Calculating PMSM dq-axis parameters with a Precision Power Analyzer and Waveform Measuring Instrument

Permanent Magnet Synchronous Motors (PMSMs) are highly efficient, with lower losses than induction motors, and they are used everywhere from air conditioners and refrigerators to EVs as demand for energy-saving performance grows. Controlling a PMSM well depends on a set of parameters expressed in the dq-axis, an orthogonal coordinate system obtained by transforming the motor's three-phase AC quantities first into αβ-axis coordinates (the Clarke transform) and then into dq-axis coordinates (the Park transform). Once the transform is complete, quantities that were sinusoidal AC in the three-phase frame become constant, DC-like values, which is what makes practical PMSM control math tractable.

The Yokogawa WT5000 Precision Power Analyzer and DL950 ScopeCorder both include functions for measuring electrical angle, and that measurement is the key input needed to calculate a motor's dq-axis parameters. This article walks through the derivation of those parameters and shows how the WT5000 and DL950 are used to obtain them from real measurements.

Orthogonal coordinate system dq-axis parameters

The derivation that follows assumes:

• Magnetic flux distribution takes a sinusoidal waveform.
• Iron loss is ignored.
• Harmonic components of the three-phase AC are ignored.

Clarke transform: three-phase to αβ-axis

The first step is the transform from symmetric three-phase AC to the αβ-axis coordinate system, known as the Clarke transform. The three-phase currents i_u, i_v, and i_w are recast as two orthogonal currents i_α and i_β, with the α-axis chosen to coincide with the u-phase current axis. The transform is a simple weighted combination of the three phase currents using sine and cosine terms spaced 120° apart. A scaling coefficient of 2/3 is used when the transform is defined so that phase current amplitude is preserved before and after the transform (a "relative" transform); if instead the transform is defined so that instantaneous power is preserved (an "absolute" transform), the coefficient becomes √(2/3). This paper uses the relative transform throughout.

Park transform: αβ-axis to dq-axis

The second step, the Park transform, rotates the αβ-axis coordinates by a leading angle θ [rad] into the dq-axis coordinate system. Combining the Clarke and Park transforms gives a single matrix that converts the three original phase currents directly into i_d and i_q.

Figure 1. Vector diagram of orthogonal coordinate system obtained by Clarke & Park transform.

If the three-phase currents are themselves sinusoidal with rms amplitude I, and the u-phase current i_u leads the d-axis by an angle γ [rad], then substituting θ = ωt into the combined transform and solving for i_d and i_q gives a strikingly simple result:

i_d = √2·I·sin(γ)
i_q = −√2·I·cos(γ)

Both quantities are constant, not time-varying, confirming that current expressed in dq-axis coordinates is DC. If instead the armature current vector's leading angle is measured from the q-axis and called β [rad] (so that γ = β + π/2), the same result can be rewritten as:

i_d = −√2·I·sin(β)
i_q = √2·I·cos(β)

(the coefficient becomes √3 instead of √2 if the absolute transform is used). The armature voltage follows the identical pattern. If V is the rms value of the u-phase voltage and δ [rad] is the armature voltage vector's leading phase from the q-axis, then:

v_d = −√2·V·sin(δ)
v_q = √2·V·cos(δ)

In short, β is the phase difference between the q-axis and the u-phase current, and δ is the phase difference between the q-axis and the u-phase voltage. Measuring these two angles is what unlocks the dq-axis parameters.

The dq-axis voltage equation and inductance

With v_d, v_q, i_d, and i_q in hand, the PMSM's voltage equation in dq-axis coordinates relates them to the motor's electrical parameters: armature winding resistance R_a [Ω], dq-axis inductances L_d and L_q [H], the armature flux linkage Ψ_a [Wb] produced by the permanent magnet, and the electrical angular frequency ω = 2πf [rad/s]. Ψ_a is obtained separately, as V/ω, from the fundamental component of the motor's no-load back electromotive force (BEMF) voltage.

Because the motor is assumed to be running in a steady state, the differential (d/dt) terms in the full voltage equation can be dropped, which leaves a purely algebraic relationship between voltage, current, resistance, inductance, and flux linkage. Rearranging that steady-state relationship for the two unknown inductances gives the two equations that are actually used in practice:

L_d = (v_q − R_a·i_q − ω·Ψ_a) / (ω·i_d)
L_q = (R_a·i_d − v_d) / (ω·i_q)

Everything on the right-hand side of these two equations, v_d, v_q, i_d, i_q, R_a, ω, and Ψ_a, is either measured directly or derived from a measured electrical angle. That is the practical payoff of the whole derivation: once the electrical angle can be measured accurately, the motor's dq-axis inductances follow directly.

