energies Article
Comparative Evaluation of a Permanent Magnet Machine Saliency-Based Drive with Sine-Wave and Square-Wave Voltage Injection Jyun-You Chen 1 , Shih-Chin Yang 1, * and Kai-Hsiang Tu 2 1 2
*
Department of Mechanical Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan;
[email protected] Hua-Chuang Automobile Information Technical Center, Zhongxing Road, New Taipei 231, Taiwan;
[email protected] Correspondence:
[email protected]
Received: 11 July 2018; Accepted: 18 August 2018; Published: 21 August 2018
Abstract: This paper improves a permanent magnet (PM) machine saliency-based drive performance based on the selection of a suitable injection signal. For the saliency-based position estimation, a persistently high-frequency (HF) voltage signal is injected to obtain a measurable spatial saliency feedback signal. The injection signal can be sine-wave or square-wave alternating current (AC) voltage manipulated by the inverter’s pulse width modulation (PWM). Due to the PWM dead-time effect, these HF voltage injection signals might be distorted, leading to secondary harmonics on the saliency signal. In addition, the flux saturation in machine rotors also results in other saliency harmonics. These nonlinear attributes cause position estimation errors on saliency-based drives. In this paper, two different voltage signals are analyzed to find a suited voltage which is less sensitive to these nonlinear attributes. Considering the inverter dead-time, a sine-wave voltage signal reduces its influence on the saliency signal. By contrast, the flux saturation causes the same amount of error on two injection signals. Analytical equations are developed to investigate position errors caused by the dead-time and flux saturation. An interior PM machine with the saliency ratio of 1.41 is tested for the experimental verification. Keywords: permanent magnet machine; encoderless machine drive; high-frequency voltage injection and self-sensing machine
1. Introduction Permanent magnet (PM) machines are widely used in applications on multi-axis machine tools due to their high dynamic response and high torque density [1–4]. Considering manufacturing using machine tools, the machining allowance should be limited within a 6~320-mm diameter. In order to overcome this demanded error allowance, closed-loop motion control is preferred by installing high-resolution position sensors in PM machines. High-resolution position sensors can be categorized into encoders with the output of digital pulse signals and resolvers with analog position signals. For these sensors, a rotating mechanism must be attached to the machine’s rotor in order to measure the instantaneous rotor position. It is noteworthy that the installation of the position sensor increases the overall machine size, leading to the reduction of torque density under the same machine size [5]. Considering motion systems with a size limitation, position sensing using separated position sensors might not be well-suited to maximize the drive power density [6,7]. Elimination of a separated position sensor by using the spatial information in the machine itself (self-sensing) has become a promising solution for next-generation machine drives [5,8]. Different from Energies 2018, 11, 2189; doi:10.3390/en11092189
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sensor-based drives, the rotor position has to be estimated from the position-dependent signal in the electromotive force (EMF) voltage or the saliency of a machine [9–11]. For EMF-based sensorless drives, the EMF voltage is obtained based on a machine model through current measurements. However, EMF voltage is a speed-dependent function; the signal-to-noise (SNR) ratio of EMF voltage decreases as the speed decreases [12]. For motion applications requiring low-speed operation, an EMF-based drive might not be stable enough for closed-loop control due to the low EMF SNR [13,14]. Currently, EMF-based drives are still limited for speeds beyond ~5% of the rated speed under load [15–17]. Unlike EMF voltage, the spatial signal inside the machine’s saliency is speed-independent, which is suited for sensorless drives at low speed [18]. However, this type of spatial signal is typically lower than the signal in EMF at normal speed. Under this effect, several implementation issues on the position estimation error might appear. In [19,20], a rotating sine-wave voltage injection in a stator frame and a pulsating sine-wave voltage injection in a rotor frame are evaluated in an interior PM (IPM) machine. It is reported that the pulsating injection results in lower estimation error at steady state. Besides this, an additional voltage sensor is used to directly measure the saliency from the machine’s neutral point voltage [21–23]. Due to an additional voltage sensor dedicated to saliency signal measurement, a better SNR of the spatial signal is achieved compared to measurement through the phase current sensors. However, the galvanic isolation between the voltage sensor and inverter power switches is still a challenge in inverters [22]. Considering the saliency-based drive at low speed, the low SNR of saliency in PM machines is still a problem which needs to be solved. In addition to the low SNR of saliency, secondary saliency harmonics also affect the position estimation accuracy. As reported in [24,25], the PWM dead-time causes error in the high-frequency (HF) voltage signal. To mitigate the dead-time effect, the injection voltage must be higher than a certain voltage to remove the capacitor clamping effect [26,27]. However, injection-reflected acoustic noises might be a potential problem for a voltage injection with high magnitude [28]. In [29–31], a third-order harmonic in the estimated position is observed when the square-wave injection voltage frequency is equal to the PWM switching frequency. The inverter dead-time distortion on the injected voltage is the primary issue. However, if the voltage frequency is lower than the PWM frequency, the third-order harmonic changes to a sixth-order harmonic where the magnitudes are reduced. Similar to dead-time, the flux saturation also results in saliency harmonics and position estimation errors [31]. For a rotating sine-wave voltage injection, these saliency harmonics can be decoupled with knowledge of harmonic magnitudes and phases [32]. However, dead-time harmonics in the saliency greatly increase when a rotating voltage is superimposed [33]. More importantly, these dead-time harmonics are usually higher than the harmonics caused by flux saturation [34]. Recently in [35], position estimation using the pulsating voltage with a square-wave and a sine-wave has been compared. The sine-wave injection demonstrates lower dead-time harmonics. The compensation for the saturation saliency harmonic further improves the sine-wave pulsating injection technique. This paper improves a PM machine’s saliency-based drive by determining a suited HF voltage signal. Both saliency harmonics and position estimation errors under different load conditions are evaluated with two different injection signals. In general, PM machines are widely used for motion control because of their high torque density and efficiency. However, considering sensorless PM machine drives, position estimation errors from inverter dead-time and flux saturation must be improved. Considering the influence of inverter dead-time, saliency-based position estimation contains a second-order and a sixth-order spatial harmonic. This saliency harmonic can be decreased by injecting a sine-wave voltage signal. At full load, saliency harmonics are decreased because the dead-time voltage error can easily compensate for high currents in the inverter. Besides this, considering the flux saturation, the second-order saliency harmonic due to the reduction of q-axis inductance is the primary issue. This harmonic results in the same estimation error between two injection signals. Analytical equations are developed to investigate the position error caused by the dead-time and flux saturation. An IPM machine with a 1.41 saliency ratio (Lq /Ld ) is built to evaluate the estimation performance using different voltage injection signals.
