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Before choosing a trim sequence, determine what each accessible resistor actually controls. A network can contain several trim regions yet provide fewer independent electrical adjustments than its acceptance requirements need. A local sensitivity matrix makes that limitation visible. It also translates a resistance increment into the units of each circuit output, so a change that looks small in ohms can be judged against the functions it disturbs.
Key design decisions
- Define one row for each independently accepted circuit output and one column for each accessible resistor adjustment.
- Normalize sensitivities to meaningful electrical scales before comparing control strength or numerical conditioning.
- Check the predicted multi-output correction against the exact loaded circuit and a measured perturbation before translating it into irreversible material removal.
List the outputs that must remain independently correct
Use terminal names and operating conditions to define the accepted quantities. Two voltage ratios at different ladder nodes are distinct outputs; two algebraically identical representations of the same ratio are not. Include a separate absolute impedance requirement if it is measured and controlled, because matching voltage ratios can leave overall resistance unconstrained. A row should correspond to a real acceptance decision rather than to every number the instrument displays.
Identify available controls from the physical array. A resistor hidden by a component, inseparable from a parallel branch during measurement or outside its permitted remaining geometry cannot be counted as an ordinary adjustable column. Document the current resistance and usable direction for each control. The matrix describes local electrical influence; the separate manufacturing review determines whether the required adjustment is physically reachable and leaves an acceptable current path.
Express the local electrical influence as a normalized matrix
For nonzero outputs, define each sensitivity as the fractional change of an output divided by the fractional change of one resistor, with other conditions held constant. Multiplying the sensitivity matrix by a vector of small fractional resistor changes predicts the vector of fractional output changes. This provides a common scale when one element is ten kilohms and another is much larger.
A zero or near-zero output needs a different normalization. For bridge imbalance, use an agreed full-scale voltage or input reference rather than dividing by the imbalance itself. Otherwise an electrically well-behaved circuit can appear numerically singular simply because its desired output is zero. Record the normalization beside every row and preserve it when calculating uncertainty, correction vectors and validation residuals.
Sij = (Rj/yi)·∂yi/∂Rj; bi ≈ Σj Sij uj; uj = ΔRj/Rj
- yi is a nonzero circuit output at the current operating point; bi is its requested fractional correction.
- Rj is the measured starting resistance of adjustable element j.
- Sij is the dimensionless local sensitivity and uj is a proposed fractional resistance change.
The circuit remains in the same operating state and perturbations are small enough for local linearization. Zero-valued outputs require a separately stated fixed normalization.
Keep receiver loads inside a two-output ladder model
Consider three series resistors R1, R2 and R3 between excitation and return. The upper output is the node below R1; the lower output is the node above R3. With receiver resistance L1 from the upper output to return and L2 from the lower output to return, first calculate Z3 = R3 parallel L2. Then calculate Z2 = L1 parallel the sum of R2 and Z3.
The upper voltage ratio is Z2 divided by R1 plus Z2. The lower ratio is that upper ratio multiplied by Z3 divided by R2 plus Z3. These equations provide a compact exact starting model for computing the derivatives. A receiver load is not another trim control unless it is deliberately adjustable and part of the supplied system. Recompute the matrix if receiver mode, diagnostic circuitry or measurement input impedance changes; unloaded sensitivity signs alone do not quantify a loaded array.
Read the direction and strength of each ladder adjustment
For a worked idealization, remove both receiver loads and assume R1, R2 and R3 are each ten kilohms. The upper output ratio is two-thirds and the lower ratio is one-third. Differentiating the two ratios produces the table below. Each entry is output fractional change per unit fractional resistor change, evaluated at this equal-value state.
