# PMSM Electrical Parameters Measurement

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1 Freescale Semiconductor Document Number:AN4680 Application Note Rev. 0, 02/2013 PMSM Electrical Parameters Measurement by: Viktor Bobek 1 Introduction The vector control, also known as the field-oriented control (FOC), of a permanent magnet synchronous motor (PMSM) is the algorithm often used in today s advanced motor control drives. Such advanced motor control algorithms require the setting of motor electrical parameters for its proper functionality. This application note deals with the measurement of electrical parameters needed for vector control of PMSM. The electrical parameters are needed to set the current PI controller gains to get the desired closed-loop performance and for BEMF observer constants. The proposed measurement techniques determine a number of pole pairs, a stator resistance, synchronous inductances, and an electrical constant with common measurement equipment. A summary of PMSM sensorless control and explanation of motor control terms can be found in [1 ]. 2 Motor parameters needed for PMSM FOC One of the possible methods to set the PI controller gains, is to calculate them from motor parameters. The current PI controller gains in time domain are calculated from the motor electrical parameters [1 ]; see the following equations. Contents 1 Introduction Motor parameters needed for PMSM FOC Motor pole pairs Background Guide Stator resistance Background Guide Synchronous inductances Background Guide Back-EMF constant Background Guide Conclusion References Acronyms and Abbreviated Terms Freescale Semiconductor, Inc.

2 Motor parameters needed for PMSM FOC Equation 1 Equation 2 Where ω 0 is the natural frequency of the current closed-loop system (loop bandwidth) and ξ is the current loop attenuation. Therefore, the PMSM vector control algorithm typically requires the following parameters. Table 1. Electrical parameters needed for FOC Electrical parameters needed for FOC current PI controller gains calculation Parameter Dimension Description Used in constant calculation R s (Ω) Resistance one of the motor phase L d (H) d-axis inductance of one motor phase L q (H) q-axis inductance of one motor phase Additional electrical parameters needed for FOC Current PI controller BEMF Observer Current PI controller BEMF Observer Current PI controller BEMF Observer K e (V.s/rad) Electrical constant BEMF Observer pp (-) Motor pole pairs Speed and position mechanical/electrical quantities recalculation The speed PI controller gains in time domain are calculated from the motor/load mechanical parameters; see Equation 1 on page 1 and Equation 2 on page 2 Equation 3 Equation 4 Where ω 0ω is the natural frequency of the speed closed-loop system (loop bandwidth) and ξ is the speed loop attenuation. Therefore, the PMSM vector control algorithm typically requires the following parameters. Table 2. Mechanical/load parameters needed for FOC speed PI controller gains calculation Parameters Dimension Description Used in constant calculation J (kg.m 2 ) Total mechanical inertia Speed PI controller B m (N.m.s) Viscous friction coefficient More precise speed PI controller gains setting The measurement of individual electrical parameters is described in the following chapters of this application note. 2 Freescale Semiconductor, Inc.

3 Motor pole pairs 3 Motor pole pairs 3.1 Background The motor pole pairs parameter defines a ratio between mechanical and electrical quantities (mechanical vs electrical rotor position/speed). The motor pole pairs represent the number of north and south segments the rotor contains. 3.2 Guide The equipment required to measure motor pole pairs depend on the method used for measurement. DC power supply Three-phase inverter, oscilloscope, hand velocity meter, and a current probe Driving motor, oscilloscope and a voltage probe Usually, the number of the motor pole pairs is written on the label of the motor. If there is no information regarding the number of pole pairs, it can be determined. See the following subsections Method to determine low number of the pole pairs Guide: The following steps describe the method to determine the low number of motor pole pairs. See Figure Connect the phase A wire to the positive potential (+) and phase B and C to negative potential (-) of the voltage source. 2. Set a current limit of the power supply to such a level so that the user is able to rotate the shaft manually, and the rotor is aligned in the stable position. Common current limit is about 10% of the rated motor current. For more powerful motor, the current limit is lower. 3. Draw a line/sign for every stable position in which the rotor is aligned. 4. Number of stable positions is equal to the motor pole pairs. Freescale Semiconductor, Inc. 3

4 Motor pole pairs Figure 1. Method for the determination of low number of pole pairs Method to determine high number of the pole pair It is possible to use two methods for determination high number of pole pairs. Selection of the method depends on available measuring equipment. An oscilloscope is required for measurement in both the methods Method A: a current probe and an inverter using Volt/Herz method to spin the motor with unknown parameters Method B: a voltage probe and driving motor, which spins the motor Guide for Method A: The following steps describe the method to determine the high number of motor pole pairs. 1. Spin the motor by an inverter using Volt/Herz method and set the frequency in such a way that the motor will spin at a constant, and preferably higher speed. 2. Measure the phase current frequency using oscilloscope current probe. The frequency of the phase current must be the same as that generated by Volt/Herz method. 3. Measure the speed of the motor by some hand velocity meter. The speed reading must be constant. 4. Calculate the number of pole pairs using the equation given below. The result should be very close to an integer number. Equation 5 4 Freescale Semiconductor, Inc.