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2.2.HF HFSine-Wave Sine-WaveVoltage VoltageInjection Injection This Thissection sectionanalyzes analyzesaasaliency-based saliency-baseddrive driveusing usingsine-wave sine-wavevoltage voltageinjection. injection. The Thepulsating pulsating sine-wave injection technique originally reported in [10] is firstly explained. Key position sine-wave injection technique originally reported in [10] is firstly explained. Key positionestimation estimation errors thethe fluxflux saturation and inverter dead-time are investigated by extending the analytical errorscaused causedbyby saturation and inverter dead-time are investigated by extending the models in [10]. analytical models in [10]. In Inorder orderto toobtain obtainaahigh-bandwidth high-bandwidthspatial spatialsaliency saliencysignal, signal,aasine-wave sine-wavevoltage voltageof ofaround aroundaafew few kilohertz can be injected along the direction of the estimated d-axis in a rotor frame. kilohertz can be injected along the direction of the estimated d-axis in a rotor frame. "
# " # 0 vedv e'd vvc ccos(ω cos(ωcct) t) = 0 0 veq v e' =
0
q
(1) (1)
where wherevvcc isisthe thecarrier carriervoltage voltagemagnitude magnitudeand andωωcc isis the theinjection injectionfrequency. frequency.Considering Considering the thesmall small signal due to HF injection, the inductive voltage drop is dominant. Under this effect, the HF signal due to HF injection, the inductive voltage drop is dominant. Under this effect, the HFmachine machine model modelisissimplified simplifiedby by[10] [10] "
0
#
ved 0 veq
"
≈p
(2θerr ) −∆L sin-ΔLsin(2θ (2θerr ) err ) ied ie'd ΣL − ∆L cos(2θ ) −∆L sin(2θ ) e' ΣL ve'd − ∆Lcos ΣL-ΔLcos(2θ err ) = jω ΣL-ΔLcos(2θerr )err -ΔLsin(2θerr ) err id . 0 ≈ psin (2θerr ) ΣL + ∆L cos(2θerr ) ΣL + ∆L cos(2θerr e'−∆L )e' ieq e' = jωcc −∆L sin(2θerr ) -ΔLsin(2θerr ) ΣL+ΔLcos(2θerr ) iq vq -ΔLsin(2θerr ) ΣL+ΔLcos(2θerr ) iq 0
#"
#
"
#"
0
ied 0 ieq
# .
(2) (2)
In (2), ΣL and ∆L are the average and differential inductance, respectively. Figure 1 illustrates ΣL In (2), ΣL and ΔL are the average and differential inductance, respectively. Figure 1 illustrates and ∆L to the inductance waveform versus position. The superscript e’ in (2) represents ΣL andwith ΔL respect with respect to the inductance waveform versus position. The superscript e’ in (2) the estimated synchronous rotor frame while θ is the position error between the estimated rotor err represents the estimated synchronous rotor frame while θerr is the position error between the frame and the actual frame. The relative location between these two rotor frames is shown in Figure estimated rotor frame and the actual frame. The relative location between these two rotor frames 2. is The differential operator p in (2) is equal to jω assuming a steady state with the sine-wave injection. c shown in Figure 2. The differential operator p in (2) is equal to jωc assuming a steady state with the Substituting (1) into (2), the voltage current is given by current is given by sine-wave injection. Substituting (1)injection into (2), induced the voltage injection induced "
0
ied 0 ieq
#
(
)
(
e'
q
c
(
err
)
Phase inductance
#−1−1"
cos ω t 1 v1ccosω ΣL + ∆Lcos vvcccosω ΣL+ΔLcos ( 2θerr )(2θ vc cos c tc . err ) . c tωc t = = jωc2ΣL2 − 2 err ) 2 ∆L 0 ∆Lsin ( 2θ 0 jωc ΣL -ΔL ΔLsin ( 2θerr ) err )
(2θerr ) 2θerr−∆Lsin (2θerr2θ ) err −1∆Lcos ΣL-ΔLcos -ΔLsin 1 id ΣL jωc e' = (2θerr ) − ∆Lsin ΣL + ∆Lcos ( 2θ ) err i ΣL+ΔLcos 2θ jω -ΔLsin 2θ "
=
(
)
Lq
#
"
#
(3) (3)
With saturation Without saturation
ΔL ΣL
Ld 0
90
Rotor position
180
270
360
Figure 1. Comparison of phase inductance versus rotor position with and without the flux saturation. Figure 1. Comparison of phase inductance versus rotor position with and without the flux saturation.
Assuming the estimated frame in Figure 2 is not aligned with the actual frame at the initial Assuming the estimated frame e’ in Figure 2 is not aligned with the actual frame at the initial 0 estimation state, a saliency current i thethe magnitude is dependent on ΔL. The q estimation state, a saliency current ieq in in(3) (3)appears appearswhere where magnitude is dependent on ∆L. e’ 0 e’0 is modulated by to to bebe zero rotor position cancan be be obtained by by regulating θerrθerr The rotor position obtained regulating zeroinini qieq. .However, However, iiqeq is modulated by cos(ω injection. In In [19,20], a signal process by multiplying the same per unit cos(ωcct) t) due dueto tothe theHF HFsine-wave sine-wave injection. [19,20], a signal process by multiplying the same per sine voltage and adding a low-pass filter (LPF) is proposed to obtain the spatial saliency signal, i unit sine voltage and adding a low-pass filter (LPF) is proposed to obtain the spatial saliency sal_sine signal,. Figure illustrates the signal of saliency-based positionposition estimation using a sine-wave voltage isal_sine. 3Figure 3 illustrates theprocess signal process of saliency-based estimation using a sine-wave voltage injection. In addition to the LPF, a high-pass filter (HPF) is applied to isolate the injectioninduced current from the fundamental current. The corresponding analytical equation of isal_sine is given by
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injection. In addition to the LPF, a high-pass filter (HPF) is applied to isolate the injection-induced ΔL The ΔL vvc of i 2 ΔL ΔL vvc c corresponding current current. -ie' e'qfundamental × sinωc t = LPF isal_sinefrom =LPFthe sin ( 2θerr ) sin2analytical ωc t = equation sinsal_sine c ( 2θerr is) given by 2 2sin ( 2θ ) sin ω t = 2 2sin ( 2θ ) q × sinωc t = LPF ω ΣL isal_sine =LPF -i err c err 2ω ΣL 2 -ΔL 2 2 -ΔL 2 ω 2ω h i c c ΣL -ΔL h ic cΣL -ΔL 0 vc vc e ∆L ∆L 2
isal_ sin e = LPF −iq × sin ωc t = LPF
ωc ΣL2 −∆L2
sin (2θerr ) sin ωΔL = ct v
sin (2θerr )
c ΣL − ∆L ΔL vc c 2ω ≈ sinv2θ (c2θsinerr) ()2θerr ). ∆L ≈ 2ω ΣL ≈2 2sin 2ωc (ΣL2 err c 2ωc ΣL . . 2
2
(4) (4) (4)
qq
e’e’ q q
Estimated Estimated frame frame
θθerrerr
e’
e’ dd
Actual Actual frame frame
dd
Figure 2. Relative location between the estimated rotor frame and the actual rotor frame. Figure frame and and the the actual actual rotor rotor frame. frame. Figure 2. 2. Relative Relative location location between between the the estimated estimated rotor rotor frame
^
IdqIdq
^
-jθ e ee-jθe
e'
Iαβ_fund I e'Idq_fund Iαβ_fund dq_fund + + + e'+ e'Idq_hf I αβ_hf Idq_hf Iαβ_hf
ωω HPF HPF
e'
e' e'iq_hf dd iq_hf qq
Idq_hf Idq_hf e'
-sin(ωt)ct) -sin(ω c
iqiq
LPF LPF
^
^
isal_sine Position θθe e isal_sine Position estimation ωω estimation
Figure3.3.Illustration Illustration of of saliency-based saliency-based position estimation by injecting aasine-wave voltage in the dFigure estimationby byinjecting injecting sine-wave voltage Figure 3. Illustration of saliency-based position estimation a sine-wave voltage in in thethe daxis. HPF, high-pass filter; LPF, low-pass filter. d-axis. HPF, high-pass filter; LPF,low-pass low-passfilter. filter. axis. HPF, high-pass filter; LPF,
(4),only only a single single saliency harmonic with respect to 2θ e is considered [10]. It is assumed that InIn(4), (4), saliencyharmonic harmonic with respect is considered [10]. is assumed e considered only aasingle saliency with respect to to 2θ2θ e is [10]. It is Itassumed that the phase inductance waveform in Figure 1 is purely sinusoidal where the frequency is 2 times with that the phase inductance waveform in Figure 1 is purely sinusoidal where the frequency is 2 times the phase inductance waveform in Figure 1 is purely sinusoidal where the frequency is 2 times with respect to the the rotor position. However, for for actual machines, flux saturation must occur due to the with respect rotor position. However, actual machines, saturation occur to respect to thetorotor position. However, for actual machines, flux flux saturation mustmust occur due due to the nonlinear relationship between flux density and magnetic field strength. Considering the flux the nonlinear relationship between flux density andmagnetic magneticfield fieldstrength. strength. Considering Considering the the flux nonlinear relationship between flux density and saturation,the thephase phaseinductance inductancewaveform waveformmight mightbe bechanged changedby bythe thered redsignal signalininFigure Figure1.1.Under Under saturation, waveform might thiseffect, effect,secondary secondary saliency harmonicsinstead insteadofof a 2θeharmonic harmonicoccur, occur,leading leading to saliency-based this saliency harmonics instead of a 2θee leading to saliency-based position estimation error. In this paper, the ideal saliency signal in (4) is to extended consider position In In thisthis paper, the ideal saliency signal in (4) isin extended considertoto saturation position estimation estimationerror. error. paper, the ideal saliency signal (4) is extended consider saturation saliency harmonics as depicted in (5). saliency harmonics as depictedasindepicted (5). saturation saliency harmonics in (5).
