Engineering & Technology · Electrical Circuits & Measurement

Circuit measurements, instrument loading and uncertainty

An unloaded divider is predicted to give 5 V. A meter displays about 4.76 V and produces nearly the same reading each time. The repeatability may be excellent while the comparison is wrong: the instrument itself can change the circuit being measured.

Forces, Energy & Systems and Engineering Measurement supply energy, units, model assumptions and measurement records. This chapter defines its own electrical terms and recalls each algebraic relationship before substitution. Exponentials and complex impedance are introduced as supplied representations; readers can follow the physical comparisons first. All numerical networks, curves and specifications are original teaching models. They are not physical build instructions, actual component ratings or approval to work on electrical equipment.

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Include a measuring instrument in the circuit model and separate loading effects, display resolution and stated uncertainty.

Specify which voltage the measurement is about

The measurand is the quantity intended to be measured. An unloaded output voltage and the voltage with a particular instrument attached are different conditions. Model the measurement connection, reference points, range, bandwidth and relevant input behaviour. A displayed value cannot reveal every omitted condition.

An ideal voltmeter has infinite input resistance and draws no current. A real voltage input can draw current, represented here by a finite resistance across its terminals. Current measurement inserts a different element into the measured path; its added voltage drop is often called burden voltage. Neither method is automatically nonintrusive.

Add the instrument as an actual branch

Use an original ideal 10 V source and two 100 kΩ resistors in series. The unloaded midpoint is 5 V. The network seen from midpoint to reference has Thévenin voltage 5 V and resistance 100 kΩ in parallel with 100 kΩ, or 50 kΩ. A model meter of 1 MΩ becomes a load on that port.

Its loaded voltage is 5×1,000,000/(50,000+1,000,000)=4.7619… V. With a 100 kΩ meter model instead, it is 3.333… V. The figure compares these explicitly assumed input resistances; they are not claims about typical meter products or settings. Loading is a circuit effect even if the instrument reports its own terminal voltage accurately.

Upper circuit: ideal 5 V source in series with 50 kΩ source resistance feeding meter resistance Rm to return. Lower: Vloaded=5 Rm/(50,000+Rm). Horizontal Rm 0.1–10 MΩ logarithmic,0.1,1,10 equally spaced. Vertical voltage 0–5 V,45 SVG units/V. At 0.1 MΩ voltage 3.3333 V;1 MΩ4.7619 V;10 MΩ4.9751 V. Dashed horizontal line is unloaded 5 V. The port represents an ideal 10 V divider of two 100 kΩ resistors. No real product input specification is implied.

Attach limits to the quantity and conditions they describe

Resolution is the smallest indicated increment in the stated range. It is not total accuracy or measurement uncertainty. A hypothetical specification of ±(0.5% of reading+2 counts), with 0.01 V per count, gives an arithmetic limit of 0.005×4.76+0.02=0.0438 V at a 4.76 V indication, under whatever conditions the specification defines.

That rule alone does not provide a probability distribution or a complete uncertainty budget for the unloaded 5 V measurand. Input-resistance knowledge, source/component variation, temperature, contacts, reference paths and other relevant effects may contribute. Correcting a modelled loading effect also requires uncertainty in the correction. Repetition can investigate variation; it does not remove a stable loading bias.

For changing signals, include the instrument’s frequency response, sampling and input capacitance when relevant. A DC input-resistance model can be insufficient for a fast transition. Retain what was connected and what was measured, together with calibration status, assumptions and the basis of any uncertainty interval.

Compare the sizes of two different effects

The predicted loading difference for the 1 MΩ input is 5−4.7619…≈0.2381 V. Relative to the unloaded 5 V this is about 4.76%. It is larger than the constructed 0.0438 V instrument-specification limit at the rounded indication.

The two numbers answer different questions: one compares unloaded and loaded model conditions; the other is a hypothetical limit on the instrument’s indicated result under its specification. Do not combine them by calling both random scatter.

If a justified model correction were used, multiplying loaded voltage by 1+50,000/1,000,000=1.05 would recover 5 V from the unrounded model value. A real corrected result must carry the uncertainties in both the observed voltage and the model parameters.

Try the reasoning

Does a stable 4.76 V indication disprove a 5 V unloaded prediction?
Show the answer and reasoning
Not necessarily. The meter may load the output; the two statements can refer to different circuit conditions.
Does 0.01 V display resolution establish ±0.01 V total uncertainty?
Show the answer and reasoning
No. Resolution is one property. Other instrument and measurement-model effects remain.
Why retain the instrument input model in the measurement record?
Show the answer and reasoning
It helps reconstruct how the measurement changed the circuit and whether a later instrument or signal condition is comparable.

What to carry forward

The measurement setup is part of the system. State the measurand, model the loading and explain what any quoted limit means.

References

Teaching resources and research supporting this lesson’s explanations and examples.