The temperature coefficient of resistance (TCR or α) for pure annealed copper is 0.00393 °C⁻¹ at 20°C. For the standard electrodeposited (ED) copper used in printed circuit boards, it is slightly lower, typically around 0.00386 °C⁻¹. This means that for every 1°C rise in temperature, copper’s electrical resistance increases by roughly 0.39%. In power electronics and high-current PCB design, this is not just an academic footnote; it is the primary trigger for thermal runaway if your thermal path from junction to ambient is under-engineered.
Copper TCR and the Thermal Runaway Loop
When you push high current through a PCB trace or a copper busbar, you generate heat via I²R losses. Because of copper's positive TCR, as the trace heats up, its resistance increases. Higher resistance at the same current generates even more heat, which drives the temperature and resistance up further. If the rate of heat generation exceeds the rate of heat dissipation to the ambient environment, the system enters thermal runaway.
To see this in practice, consider a 10mm long, 1mm wide, 1oz (35µm) copper trace on an FR4 board. At 20°C, its resistance is roughly 5 mΩ. If you pass 20A through it, the initial power dissipation is P = 20² × 0.005 = 2.0W. If poor airflow allows that trace to reach 100°C, the resistance shifts according to the formula R_T = R_20[1 + α(T - 20)]:
R_100 = 0.005 × [1 + 0.00386 × (100 - 20)] = 0.00654 Ω (6.54 mΩ)
At 100°C, your power dissipation jumps to P = 20² × 0.00654 = 2.61W. That is a 30% increase in heat generation purely from the temperature coefficient of resistance for copper, demanding 30% more thermal headroom from your PCB substrate and enclosure.
Reference Table: Copper Alloys and Thermal Limits in Electronics
| Material / Trace Type | TCR (α) at 20°C | Thermal Conductivity | Max Safe Operating Temp |
|---|---|---|---|
| Pure Annealed Copper | 0.00393 °C⁻¹ | 401 W/m·K | N/A (Bulk wire/busbar) |
| Electrodeposited (ED) PCB Copper | 0.00386 °C⁻¹ | 380 - 395 W/m·K | 105°C (Standard Tg FR4) |
| Rolled Annealed (RA) Flex Copper | 0.00390 °C⁻¹ | 398 W/m·K | 130°C (Polyimide substrate) |
| Constantan (Shunt Reference) | ±0.00002 °C⁻¹ | 22 W/m·K | 200°C+ (No TCR drift) |
Note: Data aligns with IPC-2152 standards for current-carrying capacity. Always derate for enclosed spaces.
How hot is too hot? The copper itself won't melt until 1085°C, but the PCB substrate will fail long before that. Standard FR4 has a glass transition temperature (Tg) of 130°C to 140°C. Above this, the resin softens, the Z-axis expands rapidly, and plated through-holes (vias) barrel-crack. For reliable continuous operation, keep high-current copper traces under 105°C measured at the board surface.
Thermal Path Math: Junction to Ambient (RθJA)
To prevent the copper and the silicon from cooking, you must calculate the thermal resistance from the semiconductor junction to the ambient air. The governing equation is:
T_J = T_A + P_D × (R_θJC + R_θCS + R_θSA)
- T_J: Junction temperature (°C)
- T_A: Ambient temperature inside the enclosure (°C)
- P_D: Power dissipated (Watts)
- R_θJC: Thermal resistance, Junction-to-Case (°C/W) — fixed by the silicon manufacturer.
- R_θCS: Thermal resistance, Case-to-Sink (°C/W) — determined by your thermal interface material (TIM).
- R_θSA: Thermal resistance, Sink-to-Ambient (°C/W) — determined by your heatsink and airflow.
Interpreting the Derating Curve
When reading a MOSFET datasheet (like the Infineon IRF3205), look at the power derating curve. The X-axis is case temperature (T_C), and the Y-axis is allowable power dissipation (P_D). The line starts at max power at 25°C and slopes linearly down to zero watts at the max junction temperature (usually 175°C). The absolute value of the slope of this line is exactly 1 / R_θJC. If a part is rated for 200W at 25°C and derates to 0W at 175°C, the slope is 200 / (175 - 25) = 1.33 W/°C. Therefore, R_θJC = 1 / 1.33 = 0.75 °C/W. This math is critical for validating the manufacturer's claims against your real-world thermal path.
