Sensor types, measurement models and calibration

Symbols. S is Seebeck coefficient; temperatures refer to sensing and reference junctions.

Derivation. Integrate the difference of thermoelectric fields around the two-metal circuit; constant coefficient gives the linear approximation.

Assumptions & limits. Reference-junction compensation and material-specific calibration are required.

Symbols. R0 reference resistance; α local temperature coefficient.

Derivation. Linearize the calibrated resistance curve at T0 and invert its slope to estimate temperature.

Assumptions & limits. Limited temperature interval; lead resistance and self-heating bias the result.

Symbols. B beta coefficient in kelvin; T absolute temperature.

Derivation. Integrate dlnR/d(1/T)=B for an activated-resistance approximation.

Assumptions & limits. NTC beta law over a calibrated range; use a richer fit when needed.

Symbols. r ratio of junction current densities; q elementary charge.

Derivation. Subtract two ideal diode voltage laws at different current densities to cancel the common saturation-current term.

Assumptions & limits. Matched junctions, appropriate ideality and calibrated parasitics.

Symbols. N junction pairs, Rth thermal resistance, α absorptance, A area, Tt target and Ts sensor temperatures.

Derivation. Absorbed net radiant power creates a thermal rise; series thermocouples convert that rise to voltage.

Assumptions & limits. Ideal blackbody full-view target; real bandpass, view factor and emissivity need calibration.

Symbols. Cth heat capacity, Gth conductance, αR resistance temperature coefficient.

Derivation. Balance absorber heat storage and loss, then linearize resistance with temperature.

Assumptions & limits. Small signal, fixed bath and negligible nonlinear electrothermal feedback.

Symbols. p pyroelectric coefficient; A electrode area.

Derivation. Differentiate polarization charge Q=AP(T) with respect to time.

Assumptions & limits. Detects temperature changes; a steady target needs modulation and a thermal response model.

Symbols. k thermal conductivity and L sensing-layer thickness.

Derivation. Apply Fourier conduction through the calibrated layer; a thermopile measures its temperature drop.

Assumptions & limits. Approximately one-dimensional heat flow and known contact resistances.

Symbols. GF gauge factor; ε axial strain.

Derivation. Differentiate resistance ρL/A and combine geometric and piezoresistive contributions into GF.

Assumptions & limits. Small strain, temperature compensation and appropriate bonding.

Symbols. E elastic modulus, A effective area, bridge excitation Vexc.

Derivation. An axial elastic member turns force into strain; a quarter bridge turns a small resistance change into voltage.

Assumptions & limits. Representative axial quarter-bridge model; commercial bending/full-bridge cells use different calibrated factors.

Symbols. π coefficients relate longitudinal and transverse stress to resistance change.

Derivation. Pressure bends a diaphragm; its stress field enters the piezoresistive law and a bridge reads the imbalance.

Assumptions & limits. Requires diaphragm mechanics, crystal orientation and temperature calibration.

Symbols. εd dielectric permittivity, w diaphragm displacement.

Derivation. Pressure moves a compliant electrode, changing electric-field spacing and stored charge per volt.

Assumptions & limits. Uniform-gap approximation; a real bent diaphragm requires area integration.

Symbols. dp charge coefficient, C total capacitance.

Derivation. Integrate piezoelectric polarization from applied force over the electrode.

Assumptions & limits. Dynamic or quasi-static measurement; leakage and charge-amplifier time constants limit DC response.

Symbols. Sp calibrated tube-tip sensitivity; Gm linkage gain.

Derivation. Linearize elastic tube straightening around an operating point and apply the mechanical linkage ratio.

Assumptions & limits. Geometry-dependent local calibration, hysteresis and overload limits; not a universal tube law.

Symbols. M1,M2 mutual inductances; Ip excitation phasor.

Derivation. Subtract induced secondary voltages; near the null the coupling difference is proportional to core displacement.

Assumptions & limits. Demodulated linear range, stable excitation and limited magnetic saturation.

Symbols. x wiper position along uniform track length L.

Derivation. Use the resistive voltage-divider ratio Rsegment/Rtotal=x/L.

Assumptions & limits. Negligible loading, uniform track and good wiper contact.

Symbols. d gap; A overlap area.

Derivation. Integrate a uniform plate field to obtain charge, then divide by voltage and invert.

Assumptions & limits. Parallel plates, small fringe contribution and known dielectric.

Symbols. N turns, magnetic reluctance Rm, oscillator capacitance C.

Derivation. A target changes the coil’s magnetic or loss environment, shifting inductance or oscillator damping.

