Measuring inductance of a cable

You need to focus the loop inductance per unit length.

This means that forward and return paths have to be considered together. So, if the forward path (for an arbitrarily short piece of cable) has a current travelling from left to right, the return path will have a current travelling from right to left AND, because the two wires (forward and return) are in close proximity, the combined magnetic fields near-enough cancel each other.

Given that inductance can be defined as total flux produced per amp (for a single turn), trying to measure the inductance of one of the wires is irrelevant because the total flux that wire will produce will be unaffected by what would be the virtually identical (but opposite) flux produced by the other wire.

It's the composite net flux that is important and what defines the cables inductance per metre.

I don't know if you ever studied coaxial cable but if you do then you will find that the only magnetic field produced is between the inner core and the outer shield. Outside the coax cable there is no net magnetic field because of cancellation due to forward and return currents.

Great Question!

Will the inductance measured on the left side be half of the inductance measured on the right side?

In general, no, the two are unrelated.

In the cables datasheet we often find inductance per unit length. Is the inductance per unit length measured like on the right side?

Yes, what you pictured on the right is exactly what the cable datasheet is referring to as inductance per unit length.

The first measurement pictorial is ineffective. Inductance depends, in part, on the AREA of the loop that you have made with the probe leads and the return through the wire. That significant area is a poorly controlled variable.

A two-wire transmission line can be characterized for its impedance (just terminate it and feed a pulse in, look for the reflection to be minimum when the terminator resistor is adjusted). Then you can characterize it for capacitance (with a low frequency AC input and the output end left open), and find the inductance from the formula

$$ Z_{out} = sqrt {frac {L_{per length} {C_{per length} } } $$

and a measurement of the length of the line.

Or, if you have a sufficiently low impedance test signal, just short the distal end and apply the L-meter as shown in the second pictorial, but beware, the capacitance and wire resistance is not always negligible (and an L-meter will interact with that, depending on test frequency).

Total capacitance, with the distal end open, is not resistance-sensitive, nor is it affected by skin effect (at low frequency).

LRC meters generally measure impedance, that is resistance and reactance at a given frequency. The way impedance is usually measured by such instruments is generally as follows. The instrument applies a voltage (or alternatively a current) sine wave to the DUT "device under test" (is will here be called a "device" even though it may be wire). The current through the DUT is measured, both its amplitude, and the phase angle it has with the voltage. If the instrument applied a current, rather than a voltage, then the voltage is measured, along with the phase angle.

From the ratio between the voltage and the current, and the phase angle between the voltage and current, the instrument calculates resistance and reactance. At this point, with an injected signal of a specific frequency, the capacitative reactance and inductive reactance can only be distinguished by the sign of the phase shift.

If the DUT has both capacitative and inductive reactance, the test at this point only gives the algebraic sum of those reactances. If inductance of the DUT causes phase shift in one direction, and capacitance of the DUT causes phase shift in the opposite direction, the result may appear as a capacitative reactance, or an inductive reactance depending upon which reactance has a larger magnitude. For example, if the DUT at this frequency has an inductive reactance of 4 k$Omega$ and a capacitative reactance of 3 k$Omega$. The instrument will detect 1 k$Omega$ of inductive reactance. It cannot tell that the actual inductance contributed 4 k$Omega$ and the actual capacitance 3 k$Omega$ because the same result would have obtained if the reactance contributed by the inductance were 16 k$Omega$ and the reactance contributed by the capacitance was 15 k$Omega$.

To get around this problem, measurement needs to be taken at multiple frequencies. The reactance of an ideal inductor is proportional to frequency, and the reactance of an ideal capacitor is inversely proportional to frequency. From reactance measurements at multiple frequencies, and by assuming the capacitance and inductance are simply connected in series (or alternatively in parallel) one can calculate the capacitance and inductance.

Note several things. First, the result for capacitance and inductance is a calculation, not a direct measurement. Second, to measure both capacitance and inductance in a DUT, one needs to measure reactance at multiple frequencies. Third, capacitance and inductance are combined in networks that are more complicated than simply in parallel or in series. The instrument cannot know the actual network that combines these reactances (as well as the resistances).

For these reasons, an LRC meter cannot always give an accurate measurement of inductance and capacitance, because it doesn't actually measure these, but calculates them, based upon assumptions.

My guess, and I could be wrong, is that if you attempt to measure the "inductance" of your cable by putting a probe to each wire and terminating the cable with a short, your LRC meter will not give you an accurate answer unless your LRC meter drives the wire with a signal frequency higher than the resonant frequency of the cable. The resonant frequency is inversely proportional to the length of the cable. This is because the capacitance between the wires at low frequency dominates the impedance, rather than the inductance dominating. (For reference, a CAT5 cable one meter long has a resonant frequency somewhere around 50 MHz.)

Not having done the experiment myself, I am not entirely sure, but that would be my guess.

Normally, the inductance of a pair of wires is computed based upon the diameter of the wires, and their separation. But since you are asking about measuring them, I have addressed the difficulty in making such a direct measurement.