Experimental Verification of High-Speed Permanent ... - IEEE Xplore

Experimental Verification of High-Speed Permanent Magnet Synchronous Motor Model Martin Novak, Jaroslav Novak, Jan Chysky

Φ Abstract -- This paper presents the development and experimental verification of a high-speed permanent magnet motor simulation model. The used motor has maximal speed 42000 RPM, nominal speed 25000 RPM, nominal torque 1,2 Nm and nominal current 11 A. The verification is based on the motor start-up and spin down measurement. Specific issues of high-speed machines like modeling of air friction are addressed. The achieved results show a good agreement between the modeled and experimental results.

Index Terms-- Computational modeling, Experimental verification, High-speed motors, Permanent magnet motors

I.

NOMENCLATURE

vx(t) – instantaneous voltage value in phase x [V] Rs – stator resistance [Ω] Ψ – flux linkage [Wb] ix – instantaneous current value in phase x [A] φ - rotor angle [rad] L – inductance ω – electrical speed [rad/s] ωm –mechanical speed [rad/s] Te – electrical torque [Nm] TL – load torque - mechanical [Nm] J – moment of inertia [kgm2] B – friction damping coefficient [N/rad/s] K – air friction damping coefficient [N/rad2/s2] P – power [W] B – flux density [T] pp - number of pole-pairs [-]

[1] one large turbo compressor can be replaced by a set of multiple smaller turbo compressors by maintaining the same power. A reduction to 1/4 size is mentioned. That a production of such high-speed machine is possible was shown in [2][3]. The motor developed here was a 240 000 RPM PMSM with rated output power 5 kW. Yet the motor dimensions were small, stator diameter 60 mm and rotor diameter only 20 mm. Such high efficiency applications are demanded in the industry as shown in [4]. Also high-speed machines are becoming interesting for the energy production industry as microturbine generators, co-generation units etc. are being developed [5]. To develop a controller system with good control quality, a good estimate of motor parameters is required to allow correct setting of controller. The controller setting can either be made experimentally by some known methods or simulation experiments can be made. For simulation experiments a motor model is required. Although the way of modeling PMSM's is generally well known [6]-[9] some specifics issues for high-speed machines, that are neglected for low speeds, have to be considered in theirs high-speed counterparts. Those issues will be addressed in this paper together with real PMSM parameter identification and comparison of experimental and simulated results. III. PMSM SIMULATION MODEL The PMSM model is based on the electrical properties of the stator windings and on theirs interaction with the rotor. The electrical dynamic equations of the phase voltages va, vb, vc are

II. INTRODUCTION F we look on the development in the field of electrical machines we can see some interesting moments. As in all areas of technology, the trend is to find more efficient solutions. One solution to develop smaller yet powerful machines is to use high-speed machines. Especially permanent magnet synchronous machines (PMSM) are suitable for this purpose. The reason for creating high-speed machines is in theirs high power density. As it was shown in

dψ a dt dψ b vb (t ) = Rs + dt dψ c vc (t ) = Rs + dt va (t ) = Rs +

I

Φ This work was supported in part by the Czech ministry of education, youth and sport grant no. MSM6840770035 “The Development of Environmental - Friendly Decentralized Power Engineering” and internal CTU Grant " Development of measuring, simulation and experimental methods with focus on non-traditional energy source". M. Novak is with Department of Instrumentation and Control Engineering, Faculty of Mechanical Engineering, Czech Technical University in Prague, Prague, Czech republic (e-mail: Martin.Novak2 @fs.cvut.cz). J. Novak is with Department of Instrumentation and Control Engineering, Faculty of Mechanical Engineering, Czech Technical University in Prague, Prague, Czech republic (e-mail:Jaroslav.Novak @fs.cvut.cz). J. Chysky is with Department of Instrumentation and Control Engineering, Faculty of Mechanical Engineering, Czech Technical University in Prague, Prague, Czech republic (e-mail: Jan.Chysky @fs.cvut.cz).

