need to distinguish between:
  1. Low frequency signals or DC signals.
     Read-out for low frequency signals:
     1. FM: Frequency Modulation which uses a grounded sensor.
     2. AM: Amplitude Modulation which uses an unreferenced sensor.
  2. High frequency signals.
     Read-out for high frequency signals:
     1. DC exitation.
• For low frequency signals, we will need an AC exitation.
  The frequency of the excitation ($f_{exc}$) has to be much larger than the maximum signal frequency ($f_{S_{MAX}}$), it should not be a problem since it’s a low frequecny signal.
  Since the signal $g(t)$ varies slowly, the capacitance takes a value which is constant for a long time.
  This time is enough to allow the loss of the charge which is stored in the capacitors due to the parasitic resistances.

---

SaM - Square Wave Oscillator

• Structure:
  
• Formula:$T=2RC\cdot \ln \left(\frac{1-\beta }{1+\beta }\right)$

---

SaM - FM (Frequency Modulation) Based on Oscillators

• Using a square wave oscillator:
  
• Formula:$T=2RC\cdot \ln \left(\frac{1+\beta }{1-\beta }\right)$
• Frequency of an FM modulated signal:$f(t)=f_0\left(1+\frac{f_A}{f_0}x(t)\right)$
• FM modulation using an integrator:
  (For instance if I have $x(t)$ which is a slowly growing invariant signal, then the frequency would increase a litte over time.)
  
• Bandwidth of the FM modulated signal:
  
  Where:
  • $\Delta f_{MAX}=f_A\cdot x_{MAX}$.
  • And $f_A$ comes from: $f(t)=f_0+f_{A}x(t)$.
• Since FM transforms a the physical input variation into a variation of frequency, it is a very robust solution to reject noise.
• Complete FM Circuit:
  
  • The sensor is part of the oscillator.
  • The FM Demodulator gives as output a voltage related to the frequency in input.
• Real World Measures:
  • FM is used for signals which can be static but also they have a large bandwidth up to $100\text{kHz}$
  • The typical value for $f_0$ is $2\text{MHz}$ (much larger than the maximum signal frequency).

---

SaM - AM (Amplitude Modulation) Based on Oscillators

• Simplest circuit for AM (Amplitude Modulation):
  
• Output:$V_o(t)=\frac{C_0}{C_R}(1+x)\cdot V_g$Where: $V_g=V\sin (2\pi f_{0}t+\phi )$
• General formula of the AM modulation:$V_0(t)=V_R[1+mx(t)]\sin (2\pi f_{0}t+\phi )$Where: $V_R=V\cdot \frac{C_0}{C_R}$
• Spectrum analysis:
  
• Good circuits for AM modulation:
  • Current or charge amplifiers.
  • AC bridges.
  • Voltage dividers.

---

SaM - AM Modulation with 2 Sensors • Carrier Amplifier

• Circuit:
  
• Sensors:$C_1=\frac{C_0}{1+x},C_2=\frac{C_0}{1-x}$
• Output:$V_{out}=V_g\left(\frac{Z_f}{Z_1}-\frac{Z_f}{Z_2}\right)$
• Final formula:$V_{out}=Vg\frac{R_f}{1+j\omega R_f{C}_f}(j\omega 2C_{0}x)$
• For $f_0\gg f_{S_{MAX}}$:$V_{out}\approx -\frac{2C_0}{C_f}|x|$(If I select a value for $C_f$ too large with respect to $C_0$ than I’ll have a very small signal, not good).
  (Since this is an FM modulation, we will lose sign of $x$).
• Bode graph:
  
• Complete circuit with sign-recovery:
  
• Real World Measures:
  • $x\in [0,1]$, also $x\ll 1$

---

SaM - Complete AM (Amplitude Modulation) Circuit

• Circuit:
  
• This is a differential circuit, since:
  • $C_1=\frac{C_0}{1+x}$
  • $C_2=\frac{C_0}{1-x}$
• Pre-filtered formula:$V_0=-\frac{2C_0}{C_f}|x|$
• Bode graph:
  
  • $\frac{1}{2\pi R_f{C}_f}$ is the lower frequency limit, defined by the feedback impedance, our frequency $f$ must be higher than this.

---

SaM - Carrier Amplifier

---

Link to original

SaM - Low Pass Filter

• Circuit:
  
• Circuit:
  

---

SaM - Switched Capacitors

• Circuit:
  (It can be used to increase the capacitance value, with respect to the sensor’s capacitance, without changing the actual capacitance)
  
• Workings:
  1. START : $S_3$ : closed, so that we start from $0V$.
  2. $C_S$ CHARGES: ($S_1$: ON, $S_2$: OFF, $S_3$: OFF).
  3. $C_S$ SHARE ITS CHARGE WITH $C_E$ : ($S_1$: OFF, $S_2$: ON, $S_3$: OFF).
  4. repeat (2.) and (3.), $n$ times.
• Output at each step, relative to previous step:$V_{out}^{(n)}=\frac{C_S{V}_R+C_R{V}_{out}^{(n-1)}}{C_S+C_R}$
• Absolute output formula:$V_{out}^{(n)}=n\cdot C_S\cdot \Delta Q_n$Where: $\Delta Q_n=\frac{V_R}{C_R}$
• Graph:
  (If I limit the measurement ratio to $n_{MAX}$, opprotunaly choosen, I will remain in the linear range)
  

---

SaM - Sigma-Delta Converter (AD Converter)

• Structure:
  
• For this converter you need a not-referenced capacitive sensor.
• Simplified formula (based on the coffee shop example):$V_{out}=V_{ref}\cdot \frac{\text{\# of 1s}}{\text{\# of 1s}+\text{\# of 0s}}$
• Coffee shop example:
  
  
  
  

---

Link to original

SaM - High Frequency (AC) Signals

• Pure AC input signal:
  
• DC exitation circuit:
  
• Sensors:$C_1=\frac{C_0}{1+x},C_2=\frac{C_0}{1-x}$
• Output:$V_{out}=\frac{V_{DC}}{2}(1-x)$
• ~Ex.: capacitive microphone electrical model:
  
• Needed force to move the microphone’s plate:$F\propto \frac{A}{h^2}$The needed DC exitation to get a readable signal would be really high, $V_{DC}$ usually, it’s a large value ($\sim 100V$).

---

SaM ~ Examples of Capacitive Sensors

(Skipped)