Measuring the machine

Every page on this site so far has taken four numbers as given: the stator resistance $R_{s}$ , the two axis inductances $L_{d}$ and $L_{q}$ , and the magnet flux linkage $λ_{m}$ . The dq model is written in them, the current loop sets its gains from them, and the field-weakening envelope is entirely determined by them.

They are not given. A datasheet may quote two of them, often at a temperature you are not running at and sometimes in units you did not expect. The rest have to be measured — and the measurements are easy. It is the arithmetic between the measurement and the parameter that goes wrong.

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The middle column is where the errors live. Every box in it is a conversion that is easy to forget, and none of them announces itself — a dropped factor produces a plausible number, not an error. The dashed wire is a free cross-check: the same locked-rotor step that gives you an inductance also gives you a resistance, by a completely different route, so it can be compared against the DC measurement for nothing.

$R_{s}$ , and the factor of two

Inject DC into one terminal and out of another, measure the volts and the amps, divide. The current went through two windings in series, so what you have is twice the per-phase resistance.

On the reference machine that is 0.70 V at 1.00 A across two terminals, giving 0.35 Ω per phase. Take the ratio at face value and you have 0.70 Ω — and since the current loop’s integral gain is proportional to $R_{s}$ , the loop is then tuned for a machine that does not exist.

The same factor applies to an LCR meter across two terminals: 660 µH measured is 330 µH per phase.

On the bench Use four-wire measurement, or measure the leads and subtract. A phase resistance of 0.35 Ω is the same order as a pair of test leads and a couple of connectors, so a two-wire reading can be tens of percent high without looking wrong.

$L_{d}$ and $L_{q}$ from a step, and why one step is not enough

Hold the rotor, apply a voltage step, and watch the current rise. The final value gives $R = V / I$ ; the time to reach 63.2% of it is $τ = L / R$ . One step, both parameters — which is why the diagram above draws a second arrow back to $R_{s}$ .

The catch is that the inductance a held rotor presents depends on where it is held:

L (θ) = \frac{L _{d} + L _{q}}{2} - \frac{L _{q} - L _{d}}{2} cos 2 θ

Aligned with the d-axis it reads $L_{d}$ ; a quarter of an electrical revolution away it reads $L_{q}$ . Anywhere in between it reads a number that is neither. On the reference machine, 30° from the d-axis gives 240 µH — a perfectly plausible figure, and not a parameter of anything.

So the procedure is not one step. It is a sweep, taking the minimum and the maximum across an electrical revolution.

The instrument Sample rate Noise

The machine Rotor angle Step

Top: one locked-rotor step, sampled, with the 63.2% crossing interpolated between the two samples that straddle it. Bottom: the same measurement repeated at every rotor angle — the dot is where the step above was taken. Drag Rotor angle and watch the fitted inductance walk between L_d and L_q without the fit ever getting worse: the measurement is fine, it is simply measuring something else. Then drop Sample rate until there are fewer than five samples in a time constant.

Interpolate the crossing

At 20 kHz on the reference machine’s 600 µs d-axis time constant there are twelve samples in one $τ$ . Interpolating between the two samples that straddle the 63.2% threshold recovers $L_{d}$ to 0.04%.

Take the nearest sample instead and the error is up to half a sample on $τ$ , about 4%. Take the first sample past the threshold — which is what the obvious loop does — and it rounds up rather than to nearest, so the worst case doubles to 7.3%. All three implementations look equally correct in the source.

$λ_{m}$ , and three conversions in a row

Spin the machine with something else, leave the terminals open, and measure the voltage. With no current there is no resistive or inductive drop, so the terminal voltage is the back-EMF, and

λ_{m} = \frac{v ^ _{p ha se}}{ω _{e}}

Every trap here is a unit. A scope across two terminals reads line-to-line, which is $3$ times phase. A multimeter reads RMS, which is $1/ 2$ of peak. And $ω_{e}$ is electrical, which is the mechanical speed times the pole pairs.

Three conversions, each of which produces a believable answer if you drop it:

Dropped	Result
the $3$ — treating line-to-line as phase	1.73× too big
the $2$ — treating RMS as peak	0.71×
pole pairs — using mechanical speed	4× too big on this machine

The reference machine’s 8.5 mWb corresponds to a back-EMF constant of 4.36 V/krpm line-to-line RMS, which is the form a datasheet is most likely to quote. Converting that figure back through all three factors returns 8.5 mWb exactly — which is the check worth doing before trusting either number.

Pole pairs are worth measuring too

Spin at a known mechanical speed and count electrical cycles. The reference machine at 1000 rpm produces 66.67 Hz, giving four pole pairs.

Worth doing rather than trusting a label, because getting it wrong is not subtle: the drive runs at the wrong speed by an exact integer ratio. That is a distinctive symptom, and recognising it saves an afternoon of looking at the control loop, which is not where the fault is.

All four numbers move with temperature

The measurements above are taken on a cold machine. It does not stay cold.

Copper’s resistance rises about 0.393% per kelvin. From 25 °C to 125 °C that is 0.35 Ω becoming 0.4875 Ω — 39% more.
Neodymium loses roughly 0.12% per kelvin, reversibly. At 120 °C the reference machine’s 8.5 mWb is 7.48 mWb, down 12%.

The first is why a current loop tuned on a bench at room temperature does not behave the same after twenty minutes of work: its integral gain is set from a resistance that has since moved by a third.

The second cuts both ways. Less flux means less torque per amp, which is a loss — and a higher base speed before the back-EMF reaches the DC link, which is a rare case of a thermal effect helping.

On the bench Inductance is the stable one, and it is the reason to lean on it. $L$ barely moves with temperature; it moves with current, through magnetic saturation. Measure it at a current near where you intend to operate rather than at the milliamps an LCR meter uses, or the value will be optimistic exactly where the machine is working hardest.

What to take away

A line-to-line measurement is two windings. Halve it — for resistance and for inductance both.
One locked-rotor step gives $R$ and $L$ by two different routes. Compare them; it costs nothing and it catches a bad connection.
A single inductance measurement is a number between $L_{d}$ and $L_{q}$ . Sweep the rotor and take the extremes.
Interpolate the 63.2% crossing. Picking a sample costs 4–7% on $L$ at a plausible sample rate.
Back-EMF to flux linkage is three conversions: $3$ , $2$ and pole pairs. Round-trip the answer through a datasheet’s V/krpm to check it.
Measure the pole count. Getting it wrong shows up as an exact integer speed error, nowhere near the loop you will be tempted to blame.
Everything you measured was cold. Resistance is up 39% at 125 °C and flux is down 12% at 120 °C, and the current loop is tuned from both.