The whole Sensors and Industrial Instrumentation formula sheet, to print

Delta y_max is the largest deviation of the actual output from the reference straight line.

R_p is the total track resistance. Reduces to x for R_L infinite. The error is zero at both ends. For R_L much larger than R_p: as a fraction of the ideal reading the error is about (R_p/R_L)x(1-x), largest at x = 0.5; as a fraction of V_s (full scale) it is about (R_p/R_L)x^2(1-x), largest at x = 2/3, where it is about 0.148 R_p/R_L.

nu is Poisson's ratio of the gauge material. 1 + 2nu is the dimensional effect; the last term is the piezoresistive effect. For metal foil gauges the resistivity term is small and G is about 2. Semiconductor gauges have G of order 100 or more, mainly from the piezoresistive term, but are more temperature-sensitive and less linear.

Valid for all arms initially R and ΔR much less than R; sign of V_o depends on which terminals are taken as output.

Equal-arm bridge with one active arm; the fractional error is about (ΔR/R)/2. Half and full bridges with equal and opposite changes, each pair in series across the supply, have no such error; axial-plus-transverse arrangements keep a small one.

Input-stage resistors R_1 (two equal), gain resistor R_g, difference stage with matched R_2 at the inputs and R_3 as feedback and ground resistors. Gain is set by R_g alone when R_3/R_2 is fixed.

Fringing neglected. Sensing by varying plate gap d (output C is non-linear in d), overlap area A (linear in displacement), or dielectric (varying the effective ε_r).

Exactly linear in displacement x of the centre plate; needs a ratio-measuring circuit such as a bridge or ratiometric amplifier.

C_1 and C_2 in series across the AC excitation V_s, the other half of the bridge two equal impedances, V_o measured across the detector diagonal by a high-impedance detector. The phase of V_o (0 or 180 degrees) shows the direction of x.

N turns, l_g the total air-gap length, A_g the gap area; valid when the core's reluctance is negligible beside the gap's. L is then proportional to 1/l_g, so a single-gap sensor is non-linear; a differential (push-pull) pair in a bridge improves linearity, as for the capacitor.

t is the crystal thickness, p = F/A the applied stress; g in V·m/N. Follows from q = dF and C_p = ε_0 ε_r A/t, for an open-circuit crystal with no cable or amplifier load.

Mid-band value, leakage neglected. Output changes if the cable is changed; this is the reason a charge amplifier is preferred.

Op-amp with feedback capacitor C_f; independent of cable capacitance when the op-amp gain is high. Sign is inverting.

A_ol is the op-amp's open-loop gain. When (1 + A_ol)C_f is much larger than C_t, V_o tends to −q/C_f, which is why the cable capacitance hardly matters.

R_f in parallel with C_f gives a high-pass response; at f_c the gain magnitude is 3 dB below its mid-band value 1/C_f. A steady (DC) force gives a charge that leaks away, so true static force cannot be measured.

Faraday's law for a coil of N turns linking flux Φ; Blv for a conductor of length l moving at speed v, with B, l and v mutually perpendicular. Electromagnetic velocity pickups and flowmeters use this principle.

A ferromagnetic wheel with T teeth turning at n rev/min past a magnetic pickup gives pulses at frequency f in Hz. The pulse amplitude also rises with speed, so the speed is read from the frequency.

B perpendicular to the plate and to the current; n is carrier density, q the carrier charge magnitude, so this gives the magnitude of V_H. The sign of R_H follows the carrier sign (negative for electrons), which identifies the carrier type. V_H reverses sign when B (or I) is reversed. Semiconductors give a larger V_H than metals because n is much smaller.

$R_p$ is the total track resistance and $R_L$ the load across the wiper. For $R_L\to \infty$ the output is $V_{s}k$, which is linear. A finite load makes it nonlinear. For $R_L\gg R_p$ the error (unloaded minus loaded output, as a fraction of $V_s$) is approximately $k^2(1-k)R_p/R_L$, largest near $k=2/3$, where it is about $0.148R_p/R_L$.

