Design of optimal digital controller for stable super-high-speed permanent-magnet synchronous motor L. Zhao, C.H. Ham, Q. Han, T.X. Wu, L. Zheng, K.B. Sundaram, J. Kapat and L. Chow Abstract: A new collaborative design scheme of a super-high-speed permanent-magnet synchronous motor (PMSM) and its digital controller is presented, which provides a low-cost but highly efficient motor system with guaranteed stability and performance. Since the systematic design of the PMSM can ensure its stability over the full operating speed range, a simple and reliable open-loop controller can be designed for the super-high-speed motor. With stability assurance, an optimal digital control is also designed in order to enhance the efficiency and performance of the PMSM. The unique feature in the proposed optimal V/f control is its design consideration to the stator resistor, which is generally neglected in most V/f controls but cannot be neglected in the super-high-speed motor owing to the extra small-size requirement. The simulation and stability analysis for various design options are provided. Finally, simulation and experimental results validate the design technique and its effectiveness.
1
Introduction
In recent years, the demand for super-high-speed motors has increased owing to the advanced technology and reduced cost. Several design advantages, such as low harmonics in the induced electromagnetic force, no excitation power loss in the rotor and only low eddy-current loss in the stator and rotor, allow the PMSM to have greater efficiency at high speed and become more attractive in super-high-speed applications. The most common motor control methods are scale control, vector or field-oriented control, and direct torque control (DTC) [1]. Vector control and DTC normally are used to achieve high dynamic performance in position, speed and torque. However, for applications that do not have a high dynamic performance requirement and whose load is predictable, a simple V/f scale digital control strategy can be used instead of the more complex vector or direct torque control [2]. Open-loop V/f control is an attractive solution for super-high-speed PMSMs because of its inherent lack of mechanical shaft sensors. To make motors stable in their speed range, damper windings are implemented to assure the synchronisation of the rotor with the applied stator electrical frequency. However, because of high cost, design difficulties and efficiency considerations, damper windings are not generally implemented in PMSMs [3, 4]. As a result, the stability of super-high-speed PMSMs is an important issue particularly under open loop V/f control, given that PMSMs may be unstable when applying frequencies exceeding certain value. Different approaches need to be developed in order to solve the stability problem.
A new PMSM design approach that satisfies design and stability requirements is provided in this paper. The effect of the motor parameters on dynamic characteristics, including stability and performance, is analysed. It is shown that super-high-speed PMSMs stability can be achieved over full operating speed range. Constant V/f scale control profiles for ordinary motors are straight lines with boost voltages. This occurs because stator resistors are negligible when compared with reactance and only have significant effect at low frequencies. However, for the proposed super-high-speed PMSM, due to the extra small size, the stator resistance is quite large compared with the stator reactance and cannot be neglected. No research results using the V/f control scheme application with consideration of stator resistor have been found. In this paper, the stator resistance, Rs, is taken into account to design an optimal V/f digital control. A nonlinear V/f digital control scheme is also derived. The optimal V/f control schematic is implemented with the proposed superhigh-speed PMSM with rated speeds of 50 000 rpm. With the proposed optimal digital controller, true constant V/Hz control is realised, which assures that maximum available torque per ampere of the stator current can be achieved with fast transient response [5]. The optimal method also provides better motor system performance especially at low speed.
