Resistor

100 mA flowing through 100 Ω results in a power dissipation of 1 W, not 1 mW. Even if this unit conversion is correct, you cannot simply conclude the selection by "choosing a resistor with twice the power rating": ambient temperature, pad heat dissipation, continuous voltage, and pulse loads may each become limiting factors. This article expands from ideal resistance values to actual components, explaining how various parameters affect results using voltage divider, pull-up, and current sensing examples.

Ideal Resistors vs. Actual Resistors

An ideal positive resistor satisfies $u=Ri$, absorbs power $p=i^{2}R$ according to the passive sign convention, and does not store net energy. Actual resistors form conductive paths using resistive films, metal alloys, or resistance wire, converting electrical energy primarily into heat; they also possess lead/terminal inductance, parasitic capacitance, and temperature dependence.

Therefore, "10 kΩ" describes the resistance value under specified conditions, not an impedance that holds strictly true at any frequency, temperature, or voltage. At high frequencies, parasitic parameters may affect behavior; for wirewound structures, it is particularly incorrect to broadly claim they have low inductance; low-inductance winding is a specific design that requires checking the datasheet.

Nominal Values and Coding

For four-band color codes, the first two bands are significant digits, the third band is the multiplier, and the fourth band is the tolerance; five-band codes generally use the first three bands for significant digits, the fourth for the multiplier, and the fifth for tolerance. First confirm the reading direction; the tolerance band often has a larger gap from the others and cannot be identified solely by proximity to a lead.

For example, Brown-Black-Red-Gold is $10\times 100=1\text{ k}\Omega \pm 5\%$; Yellow-Purple-Orange-Gold is $47\times 1000=47\text{ k}\Omega \pm 5\%$. Small packages may have no markings at all; do not assume a blank component is zero ohms.

Common three-digit SMD codes like 103 indicate $10\times 10^3=10\text{ k}\Omega$, and four-digit codes like 1002 indicate $100\times 10^2=10\text{ k}\Omega$; 4R7 uses 'R' as the decimal point, indicating $4.7\Omega$. EIA-96 codes consist of two digits plus a letter and require a lookup table; they cannot be read directly like standard three-digit codes. Markings for special series are defined by the manufacturer.

E-series resistors organize preferred nominal values: E12 has 12 values per decade, such as 10, 12, 15, 18, 22, 27, 33, 39, 47, 56, 68, 82, multiplied by integer powers of 10. The series number alone does not prove component tolerance; you must confirm both resistance and tolerance in actual available specifications.

Precision is an Error Budget

Tolerance describes the allowable deviation of the actual resistance from the nominal value under specified reference conditions; it is not a complete guarantee of "measurement accuracy." Within an approximately linear temperature range, $R(T)\approx R(T_0)[1+\alpha (T-T_0)]$, where $\alpha$ is the TCR. 50 ppm/°C indicates a relative change of approximately $50\times 10^{-6}$ per degree Celsius; a temperature difference of 60°C corresponds to approximately 0.3%, with the direction depending on the sign of the coefficient.

Pay attention to the actual operating temperature of the resistor, including self-heating, not just room temperature. Initial tolerance, TCR, long-term drift, and voltage coefficient are not the same metric; you cannot eliminate all subsequent variations with a single room-temperature measurement.

Divider Precision Depends on Ratio

Assume two resistors are both nominally 10 kΩ with ±1% tolerance, the input is an ideal 10 V, and the load is negligible. The output voltage is highest when the upper resistor is 9.9 kΩ and the lower is 10.1 kΩ, yielding 5.05 V; the lowest is 4.95 V. Relative to the nominal 5 V, this is ±1%. Here, the absolute changes in the two resistance values do not simply add up to a ±2% output error.

If both resistors change proportionally with temperature, the divider ratio remains constant, but the total current changes. Therefore, the ratio tolerance and tracking TCR of precision resistor networks may be more important than the absolute TCR of a single resistor. On the other hand, if the upper and lower resistors heat differently or layout causes temperature differences, ratio drift may occur even if the nominal TCR is the same.

The mean square value of thermal noise voltage under classical approximation is $4k_{B}T{R}_B$, where $T$ is absolute temperature and $B$ is observation bandwidth. Under the same $R$, $T$, and bandwidth, this noise does not disappear just because you switch to "metal film"; material and structure also affect characteristics such as excess noise under bias. Do not treat "low noise for a certain type" as "no noise."

Power is Not the Only Upper Limit

Calculate the worst-case operating point first, then compare with datasheet conditions. For a constant current of 0.1 A through 100 Ω, $P=0.1^2\times 100=1\text{ W}$; if both current and resistance vary, find the maximum case of the product $I^{2}R$. Rated power usually depends on specified ambient or terminal temperatures and mounting conditions; derate according to curves as temperature rises. PCB copper area, nearby heat sources, and package affect actual temperature rise. Vishay: Basics of Fixed Resistors

For example, assume a 1 MΩ device allows continuous power dissipation of 0.1 W; a pure power calculation gives approximately 316 V. If its maximum operating voltage is 100 V, it is actually limited by 100 V first. These values are only to illustrate constraint relationships and do not correspond to specific device recommendations.