Safety Warning: When designing high-current paths (>30A) on PCBs, a failure in the thermal path can cause the copper trace to act as a slow-blow fuse, potentially igniting the FR4 substrate or nearby wiring. Always use TI's PowerPAD guidelines or equivalent thermal vias to dump heat to internal ground planes, and protect the circuit with a properly sized fuse or breaker upstream of the PCB.
Heatsink Selection: Sizing for Copper and Silicon
Let’s run a real-world heatsink selection example. You are designing a linear power supply using a TO-220 package pass transistor (e.g., TIP35C) that must dissipate 25W. The maximum allowable T_J is 150°C, but for long-term reliability, we want to keep it under 110°C. The ambient temperature inside your enclosure (T_A) is 40°C due to other components.
We need to find the maximum allowable R_θSA:
R_θSA ≤ (T_J - T_A) / P_D - R_θJC - R_θCS
R_θSA ≤ (110 - 40) / 25 - 1.0 (Junction-to-Case) - 0.5 (Sil-pad Case-to-Sink)
R_θSA ≤ 2.8 - 1.5 = 1.3 °C/W
You need a heatsink with a thermal resistance of 1.3 °C/W or lower. If you select the Aavid Thermalloy 577302B00000G (a common extruded aluminum TO-220 vertical sink), its natural convection R_θSA is roughly 2.2 °C/W. This is insufficient; your junction temperature will soar to 147.5°C, leaving almost no margin for TCR-induced copper trace heating or ambient spikes.
What Airflow and Enclosure Changes Buy You
To fix this without buying a massive chassis-mount sink, you alter the environment:
- Add Forced Air: Mounting a 40mm fan (like a Noctua NF-A4x20 FLX pushing 10 CFM) across the Aavid 577302B00000G drops its effective R_θSA from 2.2 °C/W down to approximately 0.8 °C/W. This easily clears your 1.3 °C/W target.
- Enclosure Louvers: If your enclosure is sealed, internal T_A will rise as the heatsink warms the trapped air. Cutting louvered ventilation slots at the bottom and top of the enclosure creates a natural convection chimney effect, dropping internal T_A by 10°C to 15°C compared to a sealed box.
- Thermal Vias under Copper: If the TO-220 is mounted directly to the PCB instead of a standalone sink, use an array of 0.3mm filled thermal vias under the drain pad to couple the heat into a 2oz internal copper pour, effectively using the PCB itself as a spreader.
Failure Signatures of Thermal Stress
When thermal management fails and the TCR-driven heating loop wins, the hardware leaves specific forensic signatures. Recognizing these helps you debug whether the failure was a silicon defect or a thermal design flaw.
- Copper Trace Delamination (Measling): If the PCB exceeds its Tg repeatedly, the epoxy resin expands in the Z-axis. This shears the bond between the copper foil and the fiberglass, creating white, star-shaped micro-cracks inside the board known as 'measles'. The trace eventually lifts off the pad during solder rework or high-vibration operation.
- Solder Joint Creep and Fatigue: Lead-free SAC305 solder melts at 217°C, but it begins to suffer from thermal creep at roughly 100°C (half its absolute melting temperature). If your copper busbars and component leads run constantly at 110°C, the solder joints will undergo grain boundary sliding, leading to visible micro-cracks at the heel of the joint, eventually causing high-resistance open circuits.
- Silicon Electromigration: Before a MOSFET or IC catastrophically shorts, running it at the absolute edge of its 150°C T_J limit causes aluminum or copper interconnects inside the silicon die to physically migrate due to high electron wind and heat. The signature is parametric drift: the R_DS(on) of the MOSFET slowly increases over months, or an op-amp's offset voltage drifts out of spec, long before the magic smoke appears.
By respecting the temperature coefficient of resistance for copper and properly calculating your R_θJA thermal path, you prevent these failure modes and ensure your power electronics run cool, efficient, and reliable for years.