The first result is for a change of gap: sensitivity rises as the gap shrinks. The second is for a centre plate moved by $x$ from the middle of two fixed plates, each at gap $d_0$ when centred, so $C_1={\epsilon }_0{\epsilon }_{r}A/(d_0-x)$ and $C_2={\epsilon }_0{\epsilon }_{r}A/(d_0+x)$.

$N$ is the number of slots (pulses per revolution per channel). Resolution is the smallest angle that can be distinguished.

$\omega$ in rad/s, $k_t$ in V per rad/s, $n$ in rpm.

For an encoder, $f$ is the pulse frequency of one channel and $N$ the pulses per revolution of that channel; with $\times 4$ decoding, use the count rate and $4N$. For a variable-reluctance (magnetic) pickup facing a toothed ferromagnetic wheel, each tooth gives one pulse, so $N$ is the number of teeth: $f=Nn/60$.

$Z$ and $Y$ are amplitudes. For $r\gg 1$, $Z/Y\to 1$ and the relative motion is about 180° out of phase with the base motion (vibrometer).

$Z$ is proportional to the acceleration amplitude $Y{\omega }^2$ when $r\ll 1$, with static sensitivity $1/{\omega }_n^2$ (relative displacement per unit acceleration). For $\zeta =1/\sqrt{2}$ the second factor is $1/\sqrt{1+r^4}$.

The angle by which the relative displacement lags the input acceleration, taken between 0° and 180° (90° at $r=1$). For $\zeta \approx 0.7$ and $r<1$, $\varphi$ is nearly proportional to frequency, so a waveform with several frequency components is delayed but not distorted.

$d$ is the charge sensitivity in C/N. $C_p$ is the crystal capacitance, $C_c$ the cable capacitance and $C_a$ the amplifier input capacitance. Cable length changes the voltage sensitivity, so a charge amplifier ($V_o=-q/C_f$) is used to remove that dependence.

$g$ is the voltage coefficient in V m/N, $t$ the crystal thickness and $F/A$ the applied stress. This is the open-circuit voltage of the crystal alone, $q/C_p$ with $C_p={\epsilon }_0{\epsilon }_{r}A/t$.

For a voltage amplifier, $R$ is the combined leakage and input resistance and $C$ the total capacitance of sensor, cable and amplifier input. With a charge amplifier, the feedback resistor $R_f$ and capacitor $C_f$ set the −3 dB point.

$X$ is the displacement amplitude and $f$ the vibration frequency in Hz.

$\nu$ is Poisson's ratio of the gauge material and $\Delta \rho /\rho$ the fractional change in its resistivity (the piezoresistive effect). For metal-foil gauges $G\approx 2$; semiconductor gauges have a much larger, temperature-sensitive $G$.

Axial load $F$ on cross-section $A$ of Young's modulus $E$, within the elastic limit. A gauge bonded across the column sees the transverse strain $-\nu \epsilon$.

Equal arms, bridge balanced at zero strain, signs of the active arms arranged so their effects add. The half- and full-bridge results with equal and opposite strains are exactly linear. Temperature effects cancel when identical gauges at the same temperature sit in adjacent arms (a dummy gauge does this for one active gauge).

$d$ is the shaft diameter in metres, $T$ in N m, $G_s$ in Pa. Mechanical power transmitted is $P=T\omega$.

Gauge pressure is negative below atmospheric; absolute pressure is never negative.

SI units. p is the pressure difference between two points a vertical height h apart in a fluid of density $\rho$ at rest.

Both pressure taps at the same elevation. ${\rho }_m$ is the manometer liquid, ${\rho }_f$ the fluid above it in the limbs, h the difference of levels. For a gas, ${\rho }_f\ll {\rho }_m$ and $\Delta p\approx {\rho }_{m}gh$.

L is the reading along the inclined tube of area a, $\theta$ its angle to the horizontal, A the area of the well; a gas sits above the liquid (otherwise use ${\rho }_m-{\rho }_f$). For $A\gg a$ the vertical height is $h=L\sin \theta$, so the reading is $1/\sin \theta$ times what a vertical tube would show. The magnification rises as $\theta$ falls, so it suits small pressures.

h is the rise read on the vertical tube of area a, A the area of the well. The well level falls by $x=ah/A$; only the tube reading is taken. For $A\gg a$ it reduces to $\rho gh$.

