Resistor tolerance changes an op amp’s closed-loop gain because the external resistors set a ratio. In a two-resistor feedback network, the resistors’ errors can reinforce one another: two independent 1% resistors can produce about 2% worst-case gain error. The exact result depends on whether the circuit is inverting, non-inverting, or differential—and resistor tolerance is only one part of total gain accuracy.
How the feedback resistors set gain
For an ideal op amp, the familiar gain equations are:
- Inverting amplifier:
Av = −Rf/Rin - Non-inverting amplifier:
Av = 1 + Rf/Rg
Here, Rf is the feedback resistor, Rin is the input resistor in an inverting circuit, and Rg is the resistor from the inverting input to ground in a non-inverting circuit. The resistor ratio—not either resistor in isolation—sets the ideal closed-loop gain.
A 10 kΩ resistor marked ±1% may have an initial resistance from 9.9 kΩ to 10.1 kΩ. That tolerance does not specify how the resistance changes with temperature, time, voltage, self-heating, or frequency, nor whether another resistor will track it.
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Calculate worst-case gain for an inverting amplifier
To find the largest gain magnitude, use the highest possible feedback resistance and the lowest possible input resistance. To find the smallest magnitude, use the reverse combination. For nominal values and fractional tolerances tf and tin:
- Maximum magnitude:
|Amax| = Rf,nom(1 + tf)/[Rin,nom(1 − tin)] - Minimum magnitude:
|Amin| = Rf,nom(1 − tf)/[Rin,nom(1 + tin)]
Example: nominal gain −10 with 1% resistors
With Rin = 10 kΩ and Rf = 100 kΩ, nominal gain is −10. If both independent resistors can be anywhere within ±1%, the resistor-only extremes are:
- Largest magnitude:
−(100 kΩ × 1.01)/(10 kΩ × 0.99) ≈ −10.202 - Smallest magnitude:
−(100 kΩ × 0.99)/(10 kΩ × 1.01) ≈ −9.802
So the possible range is approximately −9.802 to −10.202. Relative to nominal magnitude, that is about −1.98% to +2.02%; the small asymmetry comes from the exact ratio calculation. For equal independent tolerances t, the exact relative extremes are −2t/(1 + t) and +2t/(1 − t). For small t, designers often approximate this as ±2t.
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Calculate worst-case gain for a non-inverting amplifier
The non-inverting equation includes a fixed 1, so resistor errors have less relative effect than they do on an otherwise comparable pure ratio. The exact limits are:
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Amax = 1 + Rf,nom(1 + tf)/[Rg,nom(1 − tg)] - Minimum gain:
Amin = 1 + Rf,nom(1 − tf)/[Rg,nom(1 + tg)]
For small resistor errors, the relative gain change is approximately ΔAv/Av ≈ [(Av − 1)/Av](ΔRf/Rf − ΔRg/Rg). With equal independent tolerances t, the worst-case first-order magnitude is approximately 2t(Av − 1)/Av.
Example: nominal gain +11 with 1% resistors
For Rg = 10 kΩ and Rf = 100 kΩ, nominal gain is 11. The resistor-only extremes are 1 + 101/9.9 ≈ 11.202 and 1 + 99/10.1 ≈ 10.802. That is approximately −1.80% to +1.84% relative to nominal. The first-order estimate is ±2 × 1% × 10/11, or about ±1.82%.
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Compare common cases
| Topology | Nominal gain | Equal resistor tolerance | Approximate worst-case resistor-only error |
|---|---|---|---|
| Inverting | −2 | ±1% | ±2% |
| Inverting | −10 | ±1% | ±2% |
| Inverting | −10 | ±0.1% | ±0.2% |
| Non-inverting | +2 | ±1% | ±1% |
| Non-inverting | +11 | ±1% | ±1.82% |
| Non-inverting | +101 | ±1% | ±1.98% |
These are approximate worst-case estimates from resistor tolerance alone. The non-inverting sensitivity factor approaches 1 as gain rises, so its resistor-tolerance effect approaches that of a two-resistor ratio.
Worst-case limits and statistical estimates answer different questions
Use worst-case analysis when a design must meet a guaranteed production limit, such as a safety, calibration, or hard accuracy requirement. Combine resistor extremes in the directions that maximize the deviation; do not assume one error will cancel another.
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1Scan for outdated or missing drivers - takes under a minute2Clear out junk files and repair common Windows errors3Fix the driver behind crashes, sound loss and screen glitchesIf resistor errors are independent and random, root-sum-square (RSS) can estimate typical ratio spread. For two equal independent distributions, the ratio variation is roughly √2 × t, compared with the worst-case approximation of 2t. RSS is not a guaranteed limit and is meaningful only when the distribution, independence, manufacturing process, and meaning of the specified tolerance are understood. Monte Carlo simulation can help explore those assumptions, but it does not replace a worst-case guarantee or physical validation.
