Formula sheet

Power and Sensor Interface Formula Sheet

Power and sensor interface formulas for power budgets, converter duty cycle, flyback energy, H-bridge power, current sensing, ADCs, EMI, and thermal limits.

Branch: Electronic Engineering
Content: Formula sheet
Updated: May 29, 2026
Revision: v1.0.9 · reviewed

This formula sheet collects first-pass relationships for power supplies, switching interfaces, sensor front ends, error budgets, thermal checks, and validation. Use it with stated operating points, tolerances, temperature range, cable conditions, layout assumptions, and fault cases.

Power and Energy

Electrical power:

P=VI

Resistive loss:

P=I^2R

Energy over time:

E=Pt

Efficiency:

\displaystyle \eta=\frac{P_{out}}{P_{in}}

Power loss:

P_{loss}=P_{in}-P_{out}

Check peak, RMS, transient, and continuous values separately.

Power Budget and Startup

Total rail current:

I_{rail}=\sum_i I_i

Rail power:

P_{rail}=V_{rail}I_{rail}

Total supply power:

P_{total}=\sum_i P_{rail,i}

Capacitor hold-up estimate:

\displaystyle C\approx \frac{2P\Delta t}{V_{start}^2-V_{end}^2}

Capacitive inrush current:

\displaystyle I_{inrush}=C\frac{dV}{dt}

Power budgets should separate startup, steady-state, sleep, peak load, sensor excitation, communication bursts, and fault cases.

Linear Regulator Loss

Linear regulator dissipation:

P_D=(V_{in}-V_{out})I_{load}

Approximate efficiency:

\displaystyle \eta\approx \frac{V_{out}}{V_{in}}

Minimum input voltage:

V_{in,min}=V_{out}+V_{dropout}

Load regulation:

\displaystyle Load\ Regulation=\frac{\Delta V_{out}}{\Delta I_{load}}

Thermal limit often determines allowable load current.

Switching Regulator Checks

Ideal buck duty cycle:

\displaystyle D\approx \frac{V_{out}}{V_{in}}

Ideal boost duty cycle:

\displaystyle D\approx 1-\frac{V_{in}}{V_{out}}

Inductor ripple current, buck approximation:

\displaystyle \Delta I_L=\frac{(V_{in}-V_{out})D}{Lf_s}

Output capacitor ripple, rough triangular approximation:

\displaystyle \Delta V_C\approx \frac{\Delta I_L}{8f_s C}

Real converters require ESR, ESL, switching loss, control stability, layout, and transient checks.

Flyback Converter

Energy stored in magnetizing inductance:

\displaystyle E_L=\frac{1}{2}L_m I_{pk}^2

Approximate power transfer:

P\approx E_L f_s \eta

Reflected voltage:

\displaystyle V_R=\frac{N_p}{N_s}(V_{out}+V_D)

Primary switch voltage stress, simplified:

V_{SW}\approx V_{in}+V_R+V_{spike}

Leakage inductance, snubbers, insulation, and thermal limits must be checked explicitly.

H-Bridge and Inverter Switching

Average PWM output voltage for a simple unipolar load model:

V_{avg}=D V_{bus}

Load power:

P_{load}=V_{rms}I_{rms}

MOSFET conduction loss:

P_{cond}=I_{rms}^2R_{DS(on)}

Approximate switching loss:

\displaystyle P_{sw}\approx \frac{1}{2}V_{bus}I(t_r+t_f)f_s

Dead-time margin:

t_{dead}>t_{off,max}-t_{on,min}

Check stall current, braking energy, diode recovery, gate drive, short-circuit response, and thermal cycling.

Sensor Sensitivity

Linear sensor model:

y=Sx+b

where $S$ is sensitivity and $b$ is offset.

Measured quantity:

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

Gain error contribution:

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

Offset error contribution:

\displaystyle e_o=\frac{\Delta b}{S}

Sensor calibration should include range, temperature, mounting, excitation, and signal-conditioning chain.

Sensor Bandwidth and Response

First-order sensor response:

y(t)=y_f+(y_0-y_f)e^{-t/\tau}

Approximate 95 percent settling time:

t_{95}\approx 3\tau

Bandwidth from time constant:

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

Measurement bandwidth should be matched to sensor dynamics, anti-alias filtering, expected transient duration, noise level, and controller or logging rate.

Current Sensing and Sensor Excitation

Shunt voltage:

V_{shunt}=IR_{shunt}

Shunt dissipation:

P_{shunt}=I^2R_{shunt}

Amplified current-sense output:

V_{out}=G V_{shunt}+V_{offset}

Ratiometric sensor output:

V_{out}=kV_{exc}x+V_{offset}

Sensor excitation power:

P_{exc}=V_{exc}I_{exc}

Check shunt tolerance, Kelvin routing, common-mode range, amplifier offset, excitation stability, self-heating, and fault current.

