The direct answer: Resistivity is a material's intrinsic opposition to current flow, while the temperature coefficient of resistance (TCR, denoted as α) defines how much that resistivity changes per degree of temperature change. In power electronics, ignoring TCR leads to catastrophic underestimation of conduction losses, because as a component heats up, its resistance increases (for most metals and MOSFETs), which generates more heat, creating a compounding thermal loop.
Designing reliable power stages requires coupling electrical TCR math with thermal resistance (Rθ) path calculations. Here is how to model, manage, and mitigate thermal stress on the bench.
The Math: Resistivity, TCR, and Self-Heating
Every conductive material has a baseline resistivity (ρ) measured at a reference temperature (usually 20°C or 25°C). When current flows, I²R losses generate heat. As the temperature rises, the resistance changes according to the linear approximation formula:
RT = Rref [1 + α(T - Tref)]
Where α is the TCR. For copper PCB traces, α is approximately +0.0039 °C-1. This means for every 1°C rise in temperature, copper's resistance increases by 0.39%.
| Material | Resistivity at 20°C (μΩ·cm) | TCR (α) at 20°C (°C-1) | Thermal Behavior |
|---|---|---|---|
| Copper (PCB Traces) | 1.68 | +0.0039 | Positive (PTC) |
| Aluminum (Heatsinks) | 2.65 | +0.0043 | Positive (PTC) |
| Silicon (MOSFET Die) | Variable (Doped) | +0.0050 to +0.0080 | Positive (PTC) for RDS(on) |
| Constantan (Shunts) | 49.0 | ±0.00002 | Near-Zero |
| NTC Thermistor Ceramic | High | -0.030 to -0.060 | Negative (NTC) |
Source: All About Circuits - Temperature Coefficient of Resistance
Datasheets typically list MOSFET RDS(on) at 25°C. At a realistic junction temperature of 100°C, the actual RDS(on) is often 1.5 to 1.8 times higher due to silicon's positive TCR. If you calculate conduction losses using the 25°C datasheet value, your thermal design will fail.
Thermal Path Math: Junction to Ambient (Rθ)
To keep a component within safe limits, you must calculate the thermal path from the silicon junction to the surrounding air. We use thermal resistance (Rθ), measured in °C/W, which acts exactly like electrical resistance in Ohm's Law, but for heat flow.
The governing equation is:
Tj = Ta + Pd × (RθJC + RθCS + RθSA)
- Tj (Junction Temp): How hot is too hot? Silicon maxes out at 150°C or 175°C, but running at the absolute limit accelerates electromigration. For high reliability, design for Tj ≤ 105°C.
- Ta (Ambient Temp): The air temperature inside the enclosure, not the room. Assume 45°C for a sealed enclosure.
- Pd (Power Dissipation): Calculated using the temperature-adjusted resistance (I²RT).
- RθJC (Junction-to-Case): Fixed by the manufacturer (e.g., 1.0 °C/W for a TO-247 package).
- RθCS (Case-to-Sink): Determined by your thermal interface material (TIM). A standard Bergquist Sil-Pad is ~0.5 °C/W; high-end thermal paste is ~0.1 °C/W.
- RθSA (Sink-to-Ambient): The heatsink's thermal resistance. This is the variable you solve for.
Heatsink Selection and Derating in Practice
Let's size a heatsink for an IRFP460 MOSFET switching a 12A continuous load.
1. Calculate True Dissipation:
Datasheet RDS(on) at 25°C is 0.26Ω. Assuming a Tj of 100°C, we apply a TCR multiplier of ~1.6. True RDS(on) = 0.416Ω.
Pd = 12A² × 0.416Ω = 59.9W.
2. Solve for Required RθSA:
Target Tj = 105°C. Ta = 40°C. RθJC = 0.83 °C/W (TO-247). RθCS = 0.3 °C/W (Arctic Silver thermal paste).
105 = 40 + 59.9 × (0.83 + 0.3 + RθSA)
65 = 59.9 × (1.13 + RθSA)
1.08 = 1.13 + RθSA
RθSA = -0.05 °C/W.
A negative RθSA is physically impossible; it implies a heatsink that cools below ambient without active refrigeration. The 12A load on a single IRFP460 is too high for this thermal envelope. Fix: Parallel two MOSFETs (halving Pd per device to ~30W) or switch to a lower RDS(on) part like the IXTN660N04T4.
3. Revised Sizing (Paralleled MOSFETs, 30W each):
105 = 40 + 30 × (1.13 + RθSA) → RθSA = 1.03 °C/W.
Looking at the Boyd Corporation (Aavid) thermal catalogs, a standard extruded aluminum heatsink like the Aavid 577302B00000G has a natural convection RθSA of about 3.5 °C/W. To hit 1.03 °C/W, we must add forced air. According to the manufacturer's derating curve, applying 400 Linear Feet per Minute (LFM) of airflow drops the RθSA of this specific sink to roughly 1.1 °C/W. We would select a slightly larger fin profile, like the Aavid 702102B00000G, which hits 0.95 °C/W at 400 LFM.
