Capacitive & Magnetic Sensors

Cheatsheet Content

### Capacitive Sensors Capacitive sensors measure changes in capacitance based on variations in plate geometry or dielectric properties. #### Parallel Plate Capacitance The capacitance of a parallel plate capacitor is given by: $$C = \frac{\epsilon_0 \epsilon_r A}{d}$$ Where: - $\epsilon_0$: Permittivity of free space ($8.85 \text{ pF m}^{-1}$) - $\epsilon_r$: Relative permittivity (dielectric constant) of the insulating material - $A$: Area of overlap of the plates ($m^2$) - $d$: Separation between the plates ($m$) **When used:** This fundamental equation is used as the basis for all capacitive sensor calculations, where changes in $A$, $d$, or $\epsilon_r$ are measured. #### Variable Separation Displacement Sensor In this type, the plate separation $d$ changes. If the displacement $x$ increases the separation to $d+x$: $$C = \frac{\epsilon_0 \epsilon_r A}{d+x}$$ **When used:** To measure displacement by detecting changes in the distance between capacitor plates. #### Differential Capacitive Displacement Sensor (Push-Pull) This sensor uses two capacitors ($C_1$, $C_2$) in a push-pull arrangement. If displacement $x$ causes one gap to decrease ($d-x$) and the other to increase ($d+x$), the capacitances are: $$C_1 = \frac{\epsilon_0 \epsilon_r A}{d+x}, \quad C_2 = \frac{\epsilon_0 \epsilon_r A}{d-x}$$ **When used:** For highly linear and sensitive displacement measurements, as it provides a differential output. #### Pressure Measurement (Capacitive Pressure Sensor) A flexible diaphragm (clamped) moves due to pressure, changing the plate separation. The deflection $y$ at any radius $r$ is given by: $$y = \frac{3(1-\nu^2)}{16Et^3}(a^2-r^2)^2 P$$ Where: - $a$: Radius of diaphragm - $t$: Thickness of diaphragm - $E$: Young's modulus - $\nu$: Poisson's ratio - $P$: Applied pressure The resulting increase in capacitance $\Delta C$ due to reduced average separation is: $$\frac{\Delta C}{C} = \frac{(1-\nu^2)a^4}{16Ed^3}P$$ Note: $C = \epsilon_0\pi a^2/d$ at zero pressure. **When used:** To measure pressure by sensing the deformation of a diaphragm, which alters the capacitance. Dielectric is typically air ($\epsilon_r \approx 1$). #### Variable Area Displacement Sensor In this type, the effective overlap area $A$ changes. If displacement $x$ decreases the overlap area by $\Delta A = wx$ (where $w$ is the width of the plates): $$C = \frac{\epsilon_0 \epsilon_r (A-wx)}{d}$$ **When used:** To measure displacement by varying the effective overlapping area between capacitor plates. #### Variable Dielectric Displacement Sensor Here, the dielectric material in the gap changes. The total capacitance is the sum of two capacitances with different dielectric constants $\epsilon_1$ and $\epsilon_2$, and areas $A_1$ and $A_2$. $$C = \frac{\epsilon_0 \epsilon_1 A_1}{d} + \frac{\epsilon_0 \epsilon_2 A_2}{d}$$ If $A_1 = wx$ and $A_2 = w(l-x)$, where $w$ is the width and $l$ is the total length: $$C = \frac{\epsilon_0 w}{d}[\epsilon_2 l - (\epsilon_2 - \epsilon_1)x]$$ **When used:** To measure displacement or detect changes in material composition by altering the dielectric filling the space between the plates. #### Capacitive Level Sensor Used to measure the level of a liquid (dielectric) in a tank. For a coaxial cylindrical capacitor partially immersed in a liquid: $$C_L = \frac{2\pi\epsilon_0}{\log_e(b/a)}[l + (\epsilon_r - 1)h]$$ Where: - $l$: Total length of the sensor - $h$: Height of the liquid - $a, b$: Radii of the inner and outer cylinders - $\epsilon_r$: Relative permittivity of the liquid **When used:** To continuously monitor liquid levels based on the dielectric difference between the liquid and air. #### Thin-Film Capacitive Humidity Sensor The capacitance