Formula sheet

Thermoelectric and Piezoelectric Devices Formula Sheet

Thermoelectric and piezoelectric formulas for Seebeck voltage, Peltier cooling, ZT, TEG matching, piezo charge, charge amplifiers, actuators, resonance, and validation.

Branch: Engineering Physics
Content: Formula sheet
Updated: Jun 23, 2026
Revision: v1.0.0 · reviewed

This formula sheet collects calculation relationships used when thermoelectric and piezoelectric effects must become working engineering devices. It focuses on device sizing, signal magnitude, power balance, interface limits, resonance checks and validation margins.

Use these formulas for screening, review and plausibility checks. Final design still requires material data over temperature, manufacturer limits, calibration evidence, installation-specific thermal and mechanical boundary conditions, environmental qualification and safety review where the device supports a critical decision.

Scope and Sign Conventions

Thermoelectric devices couple heat flow, electric current and temperature difference. Piezoelectric devices couple mechanical stress, electric field, charge and strain. Both are sensitive to boundary conditions, so the same active material can produce different results after packaging, bonding, preload, cable capacitance or heat-sink selection changes.

The formulas below use these conventions unless stated otherwise:

$T_h$ is hot-side absolute temperature in kelvin.
$T_c$ is cold-side absolute temperature in kelvin.
$\Delta T=T_h-T_c$ is positive when the hot side is warmer.
$S$ is the effective Seebeck coefficient in V/K.
$\alpha$ is the thermoelectric module Seebeck coefficient used in heat-pumping equations in V/K.
$R$ is electrical resistance in ohm.
$K$ is thermal conductance in W/K.
$d$ is an effective piezoelectric charge coefficient in C/N.
$C_p$ is piezoelectric element capacitance.
$C_f$ and $R_f$ are charge-amplifier feedback capacitance and resistance.

Core Symbols

Symbol	Meaning	Typical unit
$V_{oc}$	open-circuit thermoelectric voltage	V
$V_L$	load voltage	V
$I$	current	A
$R_i$	internal electrical resistance of module or source	ohm
$R_L$	load resistance	ohm
$P_L$	electrical load power	W
$Q_c$	heat absorbed at thermoelectric cold side	W
$Q_h$	heat rejected at thermoelectric hot side	W
$P_e$	electrical input power	W
$COP$	coefficient of performance for cooling	dimensionless
$ZT$	dimensionless thermoelectric figure of merit	dimensionless
$\sigma$	electrical conductivity	S/m
$\rho$	electrical resistivity	ohm m
$k$	thermal conductivity	W/(m K)
$R_{th}$	thermal resistance	K/W
$Q$	piezoelectric charge or heat rate, by context	C or W
$F$	force	N
$V_p$	piezoelectric element voltage	V
$E$	electric field	V/m
$\delta$	displacement	m
$k_m$	mechanical stiffness	N/m
$f_n$	natural frequency	Hz
$Q_m$	mechanical quality factor	dimensionless

Seebeck Voltage

For two thermoelectric materials with temperature-dependent Seebeck difference:

V_{oc}=\int_{T_c}^{T_h} S_{ab}(T)\,dT

For a narrow temperature range:

V_{oc}\approx S_{ab}\Delta T

For a thermocouple measurement with a local linear sensitivity:

\displaystyle T_{hot}\approx T_{ref}+\frac{V}{S_{ab}}

Use this approximation only when the temperature span is small enough for a constant local sensitivity. Production thermocouple work uses reference-junction compensation, extension-wire controls and tables or polynomial fits.

Common Mistakes

Treating thermocouple voltage as absolute temperature without reference-junction compensation.
Using a room-temperature Seebeck coefficient across a wide temperature range.
Ignoring unintended dissimilar-metal junctions in a temperature gradient.
Estimating sensor voltage without checking amplifier offset, noise and ground-loop pickup.

Thermoelectric Generator Load Model

A thermoelectric generator can be screened as a Thevenin source:

\displaystyle I=\frac{V_{oc}}{R_i+R_L}

V_L=IR_L

\displaystyle P_L=I^2R_L=\frac{V_{oc}^2R_L}{(R_i+R_L)^2}

Maximum electrical power occurs when:

R_L=R_i

and:

\displaystyle P_{max}=\frac{V_{oc}^2}{4R_i}

The electrical efficiency is:

\displaystyle \eta=\frac{P_L}{Q_h}

where $Q_h$ is heat extracted from the hot side. This efficiency cannot be assessed from the electrical circuit alone because thermal interfaces, heat-source impedance and heat rejection determine the usable $\Delta T$ across the module.

