Resistor Network Thermal Coupling: Unequal Power and Ratio Error

Separate common ambient temperature from unequal dissipation

A chamber-temperature test changes the environment around the network. A powered test also creates heat inside the resistors. These conditions can produce different temperature distributions even at the same measured ambient temperature. Matching TCR under uniform temperature does not guarantee a stable ratio when one resistor runs warmer than the other.

Both uniform-temperature characterization and application power-state characterization are needed for a printed network. One describes how the elements track a common temperature change; the other describes how their temperatures separate under the circuit's actual dissipation and mounting.

Calculate element power from the loaded circuit

Use the actual voltage across or current through each resistor, including receiver loading and switching states. In a series chain with a common current, the higher resistance dissipates more power. In parallel elements at the same voltage, the lower resistance dissipates more power. Similar physical size does not imply similar heat generation.

For a hypothetical series pair of one kilohm and nine kilohms carrying one milliampere, dissipation is one milliwatt and nine milliwatts respectively. Their nine-to-one resistance ratio remains a nominal electrical relationship, but their unequal power can create a temperature difference. Include the power of any connected active device or reference nearby, since its heat can introduce a gradient unrelated to resistor dissipation.

Represent self-heating and cross-heating separately

A small-signal steady thermal model can relate each element's temperature rise to every element's power. Diagonal coefficients describe an element's own power influence; off-diagonal coefficients describe cross-heating. This model is a useful bookkeeping tool for measured comparisons, provided the geometry and boundary conditions remain fixed.

For two elements, increasing the power in the first can warm both. If both warm equally and their TCRs match, ratio change may be small. If the first warms much more, the ratio can move even with identical TCRs. Do not assume coefficient symmetry from geometric appearance alone; attachment, conductor routing and airflow can make the effective paths different.

Translate temperature separation into ratio change

For Q equal to R2 divided by R1, the first-order fractional ratio change is approximately alpha2 times delta T2 minus alpha1 times delta T1. Both temperatures are required. Replacing them with one substrate temperature discards the effect the study is intended to reveal.

Assume both coefficients are 100 parts per million per degree Celsius, while the two elements rise by two and five degrees. The predicted ratio change is approximately 300 parts per million, or 0.03 percent. If they instead both rise by five degrees, their first-order changes cancel. These hypothetical values illustrate the mechanism; actual coefficients and element temperatures must come from the selected network and installation.

Use a power-state matrix to identify the coupling

Begin from the same initial condition and measure each state long enough to apply the chosen settling criterion. Monitor the required ratio and, where practical, individual element temperatures or resistance-derived temperature estimates. The latter require an independently established temperature characteristic and sufficiently low measurement disturbance.

Observe the ratio during transitions as well as after settling

Elements can have different thermal response times because of geometry, attachment and local surroundings. Their final temperatures may be similar while their temperatures differ during a power transition. A ratio measured only after a long dwell can miss an error that occurs while a controller is actively using the signal.

Record the ratio and power waveform on a common time base. State whether a displayed value is instantaneous, averaged or filtered, since those choices change the apparent peak. Begin repeated transitions from the same thermal condition, and include recovery. If the application changes state before full settling, reproduce that sequence instead of testing only isolated steps from room temperature.

Use thermal symmetry where the circuit needs matching

Place matched elements with attention to their thermal surroundings, not merely their geometric distance. A pair can be close while one sits beside a hot terminal, mounting screw or active component. Compare conductor widths, substrate edges, protective layers and the heat-removal path beneath each element.

Changing element area or distributing resistance can alter self-heating while preserving the nominal ratio. Such changes also affect print variation, parasitic behavior and trim access. Evaluate the full circuit trade-off. Increasing coupling may reduce a differential temperature under one state but transmit more disturbance from a heavily powered neighbor under another. The power-state matrix helps expose that dependence before layout selection.

Specify ratio performance under named power conditions

A meaningful network requirement identifies the controlled ratio, supply or current states, receiver loading, mounting and observation time. Include ambient range separately from powered temperature rise. Define whether acceptance concerns a steady ratio, a maximum transition error or a filtered system output.

Provide measured thermal comparisons when available, along with the schematic and artwork. If only uniform-temperature TCR data exists, the application power study remains a separate task. This keeps the matching claim tied to a reproducible circuit condition. Shared-substrate construction can support useful tracking, but it does not by itself establish a universal ratio-drift limit under unequal dissipation.