Passive Resistor Averaging: An Open Input Changes the Weights

System boundary

Multiple source outputs, printed averaging branches and one receiver form a coupled passive network. Source validity and redundancy are not established by the average alone.

Integration interfaces

Integration risks

Define the node without assuming an active summing amplifier

The circuit considered here connects several voltage sources through individual resistors to one common node. A high-impedance receiver measures that node against the shared return. There is no amplifier holding the node at a virtual ground. Consequently, current from one source can flow into another source through the resistor network, even when the receiver draws almost no current.

For each branch, combine the printed resistor with the source's justified series output resistance. Keep sources with nonlinear output limits, clamps or changing supply states separate from this linear model. A circuit that actively sums currents into an amplifier is a different architecture and cannot be analysed by simply copying the passive-node weights.

Derive the weights from current conservation

Write the current into the node from source i as its conductance Gi multiplied by Vi minus Vout. Summing those currents and setting the total equal to the receiver current gives the node equation. With a receiver conductance GL to a reference VL, the solution is the conductance-weighted sum of every connected voltage divided by the total connected conductance.

When there is no load and all branch resistances are equal, the result is the arithmetic mean. Equal printed resistors are insufficient if the source output resistances differ materially. A load to ground adds conductance to the denominator without adding a positive signal term, attenuating the node. A load to a nonzero reference contributes its own weighted voltage.

Compare an open source with a zero-valued source

Take three ideal sources at 1 V, 2 V and 3 V, each connected through 10 kilohms to an unloaded node. The output is 2 V. If the 3 V branch becomes a true open circuit, only the 1 V and 2 V conductances remain, and the output becomes 1.5 V. The remaining inputs now each have one-half weight instead of one-third.

If the third source instead remains connected through 10 kilohms but is held at zero, the output becomes 1 V. Zero contributes nothing to the numerator but its conductance remains in the denominator. Calling both cases a missing input conceals a full 0.5 V difference in this example. A powered-down source can resemble neither case if its output clamps or floats through leakage.

Check what each source must drive or absorb

In the healthy three-input example, the 3 V source supplies 100 microamperes through its resistor, the 1 V source absorbs 100 microamperes, and the 2 V source carries no current. The receiver can draw essentially zero current while the sources still exchange current. Confirm that every source can operate with the required current direction; an output specified only for sourcing current is not automatically an ideal sink.

Changing one source voltage changes the shared node and therefore the currents in every other branch. This interaction can disturb a high-impedance sensor or an output with a limited sink path. The printed resistors limit the interaction but do not provide galvanic isolation or one-way signal flow. If independent inputs are required, buffering or separate acquisition may be a system-level architectural decision.

Include impedance changes before specifying tighter matching

Suppose the third source adds 10 kilohms of output resistance to its 10-kilohm printed branch. With the first two branches unchanged, its conductance halves. The normalized weights become 0.4, 0.4 and 0.2, producing 1.8 V from the original 1 V, 2 V and 3 V sources. Perfect matching of the three printed resistors would not correct this source imbalance.

With all original branches restored, their parallel output resistance is approximately 3.333 kilohms. Adding a 30-kilohm receiver to ground reduces the 2 V unloaded output to 1.8 V. The same observed 1.8 V can therefore result from different causes. Preserve input voltages, individual source impedances and receiver loading in the diagnostic record instead of identifying a cause from one node value.

Decide whether the average preserves enough information

An average intentionally removes information. Different combinations of input values can produce exactly the same node voltage. In this example, three equal 2 V sources and the set 1 V, 2 V, 3 V both produce 2 V. A single node cannot establish agreement between sensors, and an in-range output cannot establish that no input has failed.

If a disconnected or implausible source must be detected, identify an independent observation or a controlled diagnostic state that makes the fault distinguishable. The system owner must assess whether such a state changes the process being measured. Do not promise redundant sensing simply because several resistors or sensors feed one node; diagnostic coverage depends on what remains observable after they are combined.

Measure the weights one source at a time

Hold all but one source at defined voltages and apply a small, known change to the remaining source within its linear range. Divide the output change by that input change to estimate its weight. Repeat for each branch without altering the receiver. Use measured input voltages at the agreed boundary so that source loading is not hidden in the stimulus setting.

Then exercise the declared open, held-zero and powered-down states using controlled low-energy fixtures appropriate to the circuit. Record the node and source currents where they are relevant. Compare the observations with the state-specific model. A sum of measured weights below one may indicate loading to a fixed reference, but leakage, measurement error and unmodelled active behaviour must be excluded before assigning the cause.

Release the weights and fault states with the network

The useful drawing package identifies each branch, its total effective resistance, normal weight, source-current direction and permitted receiver loading. Attach the fault-state output table and the required independent diagnostics. Keep resistor manufacturing tolerances separate from the system's assumptions about the connected sources.

Recalculate after changing a sensor, buffer, input connector, receiver or power sequence. If software converts the node into an engineering quantity, its conversion must match the approved connected-state model. A printed network can implement the agreed weighting accurately, but it cannot determine whether a plausible analog value originated from the intended set of valid inputs.