Formula sheet

Engineering Sensors and Instrumentation Formula Sheet

Sensor formulas for transduction, strain bridges, piezo amplifiers, thermocouples, photodiodes, vacuum/radiation instruments, bandwidth, ADC, limits, and validation.

Branch: Engineering Physics
Content: Formula sheet
Updated: Jul 03, 2026
Revision: v1.1.0 · reviewed

This formula sheet collects first-pass relationships used when a physical effect must become a defensible engineering measurement. It focuses on the installed chain: measurand, transducer, excitation, signal conditioning, filtering, sampling, conversion, calibration and validation evidence.

Use these formulas for sizing, review and plausibility checks. Do not use them as a substitute for a sensor data sheet, calibration certificate, uncertainty budget, safety standard, environmental qualification or installed-system validation.

How to Use This Formula Sheet

Use this sheet by following the full measurement chain, not by selecting a transducer formula in isolation. Start with the decision: detection, control, protection, calibration, trend monitoring, acceptance test or diagnostic investigation. Then define the measurand, range, environment, bandwidth, required uncertainty, failure consequence and allowed response time.

After the decision is clear, work forward from physics to evidence: transduction, mounting, excitation, cable, amplifier, filter, sampler, ADC scaling, software conversion, calibration and validation. A sensor with an excellent data sheet can still fail if the adhesive creeps, the cable adds capacitance, the reference junction is biased, the optical path saturates, the pressure port lags, the radiation detector loses counts or the ADC aliases the event.

Use the family-specific sections for first-pass sizing. Use the bandwidth, sampling and ADC sections when the measurement is dynamic or digital. Use the validation section before releasing any measurement that supports safety, quality, compliance, control tuning, condition monitoring or root-cause analysis.

Scope and Notation

Symbol	Meaning	Typical unit
$x$	measurand	varies
$y$	sensor or conditioned output	V, A, count
$S$	sensitivity	output unit per input unit
$GF$	strain gauge factor	dimensionless
$\varepsilon$	strain	strain or microstrain
$V_{ex}$	bridge excitation voltage	V
$Q$	generated charge	C
$d$	piezoelectric charge coefficient	C/N
$C$	capacitance	F
$C_f$	charge-amplifier feedback capacitance	F
$R_f$	charge-amplifier feedback resistance	ohm
$R_\lambda$	photodiode responsivity	A/W
$P_{opt}$	optical power at detector	W
$B$	measurement bandwidth	Hz
$f_s$	sampling frequency	Hz
$f_c$	filter cutoff frequency	Hz
$N$	ADC resolution	bit
$FS$	full-scale input span	V or engineering unit
$q$	ADC code width	V/count or unit/count
$\lambda$	gas mean free path or optical wavelength, by context	m
$Kn$	Knudsen number	dimensionless

Always state the installation boundary. A laboratory transducer, a bonded sensor, a cable, an amplifier, a filter and a controller input can have different bandwidth, offset, noise and failure modes.

Basis and Validity Limits

The formulas in this sheet are local, first-order engineering models. Linear sensitivity assumes the sensor is operated inside its calibrated range and around a defined operating point. Dynamic formulas assume a defensible bandwidth model. Noise formulas assume that the stated bandwidth and signal convention match the measurement decision. ADC formulas assume that analog range, reference stability, settling, filtering and calibration have already been checked.

Sensor physics is regime-dependent. A strain-gauge bridge depends on bonding, strain field, temperature compensation and lead wiring. A piezoelectric sensor is suited to dynamic events, not static hold measurements. A thermocouple relation is local unless tables or polynomial fits are used. Photodiode responsivity depends on wavelength, bias, temperature and optical geometry. Radiation and vacuum instruments depend strongly on energy spectrum, gas species, pressure range, detector model and geometry.

