Formula sheet

Thermal Resistance Network Formula Sheet

Formula sheet for thermal resistance networks covering conduction, contact interfaces, convection, radiation, series and parallel paths, junction temperature, transient RC models, heat spreading, and validation from measured temperatures.

Branch: Mechanical Engineering
Content: Formula sheet
Updated: Jun 20, 2026
Revision: v1.0.0 · reviewed

This formula sheet collects first-pass equations for thermal resistance networks. Use it to estimate component temperature, compare heat paths, check interface quality, screen heat-sink or cold-plate options, and plan validation measurements.

Thermal resistance models are engineering approximations. They are useful when the heat path is clear and the dominant resistances are known. They become weak when heat spreading, contact pressure, radiation, phase change, transient storage, nonuniform heat generation, local hot spots, or complex three-dimensional geometry dominate.

Basic Thermal Resistance

Thermal resistance:

\displaystyle R_\theta=\frac{\Delta T}{\dot{Q}}

where $R_\theta$ is thermal resistance, $\Delta T$ is temperature difference, and $\dot{Q}$ is heat flow.

Temperature rise:

\Delta T=\dot{Q}R_\theta

Hot-side temperature:

T_{hot}=T_{sink}+\dot{Q}R_{\theta,total}

Use consistent units. If heat flow is in watts and temperature is in degrees Celsius or kelvin difference, thermal resistance is in:

^\circ\text{C/W}

or:

\text{K/W}

Temperature differences in degrees Celsius and kelvin are numerically equal.

Series Thermal Resistances

For heat flowing through resistances in series:

R_{\theta,total}=R_1+R_2+R_3+\cdots+R_n

Temperature after resistance $i$ :

T_i=T_{sink}+\dot{Q}\sum_{j=1}^{i}R_j

Series networks are common in electronics and cooling systems:

junction -> package -> interface -> heat sink -> air

or:

cell -> module interface -> cold plate -> coolant -> heat exchanger

Reducing one resistance helps only if it is a meaningful part of the total path.

Parallel Thermal Paths

For parallel heat paths between the same two temperature nodes:

\displaystyle \frac{1}{R_{\theta,eq}}=\sum_{i=1}^{n}\frac{1}{R_i}

Heat through path $i$ :

\displaystyle \dot{Q}_i=\frac{\Delta T}{R_i}

Fraction of heat through path $i$ :

\displaystyle f_i=\frac{1/R_i}{\sum_j 1/R_j}

Parallel paths appear when heat leaves a component through a heat sink, printed circuit board copper, mounting screws, enclosure wall, or cable connection at the same time.

Plane-Wall Conduction

Thermal resistance for one-dimensional conduction through a plane wall:

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

where $L$ is thickness, $k$ is thermal conductivity, and $A$ is heat-transfer area.

Heat flow:

\displaystyle \dot{Q}=\frac{kA(T_1-T_2)}{L}

This equation assumes uniform area, one-dimensional heat flow, constant conductivity, and no internal heat generation. Real parts may require spreading-resistance or finite element checks.

Cylindrical Conduction

For radial conduction through a cylindrical wall:

\displaystyle R_{cyl}=\frac{\ln(r_o/r_i)}{2\pi kL}

where $r_i$ is inner radius, $r_o$ is outer radius, and $L$ is cylinder length.

This form is useful for pipes, insulation, sleeves, bushings, pressure vessels, and cylindrical battery or motor components.

Contact and Interface Resistance

Contact or interface resistance:

\displaystyle R_{contact}=\frac{\Delta T_{interface}}{\dot{Q}}

If a thermal interface material of thickness $t$ and conductivity $k_{TIM}$ covers area $A$ :

\displaystyle R_{TIM}\approx \frac{t}{k_{TIM}A}

This simple formula ignores bond-line variation, voids, surface roughness, pump-out, aging, and contact pressure. In real designs, interface resistance often dominates the heat path even when metal parts have high thermal conductivity.

Convection Resistance

Convection resistance:

\displaystyle R_{conv}=\frac{1}{hA}

where $h$ is convection coefficient and $A$ is exposed area.

Heat transfer:

\dot{Q}=hA(T_s-T_\infty)

Convection coefficient depends on flow speed, fluid properties, geometry, surface condition, orientation, turbulence, fouling, and temperature. Reusing $h$ outside the tested condition is a common source of thermal error.