Assumptions & limits. Representative resonance model; conductive targets also produce eddy-current losses and require calibration.

Symbols. δ skin depth, σe electrical conductivity, d target gap.

Derivation. Maxwell induction creates target currents whose secondary field changes complex probe impedance.

Assumptions & limits. Gap inference requires a calibrated geometry-dependent impedance function; skin depth alone is not a displacement calibration.

Symbols. n counted increments, N resolved increments per revolution.

Derivation. A periodic optical pattern converts each resolved transition into a known angle increment.

Assumptions & limits. Specify quadrature decoding, index reference, direction and missed-count limits.

Symbols. Vsin,Vcos offset-corrected quadrature magnetic signals.

Derivation. Resolve two orthogonal field components and recover phase from their ratio with quadrant information.

Assumptions & limits. One-pole-pair model; multipole patterns, offsets and harmonics need correction.

Symbols. Λ grating period, pe photoelastic factor, α expansion and ξ thermo-optic coefficient.

Derivation. Constructive Bragg reflection fixes wavelength; differentiate index and period changes.

Assumptions & limits. Small strain/temperature changes; temperature–strain cross-sensitivity needs separation.

Symbols. pi calibrated local pressure, Ai pixel area.

Derivation. Convert each cell’s signal to pressure and integrate over contact area.

Assumptions & limits. Cell transfer laws depend on resistive, capacitive or optical construction; saturation and cross-talk matter.

Symbols. m proof mass,b damping,k stiffness.

Derivation. Newton’s law in the accelerating sensor frame gives a driven oscillator; gap capacitances read x.

Assumptions & limits. Low-frequency approximation below resonance; measures specific force with gravity handled by navigation conventions.

Symbols. m effective proof mass and dp piezoelectric coefficient.

Derivation. Inertial force ma loads the piezoelectric element, producing proportional charge.

Assumptions & limits. Dynamic band only, below structural resonance and above leakage cutoff.

Symbols. Ω rotation rate, vd driven proof-mass velocity perpendicular to the rate axis.

Derivation. Coriolis acceleration couples the driven mode into the sense mode.

Assumptions & limits. Small angular rates and calibrated resonant transfer function; quadrature and bias need correction.

Symbols. Aeff oriented loop area summed over turns; λ vacuum wavelength.

Derivation. Counterpropagating paths acquire a Sagnac time difference and hence an optical phase difference.

Assumptions & limits. Reciprocal-path approximation; polarization and thermal nonreciprocity require control.

Symbols. A ring area, P perimeter, λ wavelength.

Derivation. Sagnac path-time asymmetry splits counterpropagating cavity resonances.

Assumptions & limits. Ideal ring; lock-in and bias require engineering compensation.

Symbols. ax,az calibrated gravity-vector components in a chosen sign convention.

Derivation. Project static gravity onto two axes and recover orientation.

Assumptions & limits. Translational acceleration invalidates a gravity-only tilt estimate.

Symbols. ρi pseudorange, ri satellite position, δt receiver clock offset.

Derivation. Convert measured code delay to length and solve simultaneous position/clock equations.

Assumptions & limits. Satellite clock, relativity, atmosphere, ephemeris and multipath corrections are required.

Symbols. si catalog star direction, bi measured body-frame direction, R attitude rotation.

Derivation. Project star images to unit vectors and fit a common rotation by weighted least squares.

Assumptions & limits. Identified stars, calibrated optics, R orthogonal with determinant one and appropriate noise weights.

Symbols. ηq quantum efficiency; R responsivity.

Derivation. Divide optical power by photon energy and multiply detected photon rate by electron charge.

Assumptions & limits. Linear spectral response; dark current, saturation and bandwidth remain separate.

Symbols. M avalanche gain.

Derivation. Multiply primary photocarrier current by mean impact-ionization gain.

Assumptions & limits. Below breakdown linear mode; gain noise, temperature and bias stability matter.

Symbols. PDE detection efficiency, τd nonparalyzable dead time.

Derivation. Detected arrivals occupy the detector for a dead interval, reducing count throughput.

Assumptions & limits. Simplified stationary model; afterpulsing, pile-up and timing response need characterization.

Symbols. Ncell microcells; Nγ short-pulse incident photons.

Derivation. Treat independent photon assignment to cells as Poisson occupancy; a cell fires if at least one photon triggers it.

Assumptions & limits. No recharge during pulse, cross-talk or afterpulsing in this approximation.

Symbols. δ mean gain per dynode,n stages.

Derivation. A photocathode converts photons to electrons and repeated secondary emission multiplies their number.