Installed validation is part of the model boundary. The formulas do not cover all error sources from cable motion, grounding, common-mode voltage, EMI, humidity, contamination, thermal gradients, mechanical resonance, pressure conductance, software scaling, drift, human setup error or maintenance state. If those effects can change the decision, the calculation should be labelled a screen until an installed check closes the gap.

Linear Sensor Chain

Small-signal sensitivity:

\displaystyle S=\frac{\Delta y}{\Delta x}

For a nonlinear sensor around an operating point:

\displaystyle S(x_0)=\left.\frac{dy}{dx}\right|_{x=x_0}

Output change from measurand change:

\Delta y\approx S\Delta x

Conditioned output with gain and offset:

V_{out}=G(Sx+V_0)

Engineering value recovered from a linear calibration:

\displaystyle \hat{x}=\frac{V_{out}/G-V_0}{S}

Use

Use a linear chain only over the calibrated range. State temperature, frequency, bias, bridge excitation, optical wavelength, pressure regime, loading, mounting and any software scaling.

Common Mistakes

Reporting a sensitivity without the operating point or bandwidth.
Calibrating the transducer alone when the installed cable, adhesive, connector, amplifier or filter dominates the error.
Treating a display value as a physical measurement without checking scaling and units.

Minimum Detectable Measurand

Input-referred noise:

\displaystyle n_x=\frac{n_y}{|S|}

Minimum detectable change for a required signal-to-noise ratio:

\displaystyle \Delta x_{min}=\frac{SNR_{min} n_y}{|S|}

Amplitude signal-to-noise ratio:

\displaystyle SNR=\frac{A_{signal,rms}}{A_{noise,rms}}

Amplitude signal-to-noise ratio in decibels:

\displaystyle SNR_{dB}=20\log_{10}\left(\frac{A_{signal,rms}}{A_{noise,rms}}\right)

Use

The noise bandwidth must be the same bandwidth used for the measurement decision. A sensor that looks adequate after heavy filtering may be unusable for a transient event.

Strain Gauge and Wheatstone Bridge

Gauge resistance change:

\displaystyle \frac{\Delta R}{R}=GF\varepsilon

Quarter-bridge small-strain output:

\displaystyle \frac{V_o}{V_{ex}}\approx\frac{GF\varepsilon}{4}

Half-bridge small-strain output with one tensile and one compressive active gauge:

\displaystyle \frac{V_o}{V_{ex}}\approx\frac{GF\varepsilon}{2}

Full-bridge small-strain output with four active gauges:

\displaystyle \frac{V_o}{V_{ex}}\approx GF\varepsilon

Bridge signal after instrumentation amplifier:

V_{amp}=G_{amp}V_o

Gauge self-heating power:

\displaystyle P_g=\frac{V_g^2}{R_g}

where $V_g$ is the voltage across the gauge element. Self-heating depends on bridge wiring and excitation.

Use

Bridge formulas assume small strain, matched arms and correct sign convention. Installed error can come from lead-wire resistance, adhesive creep, temperature compensation, transverse sensitivity, bending gradients and amplifier common-mode range.

Piezoelectric Sensor and Charge Amplifier

Generated charge:

Q=dF

Open-circuit voltage estimate:

\displaystyle V=\frac{Q}{C_s+C_c+C_{in}}

Charge-amplifier output:

\displaystyle V_{out}\approx-\frac{Q}{C_f}

Charge-amplifier low-frequency cutoff:

\displaystyle f_{L}=\frac{1}{2\pi R_f C_f}

Capacitive reactance magnitude:

\displaystyle |X_C|=\frac{1}{2\pi f C}

Use

Piezoelectric sensors are strong for dynamic force, pressure and acceleration. They are weak for true static load because leakage, insulation resistance and amplifier bias currents discharge the signal.

Common Mistakes

Using a voltage-mode piezoelectric calculation while ignoring cable capacitance.
Selecting $R_fC_f$ too small, causing low-frequency droop.
Interpreting saturation or recovery time as a mechanical event.