1
v
ΔLsin 2θ isal_sine_sat=1= 1 vc v c c2 ΔLsin ΔLsin −2ω 2ωte t )+ΔL +ΔL 6sin 2θ 4ωte t )++...... )++ΔL ( 2θ−err−2ω ( 2θerrerr−−−4ω 4 sin ( ()2θerr+err)∆L ( 2θ isal_ sinie_sat = (2θerr sin((2θerr 4ω [∆Lsin err err e te) )+ ∆L6 sin sal_sine_sat 4 (2θ e e)t) + . . . ] 4 sin 2ωcΣL 2 2 ΣL 2ω2ω c ΣL c
(5) (5) (5)
wherei isal_sine_sat is the saliency signal considering saturation saturation harmonics, harmonics, and and ∆L ΔL4 and and ∆L ΔL6 are arethe the where thesaliency saliencysignal signalconsidering considering isal_sine_sat sal_sine_sat isisthe saturation harmonics, and ΔL 44 and ΔL 66 magnitudes offourth-order fourth-orderand andsixth-order sixth-ordersaturation saturationharmonics. harmonics. These secondary harmonics are magnitudes of harmonics. These secondary harmonics are dependent on the condition of flux saturation. In (5), high-order secondary harmonics (larger than dependent on the condition of flux saturation. saturation. In (5), high-order secondary harmonics (larger than sixth-order)are arenegligible. negligible. Due to the influence of saturation harmonics, periodic position estimation sixth-order) Due to the influence of saturation harmonics, periodic position estimation errors might occur onthe theposition positionestimation estimationusing usingthe thesine-wave sine-wave voltageinjection. injection. errors might occur on sine-wave voltage voltage injection. In addition to the flux saturation, inverter dead-time also causes secondary harmonics in the In addition to the flux saturation, saturation, inverter dead-time also causes secondary harmonics in the saliencysignal signali isal_sine.. Figure illustrates the the dead-time dead-time effect effect on onthe theinverter’s inverter’sA-phase A-phaseleg. leg.In In(a), (a), saliency Figure444illustrates illustrates sal_sine. Figure the dead-time effect on the inverter’s A-phase leg. In (a), sal_sine when both gate signals V Gt and VGb suddenly turn off, the phase current flows from the low-side when both gate signals suddenly turn turn off, off, the the phase phase current current flows from the low-side signals V VGt Gt and VGb Gb suddenly diodeDDbtotothe thewindings windingswhen whenthe thecurrent currentpolarity polarityisispositive. positive.At Atthis thisinstant, instant,aanegative negativevoltage voltage diode b positive. b errorresults results onthe theA-phase A-phase PWMvoltage voltageVVa_pwm.. By contrast, in (b), once the current is negative, a error contrast, (b), once the current negative, on PWM voltage V a_pwm . ByBycontrast, inin (b), once the current is is negative, a a_pwm positive voltage error occurs during the dead-time period. Considering the influence of dead-time, positive voltage error occurs during the dead-time period. Considering the influence of dead-time, the injection voltage in (1) needs to be corrected by the injection voltage in (1) needs to be corrected by
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the dead-time period. Considering the influence of dead-time, 5 of 15 the injection voltage in (1) needs to be corrected by "
# " # 0 e' t)+v err_dt_d vedv d v cvcos(ω cos(cω t ) + v c c err_dt_d = 0 e' = v err_dt_q verr_dt_q veq v q
(6)(6)
where err_dt_d and Verr_dt_q are the andd-q-axis dead-time voltagevoltage error, respectively, which iswhich shownis whereVV Verr_dt_q ared-the and q-axis dead-time error, respectively, err_dt_d and by (7). shown by (7).
+
1 Vdc 2 − + 1 Vdc 2 −
VGt
+ −
o
Va_pwm VGb
+
1 Vdc 2 −
vcph
ia + −
M
VGa
−
o
+ 1 Vdc 2 −
vcpl
+
(a)
Va_pwm VGN
vcph
ia + −
M vcpl
(b)
Figure 4. Illustration of A-phase current in the inverter leg during dead-time when the current Figure 4. A-phase current in the inverter leg during dead-time when the current polarity polarity is Illustration (a) positiveof and (b) negative. is (a) positive and (b) negative.