A 1 W, 1 ms rectangular pulse has an energy of 1 mJ, but equal energy does not mean all pulse widths are equivalent. Local temperature of the film, maximum pulse voltage, and cumulative heating from repetition must all be checked. "Choosing twice the power" can be part of project margin but cannot replace these conditions. Vishay: Pulse Load Handling

Package names must also specify the system: the commonly referred imperial 0402 is approximately metric 1005, about $1.0\times 0.5\text{ mm}$; metric 0402 is approximately $0.4\times 0.2\text{ mm}$. Different series of the same size may have different power ratings and voltage withstand capabilities.

Pull-up Values Have Upper and Lower Bounds

Pull-up resistors bias high-impedance nodes toward the supply potential; when an open-drain output pulls low, the resistor limits the current flowing into the output transistor. It must allow the low-level current to be tolerable while ensuring the rise time after release is fast enough; you cannot simply select "commonly used 4.7 kΩ."

Here is a teaching example with explicit parameters: VDD = 3.3 V, the device guarantees $V_{OL}\le 0.4\text{ V}$ at a 3 mA sink current, bus capacitance is 100 pF, and the requirement is that the 30% to 70% rise time does not exceed 300 ns. Typical calculations using these boundaries are:

• Lower bound $R_{min}=(3.3-0.4)/0.003\approx 967\Omega$;
• Upper bound $R_{max}=300\text{ ns}/(0.8473\times 100\text{ pF})\approx 3.54\text{ k}\Omega$.

The coefficient 0.8473 comes from the time difference of the exponential charging curve from 30% to 70%, $\ln (0.7/0.3)RC$, and is not a universal constant for all rise time definitions. TI: I²C Pull-up Resistor Calculation

2.2 kΩ falls within this nominal range, with an expected rise time of approximately 186 ns; final judgment must use worst-case deviations of supply, resistance, and capacitance, as well as guaranteed conditions of the corresponding devices and protocols. If $R_{min}$ is greater than $R_{max}$, it indicates that the current drive, capacitance, and timing requirements have no feasible resistance value; this cannot be solved by arbitrarily selecting a middle value.

For ordinary static inputs, leakage must also be checked: for example, 10 μA flowing through 100 kΩ produces a 1 V offset, which may affect high-level margin. The same applies to pull-downs, just in the opposite direction. When multiple devices have built-in pull-ups, they are in parallel, and the total resistance may be much smaller than a single value.

Can the pull-up window disappear?

The horizontal lines define 30% and 70%, not zero-to-supply time. Increasing capacitance shrinks Rmax. When the interval vanishes, change capacitance, drive capability or timing requirements. Equality satisfies a bound.

$$t_{30\to 70}=RC\ln \frac{0.7}{0.3},R_{min}\le R\le R_{max}$$

Shunt Resistors and Kelvin Connections

Shunt resistors are placed in series in the measured current path to infer current via $V_{sense}=IR$. 5 A through 10 mΩ yields 50 mV and dissipates 0.25 W; changing to 1 mΩ yields only 5 mV and 25 mW, reducing loss, but the same measurement offset constitutes a significantly larger proportion.

Assume the amplifier input offset is 50 μV; this single item accounts for approximately 0.1% of a 50 mV signal, but approximately 1% of a 5 mV signal. Smaller resistors also make trace resistance and thermoelectric potentials more significant, so "the smaller the R, the better" lacks measurement resolution and error condition context.

If the target resistance is 10 mΩ, but the voltage drop of an additional 2 mΩ high-current trace is included in the reading, the calculated current will be off by approximately 20%. Kelvin connections separate current leads from voltage sensing leads, reducing shared conductor voltage drop entering the measurement; they do not eliminate resistance tolerance and self-heating TCR, nor can they remedy incorrect common-mode ranges. Recommended sensing positions for four-terminal and two-terminal devices should be followed according to the datasheet layout.

High-side sensing faces common-mode voltages close to the supply potential, while low-side sensing may raise the load ground. When choosing the location, consider amplifier input range, load reference ground, and fault observability together, not just resistor loss.

from math import log, isclose
assert isclose(.1**2*100, 1.0)
ratio_low = 9900/(10100+9900)
ratio_high = 10100/(9900+10100)
assert (ratio_low, ratio_high) == (.495, .505)
rmin = (3.3-.4)/.003
rmax = 300e-9/(log(.7/.3)*100e-12)
assert rmin < 2200 < rmax
for resistance, expected in [(.01, .001), (.001, .01)]:
    signal = 5*resistance
    assert isclose(50e-6/signal, expected)
print(f"Power 1 W; Pull-up nominal range {rmin:.1f}–{rmax:.1f} Ω; Sampling offset ratio passed")

Boundaries of Variable and Nonlinear Resistors

The NTC commonly uses the simplified B-parameter model $R(T)=R_{25}\cdot \exp [B(1/T-1/298.15)]$, where $T$ must be in Kelvin. It is an approximation for a limited temperature range and does not imply that a single B value can accurately cover all temperatures. Self-heating from measurement current ($I^{2}R$) causes the sensor temperature to deviate from the measured environment.

These components cannot be uniformly substituted into fixed R linear analysis, and MOVs are not ideal shorts once a certain voltage is crossed. Coordination of protection devices is expanded in the subsequent protection circuit chapter. Next, read Capacitor to understand energy storage, impedance, and actual decoupling networks.