Log-Ratio Sensor Networks: Separate Reference-Current Error from Logarithmic Slope

System boundary

A DC unipolar log-ratio front end with a separately selected active logarithmic converter and drawing-defined passive reference or scaling network. RF envelope detectors, bipolar compression and complete optical sensor performance are outside this model.

Integration interfaces

Integration risks

Specify a dimensionless ratio inside the logarithm

Use a transfer written as Vout = V0 plus S times log10(I/Iref), where both currents are positive and expressed in the same units. S is the output change per decade, not ordinary linear voltage gain. V0 is the output at I equal to Iref in this convention. Changing the logarithm base without changing S changes the stated transfer.

Check the selected active device's current direction, input voltage and permitted ranges before applying that relationship. A DC log-ratio channel is not automatically an RF power detector or a bipolar signal compressor. The passive thick-film network may set reference current, scale or offset, but the active logarithmic function and its conformity limits belong to the actual circuit implementation.

Calculate what a reference-current change looks like

If the reference increases by a factor of one plus epsilon while the signal current stays fixed, the output shift is minus S times log10(one plus epsilon). This is constant across signal currents in the ideal model. The change moves the intercept in the log-domain plot rather than changing its slope. A reference-programming resistor therefore needs an error allocation linked to current generation, not just a generic matching requirement.

For an assumed slope of 0.5 volt per decade and a one-percent reference-current increase, the output shifts by approximately minus 2.1607 millivolts. The same ideal shift appears at signal-to-reference ratios of ten and one thousand. These illustrative values do not describe a released network. The actual reference current may depend on a nonzero input-node voltage as well as the source voltage and resistor.

Use decade spacing to expose a slope error

An error delta S in slope contributes delta S times log10(I/Iref). It is zero at the chosen reference ratio and grows in magnitude farther from that point. If a slope intended to be 0.5 volt per decade is one percent high, the error is 5 millivolts at one decade above the reference and 15 millivolts at three decades. A constant offset cannot remove both simultaneously.

Use two or more well-separated supported current ratios when checking scale. Keep their reference current unchanged so the experiment does not mix slope with reference movement. If a calibration is performed away from I equal to Iref, its apparent pivot moves to that calibration point, but the residual still varies with log current. State the pivot explicitly when comparing before-and-after results.

Put additive current errors before the logarithm

Suppose an unwanted positive current Il adds to the signal input. Its output error is S times log10(one plus Il/I), not a fixed number of millivolts over the full range. An assumed 1 nanoampere leakage added to a 10 nanoampere signal creates approximately 20.696 millivolts of error at S equal to 0.5 volt per decade. At a 1 microampere signal, the same leakage creates approximately 0.217 millivolt.

The polarity matters. An opposing leakage can reduce the net current and eventually violate the positive-input condition, where the simple log model no longer applies. Do not extrapolate a fitted curve through zero current. Separate leakage at the reference input from leakage at the signal input, and retain the actual node potentials and environment when estimating either contribution.

Do not assume paired junctions cancel every temperature term

In a junction-based log-ratio circuit, matched junctions at the same temperature can cancel a shared saturation-current factor when their voltages are differenced. The remaining basic voltage ratio still contains the thermal voltage kT/q, so the raw slope is proportional to absolute temperature. A matched passive resistor pair alone does not remove that factor.

An integrated converter may include compensation, while a discrete design may use a separately defined temperature-dependent scaling stage. Use the actual implementation rather than applying compensation twice or assuming it exists. Temperature differences between the logging elements, reference drift and resistor-network scale drift are different mechanisms. A common ceramic substrate can help define layout but does not prove identical active-junction temperatures or a guaranteed compensated transfer.

Calibrate identifiable parameters rather than fitting every observation

A one-point adjustment can make one condition agree while leaving slope, leakage and conformity errors unresolved. Two points can identify a line in the log-domain model, but they do not establish its validity between or beyond those points. Preserve additional check currents that were not used to set the correction, including relevant low-current conditions where additive errors become more important.

Keep the source-current uncertainty, reference-current uncertainty and measurement repeatability with the calibration. A wide numerical span is not helpful if one endpoint is close to the source's leakage or the converter's operating limit. If an active circuit departs from the assumed log law, diagnose the mechanism before increasing polynomial order. A fitted display can conceal a physically invalid input or a saturated output.

Convert output errors back into the customer measurand

The inverse relation is I/Iref = ten raised to the power of (Vout minus V0) divided by S. A constant output error therefore becomes a multiplicative ratio error, not an additive current error after inversion. State whether the customer requires absolute current, optical attenuation or a dimensionless channel ratio; their acceptable errors need not use the same units.

If two optical channels form the ratio, common source intensity may cancel only to the extent that both observe the same change under the defined optical transfer. Different backgrounds, detector responses or path timing can defeat that cancellation. Keep the optical ratio question separate from the electronic reference-current error. Neither passive matching nor a logarithm guarantees rejection of every common-looking disturbance.

Specify the reference and scale networks with separate limits

The drawing review should identify which resistors affect absolute reference current and which affect output slope or offset. Include the active input-node voltage, current polarity, expected input range and available output swing. Assign reference drift, scale error and leakage separately so a component choice can be evaluated against the mechanism it actually influences.

For a custom ceramic thick-film network, ChipSimple can review the defined passive circuit and interface conditions. The analog and optical owners validate the complete logarithmic channel. Retain the decade-spaced checks and inverse-conversion convention with the release record, so a later resistor substitution or reference change does not silently turn a previously calibrated sensor into a differently scaled instrument.