Design Checks

Verify whether $\Delta T$ is across the module, not only between the external heat source and ambient air.
Check hot-side temperature, solder limit, ceramic stress and contact pressure.
Include wiring resistance when output voltage is low.
Confirm that maximum power point is compatible with the electrical load or converter input.

Peltier Cooler Heat Balance

For a simplified thermoelectric cooler:

\displaystyle Q_c=\alpha T_c I-\frac{1}{2}I^2R-K(T_h-T_c)

\displaystyle Q_h=\alpha T_h I+\frac{1}{2}I^2R-K(T_h-T_c)

Electrical input power is:

P_e=\alpha (T_h-T_c)I+I^2R

Cooling coefficient of performance is:

\displaystyle COP=\frac{Q_c}{P_e}

The first term in $Q_c$ is Peltier cooling. The second term is half of the Joule heat assumed to return to the cold side. The third term is back-conduction from hot side to cold side. Increasing current does not always improve cooling because Joule heating grows with $I^2$ .

Practical Validity

This model is useful for first-pass sizing and review. It assumes lumped properties, steady operation and approximate heat splitting. Real modules require manufacturer performance curves, derating for hot-side temperature, interface resistance, controller behavior and condensation or frost checks when cold-side temperature drops below dew point.

Thermoelectric Figure of Merit

The material figure of merit is:

\displaystyle ZT=\frac{S^2\sigma T}{k}=\frac{S^2T}{\rho k}

where $T$ is absolute temperature. The numerator rewards large Seebeck coefficient and high electrical conductivity. The denominator penalizes thermal conduction that leaks heat from hot side to cold side.

The power factor is:

PF=S^2\sigma

A high material $ZT$ does not guarantee a high-performing device. Contacts, solder layers, ceramic plates, diffusion barriers, heat spreaders, parasitic electrical resistance and thermal interface resistance can dominate module performance.

Thermal Interface and Module Temperature Difference

For a heat path:

\displaystyle Q=\frac{\Delta T}{R_{th}}

For one-dimensional conduction:

\displaystyle R_{cond}=\frac{L}{kA}

For series thermal resistances:

R_{th,total}=R_1+R_2+\cdots+R_n

If a thermoelectric module carries heat rate $Q$ , the temperature drop lost in external interfaces is approximately:

\Delta T_{interface}=Q(R_{hot}+R_{cold})

The temperature difference available across the module is then:

\Delta T_{module}\approx \Delta T_{available}-\Delta T_{interface}

This check often explains why a thermoelectric generator or cooler underperforms a catalog estimate. The catalog curve may assume module-side temperatures, while the installed system has finite heat-spreader and heat-sink resistances.

Piezoelectric Charge and Voltage

For a simplified direct piezoelectric force mode:

Q=dF

Open-circuit voltage is:

\displaystyle V_p=\frac{Q}{C_p}=\frac{dF}{C_p}

Capacitive impedance is:

\displaystyle Z_C=\frac{1}{j\omega C_p}

Capacitive admittance is:

Y_C=j\omega C_p

Generated current is:

\displaystyle i=\frac{dQ}{dt}

For sinusoidal charge amplitude $Q_{pk}$ at frequency $f$ :

I_{pk}=2\pi fQ_{pk}

Piezoelectric sensors are strong for dynamic force, pressure and acceleration. They are not ideal true static load sensors because leakage, finite insulation resistance and amplifier bias current discharge the signal over time.

Charge Amplifier Relations

For an ideal inverting charge amplifier:

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

If the sensor has charge sensitivity $S_q$ in C per engineering unit:

\displaystyle S_v=\frac{S_q}{C_f}

Low-frequency cutoff from feedback resistance and capacitance is:

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

The approximate linear charge range is:

|Q|<C_f|V_{lim}|

For a force sensor:

\displaystyle |F|<\frac{C_f|V_{lim}|}{d}

For an accelerometer with charge sensitivity $S_q$ :

\displaystyle |a|<\frac{C_f|V_{lim}|}{S_q}

Interface Checks

Confirm that overload recovery is acceptable, not only that steady-state gain is correct.
Keep cable and connector leakage compatible with the required low-frequency response.
Check ADC resolution after selecting $C_f$ ; lowering gain may prevent saturation but reduce signal resolution.
Use overload flags or waveform review when transient clipping would corrupt RMS or spectral summaries.