Measuring electrical angle

Measurement principle and correction

The WT5000's motor evaluation function (optional) performs a 1- or 8-cycle FFT calculation on the voltage or current waveform, timed from the falling edge of the encoder's Z-phase signal. The result is the phase difference between that falling edge and the rising-edge zero crossing of the fundamental component of the voltage or current waveform, and this phase difference is displayed as an electrical angle.

By definition, the electrical angle is 0° when the Z-phase falling edge coincides exactly with the rising-edge zero crossing of the voltage or current waveform. In practice the motor's mounting position rarely lines these up exactly, so the WT5000 includes a correction function that subtracts the residual offset between the Z-phase signal and true electrical angle 0°.

Measuring no-load BEMF voltage and encoder signals

Before the dq-axis parameters can be calculated, the armature flux linkage Ψ_a must be obtained by rotating the motor from the load side with no electrical load applied, measuring the resulting no-load BEMF voltage, and dividing its fundamental component by the rotation speed ω. The electrical angle measured during this no-load BEMF test carries its own phase offset between the q-axis and the Z-phase signal, which needs a separate correction described below.

Only the voltage side needs to be wired, using a three-phase three-wire (3P3W) or three-voltage three-current (3V3A) system. The wiring system matters because the electrical angle correction value depends on it; the recommended WT5000 wiring is shown below.

Figure 4. WT5000's three-voltage three-current wiring system.

The encoder's A-, B-, and Z-phase signals connect to the B, C, and D BNC terminals of the motor evaluation function. With the motor rotated from the load side, the electrical angle between the Z-phase signal and voltage U1 is measured; because this value fluctuates cycle to cycle, it is best averaged over multiple measurements (the WT5000's exponential average function is convenient for this). It is also important to measure at the same rotation speed and direction that will later be used when the motor is driven from the inverter under load to obtain the actual dq-axis parameters.

Figure 5. No-load BEMF voltage measurement.

Correcting between the Z-phase signal and motor position

The electrical angle measured above is referenced to the rising-edge zero crossing of the no-load BEMF voltage's u-w line-to-line waveform, not to the q-axis itself, so a further correction is required. Writing θ for the measured electrical angle, the correction is:

−180° ≤ θ ≤ −120°: corrected value = θ + 180° + 90° + 30°
−120° < θ ≤ 180°: corrected value = θ − 180° + 90° + 30°

Each term has a specific physical origin. The ±180° term shifts the reference from the rising-edge zero crossing to the falling-edge zero crossing; because the WT5000's correction cell only accepts values within ±180°, the sign used depends on the phase relationship between the Z-phase signal and the BEMF waveform. The +90° term accounts for the q-axis sitting 90° away from the falling edge of the no-load BEMF voltage. The final +30° term converts the measurement from a line-to-line voltage reference to a phase voltage reference, since the q-axis is defined relative to the u-phase voltage, not a line-to-line voltage. A phase difference that lags the Z-phase signal (appears to its left on the time axis) is taken as positive; one that leads it (appears to its right) is taken as negative. The corrected result is entered into the WT5000's electrical angle correction value field.

Measuring armature winding resistance

The armature winding resistance R_a needed for the calculation is measured separately as a DC resistance, typically using the four-terminal method with a digital multimeter. Because winding resistance shifts with temperature, it is good practice to compensate for temperature changes that occur during motor operation when recording R_a.

Example: from measured angles to dq-axis parameters

In practice, the equations above are entered directly as user-defined functions on the WT5000 (or the DL950's real-time math, described below). The number of pole pairs, R_a, and Ψ_a are registered as constants ahead of time. The corrected phase angles δ and β are computed from the measured electrical angles of the voltage and current waveforms respectively (applying the same Z-phase corrections described above), and v_d, v_q, i_d, and i_q are then computed from δ and β using the sine/cosine relationships derived earlier. L_d and L_q follow directly from those four values. Because a 3P4W wiring system references phase voltage directly, the extra +30° line-to-line correction term is dropped and the transform coefficients change accordingly when that wiring is used instead of 3P3W/3V3A; the same is true for absolute rather than relative transform coefficients. Once the functions are registered, the WT5000 reports pole pairs, R_a, Ψ_a, and the resulting L_d and L_q continuously during measurement.