Thermocouple and Seebeck Measurement

Local linear thermocouple voltage approximation:

V_{tc}\approx S_{AB}(T_h-T_{ref})

Hot-junction estimate from measured voltage:

\displaystyle T_h\approx T_{ref}+\frac{V_{tc}}{S_{AB}}

Reference-junction temperature error propagated into temperature error:

\displaystyle \Delta T_h\approx\Delta T_{ref}+\frac{\Delta V}{S_{AB}}

Thermal lag as first-order response:

T_m(t)=T_f+(T_0-T_f)e^{-t/\tau}

Response fraction after time $t$ :

\displaystyle \frac{T_m(t)-T_0}{T_f-T_0}=1-e^{-t/\tau}

Use

The linear Seebeck relation is only a local approximation. Production thermocouple work uses tables or polynomial fits, correct wire types, cold-junction compensation and installed thermal-contact checks.

Photodiode, Quantum Efficiency and Transimpedance

Photon energy:

\displaystyle E_{ph}=\frac{hc}{\lambda}

Photodiode responsivity from quantum efficiency:

\displaystyle R_\lambda=\eta_q\frac{q_e\lambda}{hc}

Photocurrent:

I_p=R_\lambda P_{opt}

Transimpedance amplifier output:

V_{out}\approx -I_p R_f

Shot-noise current over bandwidth:

i_{shot,rms}=\sqrt{2q_e I B}

Approximate transimpedance bandwidth dominated by total input capacitance:

\displaystyle f_c\approx\frac{1}{2\pi R_f C_{tot}}

Use

State wavelength, bias, optical geometry, bandwidth, dark current, background light, detector area, amplifier noise, saturation margin and calibration source. Responsivity is not a universal constant.

Radiation Instrument Screening

Absorbed dose:

\displaystyle D=\frac{E}{m}

Dose rate:

\displaystyle \dot{D}=\frac{D}{t}

Equivalent dose:

H=w_R D

Nonparalyzable dead-time correction:

\displaystyle R_{true}\approx\frac{R_m}{1-R_m\tau_d}

where $R_m$ is measured count rate and $\tau_d$ is detector dead time.

Use

Use the dead-time correction only when $R_m\tau_d<1$ and the detector behavior matches the nonparalyzable model. Radiation measurements require geometry, energy spectrum, shielding, detector calibration and the reported dose quantity.

Vacuum Sensor and Rarefied-Gas Check

Mean free path for a simple gas model:

\displaystyle \lambda=\frac{k_BT}{\sqrt{2}\pi d_m^2p}

Knudsen number:

\displaystyle Kn=\frac{\lambda}{L}

Pressure conversion:

1\ \text{Pa}=7.5006\times10^{-3}\ \text{Torr}

Leak throughput:

Q_L=pS_p

where $S_p$ is pumping speed at the relevant pressure and location.

Use

Gauge readings depend on gas species, gauge type, location, conductance, outgassing, contamination and pressure range. A gauge mounted far from the process volume may report pump inlet pressure rather than chamber pressure.

Bandwidth, Time Constant and Sampling

First-order cutoff and time constant:

\displaystyle f_c=\frac{1}{2\pi\tau}

Rise time approximation for a first-order system:

t_{10-90}\approx2.2\tau

Nyquist condition for a band-limited signal:

f_s>2B

Practical sampling margin:

f_s\ge M_sB

where $M_s$ is commonly greater than 5 for measurement review when filtering, phase and transient interpretation matter.

Record length:

\displaystyle T_{rec}=\frac{N_s}{f_s}

Frequency spacing in an FFT record:

\displaystyle \Delta f=\frac{1}{T_{rec}}

Use

Sampling rate, anti-alias filter, sensor bandwidth and record length must be selected together. Sampling fast does not repair an aliased signal that entered the ADC before filtering.