Piezoelectric Actuator Stroke and Blocking Force

For a simple stack actuator estimate:

\delta_{free}=Nd_{33}V

where $N$ is the number of active layers and $d_{33}$ is the longitudinal piezoelectric strain coefficient in m/V.

Electric field across one active layer of thickness $t$ is:

\displaystyle E=\frac{V}{t}

If mechanical stiffness is $k_m$ , approximate blocking force is:

F_b=k_m\delta_{free}

The delivered force at partial displacement depends on the load line. For a linear actuator and linear load:

F\approx k_m(\delta_{free}-\delta)

These estimates are first-order. Real piezoelectric actuators also have hysteresis, creep, depolarization limits, tensile stress limits, preload requirements, heating and driver current limits.

Resonance and Bandwidth

For a lumped mechanical mode:

\displaystyle \omega_n=\sqrt{\frac{k_m}{m}}

\displaystyle f_n=\frac{\omega_n}{2\pi}

Mechanical quality factor:

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

where $\Delta f$ is the half-power bandwidth.

Approximate resonant amplification for a lightly damped second-order system can be large near $f_n$ . For measurement channels intended to report input motion rather than resonant response, operate well below the first mounted resonance or use calibration that includes the mounted configuration.

Noise, Resolution and Dynamic Range

Signal-to-noise ratio:

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

In decibels:

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

ADC code width for a full-scale input span $FS$ and $N$ bits:

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

Quantization noise RMS for an ideal converter:

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

Use these checks after selecting the thermoelectric amplifier gain or piezoelectric charge-amplifier feedback capacitance. A range that avoids saturation can still be too coarse for the engineering decision.

Uncertainty and Validation Checks

For independent uncertainty components:

u_c=\sqrt{u_1^2+u_2^2+\cdots+u_n^2}

Expanded uncertainty:

U=ku_c

where $k$ is the coverage factor.

Validation should include the installed condition:

thermal contact and heat-sink temperature for thermoelectric modules;
reference-junction compensation and connector temperature for thermocouples;
preload, mounting stiffness, cable condition and charge-amplifier range for piezoelectric sensors;
stroke, blocking force, resonance and driver limit for piezoelectric actuators;
environmental exposure, aging, humidity, vacuum, sterilization or thermal cycling where relevant.

Worked Example 1: Thermocouple Voltage and Temperature Error

A type K thermocouple is reviewed with a local Seebeck sensitivity:

S=41\ \mu\text{V/K}

The hot junction is at:

T_{hot}=260\ ^\circ\text{C}

The real reference junction is at:

T_{ref}=35\ ^\circ\text{C}

The input module incorrectly assumes:

T_{ref,assumed}=25\ ^\circ\text{C}

Estimate the measured voltage and the indicated temperature.

The thermocouple voltage is:

V=S(T_{hot}-T_{ref})

V=41(260-35)=41(225)=9225\ \mu\text{V}

V=9.225\ \text{mV}

The module reconstructs temperature using the assumed reference:

\displaystyle T_{ind}=\frac{V}{S}+T_{ref,assumed}

T_{ind}=225+25=250\ ^\circ\text{C}

The indication error is:

e=T_{ind}-T_{hot}=250-260=-10\ ^\circ\text{C}

Engineering Comment

The displayed temperature is low by the same amount as the reference-junction temperature error. The calculation is simplified because real type K thermocouple response is nonlinear, but the sign and mechanism are correct: a warm uncompensated reference junction can make a hot process appear cooler than it is.

Worked Example 2: Thermoelectric Generator Load Matching

A thermoelectric generator module has:

S=45\ \text{mV/K}=0.045\ \text{V/K}

The module temperature difference is:

\Delta T=80\ \text{K}

Internal resistance is:

R_i=3.2\ \Omega

The electrical load is matched:

R_L=3.2\ \Omega

Find open-circuit voltage, current, load voltage and load power.

Open-circuit voltage:

V_{oc}=S\Delta T=0.045(80)=3.60\ \text{V}

Current:

\displaystyle I=\frac{V_{oc}}{R_i+R_L}=\frac{3.60}{3.2+3.2}=0.5625\ \text{A}

Load voltage:

V_L=IR_L=0.5625(3.2)=1.80\ \text{V}

Load power:

P_L=I^2R_L=(0.5625)^2(3.2)=1.01\ \text{W}

This equals:

\displaystyle P_{max}=\frac{V_{oc}^2}{4R_i}=\frac{3.60^2}{4(3.2)}=1.01\ \text{W}

Engineering Comment

The electrical match is correct for maximum power in the Thevenin model. It does not prove that the thermal design can sustain $\Delta T=80\ \text{K}$ . In a real installation, heat-spreader and heat-sink resistance may reduce the module temperature difference and lower power substantially.