changes with humidity due to the absorption of water molecules by a hygroscopic dielectric material. The capacitance-humidity relation is often linear: $$C = 375 + 1.7 \, \text{RH pF}$$ Where RH is Relative Humidity. **When used:** To measure relative humidity. ### Magnetic Circuits Review Deals with fundamental concepts of magnetic fields and circuits, often analogous to electrical circuits. #### Definitions - **Electromotive Force (e.m.f.):** Analogous to voltage in electrical circuits. Not explicitly related to currents and resistances in the provided text for magnetic circuits, but generally is the driving force for current ($e.m.f. = \text{current} \times \text{resistance}$). - **Magnetomotive Force (m.m.f.):** The driving force for magnetic flux. $$m.m.f. = \text{flux} \times \text{reluctance} = \Phi \mathcal{R}$$ For a coil with $n$ turns and current $i$: $$m.m.f. = ni$$ - **Flux ($\Phi$):** The total magnetic field passing through a given area. $$\Phi = \frac{ni}{\mathcal{R}} \text{ Weber}$$ - **Total Flux N linked by a coil:** For a coil of $n$ turns, the total flux linkage is $N_{total} = n\Phi$. $$N_{total} = \frac{n^2i}{\mathcal{R}}$$ - **Reluctance ($\mathcal{R}$):** Opposition to magnetic flux, analogous to resistance in electrical circuits. $$\mathcal{R} = \frac{l}{\mu \mu_0 A}$$ Where: - $l$: Length of the flux path - $\mu$: Relative permeability of the material - $\mu_0$: Permeability of free space ($4\pi \times 10^{-7} \text{ Hm}^{-1}$) - $A$: Cross-sectional area of the flux path - **Self-inductance ($L$):** The total flux per unit current. $$L = \frac{N_{total}}{i} = \frac{n^2}{\mathcal{R}}$$ **When used:** These formulas are used to characterize magnetic circuits, calculate flux, m.m.f., reluctance, and inductance in various magnetic devices. ### Inductive Displacement Sensors (Variable Reluctance Elements) These sensors measure displacement by changing the magnetic reluctance of a circuit, which in turn changes the inductance of a coil. #### Variable Reluctance Displacement Sensor Consists of: 1. A ferromagnetic core (e.g., semitoroid) 2. A variable air gap 3. A ferromagnetic plate or armature The total reluctance is the sum of reluctances of the core, air gap, and armature: $$\mathcal{R}_{TOTAL} = \mathcal{R}_{CORE} + \mathcal{R}_{GAP} + \mathcal{R}_{ARMATURE}$$ Specific formulas for each component's reluctance are derived from $\mathcal{R} = l/(\mu \mu_0 A)$. For example: - **Core:** $\mathcal{R}_{CORE} = \frac{\pi R}{\mu_0 \mu_c r^2}$ (simplified, general form is $\frac{l_{core}}{\mu_0 \mu_c A_{core}}$) - **Air Gap:** $\mathcal{R}_{GAP} = \frac{2d}{\mu_0 \pi r^2}$ (simplified for a specific geometry, general form is $\frac{d}{\mu_0 A_{gap}}$) - **Armature:** $\mathcal{R}_{ARMATURE} = \frac{2R}{\mu_0 \mu_a 2rt}$ (simplified, general form is $\frac{l_{armature}}{\mu_0 \mu_a A_{armature}}$) The inductance $L$ is related to total reluctance by $L = n^2 / \mathcal{R}_{TOTAL}$. Often, total reluctance is expressed as: $$\mathcal{R}_{TOTAL} = \mathcal{R}_0 + kd$$ Where $\mathcal{R}_0$ is reluctance at zero air gap, and $k$ is a constant. Then, inductance: $$L = \frac{n^2}{\mathcal{R}_0 + kd} = \frac{L_0}{1 + \alpha d}$$ Where $L_0 = n^2/\mathcal{R}_0$ is the inductance at zero air gap, and $\alpha = k/\mathcal{R}_0$. **When used:** To measure linear proximity or displacement. #### Differential Reluctance Displacement Sensor Uses two coils ($L_1$, $L_2$) arranged differentially. As an armature moves, the air gap for one coil decreases while for the other it increases, leading to a differential change in inductance. $$L_1 = \frac{L_0}{1 + \alpha(a-x)}, \quad L_2 = \frac{L_0}{1 + \alpha(a+x)}$$ Where $a$ is the nominal gap and $x$ is the displacement. **When used:** Provides