Circular flat diaphragm clamped at its edge, radius R, thickness t, Young's modulus E, Poisson's ratio $\nu$, uniform pressure difference p. Valid while the deflection is small compared with the thickness; the deflection is then linear in p.

The gas of total trapped volume V at unknown pressure p is compressed into a closed capillary of area a. The mercury in the open reference capillary is raised level with the closed end, so the trapped gas column length equals the level difference h. Boyle's law gives $pV=(p+h)ah$ with pressures in mmHg; use consistent units (a in mm², h in mm, V in mm³ gives p in mmHg).

m is the total mass of the piston and the weights on it, $A_p$ the effective piston area; the oil pressure that floats the piston is applied to the gauge under test. SI units.

Q volumetric flow rate, v mean velocity, $\dot{m}$ mass flow rate.

Steady, incompressible, frictionless flow along a streamline at the same elevation.

$A_2$ is the throat or orifice area, $\beta =d/D$, $\rho$ the fluid density; incompressible fluid. $1/\sqrt{1-{\beta }^4}$ is the velocity-of-approach factor, and $K=C_d/\sqrt{1-{\beta }^4}$ is called the flow coefficient. Q is proportional to $\sqrt{\Delta p}$. For a gas with an appreciable pressure drop, multiply by an expansibility factor $\epsilon <1$.

h is the manometer level difference, ${\rho }_m$ the manometer liquid, $\rho$ the flowing liquid filling the leads; from $\Delta p=({\rho }_m-\rho )gh$. The manometer reads the piezometric difference, so the same relation holds for an inclined or vertical meter.

$p_0$ stagnation pressure, $p_s$ static pressure, $C_p\approx 1$; incompressible flow (low Mach number for a gas). Gives the local velocity at the tip, not the mean velocity.

$A_a$ annular area between float and tube, $V_b$ float volume, $A_b$ float cross-section, ${\rho }_b$ float density, $\rho$ fluid density. With a small linear taper, and the bore at the zero mark equal to the float diameter, $A_a$ and so Q are nearly proportional to float height, giving a nearly linear scale. The calibration holds only for the fluid density (and viscosity) it was made for.

Uniform field B; v is the mean velocity for axisymmetric flow; SI units (B in tesla). The emf is linear in flow and, above the minimum conductivity, independent of density, viscosity, temperature and pressure.

From $t_{down}=L/(c+v\cos \theta )$ and $t_{up}=L/(c-v\cos \theta )$. Since $v\ll c$ the approximation is used, so $v=c^2\Delta t/(2L\cos \theta )$. It needs a clean fluid, and this form depends on c, which varies with temperature.

Exact, and independent of the speed of sound c, so a meter computing v this way is not affected by temperature changes in c.

$f_0$ is the transmitted frequency and $\theta$ the angle between the beam and the flow, with transmitter and receiver both at that angle. The beam is reflected by particles or bubbles moving with the fluid, so the fluid must contain scatterers.

f is the pulse frequency, K the K-factor in pulses per unit volume; the total count gives total volume. Linear over its working range; needs clean, low-viscosity fluid (K changes with viscosity), and wears in the bearings.

A change of 1 K equals a change of 1 °C and 1.8 °F. Use 273 when the question does.

T in °C, a and b depend on the pair. dE/dT is the thermoelectric sensitivity (Seebeck coefficient of the pair). For hot junction T_1 and cold junction T_2: E = E(T_1) - E(T_2).

For E = aT + bT^2 with a > 0 and b < 0. The emf is maximum when the hot junction is at T_n (where dE/dT = 0), whatever the cold-junction temperature T_c. It returns to zero at the inversion temperature T_i, which lies as far above T_n as T_c lies below it. With the cold junction at 0 °C, T_i = -a/b = 2T_n.

N identical thermocouples in series, with all hot junctions at the hot temperature and all cold junctions at the cold one. Output and sensitivity are N times those of one thermocouple. It is used to detect radiant power.