Assumptions & limits. Stage-independent mean gain model; dark counts, excess noise and saturation omitted.

Symbols. Ne collected electrons,t exposure.

Derivation. Integrate detected photon rate; Poisson arrivals give shot-noise standard deviation.

Assumptions & limits. Add dark/read noise and charge-transfer losses; avoid full-well saturation.

Symbols. Cnode conversion-node capacitance.

Derivation. Photons generate charge that is converted into a pixel voltage and read electronically.

Assumptions & limits. Representative conversion gain; read noise, rolling/global shutter and pixel circuitry differ.

Symbols. L active length; IR,IL electrode photocurrents.

Derivation. Normalize current imbalance by total intensity to infer spot position.

Assumptions & limits. Calibrated linear lateral-effect model, spot wholly inside active area.

Symbols. d grating period,m diffraction order,α incidence,β diffraction angle in the chosen convention.

Derivation. Adjacent grooves produce a path difference; integer wavelengths interfere constructively.

Assumptions & limits. Representative grating spectrometer; resolution also depends on slit, aberrations, pixel sampling and illuminated grooves.

Symbols. Δt round-trip delay,n medium refractive index.

Derivation. Light travels to the target and back at speed c/n; divide total path length by two.

Assumptions & limits. Timing offsets, multiple returns, ambient light and target reflectance matter.

Symbols. Δx mirror displacement in a double-pass path.

Derivation. Motion changes round-trip optical path by2nΔx; multiply by2π/λ.

Assumptions & limits. Phase unwrapping and environmental refractive-index correction required.

Symbols. S0,S1,S2 Stokes parameters,θ linear analyzer angle.

Derivation. Resolve the polarization coherence matrix onto the analyzer direction.

Assumptions & limits. Linear analyzers alone cannot measure circular StokesS3; add a retarder for full polarimetry.

Symbols. x diaphragm deflection.

Derivation. Acoustic pressure drives diaphragm mechanics and changes electrode spacing.

Assumptions & limits. Small motion; bias circuit and mechanical frequency response set pressure sensitivity.

Symbols. B gap flux density,ℓ active wire length,v coil velocity.

Derivation. Motion through magnetic flux induces an EMF proportional to velocity.

Assumptions & limits. Lumped moving-coil model; diaphragm/acoustic loading relates pressure to velocity.

Symbols. p applied pressure,A loaded area.

Derivation. Pressure creates forceAp; piezoelectric charge is dp times force.

Assumptions & limits. Dynamic frequency response and boundary constraints affect sensitivity.

Symbols. M complex calibrated sensitivity in V/Pa.

Derivation. Solve acoustic–mechanical–electrical transduction, then express the resulting frequency-dependent transfer function.

Assumptions & limits. Sensitivity is a calibration function, not a universal constant; pressure, temperature and direction matter.

Symbols. cs sound speed,Δt echo delay.

Derivation. Divide round-trip acoustic propagation length by two.

Assumptions & limits. Sound speed depends on medium and temperature; blanking and multiple echoes matter.

Symbols. r event position,ri sensor positions.

Derivation. Compare travel times of an elastic disturbance to estimate source location.

Assumptions & limits. Known wave mode and speed; anisotropy, dispersion and threshold timing complicate localization.

Symbols. I bias current,B perpendicular field,n carrier density,t plate thickness.

Derivation. Balance transverse Lorentz force with the Hall electric field and integrate across the plate.

Assumptions & limits. Single-carrier idealization; sign, geometry factor, offset and temperature require calibration.

Symbols. θ between magnetization and current.

Derivation. Spin-dependent scattering produces an orientation-dependent longitudinal resistance.

Assumptions & limits. Magnetization must follow the assumed field state; hysteresis and bridge geometry matter.

Symbols. P/AP parallel/antiparallel limits.

Derivation. A simple angular interpolation represents spin-dependent relative layer alignment.

Assumptions & limits. Phenomenological resistance law; actual tunneling conductance and bias dependence may differ.

Symbols. V2ω demodulated second-harmonic voltage,KB calibrated sensitivity.

Derivation. Symmetric core saturation has balanced half-cycles; a small external field breaks symmetry and generates an even harmonic.

Assumptions & limits. Small-field calibrated operating range, drive stability and core hysteresis constraints.

Symbols. I0 junction critical current,Φ loop flux.

Derivation. Add two Josephson supercurrents subject to flux-dependent phase difference.

Assumptions & limits. Symmetric low-inductance DC-SQUID idealization; flux-locked loops linearize a periodic response.

Symbols. R calibrated shunt resistance.