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Choose resistor tolerance from the gain-error budget
For an inverting stage with equal tolerances on both resistors, a useful first estimate is t ≤ Egain/2, where Egain is the allowed resistor-only gain error. This is a starting point, not a total-accuracy guarantee.
| Resistor choice | When it is a reasonable fit | Trade-off or caution |
|---|---|---|
| 5% | Exploratory or educational circuits, wide acceptance limits, or designs calibrated after assembly | A nominal −10 inverting stage could range from about −9.048 to −11.053 from resistor tolerance alone, roughly −9.5% to +10.5% relative to nominal. |
| 1% | General-purpose amplification where a few percent of resistor-only gain error is acceptable | Two independent 1% resistors can produce about 2% worst-case error in a ratio-based gain. |
| 0.1% | Applications needing roughly 0.2% or better resistor-only error in an inverting ratio, provided other errors are also controlled | Tighter initial tolerance does not automatically provide equivalent temperature tracking. |
| Matched network or calibration | Very tight ratio accuracy, differential measurements, high CMRR, or production correction | Check ratio match over the required temperature range; calibration does not remove drift, noise, nonlinearity, or every mismatch effect. |
If the allowable resistor-only error is 2%, 1% resistors are a reasonable first estimate for an inverting ratio. For 0.2%, start around 0.1%; for 0.02%, consider a network specified for approximately 0.01% ratio matching or calibration. Leave margin for temperature, aging, and op-amp errors.
Distinguish tolerance, ratio matching, and temperature coefficient
Absolute tolerance bounds an individual resistor’s initial value. Ratio matching specifies how closely two or more resistors relate to one another. Temperature coefficient (TC) describes resistance change with temperature. These are separate specifications.
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For gain ratio G = Rf/Rin, approximate relative drift with temperature is (1/G)(dG/dT) ≈ TCRf − TCRin. Similar TCs can make drift cancel when resistors track thermally. For a non-inverting amplifier, the corresponding approximate relative gain drift is weighted by (Av − 1)/Av. A matched network in one package often tracks better than discrete resistors separated on a board; matched sets can improve temperature-coefficient performance substantially when their specifications support the required conditions. See Analog Devices’ discussion of resistor matching and gain-setting drift.
Why difference amplifiers need matching
In a single inverting or non-inverting stage, the key passive concern is generally the gain-setting ratio. In a four-resistor difference amplifier, mismatched ratios also convert common-mode input into differential output, reducing common-mode rejection ratio (CMRR). Analog Devices notes that even an ideal op amp with 0.1% resistors in a four-resistor difference amplifier can have minimum CMRR of only about 54 dB; its matched-network guidance describes ratio matching as tight as 0.01%. Those figures illustrate why matching can matter more than individual absolute accuracy in this topology. See the topology discussion and matched resistor network guidance.
A network can have looser absolute resistance accuracy while offering tight ratio matching. Check that its absolute values still suit loading and bias-current requirements, and that the matching specification applies over the intended temperature range. For critical differential measurement, an integrated difference amplifier or instrumentation amplifier may offer a more controlled alternative than discrete resistors.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Check whether resistor tolerance is the dominant error
Resistor-only calculations assume ideal op-amp behavior. In a real circuit, build a total error budget before paying for tighter resistors. TI separates resistor-tolerance error from op-amp error sources in its application note on precision op-amp error sources.
- Finite open-loop gain: The ideal gain equations assume infinite open-loop gain. Finite gain creates closed-loop error, especially at high closed-loop gain or high frequency.
- Input offset voltage: Offset is amplified by noise gain, not necessarily signal gain. Noise gain is
1 + Rf/Rgfor a non-inverting circuit and1 + Rf/Rinfor an inverting circuit. Thus an inverting signal gain of −10 has noise gain 11. See Analog Devices’ explanation of noise gain. - Input bias current: Current through the resistor network creates an offset; larger resistance values generally increase its effect.
- Source resistance: Finite input impedance and source resistance can form an additional divider and change gain. See Analog Devices’ discussion of DC error sources.
- Temperature, aging, and self-heating: Initial tolerance does not describe drift over time or operating conditions.
- Frequency and parasitics: Resistor and PCB capacitance can alter the feedback ratio at higher frequencies and affect stability. See Analog Devices’ variable-gain feedback note.
- Loading and output limits: Resistor values determine feedback current and loading. Low values can increase output-current demand and dissipation; high values can increase bias-current error, thermal noise, leakage sensitivity, and capacitive pickup.
Near unity gain, the non-inverting resistor ratio contributes less relative gain error because the fixed 1 dominates, but offset and input common-mode limitations can still matter. At high gain, resistor-ratio sensitivity approaches the inverting case and finite bandwidth and open-loop gain become more consequential.
Quick Recap
A practical design sequence
- Identify the topology. Determine whether the circuit is inverting, non-inverting, or differential, then write its gain equation.
- Set the actual accuracy requirement. Decide whether it applies at initial temperature, across an operating range, over product life, or after calibration.
- Calculate resistor-only limits. Use the exact maximum and minimum formulas with the individual tolerances; use first-order estimates only for quick selection.
- Allocate the remaining error. Include op-amp offset, bias current, open-loop gain, source and load effects, temperature drift, reference and ADC errors, and any relevant parasitics.
- Choose the passive strategy. Use discrete 1% parts for forgiving general-purpose circuits, tighter thin-film parts when initial accuracy warrants them, and matched networks when ratios or thermal tracking dominate.
- Validate the design. Simulate sensitivity or statistical spread where useful, then measure representative hardware across required conditions. Calibration can correct initial gain error, but does not inherently fix drift, noise, nonlinearities, or CMRR loss from resistor mismatch.
References
- Texas Instruments: Precision Op Amp Error Sources
- Analog Devices: DC Error Characteristics of an Op Amp
- Analog Devices: Op Amp Issues—Noise Gain
- Analog Devices: Choosing a Precision Amplifier Topology
- Analog Devices: Matched Resistor Networks for Precision Amplifier Applications
- Analog Devices Application Note 42
- Analog Devices Application Note 1206
- Texas Instruments: Difference Amplifier Resistor Tolerance and CMRR
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