Electrical materials
A 1 mm copper wire has the same resistance at 50 Hz as at DC, but at 1 MHz it has four times more: the current crowds into a 66 μm layer next to the surface.
01Conducting materials
A conductor is chosen for its resistivity \( \rho \), its temperature coefficient \( \alpha \), its mechanical strength, its ease of soldering and its cost. The physics of conduction was covered in electrical properties of matter; here what matters is the material as an input to a product:
| Material | ρ at 20 °C (10⁻⁸ Ω·m) | α (1/K) | Use |
|---|---|---|---|
| Silver | 1.59 | 0.0038 | Contacts, coatings, conductive pastes |
| Annealed copper | 1.724 | 0.00393 | Cables, windings, printed circuit board traces |
| Gold | 2.44 | 0.0034 | Contacts that do not oxidize, chip bonding wires |
| Aluminum | 2.83 | 0.0040 | Overhead lines, integrated circuit metallization |
| Manganin, constantan | ≈ 45 | almost zero | Precision resistors and shunts |
| Nichrome | ≈ 110 | very low | Power resistors and heating elements |
The resistance of a conductor changes with temperature in an almost linear way:
Resistive alloys, such as manganin or constantan, have high resistivity and an almost zero temperature coefficient, and so they are used in precision resistors and in the shunts used to measure current.
02Dielectrics and insulators
A dielectric is characterized by its relative permittivity \( \varepsilon_r \), which decides the capacitance; its dielectric strength, the maximum field before breakdown; its loss factor \( \tan\delta \), the fraction of energy that is dissipated in each cycle; and its volume and surface resistivity.
| Material | Approximate εr | Use |
|---|---|---|
| Air | 1 | Variable capacitors; strength of about 3 kV/mm |
| PTFE (Teflon) | 2.1 | Coaxial cables and microwave boards, with very low losses |
| Polypropylene | 2.2 | Low-loss film capacitors |
| FR-4 (glass and epoxy) | 4.2 to 4.8 | General-purpose printed circuit boards |
| Alumina | 9.8 | Ceramic and hybrid substrates |
| Barium titanate | thousands | High-capacitance ceramic capacitors, which vary with voltage and temperature |
The maximum temperature of an insulator defines its thermal class according to IEC 60085: 105 °C (A), 120 °C (E), 130 °C (B), 155 °C (F) and 180 °C (H), among others. Above that temperature the insulator ages quickly; as a rule of thumb, its life is cut in half for every 8 to 10 °C of excess.
03The skin effect
In alternating current, the magnetic field inside the conductor induces currents that oppose the flow of current through the center, and the current concentrates near the surface. The current density falls off exponentially inward, with a penetration depth:
For a round wire the exact resistance comes from Bessel functions; when the radius \( r \) exceeds \( \delta \) several times, it is approximated by:
The ratio between the AC and DC resistance of a round wire, calculated with Bessel functions, as a function of frequency. The circle shows how the current is distributed across the cross section.
That is why at high frequency Litz wire is used, many fine insulated strands twisted so that each one occupies every position; silver-plated conductors, because the current flows only on the surface; and tubes instead of bars.
04In the lab
Measure with the four-terminal method the resistance of a copper winding at room temperature and after heating it, and compute the temperature rise from the change in resistance.
With an LCR meter, measure the series resistance of a solid-wire inductor and of a Litz one between 1 kHz and 100 kHz, and compare.
Measure the capacitance and dissipation factor of polypropylene, polyester and ceramic capacitors of the same nominal capacitance, at several frequencies.
05Common mistakes
- Calculating the losses of a high-frequency winding with the DC resistance. The skin and proximity effects multiply them.
- Forgetting the temperature coefficient of copper. A hot winding dissipates more than in the cold calculation.
- Taking the permittivity of FR-4 as constant. It varies with frequency, manufacturer and resin content.
- Running an insulator above its thermal class. It does not fail right away, but its life is greatly shortened.
06Self-assessment
What resistance do 100 m of 1.5 mm² copper have at 20 °C, and at 70 °C?
1.15 Ω at 20 °C and 1.38 Ω at 70 °C.
What is the penetration depth in copper at 10 kHz?
About 0.66 mm.
What maximum temperature does a class F insulator allow?
155 °C.
Why does Litz wire reduce the AC resistance?
Each strand is thinner than the penetration depth and the twisting makes all of them carry the same current.
07Further reading
- William D. Callister and David G. Rethwisch. Materials Science and Engineering: An Introduction. Wiley (Spanish edition: “Ciencia e ingeniería de materiales”, Reverté). The electrical and dielectric properties of materials, with tables of values.
- Charles A. Harper (ed.). Electronic Materials and Processes Handbook. 3rd ed., McGraw-Hill, 2004. The materials of the electronics industry and their processes.
- Frederick E. Terman. Radio Engineers' Handbook. McGraw-Hill, 1943. The classic tables of skin and proximity effects in conductors.