The first column moves both outputs by the same fractional amount. The other two columns produce different directions and strengths, allowing two independent ratio corrections locally. Adding all three columns gives zero in each row: increasing every resistance by the same percentage leaves both unloaded voltage ratios unchanged. This common scaling direction remains invisible unless another acceptance quantity, such as total resistance, is included.
| Output row | R1 adjustment | R2 adjustment | R3 adjustment |
|---|---|---|---|
| Upper ratio: (R2 + R3)/(R1 + R2 + R3) | −1/3 | +1/6 | +1/6 |
| Lower ratio: R3/(R1 + R2 + R3) | −1/3 | −1/3 | +2/3 |
Solve a two-output correction and check the nonlinear remainder
Using that hypothetical ladder, hold R1 fixed and request an upper-ratio increase of 0.15 percent and a lower-ratio increase of 0.20 percent. The first row gives (u2 + u3)/6 = 0.0015. The second gives minus u2/3 plus 2u3/3 = 0.0020. Solving yields u2 = 0.004 and u3 = 0.005: proposed increases of 40 ohms and 50 ohms from the ten-kilohm starting values.
Recalculate the exact ratios with R1 = 10,000 ohms, R2 = 10,040 ohms and R3 = 10,050 ohms. The upper becomes 20,090/30,090 and the lower becomes 10,050/30,090. Relative to their original values, the changes are approximately 0.14955 percent and 0.19940 percent. The small residuals are the expected finite-step difference from the linear prediction. Whether they are acceptable depends on the actual output limits; the arithmetic does not establish a trim accuracy.
Detect controls that cannot independently satisfy the outputs
If two columns are proportional, those adjustments move every monitored output in the same relative direction. They do not provide two independent correction directions. Likewise, duplicate output rows do not add an independent requirement. Check rank after scaling rows by the relevant electrical units or error allowances; otherwise a small numerical row can be overwhelmed merely by unit choice.
Nearly proportional columns are also troublesome. A mathematical inverse may exist but request large opposing corrections to produce a small output change. Measurement noise then causes a large change in the solved adjustment vector. Compare the correction with its uncertainty and permitted resistance ranges, and inspect the matrix condition using an appropriate numerical method. A least-squares answer is a compromise when requirements conflict; it must not be presented as satisfying every individual acceptance band.
Measure each column without confusing sensitivity with drift
Validate the circuit derivatives on an appropriate electrical surrogate or controlled test coupon before applying them to an irreversible final trim. Change one resistor by a known small amount, hold the other elements and receiver conditions fixed, and record all outputs after the same settling condition. Use the measured resistor change, not merely the commanded substitution value, when estimating a column.
Repeat with a second perturbation size. A consistent local slope supports the linear region; a slope that changes with amplitude indicates curvature or a changed circuit state. Where a reversible surrogate permits positive and negative perturbations, compare both directions and restore the baseline between trials. Keep a reference output or repeat baseline observations to detect thermal drift. Finally test one combined correction that was not used to estimate the columns and compare its entire output vector with the prediction.
Use sensitivity failures to revise the electrical control plan
A measured column with the wrong sign suggests mislabeled nodes, an unexpected parallel path, a receiver state change or an incorrect derivative. A column that changes with dwell may include self-heating instead of purely resistive response. Outputs that move while every resistor is held fixed point toward reference, amplifier, contact or acquisition drift. Resolve these signatures before allowing an adjustment algorithm to chase them.
A predicted solution containing a negative change cannot be executed by an upward-only resistor trim without another available degree of freedom. An extremely large correction may signal poor conditioning rather than inadequate trim length. When electrical independence is insufficient, revise the network topology, accessible controls or acceptance allocation. Pass the validated matrix, its operating point and its residual limits to the separate trim-sequence and endpoint reviews; they need that electrical mapping but still own cut direction, reachability and stopping behavior.
Provide the array's output-to-adjustment map
A sensitivity review needs the complete monitored circuit and every physically available control.
- Schematic, source and receiver impedances, named output equations, reference states and individual output acceptance bands.
- Measured starting resistances, accessible adjustable elements, permitted change directions and absolute resistance windows.
- One-at-a-time perturbation data with all outputs, measured resistance changes, settling times and measurement uncertainty.
- Proposed simultaneous correction, normalization conventions, exact-circuit check and any operating-state dependence of the response.
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