Extension of ammeter and voltmeter ranges

Why it matters

A PMMC movement typically gives full-scale deflection with only 50 μA–1 mA and a few tens of millivolts. Every practical ammeter and voltmeter — including each range on a multimeter — is that movement with a shunt or a multiplier. Designing them, and knowing how much a voltmeter disturbs the circuit it measures, are standard lab and exam tasks.

Key ideas

Ammeter: the shunt. A low resistance Rsh is connected in parallel with the meter (resistance Rm, full-scale current Im). Meter and shunt see the same voltage, so the current divides in inverse ratio of resistance. The multiplying power m = I/Im is the factor by which the range is extended.

• The shunt must be of a material with a very low temperature coefficient (manganin) and stable resistance, because the copper coil's resistance rises with temperature and would change the current split. A manganin swamping resistor in series with the coil reduces this error further.
• Heavy-current shunts are four-terminal: current terminals carry the load current, potential terminals connect the meter, so contact resistance is excluded.
• An ammeter must have very low total resistance so it does not change the current it measures.

Multirange ammeters. Separate shunts selected by a switch need a make-before-break switch, otherwise the meter is momentarily unshunted and burns out. The Ayrton (universal) shunt avoids this: one tapped shunt R1 + R2 + R3 is permanently across the meter, and the switch only changes which part is in parallel with the meter and which part in series with it.

Voltmeter: the multiplier. A high resistance Rs in series with the meter limits the current to Im at the desired full-scale voltage V. Multiplying factor m = V/v, where v = Im·Rm is the meter's own full-scale voltage.

• Multipliers are wire-wound manganin or metal film, non-inductive.
• Sensitivity S = 1/Im (Ω/V). The total resistance of a voltmeter on any range is S × range. A 50 μA movement gives 20 kΩ/V.
• Multirange voltmeters use either separate multipliers per range or a series string (each range adds a resistor to the previous total).

Loading effect. A voltmeter of resistance Rv placed across a high-resistance part of a circuit draws current and lowers the voltage it reads. Model the circuit as its Thevenin equivalent (Vth, Rth): the reading is Vth·Rv/(Rv + Rth). Keep Rv at least 10–100 times Rth, or use a higher-sensitivity meter, a higher range (larger Rv but poorer resolution) or an electronic voltmeter. Similarly an ammeter's resistance in series reduces the current in a low-resistance circuit.

Moving-iron and AC meters. MI ammeters are normally extended by changing the number of coil turns (shunts are frequency-sensitive because the coil is inductive); on AC, current transformers are used for large currents and potential transformers for high voltages.

Formulas

m = I / Im ; Rsh = Rm / (m − 1) = Im·Rm / (I − Im) (shunt) I = required full-scale current (A); Im = meter full-scale current (A); Rm = meter resistance (Ω); Rsh in Ω.

m = V / v = V / (Im·Rm) ; Rs = (m − 1)·Rm = V/Im − Rm (multiplier) V = required full-scale voltage (V); Rs in Ω.

S = 1 / Im (Ω/V) ; Rv = S × V_range S = voltmeter sensitivity; Rv = total voltmeter resistance on that range (Ω).

Ayrton shunt: Rx = Im·(Rm + Rt) / I Rt = total shunt resistance (set by the lowest range, Rt = Rm/(m₁ − 1)); Rx = portion of the shunt across which the meter-plus-remaining-shunt path is connected for range I.

V_read = Vth·Rv / (Rv + Rth) ; % loading error = (V_read − Vth) / Vth × 100 Vth, Rth = Thevenin voltage (V) and resistance (Ω) seen by the meter.

Worked examples

Example 1 — shunt and multiplier. A PMMC movement has Rm = 100 Ω and Im = 1 mA. Convert it to (a) a 0–5 A ammeter and (b) a 0–100 V voltmeter.

1. (a) Rsh = Im·Rm / (I − Im) = 0.001 × 100 / (5 − 0.001) = 0.1 / 4.999 = 0.02000 Ω.
2. (b) Rs = V/Im − Rm = 100/0.001 − 100 = 100 000 − 100 = 99 900 Ω.

Answer: (a) Rsh ≈ 0.020 Ω (20.0 mΩ) in parallel; (b) Rs = 99.9 kΩ in series. Sensitivity of the voltmeter is 1000 Ω/V.

Example 2 — loading error (GATE level). Two 100 kΩ resistors in series are across a 10 V source. A voltmeter of sensitivity 20 kΩ/V on its 10 V range is connected across the lower resistor. Find its reading and the percentage error.

1. True voltage: 10 × 100/200 = 5 V. Thevenin: Vth = 5 V, Rth = 100 ∥ 100 = 50 kΩ.
2. Rv = 20 kΩ/V × 10 V = 200 kΩ.
3. V_read = 5 × 200/(200 + 50) = 4.0 V.
4. Error = (4.0 − 5)/5 × 100 = −20 %.

Answer: the meter reads 4.0 V, an error of −20 %. A 1 MΩ electronic voltmeter would read 5 × 1000/1050 = 4.76 V (−4.8 %).

Example 3 — Ayrton shunt. Design a three-range Ayrton shunt (10 mA, 100 mA, 1 A) for a movement with Rm = 50 Ω, Im = 1 mA.

1. Lowest range sets the total: Rt = Rm/(m − 1) = 50/(10 − 1) = 5.556 Ω.
2. 100 mA tap: Rx = 0.001 × (50 + 5.556)/0.1 = 0.5556 Ω.
3. 1 A tap: Rx = 0.001 × 55.556/1 = 0.05556 Ω.
4. Sections: R3 = 0.0556 Ω, R2 = 0.5556 − 0.0556 = 0.5 Ω, R1 = 5.556 − 0.5556 = 5.0 Ω.

Answer: R1 = 5.0 Ω, R2 = 0.5 Ω, R3 = 0.0556 Ω.

Common mistakes

• Using Rsh = Rm/m instead of Rm/(m − 1); the meter still carries Im.
• Writing the multiplier as m·Rm; the meter's own resistance is part of the total, so it is (m − 1)·Rm.
• Forgetting that voltmeter resistance depends on the range selected (S × range).
• Ignoring the Thevenin resistance of the circuit when estimating loading; the error depends on Rv relative to Rth, not to the resistor being measured alone.
• Switching between separate shunts with a break-before-make switch — the meter is momentarily unprotected.
• Using an ordinary shunt with an MI meter on AC; the inductive coil makes the current split frequency-dependent.

For GATE IN

Expect NAT questions on shunt and multiplier values, voltmeter sensitivity and resistance on a range, and the reading or percentage error caused by voltmeter loading in a divider or bridge circuit. MCQs test why shunts are manganin, four-terminal shunts and the Ayrton shunt. Practise reducing the circuit to its Thevenin equivalent before applying the loading formula.

Quick check

1. A 50 μA movement has what sensitivity?
2. A 2 mA, 25 Ω movement is to read 10 A. Shunt resistance?
3. Total resistance of a 20 kΩ/V meter on its 50 V range?
4. Why is the multiplying factor of a shunt m and not m − 1 in m = I/Im? Answers: 1. 20 kΩ/V; 2. 0.002 × 25/(10 − 0.002) ≈ 5.0 mΩ; 3. 1 MΩ; 4. m is defined as the ratio of total to meter current; the shunt carries (m − 1)·Im, which is where the (m − 1) in the resistance comes from.