T in °C, α referred to 0 °C. Good over a modest range.

α_0 is referred to 0 °C and α_{T1} to T_1 (°C). Both describe the same linear law. For example, α_0 = 0.00385 /°C gives α_20 = 0.00385/1.077 = 0.003575 /°C.

For industrial platinum A = 3.9083 x 10^-3 /°C, B = -5.775 x 10^-7 /°C^2, C = -4.183 x 10^-12 /°C^4. Sensitivity falls slightly as T rises because B is negative.

R_L is the resistance of each lead. The lead resistance is assumed constant and the linear law is used.

Bridge with supply V_s, the RTD in one arm and the other three arms equal to R_0, so it balances at 0 °C. The approximation needs αT much less than 2. The exact output is slightly non-linear in T.

R_0 is the resistance at reference temperature T_0 (kelvin), β is the material constant in kelvin. R falls as T rises.

T_1, T_2 in kelvin, R_1 and R_2 the resistances at them.

Negative for NTC, and its magnitude falls as T rises. At room temperature it is a few per cent per kelvin.

T in kelvin, R in ohms. A, B and C are found from three calibration points. It fits a thermistor more closely over a wide range than the single-β law.

For two matched transistors at current-density ratio n. kT/q is about 25.85 mV at 300 K. The output is linear in absolute temperature.

T in kelvin. With surroundings at T_s, the net emission is εσ(T^4 - T_s^4).

T_a is the black-body (apparent) temperature the pyrometer shows, T is the true surface temperature in kelvin. Valid when radiation from the surroundings is neglected and ε is the total emissivity.

The wavelength at which a black body's spectral emission peaks falls as its temperature rises. T in kelvin.

T_a is the brightness temperature read, ε_λ the spectral emissivity at wavelength λ, temperatures in kelvin. Since ln ε_λ < 0, the true temperature is above the reading.

Ratio of spectral radiances at two wavelengths, under the Wien approximation. For a grey body (ε_1 = ε_2) the emissivity cancels, so the temperature follows from the ratio alone.

T_m is the sensor temperature, T_f the fluid temperature, m the mass, c the specific heat, h the film coefficient, A the surface area. A small, light sensor has a small τ.

The fluid steps from the initial temperature T_0 to T_f at t = 0. The sensor reaches 63.2 % of the step at t = τ, 90 % at 2.303τ and about 98 % at 4τ.

For a fluid temperature rising at a steady rate r (°C/s), after the transient dies away the reading lags the fluid by rτ in temperature, which is a time lag of τ.

Open tank, high side to the tank, low side to atmosphere; $h$ is the liquid height above the tapping point. If the transmitter sits a height $h_0$ below the tapping point (impulse line full of the same liquid), $\Delta P=\rho g(h+h_0)$, and the transmitter is given a zero suppression of $\rho g{h}_0$ so that empty still reads 4 mA.

$A$ is the displacer's uniform cross-sectional area and $h$ the submerged length (partly submerged); the buoyancy of the vapour above the liquid is neglected. Apparent weight $=W-A\rho gh$, where $W$ is the displacer's weight in air.

Insulating liquid, $0\le h\le L$, gas above the liquid taken as ${\epsilon }_r=1$, end (fringing) effects neglected. Capacitance is linear in $h$; at $h=0$ it is the empty value $2\pi {\epsilon }_{0}L/\ln (b/a)$.

$t$ is the round-trip time and $v$ the wave speed in the space above the liquid: speed of sound for ultrasonic, speed of light for radar.

$R=8.314$ J/(mol K), $F=96485$ C/mol, $T$ in kelvin. $E_0$ is the potential at pH 0 and combines the reference and internal potentials. An ideal electrode with its isopotential point at pH 7 and zero offset gives $E=S(7-\mathrm{pH})$, that is, 0 mV at pH 7.

$T$ in kelvin. At 25 °C ($298.15$ K) the slope is about 59.16 mV per pH; at 0 °C about 54.2 mV; at 50 °C about 64.1 mV.