Interpreting the Derating Curve
Every power datasheet includes a Power Derating Curve. It shows that at a case temperature (Tc) of 25°C, the part can dissipate its max rated wattage (e.g., 280W). But the curve slopes downward linearly, hitting 0W at 150°C. If your thermal math dictates a case temperature of 100°C, you must read the Y-axis at that X-axis point to find your actual allowable dissipation limit. Never assume the headline wattage applies at operating temperatures.
Failure Signatures of Thermal Stress
When thermal management fails, components don't just stop working; they exhibit specific physical failure signatures:
- Electromigration: High current density combined with high temperature causes metal atoms in the silicon interconnects to physically migrate, eventually creating an open circuit. Governed by Black's Equation, the time-to-failure drops exponentially as Tj rises.
- Die Attach Delamination: The solder or epoxy bonding the silicon die to the copper leadframe expands and contracts at different rates (Coefficient of Thermal Expansion mismatch). Over hundreds of thermal cycles, this creates microscopic voids, spiking RθJC and causing sudden thermal runaway.
- Bond Wire Lift-Off: The aluminum wire bonds connecting the die to the package pins shear off at the heel due to thermomechanical fatigue.
- BJT Thermal Runaway vs. MOSFET Current Hogging: In BJTs, VBE has a negative TCR (drops ~2mV/°C). As a BJT heats up, it draws more current, heating up further—a classic thermal runaway. MOSFET RDS(on) has a positive TCR, which naturally balances current in parallel setups. However, the MOSFET's gate threshold voltage (VGS(th)) has a negative TCR. If paralleled MOSFETs are operated in the linear (sub-threshold) region rather than fully switched, the hotter device will turn on earlier and hog current, leading to localized melting.
Airflow and Enclosure Strategies
What do enclosure changes actually buy you? Moving from natural convection to forced air doesn't just double your cooling; it fundamentally changes the boundary layer physics.
| Cooling Method | Typical Airflow (LFM) | Relative RθSA Reduction | Best Application |
|---|---|---|---|
| Natural Convection | 0 - 50 | Baseline (100%) | Low power, silent audio, outdoor sealed |
| Low Forced Air | 100 - 200 | ~40% reduction | Desktop PCUs, motor drives |
| High Forced Air | 400 - 800 | ~65% reduction | Server racks, high-power inverters |
| Liquid Cold Plate | N/A | ~90% reduction | EV traction inverters, RF amplifiers |
Enclosure Tip: If you seal an enclosure to achieve an IP67 rating, Ta inside the box will rise significantly above room ambient. You must calculate the total internal wattage, multiply it by the enclosure's thermal resistance (often provided by the manufacturer in °C/W per square inch of surface area), and add that delta to your external ambient temperature before starting your Rθ math.
Frequently Asked Questions
How does the temperature coefficient of resistance affect PCB trace sizing?
When sizing PCB traces using the IPC-2221 standard, the calculators assume a specific temperature rise (usually 10°C or 20°C above ambient). Because copper has a positive TCR (+0.0039 °C-1), a trace that heats up by 40°C will see its resistance increase by roughly 15%. This higher resistance causes more I²R heating. If you are pushing high currents through narrow internal layers (which have worse thermal conductivity than external layers), you must oversize the trace width by 10-15% beyond the standard calculator output to compensate for the TCR-induced resistance bump.
Why do NTC thermistors have a negative temperature coefficient while copper is positive?
In pure metals like copper, heat causes the metal lattice to vibrate more intensely, which scatters the free-flowing electrons and increases resistance (positive TCR). NTC thermistors are made of sintered semiconductor ceramics (like manganese or nickel oxides). In these materials, thermal energy excites electrons across the bandgap, creating vastly more charge carriers. The explosion in available charge carriers overwhelms the lattice scattering effect, causing the overall resistance to drop dramatically as temperature rises (negative TCR).
Can I ignore the temperature coefficient when calculating power dissipation for short pulses?
Yes, but only if the pulse duration is shorter than the thermal time constant of the silicon die. For a typical TO-247 MOSFET, the die's thermal time constant is in the range of 1 to 10 milliseconds. If your pulse is 100µs (like in a switching power supply or a short-circuit protection event), the silicon physically does not have time to heat up, so the resistance remains at the 25°C baseline. For pulses longer than 50ms, you must consult the Safe Operating Area (SOA) curve and apply TCR derating, as the die temperature will have spiked.
How do I measure the actual temperature coefficient of a custom wirewound resistor?
Place the resistor in a temperature-controlled chamber or a silicone oil bath (for uniform heat transfer). Measure the baseline resistance at 20°C using a 4-wire Kelvin measurement to eliminate lead resistance. Raise the temperature to 70°C, allow it to soak for 30 minutes to ensure thermal equilibrium, and measure again. Use the formula α = (R2 - R1) / [R1 × (T2 - T1)]. For precision shunts, expect α to be in the single-digit ppm/°C range, requiring a 6.5-digit multimeter to resolve accurately.
