Formula sheet

Biomedical Instrumentation Formula Sheet

Biomedical instrumentation formulas for sensitivity, SNR, CMRR, sampling, quantization, filters, bridges, photodiodes, leakage current, uncertainty, and validation.

Branch: Biomedical Engineering
Content: Formula sheet
Updated: May 29, 2026
Revision: v1.0.10 · reviewed

This formula sheet collects common first-pass calculations used in biomedical signal acquisition and instrumentation. It is intended for engineering screening, design review, calibration planning, and test interpretation. Medical-device design also requires risk management, applicable standards, regulatory evidence, clinical validation, and intended-use controls.

State the measurand, body interface, bandwidth, sampling rate, operating range, safety boundary, and intended use before applying the equations.

Transducer sensitivity

Sensitivity:

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

where $x$ is the measurand and $y$ is the output.

Output estimate:

y=y_0+Sx

where $y_0$ is offset.

Percent nonlinearity can be estimated as:

\displaystyle NL=\frac{y_{actual}-y_{fit}}{y_{FS}}\times 100\%

where $y_{FS}$ is full-scale output.

Sensitivity is not total accuracy. Drift, hysteresis, mounting, temperature, noise, cross-sensitivity, and calibration error must be included.

Signal-to-noise ratio

Power ratio:

\displaystyle SNR=\frac{P_{signal}}{P_{noise}}

Decibel form for power:

SNR_{dB}=10\log_{10}(SNR)

For equal-impedance voltage ratios:

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

SNR must state bandwidth and measurement location. Integrated noise usually changes when bandwidth changes.

Differential Gain and CMRR

Instrumentation amplifier output:

V_{out}=G(V_+-V_-)+V_{ref}

Common-mode rejection ratio:

\displaystyle CMRR=\frac{A_d}{A_{cm}}

Decibel form:

CMRR_{dB}=20\log_{10}(CMRR)

Input-referred common-mode error, screening approximation:

\displaystyle V_{err,cm}\approx \frac{V_{cm}}{CMRR}

Input bias current error:

V_{err,bias}=I_bR_{source}

Biomedical front ends should check electrode impedance imbalance, cable motion, common-mode voltage, shielding, driven-reference circuits, isolation, and amplifier saturation.

Root-mean-square signal level

RMS value of samples:

\displaystyle x_{rms}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}x_i^2}

Mean value:

\displaystyle \bar{x}=\frac{1}{N}\sum_{i=1}^{N}x_i

Standard deviation:

\displaystyle s=\sqrt{\frac{1}{N-1}\sum_{i=1}^{N}(x_i-\bar{x})^2}

For noise around a zero-mean signal, RMS and standard deviation are often closely related. State whether offset has been removed.

Sampling

Nyquist screening condition:

f_s>2f_{max}

Sampling interval:

\displaystyle \Delta t=\frac{1}{f_s}

Record duration:

T=N\Delta t

Frequency resolution for an $N$ -point record:

\displaystyle \Delta f=\frac{f_s}{N}=\frac{1}{T}

Practical systems need margin beyond the ideal Nyquist condition because filters have finite roll-off and physiological features may require timing precision.

Quantization

ADC step size:

\displaystyle \Delta=\frac{V_{FS}}{2^N}

where $V_{FS}$ is full-scale input range and $N$ is number of bits.

Ideal quantization noise RMS:

\displaystyle q_{rms}=\frac{\Delta}{\sqrt{12}}

Ideal ADC SNR for a full-scale sine wave:

SNR_{dB}\approx 6.02N+1.76

This ideal value excludes analog noise, distortion, jitter, reference error, nonlinearity, and input-range mismatch.

Low-pass filter

First-order low-pass cutoff frequency:

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

Magnitude response:

\displaystyle |H(f)|=\frac{1}{\sqrt{1+(f/f_c)^2}}

Time constant:

\tau=RC

Rise time estimate for a first-order response:

t_r\approx 2.2\tau

Filtering can reduce noise but can also distort timing, amplitude, and waveform morphology.

Bridge sensors

Strain-gauge relation:

\displaystyle GF=\frac{\Delta R/R}{\epsilon}

so:

\displaystyle \frac{\Delta R}{R}=GF\,\epsilon

For a small-change quarter-bridge approximation:

\displaystyle V_o\approx \frac{V_{ex}}{4}\frac{\Delta R}{R}

where $V_{ex}$ is bridge excitation.

Bridge output depends on bridge configuration, temperature compensation, lead resistance, excitation stability, gauge placement, and mechanical loading.