5 Stator resistance Figure 2. Current waveform for the determination of high number of the pole pair Guide for Method B: The following steps describe the method to determine the high number of motor pole pairs. 1. Spin the motor by an external driving motor at a constant speed. 2. Measure the generated voltage frequency. 3. Measure the speed of the motor by some hand velocity meter. 4. Calculate the motor pole pairs using Equation 5 on page 4. 4 Stator resistance 4.1 Background A resistance of the stator winding Rs is defined as a resistance between a phase terminal and the center of the winding. The winding resistance is temperature dependent. Usually the resistance value at 25 C or specified temperature is listed in the motor s datasheet. Calculate resistance R at operational temperature t ( C) of stator winding (if the temperature is known), using the resistance value measured at temperature t 0 ( C). Equation 6 where α is the constant determined by the material (for copper, α = K -1 ) Equation 7 For 50 C temperature difference, R can be calculated as given below. Equation Guide The equipment required to measure stator resistance depend on the method used for measurement: Digital multimeter RLC meter Freescale Semiconductor, Inc. 5

6 Stator resistance Digital multimeter Higher values of stator resistance ( > 10 Ω) can be measured by a digital multimeter. The usual stator winding configuration is the wye, so the final stator resistance is half of the measured resistance. The following figure shows the stator resistor measurement using a digital multimeter. Figure 3. Stator resistance measurement by a digital multimeter RLC meter Lower values of stator resistance can be measured by an RLC meter, for example MOTECH MT 4080A. The four-terminal measurement reduces the effect of the test lead resistance. See Figure 4. Usual measurement range is between (10 mω 10 kω). Before the measurement, calibrate the RLC meter (open-circuit, and short circuit). The usual stator winding configuration is the wye, so the final stator resistance is half of measured resistance. 6 Freescale Semiconductor, Inc.

7 Synchronous inductances Figure 4. Four-terminal measurement schematic 5 Synchronous inductances 5.1 Background The synchronous inductances of Interior Permanent Magnet Synchronous Motor (IPMSM) winding are different (L d <L q ), because of lower reluctance in q-axis. The synchronous inductances of Surface Mounted Permanent Magnet Synchronous Motor (SMPM) motor are almost equal, because the permanent magnets are surface mounted and reluctance is the same in every position, that is: μ PM μ air L d L q, where μ PM is the relative permeability of the permanent magnet, and μ air is the relative permeability of the air. See the following figure depicting the reluctance paths of d- and q-axis in IPMSM. Figure 5. Reluctance paths in d- and q-axis of IPMSM Freescale Semiconductor, Inc. 7

8 Synchronous inductances In practice, magnetic circuits are subject to saturation as the current increases. Especially, when current I q is increased, the value of L q is decreased. Since I d is maintained to zero or negative value (demagnetizing) in most operating conditions, saturation of L d rarely occurs. The flux linkage λ m and L d are subject to armature reaction. See the following figure. Figure 6. Typical inductance characteristic of PMSM NOTE Majority of the applications use single value; however the determination of inductances depends on selected working conditions. In order to measure synchronous inductance, the users must maintain balanced three-phase current condition. When the rotor is aligned with the center of phase A winding, L d (L q ) can be derived from the measured equivalent inductance L of the circuit, as shown in the following figure. 8 Freescale Semiconductor, Inc.

9 Synchronous inductances Figure 7. Inductance measurement circuit Depending on the rotor angle θ el, it is possible to measure inductance in d-axis or q-axis, where L is the total inductance for serial-parallel connection of the stator winding: Equation 9 Equation 10 When the rotor is aligned with phase A (θ el = 0 ) and locked, then the current response is first order RL circuit. Where τ is a time constant of the circuit Equation 11 Equation 12 After measuring τ, the inductance L d can be calculated as follows. Equation 13 Freescale Semiconductor, Inc. 9

10 Synchronous inductances Figure 8. Equivalent phase model of PMSM in d/q axis for a locked rotor shaft Since V d = (2/3)V, V q = 0, and I is the same as I d and the total resistance of the circuit is (3/2)R s, the equivalent inductance seen from the supply source is (3/2)L d. Similar explanation can also be applied to L q when the rotor is locked at 90 electrical Q-axis alignment To measure the inductance in q-axis without an inverter, an alignment has to be done into the q-axis. The alignment into d- axis is done by phase A connected to the positive potential (+) and phase B and C are grounded (-). It can be seen from the following figure that 90 electrical shifted position is when phase B terminal is connected to the positive potential (+) of the voltage source, phase C is grounded (-), and phase A is floating (NC). Figure 9. Explanation of q-axis alignment 10 Freescale Semiconductor, Inc.