$K$ in m${}^{-1}$, $R$ the measured cell resistance in ohms, $\kappa$ in S/m. Unit conversion: 1 S/m $=10^4\mu$S/cm.

Steady, fully developed laminar flow of a Newtonian liquid in a tube of radius $r$ and length $L$ under pressure drop $\Delta P$; $Q$ is the volume flow rate.

Sphere radius $r$ and density ${\rho }_s$ falling at constant terminal speed $v_t$ in an unbounded Newtonian liquid of density ${\rho }_f$; valid for small Reynolds number (${\rho }_f{v}_t(2r)/\mu$ below about 1, creeping flow). Near a container wall the sphere falls slower, so a wall correction is needed.

Linear transmitter, 4 mA at LRV and 20 mA at URV; I in mA.

Inverse of the relation above; valid for 4 mA to 20 mA.

0 % at 4 mA, 50 % at 12 mA, 100 % at 20 mA.

A 250 Ω receiver turns 4 mA to 20 mA into 1 V to 5 V.

I_max = 20 mA unless an over-range current is stated; all series resistances are included.

R_L is wire plus receiver resistance in series; with V in volts and I in amperes, R is in ohms.

Head-type primary element, turbulent flow, constant fluid density; Q_max is the flow at full-scale differential pressure.

Same conditions as the flow law; with a square-root extractor the relation becomes linear, I = 4 + 16 Q/Q_max mA.

The emitted wavelength of an LED is set by the band gap, so $E\approx E_g$. SI: $h=6.63\times 10^{-34}$ J s, $c=3\times 10^8$ m/s, $1\text{eV}=1.6\times 10^{-19}$ J.

$\Delta \lambda$ is the spectral width of the source (order-of-magnitude relation). Clear interference fringes need the path difference to be smaller than $L_c$.

Light of wavelength longer than ${\lambda }_c$ is not absorbed and gives no photocurrent. Silicon ($E_g\approx 1.12$ eV): ${\lambda }_c\approx 1.1\mu$m.

For a detector without internal gain, $0\le \eta \le 1$. Avalanche gain is counted separately.

$I_p$ is the photocurrent and $P$ the incident optical power. Applies for $\lambda <{\lambda }_c$; responsivity falls to zero beyond the cut-off. $M$ is the multiplication factor of an avalanche photodiode ($M=1$ for an ordinary photodiode).

The photocurrent follows optical power, not field amplitude, so the electrical power in the load $R_L$ varies as the square of the optical power. The bar denotes an average over many optical cycles.

$N$ fringes counted; $\lambda$ is the wavelength in the medium of the path; the mirror moves along the beam (Michelson, light passes twice).

A transparent plate of index $n$ and thickness $t$ (or a gas cell of length $t$ whose index changes from 1 to $n$) placed normal to the beam in one arm shifts $N$ fringes. Michelson double pass, surrounding medium of index 1, $\lambda$ the vacuum wavelength.

Step-index fibre. $n_1$ core index, $n_2$ cladding index, $n_0$ index of the outside medium (1 for air); ${\theta }_{\max }$ is the acceptance half-angle measured from the fibre axis; ${\theta }_c$ is the critical angle at the core-cladding boundary.

$a$ is the core radius. Step-index fibre.

$n_{\text{eff}}$ is the effective index of the guided mode and $\Lambda$ the grating period. Strain or temperature changes $\Lambda$ and $n_{\text{eff}}$, which shifts the reflected wavelength ${\lambda }_B$.

$\epsilon$ is the molar absorptivity at the chosen wavelength (usually L mol${}^{-1}$ cm${}^{-1}$), $c$ the concentration (mol/L) and $l$ the path length (cm). Holds for a dilute, non-scattering solution, monochromatic light, and no association or reaction of the absorbing species.

$T$ as a fraction; $T\%$ is the percentage transmittance.

The ion starts from rest, is accelerated through potential $V$, then moves on a circle of radius $r$ in a uniform magnetic field $B$ perpendicular to its velocity. SI units.

The ion starts from rest, is accelerated through $V$ and drifts a field-free length $L$. Flight time grows as $\sqrt{m/q}$.