Measurement of low, medium and high resistance

Why it matters

The method for measuring a resistance depends on its size. A 50 μΩ busbar joint, a 2 kΩ strain gauge and the 500 MΩ insulation of a motor winding need completely different techniques, because lead and contact resistance swamp low values while leakage currents swamp high ones. Choosing the wrong method gives a confident but wrong number.

Key ideas

Classification (conventional).

• Low: below about 1 Ω — armature and series-field windings, ammeter shunts, contacts, cable lengths.
• Medium: about 1 Ω to 100 kΩ — most circuit resistors, relay coils.
• High: above about 100 kΩ — insulation, high-value resistors, cable insulation.

Low resistance. Lead and contact resistances (milliohms) are comparable to the unknown, so they must be kept out of the result. The cure is the four-terminal resistor: two current terminals carry the test current; two potential terminals, inside them, define the resistance actually measured. Methods:

• Ammeter–voltmeter with the voltmeter across the potential terminals.
• Kelvin's double bridge (next topic covers bridges): a second set of ratio arms (p, q) cancels the effect of the link resistance r joining the unknown to the standard. Range about 1 μΩ to 1 Ω.
• Potentiometer method: the unknown and a standard of similar value carry the same current, and their voltage drops are compared on a DC potentiometer.
• Micro-ohmmeter (ducter): a four-wire instrument that reads directly.

Medium resistance.

• Ammeter–voltmeter: simple but the meters' resistances introduce error, depending on connection (see formulas).
• Substitution: replace the unknown with a calibrated variable resistor until the meter reading is the same; accuracy depends on the standard and on constant supply.
• Wheatstone bridge: the standard laboratory method, 1 Ω–100 kΩ (beyond about 100 kΩ leakage and galvanometer sensitivity limit it).
• Ohmmeter: series type (scale reversed, zero at the right) or shunt type for low values. Quick, not precise.

High resistance. The current is tiny, so surface leakage over the insulation can be larger than the volume current you want. Problems and remedies:

• Guard ring / guard terminal: a conductor around the measuring electrode, held at the same potential, diverts surface-leakage current away from the meter.
• Electrostatic effects and the absorption current in insulation; readings are taken after a fixed time (commonly 1 min).
• Methods: direct deflection (sensitive galvanometer in series, with guard), loss of charge (a capacitor discharges through the unknown and the time and voltage fall are measured), megohm bridge, and the megger (insulation tester), a hand-driven or electronic source of 500 V–5 kV with a ratio-type indicator that reads MΩ directly.
• Polarisation index (10-min / 1-min reading) is used for machine insulation; values above about 2 indicate dry, clean insulation (check the applicable standard for limits).

Formulas

Rm = V / I (measured value, ammeter–voltmeter) Connection A (voltmeter directly across R, ammeter outside): Rm = R·Rv / (R + Rv) — reads low; best for low R. Connection B (voltmeter across ammeter + R): Rm = R + Ra — reads high; best for high R. R_crossover = √(Ra·Rv) — below this use A, above it use B. R = true resistance; Ra = ammeter resistance; Rv = voltmeter resistance (all Ω).

R = (P/Q)·S + [q·r / (p + q + r)]·(P/Q − p/q) (Kelvin double bridge) P, Q = outer ratio arms; p, q = inner ratio arms (Ω); S = standard (Ω); r = link resistance (Ω). With P/Q = p/q, R = (P/Q)·S exactly.

R = t / (C·ln(V/v)) = 0.4343·t / (C·log₁₀(V/v)) (loss of charge) t = time (s); C = capacitance (F); V = initial voltage, v = voltage after time t (V). If the capacitor has its own leakage R1, the measured value is R∥R1 and a second test with the unknown removed separates them.

Worked examples

Example 1 — ammeter–voltmeter connections. A 10 Ω resistor is measured with an ammeter of 0.1 Ω and a voltmeter of 5 kΩ. Find the measured value and percentage error with each connection, and the crossover resistance.

1. Connection A: Rm = 10 × 5000/(10 + 5000) = 9.980 Ω → error = −0.20 %.
2. Connection B: Rm = 10 + 0.1 = 10.1 Ω → error = +1.0 %.
3. R_crossover = √(0.1 × 5000) = 22.4 Ω.

Answer: A gives 9.98 Ω (−0.2 %), B gives 10.1 Ω (+1.0 %); since 10 Ω < 22.4 Ω, connection A is better.

Example 2 — loss of charge (GATE level). A cable's insulation resistance is measured by the loss-of-charge method. The cable capacitance is 300 pF. The voltage falls from 250 V to 92 V in 1 minute. Find the insulation resistance.

1. R = t / (C·ln(V/v)).
2. ln(250/92) = ln(2.717) = 0.9997.
3. R = 60 / (300 × 10⁻¹² × 0.9997) = 2.0 × 10¹¹ Ω.

Answer: R ≈ 2.0 × 10¹¹ Ω = 200 GΩ.

Example 3 — Kelvin bridge with mismatched ratios. In a Kelvin double bridge, P = 100 Ω, Q = 1000 Ω, p = 102 Ω, q = 1000 Ω, S = 0.05 Ω and the link resistance r = 0.1 Ω. Find R and the error made by using R = (P/Q)·S.

1. Main term: (P/Q)·S = 0.1 × 0.05 = 5.000 mΩ.
2. Correction: q·r/(p + q + r) = 1000 × 0.1/1102.1 = 0.09074 Ω; (P/Q − p/q) = 0.100 − 0.102 = −0.002.
3. Correction term = 0.09074 × (−0.002) = −0.181 mΩ.
4. R = 5.000 − 0.181 = 4.819 mΩ.

Answer: R = 4.82 mΩ; ignoring the correction gives 5.00 mΩ, about 3.8 % high. This is why the two ratio pairs are ganged so that P/Q = p/q.

Common mistakes

• Using the Wheatstone bridge for a sub-ohm resistor: lead and contact resistance appear directly in the answer.
• Swapping the two ammeter–voltmeter connections: voltmeter directly across R (A) for low R, ammeter directly in series with R (B) for high R.
• Taking log₁₀ in R = t/(C·ln(V/v)) without the 0.4343 factor (or the reverse).
• Forgetting the guard terminal in insulation tests, so surface leakage makes the insulation look worse than it is.
• Using a megger on a low-resistance path and expecting a meaningful value; it is designed for MΩ ranges and high test voltage.

For GATE IN

Expect NAT questions on ammeter–voltmeter errors and the crossover resistance, loss-of-charge insulation resistance, and the Kelvin double bridge balance. MCQs ask which method suits which resistance range, why four-terminal resistors and guard rings are used, and what a megger measures. Practise the logarithmic discharge formula until you can do it quickly.

Quick check

1. Which ammeter–voltmeter connection suits a 1 MΩ resistor?
2. What is the purpose of the potential terminals of a four-terminal resistor?
3. In loss of charge, the voltage halves in 20 s with C = 1 μF. What is R?
4. What does a guard ring remove from a high-resistance measurement? Answers: 1. Connection B (voltmeter across ammeter and resistor); 2. they define the resistance between them and keep lead and contact resistance out of the measured voltage; 3. R = 20/(10⁻⁶ × ln 2) = 28.9 MΩ; 4. surface-leakage current.