978-1-4673-0142-8/12/$26.00 ©2012 IEEE

(1) (2) (3)

The motor model in dq coordinate system is obtained through Park and Clarke transformations of (1) (3)

diq

+ ω Ld id + ωψ m dt di vd = Rs id + Ld d − ω Lq iq dt

vq = Rs iq + Lq

(4) (5)

The electric torque produced by the PMSM is

Te =

3 p p ⎡ψ miq + ( Ld − Lq ) id iq ⎤⎦ 2 ⎣

(6)

The above described electrical model is completed with a mechanical model

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Te = TL +

K J dω B + ω + air _ friction ω 2 p p dt p p pp

IV. REAL PMSM MODEL PARAMETER IDENTIFICATION To set correctly the PMSM model it was necessary to estimate some model parameters. Some of them could be determined directly from the manufacturer’s data while others required some experiments.

(7)

The air friction can be calculated by equations in [10]. Rotor mechanical speed and rotor angle is

ωm =

ω

pp

ϕ ( t ) = ∫ ωm dt

(8)

TABLE I MOTOR DATA

Motor type: 2AML406B-090-10-170 Manufacturer: VUES Brno Vdc = 560 V In rms = 11 A Tn = 1,2 Nm nn = 25 000 min-1 , nmax = 42 000 minK_E = 7,3 V/ kRPM

Hence the PMSM space state model is

dX = A ⋅ X + Bu + N1 ⋅ X ⋅ ω + X T ⋅ N 2 ⋅ X dt

(9)

where

⎡ did ⎤ ⎢ dt ⎥ ⎢ ⎥ dX ⎢ diq ⎥ dt ⎢ dt ⎥ ⎢ ⎥ ⎢ dω ⎥ ⎢⎣ dt ⎥⎦

⎡ −R ⎢ ⎢ Ld ⎢ A=⎢ 0 ⎢ ⎢ ⎢ 0 ⎢⎣

⎡id ⎤ X = ⎢ iq ⎥ ⎢ ⎥ ⎢⎣ω ⎥⎦

0 −R Lq 2 3 pp ψm 2 J

To estimate other motor parameters two different methods were used and the results compared. Method one consisted of a direct measurement with LCRG Meter Tesla BM591 where both winding resistance and inductance was measured. The disadvantage of this method is that the measuring current is not known and that inductance will be measured with a small unknown current. As inductance is a function of current, it is expected that the real inductance value for nominal current will be higher. In method two, resistance was measured with Ohm's method with a Diametral Q130R50D power supply, 2x Pro's kit digital multimetr MT1232. The inductance in method two was measured from the time constant of a transient characteristics with oscilloscope GDS-806C with probe GTP-060A. An example of this measurement is on Fig. 1. The resulting measured PMSM inductance as a function of stator current is shown in Fig. 2. It can bee seen that the inductance is changing somewhat with current as it was expected. It is rising with increasing current. For nominal current, the inductance is 1,1 mH. It is also obvious that the measurement has a relatively high error as the values are fluctuating. Nevertheless the trend is visible

(10)

⎤ ⎥ ⎥ −ψ m ⎥ ⎥ Lq ⎥ ⎥ − pp F ⎥ J ⎥⎦ 0

⎡1 ⎤ 0 0 ⎥ ⎢ ⎢ Ld ⎥ ⎢ ⎥ 1 0 ⎥ B=⎢ 0 Lq ⎢ ⎥ ⎢ − pp ⎥ ⎢0 ⎥ 0 J ⎦ ⎣ Lq ⎡ ⎤ 0⎥ ⎢ 0 Ld ⎢ ⎥ ⎢ − Ld ⎥ N1 = ⎢ 0 0⎥ ⎢ Lq ⎥ ⎢ 0 0 0⎥ ⎢ ⎥ ⎣ ⎦ 2 ⎡ ⎤ 3 pp Ld − Lq ) 0 ( ⎢0 ⎥ 2 J ⎢ ⎥ N 2 = ⎢0 0 0 ⎥ ⎢ ⎥ p p K air _ friction ⎥ ⎢0 − 0 ⎢⎣ ⎥⎦ J

(11)

(12)

(13)

(14) Fig. 1. Example of PMSM inductance measurement from current step response

The PMSM model based on (10) - (14) was used to create a Simulink model of the system and to identify and verify it with experimental results obtained from the real system.