a more linear output than single coil sensors and improved sensitivity for displacement measurements. Often incorporated into AC deflection bridges. ### Linear Variable Differential Transformer (LVDT) An LVDT measures linear displacement by converting it into an electrical signal. #### LVDT Operation - **Construction:** Consists of a primary winding, two secondary windings (wound in opposition), and a ferromagnetic core (plunger) that moves inside. - **Principle:** The primary winding is energized by an AC voltage. The moving core changes the magnetic coupling between the primary and the two secondary windings, inducing differential voltages. - **Output Voltage:** The output is the difference between the voltages induced in the two secondary windings ($V_{OUT} = V_1 - V_2$). - **Core Position:** - **Centered (null position, x=0):** $V_1 = V_2$, so $V_{OUT} = 0$. - **Moved to one side (e.g., +x):** $V_1 > V_2$, so $V_{OUT}$ is positive. - **Moved to the other side (e.g., -x):** $V_1 #### LVDT after Rectifier To obtain a DC voltage proportional to displacement, the AC output of the LVDT is rectified and filtered. - **Rectifier and Low Pass Filter:** Converts the AC output into a DC voltage. - **DC Characteristics:** The output voltage magnitude changes linearly with displacement across a range, with the sign (or presence of signal) indicating direction. #### LVDT after Phase Sensitive Detector (PSD) and LPF Using a phase-sensitive detector (PSD) before filtering allows for detection of both magnitude and direction of displacement. - **PSD:** Compares the LVDT output phase with the primary excitation phase. - **Output:** The DC output voltage from the LPF is directly proportional to displacement and has a sign indicating direction over the full range. **When used:** To provide a DC output that is both linear and bidirectional, making it suitable for feedback control systems. ### Electromagnetic Sensing Elements Sensors whose operation is based on Faraday's law of electromagnetic induction. #### Faraday's Law of Electromagnetic Induction If the magnetic flux $N_{total}$ linked by a conductor is changing with time, then a back e.m.f. $E$ is induced in the conductor, with magnitude equal to the rate of change of flux. $$E = -\frac{dN_{total}}{dt}$$ **When used:** Fundamental law for all electromagnetic sensors and generators, including tachogenerators. #### Variable Reluctance Tachogenerator Measures angular velocity by generating an e.m.f. proportional to the rate of change of magnetic flux. - **Construction:** Typically consists of a toothed ferromagnetic wheel, a permanent magnet, and a coil. - **Principle:** As the toothed wheel rotates, the reluctance of the magnetic circuit changes cyclically, causing the magnetic flux linked by the coil to vary. This varying flux induces an e.m.f. in the coil. - **Flux Linkage:** The total flux linkage $N_{total}$ can be approximated as: $$N_{total}(\theta) = a + b \cos(m\theta)$$ Where: - $a$: Mean flux - $b$: Amplitude of flux variation - $m$: Number of teeth - $\theta$: Angular position - **Induced e.m.f.:** $$E = -\frac{dN_{total}}{dt} = -\frac{dN_{total}}{d\theta} \frac{d\theta}{dt}$$ With $\frac{d\theta}{dt} = \omega_r$ (angular velocity) And $\frac{dN_{total}}{d\theta} = -bm \sin(m\theta)$ Substituting, if $\theta = \omega_r t$: $$E = -(-bm \sin(m\omega_r t)) \omega_r = bm\omega_r \sin(m\omega_r t)$$ - **Output Signal Amplitude:** Proportional to $bm\omega_r$. - **Output Signal Frequency:** $f = \frac{m\omega_r}{2\pi} = \frac{m \times \text{RPM}}{60}$ (since $\omega_r = 2\pi \times \text{RPM}/60$) Where RPM is revolutions per minute. **When used:** To measure rotational speed (angular velocity) through the frequency and amplitude of the induced voltage.