Photodiode Interface

Photodiode current:

I_{ph}=R_\lambda P_{opt}

Responsivity from quantum efficiency:

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

Transimpedance amplifier output:

V_{out}=-I_{ph}R_f

Shot noise RMS current over bandwidth:

i_{n,rms}=\sqrt{2qI B}

Equivalent one-sided current noise density:

i_{n,density}=\sqrt{2qI}\ \text{A}/\sqrt{\text{Hz}}

where $B$ is bandwidth. Include dark current, capacitance, op-amp noise, feedback capacitance, leakage, ambient light, and saturation.

Thermocouple Interface

Small-range linearized thermocouple voltage:

V_{TC}\approx S_{TC}(T_{hot}-T_{cold})

Temperature estimate:

\displaystyle T_{hot}\approx T_{cold}+\frac{V_{TC}}{S_{TC}}

Amplified output:

V_{out}=G V_{TC}+V_{offset}

Cold-junction compensation is required because the thermocouple measures a temperature difference, not absolute temperature by itself.

ADC, Sampling, and Quantization

ADC code width:

\displaystyle q=\frac{V_{ref,high}-V_{ref,low}}{2^N}

Ideal input estimate:

V_{in}\approx Code\cdot q+V_{ref,low}

Input-referred quantization noise, RMS approximation:

\displaystyle V_{q,rms}\approx \frac{q}{\sqrt{12}}

Nyquist sampling condition:

f_s \ge 2f_{max}

Ratiometric conversion can cancel excitation drift when the sensor and ADC reference share the same stable source. Anti-alias filtering is still required before sampling.

Operational Amplifier Gain

Non-inverting amplifier:

\displaystyle G=1+\frac{R_f}{R_g}

Inverting amplifier:

\displaystyle G=-\frac{R_f}{R_{in}}

Input bias current error:

V_{error}=I_b R_{source}

Slew-rate requirement:

SR \ge 2\pi f V_{peak}

Check input common-mode range, output swing, gain-bandwidth, stability, noise, and capacitive loading.

Filters and Q-Factor

First-order low-pass cutoff:

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

RC time constant:

\tau=RC

Capacitor impedance magnitude:

\displaystyle |X_C|=\frac{1}{2\pi fC}

Inductor impedance magnitude:

|X_L|=2\pi fL

Resonant frequency:

\displaystyle f_0=\frac{1}{2\pi\sqrt{LC}}

Q-factor:

\displaystyle Q=\frac{f_0}{\Delta f}

High Q can improve selectivity but increase ringing and sensitivity.

Error Budget

Worst-case error sum:

e_{WC}=\sum_i |e_i|

Root-sum-square error:

e_{RSS}=\sqrt{\sum_i e_i^2}

Relative error:

\displaystyle e_{rel}=\frac{|x_{measured}-x_{reference}|}{|x_{reference}|}

Signal-to-noise ratio:

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

State whether errors are systematic, random, bounded, calibrated, temperature-dependent, or drift-related.

Input Protection and Leakage

Series resistor fault current:

\displaystyle I_{fault}\approx \frac{V_{fault}-V_{clamp}}{R_{series}}

Input RC pole:

\displaystyle f_c=\frac{1}{2\pi R_{source}C_{in}}

Leakage-induced voltage error:

V_{leak}=I_{leak}R_{source}

Divider output with load:

\displaystyle V_{out}=V_{in}\frac{R_2\parallel R_L}{R_1+(R_2\parallel R_L)}

Protection parts must be checked for clamp voltage, leakage, capacitance, surge energy, recovery, and how they interact with measurement accuracy.

Junction Temperature

Junction temperature from ambient:

T_J=T_A+P_D\theta_{JA}

Junction temperature from case:

T_J=T_C+P_D\theta_{JC}

Thermal margin:

M_T=T_{J,max}-T_J

Transient temperature rise, first-order approximation:

\Delta T(t)=P_D Z_\theta(t)

Use board-specific thermal data where possible. Datasheet thermal resistance may not match the actual layout.

EMI and Layout Screening

Loop-induced voltage:

\displaystyle V_L=L\frac{di}{dt}

Capacitive coupling current:

\displaystyle I_C=C\frac{dv}{dt}

Decoupling capacitor charge relation:

\displaystyle \Delta V=\frac{\Delta Q}{C}=\frac{I\Delta t}{C}

Switching edge frequency scale:

\displaystyle f_{edge}\sim \frac{1}{t_r}

Fast edges require small current loops, low-inductance decoupling, controlled return paths, and suitable filtering.

Reliability and Validation

Reliability for constant failure rate:

R(t)=e^{-\lambda t}

Mean time between failures:

\displaystyle MTBF=\frac{1}{\lambda}

Load-step output deviation:

\Delta V_{out}=V_{out,max}-V_{out,min}

Pass rate:

\displaystyle P_{pass}=\frac{N_{pass}}{N_{tests}}

Validation should include nominal, corner, transient, fault, thermal, EMI, and long-duration cases.

REF

Disciplines