2. DC Voltmeters

2.1 The galvanometer-based voltmeter

The earliest voltmeter is a moving-coil galvanometer: a coil suspended in a magnet, with a needle attached. Current through the coil produces a torque proportional to current; a spring opposes the torque; the deflection settles where torque equals spring force, giving deflection proportional to current. A typical galvanometer (the d'Arsonval movement) might have full-scale deflection at 50 microamps with a coil resistance of 1 k$\Omega$.

To turn this current meter into a voltmeter, put a series resistor (called a multiplier) in front. To get full-scale deflection (50 $\mu$A) at 10 V, you need a total resistance of $10\,\text{V}/50\,\mu\text{A} = 200\,\text{k}\Omega$, so the multiplier is $200\,\text{k}\Omega - 1\,\text{k}\Omega = 199\,\text{k}\Omega$.

        R_mult
   o───/\/\/\───┬───────o
                │
              ─┴─ Galvanometer (50 μA FS, 1 kΩ)
                │
   o────────────┴───────o

The sensitivity is $1/I_{FS}$. For 50 $\mu$A, that is 20 k$\Omega$/V: on the 10 V range you load the circuit with 200 k$\Omega$. On the 1 V range, only 20 k$\Omega$, which is heavy loading. This is why analog VOMs had to be used with care on high-impedance circuits: they could pull a node down by reading it.

2.2 Multi-range and range extension

A multi-range voltmeter switches between multiplier resistors. A common arrangement uses a selector switch that engages different series resistors:

          R1   R2   R3   R4
     o──/\/\─/\/\─/\/\─/\/\──┐
        │    │    │    │     │
        ●────●────●────●     │  Range switch picks tap point
                  │          │
                  └──────────┴── Galvanometer

The "Ayrton shunt" arrangement for multi-range ammeters and "ring" arrangement for voltmeters share this idea: a chain of resistors with taps for each range. Range extension simply means adding a higher-value series resistor for higher voltage ranges.

2.3 Solid-state DC voltmeter

Modern DMMs replace the galvanometer with op-amps and an ADC. A typical front end:

                  R_in (10 MΩ)
   Vin o─/\/\─┬──────┬───────────► to ADC
              │      │
           Z_in   Buffer (FET-input op-amp)
              │      │
              └──────┴── (negligible bias current)

The FET-input op-amp presents virtually infinite impedance (gigaohms), so the voltmeter no longer loads the circuit. The op-amp drives an ADC (often a sigma-delta or dual-slope integrating type, see section on DMMs below). Range switching is done with internal solid-state attenuators or by changing the gain of the front-end stage. There is no needle to deflect; the ADC count is processed by a microcontroller and rendered on an LCD.

This shift from coil-and-spring to op-amp-and-ADC is one of the great quiet revolutions in instrumentation. It also moved measurement from analog skill (interpolating the needle, accounting for parallax) to digital reading and software.

2.4 Differential voltmeter

A differential voltmeter measures the difference between an unknown voltage and a precision reference. The classic configuration: a Kelvin-Varley voltage divider gives a precisely known voltage; a galvanometer detects null between this and the unknown. When the galvanometer reads zero, the unknown equals the divider setting.

This is the lab equivalent of weighing with balance scales: you adjust known masses until balance is achieved, and the unknown equals the sum of the knowns. Because at null no current flows through the detector, the source impedance of the unknown does not affect the result. This is how the highest-precision DC voltmeters of the pre-DMM era achieved parts-per-million accuracy: they reduced measurement to a comparison against a known. Modern equivalents use Josephson voltage standards in national metrology labs.