Measurement of unknown resistance by a Wheatstone bridge
Wheatstone bridge experiment: the balance condition, measuring an unknown resistance by the null method, bridge sensitivity, why the result is independent of supply voltage, sources of error, and viva questions with answers.
Aim
To measure an unknown resistance using a Wheatstone bridge by the null-deflection method, to verify the balance condition against the marked value of the resistor, and to examine how the sensitivity of the bridge depends on the ratio arms and the supply voltage.
Apparatus required
| Apparatus | Specification | Qty |
|---|---|---|
| Wheatstone bridge kit or plug-type ratio box | Ratio arms 1, 10, 100, 1000 Ω | 1 |
| Decade resistance box | 1 Ω to 10 kΩ in 1 Ω steps | 1 |
| Galvanometer | Centre-zero, sensitive, with a shunt or protection key | 1 |
| Unknown resistors | Around 470 Ω and 4.7 kΩ, ±5 % | 2 |
| DC supply or cell | 2–6 V with a series key | 1 |
| Digital multimeter | For an independent check of the unknown | 1 |
Theory
The Wheatstone bridge is four resistances in a diamond: two ratio arms P and Q in one pair of adjacent arms, a known variable standard S, and the unknown R. A supply is connected across one diagonal and a galvanometer across the other. When the potential at the two ends of the galvanometer is the same, no current flows through it, and the bridge is said to be balanced.
At balance the two halves divide the supply in the same proportion, which gives P/Q = R/S, so the unknown is R = (P/Q) × S. The important feature of this result is what it does not contain: the supply voltage, the galvanometer resistance and the galvanometer's calibration have all cancelled. The measurement depends only on the accuracy of three resistors and on the ability to detect zero, which is why null methods are far more accurate than deflection methods.
Sensitivity is the other half of the story. Being able to detect a small departure from balance decides how precisely the null can be set. For a small fractional unbalance the voltage appearing across the galvanometer terminals is approximately e = V·(ΔR/R)·k/(1 + k)², where k = P/Q is the ratio. That expression is greatest at k = 1, so equal ratio arms give the most sensitive bridge; a ratio of 100 or 1000, chosen to bring a large unknown within range of the standard, costs sensitivity.
Sensitivity also rises with the supply voltage, but only until the power rating of the arms or the safe current of the galvanometer is reached — which is why the galvanometer is protected by a shunt or a tapping key while the balance is being approached, and only put at full sensitivity for the final adjustment.
The bridge is not universal. Below about 1 Ω the resistance of the leads and the contacts becomes comparable with the unknown and swamps it, which is what the Kelvin double bridge with its extra pair of ratio arms exists to solve. Above a few megohms, insulation leakage and surface currents dominate instead, and a megohmmeter is the right instrument.
Circuit connections
Check every point below against your board before switching on. There is no diagram here on purpose — a wrong diagram is worse than none, and this is the list a demonstrator actually walks through with you.
- The four arms in a closed diamond: P and Q forming one adjacent pair, the standard S and the unknown R the other, so that P/Q and R/S are the two divider ratios.
- The supply, with its key, across one diagonal — between the junction of P and R and the junction of Q and S.
- The galvanometer, with its protective shunt or tapping key, across the other diagonal — between the junction of P and Q and the junction of R and S.
- The unknown resistor clamped in its terminals with short, tight leads; loose contacts add resistance directly into the arm being measured.
- Battery key closed first and galvanometer key tapped second, so the bridge is energised before the detector is connected.
Procedure
- 1Measure the unknown with a multimeter first, so the approximate value is known and the ratio arms can be chosen sensibly.
- 2Choose the ratio P/Q so that the standard S can be adjusted to a value using most of its decades — a ratio of 1 where possible, since it gives the best sensitivity.
- 3With the galvanometer shunted, close the battery key and tap the galvanometer key. Note the direction of the deflection.
- 4Adjust the standard S to reduce the deflection, changing direction of adjustment whenever the deflection reverses, and work down through the decades from coarse to fine.
- 5Remove the shunt for the final adjustment and set S for exact zero deflection. Record P, Q and S at balance.