v err_dt_d
T +T +T
T
on off " # = - dt v dc × sign(i) ≈ - dt v dc × sign(i) (7) v err_dt_q Tpwm Tdt TdtT+ T + T verr_dt_d off pwmon =− vdc × sign(i) ≈ − v × sign(i) (7) Tpwm Tpwm dc verr_dt_q In (7), Tpwm and Tdt represent the period of PWM and dead-time, respectively. In addition, Ton represent period PWM dead-time, In addition, pwm and Tdtthe and TIn off (7), are Trespectively switch the turn-on andofoff timeand period. In thisrespectively. paper, Ton and Toff are T and T are respectively the switch turn-on and off time period. In this paper, T and on on assumed tooffbe sufficiently smaller than Tdt for simplicity. In addition, sign(i) is the polarity ofTthree off are assumed to be sufficiently smaller than T for simplicity. In addition, sign(i) is the polarity of three phase currents during internal switching, as dt given by phase currents during internal switching, as given by 0 0 cos cosθ # e " cos ( θe − 120 ) # " ( θe − 240 ) # sign(i 0 0) c ) . sign(i) sign(ia )+ cos (θe − 120 ) sign(ib )+ (8) cos=θe cos (θe − 240 0 0 -sinθsign -sin ( θe − 120 0) sign(ib )+ 0 sign(ic ). (8) sign(i) = -sin θ − 240 e (ia ) + ( ) e − sin (θ − 120 ) − sin θ − sin (θ −240 )
"
e
e
e
Figure 5 illustrates the actual A-phase high-side and low-side PWM signal considering the deadFigure 5 illustrates the actual A-phase high-side and low-side PWM signal considering the 0 e’ 0 time effect. Substituting (6) into(6) (2),into the(2), injection-reflected HF current, ie’ dead-time effect. Substituting the injection-reflected HF current, ied iand in is (3), is d and q inieq(3), −1
"
=
0
ied 0 ieq
vc cos(ωc t)+verr_dt_d# #i e'd "1 ΣL-ΔLcos ( 2θerr ) -ΔLsin ( 2θ err#)−1 " v cos (ωc t) + verr_dt_d e'= p1= ΣL − ∆Lcos(2θerr ) −∆Lsin(2θerr ) c verr_dt_q -ΔLsin ΣL+ΔLcos 2θ verr_dt_q (2θerr( 2θ ) err ) ΣL + ∆Lcos(2θerr ) err ) ( i q p−∆Lsin # " ( "
1 12 1 2 jω = c ΣL − ∆L
#" # ) verr_dt_d ΣL + ∆Lcos (2θerr ) ΔLsin ∆Lsin (2θerr ) v ∆Lcos(2θ(err ΣL+ΔLcos 2θ)err ) + R ΣL+ΔLcos ( 2θ ( 2θ 1 vc cos (ωc t) ΣL+ . dt . err ) err ) err_dt_d × × vc cos(ωc t) ∆Lsin(2θerr ) + ) verr_dt_q dt ∆Lsin(2θerr ) ΣL − ∆Lcos(2θerr 2 2
jωc ΣL -ΔL
ΔLsin ( 2θ err )
ΔLsin ( 2θerr )
ΣL-ΔLcos ( 2θ err ) verr_dt_q
(9) (9)
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Tpwm
VGt
(a) Tpwm /2
VGb
Tpwm /2 Td
VGt Td
(b)
VGb Ts Figure 5. Voltage gate signals VGt and VGb in an inverter (a) with and (b) without the dead-time. Figure 5. Voltage gate signals VGt and VGb in an inverter (a) with and (b) without the dead-time.
By using the same signal process in (4) with a dead-time effect, the saliency signal is changed by By the same signal process in (4) with a dead-time effect, the saliency signal is changed by isal_sine_dt using in (10). isal_sine_dt in (10). e' isal_sine_dt =LPF -iq × sinωc t i h sin ( ω t ) ΔLe0 vc ωc t isal_ sin e_dt ==LPF LPF −iq × 2sin sin ( 2θerr ) sin 2ωc t + h2 c 2 ΔLverr_dt_dsin ( 2θerr ) + ΣLverr_dt_q -ΔLverr_dt_q cos ( 2θerr )dt i i 2 h ω ΣL -ΔL ΣL -ΔL R (10) c sin (ω t) vc ∆L = LPF ω sin (2θerr ) sin2 ωc t + 2 c 2 ∆Lverr_dt_d sin (2θerr ) + ΣLverr_dt_q −∆Lverr_dt_q cos (2θerr ) dt (10) c ΣL2 − ∆L2 ΣL − ∆L verr_dt_q sin ΔL vc hv i ( ωc t ) sin ( ω t ) sin 2θ LPF ΔT ≈ + c ( ) err_dt_q ∆L vc 2 LPF err ≈ 2ωc ΣL2 sin (2ω 2θerr )+ ∆T ΣL ΣL c ΣL
As shown shownin in(10), (10),dead-time dead-timeleads leadstotoadditional additionalharmonics harmonicsmodulated modulatedbyby sin(ω isal_sine_dt. As sin(ω inin isal_sine_dt c t)ct) These harmonics are proportional to the dead-time voltage error V err_dt . However, it is noted that These harmonics proportional to the dead-time voltage error Verr_dt . However, it is notedthese that harmonics can be filtered based on the for the signal process in Figure 3. Thus, thethe selection of these harmonics can be filtered based on LPF the LPF for the signal process in Figure 3. Thus, selection LPF bandwidth is is the of LPF bandwidth thetrade-off trade-offbetween betweenthe theposition positionestimation estimationbandwidth bandwidthand and the the estimation estimation error. In (10), (10), only only the the dead-time dead-time error errorisisconsidered. considered. For For actual actual inverter inverter circuits, circuits, device device parasitic parasitic In capacities also also cause cause aa HF HF voltage voltage error error for for the the instant instant when when the phase current flows across zero. capacities According to the analysis in [5,6], this error causes a second-order saliencyharmonic harmonic with with respect respect to to According to the analysis in [5,6], this error causes a second-order saliency the rotor rotorposition. position. the 3. 3. HF HF Square-Wave Square-WaveVoltage VoltageInjection Injection The The flux flux saturation saturation and and inverter inverter dead-time dead-time on on aasquare-wave square-wave voltage voltage saliency-based saliency-based drive drive are are investigated thisthis section. As reported in [29–31], a square-wave voltage in (11) can beinsuperimposed investigatedinin section. As reported in [29–31], a square-wave voltage (11) can be in the estimatedind-axis. superimposed the estimated d-axis. " 0 # " # ved ±vsqu (11) 0 e' = 0 veq v d ± vsqu = (11) e' v q By0injecting where vsqu is the magnitude of square-wave voltage. a square-wave voltage, the HF model is given by where vsqu is the magnitude of square-wave voltage. By injecting a square-wave voltage, the HF
model is given "by e0 # vd 0 veq
# " 0 # ied ΣL − ∆L cos(2θerr ) −∆L sin(2θerr ) d = 0 e' e' dt − sin vd∆L ) err ΣL-ΔL d) i d (2θerrcos(2θ ) + ∆L coserr(2θ ie err ) ΣL -ΔLsin(2θ " e' = e' #" q # -ΔLsin(2θ err ) ΣL+ΔL cos(2θ err ) dt i q ∆L e0 vq ΣL − ∆i cos ( 2θ ) − ∆L sin ( 2θ ) err err 1 d ≈ ∆T . 0 −1∆LΣL-ΔL sin(2θcos(2θ ΣL-ΔLsin(2θ + ∆L cos() 2θerr )ie' ∆ieq err ) ) Δ err err d ≈ e' . "
ΔT -ΔLsin(2θerr )
(12)
(12)
ΣL+ΔLcos(2θerr ) Δiq
e'
e'
In (12), ΔT is the time period of the square-wave voltage. Besides this, Δi d and Δi q represent the d-axis and q-axis current difference, respectively, based on the current state T1 and the last state T0. They are expressed by
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0