2
PMSM stability analysis
The steady-state equations for PMSMs can be expressed as [6, 7]: r IEE, 2006 IEE Proceedings online no. 20045266 doi:10.1049/ip-epa:20045266 Paper first received 13th December 2004 and in final revised form 18th July 2005 L. Zhao, Q. Han, T.X. Wu, L. Zheng and K.B. Sundaram are with the Department of Electrical and Computer Engineering, University of Central Florida, Orlando, FL 32816, USA C.H. Ham, J. Kapat and L. Chow are with the Department of Mechanical, Materials and Aerospace Engineering, University of Central Florida, Orlando, FL 32816, USA E-mail:
[email protected] IEE Proc.-Electr. Power Appl., Vol. 153, No. 2, March 2006
diq dt did dt dor dt dy dt
Rs Ld lm V iq id or or þ cos y Lq Lq Lq Lq Lq Rs V ¼ id þ iq or sin y Ld Ld Ld N 3N 2 B ¼ lm iq þ ðLd Lq Þiq id TL or J J 2J ¼
ð1Þ
¼ o or 213
where id and iq are the d- and q-axes currents, respectively; Ld and Lq are the inductance values for the d- and q-axes, respectively; V is the supplied voltage, Rs is the stator resistance, y is load angle, o is supply electrical frequency, or is the rotor speed, N is the number of pole pairs, and lm is permanent-magnet flux, J is the moment of inertia, B is the coefficient of viscous friction, and TL is the load torque. The linearised version of system equations (1) is given by 2 Rs Ld 2 3 or 6 Lq Lq Di 6 q 7 6 6 7 6 L Rs q 6 6 Did 7 6 or 7 ¼6 6 L L d d 6 7 6 6 Dor 7 6 2 3N 2 5 6 3N 4 ðLd Lq Þid 4 2J lm þ ðLd Lq Þid 2J Dy 0 0 3 l m þ Ld i d V sinðyÞ 7 Lq Lq 72 Di 3 q 7 76 Lq V 7 Di or cosðyÞ 7 d 7 76 6 Ld Ld ð2Þ 74 Do 7 r5 7 B 7 7 Dy 0 5 J 1 0 The eigenvalues of the matrix provide complete information to predict the stability properties of the PMSM at any balanced operating condition. The eigenvalues form two groups. One group is composed of well-damped complex conjugate pairs. The other group consists of either lightly damped or negatively damped complex conjugate pairs. The well-damped group has no effect on the stability analysis, so the stability study was focused on the second group, the under-damped or negative-damped pairs, which has a substantial influence on the stability characteristics. As shown in results presented in [8], it is clear that the eigenvalues are functions of motor parameters and the stability depends only on the time constant of the motor, Ls/Rs. The higher the value of stator resistor Rs, the faster the motor will run to the stability boundary. So, Rs and Ls are considered as design variables within certain lower and upper limits, while the design requirement and performance of the super-high-speed PMSMs are satisfied. As an example, the motor stability for different Rs with Ls ¼ 1.6 mH at no-load condition are shown in Fig. 1. It shows that resistor Rs has a strong effect on the eigenvalue
location, i.e. the stability. A larger stator resistance is favourable for stability, but results in higher copper loss. In our design, 0.06 O stator resistance is chosen considering stable margin and other design requirements. Also a nonsalient-pole machine design with inductance Ld ¼ Lq ¼ Ls is taken in this design. Note that a comprehensive stability analysis can be found in [8], which presents theoretical stability criteria for hybrid stepping motors using Rs/Ls in detail. 3 Collaborative PMSM design utilising stability analysis The PMSM stability analysis is employed to evaluate suitable motor design parameters. Owing to the flexibility of some design parameters, it is possible that the super-highspeed PMSM can satisfy the design requirements and stability at the same time. A stable slotless PMSM is designed that demonstrates the effectiveness of the proposed design method. FEM simulation and tradeoffs are used in the design to gauge and optimise the performance. The structure of the proposed motor is shown in Fig. 2. It is a 100 W and threephase super-high-speed permanent-magnet synchronous motor. The permanent magnets are made of Nd–Fe–B to provide high energy density. The motor configuration is composed of a single rotor and a single stator. The stator core is slotless and the winding is fitted on the surface of the stator back sintered powdered iron to reduce eddy-current loss. Four axially magnetised permanent magnets are mounted on the steel back iron in the rotor. The basic specifications are shown in Table 1.