Photodiode signal

Photodiode current:

I_p=R_\lambda P_{opt}

where $R_\lambda$ is responsivity and $P_{opt}$ is optical power.

Transimpedance output:

V_o=-I_pR_f

where $R_f$ is feedback resistance.

Optical biomedical sensors must also account for ambient light, tissue scattering, motion, perfusion, skin tone, path length, detector saturation, and source stability.

Temperature sensors

Thermocouple voltage is commonly approximated locally as:

V\approx S_T(T_{hot}-T_{ref})

where $S_T$ is Seebeck coefficient over the local range.

Thermal response can often be screened as a first-order system:

T(t)=T_\infty+(T_0-T_\infty)e^{-t/\tau}

Sensor temperature may lag tissue or fluid temperature because of thermal mass, contact, perfusion, airflow, insulation, and mounting.

Leakage current

Ohm’s law screening:

\displaystyle I=\frac{V}{R}

Power dissipated:

\displaystyle P=VI=I^2R=\frac{V^2}{R}

Leakage-current safety evaluation must follow the applicable device standard, patient connection type, isolation design, frequency content, fault condition, and measurement network. Simple Ohm’s law is only an engineering screen.

Timing and latency

End-to-end acquisition delay:

T_{end}=T_{sensor}+T_{filter}+T_{ADC}+T_{processing}+T_{display}+T_{communication}

Sampling-related delay for block processing:

\displaystyle T_{block}=\frac{N}{f_s}

Latency matters for alarms, closed-loop control, synchronization, and event timing. Report average and worst-case timing when the system is time-critical.

Error budget

Independent uncertainty combination:

u_c=\sqrt{\sum_i u_i^2}

Expanded uncertainty:

U=ku_c

where $k$ is coverage factor.

Relative error:

\displaystyle e_r=\frac{x_{measured}-x_{reference}}{x_{reference}}\times 100\%

Biomedical uncertainty should include sensor, calibration, repeatability, drift, noise, resolution, environment, body interface, and algorithmic effects.

Calibration and Agreement

Linear calibration model:

y=Sx+b

Estimated measurand:

\displaystyle \hat{x}=\frac{y-b}{S}

Bias against a reference:

\displaystyle Bias=\frac{1}{N}\sum_{i=1}^{N}(x_i-r_i)

Difference standard deviation:

\displaystyle s_d=\sqrt{\frac{1}{N-1}\sum_{i=1}^{N}(d_i-\bar{d})^2}

Agreement limits, screening form:

LOA=\bar{d}\pm 1.96s_d

Calibration evidence should state the reference method, range, population or sample type, environmental conditions, repeated measurements, drift interval, and acceptance limits.

Alarm and Threshold Screening

Z-score:

\displaystyle z=\frac{x-\mu}{\sigma}

Upper threshold from baseline statistics:

x_{high}=\mu+z\sigma

Lower threshold:

x_{low}=\mu-z\sigma

Alarm rate estimate:

\displaystyle R_{alarm}=\frac{N_{alarms}}{T_{monitoring}}

Thresholds should be evaluated against false alarms, missed events, latency, sensor dropout, patient variability, clinical workflow, and intended-use risk.

Reliability screening

If independent component availabilities are $A_i$ , series availability is:

A_{series}=\prod_i A_i

For two independent redundant paths:

A_{parallel}=1-(1-A_1)(1-A_2)

This is a simplified screen. Common-cause failures, shared software, shared power, sensor placement, maintenance, and user workflow can dominate real reliability.

Validation metrics

Mean error:

\displaystyle ME=\frac{1}{N}\sum_{i=1}^{N}(x_i-r_i)

Mean absolute error:

\displaystyle MAE=\frac{1}{N}\sum_{i=1}^{N}|x_i-r_i|

Root-mean-square error:

\displaystyle RMSE=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_i-r_i)^2}

where $x_i$ is device output and $r_i$ is reference value.

Validation metrics must be interpreted against intended use, reference uncertainty, population range, operating conditions, and safety risk.

Practical checklist

Use these formulas with a short biomedical instrumentation checklist:

Define measurand, intended use, body interface, and operating environment.
Check transducer range, sensitivity, bandwidth, drift, and failure modes.
Set gain, filtering, isolation, and ADC range from the expected signal and artefacts.
Check sampling rate, quantization, timing, and data integrity.
Build a full uncertainty and error budget.
Test noise, motion artefacts, saturation, disconnection, and fault conditions.
Validate with appropriate references and intended-use conditions.

The formulas support engineering review. They do not by themselves establish clinical suitability, regulatory compliance, or safety for patient use.

REF

Disciplines