ADC Scaling and Quantization

Code width:

\displaystyle q=\frac{FS}{2^N}

Input estimate from ADC count:

V_{in}\approx V_{min}+qC_{ADC}

Quantization RMS noise approximation:

\displaystyle v_{q,rms}=\frac{q}{\sqrt{12}}

Engineering-unit resolution after linear scaling:

\displaystyle x_q=\frac{q}{G|S|}

Ideal ADC signal-to-quantization-noise ratio for a full-scale sine:

SNR_q\approx6.02N+1.76\ \text{dB}

Use

ADC resolution matters only after input range, reference accuracy, analog noise, settling, source impedance, timing jitter, filter delay and calibration uncertainty have been checked.

Installed Validation Checks

Sensor chain release should include:

Check	Formula or evidence	Engineering purpose
Signal magnitude	$\Delta y=S\Delta x$	Confirms measurable output
Noise margin	$SNR=A_s/A_n$	Confirms decision confidence
Bandwidth	$f_c=1/(2\pi\tau)$	Confirms dynamic response
Sampling	$f_s>2B$ with margin	Prevents aliasing
Scaling	$\hat{x}=(V/G-V_0)/S$	Confirms units and conversion
Saturation	compare $V_{out}$ with input/output limits	Protects against clipped evidence
Installation	calibration or check in final mounting	Captures cables, adhesive, optics, pressure conductance and thermal paths
Failure modes	open circuit, short, drift, saturation, bias, noise, environmental coupling	Prevents plausible but wrong measurements

Common Formula Mistakes

The most common mistake is validating the sensor element while ignoring the installed chain. A transducer sensitivity does not include adhesive, bracket compliance, cable capacitance, shielding, amplifier common-mode range, filter phase delay, sampling clock, ADC reference, software scaling or operator setup unless those items were included in the calibration or validation method.

Do not use static formulas for dynamic measurements without checking bandwidth and phase. A thermocouple, pressure port, strain bridge, accelerometer, photodiode amplifier or vacuum gauge can show the correct steady value while missing a transient, phase-shifting a control signal or aliasing a vibration component into a false low-frequency feature.

Do not confuse resolution with uncertainty. A high-bit ADC, small display increment or clean digital trend does not prove accuracy. Calibration uncertainty, drift, repeatability, linearity, hysteresis, noise floor, environmental sensitivity and installation bias can dominate the error budget.

Do not treat validation as a single pass/fail reading. A defensible sensor decision should link the formula result to calibration status, installed check, known input or reference comparison, dynamic response, failure diagnostics and uncertainty margin. If the measurement supports safety or release, the evidence must show what happens when the sensor is wrong.

Worked Example 1: Strain Gauge Bridge Signal

A foil strain gauge is bonded to a bracket. Use:

GF=2.1

\varepsilon=400\ \mu\varepsilon=400\times10^{-6}

V_{ex}=5.0\ \text{V}

The gauge is in a quarter bridge. Estimate bridge output and the amplifier output for:

G_{amp}=800

Bridge ratio:

\displaystyle \frac{V_o}{V_{ex}}\approx\frac{GF\varepsilon}{4}=\frac{2.1(400\times10^{-6})}{4}=2.10\times10^{-4}

Bridge output:

V_o=(5.0)(2.10\times10^{-4})=1.05\times10^{-3}\ \text{V}=1.05\ \text{mV}

Amplifier output:

V_{amp}=800(1.05\ \text{mV})=0.84\ \text{V}

Engineering comment: the signal is large enough for ordinary data acquisition after amplification, but the review is incomplete without bridge completion tolerance, lead-wire compensation, temperature drift, amplifier common-mode range and a shunt-calibration check.