casing stator
winding shaft
PM
120
3300 Hz 2500 Hz 1500 Hz
100
Fig. 2
imaginary axis
80 500 Hz 60
4
40 0.003 Ω 20
0.06 Ω
0.006 Ω 0.03 Ω 0.3 Ω 100 Hz
0 − 1.0 − 0.8 − 0.6 − 0.4 − 0.2
0 0.2 real axis
0.4
0.6
0.8
1.0
Fig. 1 Loci of super high-speed PMSM under different driving frequencies and Rs, with Ls ¼ 1.6 mH 214
PMSM structure
Derivation of optimal V/f control strategy
The ordinary V/f control method design is based on the effect of the stator resistance becoming smaller than reactive impedance when frequency increases. The stator resistor effect can be neglected and only compensated for by a boost voltage. However, for the proposed motor, within the normal operation speed range (below 50 000 rpm), the stator resistance is always larger than the reactive impendence. These two values are the same only after an electrical frequency increase of over 5 kHz (150 000 rpm) (Table 2). IEE Proc.-Electr. Power Appl., Vol. 153, No. 2, March 2006
Table 1: Specifications of three-phase axial flux PMSM Rated output power
100 W
Rated speed
50 000 rpm
Rated voltage
28 VDC
Xs
Rs
Rs Iq
Iq Xs
Is
Id Xs V
Rotor diameter
54.6 mm (2.15 in)
Phase resistance Rs
0.06 O
Ef
Es V
Rated frequency
1667 Hz
lmf
1.4 104 Wb
Ld
1.6 mH
Lq
1.6 mH
J
36.52 mkg m2
Ef
δ
Is
Iq
Id
Fig. 3
In the constant V/f control method, the command phase voltage V* is generated from a function generator, in which the stator flux remains constant, which permits for nearly maximum available torque per ampere of the stator current and fast transient response for the V/f control [5]. The steady-state equivalent circuit and phasor diagram of the PMSM motor, neglecting the core loss, is shown in Fig. 3. Is is stator current, Xs is the synchronous reactance, d is the electrical angle between Vs and the induced EMF Ef from the magnet, which is positive for motoring. Es is the stator voltage, neglecting Rs. The steady-state equation [5] can be derived as Vd 0 Rs oLs Id ¼ þ ð3Þ Ef Vq Iq oLs Rs where Vd and Vq are the d- and q-axes stator voltages, respectively. Substituting Vd ¼ Vs sin d and Vq ¼ Vs cos d, we obtain: Iq ¼
q
Rs Id
Rs V cos d Rs Ef þ oLs V sin d R2s þ o2 L2s
ð4Þ
Then, the developed torque is 3NEf Iq 3Pout ¼ or o 3NEf Rs V cos d Rs Ef þ oLs V sin d ¼ R2s þ o2 L2s o
Te ¼
where, Pout is the motor developed power.
ð5Þ
d
PMSM steady-state equivalent circuit and phasor diagram 1
From Fig. 4, we obtain: V Ef ffd Is ¼ Rs þ joLs V ðEf cos d þ jEf sin dÞ ¼ Rs þ joLs ¼ jIs jfff where
ð6Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðV Ef cos dÞ2 þ ðEf sin dÞ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jIs j ¼ R2s þ o2 L2s
and f ¼ tan1
oLs V þ Rs Ef sin d oLs Ef cos d Rs V Rs Ef cos d oLs Ef sin d
where f is the power factor angle. To find dmax (when maximum available torque per ampere of the stator current occurs), we obtain Te/7Is7 as: 3NEf Rs V cos d Rs Ef þ oLs V sin d o Te R2s þ o2 L2s qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ jIs j ðV Ef cos dÞ2 þ ðEf sin dÞ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2s þ o2 L2s 3NEf Rs V cos d Rs Ef þ oLs V sin d ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o R2s þ o2 L2s ðV Ef cos dÞ2 þ ðEf sin dÞ2 ð7Þ Differentiating the (7) with respect to d and setting it to zero leads to the following equation, from which the
Table 2: Various PMSM motor parameters Parameters
Motor 1
Motor 2
Motor 3
Motor 4
Proposed motor
Number of poles
4
10
6
2
4
Rated voltage
100 V
50 V
380 V
360 V
28 V
Rated current
3.6 A
4.3 A
4.1 A
237 A
5A
Rated speed
N/A
N/A
