The relationship between electrical resistance and temperature dictates the reliability of every electronic system you build. In pure conductors like copper, resistance increases as temperature rises (a positive temperature coefficient). In semiconductors like silicon, resistance drops as temperature rises (a negative temperature coefficient). Managing this divergence is the entire foundation of thermal design. If you ignore the math, your copper traces will sag under voltage drop, and your silicon will melt into thermal runaway.
The Physics: How Temperature Alters Electrical Resistance
To design a reliable circuit, you must first understand how your materials react to heat. The behavior splits cleanly into two categories: Positive Temperature Coefficient (PTC) and Negative Temperature Coefficient (NTC).
| Material Type | Examples | Temp Coefficient | Physical Mechanism | Circuit Impact |
|---|---|---|---|---|
| Conductors (PTC) | Copper, Aluminum, Gold | Positive (+) | Lattice vibrations scatter electrons | Increased I²R losses, voltage drop |
| Semiconductors (NTC) | Silicon, Germanium | Negative (-) | Heat frees bound charge carriers | Thermal runaway, leakage current |
| Alloys (Zero/Low) | Manganin, Constantan | Near Zero | Competing effects cancel out | Stable shunt resistors, sense lines |
For copper PCB traces and wiring, the resistance at a given temperature is calculated using the linear approximation formula:
RT = R0 [1 + α(T - T0)]
Where α (alpha) is the temperature coefficient of resistance. For copper, α is approximately 0.00393/°C at 20°C. If you have a 10mΩ copper trace carrying 20A at 20°C, it dissipates 4W. If poor ventilation allows that trace to reach 100°C, its resistance climbs to roughly 13.1mΩ. Your dissipation jumps to 5.24W, which generates more heat, raising the resistance further. This is why oversizing traces and managing ambient heat are non-negotiable in high-current DC designs.
Thermal Path Math: Junction to Ambient (RθJA)
Heat flows through a semiconductor package exactly like current flows through a series of resistors. We use thermal resistance (Rθ), measured in °C/W, to model this path. The thermal equivalent of Ohm’s Law is:
TJ = TA + PD × (RθJC + RθCS + RθSA)
- TJ: Junction Temperature (the silicon die)
- TA: Ambient Temperature (the air around the heatsink)
- PD: Power Dissipation (Watts)
- RθJC: Junction-to-Case resistance (fixed by the manufacturer)
- RθCS: Case-to-Sink resistance (determined by your thermal interface material)
- RθSA: Sink-to-Ambient resistance (determined by your heatsink and airflow)
Understanding this chain is how you determine what airflow or enclosure changes will buy you. For example, a standard TO-220 package in still air has an RθJA of about 62°C/W. Adding a basic extruded aluminum heatsink might drop the RθSA component to 15°C/W. Strapping a 40mm 12V fan to that same heatsink (forced convection) can plummet the RθSA to 4°C/W. Conversely, sealing the project in a plastic enclosure traps hot air, artificially raising your local TA from a room-temperature 25°C to an internal 45°C, severely cutting into your thermal headroom.
Heatsink Selection: A Real-World Sizing Example
Let’s size a heatsink for a real application. We are switching a 12V, 30A continuous load using an IRF3205 N-channel MOSFET in a TO-220 package.
Step 1: Calculate True Power Dissipation
The datasheet lists the RDS(on) at 8mΩ, but that is at a 25°C case temperature. In reality, the die will be hot. The RDS(on) temperature coefficient for this silicon is roughly 2.0x at 120°C. We must design for the worst-case hot resistance: 16mΩ (0.016Ω).
PD = I² × RDS(on) = 30² × 0.016 = 14.4W.
Step 2: Define Temperature Limits
The absolute maximum TJ for the IRF3205 is 175°C. However, running silicon at its absolute limit guarantees early failure. We will set a practical target TJ of 115°C. Our enclosure’s internal ambient air (TA) is measured at 40°C.
Step 3: Calculate Maximum Allowed RθJA
Max RθJA = (TJ - TA) / PD
Max RθJA = (115 - 40) / 14.4 = 5.2°C/W.
