Voltage measurement, input resistance & loading

Voltage is defined between two electrical points. In DC voltage mode the meter is connected across those two points, so its input becomes a branch between them. For a linear resistive source network, replace everything behind the port by its Thévenin equivalent: open-circuit voltage V_TH and source resistance R_TH.

The first useful DC model of the meter is a finite input resistance R_IN. The meter therefore forms a divider with the source resistance.

The same current flows through the source resistance and the meter input:

The voltage displayed by the idealized meter model is the voltage across R_IN:

Normalize by the open-circuit voltage:

For a passive positive source resistance and finite positive meter resistance, the ratio is below one. The meter therefore reads below the unloaded Thévenin voltage in this simple model.

Define the relative loading error with respect to the open-circuit voltage:

Substitute the divider result:

If R_IN≫R_TH, then the denominator is approximately R_IN and

The approximation is useful for mental estimates, but the exact relation should be used whenever the ratio is not very large.

Suppose the allowed loading error is ε_max. Require

Solving for the meter input resistance gives

This is more useful than the vague advice “use a high-impedance voltmeter.” For a source with R_TH=20 kΩ:

The required instrument changes by two orders of magnitude when the allowed loading error changes from 1% to 0.01%.

Assume a meter input resistance of 10 MΩ only as a numerical example—not as a universal DMM property. For a 5 V source:

There is no context-free statement that “10 MΩ is high.” What matters is R_IN/R_TH together with the required accuracy.

Instrument input behavior is a specification, not a universal constant. A particular meter may provide a nominal resistance on some DC voltage ranges, a much higher input resistance on other ranges or modes, or a deliberate low-impedance voltage mode intended to suppress ghost voltages. The exact implementation is model dependent.

Therefore a rigorous workflow is:

1. Identify the exact instrument model.
2. Identify the selected function and range.
3. Read the input-resistance/input-impedance specification for that condition.
4. Use the specification in the loading model.
5. Record the function/range when the result depends materially on it.

The term input resistance is appropriate for the purely DC resistive model used here. Later, with time-varying signals, the more general input impedance can depend on frequency.

If R_TH and R_IN are known, invert the divider relation:

Example: a measured value of 2.273 V with R_TH=1.00 MΩ and R_IN=10.0 MΩ gives

But the correction factor itself depends on the values used. If the source resistance is poorly known, or the meter input differs from its assumed value, the corrected result inherits that uncertainty. A large correction is therefore not equivalent to a good measurement.

A better engineering strategy is often to choose a measurement method that perturbs the source much less in the first place.

The finite-resistance model is powerful, but it is not universal. It can become incomplete when:

• the source resistance is so high that input leakage currents, connector contamination or board-surface leakage produce comparable voltage errors;
• the measured quantity is changing fast enough that the instrument input can no longer be represented by one DC resistance;
• the meter uses a special input mode whose behavior is not equivalent to the assumed simple resistance;
• the source itself is nonlinear, time dependent or changes operating point when loaded.

At that point the model should become more detailed, not the algebra more confident. The rule is: use the simplest model that has been justified by the circuit and the instrument specification.

1. A 3.300 V source has RTH=2 kΩ and is measured by RIN=10 MΩ. Find Vmeas and loading error.
2. A 2.50 V sensor has RTH=1 MΩ. Compare readings from 10 MΩ and 100 MΩ inputs.
3. For RTH=47 kΩ, find the minimum RIN for loading errors below 1%, 0.1% and 0.01%.
4. A meter reads 1.800 V from a source whose RTH=500 kΩ. If RIN=10 MΩ, estimate VTH by correcting loading.
5. Two nominally accurate meters disagree on a high-resistance node. List three checks to make before declaring either meter defective.
6. Why is an input-resistance value from one voltage range not automatically valid for every other range?
7. A node has RTH=20 MΩ and the available meter has RIN=10 MΩ. Is numerical correction alone a satisfactory strategy? Explain.
8. Derive the approximation εL≈RTH/RIN from the exact expression and state its required condition.

Chapter 03 will move to current measurement and treat shunts, burden voltage, range choice, fuse/protection consequences and measurement-induced current error at the same depth.