4. Ohmmeters

4.1 Series ohmmeter

A series ohmmeter places the unknown $R_x$ in series with a battery, a calibration resistor, and a meter:

   +──Battery──R_cal──[Meter]──[R_x]──−

When $R_x = 0$ (probes shorted), maximum current flows; the meter reads full scale. As $R_x$ grows, current drops; the meter's deflection drops. The scale is therefore inverted (zero ohms at full deflection, infinity at zero deflection) and non-linear (logarithmic-ish): the resolution is best for mid-scale resistances and poor at the extremes. As the battery ages, the full-scale point drifts: hence the "zero-ohm adjust" knob on classic VOMs, which lets you trim the meter back to full scale when the leads are shorted.

4.2 Shunt ohmmeter

The shunt ohmmeter places the unknown in parallel with the meter:

   +──Battery──R_cal──┬──[Meter]──−
                       │
                       └──[R_x]──

For low $R_x$, current is bypassed around the meter and the reading drops. For high $R_x$, all current goes through the meter. Used for low resistance measurement (milliohms to tens of ohms), where a series ohmmeter would deflect almost nothing.

4.3 Four-wire (Kelvin) resistance measurement

For very low resistances (say, a 10 m$\Omega$ shunt resistor or the contact resistance of a connector), the resistance of your test leads (typically 50-200 m$\Omega$) is comparable to or larger than what you're trying to measure. Four-wire measurement separates the current path from the voltage-sensing path:

   I_source o──┬──── R_x ────┬──o I_return
               │             │
               ●             ●     ← Voltage-sense leads
               │             │
   V+ o────────┘             └──────o V−
                  (no current here)

Two leads carry the current; two more leads sense the voltage at the resistor terminals themselves. Because the voltmeter has high impedance, no current flows through the sense leads, so their resistance contributes no error. This is called the Kelvin connection after Lord Kelvin. Bench DMMs (Keithley DMM7510, Keysight 34465A) and dedicated micro-ohmmeters offer 4-wire mode with separate banana jacks or Kelvin clips. Essential for measuring battery internal resistance, PCB trace resistance, weld quality, and any sub-ohm value with confidence.