Worked Example 2: Piezoelectric Charge Amplifier

A piezoelectric force sensor has:

d=250\ \text{pC/N}

The expected dynamic force is:

F=8.0\ \text{N}

The charge amplifier uses:

C_f=5.0\ \text{nF}

Generated charge:

Q=dF=(250\ \text{pC/N})(8.0\ \text{N})=2000\ \text{pC}=2.0\ \text{nC}

Charge-amplifier output magnitude:

\displaystyle |V_{out}|=\frac{Q}{C_f}=\frac{2.0\times10^{-9}}{5.0\times10^{-9}}=0.40\ \text{V}

If:

R_f=100\ \text{M}\Omega

then the low-frequency cutoff is:

\displaystyle f_L=\frac{1}{2\pi R_f C_f}=\frac{1}{2\pi(100\times10^{6})(5.0\times10^{-9})}=0.318\ \text{Hz}

Engineering comment: the amplitude is comfortable, and the low-frequency cutoff is suitable for dynamic events above a few hertz. It is not suitable for static force hold measurement; leakage and amplifier recovery must be validated.

Worked Example 3: Thermocouple Cold-Junction Effect

A type-like thermocouple is approximated locally by:

S_{AB}=41\ \mu\text{V/K}

The hot junction is at:

T_h=300^\circ\text{C}

The reference junction is at:

T_{ref}=30^\circ\text{C}

Voltage estimate:

V_{tc}=S_{AB}(T_h-T_{ref})=(41\ \mu\text{V/K})(270\ \text{K})=11070\ \mu\text{V}=11.07\ \text{mV}

If the cold-junction sensor is biased by:

\Delta T_{ref}=2.0\ \text{K}

the indicated hot-junction temperature shifts by approximately:

\Delta T_h\approx2.0\ \text{K}

Equivalent voltage error:

\Delta V=S_{AB}\Delta T_{ref}=(41\ \mu\text{V/K})(2.0\ \text{K})=82\ \mu\text{V}

Engineering comment: an 82 microvolt error is small electrically but significant thermally. The installation needs cold-junction measurement, correct extension wire, connector gradient control and a known-temperature check.

Worked Example 4: Photodiode Signal and Shot Noise

A photodiode receives:

P_{opt}=80\ \mu\text{W}

At the wavelength of interest, responsivity is:

R_\lambda=0.55\ \text{A/W}

Photocurrent:

I_p=R_\lambda P_{opt}=(0.55)(80\times10^{-6})=44\ \mu\text{A}

With a transimpedance resistor:

R_f=20\ \text{k}\Omega

output magnitude is:

|V_{out}|=I_pR_f=(44\times10^{-6})(20\times10^3)=0.88\ \text{V}

For current:

I=44\ \mu\text{A}

and bandwidth:

B=10\ \text{kHz}

shot-noise current is:

i_{shot,rms}=\sqrt{2q_eIB}=\sqrt{2(1.602\times10^{-19})(44\times10^{-6})(10^4)}=3.75\times10^{-10}\ \text{A}

Current-only shot-noise signal-to-noise ratio:

\displaystyle SNR=\frac{44\times10^{-6}}{3.75\times10^{-10}}=1.17\times10^5

SNR_{dB}=20\log_{10}(1.17\times10^5)=101\ \text{dB}

Engineering comment: this is only a lower-bound noise screen. Real performance may be set by dark current, background light, op-amp current noise, resistor noise, capacitance, bandwidth compensation, optical alignment, contamination or ADC range.

Worked Example 5: Vacuum Gauge Regime Check

A vacuum instrument has a characteristic gap:

L=2.0\ \text{mm}=2.0\times10^{-3}\ \text{m}

Air is approximated with molecular diameter:

d_m=0.37\ \text{nm}=0.37\times10^{-9}\ \text{m}

At:

T=293\ \text{K}

and pressure:

p=10\ \text{Pa}

mean free path is:

\displaystyle \lambda=\frac{k_BT}{\sqrt{2}\pi d_m^2p}

\displaystyle \lambda=\frac{(1.381\times10^{-23})(293)}{\sqrt{2}\pi(0.37\times10^{-9})^2(10)}=6.65\times10^{-4}\ \text{m}=0.665\ \text{mm}

Knudsen number:

\displaystyle Kn=\frac{\lambda}{L}=\frac{0.665}{2.0}=0.333

Engineering comment: $Kn=0.333$ is in a transitional regime. A continuum assumption and a gauge calibration from a different pressure range may be misleading. Gauge location, gas species and conductance between chamber and pump should be checked.

