Advanced: physics, mathematics, and applications
An ideal resistor is defined by a constant ratio of voltage to current, but real resistors also have temperature dependence, noise, parasitic inductance and capacitance, voltage limits, pulse limits, aging, and construction-specific behavior. Engineering selection requires more than choosing the nearest resistance value.
Resistivity and geometry
For a uniform material:
- R is resistance in ohms.
- ρ is material resistivity in ohm-metres.
- L is conductor length.
- A is cross-sectional area.
Longer current paths increase resistance, while greater cross-sectional area reduces it. Film resistors exploit geometry by cutting a long path into a thin resistive layer. Current-sense resistors use low-resistivity alloys and carefully controlled geometry to obtain small, stable values.
Conductance
Conductance is measured in siemens. Parallel networks are often easier to analyze by adding conductances:
Temperature coefficient
Over a limited temperature range, resistance is often approximated by:
The coefficient α is often given in parts per million per degree Celsius (ppm/°C). A 100 kΩ resistor with a coefficient of 50 ppm/°C changes by about 0.005% per degree Celsius, before other errors are considered.
Fractional change = 50 × 10−6 × 40 = 0.002
Resistance change ≈ 200 Ω
Self-heating changes the resistor's own temperature. Precision circuits must consider ambient temperature, thermal resistance, board heating, airflow, nearby power devices, and the time required to reach thermal equilibrium.
Power, energy, and pulse loading
Energy during a pulse = ∫ P(t) dt
A steady-state wattage rating does not fully describe short pulses. Different constructions store and spread heat differently. Pulse-capability graphs may limit peak power, pulse width, repetition rate, and total energy. A resistor that survives one pulse may fail under repeated pulses because it cannot cool between events.
Voltage coefficient and maximum working voltage
Some resistor materials change value slightly with applied voltage. Even when that effect is negligible, insulation and geometry impose a maximum continuous working voltage. The allowable operating voltage is therefore the lower of the voltage set by power rating and the manufacturer's absolute voltage limit.
Johnson-Nyquist noise
Any resistor above absolute zero produces thermal voltage noise. For an ideal resistor over bandwidth B:
- k is Boltzmann's constant.
- T is absolute temperature in kelvins.
- R is resistance.
- B is noise bandwidth in hertz.
Higher resistance, temperature, or bandwidth increases thermal noise. Real resistors may also produce excess noise when DC current flows; this depends strongly on construction and operating level.
Parasitic inductance and capacitance
At low frequency, a resistor may behave almost ideally. At higher frequency, lead length and spiral construction add inductance, while the body, terminals, and circuit board add capacitance. Wirewound resistors can be especially inductive unless wound by a non-inductive method. Surface-mount film resistors usually have smaller parasitics, but package size and layout still matter.
A practical high-frequency model may include an ideal resistance with series inductance and shunt capacitance. At sufficiently high frequency, impedance is no longer equal to the marked DC resistance.
Tolerance, drift, and error budgets
Initial tolerance is only one term in a precision error budget. Others may include:
- Temperature coefficient and temperature gradients.
- Long-term aging and humidity effects.
- Soldering and mechanical stress.
- Voltage coefficient.
- Self-heating.
- Contact and lead resistance.
- Thermoelectric voltages in low-level measurements.
Ratio circuits can be more accurate than their individual resistor tolerances when a matched network provides close ratio tracking and similar temperature. This is one reason precision amplifiers and converters often use integrated resistor networks.
Preferred-number series
Standard resistor values are commonly arranged in E-series. Each decade is divided into approximately equal logarithmic steps. E6, E12, and E24 suit broader tolerances; E48, E96, and E192 provide finer choices for tighter tolerances. The series reduces inventory while ensuring that neighboring nominal values overlap sensibly with their tolerances.
Current sensing and Kelvin connections
A low-value shunt converts current into a measurable voltage. At milliohm values, trace, lead, and contact resistances can be comparable to the shunt itself. A four-terminal or Kelvin connection separates the current path from the voltage-sense path so the measuring circuit does not include most of those unwanted drops.
Feedback and gain setting
Resistor ratios set the gain of many operational-amplifier circuits. For an ideal non-inverting amplifier:
The absolute values affect input bias-current error, noise, loading, stability, and power, while the ratio primarily sets gain. Very high values reduce loading but increase thermal noise and sensitivity to leakage. Very low values consume current and load the amplifier.
Digital pull resistors
A pull-up or pull-down resistor establishes a known logic state when no active device is driving the node. Selection balances current consumption, switching speed, noise immunity, leakage, input thresholds, and bus capacitance. Open-collector and open-drain outputs depend on an external pull-up to create the high state.
Transmission-line termination
When conductor length and signal rise time make a connection behave as a transmission line, a resistor may be chosen to match or approximate the line's characteristic impedance. Correct source, parallel, or network termination reduces reflections, ringing, and false switching. The required value is based on the interconnect, driver, receiver, and topology rather than on DC current alone.
Precision measurement
Two-wire resistance measurement includes test-lead and contact resistance. Four-wire measurement uses one pair of leads to force current and another pair to measure voltage. For very high resistance, leakage, shielding, insulation, humidity, and meter input current become dominant concerns. Laboratory standards use temperature-controlled reference resistors and comparison bridges, with traceability ultimately linked to national resistance standards.
Choosing a resistor construction
| Construction | Typical strengths | Possible limitations |
|---|---|---|
| Carbon composition | Useful pulse behavior in certain applications; historically common. | Broader tolerance, drift, and excess noise compared with many film types. |
| Carbon film | Low cost and general-purpose use. | Usually less precise and stable than good metal film. |
| Metal film | Low noise, tighter tolerance, good stability. | Pulse and overload capability depends on construction. |
| Metal oxide | Useful temperature and overload performance. | Not automatically the best choice for precision or high frequency. |
| Wirewound | Power handling, low values, and precision options. | Can be inductive and physically larger. |
| Thick-film chip | Small, inexpensive, and widely available. | Greater excess noise and voltage coefficient than some thin-film parts. |
| Thin-film chip | Precision, low noise, close ratios, good stability. | Higher cost and construction-specific pulse limits. |
| Metal strip or foil shunt | Very low resistance, current sensing, low inductance designs available. | Layout and thermal design are critical. |
Sources and further reading
- NIST — SI units and the ohm
- NIST Technical Note 1458 — standard resistor calibration
- Vishay — resistor color-code chart
- Vishay — precision thin-film resistor example and specifications
- For actual design limits, use the exact resistor manufacturer's current datasheet and derating information.