1750 rpm
70 000 rpm
50 000 rpm
Rated fe
N/A
N/A
58 Hz
2333 Hz
1667 Hz
lmf
281 mV s
108 mV s
0.4832 V s rad1
N/A
1.4 104 Wb
Ld
109 mH
4.9 mH
41.59 mH
28 mH
1.6 uH
Lq
35 mH
4.9 mH
50.07 mH
28 mH
1.6 uH
Rs
2.4 O
0.75 O
3.3 O
0.0055 O
0.06 O
J
2.9 Nm s2
2.9 Nm s2
12 Nm s2
17 Nm s2
36.52 mkg m2
f (Rs ¼ Xds)
10.9 Hz
24.3 Hz
12.5 Hz
31.2 Hz
5968.3 Hz
IEE Proc.-Electr. Power Appl., Vol. 153, No. 2, March 2006
215
Xs = ω Ls
8
Rs
I
7 6
voltage
5 4 3
Es V ∠ 00
2
Ef ∠ δ0
1 0 200
I Rs
600
800 frequency
1000
1200
1400
Linear V/f control with boost voltage (Rs ¼ 0.001 O)
Fig. 5
j IX s
400
15 Es
14 13
V
12 11 I
voltage
I Rs
Ef
10 9 8 7
6
5
Fig. 4
4
PMSM steady-state equivalent circuit and phasor diagram 2
0
Fig. 6
500
1000
2000 1500 frequency
2500
3000
Optimal V/f curve for proposed motor
corresponding dmax may be solved by Simulink: V can also be presented as V ¼ jIs jRs cos f þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jEs j2 ðjIs jRs sin fÞ2
ð8Þ
ð9Þ
At any frequency f, the required value of Es is set as [9]: Vso Es ¼ f ð10Þ fr where Vso is the magnitude of Es at rated frequency fr, which is a constant defined by rated conditions. Ef also can be substituted by Ef ¼ Kf, where K is a constant determined by the system. So, we can rewrite V as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ¼ jIs jRs cos f þ ððVso =fr Þf Þ2 ðjIs jRs sin fÞ2 ð11Þ Using d ¼ dmax, we obtain the optimal V/f control when stator resistance cannot be neglected. When Rs is small enough, its effect is small and linear V/f control only with a boost voltage can be used (Fig. 5). Figure 6 is the proposed optimal V/f control profile for the super-high-speed motor. Figure 7 shows the effect of different stator resistor values. The optimal V/f control profiles are quite different from the originally linear V/f control profile. The stator resistance affects the schematic 216
15
10 voltage
oLs V cos dmax ¼ oLs Ef þ Rs V sin dmax
Rs = 0.09 Ω Rs = 0.06 Ω Rs = 0.03 Ω Rs = 0.01 Ω Rs = 0.001 Ω
5
0
0
500
1000
2000 1500 frequency
2500
3000
Fig. 7 Optimal V/f control curves for different stator resistance and Ls ¼ 1.6 mH, K ¼ 0.0045
substantially during the entire proposed motor speed range. Its effect fades with increasing frequency and the scheme becomes linear when the frequency passes some value. The IEE Proc.-Electr. Power Appl., Vol. 153, No. 2, March 2006
proposed curves have a clear decreasing trend at low frequencies because the voltage drop on the stator resistor is faster than that increment on the reactance at low frequency. When increasing stator resistance, boost voltage increases quickly. At the same time, the optimal control scheme has a much sharper curve and a larger nonlinear frequency range, which can be found from Fig. 7. The slope of linear part also decreases. The inductance Ls also plays an important role in the optimal curve. When inductance increases, the boost voltage drops in an opposite manner because the stator resistor effect becomes smaller (Fig. 8). Otherwise, if the inductance voltage increases, the boost voltage will increase dramatically and the bending trend becomes gentle (Fig. 9). The relationship of the stator resistance and the voltage drop effect are nonlinear. The effect weakens quickly when the frequency increases. Simulation results also show that stator resistance critically affects the dynamic stability. An almost linear V/f control scheme happens far before the cutoff frequency, where it was originally thought that the stator resistance effect disappears.