Worked Example 3: Peltier Cooler Heat Pumping and COP

A thermoelectric cooler is screened at:

T_c=293\ \text{K}

T_h=308\ \text{K}

I=3.0\ \text{A}

Module parameters are:

\alpha=0.055\ \text{V/K}

R=2.1\ \Omega

K=0.42\ \text{W/K}

Estimate cold-side heat absorption, electrical input power and cooling COP.

Temperature difference:

\Delta T=308-293=15\ \text{K}

Cold-side heat absorption:

\displaystyle Q_c=\alpha T_cI-\frac{1}{2}I^2R-K\Delta T

\displaystyle Q_c=0.055(293)(3.0)-\frac{1}{2}(3.0)^2(2.1)-0.42(15)

Q_c=48.345-9.45-6.30=32.595\ \text{W}

Electrical input power:

P_e=\alpha\Delta T I+I^2R

P_e=0.055(15)(3.0)+(3.0)^2(2.1)=2.475+18.90=21.375\ \text{W}

Cooling COP:

\displaystyle COP=\frac{Q_c}{P_e}=\frac{32.595}{21.375}=1.53

Engineering Comment

The cooler can absorb about $32.6\ \text{W}$ in this simplified operating point, but the hot side must reject:

Q_h=Q_c+P_e=32.595+21.375=53.970\ \text{W}

If the heat sink cannot reject roughly $54\ \text{W}$ while keeping $T_h$ near $308\ \text{K}$ , the hot side warms, $\Delta T$ increases and the cooling capacity falls.

Worked Example 4: Thermal Interface Loss in a Thermoelectric Installation

A thermoelectric generator is placed between a hot plate and a heat sink. The external source-to-sink temperature difference is:

\Delta T_{available}=120\ \text{K}

The estimated heat flow through the stack is:

Q=18\ \text{W}

The hot-side interface resistance is:

R_{hot}=1.1\ \text{K/W}

The cold-side interface resistance is:

R_{cold}=1.4\ \text{K/W}

Estimate the temperature difference lost in the interfaces and the remaining module temperature difference.

Interface temperature loss:

\Delta T_{interface}=Q(R_{hot}+R_{cold})

\Delta T_{interface}=18(1.1+1.4)=18(2.5)=45\ \text{K}

Module temperature difference:

\Delta T_{module}=120-45=75\ \text{K}

Engineering Comment

More than one third of the available temperature difference is lost outside the module. A power estimate based on $120\ \text{K}$ would be optimistic. The practical design action is not only to choose a better thermoelectric material; it is also to reduce contact resistance, improve clamping, use better heat spreaders or lower heat-sink thermal resistance.

Worked Example 5: Piezoelectric Force Sensor and Charge Amplifier

A piezoelectric force sensor has:

d=350\ \text{pC/N}=350\times 10^{-12}\ \text{C/N}

Applied dynamic force is:

F=18\ \text{N}

Element capacitance is:

C_p=12\ \text{nF}=12\times 10^{-9}\ \text{F}

The charge amplifier uses:

C_f=2.2\ \text{nF}=2.2\times 10^{-9}\ \text{F}

Find generated charge, open-circuit voltage and charge-amplifier output.

Generated charge:

Q=dF=(350\times 10^{-12})(18)=6.30\times 10^{-9}\ \text{C}

Q=6.30\ \text{nC}

Open-circuit voltage:

\displaystyle V_p=\frac{Q}{C_p}=\frac{6.30\times 10^{-9}}{12\times 10^{-9}}=0.525\ \text{V}

Charge-amplifier output:

\displaystyle V_{out}=-\frac{Q}{C_f}=-\frac{6.30\times 10^{-9}}{2.2\times 10^{-9}}=-2.86\ \text{V}

Engineering Comment

The open-circuit voltage is not a stable design target by itself because cable capacitance and leakage can change it. The charge amplifier makes sensitivity primarily depend on $C_f$ , which is why charge-mode interfaces are common for dynamic piezoelectric measurements. The output range still needs overload and recovery checks.