Step 4: Subtract Known Resistances to Find Heatsink Requirement
| Path Segment | Value (°C/W) | Source / Material |
|---|---|---|
| RθJC (Junction-to-Case) | 1.0 | IRF3205 Datasheet |
| RθCS (Case-to-Sink) | 0.5 | Arctic MX-4 Paste + Mica Insulator |
| Required RθSA | 3.7 | 5.2 - 1.0 - 0.5 |
Step 5: Select the Hardware
We need a heatsink with an RθSA of 3.7°C/W or lower in natural convection. A standard cheap stamp-metal clip will not work (typically 20°C/W). We select the Wakefield Engineering 637K TO-220 heatsink, which is rated at approximately 4.0°C/W in natural convection, but drops to roughly 2.5°C/W with minimal chassis airflow. This gives us the 1.2°C/W safety margin we need to keep the junction safely under 115°C.
Derating Curves, Enclosures, and Failure Signatures
Every power semiconductor datasheet includes a power derating curve. This graph shows a flat line of maximum power dissipation up to 25°C, followed by a linear slope down to zero watts at the maximum junction temperature (usually 150°C or 175°C). This curve is the manufacturer explicitly telling you how hot is too hot for this part. If your case temperature hits 100°C, the derating curve might dictate that you can only safely dissipate 50% of the component's nominal wattage.
Derating curves assume the ambient air (TA) is actively being replaced. If you mount a properly heatsunk power supply inside a sealed NEMA enclosure without ventilation, the internal TA will rise until the heat escaping the enclosure walls equals the heat generated inside. A 40W loss inside a small sealed box can easily push internal ambient to 70°C, instantly invalidating your heatsink math and cooking the components.
When thermal limits are repeatedly exceeded, components do not just instantly vaporize; they exhibit specific failure signatures:
- Solder Creep and Fatigue: Repeated thermal cycling (heating up under load, cooling down at idle) causes the die-attach solder and PCB pads to expand and contract at different rates. Over time, microscopic cracks form, leading to intermittent connections or sudden open circuits.
- Electromigration: At high current densities and elevated temperatures (typically >105°C), metal atoms in the silicon interconnects are physically pushed by the electron wind. This causes voids (opens) or hillocks (shorts) inside the IC.
- Thermal Runaway (BJTs): Unlike MOSFETs, Bipolar Junction Transistors have a negative temperature coefficient for VBE. As they get hot, they turn on harder, pulling more current, generating more heat, until the silicon literally melts a hole through the package.
Frequently Asked Questions
Why does electrical resistance increase with temperature in copper but decrease in silicon?
In copper (a conductor), the outer electrons are already free to move. When heat is applied, the copper atoms vibrate more violently, creating a physical obstacle course that scatters the flowing electrons, increasing resistance. In silicon (a semiconductor), the electrons are largely bound to their atoms at room temperature. Adding thermal energy breaks these atomic bonds, freeing vastly more charge carriers. The massive increase in available carriers overwhelms the scattering effect, resulting in a net drop in resistance.
How much does adding a fan change the thermal resistance of a heatsink?
Forced convection drastically alters the boundary layer of stagnant air clinging to the heatsink fins. A standard extruded aluminum heatsink rated for 12°C/W in natural (still air) convection will typically drop to between 3°C/W and 5°C/W when subjected to just 1 to 2 meters per second of airflow from a standard 40mm or 60mm cooling fan. However, this introduces a mechanical point of failure; if the fan dies and the system relies on the 3°C/W rating, the silicon will rapidly overheat.
What are the visual or electrical signs of thermal stress on a PCB?
Visually, look for 'tombstoning' or micro-cracks in the solder joints around large power components, yellowing or delamination of the FR4 fiberglass near heat sources, and electrolytic capacitors bulging at the vent cross (heat accelerates electrolyte evaporation). Electrically, thermal stress manifests as unexplained voltage drops (due to increased trace resistance), erratic sensor readings, or a system that works perfectly on the bench but reboots randomly once installed in a warm, enclosed chassis.