Worked Example 6: ADC Resolution for a Conditioned Pressure Sensor

A conditioned sensor output spans:

0\ \text{V}\ \text{to}\ 5.0\ \text{V}

for:

0\ \text{bar}\ \text{to}\ 100\ \text{bar}

The ADC has:

N=16\ \text{bit}

Voltage code width:

\displaystyle q_V=\frac{5.0}{2^{16}}=7.63\times10^{-5}\ \text{V}=76.3\ \mu\text{V}

Conditioned sensitivity:

\displaystyle S_V=\frac{5.0\ \text{V}}{100\ \text{bar}}=0.050\ \text{V/bar}

Engineering-unit code width:

\displaystyle q_p=\frac{q_V}{S_V}=\frac{7.63\times10^{-5}}{0.050}=1.53\times10^{-3}\ \text{bar}

Quantization RMS pressure noise:

\displaystyle p_{q,rms}=\frac{q_p}{\sqrt{12}}=4.41\times10^{-4}\ \text{bar}

Engineering comment: the ADC is probably not the limiting element for most industrial pressure decisions. Calibration uncertainty, temperature drift, pressure port dynamics, overpressure history and electrical noise should be checked before increasing bit depth.

Validation Evidence Package

Before accepting a sensor chain, assemble a validation evidence package that proves the measurement is fit for its decision. The package should identify the measurand, sensor physics, installed boundary, calculation assumptions and independent checks.

Include:

The measurand, range, environment and decision threshold are explicit.
Sensor physics is valid in the installed temperature, pressure, wavelength, strain, frequency or radiation regime.
Signal magnitude, noise, bandwidth, sampling and ADC range are calculated together.
Saturation, aliasing, thermal lag, leakage, cable loading, electromagnetic interference and common-mode limits are checked.
Calibration covers the installed chain or a justified transfer path from laboratory calibration to installed use.
The validation evidence matches the decision risk: reference check, shunt calibration, known-temperature check, optical calibration, leak check, source check, dynamic comparison or uncertainty budget.
Failure modes produce detectable diagnostics or conservative decisions.

Also include calibration certificate identifiers or traceability statements, as-found/as-left status where relevant, environmental conditions, cable and connector configuration, software scaling revision, sampling/filter settings, reference standard used, acceptance limits, uncertainty or guard band, deviations from the intended setup and the retest interval. For safety or release measurements, record the action taken on open circuit, short circuit, saturation, stale value, communication loss and out-of-range diagnostics.

REF

Disciplines

Engineering Sensors and Instrumentation Formula Sheet

How to Use This Formula Sheet

Scope and Notation

Basis and Validity Limits

Linear Sensor Chain

Use

Common Mistakes

Minimum Detectable Measurand

Use

Strain Gauge and Wheatstone Bridge

Use

Piezoelectric Sensor and Charge Amplifier

Use

Common Mistakes

Thermocouple and Seebeck Measurement

Use

Photodiode, Quantum Efficiency and Transimpedance

Use

Radiation Instrument Screening

Use

Vacuum Sensor and Rarefied-Gas Check

Use

Bandwidth, Time Constant and Sampling

Use

ADC Scaling and Quantization

Use

Installed Validation Checks

Common Formula Mistakes

Worked Example 1: Strain Gauge Bridge Signal

Worked Example 2: Piezoelectric Charge Amplifier

Worked Example 3: Thermocouple Cold-Junction Effect

Worked Example 4: Photodiode Signal and Shot Noise

Worked Example 5: Vacuum Gauge Regime Check

Worked Example 6: ADC Resolution for a Conditioned Pressure Sensor

Validation Evidence Package

See also