- 6Compute R = (P/Q) × S and compare it with the multimeter reading and the colour-code value.
- 7Repeat with the ratio arms interchanged and with a different ratio, and repeat the whole measurement for the second unknown resistor.
- 8To measure sensitivity, offset S by a known small amount from balance and record the resulting galvanometer deflection.
Work out your readings
Type in the numbers off the meters. This fills the tabular column, works the calculation through step by step, plots the characteristic — and tells you when a reading cannot physically be right, which is the part a manual can't do. Everything stays on this device, and it works with the network off.
Nameplate and machine data
Fill in the machine data to see the results, the worked calculation and the curve.
Precautions
- Keep the galvanometer shunted or use the tapping key while approaching balance; an unprotected sensitive galvanometer can be damaged by the current at large unbalance.
- Close the battery key before the galvanometer key and open it after, so the detector never sees a switching transient.
- Do not leave the supply connected for longer than needed. Current heats the arms, their resistance drifts, and the balance point moves.
- Ensure all plugs in the resistance boxes are clean and firmly seated; a loose plug adds an unknown resistance in series with that arm.
- Use short, thick leads for the unknown, and account for lead resistance if the unknown is small.
- Check that the supply is within the power rating of the smallest arm before closing the key.
Sources of error
Every record asks for these, and every record gets the same three lines copied from the one before. These are the errors this particular experiment actually has.
- Contact and lead resistance in series with the unknown, which is the dominant error for low resistances and is the reason the plain bridge is unsuitable below about an ohm.
- The tolerance of the ratio arms and the standard resistance box, which sets the floor on accuracy no matter how carefully the null is found.
- Limited sensitivity near balance: if the galvanometer cannot resolve the last small unbalance, a range of values of S all appear to give zero.
- Self-heating of the arms if the key is kept closed, which changes their resistance during the measurement.
- Thermoelectric EMFs at junctions of dissimilar metals, which add a small offset to the detector reading and matter most in sensitive low-resistance work.
Viva questions with answers
State the balance condition of a Wheatstone bridge and derive the unknown resistance.
At balance no current flows through the galvanometer, so the potential drops across the two arms of each branch are in the same ratio: P/Q = R/S. Rearranging gives R = (P/Q) × S. The condition is reached by adjusting the standard S until the detector reads zero.
Why is the measured value independent of the supply voltage?
Because balance is a condition of equal potential at the two detector terminals, not a particular current. Changing the supply changes both divider outputs in the same proportion, so the null occurs at the same setting of S. The supply voltage affects only how sensitively the null can be detected, not where it lies.
Why is a null method more accurate than a deflection method?
A deflection method depends on the calibration, linearity and zero of the indicating instrument, so every one of those becomes an error. A null method requires the detector only to distinguish zero from not-zero, so its calibration is irrelevant and the accuracy rests on the three resistors, which can be made and certified far more precisely than a meter movement.
How does the choice of ratio arms affect sensitivity?
Sensitivity is proportional to k/(1 + k)² where k = P/Q, which is greatest when k equals one. Equal ratio arms therefore give the sharpest null. Large ratios are used only when they are needed to bring the unknown within the range of the standard, and they cost sensitivity in exchange for range.
Why can a Wheatstone bridge not measure very low or very high resistances?
At very low values the resistance of leads and contacts is comparable with the unknown and is measured along with it; the Kelvin double bridge adds a second pair of ratio arms to eliminate that. At very high values leakage across insulation and surface films carries a significant part of the current, so a megohmmeter with guarded terminals is used instead.
What is the purpose of the galvanometer shunt or tapping key?
Far from balance the detector diagonal can carry enough current to damage a sensitive galvanometer or drive its pointer hard against the stop. The shunt diverts most of that current while the balance is approached coarsely, and is removed only for the final fine adjustment when the unbalance is already small.
Why is the battery key closed before the galvanometer key?
Closing the supply causes a transient as the circuit capacitances charge. If the detector were already connected it would see that transient as a large momentary deflection, which is both misleading and potentially damaging. Energising first and tapping the detector afterwards lets the circuit settle before it is observed.