Derivation. Ohm’s law converts current to differential voltage; Joule heating changes temperature.

Assumptions & limits. Four-wire sensing, bandwidth, thermoelectric offsets and resistance drift matter.

Symbols. Np,Ns turns; winding current signs follow flux convention.

Derivation. Ampere-turn balance under high core permeability gives the secondary current ratio.

Assumptions & limits. AC only for ordinary transformers; burden, saturation and magnetizing current create errors.

Symbols. M mutual inductance.

Derivation. Faraday induction measures changing current; an integrator reconstructs current up to an initial constant.

Assumptions & limits. Cannot recover DC from induction alone; integrator drift and bandwidth matter.

Symbols. E normal electric field,A effective electrode area.

Derivation. Induced electrode charge follows electric displacement flux; read charge or its time derivative.

Assumptions & limits. Representative capacitive probe; field disturbance and geometry factor need calibration.

Symbols. vr radial closing speed,λ wavelength.

Derivation. Round-trip delay gives range and two-way phase rate gives monostatic Doppler.

Assumptions & limits. Signs depend on convention; FMCW range needs chirp-slope processing and Doppler separation.

Symbols. RH relative humidity; εr calibrated sorption-dependent permittivity.

Derivation. Water uptake changes dielectric polarization and capacitance.

Assumptions & limits. Material-specific temperature, hysteresis and contamination dependence; not a universal linearRHlaw.

Symbols. Td dew point,Ta ambient temperature,psat saturation pressure.

Derivation. Condensation identifies vapor partial pressure as saturation pressure atTd; normalize to ambient saturation.

Assumptions & limits. Distinguish dew from frost and include pressure/composition corrections where relevant.

Symbols. mdot mass flow,cp heat capacity.

Derivation. A calorimetric energy balance relates added heat to transported enthalpy plus losses.

Assumptions & limits. Representative calorimetric mode; bypass geometry, fluid properties and heat losses require calibration.

Symbols. Ao orifice area,β diameter ratio,Cd discharge coefficient.

Derivation. Combine continuity and Bernoulli across a restriction, then correct contraction/losses empirically.

Assumptions & limits. Incompressible calibrated orifice regime; compressible flows need expansion correction.

Symbols. B transverse field,D electrode spacing.

Derivation. Conducting fluid motion induces a transverse motional EMF.

Assumptions & limits. Full conductive pipe, suitable electrode contact and calibrated velocity profile.

Symbols. Δφ measured tube-motion phase imbalance,Km calibrated sensitivity.

Derivation. Flow through a driven vibrating tube produces Coriolis coupling proportional to mass throughput in the linear regime.

Assumptions & limits. Tube geometry, stiffness, density and damping setKm; no universal phase constant.

Symbols. L acoustic path; tplus downstream,tminus upstream.

Derivation. Subtract reciprocal transit times to cancel sound speed in a uniform-flow model.

Assumptions & limits. Known path angle; cross-sectional mean flow needs a profile correction.

Symbols. N illuminated particle count; differential scattering cross section depends on size and refractive index.

Derivation. Sum single-particle scattered power into the detector’s collection angle.

Assumptions & limits. Dilute single scattering; inferred mass needs particle size, density and optical calibration.

Symbols. κ absorption coefficient per concentration,cg concentration,L path.

Derivation. Integrate Beer–Lambert attenuation and invert the logarithm.

Assumptions & limits. Band-integrated response, temperature, pressure and overlapping gases need compensation.

Symbols. αabs absorption coefficient,Pmod modulated optical power,K cell response.

Derivation. Absorbed periodic heat drives gas expansion and an acoustic response read by a microphone.

Assumptions & limits. Small absorption and a calibrated acoustic/thermal transfer function.

Symbols. A,b empirical fit parameters,cg target concentration in stated units.

Derivation. Surface reaction and carrier depletion yield a calibrated log–log response over a limited range.

Assumptions & limits. Not universal kinetics; humidity, temperature, mixed gases and aging affect selectivity.

Symbols. ne electrons per reaction,F Faraday constant,Dg diffusivity.

Derivation. Diffusion-limited transport through a layer gives fluxDgcg/L; Faraday’s law converts molar flux to current.

Assumptions & limits. Representative steady diffusion-limited sensor; membrane partition and electrode kinetics can dominate.

Symbols. θv volumetric water fraction,εeff effective permittivity,EC conductivity.

Derivation. Water’s dielectric response changes capacitance or propagation; invert a soil-specific mixing/calibration relation.

Assumptions & limits. Texture, salinity, density and bound water prevent one universal conversion.

