C.29.3:5.2 - Read an analog sum with the output scale it needs

An existing analog channel is intended to supply the sum of two numbers x and y in the range 0 to 10. Its input preparation sets voltages:

Vx = x / 10 volts
Vy = y / 10 volts

Each source is connected through an equal resistor R to one common node. The supplied electrical model has ideal voltage sources, equal resistances, settled behavior and a readout drawing negligible current. The resistor-current relation and current balance at the node give:

(Vx − Vout) / R + (Vy − Vout) / R = 0
Vout = (Vx + Vy) / 2

The arrangement physically produces an average voltage. To obtain the intended sum, derive its readout:

decoded result = 20 × [Vout expressed in volts]
               = x + y

For x = 8 and y = 6, preparation gives 0.8 V and 0.6 V. The node gives 0.7 V. Decoding returns 14. Reusing the input scale, ten units per volt, would return 7.

The abstract and physical routes now agree. Input preparation and output interpretation use different factors because the intervening operation halves the voltage sum.

Now allow each prepared input voltage an error of at most 0.001 V. Suppose the readout indication has an additional error of at most 0.002 V under the same circuit model. The worst-case error in the indicated output is:

0.5 × 0.001 V + 0.5 × 0.001 V + 0.002 V = 0.003 V
decoded-result error ≤ 20 × 0.003 = 0.06

An absolute tolerance of 0.1 is therefore met under these bounds. A tolerance of 0.01 is not established by them. Better input setting or readout, a different realization, or a weaker receiving requirement would need comparison. The bound is deterministic; no cancellation or probability distribution has been assumed.

If the instrument loads the node or the resistor values differ, the stated averaging relation needs revision. Those changes belong in the circuit and measurement account, through C.16 where appropriate. The common realization method supplies the preparation/execution/readout comparison and the receiving error calculation. Electrical design supplies the physical law and actual component behavior.