Calculating Temperature Rise in a Series Thermal-Resistance Network

Hello! Last lesson, you established that the components in a sealed enclosure generate heat that must ultimately leave through the enclosure’s exterior. That balance fixes the watts to be rejected, but it cannot tell you how much hotter a particular junction is than ambient. Today you will estimate that temperature rise by representing one heat-flow path as thermal resistances in series. The result will be a useful hand-calculation benchmark for a later Fluent model.

Give each temperature drop a resistance

For a steady heat-flow path carrying $P$ watts between a hotter point and a cooler point, define its thermal resistance as

Its units are $K/W$, equivalently ${}^{\circ }\mathrm{C}/\mathrm{W}$ for a temperature difference. Rearranging gives the working rule:

This is a lumped description of a physical region, not a new heat-transfer mechanism. A resistance might represent conduction through a package, an imperfect interface, or heat transfer from a surface to air. Its value is meaningful only for the two temperature locations and the operating conditions it describes.

Watch the selected portions of MIT OpenCourseWare’s Lecture 16: Thermal Modeling and Heat Sinking. The first develops the temperature-drop analogy; the second shows how convection can be represented by a resistance without yet requiring you to calculate a convection coefficient.

The circuit analogy is useful if you keep its physical meaning in view: temperature difference plays the role of voltage difference, heat-transfer rate plays the role of current, and thermal resistance relates the two. For example, if $4\mathrm{W}$ crosses an interface with $R_{\mathrm{th}}=0.5K/W$, the interface accounts for a $2\mathrm{K}$ temperature drop.

The supplied schematic shows a particularly clear series path:

Here, the same modeled heat rate passes through each successive element. That is the defining condition for series resistances. At steady state, if no heat is added, removed, or diverted at an intermediate node, conservation requires the rate entering that node to equal the rate leaving it.

Add the drops, then recover the temperatures

Consider the schematic’s junction, case, heat-sink, and ambient nodes. Write a drop across each resistance:

Here $h$ labels the heat sink, while $a$ labels ambient. Adding the equations cancels the intermediate temperatures:

Thus, for a path genuinely carrying the same $P$ through every element,

Watch the lecture build this junction-to-ambient path. Its package, interface, and heat sink are distinct physical parts, so their resistances are added rather than substituted for one another.

For a compact worked example, suppose one device dissipates $4\mathrm{W}$ into the following modeled path, with ambient air at $25^{\circ }\mathrm{C}$:

The total resistance is $5.0K/W$, so the junction rise is $4(5.0)=20^{\circ }\mathrm{C}$. Working from the known ambient temperature, the modeled heat sink is at $37^{\circ }\mathrm{C}$, the case at $39^{\circ }\mathrm{C}$, and the junction at $45^{\circ }\mathrm{C}$. Working backward from the junction gives the same temperatures—a useful arithmetic check.

The location of “ambient” matters. An internal heat sink in a sealed enclosure does not reject heat directly to outdoor or room air. Its immediate air surroundings are inside the enclosure, and heat must still reach the enclosure wall and leave its exterior. A heat-sink-to-ambient rating measured for an exposed heat sink cannot simply be assigned to an internal one.

Adapt the path to a PCB inside a sealed enclosure

A PCB-mounted device may send heat through its package into the board, then toward the enclosure. Watch the short PCB portion of the same lecture to see why a board and its thermal connections may need to appear in the path, rather than assuming that every component is mounted directly to a heat sink.

For a single-source approximation, you could name successive nodes junction, case, board, enclosure exterior, and external ambient. If all the device’s heat follows that one path, its estimated junction temperature is

where $R_{cb}$ spans case to board, $R_{bw}$ spans board to enclosure exterior, and $R_{wa}$ spans exterior surface to external ambient. These are defined node-to-node resistances: $R_{bw}$, for example, must account for whatever internal transfer and wall conduction the simplified model places between its named endpoints.

Return now to the multiple-component enclosure from last lesson. Its $3\mathrm{W}$, $2\mathrm{W}$, and $1\mathrm{W}$ components together reject $6\mathrm{W}$. Under that lesson’s illustrative exterior conditions, the enclosure surface was about $35^{\circ }\mathrm{C}$ when external ambient was $25^{\circ }\mathrm{C}$.

Suppose, purely for an initial path estimate, the $3\mathrm{W}$ component has the following assumed resistances on its route above that shared exterior surface:

If its $3\mathrm{W}$ traverses all three, its rise above the exterior surface is $3(1.0+0.5+1.0)=7.5^{\circ }\mathrm{C}$. The estimated junction temperature is therefore

Notice why we started at the already warmed enclosure surface. The exterior must reject all $6\mathrm{W}$, not just this device’s $3\mathrm{W}$. Multiplying every resistance from this junction to external ambient by $3\mathrm{W}$ would undercount the loading of that shared downstream path. We have assumed the other components reach the wall by routes outside the three resistances assigned to this device; a real PCB may mix their heat earlier. The next lesson will handle heat shared among different paths more explicitly.

A manufacturer’s junction-to-ambient figure also needs care. Read the short series relation in Renesas’s Thermal Resistance and its Applications, then distinguish that complete-path figure from the individually defined segments above.

The algebraic sum in that reading is valid for its stated series model. A published overall junction-to-ambient value, however, reflects its measurement arrangement and cooling conditions; it is not automatically the resistance of your PCB and sealed enclosure. Nor should you add an overall junction-to-ambient value to junction-to-case resistance: the former already spans that part of the path.

What this estimate can—and cannot—check

A series network is most defensible when you can name each endpoint, estimate its resistance for the conditions of interest, and justify approximately the same heat rate through every included element. Heat splitting from a package into both the board and nearby air breaks that last assumption. So does heat injected by another component into a shared downstream element. Convection-related and other effective resistances can also change with temperature or operating conditions, making a fixed-resistance result an estimate rather than an exact prediction.

That limitation does not make the calculation disposable. Record its assumed powers, ambient temperature, resistance values, and temperature nodes. Later, when you model the package, PCB, internal air, and enclosure in Fluent, you can compare temperatures at corresponding locations. A large difference is a prompt to investigate the assumed heat paths and conditions—not to declare either result correct solely because it produced a number.

Closing perspective

For a true series path, the same heat rate crosses each resistance, each temperature drop equals that rate times its resistance, and the drops add to the total rise above a known reference temperature. Working from ambient toward the junction reveals intermediate temperatures as well as the final junction estimate. In a sealed enclosure, be especially careful when a path reaches a wall shared by several heat sources: the wall’s rise is governed by the heat it receives from all of them.

Next, you will extend the network to parallel paths, where heat can divide between routes instead of passing through every resistance in sequence.