In (12), ∆T is the time period of the square-wave voltage. Besides this, ∆ied and ∆ieq represent the d-axis and q-axis current difference, respectively, basede' on the current state T1 and the last state T0 . e' e' Δi = i d d_T1 − i d_T0 They are expressed by . (13) 0 e0 e' e0e' e' ∆i − iied_T0 Δid0 q==id_T1 i q_T1 − q_T0 . (13) 0 0 ∆ieq = ieq_T1 − ieq_T0 e' e' Substituting (11) into (12), Δei0d and eΔ0 i q are given by Substituting (11) into (12), ∆id and ∆iq are given by −1
-ΔLsin ( 2θ err ) " ΣL-ΔLcos ( 2θ err ) e' = ΔTΣL − ∆Lcos(2θerr ) −∆Lsin(2θ ( 2θerr ) ΣL+ΔLcos ( 2θerrerr )) Δi= q ∆T −-ΔLsin ∆Lsin(2θerr ) ΣL + ∆Lcos(2θerr )
# Δi d e'
"
0 ∆ied 0 ∆ieq
# # ±−v1 "squ 0 ± vsqu 0
(14) (14)
± v ΣL 2θerrerr)) ΣL+ΔLcos + ∆Lcos((2θ = ΔT±2vsqu2 2squ 2 = ∆T .. ΣL ΣL −∆L-ΔL ∆Lsin ( 2θ ) ΔLsin 2θ ( ) err err "
#
e'0
By injecting a square-wave square-wave voltage, voltage, aa position position signal signaldependent dependenton onΔL ∆Lappears appearsinin Δ ∆ii qeq once the estimated not the same as as thethe actual frame as shown in Figure 2. Different to sine-wave estimatedframe frameininFigure Figure2 2isis not the same actual frame as shown in Figure 2. Different to sine0 e' injection in (4), ∆ieq is modulated by a ± sign due to the square-wave property. A signal process by wave injection in (4), Δi q is modulated by a ± sign due to the square-wave property. A signal process multiplying the ∓ sign was proposed to obtain the saliency signal, isal_squ [30], as seen in (15). by multiplying the sign was proposed to obtain the saliency signal, isal_squ [30], as seen in (15). vsqu vsqu 0 ∆Lsin(2θerr ) ≈ ∆T∆L isal_squ = (∓) × ∆ieqe' = ∆T vsqu (15) vsqu 2 sin (2θerr ) 2 2 −∆L (15) isal_squ = ( ) × Δiq = ΔT ΣL ΔLsin ( 2θerr ) ≈ ΔTΔL 2ΣL sin ( 2θerr ) 2 2
ΣL -ΔL
ΣL
Similar to isal_sine in (4), the rotor position is estimated by regulating θerr in isal_squ to be zero. Similar to isal_sine in (4), the rotor position is estimated by regulating θerr in isal_squ to be zero. Figure Figure 6 illustrates the signal processing for the position estimation with square-wave injection. 6 illustrates the signal processing for the position estimation with square-wave injection. Compared Compared to Figure 3 using sine-wave injection, ± sign correction in (15) instead of an LPF in (4) to Figure 3 using sine-wave injection, ± sign correction in (15) instead of an LPF in (4) is applied. By is applied. By removing the LPF in Figure 6, a higher estimation bandwidth can be of benefit to removing the LPF in Figure 6, a higher estimation bandwidth can be of benefit to a sensorless drive. a sensorless drive. Idq
e' d iq_hf
^
e -jθe Iαβ_fund + Iαβ_hf
ω
Idq_fund + e' Idq_hf
e'
q
Idq_hf
HPF
Current difference in (13)
isal_squ
^
Position θe estimation
-sign(±vsqu)
Figure 6. position estimation by by injecting a square-wave voltage in the Figure 6. Illustration Illustrationofofsaliency-based saliency-based position estimation injecting a square-wave voltage in d-axis. the d-axis.
It is noted that secondary saliency harmonics caused by the flux saturation also occur on the It is noted that secondary saliency harmonics caused by the flux saturation also occur on the saliency signal with square-wave injection. Considering saturation-reflected saliency harmonics, the saliency signal with square-wave injection. Considering saturation-reflected saliency harmonics, saliency signal, isal_squ_scnd, should be modified from (15). the saliency signal, isal_squ_scnd , should be modified from (15). vsqu (16) isal_squ_sat = ΔTΔL vsqu 2 ΔLsin ( 2θ err ) + ΔL 4sin ( 2θ err − 2ωe t ) +ΔL6sin ( 2θ err − 4ωe t ) + ... ∆Lsin(2θerr ) +∆L4 sin (2θerr − 2ωe t)+∆L6 sin (2θerr − 4ωe t) + . . .] (16) isal_squ_sat = ∆T∆L 2 [ΣL ΣL By comparing isal_squ_scnd in (16) and isal_sine_sat in (5), the flux saturation effect is the same on both By comparing isal_squ_scnd in (16) and isal_sine_sat insquare-wave (5), the flux saturation isal_squ_scnd and isal_sine_sat using either the sine-wave or the voltage. effect is the same on both The isal_squ_scnd and either the sine-wave or is the square-wave influence ofisal_sine_sat dead-timeusing on the square-wave voltage also analyzed involtage. this paper. Similar to The influence of dead-time on the square-wave voltage is also analyzed in this paper. the sine-wave voltage in (6), the HF square-wave voltage in (11) should be modified by Similar to the sine-wave voltage in (6), the HF square-wave voltage in (11) should be modified by "
v#e'd " ± v squ + v err_dt_d # 0 . ved v e' = ±vsqu + verr_dt_d v err_dt_q = . e0 q v
(17) (17)
vq err_dt_q In (17), the dead-time causes the same voltage error, verr_td, mentioned in (7). By replacing (17) e'
e'
into (12), the current differences, Δi d and Δi q , are shown by
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In (17), the dead-time the same + verr_dt_d in (7). By replacing (17) Δide' causes -ΔLsinerror, ΣL-ΔLcos ±v,squmentioned ( 2θerr )voltage ( 2θerr ) verr_td e' = ΔT ∆ie0 and ∆ie0 , are shown by into (12), the current differences, verr_dt_q d ( 2θerr ) q ΣL+ΔLcos ( 2θerr ) Δiq -ΔLsin (18) " # " # −1 " # v e0 ± v ΣL+ΔLcos 2θ ΣL+ΔLcos 2θ ΔLsin 2θ . ( ) ( ) ( ) err_dt_d err err ∆id ±ΔvTsqu + verr_dt_d ΣL − ∆Lcos(2θerr ) squ −∆Lsin(2θerr ) err = ΔT 2 + 2 ΔLsin(2θ ΣL-ΔLcos ( 2θerr ) verr_dt_q ( 2θerrerr)) ΣL2 -ΔLv2err_dt_q −∆Lsin(2θerr )ΣL -ΔL ΣL + ∆Lcos ΔLsin ( 2θerr ) (18) " " # #" # verr_dt_d ΣL + ∆Lcos(2θerr ) ΣL + ∆Lcos(2θerr ) ∆Lsin(2θerr ) ±vsqu ∆T = ∆T + . For the position estimation Figure 6 ΣL the saliency signal based demodulation 2 2 2 − ∆L ΣL2 −∆L verr_dt_q ∆Lsin(in 2θerr ) ∆Lsin (2θerr )is obtained ΣL − ∆Lcos (2θerr )on the 0
∆ieq
= ∆T
e'
of Δi q , which can be derived by For the position estimation in Figure 6 the saliency signal is obtained based on the demodulation 0 vsqu of ∆ieq , which by ( 2θ ) + ( ΔT ) ΔLv i can=be ΔT derivedΔLsin sin ( 2θ ) + ΣLv - ΔLv cos ( 2θ ) sal_squ_dt
isal_squ_dt
= ≈
ΣL2 -ΔL2
err
ΣL2 -ΔL2 h
err_dt_d
err
err_dt_q
v ∆T 2 squ 2 ∆Lsin(2θerr ) + (∓ 2∆T 2 ) ∆Lverr_dt_d sin (2θerr ) + ΣLverr_dt_q − ΣL −∆L ΣL −∆L squ err_dt_q v v err (θerr ≈ 0) err ∆T∆L squ2 ∆Lsin(2θ2err ) ∓ ∆T err_dt_q ΣL ΣL
≈ ΔTΔL
v
ΣL
ΔLsin ( 2θ
) ΔT
v
ΣL
( θ ≈ 0)
err_dt_q
err
∆Lverr_dt_q cos (2θerr )
i
(19) (19)
