Potentiometers and Q-meter

Why it matters

The DC potentiometer measures EMF by null comparison with a standard cell, drawing no current from the source at balance. It was for decades the reference for calibrating voltmeters, ammeters and wattmeters, and its principle survives in every self-balancing recorder and in thermocouple instruments. The Q-meter does the same job for radio-frequency coils and capacitors, where bridges become awkward.

Key ideas

DC potentiometer principle. A steady working current I from an auxiliary battery flows through a uniform slide wire (or a series of precision dial resistors plus a slide wire). The voltage drop per unit length is constant. An unknown EMF is connected, through a galvanometer, between one end of the wire and the sliding contact, in opposition to the drop. At balance the galvanometer reads zero, so the unknown equals the drop across the balancing length and no current flows from the unknown source — its internal resistance causes no error.

Standardisation. The working current is set by first balancing a standard cell (saturated Weston cadmium cell, about 1.0186 V at 20 °C — take the exact value from its certificate) at the length or dial setting corresponding to its EMF, adjusting the rheostat in the working-current circuit. The potentiometer then reads directly in volts. Standard cells must not supply more than a few microamperes, so a high protective resistance is kept in the galvanometer circuit until near balance.

Crompton potentiometer. A practical laboratory instrument: a dial of 15 precision coils (0.1 V each) in series with a circular slide wire (0.1 V total) gives readings to about 0.1 mV at 1.6 V full scale.

Applications.

• Voltage: directly up to about 1.5–2 V; higher voltages through a volt ratio box (precision divider).
• Current: pass it through a standard resistor and measure the voltage drop: I = V/Rs.
• Resistance: put the unknown in series with a standard resistor carrying the same current and compare the two drops.
• Calibration: of voltmeters, ammeters and wattmeters on DC, with the potentiometer as the reference.
• Thermocouple EMF measurement.

AC potentiometers measure both magnitude and phase. The working current must have exactly the frequency and waveform of the unknown and be standardised with a transfer instrument. Polar type (Drysdale): reads magnitude and phase angle directly. Coordinate type (Gall): reads in-phase and quadrature components.

Q-meter. Based on series resonance. An oscillator injects a small known voltage E across a very small shunt resistance (a few milliohms) in series with the coil under test and a calibrated variable capacitor C. When C is tuned to resonance, the current is maximum and the voltage across C is Q times E. With E held at a fixed value, an electronic voltmeter across C is calibrated to read Q directly.

• Measures Q, inductance (from f and C at resonance), effective resistance, self-capacitance of coils, and capacitance by substitution.
• Sources of error: the shunt resistance adds to the circuit resistance (indicated Q is lower than the true value); the coil's distributed (self) capacitance Cd is in parallel with the tuning capacitor and makes the indicated Q lower; residual inductance and the voltmeter's loading.
• Distributed capacitance is found by resonating the coil at f₁ with C₁, and at 2f₁ with C₂.

Formulas

k = V_std / l_std (V/m or V/cm) ; Vx = k·lx V_std = standard-cell EMF (V); l_std = its balancing length; lx = balancing length for the unknown.

I = V / Rs (current through a standard resistor Rs, Ω)

Q = ω₀L / R = 1 / (ω₀CR) = Vc / E (series resonance) ω₀ = 2πf₀ (rad/s); L = coil inductance (H); R = series resistance (Ω); C = tuning capacitance (F); Vc = voltage across C at resonance (V); E = injected voltage (V).

f₀ = 1 / (2π·√(L(C + Cd)))

Cd = (C₁ − 4C₂) / 3 (resonance at f₁ with C₁ and at 2f₁ with C₂)

Q_true = Q_indicated × (C + Cd) / C ; Q_true = Q_indicated × (1 + Rsh/R) Corrections for self-capacitance and for shunt resistance Rsh respectively.

Worked examples

Example 1 — calibrating an ammeter with a potentiometer. A slide-wire potentiometer is standardised with a 1.0186 V standard cell balancing at 101.86 cm. (a) A voltage balances at 64.3 cm; find it. (b) The drop across a 0.1 Ω standard resistor carrying an ammeter's current balances at 45.6 cm while the ammeter reads 4.5 A. Find the ammeter error.

1. k = 1.0186/101.86 = 0.01 V/cm.
2. (a) Vx = 0.01 × 64.3 = 0.643 V.
3. (b) V = 0.01 × 45.6 = 0.456 V; I = 0.456/0.1 = 4.56 A.
4. Error = (4.5 − 4.56)/4.56 × 100 = −1.32 %.

Answer: (a) 0.643 V; (b) true current 4.56 A, so the ammeter reads 1.32 % low.

Example 2 — Q-meter with self-capacitance (GATE level). A coil resonates at 1 MHz with C₁ = 450 pF, and the indicated Q is 120. At 2 MHz it resonates with C₂ = 105 pF. Find the distributed capacitance, the inductance and the true Q at 1 MHz.

1. Cd = (C₁ − 4C₂)/3 = (450 − 420)/3 = 10 pF.
2. L = 1/(ω₀²(C₁ + Cd)) = 1/((2π × 10⁶)² × 460 × 10⁻¹²) = 55.1 μH.
3. Check at 2 MHz: 1/((4π × 10⁶)² × 115 × 10⁻¹²) = 55.1 μH ✓.
4. Q_true = 120 × (450 + 10)/450 = 122.7.

Answer: Cd = 10 pF, L ≈ 55.1 μH, true Q ≈ 122.7.

Common mistakes

• Forgetting to restandardise after the working current drifts; every reading assumes the same volts per centimetre.
• Drawing current from the standard cell (no protective resistance near balance) — its EMF falls and it can be damaged.
• Measuring a voltage above the potentiometer range without a volt ratio box; no balance point exists.
• Using C₁ − 2C₂ instead of C₁ − 4C₂ for distributed capacitance; frequency doubling needs a quarter of the total capacitance.
• Taking the Q-meter reading as the true Q when the coil's self-capacitance is not small compared with C.

For GATE IN

Expect NAT questions on potentiometer balancing lengths, current and resistance measurement with standard resistors, and Q-meter calculations: inductance from resonance, distributed capacitance by the two-frequency method and corrected Q. MCQs test why a potentiometer draws no current at balance, standardisation and the sources of Q-meter error. Practise the self-capacitance method; it is short and frequently tested.

Quick check

1. Why is the potentiometer reading independent of the source's internal resistance?
2. A standard cell of 1.0183 V balances at 509.15 mm. What voltage balances at 750 mm?
3. In a Q-meter, injected voltage is 20 mV and the capacitor voltage at resonance is 2.4 V. Q?
4. A coil resonates with 300 pF at f and 60 pF at 2f. Self-capacitance? Answers: 1. at balance no current flows from the source; 2. 1.5 V (2 mV per mm); 3. 120; 4. (300 − 240)/3 = 20 pF.