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PMSM startup with Vdc = 500 V _Id

_Iq

n (min-1)

Iq (A*1000), Iq (A*1000), n (min-1)

50000

40000

30000

20000

10000

0 0,0

0,1

0,2

0,3

0,4

0,5

0,6

-10000 time (s)

Fig. 3. PMSM start-up to speed 42 000 RPM Fig. 2. Measured stator PMSM inductance as function of stator current

The obtained results from both methods are the following. TABLE II PSMS RESISTANCE AND INDUCTANCE

R ( mΩ ) L (μH)

Direct measurement with BM591 310 880

Indirect measurement 210 (for I = 1 A) 800 (for I = 1 A) 1100 (for I = In = 11 A)

As can be seen from the comparison in the table above, the values obtained from both methods are similar but with a relatively high error. The error in stator winding resistance is large because the value of stator winding resistance is small and its size is comparable with the contact resistance in the circuit. The inductance values are comparable and it is expected that a more precise value is the one obtained with indirect measurement as nominal current 11 A is used here. The permanent magnet flux linkage ψm is calculated from the back-emf constant K_E = 7,3V/kRPM

ψm =

vi

ω

⇒ ψ m = 0, 072

[Wb]

Moment of inertia is

Δt 0, 4 = 0,98 ⋅ = 0,11⋅10−3 Δω 2π ⋅ 572

As the machine used for this research has maximal speed 42 000 RPM, it can be expected that air friction losses will be much lower. The reason for this is that according to power losses caused by air friction are given

Pf _ air = c f πρ airω 3 r 4l

(17)

Where cf is friction coefficient, ρair is density of air at given temperature and pressure, ω is rotor angular speed, r is rotor radius and l is rotor length. Air friction torque is then

T f _ air = c f πρairω 3 r 4l / ω

(15)

Motor’s moment of inertia was calculated from a motor startup on Fig. 3. In this experiment the motor was powered with a given current Iq = 9 A until field weakening started. The start time was 0,4 s and the reached speed was 572 Hz (34 340 min-1). The startup current 9 A corresponds to torque 0,98 Nm. Considering the relatively high current and torque, mechanical losses were neglected in the calculations.

J =T

As can be seen from the model, air friction losses are considered. This is important for high-speed machines as the losses caused by air friction can have the same size as friction losses in bearings. According to [11], where speed was 500 000 RPM, friction losses caused by air friction were 8 W, where as bearing losses were 10 W for two bearings.

(18)

And therefore the friction torque is a function of ω2 as it is used in the model. The friction coefficient itself is dependent on the size of air gap and air flow in the air gap given by Reynolds and Taylor numbers. Unfortunately, none of those parameters could not be measured or determined precisely analytically. For this reason an attempt was made to at least estimate bearing and air friction with an experiment. It consists of accelerating the motor to maximal speed and turning off the inverter. The rotor will spin down naturally. The deceleration of the rotor is measured as a function of time.

(16)

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It is obvious that when the motor is un-powered and unloaded, its mechanical speed will decrease with losses until a complete halt. This is described by the following dynamic equation

p B p K dω = − p ω − p air _ friction ω 2 dt J J

(19)

It has to be noted that (19) does not represent all losses in the motor. The solution is given by (20)

ω (t ) = B⋅e

⎛ B ⋅t ⎞ ⎜ ⎟ ⎝ J ⎠

B ⋅ ω ( 0) − K air _ friction ⋅ ω ( 0 ) + K air _ friction ⋅ ω ( 0 ) ⋅ e

⎛ B⋅t ⎞ ⎜ ⎟ ⎝ J ⎠

(20) Equation (20) was used to find coefficients B and Kair_friction from experimentally measured PMSM rotor deceleration. The search was done with a least square method in Matlab. The comparison on Fig. 4 is a best match that could be achieved by varying just parameters B and Kair_friction . As can seen, there are some significant differences especially for lower speeds. The reason is that other losses in the motor have been neglected. One is the loss caused by eddy currents. As the permanent magnet is rotating, it induces currents to the stator windings and stator iron. Also there is an interaction between the permanent magnet and those currents. Stator core losses could be determined by the Steinmetz equation [12][13]

Pcore = Cm ⋅ f α ⋅ Bmβ

(21)

