Thermal superposition lets you estimate the temperatures at several points on a board by adding the effects of its individual heat sources. Instead of rerunning a thermal simulation for every power combination, build a matrix of source-to-temperature coefficients once, then multiply it by a new vector of component powers. The method, described by Roger Stout in a January 2007 Electronic Design article, is useful when the system is sufficiently linear and its thermal boundary conditions stay consistent.
What thermal superposition calculates
Suppose a board has m heat sources and you want temperatures at n locations. The sources might be two FETs and a coil; the locations might include the FET junctions, a device case, an IC ground pin, and a board point. Store the temperature-rise effect of each source at each location in an n-by-m matrix, Θ. Multiply that matrix by a vector of source powers, P, to get the predicted temperature rises, ΔT:
ΔT = ΘP
Each matrix entry is a temperature rise per unit of power, commonly expressed in °C/W or K/W. A coefficient for a source’s effect at its own measurement location represents self-heating; coefficients at other locations represent thermal interaction. The matrix can be rectangular because the number of temperatures you care about need not equal the number of heat sources.
The result is a vector of temperature rises, not absolute temperatures. Add the appropriate ambient or reference temperature to each predicted rise to estimate the corresponding absolute temperature. Keep the reference consistent with how the coefficients were obtained.
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The simplest calibration isolates each source so its column of Θ can be measured directly. For every source, apply known power while the other sources are off, measure every temperature location of interest, and divide each location’s rise above the chosen reference by the actual power delivered to that source. Repeat for all sources.
- Fix the conditions. Set the ambient reference, airflow, enclosure, mounting, and other relevant thermal boundaries to match the intended use.
- Excite one source. Apply a known, stable power to one component while keeping the remaining modeled sources off.
- Measure all selected locations. Record their temperatures after reaching the steady-state condition you intend to model.
- Calculate one matrix column. For each location, divide its measured temperature rise by the applied source power.
- Repeat and predict. Perform the same procedure for every source, then multiply the completed matrix by any later power vector.
A thermal simulator can act as the test environment: excite one modeled source at a time under the same boundary conditions, record the resulting temperatures, and calculate the coefficients in the same way. This avoids hardware limits on how independently a source can be powered, but the resulting model remains tied to the simulator’s assumptions and setup.
Choose a calibration route that fits the hardware
| Approach | How it works | Best fit and limitation |
|---|---|---|
| Isolated source tests | Measure every location with one source powered at a time; each test directly supplies one matrix column. | Use when individual sources can be excited safely and independently. A source that cannot dissipate enough power in a steady test may make this impractical. |
| Simulation-generated tests | Use a simulator to excite modeled sources independently and calculate the same source-to-location coefficients. | Use when hardware excitation is constrained or repeated physical testing is burdensome. The coefficients depend on the model and its boundary conditions. |
| Independent combined tests | Apply several distinct power combinations, measure the temperatures, and solve the resulting equations for the unknown coefficients. | Use when isolated excitation is impractical. The power combinations must be linearly independent; more measurements than unknowns allow a least-squares fit. |
For example, if a coil cannot dissipate enough DC power for an isolated test, Stout suggests temporarily substituting a resistor at the same footprint, using simulation, or using multiple independent power vectors. A substitute component is useful only to the extent that its placement and thermal coupling represent the source being modeled.
Recover coefficients from combined tests
Let each test use a different power vector, with the measured temperature rises collected as columns of a matrix. In matrix form, the measurements satisfy Y = ΘX, where X contains the test power vectors and Y contains the corresponding measured rises. If there are exactly as many independent tests as sources, X is invertible and the coefficients can be recovered as Θ = YX−1. If tests outnumber sources, fit each measured location’s rises against the source powers with least-squares regression instead of forcing an exact solution.
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In Excel, MMULT performs the matrix-by-vector prediction; the dimensions must align, with one row in Θ per temperature location and one column per source. MINVERSE and TRANSPOSE support the square-system coefficient-recovery workflow, while LINEST can fit overdetermined data. Check the regression fit statistics, including R-squared, and inspect residuals rather than treating a fitted matrix as proof that the system is linear.
Check linearity and keep boundaries consistent
Superposition assumes that each source’s coefficient remains stable as power changes. Under that assumption, doubling every source’s power doubles the predicted temperature rises, and the individual contributions add. Thermal resistance and capacitance can change with temperature; airflow, enclosure conditions, ambient reference, and mounting can also change the heat flow. If these conditions differ between calibration and prediction, the old coefficients may no longer represent the system.
When the system is nonlinear, use the matrix as a local approximation. Choose a nominal operating point, perturb each source around that point, and calculate incremental coefficients from the resulting temperature changes. Predictions are most defensible near that operating point; accuracy declines as operating conditions move farther away. Validate against an independent combination of source powers and examine residuals and repeatability to see whether the approximation is adequate for the decision at hand.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Extend the model to changing loads
For time-varying power, replace each steady-state coefficient with a time-dependent response curve. A step increase in a source’s power contributes a scaled, time-shifted curve to the temperatures; a power decrease subtracts its corresponding contribution. Add the contributions from each change in each source’s load to estimate the overall transient response.
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Stout’s February 2007 companion article describes using Foster ladder networks because they are convenient to analyze, while Cauer networks more directly represent physical thermal structure. In an ideal linear network, source-to-source interaction curves are theoretically reciprocal, or symmetric. If reciprocity is uncertain in the actual setup, measure both directions rather than assuming the curves match.
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