Resistor Networks: Separating Absolute and Ratio Error Budgets

Identify which resistance relationship the circuit uses

Begin with the transfer function rather than the component names. An inverting amplifier uses a feedback-to-input resistance ratio, while a passive divider uses the lower resistance divided by the sum. A bridge or difference amplifier can depend on agreement between two separate ratios. An instruction such as all resistors matched to 0.1 percent leaves these mathematically different requirements unresolved.

Write the required transfer function and name its terminals. Then identify which resistors can be measured independently in the finished circuit. Parallel paths, input protection and connected semiconductor devices can change an apparent ohmmeter reading. A specification should distinguish isolated resistor measurements from powered circuit calibration and explain which result decides acceptance when the two are not numerically identical.

Separate common movement from differential movement

Let two nominally equal elements have fractional errors e1 and e2. Their common component is the average of the two errors; the differential component is their difference. If both resistors are five percent high, their ratio remains one. Nevertheless, their common shift changes the current in a divider, its output impedance and the loading imposed on the source. Ratio cancellation is useful, but it does not make absolute resistance irrelevant.

For small errors, the fractional error of R2 divided by R1 is approximately e2 minus e1. This relation explains why a common process shift can cancel while an opposite-signed shift doubles the error. Use the exact ratio when the errors are large. Keeping both representations in the analysis prevents a favorable matching statistic from hiding a network that is outside its permitted impedance range.

Translate matching error into circuit error

Consider a hypothetical equal-resistor divider supplied by an ideal source. Suppose one element is 0.2 percent high and the other is 0.2 percent low. Dividing the higher resistance by the lower gives a component ratio about 0.4008 percent above unity, but the divider output differs from one half of the supply by 0.2 percent relative to that intended output. Quoting the ratio error as the output error would therefore misstate this circuit's sensitivity.

Now move both elements upward by three percent while maintaining equality. The unloaded midpoint remains one half, yet the total current decreases and the output resistance increases. A finite receiver impedance will respond to that second change. Calculate the actual circuit transfer at the absolute-value extremes as well as at the ratio extremes; the combination, rather than either number in isolation, defines the allowable network envelope.

Build two linked budgets, not one tolerance column

Use one budget for impedances and dissipations and another for the signal transfer function. Their entries may share measurements, but their acceptance consequences differ. The table shows how to assign observations without double counting a single physical mechanism. A final system budget also includes errors outside the resistor network, such as reference, amplifier and conversion effects.

Keep correlation when combining variation

A root-sum-square calculation is not automatically justified by calling a quantity a tolerance. Tolerances are acceptance bounds; standard deviations describe observed distributions. A worst-case assessment combines adverse permitted limits, whereas a statistical estimate requires a distribution model and an understood relationship between variables. Mixing the two can create an optimistic number with no defensible interpretation.

For measured fractional errors, the variance of their difference equals the sum of the individual variances minus twice their covariance. Positive correlation can improve ratio consistency; negative correlation can worsen it. Retain paired readings from the same network instead of shuffling them into two independent columns. Analyze within-panel and between-run movement separately so an apparent correlation is not merely an artifact of the sampling arrangement.

Evaluate the powered network, not just an isothermal pair

A matched temperature coefficient is meaningful when the temperature history is defined. If R1 and R2 operate at different temperatures, their individual coefficients multiply different temperature changes. Close placement may reduce a gradient but does not prove equal temperatures under every power combination. A nearby power device, mounting clamp or copper attachment can make otherwise similar elements thermally unequal.

Measure the network at low test power first, then under the relevant supply and load states. Change one power source at a time and observe both resistances or the actual ratio. Keep enough settling time to distinguish immediate loading from slower heating. When a transient specification matters, capture the ratio during warm-up as well as after equilibrium; a stable final value does not bound the entire switching interval.

Make the verification uncertainty smaller than the decision

Ratio measurements can reject some common instrument errors when the same instrument and conditions are used appropriately. They do not automatically cancel switching resistance, thermoelectric offsets, time drift or different measurement currents. If one element is read much later than the other while the substrate is warming, the calculated ratio includes a timing error that looks like poor matching.

Define the connection sequence and repeat it in reverse order. Compare direct ratio measurements with independently measured values on a retained subset. Use Kelvin sensing where terminal resistance is significant and check the receiver loading when measuring a powered divider. Report the uncertainty at the required ratio, not merely the number of display digits, and establish a consistent decision rule for results close to the acceptance boundary.

Express the final requirement in circuit terms

A complete network callout names the absolute windows, specific ratios, reference temperature, operating temperature range, allowed power states and electrical connection points. It also states whether the ratio limit applies initially, after defined processing, or throughout a specified environmental test sequence. These are different obligations and should not be compressed into a single matching figure.

For networks with several functions, attach a small matrix relating each circuit output to its controlling resistor pair. This exposes conflicting trim objectives before manufacturing: changing an element to improve one gain may disturb another ratio. Where independent adjustment is required, reserve the necessary terminals and accessible trim regions in the layout. The result is an actionable network requirement rather than an attractive but incomplete precision claim.