Resistance

Resistance is how strongly a material opposes current. Learn resistivity and the R = ρL/A formula, conductors vs insulators vs semiconductors, temperature coefficient, resistor types, the colour code and E-series values, tolerance, power rating, and series/parallel combinations.

Every wire, track and component has resistance. Engineers use it deliberately in resistors to set currents, divide voltages and bias transistors — and fight it in cables and PCB traces where it wastes power as heat.

Resistance depends on the material (resistivity), its length and its cross-section, and changes with temperature — which is also how thermistors and RTDs measure temperature.

An analogy

Resistance is like a narrow, rough pipe in a water system: a longer, thinner or rougher pipe lets less water through for the same pressure.

Materials

Conductors (copper ρ = 1.72 × 10⁻⁸ Ω·m, aluminium 2.8 × 10⁻⁸) have many free electrons. Insulators (glass, PTFE) have almost none. Semiconductors (silicon) sit between and can be controlled by doping, light and temperature — the basis of all electronics.

Resistor colour code

Four-band resistors: two digits, a multiplier and tolerance. Brown–black–red–gold = 1, 0, × 100 = 1 kΩ ± 5 %. Values come from E-series (E12, E24, E96) so any value is within tolerance of a standard part.

• Black 0, brown 1, red 2, orange 3, yellow 4, green 5, blue 6, violet 7, grey 8, white 9
• Gold ± 5 %, silver ± 10 %, brown ± 1 %

Temperature and self-heating

Metals have positive α (copper 0.00393/°C): resistance rises when hot. NTC thermistors fall sharply with temperature. Resistors must be rated for dissipated power (0.125 W, 0.25 W SMD, several watts for power parts) and derated at high ambient.

Real resistors

Parasitic inductance and capacitance matter at RF; thin-film resistors have low noise and tempco (±25 ppm/°C); current-sense shunts use milliohm values with Kelvin connections; PCB trace resistance is ≈ 0.5 mΩ per square for 1 oz copper.

Formulas

• Resistance of a conductor: R = ρ L / A
• Temperature dependence: R_T = R₀ [1 + α (T − T₀)]
• Series: R_eq = R₁ + R₂ + …
• Parallel: 1/R_eq = 1/R₁ + 1/R₂ + … (two: R₁R₂/(R₁ + R₂))
• Power rating check: P = I² R = V² / R ≤ rated power

Worked example: Cable resistance

Given: Copper, 10 m long, 1.5 mm² cross-section

Find: R.

• R = 1.72 × 10⁻⁸ × 10 / 1.5 × 10⁻⁶
• = 0.115 Ω

Answer: 0.115 Ω (0.23 Ω for the round trip)

Worked example: Hot copper coil

Given: Coil 100 Ω at 20 °C, α = 0.00393; Operating at 80 °C

Find: R at 80 °C.

• R = 100 × (1 + 0.00393 × 60) = 123.6 Ω

Answer: 123.6 Ω (+24 %)

Common mistakes

• Wrong: Ignoring power rating. — Right: Check P = I²R and choose a resistor rated with margin.
• Wrong: Using mm² directly in R = ρL/A. — Right: Convert area to m² (1 mm² = 10⁻⁶ m²).

Applications

• Pull-up / pull-down resistors: Define logic levels on MCU inputs.
• Current limiting: Series resistors protect LEDs and inputs.
• Temperature sensing: RTDs and thermistors convert temperature to resistance.

Key takeaways

• R = ρL/A.
• R rises with temperature in metals.
• Series add; parallel reciprocals add.
• Always check power rating.

Going further

• Sheet resistance: Ω/□ for PCB traces and IC resistors.
• Kelvin sensing: Four-wire measurement of milliohm shunts.
• Thermal noise: Johnson noise v = √(4kTRB).