where iisal_squ_dt thesaliency saliency current current considering considering the the dead-time. dead-time. By By comparing comparing the square-wave sal_squ_dt isisthe saliency signal signal iisal_square_dt in (19) sine-wave saliency signal isal_sine_dt (10), isal_square_dt in (19) andand the the sine-wave saliency signal isal_sine_dt in (10),inisal_square_dt contains sal_square_dt contains dead-time harmonics where the magnitudes are directly proportional to .vBy . By contrast, dead-time harmonics where the magnitudes are directly proportional to verr_dt_q contrast, isal_sine_dt err_dt_q imight might achieve negligible dead-time error because of the LPF used in Figure 3. As a result, achieve negligible dead-time error because of the LPF used in Figure 3. As a result, the position sal_sine_dt the positionusing estimation using the sine-wave voltage injection reduces the dead-time error, can estimation the sine-wave voltage injection reduces the dead-time error, which can bewhich of benefit be benefit to a saliency-based sensorless drive. to aofsaliency-based sensorless drive. 4. Experimental Results Results 4. Experimental An is built built to to verify verify the the saliency-based An IPM IPM machine machine with with aa 1.41 1.41saliency saliencyratio ratio(L (Lqq/L /Ldd)) is saliency-based position position estimation analyzed in Sections 2 and 3. Table 1 lists the specifications of this test machine. A two-level estimation analyzed in Sections 2 and 3. Table 1 lists the specifications of this test machine. A twoinverter with 150-V DC busDC voltage is used is to used generate PWM voltages. The PWM frequency is designed level inverter with 150-V bus voltage to generate PWM voltages. The PWM frequency is at 10-kHz to balance the dynamic response and switching losses. Figure 7 illustrates the test setup of designed at 10-kHz to balance the dynamic response and switching losses. Figure 7 illustrates the test the saliency-based drive for drive the test A hysteresis brake is back-to-back connected connected to the test setup of the saliency-based formachine. the test machine. A hysteresis brake is back-to-back machine formachine operationfor under load. All the control and position-sensing algorithms are implemented in to the test operation under load. All the control and position-sensing algorithms are aimplemented fixed-point 32-bit microcontroller TI-TMS320F28069. For the following experiments, the magnitudes in a fixed-point 32-bit microcontroller TI-TMS320F28069. For the following and frequencies the two injection voltages are thetwo same at 30 Vvoltages and 1 kHz the experiments, theof magnitudes and frequencies of the injection areto theeasily sameevaluate at 30 V and performance difference. 1 kHz to easily evaluate the performance difference.
IPM Motor
Dyno
Inverter Figure 7. Test bench of the interior permanent magnet (IPM) machine with the saliency-based Figure 7. drive. Test bench of the interior permanent magnet (IPM) machine with the saliency-based sensorless sensorless drive.
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Table 1. Test Machine Characteristics. Table 1. Test Machine Characteristics.
Characteristics Characteristics RotorRotor polespoles Rated torque Rated torque Rated current Rated current Rated speed RatedResistance speed Resistance Inductance Inductance DC bus voltage DC bus voltage
A.
Values Values 6 poles 6 poles 0.58 Nm 0.58 Nm 2A 2A 6000 rpm 6000 1.15 Ω rpm 1.15 Ω q) 4.6 mH(Ld )/6.5 mH(L 4.6 mH(L 310 V d)/6.5 mH(Lq) 310 V
Dead-Time Harmonics: A. Dead-Time Harmonics:
This part evaluates the the dead-time effect ononboth signal,isal_sine_dt isal_sine_dt , inand (9) and This part evaluates dead-time effect boththe thesine-wave sine-wave saliency saliency signal, , in (9) the square-wave signal, i sal_squ_dt , in (18). The dead-time is set at 2 μs during a 100-μs PWM period. the square-wave signal, isal_squ_dt , in (18). The dead-time is set at 2 µs during a 100-µs PWM period. The rotor position is locked 0 degreesby byapplying applying an Because the the rotorrotor is locked, The rotor position is locked at at 0 degrees anexternal externalload. load. Because is locked, i sal_sine_dt and isal_squ_dt should both be maintained at zero considering an ideal inverter without deadisal_sine_dt and isal_squ_dt should both be maintained at zero considering an ideal inverter without ' e' 0 0 time. Figure 8a8a illustrates waveforms of v edofinve(6) ct) and isal_sine_dt in (10). In this figure, q sin(ω dead-time. Figure illustrates waveforms inand (6) iand ie sin(ω c t) and isal_sine_dt in (10). In this '
0
q
d
0
'
e e figure,ieq isin(ωct) t) instead is shown to clearly demonstrate the dead-time distortion on the HF of iqofisieqshown to clearly demonstrate the dead-time distortion on the HF current. q sin(ωcinstead
0
0
e ' ' current. By contrast, 8b compares ved and in (16) in (18). the Considering q andinisal_squ_dt By contrast, FigureFigure 8b compares and∆i isal_squ_dt (18). Considering sine-wave the Δi eq and v ed in (16) sine-wave injection signal process in Figure 3, the bandwidths of the LPF and the HPF are respectively injection signal process in Figure 3, the bandwidths of the LPF and the HPF are respectively designed designed at 100 Hz and 20 Hz for the experimental tests. In addition, for the square-wave signal at 100 Hz and 20 Hz for the experimental tests. In addition, for the square-wave signal process, the process, HPF in Figure 6 isatthe same 20 Hz. Itthat is observed that isal_sine_dt inharmonic (a) achieves lower HPFthe in Figure 6 is the same 20 Hz. It isatobserved isal_sine_dt in (a) achieves lower errors harmonic than (b)indue to the LPF in Figure This result model verifies the analytical sal_squ_dt than ierrors sal_squ_dt in (b) idue to the in LPF Figure 3. This result verifies3.the analytical explained in modelSection explained Section that 2 It the is concluded that the dead-time effect is reduced forusing the position 2 It is in concluded dead-time effect is reduced for the position estimation the sine-wave voltage injection. voltage injection. estimation using the sine-wave
vde' 0 iqe ' sin (ωc t )
1-kHz
30-V 67-mA
0
isal _ sine
33-mA
0
t 0
4.5-ms
9-ms
(a) vde '
30-V
0 1-kHz
Δiqe '
133-mA
0
isal _ squ 0
266-mA
t 0
4.5-ms
9-ms
(b) Figure 8. Comparison of saliency signals by injecting (a) the sine-wave voltage signal and (b) the Figure 8. Comparison of saliency signals by injecting voltage signal and (b) square-wave square-wave voltage signal (V C = Vsqu = 30 V, fC = (a) fsqu the = 1 sine-wave kHz and the machine is locked atthe standstill).
voltage signal (VC = Vsqu= 30 V, fC = fsqu = 1 kHz and the machine is locked at standstill).