Where Cm, α and β are material constant, Bm is peak flux density and f is frequency of the current. It can be seen that losses are a function of frequency i.e. rotational speed. The function is non linear. Unfortunately this calculation is impossible as the stator material is not known for our motor. Another loss that has been neglected is reluctance loss the interaction between the permanent magnet and stator iron. In other words, the permanent magnet is attracted to the stator iron and is braked by it. This is visible when the rotor is turned by a bare hand. It takes then preferably one of four positions. This effect seems to be significant for lower rotor speeds. Considering those simplification, the presented fit can be considered an approximate model. It is obvious that further model improvements and precisions would be required in the future. V.

DESCRIPTION OF THE EXPERIMENTAL SYSTEM

As can be seen form the block diagram on Fig. 5, the experimental system is composed of several blocks. The power network voltage is rectified with an inverter to DC voltage. The inverter has to be able to transfer the energy not only from the power network, but also back. For this reason a standard diode rectifier can not be used, but the structure has to incorporate transistor switches. At the present time this block has not yet been implemented, therefore it is dashed in the diagram. The DC voltage is supplying an insulated gate bipolar transistor (IGBT) inverter build for this purpose. The inverter is using power module SKM75GD124D and IGBT/MOSFET driver SKHI61 both from Semikron. The whole system is being prepared to work as a microturbine generator, the turbine is created by means of a standard car turbocharger. The experimental setup is shown on Fig. 6.

Fig. 4. Simulation and experimental result comparison for PMSM spin down

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Fig. 5.Block diagram of the experimental system

VI. CONCLUSIONS This paper has shown results of high-speed PMSM modeling, experimental parameter identification and verification between the measured and simulated data. The obtained results show agreement although there are visible differences especially for lower speeds. The model was verified by measuring the motor spin down from maximal speed 42 000 RPM. The possible reasons for differences between the model and experiments are discussed and could lead to further improvements in the future. Fig. 6. Coupled high-speed PMSM with turbocharger

VII.

This has the advantage of good availability, high reliability and low price. For the purpose of system testing the turbocharger is connected to a compressed air supply, in the future a combustion chamber will be build. Also some other alternatives for the turbine have been considered like a model aircraft turbine or a real aerospace turbine.

[1]

[2]

The final identified model parameters are summarized in Table III

[3]

TABLE III PSMS MODEL PARAMETERS

[4]

Model paramete r R ( mΩ )

Value and units

note

260 [mΩ]

L (μH)

1100 [μH] 0.072 [Wb] 0.11 . 10-3 [kgm2] 8.2. 10-5 [N/rad/s] 1,3. 10-10 [N/rad2/s2]

Average from direct and indirect measurement 1100 (for I = In = 11 A)

ψ

m

J B Kair_friction

[5]

[6] [7]

Very small, could be significant for higher speeds

[8]

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REFERENCES

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VIII.

Martin Novák received the MSc. degree in Instrumentation and Control Engineering in 2003 from the Faculty of Mechanical Engineering at the Czech Technical University in Prague, Czech Republic. He continued with a PhD. degree (2008) at the same university. Presently he works as an assistant professor. His main areas of interest are microcontrollers, signal processing and measuring methods and instruments. Jaroslav Novák received the MSc. degree in Power Electronics in 1989 from the Faculty of Electrical Engineering at the Czech Technical University in Prague, Czech republic. In year 1992 he has received a CSc. degree (Ph.D.) from the department of Electrical machines and traction on the same faculty. He was working since 1992 as an assistant professor, since 2003 as associated professor on the Department of Instrumentation and Control Engineering on the Faculty of Mechanical Engineering, CTU in Prague. Between 1995 and 2001 he has been working in the company Electrosystem in the field of controller system design for industrial traction applications. Since 2010 he is a full time professor. Jan Chyský, head of the Department of Instrumentation and Control Engineering, received the MSc. degree in Control Engineering in 1979 from the Faculty of Electrical Engineering at the Czech Technical University in Prague, Czech republic. In year 1987 he has received a CSc. degree (Ph.D.) on the faculty of mechanical engineering, CTU Prague. Since 1992 he is an associated professor of the Department of Instrumentation and Control Engineering.

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