Symbols. R gas constant,T kelvin,F Faraday constant.

Derivation. Nernst potential depends on hydrogen-ion activity; pH=−log10aH converts the logarithm.

Assumptions & limits. Junction potentials, reference drift and temperature calibration matter; activity is not simply concentration.

Symbols. zi ion charge,ai activity.

Derivation. Electrochemical equilibrium across the selective membrane gives the Nernst potential.

Assumptions & limits. Interfering ions and nonideal selectivity need additional terms.

Symbols. Kcell geometric cell constant,G conductance.

Derivation. Combine Ohm’s law with electrode geometry to recover material conductivity.

Assumptions & limits. AC excitation reduces polarization; temperature and electrode contamination require control.

Symbols. J molar flux,D diffusivity,c concentration.

Derivation. Faraday’s law converts electrochemical reaction flux into current.

Assumptions & limits. Steady diffusion-limited planar approximation; kinetics and transport regime must be established.

Symbols. KSV Stern–Volmer constant,c quencher concentration.

Derivation. Add collisional quenching to the excited-state decay rate and compare with the unquenched lifetime.

Assumptions & limits. Dynamic single-species quenching model; static quenching and heterogeneous environments change the law.

Symbols. f0 resonant frequency,ρq quartz density,μq shear modulus.

Derivation. A thin attached mass changes the effective oscillating inertia, shifting resonance.

Assumptions & limits. Sauerbrey limit: thin rigid uniform film; liquid loading and viscoelastic films need more complete models.

Symbols. εm metal andεd dielectric permittivity,k0 vacuum wave number.

Derivation. Match tangential light momentum to the surface mode; binding changes nearby dielectric response and the resonance position.

Assumptions & limits. Planar interface model; losses and multilayer films require complex transfer calculations.

Symbols. W mean energy per ion pair,Edep deposited energy.

Derivation. Divide deposited energy by ion-pair creation energy and collect charge.

Assumptions & limits. Collection efficiency, recombination and gas calibration matter; no avalanche gain assumed.

Symbols. M gas multiplication factor.

Derivation. Primary ionization creates pairs and avalanche multiplication amplifies their collected charge.

Assumptions & limits. Proportional region only; gain fluctuations and space charge limit spectroscopy.

Symbols. τd nonparalyzable dead time.

Derivation. A full discharge makes the tube insensitive during recovery, reducing count throughput.

Assumptions & limits. Simplified count correction, not an energy measurement or a universally valid high-rate model.

Symbols. Y photons per deposited energy,ηopt collection,ηq detector efficiency.

Derivation. Convert energy to light, then collected photons to photoelectrons.

Assumptions & limits. Quenching, nonproportional yield and photon statistics affect energy resolution.

Symbols. w pair-creation energy,Fano sub-Poisson factor.

Derivation. Deposited energy generates electron–hole pairs; collect charge and account for creation fluctuations.

Assumptions & limits. Trapping, incomplete collection, leakage and electronics add errors.

Symbols. β particle speed/c,n phase refractive index.

Derivation. Constructive interference of the moving charge’s emitted field forms the Cherenkov cone.

Assumptions & limits. Dispersion, optical acceptance and finite track length affect the observed ring.

Symbols. N target number density,σ reaction cross section,L thickness.

Derivation. Exponential attenuation gives conversion probability; multiply by probability that reaction products are collected.

Assumptions & limits. Energy-dependent cross sections, scattering and self-shielding may require transport simulation.

Symbols. c substrate concentration,KM Michaelis constant.

Derivation. Quasi-steady enzyme binding gives a saturating reaction rate; electrochemical readout converts product turnover to current.

Assumptions & limits. Transport, cofactors, interference and enzyme aging can invalidate a simple reaction-limited calibration.

Symbols. θ occupied binding-site fraction,Kd dissociation constant.

Derivation. Equate association and dissociation rates for independent sites, then map occupancy to signal.

Assumptions & limits. Equilibrium single-site Langmuir model; nonspecific binding and transport kinetics need controls.

Symbols. Ehc half-cell offset,Ib input bias current,Ze electrode impedance.

Derivation. Model tissue potential plus electrochemical interface and amplifier loading.

Assumptions & limits. Differential measurement, contact impedance, motion artifact and electrical isolation are essential; this is not a clinical interpretation model.

Symbols. μeff effective tissue attenuation,L optical path.

Derivation. Linearize optical attenuation as pulsatile blood changes the effective path and absorption.

Assumptions & limits. Scattering tissue makes this a simplified model; oxygen saturation requires multiple wavelengths and validated calibration.