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B. Saliency-BasedPosition PositionEstimation: Estimation: B. Saliency-Based This This part part compares compares the the saliency-based saliency-based position position estimation estimation using using two two different different voltage voltage injection injection signals. Both position estimation at no load and full load are evaluated to verify secondary signals. Both position estimation at no load and full load are evaluated to verify secondary saliency saliency harmonics operating conditions. FigureFigure 9 shows saliency-based position estimation harmonics under underdifferent different operating conditions. 9 the shows the saliency-based position at no load using sine-wave andsine-wave (b) square-wave injection. The injection frequency is set estimation at no(a) load using (a) and (b)voltage square-wave voltage injection. The injection at 1 kHz to clearly evaluate the position error on the two injection signals. The speed is maintained frequency is set at 1 kHz to clearly evaluate the position error on the two injection signals. The speed at 100 rpm (ωeat=100 2π rpm × 5 rad/s). in (a) voltage, the position is around is maintained (ωe = 2πAs × 5seen rad/s). As using seen insine-wave (a) using sine-wave voltage, theerror position error 6isdegrees consisting of a secondand a sixth-order harmonic. By contrast, in (b) using square-wave around 6 degrees consisting of a second- and a sixth-order harmonic. By contrast, in (b) using voltage injection, the error increases to 8.5increases degrees where second-order harmonic dominates. It is square-wave voltage injection, the error to 8.5 the degrees where the second-order harmonic shown that It the usingharmonic the square-wave is higher than that usingthan the dominates. is second-order shown that theharmonic second-order using theinjection square-wave injection is higher sine-wave injection. The dead-time distortion on HF voltages is the primary issue especially when that using the sine-wave injection. The dead-time distortion on HF voltages is the primary issue phase currents zero. especially whencross phase currents cross zero.
ia
2-A
θe
^ θe
3600 60
^ θe - θe 3.20
2fe
0.90
6fe
Fast Fourier Transform (FFT) of θe - θ^e
50-ms/div
(a) ia
2-A
θe
^ θe
3600 60
5.10
2fe
^ θe - θe ^ FFT of θe - θe
50-ms/div
(b) Figure 9. Saliency-based position estimation at no load using (a) sine-wave and (b) square-wave Figure Saliency-based estimation at no load using (a) sine-wave and (b) square-wave voltage squ = 30 V, fC = fsqu = 1 kHz and 100 rpm speed control where ωe = 2π × 5 voltage9.injection (VC = Vposition injection (V = V = 30 V, f = f = 1 kHz and 100 rpm speed control where ωe = 2π × 5 rad/s). squ squ C C rad/s).
Figure 10 shows the same position estimation comparison at full load. Due to additional fundamental currents for load operation, the dead-time-reflected voltage distortion can be easily compensated for. Under this effect, the position estimation error decreases as the fundamental
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Figure 10 shows the same position estimation comparison at full load. Due to additional fundamental currents for load operation, the dead-time-reflected voltage distortion can be 11 easily Energies 2018, 11, x FOR PEER REVIEW of 15 compensated for. Under this effect, the position estimation error decreases as the fundamental current increases. For the sine-wave injection in (a), thein position is approximately the same asthe thesame error as at current increases. For the sine-wave injection (a), theerror position error is approximately no load because the dead-time harmonic has been filtered under the demodulation process with an LPF. the error at no load because the dead-time harmonic has been filtered under the demodulation However, for the in Figure 10b, the second-order harmonic as the fundamental process with an square-wave LPF. However, for the square-wave in Figure 10b, decreases the second-order harmonic current increases. This result is consistent with the analysis in Section 4 part B. The dead-time harmonic decreases as the fundamental current increases. This result is consistent with the analysis in Section cannot be The filtered for the square-wave injection if the current demodulation Figure 6 is 4 part B. dead-time harmonic cannot be filtered for thedifference square-wave injection ifinthe current applied. Considering the saliency-based drive using square-wave injection, this visible second-order difference demodulation in Figure 6 is applied. Considering the saliency-based drive using squareharmonic at no load can be compensated if the harmonic magnitude and phase are based on wave injection, this visible second-order for harmonic at no load can be compensated forknown if the harmonic off-line measurement. In addition, high-voltage injection is also In useful to reduce dead-time magnitude and phase are known based onmagnitude off-line measurement. addition, high-voltage harmonics for square-wave injection. By contrast, at full load, the second-order harmonic magnitudes magnitude injection is also useful to reduce dead-time harmonics for square-wave injection. By are almostatthe the two different injection signals.are This harmonic component is primary contrast, fullsame load,between the second-order harmonic magnitudes almost the same between the two caused byinjection the machine’s secondary saliency harmoniciswhich cannot be compensated for bysecondary selecting different signals. This harmonic component primary caused by the machine’s different injection signals. saliency harmonic which cannot be compensated for by selecting different injection signals. ia
2-A
θe
^ θe
3600 60
^ θe - θe 2.90
50-ms/div
1.10
2fe
FFT of θe - θ^e
6fe
(a) 2-A
ia
^ θe
θe
3600 60
^ θe - θe 2.80
2fe
0.30
6fe
50-ms/div FFT of θe - θ^e
(b) Figure 10. 10. Saliency-based Saliency-basedposition position estimation at full (a) sine-wave (b) square-wave Figure estimation at full loadload usingusing (a) sine-wave and (b)and square-wave voltage voltage injection (VC==30 Vsqu = 1and kHz100 and 100speed rpm speed ωe×= 52π × 5 rad/s). injection (VC = Vsqu V, f=C30 = V, fsqufC==1fsqu kHz rpm controlcontrol where where ωe = 2π rad/s).