Radiation Resistance

Radiation heat transfer between a surface and large surroundings:

\dot{Q}_{rad}=\epsilon\sigma A(T_s^4-T_{sur}^4)

For small temperature differences, a linearized radiation coefficient may be used:

h_{rad}\approx \epsilon\sigma(T_s+T_{sur})(T_s^2+T_{sur}^2)

Radiation resistance:

\displaystyle R_{rad}\approx \frac{1}{h_{rad}A}

Temperatures must be in kelvin. Radiation can matter for hot surfaces, vacuum, low-airflow enclosures, furnaces, spacecraft, and high-emissivity coatings.

Junction Temperature

Junction temperature from ambient:

T_j=T_a+P_D R_{\theta JA}

Junction temperature from measured case temperature:

T_j=T_c+P_D R_{\theta JC}

Junction-to-ambient path:

R_{\theta JA}=R_{\theta JC}+R_{\theta CS}+R_{\theta SA}

where:

$R_{\theta JC}$ is junction-to-case resistance;
$R_{\theta CS}$ is case-to-sink or interface resistance;
$R_{\theta SA}$ is sink-to-ambient resistance.

Datasheet thermal resistance must match the installed board, copper area, airflow, enclosure, mounting, and duty cycle. Otherwise the calculated junction temperature may be misleading.

Heat Flux

Average heat flux:

\displaystyle q''=\frac{\dot{Q}}{A}

If heat flux is nonuniform, a local peak may control the design:

\displaystyle q''_{peak}>\frac{\dot{Q}}{A}

Check heat flux for semiconductor dies, battery tabs, brake surfaces, cold plates, reactor walls, laser diodes, and compact heat exchangers. Average heat flux can hide local thermal failure.

Thermal Capacitance and RC Time Constant

Thermal capacitance:

C_\theta=mc_p

where $m$ is mass and $c_p$ is specific heat capacity.

First-order thermal time constant:

\tau=R_\theta C_\theta

For a step in heat generation with sink temperature held constant:

T(t)=T_{sink}+\dot{Q}R_\theta\left(1-e^{-t/\tau}\right)

For cooling after heat generation stops:

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

Single-time-constant models are screening tools. Real systems often have multiple thermal masses and multiple heat paths.

Transient Thermal Impedance

Transient thermal impedance:

\displaystyle Z_\theta(t)=\frac{\Delta T(t)}{P}

For a rectangular power pulse:

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

For repeated pulses, superposition may be used if the system is approximately linear:

\Delta T(t)=\sum_i P_i Z_\theta(t-t_i)

Power electronics, pulsed lasers, motor drives, brakes, and cyclic machinery often require transient checks. A steady-state resistance may overstate or understate temperature rise for short pulses.

Thermal Margin

Temperature margin:

M_T=T_{limit}-T_{predicted}

Percent margin relative to allowable rise:

\displaystyle M_{\%}=100\frac{T_{limit}-T_{predicted}}{T_{limit}-T_{sink}}

A positive margin is not automatically acceptable. The margin must cover uncertainty in power dissipation, ambient temperature, thermal resistance, fouling, aging, sensor error, and degraded operation.

Measured Resistance from Test Data

Measured thermal resistance:

\displaystyle R_{\theta,meas}=\frac{T_{hot,meas}-T_{sink,meas}}{\dot{Q}_{meas}}

Deviation from model:

\displaystyle e_R=100\frac{R_{\theta,meas}-R_{\theta,model}}{R_{\theta,model}}

Use measured data only when the system is in the intended operating mode and sensors are located at comparable thermal nodes. A thermocouple on a heat sink base is not the same as junction temperature.

Uncertainty Propagation

If:

T=T_s+\dot{Q}R_\theta

and heat-flow and resistance uncertainty are independent, a first uncertainty estimate is:

\sigma_T\approx \sqrt{\sigma_{T_s}^2+(R_\theta\sigma_{\dot{Q}})^2+(\dot{Q}\sigma_R)^2}

This is a linearized estimate. Use broader analysis or testing when operation is near the temperature limit, when variables are correlated, or when properties vary strongly with temperature.

Practical Review Checklist

For a thermal resistance network, check:

heat source magnitude and duty cycle;
all meaningful heat paths;
series and parallel assumptions;
contact pressure and interface material;
convection coefficient source and validity;
radiation if airflow is weak or temperature is high;
local hot spots and heat spreading;
transient thermal impedance for pulsed loads;
margin to the controlling temperature limit;
validation measurements at the correct thermal nodes.

Thermal resistance networks are valuable because they make the heat path explicit. They are dangerous when the network is treated as the real system without checking installation, geometry, flow, contact, transient behavior, and measurement evidence.

REF

Disciplines