Worked Example 6: Charge-Amplifier Low-Frequency Limit and Force Range

Use the same sensor coefficient:

d=350\ \text{pC/N}

The charge amplifier has:

C_f=2.2\ \text{nF}

R_f=1.0\ \text{G}\Omega

The linear output limit is:

|V_{lim}|=10\ \text{V}

Find the low-frequency cutoff and approximate maximum force before clipping.

Low-frequency cutoff:

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

\displaystyle f_c=\frac{1}{2\pi(1.0\times 10^9)(2.2\times 10^{-9})}=0.072\ \text{Hz}

Maximum charge before clipping:

Q_{max}=C_f|V_{lim}|=(2.2\times 10^{-9})(10)=22\times 10^{-9}\ \text{C}

Q_{max}=22\ \text{nC}

Maximum force:

\displaystyle F_{max}=\frac{Q_{max}}{d}=\frac{22\times 10^{-9}}{350\times 10^{-12}}=62.9\ \text{N}

Engineering Comment

The low-frequency cutoff is well below $1\ \text{Hz}$ , so the channel can preserve slow dynamic content better than a shorter time-constant interface. It is still not a true static force measurement. The force range is about $63\ \text{N}$ before clipping; startup shocks or impacts above that value require a larger $C_f$ , lower sensitivity sensor or overload-tolerant acceptance procedure.

Worked Example 7: Piezoelectric Stack Stroke and Blocking Force

A multilayer piezoelectric stack has:

d_{33}=500\ \text{pm/V}=500\times 10^{-12}\ \text{m/V}

Number of active layers:

N=120

Drive voltage:

V=100\ \text{V}

Actuator stiffness:

k_m=18\ \text{N}/\mu\text{m}

Estimate free stroke and approximate blocking force.

Free stroke:

\delta_{free}=Nd_{33}V

\delta_{free}=120(500\times 10^{-12})(100)=6.0\times 10^{-6}\ \text{m}

\delta_{free}=6.0\ \mu\text{m}

Blocking force:

F_b=k_m\delta_{free}

Using $k_m=18\ \text{N}/\mu\text{m}$ :

F_b=18(6.0)=108\ \text{N}

Engineering Comment

The actuator can produce fine displacement with high force, but this simple estimate does not include hysteresis, creep, driver current, heat generation, preload or tensile stress limits. The blocking force is not the normal operating force; it is the force at zero displacement. A real load line should be checked against the required stroke and force at the same time.

Field Review Checklist

Before accepting a thermoelectric or piezoelectric design, confirm:

the calculated signal, heat flow, stroke or power is tied to the installed boundary condition;
the material coefficient is valid over the actual temperature, stress, field and frequency range;
the selected electrical interface has enough range, resolution, noise margin and recovery behavior;
the thermal path, contact pressure, bonding, preload, cable routing and shielding are specified;
the verification test reproduces the condition that controls the engineering decision;
the uncertainty budget includes drift, nonlinearity, hysteresis, cross-sensitivity and environmental effects.

The recurring engineering failure is not misunderstanding the physical effect. It is assuming that the effect survives packaging, interfaces and validation unchanged.

REF

Disciplines

Thermoelectric and Piezoelectric Devices Formula Sheet

Scope and Sign Conventions

Core Symbols

Seebeck Voltage

Common Mistakes

Thermoelectric Generator Load Model

Design Checks

Peltier Cooler Heat Balance

Practical Validity

Thermoelectric Figure of Merit

Thermal Interface and Module Temperature Difference

Piezoelectric Charge and Voltage

Charge Amplifier Relations

Interface Checks

Piezoelectric Actuator Stroke and Blocking Force

Resonance and Bandwidth

Noise, Resolution and Dynamic Range

Uncertainty and Validation Checks

Worked Example 1: Thermocouple Voltage and Temperature Error

Engineering Comment

Worked Example 2: Thermoelectric Generator Load Matching

Engineering Comment

Worked Example 3: Peltier Cooler Heat Pumping and COP

Engineering Comment

Worked Example 4: Thermal Interface Loss in a Thermoelectric Installation

Engineering Comment

Worked Example 5: Piezoelectric Force Sensor and Charge Amplifier

Engineering Comment

Worked Example 6: Charge-Amplifier Low-Frequency Limit and Force Range

Engineering Comment

Worked Example 7: Piezoelectric Stack Stroke and Blocking Force

Engineering Comment

Field Review Checklist

See also