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C. Sensorless Position Control: Position control based on saliency-based position estimation using different injection signals is C. Sensorless Position Control: compared in Figure 11. As seen in (a), the saliency-based drive with sine-wave voltage injection is Position control based on saliency-based position estimation using different injection signals is * = 0 rad to 30π rad. applied. The from the mechanical θm compared in machine’s Figure 11. position As seen is inmanipulated (a), the saliency-based drive withposition sine-wave voltage injection is ∗ ^m applied. The speed machine’s position is manipulated the mechanical θm =error 0 radθm to 30π rad. During the transition, the peak-to-peak from mechanical positionposition estimation –θ is ˆ During transition, thethis peak-to-peak mechanical estimation erroratθm –θm isstate. around aroundthe 4.2 speed degrees. However, error can be negligibleposition once the drive arrives steady By 4.2 degrees. errorerror can beθ negligible the drive arrivesat at steady contrast, * –θ ^ m is once contrast, theHowever, position this control around 1.4 degrees steady state. state. By Figure 11b m ∗ ˆ the position control error θm –θm is around 1.4 degrees at steady state. Figure 11b compares the similar compares the similar position control with the saliency-based drive while the square-wave voltage is position control with the saliency-based drive while the square-wave voltage is used. At steady state, ^ m increases to 2 degrees.∗ The * ^m used. At steadyerror state,θthe error θm–θ error θm ˆestimation ˆ m control the estimation also increases to–θ 2.3 m increases to 2 degrees. The control error θm –θ m –θ also increases to 2.3 degrees, resulting in degraded position control accuracy. Similar to prior degrees, resulting in degraded position control accuracy. Similar to prior experimental results, the experimental results, the of dead-time onsignal the square-wave voltage is the influence of dead-time oninfluence the square-wave voltage is the primary issuesignal to cause thisprimary higher issue to cause this higher estimation error. estimation error. is noted noted that that the is is selected at ItIt is the injection injection frequency frequencyof ofboth boththe thesine-wave sine-waveand andthe thesquare-wave square-wave selected 1 kHz voltage, the the injection injection at 1 kHztotocompare comparethe theposition positionestimation estimation performance. performance. For For the the sine-wave sine-wave voltage, frequency is limited by the PWM frequency. High injection frequency might lead to considerable frequency is limited by the PWM frequency. High injection frequency might lead to considerable discretizedharmonics harmonicswhich whichis is recommended position estimation. By contrast, for squarediscretized notnot recommended for for position estimation. By contrast, for square-wave wave voltage, the injection frequency canincreased be increased near PWM frequencywithout withoutaadiscretized discretized voltage, the injection frequency can be near thethe PWM frequency effect. Figure 12 shows the same sensorless position control with 5 kHz square-wave voltage effect. Figure 12 shows the same sensorless position control with 5 kHz square-wave voltage injection. injection. In this this experiment, experiment, the the injection injection flux flux is is maintained maintained the the same same as as in in Figure Figure 11b 11b for for the the comparison comparison of of In square-wave injection estimation performance using 1 kHz and 5 kHz frequencies. It is observed that square-wave injection estimation performance using 1 kHz and 5 kHz frequencies. It is observed the position estimation errorerror decreases as the frequency increases. Under the same flux that the position estimation decreases as injection the injection frequency increases. Under the same injection, the voltage magnitude is higher for the kHz5voltage signal.signal. In general, a high voltage flux injection, the voltage magnitude is higher for5 the kHz voltage In general, a high magnitude is useful to reduce the dead-time voltage effect, leading to a reduction of position voltage magnitude is useful to reduce the dead-time voltage effect, leading to a reduction of position estimation errors. errors. estimation θ m* , θˆm
30π rad
0
ωm
1100 rpm
0
iqsre
4A
0
θ m* − θ m 2° 0
θ m* − θˆm
2°
0
0
1.7°
1s
1.4°
2.5 s
(a) Figure 11. Cont.
2.1°
4s
t 5s
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θ m* , θˆm θ m* , θˆm 0
30π-rad 30π-rad
0
ωm ωm
1100-rpm 1100-rpm
0 0
iq iq 0
4-A 4-A
0
θm − θˆm θm − θˆm
2° 2°
0 0
θ m* − θˆm θ m* − θˆm
2° 2°
0 0 0 0
1s 1s
2.7° 2.7°
2.3° 2.3°
2.4° 2.4°
2.5 s 2.5 s (b)
4s 4s
t 5 ts 5s
(b) Figure 11. Sensorless position control using (a) sine-wave and (b) square-wave voltage injection (VC Figure using (a)(a) sine-wave andand (b) (b) square-wave voltage injection (VC Figure Sensorless position control using sine-wave square-wave voltage injection 30 Sensorless V, and fc =position fsqu = 1control kHz). = Vsqu =11. = squ 30 V, and = fCsqu= =fsqu 1 kHz). =V (V =V = 30 V, fc and = 1 kHz). Csqu θ m* , θˆm θ m* , θˆm
30π rad 30π rad
0 0
ωm ωm
1100 rpm 1100 rpm
0 re0
iqs iqsre
4A 4A
0 0
θm − θˆm θm − θˆm
2° 2°
0 0
θ m* − θˆm θ m* − θˆm
0 0
0 1s s position0 control1using
3.1° 3.1°
2° 2°
2.5 s 2.5 s square-wave
3.2° 3.2° t 4s 5 ts s 5(V s squ voltage 4injection
= 150 V and fsqu = 5 Figure 12. Sensorless Figure kHz). 12. Sensorless position control using square-wave voltage injection (Vsqu = 150 V and fsqu = 5 Figure kHz). 12. Sensorless position control using square-wave voltage injection (Vsqu = 150 V and fsqu = 5 kHz).
5. Conclusions 5. Conclusions 5. Conclusions This paper compares saliency-based position estimation for PM machines using either sine-wave This saliency-based position estimation PM machines using either sine-wave sine-wave This paper paper compares compares position estimation estimation for for PM in machines using either or square-wave injection saliency-based signals. Key position errors different saliency signals are or square-wave injection signals. Key position position estimation estimation errors errors in in different different saliency signals are or square-wave injectionresults signals. Key saliency signals are analyzed. Experimental also verify the position estimation errors from the analytical model. analyzed. Experimental results also verify the position estimation errors from the analytical model. analyzed. Experimental also verify the position estimation errors from the analytical model. The key conclusions are results summarized as follows: The The key key conclusions conclusions are are summarized summarized as as follows: follows: Machine flux saturation causes primary second-order harmonics on the position estimation. This fluxsaturation saturationcauses causesprimary primary second-order harmonics on position the position estimation. ••• Machine Machine flux second-order harmonics on the estimation. This second-order saliency harmonic is the same for an injection using either sine-wave or squareThis second-order saliency harmonic the for same an injection usingsine-wave either sine-wave or second-order saliency harmonic is the is same an for injection using either or squarewave voltage.voltage. square-wave voltage. • wave Inverter dead-time causes the same error between these two injection voltage signals. However, •• Inverter dead-time causes the same error between between these these two two injection injection voltage voltage signals. signals. However, However, Inverter dead-time causes the is same error the position estimation error reduced for the sine-wave injection because of the LPF used for the position estimation error is reduced for the sine-wave injection because of the LPF the position estimation error is reduced for the sine-wave injection because of the LPF used used for for saliency signal demodulation. saliency signal demodulation. signal demodulation. • saliency For the square-wave voltage injection, the dead-time causes a second-order harmonic and a • For the square-wave injection, the dead-time causes a second-order harmonicisand a sixth-order harmonic voltage in the position estimation at no load. Since the current difference used sixth-order harmonic in the position estimation at no load.beSince the current difference is used for signal demodulation, these saliency harmonics cannot removed. for signal demodulation, these saliency harmonics cannot be removed.
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For the square-wave voltage injection, the dead-time causes a second-order harmonic and a sixth-order harmonic in the position estimation at no load. Since the current difference is used for signal demodulation, these saliency harmonics cannot be removed. Although the square-wave injection is sensitive to the dead-time, this voltage signal provides a higher estimation of bandwidth if the injection frequency is increased to the PWM frequency [23,24]. The development of dead-time harmonic compensation is required to further improve saliency-based drives using square-wave injection.
Author Contributions: J.-Y.C. and S.-C.Y. wrote the paper. S.-C.Y. developed saliency-based drive methods. J.-Y.C. implemented and verified all the theories and performed all the experiments. K.-H.T. contributed analysis tools. Funding: The authors gratefully acknowledge the financial and equipment support from China Engine Corporation and Hua-Chuang Automobile Information Technical Center, Taiwan, China. Conflicts of Interest: The authors declare no conflict of interest.
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