11 Synchronous inductances 5.2 Guide The equipment required to measure inductances in q-axis and d-axis are as follows. DC power supply, oscilloscope, current and voltage probe Figure 10. Set up to measure inductance in q-axis Guide to measure d-axis inductance (non-saturated inductance measurement): Follow the steps given below to measure the d-axis inductance L d. 1. Align the rotor to phase A. Phase A is connected to the positive potential (+) and phase B and C are grounded (-). 2. Lock the rotor shaft. 3. Apply negative step voltage. Phase A is grounded (-) and phases B and C are connected to the positive potential (+). Usual level of the current is about 10% of the rated phase current. 4. Measure the step response of the current by a current probe. See Figure Calculate inductance L d. Freescale Semiconductor, Inc. 11

12 Back-EMF constant Figure 11. Current step response waveform Guide to measure q-axis inductance: Follow the steps given below to measure the q-axis inductance L q. 1. Align the rotor to the q-axis. Connect the phase B terminal to the positive potential (+) of the voltage source and phase C is grounded (-). Phase A terminal is floating. 2. Lock the rotor shaft firmly because current step response in q-axis creates torque. 3. Generate a current step response in this configuration: phase A is connected to the positive potential (+) of the voltage source and phases B and C are grounded. 4. Calculate inductance L q in the same way as L d. 6 Back-EMF constant 6.1 Background The back-emf (BEMF) constant (flux linkage of the PM denoted by λ m ) can be obtained by measuring the no-load line voltage V pk of the motor while it is driven through the shaft at a constant speed of ω m. The constant gives a ratio between BEMF voltage and the angular electrical frequency/speed. 12 Freescale Semiconductor, Inc.

13 Back-EMF constant Figure 12. Three-phase measurement of the BEMF constant 6.2 Guide The equipment required to measure the BEMF constant are listed below. Oscilloscope and at least one voltage probe Driving motor or hand drill machine The steps given below must be followed to determine the BEMF constant. 1. Spin the motor by an external driving motor or a hand drill machine at a constant speed. Higher speed is preferred, because the voltage measurement error is lower. 2. One-phase measurement: Measure the generated phase voltage (between one phase terminal and neutral point of the motor). Usually the neutral point is not accessible; then measure the line-to-line voltage. Three-phase measurement: If the neutral point is not accessible, it s possible to create the artificial neutral point from all three voltage probe clips connected together. See Figure Calculate the Back-EMF constant according to Equation 14 on page 13. Single phase measurement (line-to-line voltage measurement): Three-phase measurement (phase voltage measurement): Equation 14 Equation 15 Freescale Semiconductor, Inc. 13

14 Conclusion Figure 13. Three-phase oscilloscope measurement of the electrical constant 7 Conclusion The application note summarizes methods for determining electrical parameters of PMSM. The precise parameters determination is needed for sensorless control applications and desired closed-loop performance. The proposed measurement techniques determine the number of pole pairs, the stator windings resistance, the synchronous inductances, and the electrical constant with common measurement equipment. The parameters are determined using measured applied voltages and responding currents. A single-phase DC voltage power supply can be used to determine the synchronous inductances of three-phase PMSM with sufficient accuracy. 8 References 1. DRM110: Sensorless PMSM Control for an H-axis Washing Machine Drive, available on freescale.com. 2. MOTECH 4080A Operation Manual, available at motech.com.tw 14 Freescale Semiconductor, Inc.

15 Acronyms and Abbreviated Terms 9 Acronyms and Abbreviated Terms The following table contains acronyms and abbreviated terms used in this document. Table 3. Acronyms and Abbreviated Terms Term d-axis f IPMSM n NC PMSM q-axis SMPM θ el μ Air μ PM Meaning Direct axis Frequency (Hz) Interior Permanent Magnet Synchronous Motor Mechanical speed (rpm) Not Connected Permanent Magnet Synchronous Motor Quadrature axis Surface Mounted Permanent Magnet Synchronous Motor Electrical rotor position Relative permeability of the air Relative permeability of the permanent magnet Freescale Semiconductor, Inc. 15

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