Notes on electrical angle measurement

Recommended instrument settings

Electrical angle is measured using the WT5000's harmonic measurement function, and a few settings materially affect measurement quality:

• FFT points: 1024 points (a one-wave window) is recommended by default, since measurement period grows with window width. If fluctuations are large, for example due to the inverter's carrier frequency or drive frequency, increase to 8192 points (an eight-wave window) and compare results, keeping whichever setting gives the more stable reading.
• Frequency filter: the harmonic measurement function relies on a PLL, so accurate period detection of the PLL source signal matters. Turning on the frequency filter removes unwanted signal content ahead of the PLL; a cutoff around 1/10 of the inverter's switching frequency is recommended.
• Line filters: the anti-aliasing filter should be enabled (cutoff of 1 MHz is effective for both normal and harmonic measurement) to satisfy the sampling theorem. A digital filter cutoff of roughly 100 times the motor drive fundamental frequency is recommended for harmonic measurement, though it should be adjusted since the line filter setting directly affects the measured values. The same cutoff frequency should be used for both the loaded measurement and the no-load BEMF measurement, since changing it can shift the electrical angle reading.
• Pulse noise filter: a cutoff around 100 kHz on the encoder's A-, B-, and Z-phase signals removes inverter carrier-frequency noise without corrupting the electrical angle measurement. As with the line filter, this setting should be kept consistent between loaded and no-load BEMF measurements.

Choice of three-phase wiring system

v_d and v_q are defined in terms of phase voltage amplitude and phase, but a motor's neutral point is generally inaccessible, so phase voltage cannot be measured directly. Three practical workarounds exist: measuring line-to-line voltage with 3P3W or 3V3A wiring and correcting the amplitude and phase for an assumed balanced system (the approach used throughout this paper); measuring line-to-line voltage and computing phase voltage from the centroid of the triangle it forms; or grounding the motor case and inverter ground terminal and measuring phase voltage directly with 3P4W wiring, treating ground potential as the midpoint. None of the three is exact if the motor is genuinely unbalanced, and no one method is clearly superior to the others; where the measured motor's real behavior departs significantly from the assumptions in the derivation above, the calculated dq-axis parameters will depart from their theoretical values accordingly.

dq-axis parameter calculation using high-speed waveform computation

The WT5000 reports values averaged over its data update period, which can range from roughly 100 ms to 20 s. For applications that need to see more instantaneous behavior, the DL950 ScopeCorder's high-speed waveform computation function can calculate dq-axis parameters at the waveform sampling rate instead, by entering the same matrix equations described above directly into the DL950's user-defined MATH function. The encoder's A-, B-, and Z-phase signals feed the DL950's real-time math to derive rotation speed, and the same constants (pole pairs, R_a, Ψ_a) are registered as coefficients ahead of time. The offset between the Z-phase signal and the q-axis is again derived from the measured waveform and applied as a correction, and the line-to-line-to-phase-voltage conversion is folded into the matrix coefficients themselves.

With this setup the DL950 computes dq-axis parameters for every sample of voltage, current, and phase difference and displays them as continuous waveforms; placing cursors at the start and end of a single motor rotation converts those waveforms into numerical values for that rotation. The table below summarizes how the two approaches differ in practice.

Conclusion

This paper described how to calculate orthogonal coordinate system dq-axis parameters using the WT5000's and DL950's electrical angle measurement functions on a PMSM. In motor control, i_d and i_q are derived from the rotation angle, current, and voltage measured via the encoder, and the line-to-line voltage is then controlled so the motor produces the required rotation speed and torque. Calculating time-averaged dq-axis parameters with guaranteed measurement accuracy using the WT5000, alongside real-time parameter calculation using the DL950, are both considered highly effective for PMSM development and control work.

The methods described here rely on functions already built into the two instruments. Yokogawa will continue developing new functions and measurement techniques aimed at improving measurement accuracy, real-time performance, and operability to meet the needs of engineers working in this field.

References

• Morimoto, S., & Sanada, M. (2013). Drive System for Energy Saving Electric Motors. Japan: Kagakujyoho Shuppan Co., Ltd.
• Morimoto, S., Takeda, Y., & Hirasa, T. (1993). Parameter Measurement of PM Motor in dq Equivalent Circuit. IEEJ Transactions on Industry Applications, 113-D(11), 1330-1331.