5
Experimental results
The proposed super-high-speed PMSM and digital controller system configuration is shown in Fig. 10. The digital controller was based on a low-cost DSP, TI TMS320LF2407A, which runs with a 40 MHz clock. The optimal V/f control scheme is realised by software that is operated by a DSP chip. The test was conducted with the motor operating up to 60 krpm in order to demonstrate the stability over the full operating speed range with 20% speed margin. With the proposed optimal constant V/f control strategy, the motor has a much better dynamic performance, as shown in Fig. 11, compared to motor phase current under the original constant V/f control scheme and the optimal V/f control scheme. Figure 12 shows excellent motor phase voltage and phase current sinusoidal waveforms at 50 000 rpm. The FFT analysis of the phase current Ic harmonics is shown in Fig. 13. The most prominent
f∗
acceleration
f
V
deceleration
16
programmable SVPWM signal generator
f
∫
interface
threephase VSI
PMSM motor
14
10
Fig. 10
Proposed super-high-speed motor system configuration
8 6
0.25
4
0.20 Ls = 1.3 µH Ls = 1.6 µH Ls = 1.9 µH
2
error, %
voltage
programmable dead time generator
DSP
12
original V/f control scheme proposed V/f control scheme
0.15 0.10
0 0
500
1000
1500 2000 frequency
2500
3000
0.05 0
Fig. 8 Optimal V/f control curves for different inductance and Rs ¼ 0.06 O, K ¼ 0.0045
0
10
20
30 speed, krpm
40
50
60
Fig. 11 PMSM phase current ring error with mechanical speed (normalised) 16 14 12 a voltage
10 8
b T
6 4
K = 0.00445 K = 0.0045
2
K = 0.00455 0 0
500
1000
1500 2000 frequency
2500
3000
Fig. 9 Optimal V/f control curves for different K and Rs ¼ 0.06 O, Ls ¼ 1.6 mH IEE Proc.-Electr. Power Appl., Vol. 153, No. 2, March 2006
c 1) [TDS 3014 oscilloscope] .CH1 10 V 400 µs 2) [TDS 3014 oscilloscope] .CH2 10 V 400 µs 3) [TDS 3014 oscilloscope] .MATH1 20 V 400 µs
Fig. 12 Motor phase voltage and phase current waveforms at 50 000 rpm a PMSM phase voltage Va waveforms b PMSM phase voltage Vb waveforms c PMSM phase current Ic waveforms 217
the optimal V/f control profile was introduced based on the consideration of motor stator resistance. The stator resistance and inductance effects were also analytically derived. The proposed scheme provides dynamic stability and high performance of the super-high-speed PMSM. The proposed design approach was verified analytically and empirically with the super-high-speed PMSM up to 50 krpm.
7 6
%
5 4 3 2 1 0 1
Fig. 13
2
3 5 4 6 7 8 9 10 11 12 phase current harmonic RMS value (normalised)
13
Phase current Ic harmonic FFT analysis (normalised)
harmonic is the 5th, whose normalised RMS value is only 5.88% and is negligible. The experimental results validate the effectiveness of the proposed scheme. 6
Conclusions
This paper has presented the design of a stable super-highspeed PMSM under an optimal open V/f scale control method. The Rs and Ls of the motor are considered as design variables chosen to satisfy the performance requirements, with proper stability margin. It has been verified that the stability of the super-high-speed PMSM can be guaranteed over its operating speed range without damper windings on the rotor. This stability made an open-loop V/f controller possible. An optimal design of V/f scale control for the super-highspeed PMSM was also presented. A new approach to derive
218
7
References
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IEE Proc.-Electr. Power Appl., Vol. 153, No. 2, March 2006
